meta: {"file": "/data/tomer/Sefaria-Export/txt_done/Modern Works/Contemporary Halakhic Problems, Vol VI/English/Contemporary halakhic problems, by J. David Bleich, 1977-2005.txt"}
text: Contemporary Halakhic Problems, Vol VI
  בעיות הלכתיות עכשוויות, כרך ו
  Contemporary halakhic problems, by J. David Bleich, 1977-2005
  
  
  Contemporary Halakhic Problems, Vol VI
  
  Preface
  
  As was the case with the earlier volumes in this series, much of the material presented in this volume originally appeared in my "Survey of Recent Halakhic Periodical Literature" which is regularly featured in the columns of <i>Tradition</i>. Many of those items have been expanded and amplified for presentation in their present form. Portions of this work served as the subject matter of <i>shi'urim</i> and seminars on behalf of the students of the Rabbi Isaac Elchanan Theological Seminary and its <i>Kollel le-Hora'ah</i> as well as of the Benjamin N. Cardozo School of Law.
  This work is not intended to serve as a practical halakhic guide and, indeed, no attempt has been made to present definitive <i>psak halakhah</i>. It is devoted to an analysis of Halakhah and halakhic reasoning rather than to definitive statements of halakhic determinations. As such, it is directed primarily to those who have at least some background in the study of rabbinic literature but lack the requisite skills or the leisure to assimilate and analyze the maze of responsa pertaining to the topics treated in this volume. It is intended as an invitation to the reader to join in the noblest of Jewish activities and the supremest of joys –– the study of Torah.
  I wish to express my thanks to my brother-in-law, Rabbi Mordecai Ochs, for his painstaking reading of the manuscript; to my son, Rabbi Dr. Moshe Bleich, for drawing my attention to sources that otherwise would have eluded me and for his many valuable insights; to Rabbi Moshe Shapiro, Mr. Zalman Alpert and Mr. Zvi Erenyi of the Mendel Gottesman Library for their constant helpfulness; to Mrs. Racheline Habousha of the library of the Albert Einstein College of Medicine for her unfailing graciousness in expediting my many requests; to Professor Richard White for making his scholarly expertise available at all times; to my secretary at the Benjamin N. Cardozo School of Law, Miss Kaaron Saphir, for her patience and understanding; to Mr. Erik L. Wilson for his assiduous assistance in my research and for his work on the index of this volume; to my esteemed friend, Mr. Ernest Grunebaum, for noting typographical errors in earlier publications of this material; and most especially to my granddaughter Hadassah Gurwitz whose thorough and meticulous proofreading and incisive observations have spared this work from many inadvertent errors; and last, but certainly not least, to my students for their relentless and provocative questioning.
  My gratitude to the publisher of this series, the indefatigable Mr. Bernard Scharfstein, for his unfailing indulgence, good humor and patience as well as for his warm friendship spanning the decades of our relationship; my thanks also to Adam Bengal of Ktav Publishing House and Janice Weiss of TypeWorks for their painstaking efforts in shepherding the manuscript through the various stages of publication.
  Above all, I am grateful to the Almighty for my cherished collaborators –– the members of my family. Our prayer to the Almighty is that we continue to be numbered among the <i>mashkimim le-divrei Torah</i> and, to paraphrase the words of the <i>hadran</i>, <i>ke-shem she</i>-'<i>azartanu le-sayyem sefer zeh, ken ta'azrenu le-hatḥil sefarim aḥerim u-le-sayyemam, lilmod u-le-lamed, lishmor ve-la'asot u-le-kayyem</i>.
  
  Introduction
  
  
  
  Prelude
  
  <i>"Now these are the ordinances which you shall set before them." It should not enter your mind to say, "I shall teach them a section of Torah or a halakhah twice or thrice… but I shall not trouble myself to cause them to understand the reasons for the matter and its explanation.…"</i>
  RASHI, EXODUS 21:1
  
  Chapter I Torture and the Ticking Bomb
  
  
  
  Chapter 2 Sacrificing the Few to Save the Many
  
  
  
  Chapter 3 In the Wake of Birkat ha Hammah 5769
  
  <i>Lift up your eyes on high and see: who created these? He who brings out their host by number, calling them all by name; by the greatness of His might and because He is strong in power not one is missing.</i>
  ISAIAH 40:26
  "There is nothing new under the sun" (Ecclesiastes 1:9). The world in which the sun began its 207th 28-year cycle is not new, but it is certainly vastly different from the world as it was at the beginning of earlier cycles within both human memory and recorded history. Heretofore, <i>Birkat ha-Hammah</i> was a little-noted event, marked by the otherwise observant but of only passing curiosity—if that—to others, and of no interest at all beyond the Jewish community. This time the event was celebrated in communities throughout the world by large assemblies of Jews and, in many locales, by throngs numbering in the thousands. Unlike former occasions, the media, including wire services, conducted interviews and published articles heralding the event.1<i class="footnote">See, for example, <i>JTA Condensed Edition</i>, February 20, 2009, p. 3; <i>Jewish Week,</i> March 20, 2009, p. 3 and March 27, 2009, pp. 1 and 18-20; “Love the Earth? Bless the Sun,” <i>Wall Street Journal</i>, April 3, 2009, p. W11; and “A Jewish Holiday, Once Every 28 Years,” <i>New York Times</i>, April 3, 2009, p. A12, and “For Jews, Another 28 Years, Another Blessing of the Sun,” <i>New York Times</i>, April 7, 2009, p. A26.</i> In preparation for the last such event in 1981 more Hebrew-language publications dealing with <i>Birkat ha-Hammah</i> appeared than in all of prior Jewish literary history. This time, in addition to expanded and enhanced editions of many of those works, a host of compendia and monographs were published well in advance of the event. Clearly, both interest in, and observance of, Jewish ritual in general, and of <i>Birkat ha-Hammah</i> in particular, are far more intense than in years gone by.
  <i>Birkat ha</i>-<i>Hammah</i> 5769 was an event virtually without precedent. The sheer number of individuals who gathered in synagogues, plazas, public places and even beaches to recite the blessing was phenomenal. From the earliest recitation of <i>Birkat ha</i>-<i>Hammah</i> in New Zealand2<i class="footnote">In point of fact, there is significant controversy with regard to the location of the halakhic dateline. For a brief reference to that controversy see R. Shlomoh Yosef Zevin, <i>Ishim ve-Shittot</i>, 2nd ed. (Tel Aviv, 5718), pp. 333 f. A detailed analysis of the multiple opinions is presented by R. Menachem Kasher, <i>Kav ha-Ta’arikh ha-Yisra’eli</i> (Jerusalem, 5737) and in an abridged version, <i>Kav ha-Ta’arikh ha-Yisra’eli ha-kaẓer</i> (Jerusalem, 5738). Earlier sources have been compiled and republished by R. Judah Aryeh Bloom in a work titled <i>Shittot Kav ha-Ta’arikh be-Kadur ha-Areẓ</i> (Jerusalem, 5750). A review of halakhic problems surrounding the dateline is published in R. Yisrael Taplin, <i>Ta’arikh Yisra’el</i> (Lakewood, 5759). A concise English-language synopsis is presented by Zalman Tropper, <i>The Date Line in Halacha</i> (Lakewood, 1999). According to the authorities who locate the halakhic dateline west of New Zealand, <i>Birkat ha-Ḥammah</i> should be recited on the following day, identified as Thursday in the civil calendar. Thus, the correct day for recitation of <i>Birkat ha-Ḥammah</i> in New Zealand is a matter of doubt.</i>, the event was observed over a period of close to twenty-five hours both in metropolitan centers and in far-flung communities throughout the world.
  The tremendously enhanced interest in, and observance of, <i>Birkat ha</i>-<i>Hammah</i> as compared even to the last such event twenty-eight years ago may be attributed to a number of technological, scholarly and socioreligious developments:
  (1) <i>Technology</i>. The impact of computer and internet technology was forcefully impressed upon me during the period that elapsed between the appearance of the first impression of my revised and expanded <i>Bircas haChammah</i> and the second impression published only a few weeks later. Following the publication of <i>Bircas haChammah</i> in 5741, I was pleasantly surprised by the number of written responses I received during the ensuing months and years. Readers noted typographical errors, academics wrote with queries, rabbinic scholars shared insights, and bibliophiles pointed out additional obscure sources. But I was unprepared for the avalanche of communications that followed the appearance of <i>Bircas haChammah</i> in 5769. This time the medium was not the postal service but the computer. The sheer volume was overwhelming and, unfortunately, time did not permit an immediate response to each interlocutor. Undoubtedly, the ease and rapidity of e-mail played a significant role in fostering such extensive interaction.
  (2) <i>Scholarship</i>. Remarkable was not only the increased volume of communications but also the marked change in quality and content. I received a significant number of lengthy and detailed questions and comments from individuals, and several from groups of <i>kollel</i> scholars both in this country and in Israel, engaged in collaborative study of the topic. Both the tenor and content of those communications testify to the worldwide burgeoning of a community of serious and erudite "<i>lomdishe</i>" scholars. I was awed and impressed. <i>Akhshar dara</i>. <i>Barukh she</i>-<i>higi'anu la</i>-<i>zman ha</i>-<i>zeh</i>!
  (3) <i>Socio-religious phenomena.</i> The phenomenally large numbers of people who participated in gatherings for recitation of <i>Birkat ha</i>-<i>Hammah</i> was recorded in the media and confirmed by photographic illustrations.3<i class="footnote">For a colorful report and photographic illustrations see, for example, <i>Hamodia</i>, April 22, 2009, pp. B16-17, 22-24 and C17-19, 21-24, 32 and 39; and <i>Dos Yiddishe Vort</i>, Sivan 5769, pp. 4-8. Cf., <i>JTA Condensed Edition</i>, April 7, 2009, p. 6.</i> The media also focused upon non-traditional, deviationist and secular environmentalist groups who, seemingly, sought to turn this religiously mandated observance into a happening.4<i class="footnote">See, for example, <i>JTA Condensed Edition</i>, February 20, 2009, p. 3, for a report concerning 14 Jewish organizations who joined to launch BlessTheSun.org, a website with links to various educational materials designed for April 8 activities as well as for a study text prepared by the United Synagogue for Conservative Judaism emphasizing environmental themes. Cf., <i>Jewish Week</i>, March 20, 2009, p. 3 and March 27, 2009, pp. 1 and 18-19; and the <i>Forward</i>, March 27, 2009, p. 7.</i> The initial reaction of many—myself included—was to dismiss such efforts as sheer trivialization, bandwagoning, or a public relations ploy. But that is far too simplistic. <i>Birkat ha-Hammah</i> may sometimes have been trivialized in those quarters but it was not entirely misunderstood. The advent of the event struck a responsive cord in so many who long both for authentic religious experience and for expression of such experience in meaningful ritual. Responsible religious leaders should recognize those responses as reflective of a deep yearning and redouble their efforts to transmute those cravings into authentic observance of Torah and <i>mizvot</i>.
  I. Birkat ha-Hammah Literature: Old and New
  The months leading to <i>Birkat ha-Hammah</i> 5769 saw the republication of the classic monographs authored by R. Pinchas Zelig Schwartz, <i>Yizraḥ Or</i> (Nagyvarad, 5685; reprinted, New York, 5712, 5741, and 5769) as well as by R. Yechiel Michel Tucatzinsky, <i>Tekufat ha</i>-<i>Hammah u</i>-<i>Birkhatah</i> (Jerusalem, 5685; revised editions, 5713, 5741 and 5769). R. David Meldola's <i>Boker Yizraḥ</i> (London, 5601; reprinted Czernowitz, 5627 and 5652; and Jerusalem 5679) was also reprinted together with R. Yekuti'el Kamelhar's <i>Sha'arei Mizraḥ</i> (Cracow, 5656). <i>Or ha-Hammah</i>: <i>Nizozei Or</i> by R. Mordecai Kahane was republished in a pamphlet detailing the customs of the Spinka dynasty bearing the title <i>Seder Birkat ha</i>-<i>Hammah</i>. The encyclopedic compendia <i>Birkat ha-Hammah ke-Hilkhatah</i> (Jerusalem, 5741; revised edition, 5769) authored by R. Menachem Gerlitz and <i>Birkat ha-Hammah</i>: <i>Halakhot u-Minhagim ha-Shalem</i> (Jerusalem 5741; revised edition, 5769) authored by R. Zevi Cohen appeared in greatly enhanced and expanded editions. R. Gavriel Zinner also published an enlarged edition of his <i>Nit'ei Gavri'el</i>: <i>Hilkhot Birkat ha-Hammah</i> (New York, 5741; revised edition 5769). This author's English-language <i>Bircas haChammah</i> (Brooklyn, 5741; revised edition 5769) was republished in an expanded edition.
  In addition, <i>Tovim Me'orot she-Bara</i> (New York, 5741 and 5769), a collection of responsa on <i>Birkat ha-Hammah</i> culled from the works of latter-day rabbinic authorities by R. Shmu'el Yitzchak Roth, was reprinted. Also republished as a separate pamphlet under the title <i>Seder Birkat ha-Hammah</i> was the introduction of R. Betzalel ha-Kohen of Vilna to his <i>Reshit Bikkurim</i> with additional novellae on this subject by his brother, R. Shlomoh Ha-Kohen of Vilna, author of <i>Azei Beroshim</i>.
  Several major new volumes appeared for the first time. Of particular interest are R. Aaron M. Brisk, <i>Ozar ha-Zemanim</i>: <i>Tekufat ha-Hammah u-Birkhatah</i> ( Monsey 5769); R. Yonah Buchsbaum, <i>Kiddush ha-Hammah im Ẓizeha u-Peraḥeha</i> (Jerusalem, 5769); R. Simchah Pinchas Strohli, <i>Seder Kiddush ha-Hammah</i> (Brooklyn, Shevat 5769; revised editions, Adar and Nisan 5769); R. Pinchas Zavichi, <i>Ateret Paz</i> (Jerusalem, 5768); R. Yehuda Herskowitz, <i>The Sun Cycle</i> (Monsey, 5769); and R. Chaim Rappoport, <i>Seder Birkat ha-Hammah al pi Minhag Habad</i> (Brooklyn, 5769). Recently published works on the <i>tekufot</i> and calendrical matters, <i>Va-Yikhtov Mordekhai</i> (Brooklyn, 5769), authored by R. Mordecai Schwimmer and <i>Sha'arei Zemanim</i> (Baltimore, 5768), authored by R. David Heber, include material pertinent to the cyclical nature of <i>Birkat ha-Hammah</i>.
  Historical material, including reports of observances of <i>Birkat ha-Hammah</i> and discourses delivered on those occasions over a span of centuries, as well as astronomical and calendrical information, was compiled by R. Mordecai Genot in his intriguing <i>Birkhat ha-Hammah bi-Tkufata</i> (Bnei Brak, 5769). Another volume of historical interest, <i>Ro'ei Shemesh</i> (Jerusalem, 5769) is a compilation by R. Yehudah A. Horowitz of sermons delivered on the occasion of <i>Birkat ha-Hammah</i> by German rabbinic personalities.
  Nor was there a dearth of articles, pamphlets and briefer discussions. A section of a volume of <i>shi'urim</i> of R. Nissim Karelitz, <i>Hut ha-Shani</i> (Bnei Brak, 5769), is devoted to a discussion of <i>Birkat ha-Hammah</i>. The excellent Torah journals <i>Or Yisra'el</i> (Nisan 5769 and Tammuz 5769) and <i>Kol ha-Torah</i> (Nisan 5769) featured articles pertaining to <i>Birkat ha-Hammah</i>. Pamphlets containing novellae regarding <i>Birkat ha-Hammah</i> were authored by R. Yitzchak Raitport and included in his <i>Kuntres be-Geder Hovat Birkat ha-Pesaḥ</i>; by R. Abraham E. Waxman in his <i>Kuntres Tovim Me'orot</i>; and by the late <i>Klausenberger Rebbe</i> in <i>Be-Or ha-Hayyim: Birkat ha-Hammah</i>. The booklet <i>Birkat ha-Hammah</i> published by Machon Barah ke-Chammah of Jerusalem, includes a précis of the laws of <i>Birkat ha-Hammah</i> according to <i>Mishnah Berurah</i> and <i>Hazon Ish</i> with the approbation of R. Chaim Kanievsky and a pamphlet entitled <i>Yir'aukha im Shemesh</i> published in Jerusalem by the <i>Yated Ne'eman</i> includes the practices of R. Yosef Shalom Eliashiv as reported by R. Yosef Efratti. A useful pamphlet, <i>Sefer Yir'aukha im Shemesh</i>, by the late R. Abraham Blumenkrantz contains astronomical information. <i>Ha-Shemesh bi-Gevurato</i> (Monsey, 5769) authored by R. Yosef Zalman Bloch presents a discussion of the astronomical premises postulated by the Sages. An extensive analytic discussion of the principles governing the Jewish calendar entitled "Bircas HaChamah and Calendar Mathematics: Precision, Simplicity and Conflict," by Sheldon Epstein, Bernard Dickman and Yonah Wilamowsky appears in the online journal <i>Hakirah</i>, <i>the Flatbush Journal of Jewish Law and Thought</i> (Summer, 2008), pp. 147-195.5<i class="footnote">For a description of some of the new literature see E. Brodt and Ish Sefer, “A Preliminary Bibliography of Recent Works on Birkat ha-Chamah,” Tradition Seforim Blog, March 27, 2009 and Akavya Shemesh, “ Al Sefer Kiddush ha-Ḥammah le-R. Yonah be-Ri Buchsbaum,” Tradition Seforim Blog, March 30, 2009,  Noteworthy also is a website of the Jewish National and University Library in Jerusalem,  featuring postings of historical material related to <i>Birkat ha-Ḥammah.</i></i>
  English-language inspirational literature includes <i>Tomorrow's Sun</i> by R. David Goldwasser and <i>Once in Twenty-Eight Years</i> by R. Moshe Goldberger. Indicative of the widespread interest aroused by observance of <i>Birkat ha-Hammah</i> this year was a publication addressed to children by R. Shimon Grama of Lakewood, N.J. entitled <i>Understanding Birkat ha-Hammah its Laws and Customs</i>.6<i class="footnote">Unavailable to this author was a children’s book, <i>The Sun’s Special Blessing</i>, authored by Sandra Wasserman, published by Pitspopany Press.</i> A 3D presentation depicting the solar system and clarifying details regarding <i>Birkat ha-Hammah</i> was prepared by R. Chaim S. Friedman. A version of this production was widely disseminated by the outreach organization Oorah. A series of Yiddish-language tapes dealing with both theoretical and practical aspects of <i>Birkat ha-Hammah</i> was distributed by Satmar institutions in Monsey, New York.
  II. The Meaning of Birkat ha-Hammah7<i class="footnote">The following discussion is based upon a lecture delivered by the author at Yeshiva University on March 15, 2009.</i>
  1. The Commandment to Love God
  In the course of being interviewed by reporters, a significant number posed a single question to this writer: "How is this ritual distinguishable from sun-worship?" The question is age-old, yet it belies a shocking lack of understanding. As reported in the Tosefta, <i>Berakhot</i> 6:10, R. Judah decried recitation of <i>Birkat ha-Hammah</i> as "<i>derekh aḥeret</i>," i.e., a form of idolatry. Centuries later, R. Chaim Joseph David Azulai, known as the <i>Hida</i>, <i>Moreh be-Ezba</i>, no. 6, sec. 190, as well as a number of subsequent authorities admonished either that <i>aleinu</i> not be recited or that care be taken to turn away from the sun lest the onlooker construe bowing in the course of the prayer as an act of obeisance to the sun.
  Most certainly, <i>Birkat ha-Hammah</i> is not an act of sun-worship. But in the words of the simple son (no pun intended) voiced in the course of the Passover <i>seder</i> which in 5769 occurred on the evening following <i>Birkat ha-Hammah</i>: "<i>Mah zot</i>—What is this?" What is its import and why is there so much fanfare?
  In its essence, <i>Birkat ha-Hammah</i> is a very simple blessing. In addition to the initial phrase, "Blessed are You, O Lord, our God, King of the universe," the crucial formula that is the <i>sine qua non</i> of every <i>berakhah</i>, <i>Birkat ha-Hammah</i> consists of exactly three additional words: "<i>oseh ma'aseh bereshit</i>—who makes the work of creation." That blessing, recited in a variety of different contexts, acknowledges God as the author of various extraordinary natural phenomena representing, as they do, divine creation of the universe in its entirety. In the case of <i>Birkat ha-Hammah</i>, it is a solar event that occasions the blessing.
  Judaism exhibits a fascination bordering on a fixation with regard to matters of astronomy, astrology, and celestial phenomena. The vagaries of the Jewish calendar are only the tip of the iceberg. The prophet Isaiah exclaims, "Lift up your eyes on high and see: who created these? He who brings out their host by number, calling them all by name; by the greatness of His might and because He is strong in power, not one is missing" (Isaiah 40:26). The prophet marvels at the celestial creations, at the myriad bodies that were created in the heavens, each one having an assigned task and an assigned place. Isaiah notes that their Creator knows each and every one "by name," i.e., by role and function, and pointing to that phenomenon, he constructs a cryptic, but succinct and compelling, proof for the existence of the Deity. Isaiah points to the continued existence of the celestial bodies in formulating an Argument from Design. The argument is predicated upon an awareness that the ongoing existence of the universe is directly contingent upon the size, position and velocity—not to speak of the centrifugal force—of each of the heavenly bodies. Had any one of those bodies been created possessing different mass or shape, or located in an alternate position within the firmament, or were the laws of nature not operative, the universe would long since have ceased to exist.
  R. Isaac Abarbanel encapsulates this insight in a brief but insightful comment on the verse that declares, "God beheld the universe and it was good" (Genesis 1:18). If one may be so bold as to employ anthropomorphic language in paraphrasing Abarbanel: God looked at His handiwork and exclaimed "Eureka! It works!" Everything in creation fits together like the pieces of a cosmic jigsaw puzzle. If those components did not fit together precisely in the way they do, there would not be a universe. Thus, Judaism's fixation upon celestial matters, as I called it, is actually a fixation upon God as the <i>bore olam</i>—as the Creator of the universe.
  Evidence of creation can be found simply by gazing at the heavens. "Lift up your eyes on high and see," declares the prophet. That verse is not a declarative proposition; it is not a statement of fact. "Lift up your eyes" is an imperative; it is a command. The prophet is exhorting each and every one of us to look up at the sky; to ponder what we see; to ask "who created these;" to reflect upon their origin and nature; and bids us to recognize the greatness and majesty of God manifest in those phenomena.
  The Sages expanded upon this relatively simple motif. The Gemara, <i>Shabbat</i> 75a, declares, "Any person who is capable of calculating the equinoxes and the solstices, to plot and chart the movement of the celestial bodies comprising the zodiac but fails to engage in such calculations, of him the verse says, 'but the work of the Lord they do not regard and the work of His hands they do not see' (Isaiah 5:12)."8<i class="footnote">Indeed, R. Moshe of Coucy, <i>Sefer Miẓvot Gadol</i>, <i>miẓvot aseh</i>, no. 46, enumerates an obligation to engage in those calculations as one of the 613 commandments. That is also the position of <i>Sefer Yere’im</i> (Vilna, 5659), no. 260. That position is based upon the interpretation presented by the Gemara, <i>Shabbat</i> 75a, of the verse “You shall observe and fulfill, for it is your wisdom and understanding in the eyes of the nations” (Deuteronomy 4:6). “Which wisdom is it that [is regarded as such] in the eyes of the nations?” queries the Gemara. The immediate answer of the Gemara is: “That is, calculation of <i>tekufot</i> and <i>mazalot</i>.” The Gemara’s statement is formulated in support of the dictum of R. Samuel bar Naḥmani in the name of R. Yoḥanan: “Whence [is it derived] that it is incumbent upon a person to calculate <i>tekufot</i> and <i>mazalot</i>? As it is said ‘and you shall observe and perform etc.’” The talmudic interpretation establishing an obligation is based upon employment of the phrase “and you shall observe and perform” in the biblical verse, a term that connotes a binding imperative.Rambam also considers the exhortation rooted in this verse to be a mandatory obligation but does not include it in his enumeration of the 613 commandments. In the introduction to his <i>Sefer ha-Miẓvot, shoresh bet</i>, Rambam formulates the principle that the number 613 includes only commandments that are explicitly formulated in the Pentateuch. Excluded from that catalogue of <i>miẓvot</i>, asserts Rambam, are all commandments derived on the basis of rabbinic hermeneutics for, declares Rambam, were these also to be included, “the number of <i>miẓvot</i> would equal many thousands.”In his responsa collection, <i>Pe’er ha-Dor</i>, ed. R. Abraham Chaim Freimann (Jerusalem, 5694), no. 347, Rambam responds to a query regarding the purpose of the commandment with a citation of a dictum of R. Meir, “Ponder His works for from that you will recognize He who spoke and the universe came into being.” Rambam also cites that statement in <i>Hilkhot Yesodei ha</i>-<i>Torah</i> 2:2 and in <i>Sefer ha</i>-<i>Miẓvot</i>, <i>miẓvot aseh</i>, no. 3. The source of the dictum is probably <i>Sifri</i>, Deuteronomy 6:6. Cf., editor’s note, <i>Pe</i>’<i>er ha</i>-<i>Dor</i>, no. 347.Ramban, in his glosses on Rambam’s <i>Sefer ha-Miẓvot</i>, <i>shoresh alef</i>, s.v. <i>ve-hateshuvah ha-revi’i</i>, understands that, for Rambam, the commandment is rabbinic in nature. That position is also espoused by R. Betzalel Ze’ev Shafran, <i>Teshuvot Ravaz</i>, <i>Oraḥ Ḥayyim</i>, no. 32, who asserts that Rambam omits this commandment from his list of the 613 <i>miẓvot</i> because he regarded it to be rabbinic in nature. That understanding of Rambam is contradicted by the comments of Rambam both in his <i>Sefer ha-Miẓvot</i> and in his <i>Pe’er ha-Dor</i>. See also R. Moshe Sofer, <i>Teshuvot Ḥatam Sofer</i>, VIII, <i>Koveẓ Teshuvot</i> (Jerusalem, 5742), no. 26.Ramban himself asserts that the study of <i>tekufot</i> and <i>mazalot</i> is not mandatory even as a rabbinic command. Nevertheless, he regards such activity as a desideratum and explains that the statement of the Gemara is hortatory in nature urging those who are capable of doing so to engage in those calculations and to announce forthcoming astronomical occurrences to the nations of the world.</i> A person who is conversant with the fundamentals of astronomy but who fails to explore the ordered nature of the universe is remiss. This verse, as interpreted by the Sages, censures him for not engaging in that enterprise. What is it that the Sages are underscoring in this dictum? What halakhic obligation has this individual failed to fulfill?
  The underlying notion is codified by Rambam, <i>Hilkhot Yesodei ha-Torah</i> 2:2, in a totally different context. In defining the thrust of the commandment "And you shall love the Lord your God" (Deuteronomy 6:5), Rambam, at least by implication, informs us that the <i>mizvah</i> does not command an emotion. It is exceedingly, and probably impossibly, difficult to command emotions directly. S<i>efer Haredim</i> had a similar difficulty explaining the cogency of commanding love of a fellow human being. Accordingly, <i>Sefer Haredim</i> 1:28-29, defines the commandment requiring one to love his fellow as well as the commandment to love the proselyte as commandments requiring concrete physical acts. Rambam, <i>Sefer ha-Mizvot</i>, <i>mizvot aseh</i>, no. 3, as well as <i>Hilkhot Yesodei ha-Torah</i> 2:2, defines the <i>mizvah</i> "And you shall love the Lord your God," in intellectual, rather than emotional, terms.9<i class="footnote">See also Rambam, <i>Guide of the Perplexed</i>, Part III, chap. 28.</i> A person fulfills the <i>mizvah</i> by intellectually recognizing the majesty of the Deity and the grandeur of His creation. Moreover, as elucidated by R. Ovadiah ben David, author of the unidentified commentary published together with that section of the <i>Mishneh Torah</i>, love is directly commensurate with knowledge: the greater the intellectual apprehension, the greater the love. <i>Ahavah</i> and <i>yedi'ah</i>, love and knowledge, become conflated into a single concept. As Rambam, <i>Hilkhot Teshuvah</i> 10:6, declares:
  One loves the Holy One, blessed be He, only through the knowledge with which one knows Him. According to the knowledge is the love, if little, little and if great, great.
  Rambam, <i>Hilkhot Teshuvah</i> 10:3, followed by <i>Sefer Haredim</i> 1:5, declares that such intellectual awareness generates an emotional state akin to lovesickness as described by King Solomon, Song of Songs 2:5. How does one come to know, and hence love, God? By apprehending His glory, grandeur and majesty. And how does one achieve such an awareness? By investigating the nature of the created universe, by studying the laws of nature that guarantee the cohesiveness of the universe. As stated by Rambam, <i>Hilkhot Yesodei ha-Torah</i> 2:2:
  At the time that a person ponders His works and His great, wondrous actions and creations and sees from them His wisdom which has no measure or end, he immediately loves, praises and glorifies [God] and experiences a great yearning to know the great God, as David stated, "My soul thirsts for the Lord, the living God" (Psalms 42:3).
  We can repeat like poll parrots that God is the Creator of the universe, that He is possessed of this, that and the other attribute, that the Deity is capable of x, y and z, but the fulfillment of the <i>mizvah</i> does not lie therein. A student can memorize the Pythagorean Theorem: a2 + b2 = c2. He can then employ that proposition in solving various mathematical problems involving triangles. But does the student necessarily understand why it is that a2 plus b2 equals c2? Does he further understand that not only does a2 + b2 equal c2 but that a2 + b2 <i>must</i> equal c2? The student has committed the proposition to memory. It invariably yields the correct answer. If it works, then good enough! That is the sum total of his knowledge. A bright, attentive, tenth grade geometry student who sits in class as each succeeding step in the proof of the Pythagorean Theorem is painstakingly demonstrated by the teacher suddenly experiences an exhilarating intellectual <i>simḥah</i>, an intense intellectual pleasure which comes, not from absorbing the factual information that a2 + b2 = c2, but from a sudden recognition that the Pythagorean Theorem is a necessary proposition, that a2 + b2 <i>must</i> equal c2, that the universe could not be ordered otherwise. That is far, far different, qualitatively and epistemologically, from merely being able to recall and repeat the bare fact. The latter ability represents a very low level of knowledge; understanding why it <i>must</i> be the case, is an entirely different level of knowledge.
  As stated by Rambam, <i>Sefer ha-Mizvot</i>, <i>mizvot aseh</i>, no. 3, with regard to the <i>mizvah</i> commanding us to love God:
  That is, that we reflect upon and ponder his <i>mizvot</i> and dicta10<i class="footnote">In this exposition in <i>Sefer ha</i>-<i>Miẓvot</i> Rambam includes understanding of Torah as integral to knowledge, and hence love, of God. That is consistent with Rambam’s view that the commandments are the product of divine reason (see <i>Guide of the Perplexed</i>, Part III, chap. 26) and his doctrine of negative attributes. Accordingly, Torah, as the manifestation of divine reason, is integral to the essence of God. Hence, knowledge of Torah is knowledge of God.</i> and his works until we apprehend Him and delight in the ultimate degree of pleasure in apprehending Him. This is the mandatory love. In the words of <i>Sifri</i>: "For it says, 'You shall love the Lord your God' (Deuteronomy 6:5). I do not know how to love God. Therefore, [Scripture] teaches, 'And these matters which I command you this day shall be upon your heart' (Deuteronomy 6:6). From that you will recognize He who spoke and the universe came into being." Behold we have explained to you that through reflection you will succeed in apprehension and achieve pleasure, and love will come necessarily.
  Rambam declares that it is not belief in the existence of God or of His majesty and glory on the basis of faith that constitutes fulfillment of the commandment "And you shall love the Lord your God"; rather, it is the intellectual pleasure that is born of rational apprehension in which lies fulfillment of the <i>mizvah</i>. Understanding, and hence the ability to calculate, the precision of astronomical phenomena is one means of achieving such awareness.
  Any person can tell you when the <i>molad</i> will occur, i.e., when the next new moon can be sighted. Consult a <i>shul</i> calendar; the information is succinctly presented, in black and white.11<i class="footnote">R. Zevi Elimelech Shapiro of Dinov, <i>Derekh Pekudekha</i>, <i>miẓvah</i> 4, <i>ḥelek ha</i>-<i>ma</i>’<i>aseh</i>, sec. 10, declares that the statement of the Gemara, <i>Shabbat</i> 75a, concerning calculating <i>tekufot</i> and <i>mazalot</i> reflects a mandatory obligation and hence, at the very minimum, every person should calculate the <i>molad</i> each successive month rather than rely upon a calendar for that information.</i> If you seek information regarding the movement of the constellations of the zodiac, charts are readily available; look at the chart and the chart will provide the desired information. Does that mean that a person who uses a <i>shul</i> calendar or consults a chart understands why the <i>molad</i> occurs when it does or why the constellations move in certain patterns? Of course not. The factors governing such phenomena are quite difficult to grasp. A person must invest time and effort in an endeavor to understand such matters. When the Sages declare that a person who is a <i>yode'a le-ḥashev tekufot u-mazalot</i>, a person who is capable of making astronomical calculations but yet fails to do so is remiss, they refer to people who rely upon knowledge of the bare facts. That information can be found in books. Information regarding movements of the constellations, the dates of the solstices and equinoxes and the reappearance of the moon can be memorized. Does that mean that a person who has done so understands why it is that those phenomena occur when they do and why they could not occur at different times? Does that mean that he knows how to calculate the next <i>tekufah</i> himself? Or that he will be able to determine the day and hour of the <i>molad</i> of the next month and of the month subsequent and of the month after that? Not at all! Memorization of recorded facts pertaining to the past and to the present is not sufficient to predict the future. Inferences regarding the future can be extrapolated only if astronomical patterns of the past are analyzed and underlying rules are discovered. That understanding is the key to ascertaining future astronomical events. The Sages declared that the intellectually gifted have an obligation to study astronomy in all its aspects, to understand the phenomena upon which the calendar is based, and to apprehend the regularity of the natural order.
  A person capable of mastering those principles is well-positioned to formulate Abarbanel's Argument from Design. The orderliness of the universe did not arise by chance; intelligent design points to the existence of a rational Creator. A person who arrives at a recognition of the existence of the Deity because he recognizes the handiwork of God in the rational structure of the complex order of nature knows that God exists, not because someone told him that such is the case, and not because he possesses a <i>mesorah</i> or tradition going back to <i>Avraham avinu</i>, but because he himself has demonstrated God's existence anew. Such a person has demonstrated to himself the existence of an intelligent First Cause on the basis of the regularity of the <i>tekufot</i> and <i>mazalot</i>. The Sages regarded that awareness as the optimum, the ideal. A person who has the requisite intellectual ability is required to engage in the scientific inquiry that will lead to such awareness. By no means is such inquiry a purposeless exercise; it is integral to the fulfillment of the <i>mizvah</i> "And you shall love the Lord, your God." 12<i class="footnote">Rambam, <i>Pe</i>’<i>er ha</i>-<i>Dor</i>, no. 347, makes this point explicitly in responding to an interlocutor who questioned the need to engage in such an exercise. See <i>supra</i>, note 8.</i>
  II. Three Categories of the Blessing
  But how does this relate to the blessing "<i>oseh ma'aseh bereshit</i>—who makes the work of creation"? <i>Shulḥan Arukh</i>, <i>Oraḥ Hayyim</i> 227:1, rules that the blessing <i>oseh ma'aseh bereshit</i> is recited upon witnessing meteors or lightning and upon experiencing an earthquake or extraordinary winds. That is followed by <i>Shulḥan Arukh</i>'s codification in <i>Oraḥ Hayyim</i> 228:1 of the provision of Halakhah requiring recitation of the same blessing upon sighting tall mountains, deep canyons or certain rivers and, finally, in the ruling in <i>Oraḥ Hayyim</i> 229:2, providing for recitation of the blessing upon sighting the sun at the beginning of every 28-year solar cycle as well as upon the occurrence of a number of other astronomical events. The blessing is recited upon observing various diverse phenomena but those phenomena seem to be divided by <i>Shulḥan Arukh</i> into different categories. Although the <i>berakhah</i> has but a single formula, it seems to this writer that there are three separate and distinct types of phenomena for which the blessing was ordained, each of which occasions the blessing for a different reason. Each category evokes praise of God as the Creator of the universe as evidenced in a different and distinct manner. In effect, although there is but one text, connotatively speaking, there are three distinct <i>berakhot</i>. Thus, <i>Shulḥan Arukh</i>'s codification of the blessings in three separate chapters becomes readily understandable.
  (1) A Historical Blessing
  Perhaps the most self-evident is the category that includes high mountains and deep canyons. The threshold elevation of a "high mountain" or the threshold depression of a "deep canyon" is unclear. However, when I visited the Grand Canyon, I remember commenting that, although I do not know the precise depth of a canyon that serves to trigger the obligation, nevertheless, the Grand Canyon has a <i>Brisker shi'ur</i>, i.e, its depth is so extraordinary and wondrous that it serves to establish the obligation beyond cavil.
  Why does one recite the blessing upon beholding the Grand Canyon, Mount Everest or the like? Neither the Grand Canyon nor Mount Everest was created recently. The National Parks Service did not engage in a project to gouge out a huge canyon in order to provide us with an opportunity for a <i>berakhah</i>. Halakhah presumes that the Grand Canyon has been <i>in situ</i> since creation and that is precisely why the <i>berakhah</i> was ordained. The <i>berakhah</i> recited upon beholding such a natural phenomenon is a blessing acknowledging what may be termed the historical phenomenon of <i>ma'aseh bereshit</i>. In observing the Grand Canyon one observes creation in all its wondrous grandeur. Of course, we constantly observe the phenomenon of creation. But when we gaze at the world about us we are jaded because we have observed exactly the same phenomena yesterday and the day before and, consequently, today's observation does not give us pause or really make an impression upon us. But when we see something extraordinary, a phenomenon that we do not experience on a regular basis, we are required to praise the Almighty for having created the universe at the very beginning of time because that extraordinary encounter causes us, or should cause us, to reflect upon its source. When pronounced on such occasions, the <i>berakhah</i> is a historical <i>berakhah</i>, i.e., a <i>berakhah</i> acknowledging an event that took place at the beginning of history.
  The obvious parallel is the blessing <i>zokher ha-brit</i> that is recited upon witnessing the appearance of a rainbow. The blessing pronounced over the rainbow is clearly a "historical" <i>berakhah</i>, i.e., it commemorates a singular historical event. The rainbow seen today is hardly novel but it is a phenomenon identical to the rainbow that Noah beheld when he exited from the Ark after the waters of the Deluge receded. God swore to Noah that he would never again visit a flood upon mankind that would destroy all of human civilization and pointed to the rainbow in the sky as a sign of that covenant. The rainbow is designed to make man mindful of God's solemn promise not to permit rains to develop into a second Deluge. The blessing is "historical" in the sense that it commemorates God's oath to Noah.
  This analysis of the nature of the blessing <i>oseh ma</i>'<i>aseh bereshit</i> on such occasions serves to illuminate one aspect of the halakhah governing its recitation. The blessing is recited each time one visits the Grand Canyon or Mount Everest—but only if thirty days have elapsed since it was last seen by that individual. A person who lives in a home overlooking the Grand Canyon and wakes up every morning to the sight of this deep and wondrous chasm becomes desensitized very quickly. We walk into the street and look up at the sky, glance down at the grass growing in the lawn or observe the water flowing in a brook and it makes no impression upon us. We do not give such an experience a first, let alone a second, thought. The commonplace is unremarkable. A person who has the Grand Canyon as his neighbor reacts no differently. But if a person has not seen the Grand Canyon for thirty days, and suddenly it appears before him, he becomes, or should become, overwhelmed by the realization of the wondrous nature of the feat God performed in fashioning the canyon as part of His majestic universe.
  (2) A Contemporaneous Blessing
  There is also a second category of phenomena that occasion the blessing <i>oseh ma'aseh bereshit.</i> The text of the blessing is the same, but it is pronounced upon witnessing an entirely different type of phenomenon and acknowledges an entirely different aspect of creation. Upon witnessing lightning one is required to praise God as He "who makes the work of creation." On such occasions the blessing praises God, but not for having created lightning in the primordial era. The lightning we observe today was not created 5772 years ago. The lightning is created precisely when we observe it; it is being created before our very eyes. The blessing we pronounce acknowledges the present nature of the creation of that phenomenon. The Sages anticipated that when we witness such an event it will evoke in us the realization that only the Almighty is capable of such a feat. The realization is a form of religious experience and it can be an extremely intense religious experience.
  A type of experience that William James does not describe in his <i>Varieties of Religious Experience</i> is one that I had in my youth, an experience that was one of the most powerful spiritual moments of my life. I was all of sixteen years old taking a freshman course in college biology. Following the directions of the laboratory instructor, I took a little piece of a leaf, placed it on a slide and inserted the slide under a microscope. Lo and behold, there, under the microscope, chloroplasts were chasing each other, round and round, back and forth, in the leaf. The experience was vivid and exhilarating. Photosynthesis was no longer an intellectual abstraction accepted on the authority of a textbook or on the basis of the testimony of botanists. The microscope became a window through which one could observe <i>ma'aseh bereshit</i>.
  The Sages did not ordain that the blessing <i>oseh ma'aseh bereshit</i> be pronounced upon observing such a phenomenon under a microscope. It is questionable whether they did not do so because of the general principle that Halakhah does not take cognizance of sub-clinical phenomena or simply because they had no microscopes. I have no doubt whatsoever that, logically, absent such considerations, if one should be prompted to recite a <i>berakhah</i> upon witnessing lightning, <i>a fortiori</i>, one should feel compelled to recite a blessing when observing that phenomenon as well. The spiritual thrill, the intellectual joy, experienced in gazing upon chloroplasts dancing within a leaf is certainly much, much more powerful than is experienced in watching the zigzag of lightning in the sky. A person who has had no first-hand experience of such an event is simply missing an <i>Erlebnis</i>; it is something that one must experience in order to appreciate. The blessing pronounced upon witnessing lightning and the like may be categorized as a "contemporaneous" blessing in the sense that it is a blessing acknowledging God as Creator in the here and now.
  In both of those guises the blessing <i>oseh ma'aseh bereshit</i> is designed to underscore the notion that there is no sharp distinction between the miraculous and the natural. <i>Nes</i>, or miracle, and <i>teva</i>, or nature, are two sides of the same coin. Ramban, in numerous passages in his <i>Commentary on the Pentateuch</i>,13<i class="footnote">See, for example, Genesis 7:17, 35:13 and 46:15; Exodus 6:12; Leviticus 18:29 and 26:11; and Deuteronomy 11:13.</i> eloquently formulates a distinction between a <i>nes nistar</i>, or a hidden miracle, and a <i>nes nigleh</i>, or an open miracle. The miraculous nature of hidden miracles is obscured by virtue of the fact that to the beholder the result appears simply as the manifestation of natural occurrences. Open miracles are perceived as the suspension of the natural order. Perhaps even more emphatic are the comments of R. Meir Simchah of Dvinsk, <i>Meshekh Hokhmah</i>, <i>Parasḥat Beḥukotai</i>. <i>Meshekh Hokhmah</i> comments that miracles are not designed as ends in themselves; on the contrary, a <i>nes</i> is entirely instrumental. Miracles are designed to impress upon us that all of <i>teva</i> is a <i>nes</i>; nature is miraculous; the natural order is the greatest of all miracles. The temporary suspension of that order is designed to make us realize that the order and regularity of nature is born of divine decree and subject to divine will. The problem for mankind is that we have become desensitized. Every day the sun rises in the east and sets in the west. We take that for granted, with the result that such phenomena no longer strike us as wondrous. From time to time it is necessary for us to be jolted out of our intellectual complacency. Observation of an extraordinary phenomenon is an occasion to be reminded that miracles are built into the principles governing the universe as variations in the regularity that is the hallmark of nature.14<i class="footnote">See <i>Avot</i> 5:6; <i>Bereshit Rabbah</i> 5:4; and <i>Shemot Rabbah</i> 21:16. See also Rambam, <i>Commentary on the Mishneh</i>, <i>Avot</i> 5:6; <i>idem</i>, <i>Guide to the Perplexed</i>, Part II, chaps. 25 and 29; and Ḥasdai Crescas, <i>Or ha-Shem,</i> Part II, p. 5.</i> Miracles are designed to impress upon us that, in reality, <i>teva</i> is the greatest of all miracles. That is why, declares <i>Meshekh Hokhmah</i>, "one who recites <i>hallel ha-gadol</i> every day commits blasphemy" (<i>Shabbat</i> 118b). If a person recites <i>hallel</i> every day it is because he feels he must give thanks for what he perceives as miracles performed on his behalf on a daily basis. But God does not perform overt miracles on a daily basis; He does not disturb the laws of nature with any frequency. To presume that He does so is a form of blasphemy. But at the same time we are charged with recognizing that nature itself is miraculous. That is why, explains <i>Meshekh Hokhmah</i>, a person who recites <i>ashrei</i> thrice daily is assured of a share in the World to Come (<i>Berakhot</i> 4b). The omnipotence, grandeur and majesty of God is manifest in the ordinary, but regular and ongoing, phenomena described in <i>ashrei</i>: "You give them their food in due season. You open your hand and satiate every living creature with favor" (Psalms 145:15-16). Recitation of <i>ashrei</i> serves to acknowledge that the phenomena we regard as natural, ordinary and run of the mill are really miracles wrought by God.
  <i>Ba</i>'<i>alei mussar</i>, masters of the <i>mussar</i> movement, seized upon that notion in offering a resolution of the classic question posed by <i>Bet Yosef</i> with regard to the observance of Chanukkah. Why do we kindle the <i>menorah</i> each of the eight days of Chanukkah? The Maccabees found suitable oil sufficient for one day. If we commemorate the miracle of the cruse of oil it should be sufficient to kindle the <i>menorah</i> on seven consecutive evenings. Since there was enough oil for the first day, the lamps of the <i>menorah</i> in the Temple burned quite naturally on the first day. If so, what miracle is commemorated in lighting the Chanukkah <i>menorah</i> on the first day? To this the <i>ba'alei mussar</i> answered that, of course, the Sages ordained the days of Chanukkah in commemoration of the miracle of the cruse of oil. But lest one think that it is only a miracle that occurs as an idiosyncratic event, as an ostensive violation of natural law, that requires celebration and acknowledgement of the power of God, they ordained an additional day designed to celebrate <i>teva</i>.
  "Let He who told oil to burn come and tell vinegar to burn" (<i>Ta</i>'<i>anit</i> 25a). There is no inherent reason why oil must burn or why vinegar should not burn. Vinegar might burn as well as oil, or oil might not burn just as vinegar does not burn. God established the laws of chemistry and physics that make it possible for oil to support combustion and render it impossible for vinegar to do so. Those laws assure that both oil and vinegar will behave in an orderly, consistent and predictable manner. We take it for granted that oil will burn and vinegar will not. We tend to forget that the regularity of nature is nothing short of miraculous. Hence, we are in need of suitable occasions to have that verity impressed upon us. The kindling of the <i>menorah</i> on the first day of Chanukkah, according to the <i>ba'alei mussar</i>, was designed to convey that message globally, i.e., not only with regard to the combustible nature of oil but with regard to the laws of nature in their totality. The blessing <i>oseh ma'aseh bereshit</i>, when pronounced upon witnessing lightning, meteors, etc., forces us to reflect upon the nature of those infrequent occurrences and to recognize that those phenomena, novel and infrequent as they may be, are entirely attributable to the laws instilled in the universe by God.
  Thus, there is but one formula in which the blessing is couched, but on differing occasions the blessing is pronounced to express recognition of one or another of the two distinct roles fulfilled by God as the Creator of the universe: 1) recognition of God as the author of creation as evidenced by remarkable primordial artifacts available for our observation; 2) acknowledgment of God as the source of the laws of nature which at times give rise to unusual and seemingly spontaneous phenomena.
  (3) An Evocative Blessing
  The blessing <i>oseh ma'aseh bereshit</i> is employed in yet a third manner. The third guise of the blessing is neither an acknowledgment of the wondrous, unparalleled historical event that the creation of the universe represents, nor is it a blessing acknowledging a present novel occurrence in the realm of nature; indeed, in its third form, the blessing is not occasioned by any extraordinary phenomenon at all. The blessing in its third form is employed to teach, to underscore and to reinforce the notion that God is, as He is described in the liturgy of the morning prayer, He who "each day continuously renews the work of creation."
  In order to appreciate the purpose and function of this third form of the blessing <i>oseh ma'aseh bereshit</i>, it is necessary to provide some elucidation of the doctrine reflected in the blessing. It seems to me that this description of God as the "<i>oseh ma'aseh bereshit</i>" is designed to teach what philosophers term the "Doctrine of Continuous Creation" or, sometimes, the "Doctrine of Constant Conservation." At times, those terms are used interchangeably but they are not really synonymous. Rambam presents that doctrine in two separate places, in <i>Hilkhot Yesodei ha-Torah</i> 1:1-2 and in his formulation of the Thirteen Principles of Faith in his <i>Commentary on the Mishnah, Sanhedrin</i>, Introduction to <i>Helek,</i> the First Principle.
  In <i>Hilkhot Yesodei ha-Torah,</i> Rambam writes:
  The principle of all principles and the pillar of all wisdom is to know that there is a First Being; He brings every existing thing into existence; and all things that exist in heaven, on earth and in what is between them did not come into existence other than through the veracity of His existence. And if it could be conceived that He does not exist, no other thing could exist.
  The first two clauses of that statement are entirely straightforward: God is the First Being and the Prime Cause of all that exists in the created universe. In the second of these clauses Rambam emphasizes that nothing in the celestial spheres or on earth could have come into existence other than through God's creative power. In his final statement Rambam makes an even more profound assertion: there can be no ongoing existence of created entities other than through the medium of God's eternal existence. Hence, were God not to exist, the continued existence of all things would be impossible. Rambam is declaring that creation is not a once in an eternity phenomenon. In a very real sense, creation takes place on a continual, continuous and ongoing basis. Were God to have created the universe and then gone on an eternal sabbatical by removing His existence, the universe could not have survived. The presence of God is needed in order to guarantee the continued existence of the universe.
  In writing "if it could be conceived that He does not exist," Rambam is not quite willing to concede that such a notion is actually conceivable. Rambam is asserting that, since God is a necessary Being, we cannot even conceive His non-existence because the non-existence of the Deity is, in effect, an oxymoron. Although that argument is not identical to Anselm's Ontological Argument for the existence of God, it certainly brings that proof to mind. Be that as it may, were we able to conceive of the eventuality of the non-existence of the Deity coming to pass, the universe itself would also cease to exist. It is the presence of God that makes possible the ongoing existence of the universe.
  Rambam makes this point even more forcefully in his <i>Commentary on the Mishnah</i> in the course of developing the first of his Thirteen Principles. In addition to telling us simply that belief in the existence of the Deity is a fundamental principle of faith he continues with the statement:
  He is the cause of the existence of all existing things; In Him is the continuity of their existence and from Him comes their endurance. Let the absence of His existence not occur to the mind, for in the absence of His existence the existence of all created things would be nullified and there would not remain any existing thing whose existence would endure.
  That comment is simply an amplification of what Rambam says much more succinctly in the <i>Mishneh Torah</i>. Rambam here emphasizes that the existence of the Deity is necessary, not only because there must be a First Cause, but also because of God's role in the nature of a dynamo infusing the universe with existence.
  One can perhaps best understand that concept in terms of modern phenomena. Consider electricity. I am neither a physicist nor an electrical engineer. Nevertheless, I have a rudimentary understanding of the nature of electricity. The generator must operate constantly if there is to be electricity. If one shuts down the generator, there is no longer an electrical current. One cannot simply create a quantity of electricity and then go on vacation expecting the stored electricity to continue to flow. A generator is necessary in order continuously to produce new electricity. "Existence" is created by the Almighty. Thus, according to Rambam's depiction of the role of God, God is not merely the First Cause of all existence; He is the <i>ongoing</i> cause of existence. Without His ongoing presence there could not be any existence whatsoever.
  The Kalam philosophers presented the Doctrine of Continuous Creation in a very particular way. Rambam, in his <i>Guide of the Perplexed</i>, Part I, chapter 73, sixth proposition, reports that Kalam philosophers employed a notion similar to that of continuous creation in constructing a proof for the existence of the Deity. They asserted that the universe was not created from a substance that Jewish philosophers term "<i>ḥomer hiyuli</i>," i.e., hylic substance or primordial matter, that is endowed with accidents. Those philosophers maintained that matter cannot exist without accidents and hence no entity could exist if God did not recreate accidents during every moment of existence. As a consequence of assuming that substance cannot exist without accidents, the Kalam philosophers concluded that, since by definition a hylic substance lacks accidents, the existence of a hylic substance is impossible.
  The Kalam also accepted what the translators call an "atomic" notion of time, i.e., the notion that time is composed of discrete particles, or "atoms." I prefer to use the term "quantum" notion of time rather than "atomic" notion of time. Based upon those two notions, Kalam philosophers asserted that cloth that is dyed black does not retain its color because the dye has penetrated the cloth; rather, the "blackness" of the cloth is an accident created by God when the black pigment comes into contact with the cloth. But the "blackness" is of no duration; God must repeatedly recreate blackness during every moment of time. Absent God's re-creation of accidents there would be nothing in existence except the underlying hylic substance, a notion they considered to be incomprehensible since they asserted that one cannot conceive of a material substance having no quality other than sheer existence.
  The Kalam have a latter-day counterpart in the philosophy of René Descartes. Descartes presented a similar proof for the existence of the Deity predicated upon his conception of time.15<i class="footnote">See René Descartes, <i>Principles of Philosophy</i>, Part I, chap. 21. See also <i>Ouevres de Descartes</i>, ed. C. Adam and P. Tannery, V, (Paris, 1903), 530.</i> Descartes regarded time, not as a kind of wave or flow, or as a measure of motion, but as, literally, a fourth dimension made up of infinitesimal particles which have been termed "quanta." The Cartesian notion of time gives rise to a serious problem. If the universe exists in time and if time is a series of discrete and unconnected quanta, how does any entity traverse time by moving from one time quantum to the next? A transportation system would be required to move physical entities from one such time quantum to the next. That transportation system is supplied by God. But how does He accomplish that feat? The notion that God drags physical entities across nothingness in incomprehensible. So what does God do in order to provide for the durability of the universe? God's solution, declares Descartes, is to recreate the universe in every moment of time. God recreates the universe in every quantum of time; He creates the universe anew every microsecond. The continued existence of the universe would be impossible without continuous divine re-creation. Thus, the very existence of the universe is a proof of the existence of God.
  That proof of the existence of God is quite similar to the Kalam argument rejected by Rambam. Although not spelled out in that discussion in the <i>Guide</i>, Rambam was compelled to reject the Kalam argument, in part, because Rambam had a totally different notion of time. Rambam does not regard time as a created entity and certainly not as composed of atoms or quanta. According to Rambam, time is an accident of motion, and motion is an accident of matter. Even more fundamentally, as Rambam argues in his discussion of the Kalam theory, God, although he is the First Cause, created causal laws which, in turn, serve as a sufficient explanation for the duration of accidents. Accordingly, Rambam derisively dismisses the Kalam proof out of hand.
  Rambam's rejection of the Kalam notion of constant creation is not a contradiction to his assertion, both in the <i>Hilkhot Yesodei ha-Torah</i> and in his <i>Commentary on the Mishnah</i>, of God's ongoing role in the preservation of the universe. First, and perhaps foremost, Rambam decries only denial of causality and the accompanying notion that accidents must be recreated, but makes no such statement with regard to <i>ḥomer hiyuli</i>.
  Moreover, as I understand Rambam, he insists upon the Doctrine of Continuous Conservation while rejecting a Doctrine of Constant Creation. Although philosophers may obscure the issue by using the terms interchangeably—and I am also guilty of having done so in my book <i>Bircas ha-Chammah</i>—these are fundamentally two different and distinct doctrines. Rambam, in his <i>Commentary on the Mishnah</i> and in <i>Hilkhot Yesodei ha-Torah</i>, does not formulate a Doctrine of Continuous Creation; rather, he advances a Doctrine of Constant Conservation. According to Rambam, the universe may well have been created but once in all of eternity, but it requires the same amount of force, energy or "creative power" to maintain the universe in existence and to sustain it as was required in order to create it in the first instance. Existence, according to Rambam, is an ephemeral quality that must constantly be renewed. The infusion of existence needed to sustain the created universe can come only from divine existence. Hence, were divine existence, which Rambam refers to in <i>Hilkhot Yesodei ha-Torah</i> as "<i>amitat mezi'uto—</i>the veracity of His existence," to be removed from the universe, the universe could no longer exist and would return to sheer nothingness.
  This leads directly to what I have referred to as the third guise of the blessing <i>oseh ma'aseh bereshit</i>. This third pronouncement of the blessing employs the same formula as the first two but it differs in that it is recited in recognition and affirmation of the Doctrine of Constant Conservation of the universe. It is a blessing through which we acknowledge that God's presence is needed on a continual and ongoing basis for the preservation of the created universe with the result that, absent His presence, there could not be a world for us to inhabit.
  But how is that doctrine reflected in the words of the blessing "<i>oseh ma'aseh bereshit</i>?" When we went out on the morning of April 8th, on <i>erev Pesaḥ</i>, and looked up at the sun, did the sun look any different from the way it looked a day earlier on April 7th? If one were to have gazed heavenward on the first day of <i>Pesaḥ</i>, on April 9th, would one have missed anything in one's observation of the sun that one was able to see a day earlier? Exactly the same visual sensation was available on April 7th, April 8th and April 9th as well on any other day of any year. Observation of the sun on April 8th, 5769 was entirely unlike witnessing lightning, meteors or the like. The latter are phenomena that one does not witness with any degree of regularity. In contradistinction, the sun can be observed every single day. Nothing of significance changes in the perception of the sun from one day to the next. If so, why was a blessing required at the beginning of a new solar cycle on April 8th more so than on any other day?
  The blessing, when pronounced on the occasion of <i>Birkat ha-Hammah</i> represents a third type of <i>berakhah,</i> one that may be termed an "evocative" <i>berakhah</i>. That pronouncement of the <i>berakhah</i> is not responsive in the sense that we now see and marvel at some phenomenon that the Almighty created five thousand seven hundred and sixty-nine years ago or in the sense that we are responding to a phenomenon that He is creating here and now before our very eyes, in that a moment ago there was no lightning and now there is lightning. The <i>berakhah</i> as recited on the occasion of <i>Birkat ha-Hammah</i> is evocative rather than responsive. It is meaningful only when pronounced upon internalizing what is perceived and reflecting thereupon. One cannot appreciate the purpose of the blessing unless one internalizes the experience. The blessing is designed to promote reflection and introspection, which, in turn, serve to give rise to a feeling that one must express gratitude to the Almighty for continuously conserving the universe. It is through recitation of the blessing that such awareness is evoked.
  And what is the phenomenon that occasions this reflection? It is the sun's return to the place in the firmament which it occupied on the same day of the week and at the same hour of the day on which it was created during the six days of creation. As such, it represents the beginning of a new solar cycle. But one could not possibly know that this has occurred simply by gazing at the sun. It is impossible to realize what has occurred without preparing for the event. A person who is intellectually prepared for the event recognizes what it is that occurs on April 8th upon the culmination of one twenty-eight year cycle and the beginning of another; perforce, he must recognize that the orbit of each of the heavenly spheres is a <i>galgal ha-ḥozer</i>, cyclical and never-ending, going back and forth, back and forth, over and over again. Furthermore, he must recognize that this is so only because God guarantees the existence of the created universe through <i>amitat mezi'uto,</i> the veracity of His existence; otherwise, the universe would not exist at all. Pronouncement of the blessing is designed to spur reflection upon this phenomenon; praise of God as the ongoing "Maker of the work of Creation" evokes recognition of God's constant conservation of the universe.
  III. Other Occasions for an Evocative Blessing
  The commencement of the 28-year solar cycle is not the only event for which the Sages ordained the blessing <i>oseh ma'aseh bereshit</i> in its evocative guise. The <i>Beraita</i> cited by the Gemara, <i>Berakhot</i> 59b, also declares that the blessing must be recited upon observing a variety of celestial events. As understood by early-day authorities, the <i>Beraita</i> describes the return of celestial bodies to the places in the firmament they occupied at the time of creation.16<i class="footnote">For a more complete discussion of the <i>Beraita</i> and of the nature of those phenomena see this writer’s <i>Bircas haChammah</i> (New York, 5769), pp. 78-81.</i> Such events are similar in nature to the event that occasions <i>Birkat ha</i>-<i>Hammah</i>. The occurrence of such events certainly cannot be discerned by observation. Indeed, in practice, we do not recite the blessing on the other occasions enumerated in the <i>Beraita</i> because we do not know how to determine their occurrence. 17<i class="footnote">See <i>ibid</i>., pp. 81-83.</i> Nevertheless, the blessing on those occasions is in the same guise as <i>Birkat ha</i>-<i>Hammah</i>; the blessing is designed to evoke reflection upon the ongoing cyclical nature of the celestial spheres.
  I believe that it is possible to demonstrate that recitation of <i>oseh ma'aseh bereshit</i> on the occasion of <i>Birkat ha-Hammah</i> and on other occasions enumerated in the <i>Beraita</i> are not the only occasions regarding which the Sages at least considered ordaining recitation of the blessing in its third guise. <i>Birkat ha-Hammah</i> occurs only once in twenty-eight years and presumably the other events occur with even less frequency. But another event was considered as a candidate for recitation of the same type of blessing. The Gemara, <i>Berakhot</i> 59a, states:
  R. Joshua ben Levi said, "If one sees the sky in all its purity, he says, 'Blessed is He who has wrought the work of creation.'" When does he do so? Abaye said, "When there has been rain all the night and in the morning the north wind comes and clears the heavens." And they differ from Rafram bar Papa quoting R. Hisda. For Rafram bar Papa said in the name of R. Hisda, "Since the day when the Temple was destroyed there has never been a perfectly clear sky, since it says, 'I clothe the heavens with blackness and I make a sackcloth their covering'" (Isaiah 50:3).
  Why should that phenomenon occasion a blessing? Rashi comments, "For such was its creation but afterward it was covered by clouds." In our day, even in the absence of cloud cover, it is certainly impossible to behold the sky in its pristine glory in any urban locale. There is simply too much industrial pollution that obscures the sky, not to speak of light pollution. However, Abaye declares that one who wishes to observe the sky as it appeared at the time of creation before it was obscured by clouds may do so by waiting for a rainfall that lasts all night long. Then, if the rain is blown away by a strong wind, the completely unobscured sky can be seen in all its purity and the blessing o<i>seh ma'aseh bereshit</i> is pronounced.
  According to Rashi's interpretation of the Gemara, this is not a novel guise of the blessing. The blessing on such occasions, as categorized by Rashi, is no different from the blessing recited upon observing high mountains or deep chasms. The blessing is historical in nature in the sense that it acknowledges a primordial phenomenon present at creation. In the atmospheric conditions described, the sky is revealed in its pristine glory and appears as it did in its primordial state. That is not an everyday phenomenon. Ordinarily, the sky is covered by pollution, smoke, clouds, etc. On the rare occasion that it appears in its original state, one is required to praise God for having created a glorious universe.
  If Rashi's interpretation is accepted, it is the <i>berakhah</i> in its historical guise that is required. If the blessing is identical in nature to the blessing recited upon sighting a tall mountain, a deep canyon or the <i>yam ha-gadol</i>, the blessing is not to be recited if the phenomenon is witnessed a second time within a thirty-day period. There does not appear to be any explicit evidence of such a limitation.
  But for Rashi's comment, I would offer an entirely different analysis of the nature of the blessing. It may even be possible to read that analysis into Rashi's words as well. The phenomenon described by the Gemara seems to me to be comparable, not to a tall mountain or a deep canyon, but to the phenomenon of lightning. Lightning appears <i>sua sponte</i>; it is a novel phenomenon created before our very eyes. It seems to me that the Gemara is telling us that, when there is a prolonged rainfall and then a strong wind that blows, the resultant phenomenon appears as a new creation comparable to lightning. The sky in a place such as Bonaire is not at all comparable to the sky we are accustomed to seeing. The biblical phrase that speaks of the "multitude of stars" cannot be appreciated unless one has had such an experience. The sky in such far-flung areas is covered "wall to wall" by a blanket of light. Imagine a rain followed by a wind that would turn our sky into the sky over Bonaire. A person witnessing such an event would enjoy an exhilarating experience in the nature of an impression of a wondrous phenomenon being created before his eyes. Viewing the sky upon alighting from a plane at night in Bonaire certainly gives rise to an even greater thrill than watching lightning zigzag across the sky.
  In metropolitan areas that experience is possible only when the atmospheric conditions described by the Gemara have occurred. On such occasions, the blessing <i>oseh ma'aseh bereshit</i> is in the nature of an acknowledgement of a contemporaneous creation occurring in the here and now.
  How frequently is the blessing ordained for that phenomenon recited? Whenever the event occurs; whenever cleansing rain falls and the clouds are dispersed with the result that the sky is completely clear. If new clouds appear and, in turn, are swept away, the result is a new "creation" mandating a new <i>berakhah,</i> just as a second bout of lightning after clouds have dissipated is a new "creation" occasioning a new blessing. According to this analysis, the blessing required on perceiving such a phenomenon is in the guise of the earlier-described second form of the blessing. According to either analysis, the blessing upon the appearance of the heaven "in its purity" is in the nature of one or the other of the usual forms of the blessing <i>oseh ma'aseh bereshit</i>.
  There is, however, a parallel text in the <i>Yerushalmi</i>, the Palestinian Talmud, <i>Berakhot</i> 9:2. The <i>Yerushalmi</i> also records that the blessing <i>oseh ma'aseh bereshit</i> is pronounced upon seeing "the firmament in its purity." But, in the <i>Yerushalmi</i>'s discussion, R. Huna explains that the blessing is recited "only in the rainy season after three days, as it is written, 'And now they did not see light that is bright in the skies but a wind passes and cleanses them'" (Job 37:4), i.e., the blessing is recited after incessant rain spanning a three-day period. R. Moshe Sofer, <i>Teshuvot Hatam Sofer, Oraḥ Hayyim</i>, no. 55, expresses bewilderment: Why is the blessing occasioned only by three days of rainfall? <i>Hatam Sofer</i> amends the text to read "after thirty days" but fails to explain why the blessing is recited only after a thirty-day period of rainfall.
  It seems to me that the <i>Yerushalmi</i> describes a phenomenon totally different from that described by the Babylonian Talmud and that it is describing a blessing of a different genre. As evidenced by the citation of the verse in Job, the phenomenon described occurs after a three-day period of total darkness. It is not the rain <i>per se</i> that causes the phenomenon, but the darkness that occurs because of rainfall. When was the first occasion on which the sun was not visible for three days and then suddenly appeared? During the period of creation the sun suddenly came into being at the beginning of the fourth day. The reenactment of that event necessitates the blessing. But what is the nature of the blessing? It is not praise of God occasioned by the creation of something before our eyes nor is it an occasion on which we behold a vestige of primordial creation. Rather, it is a reenactment of <i>ma'aseh bereshit</i>. The blessing is evocative in that it calls upon us to reflect upon <i>ma'aseh bereshit</i> and to recognize that creation is an ongoing phenomenon. It calls upon us to remember that the sun originally came into existence only on the fourth day; the reappearance of the sun after darkness is a form of re-creation. Our observation of the sun being recreated should impress upon us the verity that the sun—and indeed all of creation—must continually be infused with existence. That reflection should evoke praise of God who constantly functions as the <i>oseh ma'asheh bereshit</i>.
  The statement of the <i>Yerushalmi</i> is not recorded either by Rambam or by <i>Shulḥan Arukh</i>. But it is an example of a proposed blessing in the nature of the third category earlier described. The only blessing "<i>oseh ma'aseh bereshit</i>" of the third category that is normative is <i>Birkat ha-Hammah</i>. In that guise it is a novel blessing, a rare expression of thanksgiving to the Almighty for His constant conservation of the universe. It is a form of praise that can properly be uttered only upon reflection; it is the product of intellectual awareness.
  IV. Evidence of the Evocative Nature of Birkat ha-Hammah
  It is possible to demonstrate that the blessing <i>oseh ma'aseh bereshit</i> pronounced on the occasion of <i>Birkat ha-Hammah</i> is different from every other recitation of the blessing in a very significant way. The determination of the 28-year solar cycle is based upon the calculation of the day and hour of the vernal equinox as calculated by Shmu'el. Those calculations, in turn, presume that the length of the solar year is the same as that assumed by the Julian calendar. But we know very well that the Julian calendar was inaccurate. The Sages of the Gemara knew it to be inaccurate. Rav Adda, one of the <i>Amora'im</i>, is reported to have established different and more accurate <i>tekufot</i> which we use for other purposes of Halakhah. Rav Adda realized that Shmu'el's assumption with regard to the length of the solar year was imprecise. According to Rav Adda's more accurate calculation, the dates on which equinoxes and solstices occur are different from those of Shmu'el. We employ Shmu'el's <i>tekufot</i> only for <i>Birkat ha-Hammah</i>, for determination of the commencement of recitation of the <i>tal u-matar</i> prayer for rain and possibly for one other arcane area of Jewish practice, <i>viz</i>., refraining from drinking water drawn on the evening of the <i>tekufah</i>. But if the Sages of the Talmud realized that Shmu'el's calendar was inaccurate why, in ordaining <i>Birkat ha-Hammah</i>, did they ordain a blessing to commemorate a phenomenon that does not occur at the time it is purported to occur?
  The reason they did so with regard to recitation of <i>tal u-matar</i> is not difficult to apprehend. Not everyone is a <i>yode</i>'<i>a le</i>-<i>ḥashev tekufot u-mazalot</i>; not everyone is sufficiently proficient in the science of astronomy to be able to calculate the autumn equinox so that he will be able to determine the appropriate date to commence recitation of <i>tal u</i>-<i>matar</i>. What is a simple, untutored, unlettered Jew supposed to do? Everyone is capable of calculating Shmu'el's <i>tekufah</i>. According to Shmu'el, each season is exactly 91¼ days in length. Once a person is apprised of the date and hour of the beginning of any season, calculating the beginning of any future season is extremely simple. Accordingly, the Sages adopted Shmu'el's calculations, inaccurate though they were, in ordaining the prayer for rain.
  <i>Birkat ha-Hammah</i> is much more difficult to understand. The blessing seems to be occasioned by a non-occurring event. Since the solar year is decidedly not precisely 365 days in length, the sun does not return to its place of origin on the day and hour of the original creation every twenty-eight years. In fact, in the 5772 years that have elapsed since creation the actual <i>tekufat Nisan</i> has never recurred on its original day and hour. One authority, <i>Teshuvot Mas</i>'<i>et Binyamin</i>, no. 101, declares that that phenomenon will never recur. Yet the Sages did seek an opportunity to impress upon people the Doctrine of Constant Conservation of the universe. The accepted popular view was that the solar year is 365 days in length. That view was independently known and accepted by the populace at large for other liturgical purposes, <i>viz</i>., for recitation of the <i>tal u-matar</i> prayer. Since the blessing is born of reflection and is designed to evoke acknowledgment of a basic verity, they seized upon this popular assumption and harnessed it as a vehicle for disseminating and reinforcing a fundamental principle of faith.
  This becomes even more obvious if we analyze <i>Birkat ha</i>-<i>Hammah</i> more closely. Why did the Sages not ordain that the blessing be recited Tuesday night? The <i>tekufah</i> of Shmu'el occurs at six o'clock Tuesday evening but we do not recite the <i>berakhah</i> until the sun rises on Wednesday morning. The <i>tekufah</i> in <i>Erez Yisra'el</i> on Tuesday evening, April the 7th, occurs at 5:39 P.M., give or take a few seconds. It occurs, not at 6:00 P.M., but at 5:39 P.M., because solar time depends upon the precise degree and minute of longitude. We have adopted a convention of establishing time zones varying by intervals of one hour. Accordingly, to speak of the time as 6:00 P.M. is accurately to identify the time only at the geographic midpoint of the time zone. True solar time will vary up to a half-hour earlier or a half-hour later depending on particular location east or west of the midpoint of the time zone. So, to speak of <i>tekufat Nisan</i> occurring at 6:00 P.M. is really to say that it occurs at 6:00 P.M. solar time which is 5:39 P.M. Jerusalem time on our clock. Sunset in Jerusalem does not occur until 5:50 P.M. local time. The sun remains visible during the period of time between the occurrence of the <i>tekufah</i> at 5:39 P.M. and sunset at 5:50 P.M.
  If so, why did the Sages ordain that <i>Birkat ha</i>-<i>Hammah</i> be delayed until sunrise Wednesday morning which is many hours after the actual <i>tekufah</i>? True, the blessing may be recited only if one actually sees the sun. But, if the <i>berakhah</i> celebrates the return of the sun to its position at the time of creation, the Sages could have ordained that the blessing be recited Tuesday evening during the eleven-minute window between 5:39 P.M. and 5:50 P.M.
  The answer, I believe, lies in the fact that at the time of creation the sun was placed in the firmament at 5:39 P.M. but, since the sun was created, not at the close of the third day, but at the beginning of the fourth day, sunset on the first Tuesday evening did not occur after the <i>tekufah</i> at 5:50 P.M., but at 5:39 P.M., i.e., the sun was created at sunset. In our day, sunset occurs eleven minutes after the <i>tekufah</i> but that was not the case at the time of the sun's creation.
  Were one to have been present at the creation of the sun it would have been impossible to recite a blessing because the sun had already set. For the Sages to ordain recitation of <i>Birkat ha</i>-<i>Hammah</i> during the period of time after the <i>tekufah</i> that the sun remains visible on Tuesday evening would be incongruous and even self-contradictory. Ostensively, we witness a reenactment of a solar phenomenon that occurred during <i>ma</i>'<i>aseh bereshit</i> and thereupon recite <i>Birkat ha</i>-<i>Hammah</i>. But at <i>ma</i>'<i>aseh bereshit</i> the sun was not visible! So how is it comprehensible that we pronounce a blessing acknowledging our perception of the sun's presence in a certain position at the time of <i>ma</i>'<i>aseh bereshit</i> if, at <i>ma'aseh bereshit</i>, the sun was not visible at that hour? Therefore, the Sages ordained that the blessing be pronounced at an hour commemorating the first visual appearance of the sun subsequent to its creation. Were one to have been present at <i>ma</i>'<i>aseh bereshit</i>, the earliest opportunity to recite the blessing would have been at the first appearance of the sun at sunrise on Wednesday morning.18<i class="footnote">The same point may be demonstrated in yet another manner. The <i>tekufah</i> is an astronomical event that occurs at a specific moment in time. But the hour of that occurrence must be described in local time which is dependent upon longitude. In light of the time difference between Israel and the United States the moment in time described as 6:00 P.M. in Jerusalem is 11:00 A.M. in New York City. Although the <i>tekufah</i> occurs at 6:00 P.M. solar time in Jerusalem, at that moment, the clock in New York reads 11:00 A.M. If so, why do we delay <i>Birkat ha-Ḥammah</i> until Wednesday morning even in New York? The issue was first raised by R. Elijah David Rabinowitz-Te’umim (known as the <i>Aderet</i>), Chief Rabbi of Mir and later of Jerusalem, who ruled that in Russia, where the sun was visible after the time of the <i>tekufah</i>, the blessing be recited Tuesday afternoon at the time of the actual <i>tekufah</i> rather than be delayed until Wednesday morning. A Jerusalem scholar, the late R. Moshe Nachum Spira, accepted that position and argued that in Europe and the United States the blessing must be recited on Tuesday and may not be recited on Wednesday. For a discussion and critique of those views see this writer’s <i>Bircas haChammah</i>, pp. 95-96 and 237-242.Nevertheless, it is the accepted practice throughout the world to delay <i>Birkat ha</i>-<i>Ḥammah</i> until sunrise of the morning following the <i>tekufah</i>, i.e., Wednesday morning. The Sages ordained that the blessing be recited at the hour that the sun became visible in the Land of Israel at the time of creation, i.e., at sunrise of the day following the <i>tekufah</i>. They similarly ordained that Jews all over the world recite the blessing when the sun becomes visible on the morning following the <i>tekufah</i>. This may serve to explain why Rambam, <i>Hilkhot Berakhot</i> 10:18, adds the phrase “in the morning,” which is not found in the Gemara, in ruling that one should pronounce the blessing following the <i>tekufah</i> “when one sees [the sun] on Wednesday <i>in the morning</i>.” Rambam’s emphasis is upon the point that it is not sufficient to observe the sun after the <i>tekufah</i>, but that the blessing is to be delayed until the following morning. If it is understood that the blessing is evocative and does not precisely commemorate an astronomical event, it is quite understandable that the Sages sought to promulgate uniformity of observance and based themselves upon the <i>realia</i> of the Land of Israel in establishing sunrise as the time for recitation of the blessing throughout the world.</i>
  The point is clear. No novel astronomical event occurred at the time of the first sunrise on Wednesday morning. And, of course, no novel astronomical event took place on April 8th at sunrise. The blessing is neither historical nor contemporaneous, using the terms as they have been defined earlier; the blessing is evocative in the sense that the Sages desired that appearance of the sun on April 8th spur us to reflect upon <i>ma'aseh bereshit</i> and to acknowledge the role of <i>amitat mezi</i>'<i>uto</i> in the constant conservation of the universe.
  
  Chapter 4 The Hetter Iska and American Courts
  
  <i>"Your money shall you not give him upon interest…. I am the Lord, your God who brought you out of the land of Egypt" (Leviticus 25:37-38). Upon this condition did I bring you out of the land of Egypt, on condition that you accept the commandment concerning interest.</i>
  SIFRA 22:77
  During the past years, the enforceability of <i>hetter iska</i> agreements has repeatedly been the subject of litigation in American courts. The rising number of challenges to the validity of <i>hetter iska</i> agreements is unwelcome but nevertheless represents eloquent testimony to the increased use of such instruments within the Jewish community. Recognition of the <i>hetter iska</i> by courts of competent jurisdiction as creating a joint venture is a matter of both religious and economic significance to observant Jews. The importance of that issue prompts this writer to revisit the topic1<i class="footnote">Earlier discussions of the basic nature of the <i>hetter iska</i> and its employment in various contexts appear in this writer’s <i>Contemporary Halakhic Problems</i>, II (New York, 1983), 376-396 and IV (New York, 1995), 378-384.</i> in order (1) to examine the propriety of the <i>hetter iska</i> as a means of avoiding the biblical prohibition against paying or receiving interest; (2) to explicate the nature of the <i>hetter iska</i> as a valid halakhic construct; (3) to investigate the question of whether lack of enforceability in a civil court impacts upon the halakhic validity of a <i>hetter iska</i>; (4) to analyze the relevant judicial decisions regarding the efficacy of a <i>hetter iska</i> in American law; and (5) to delineate the manner in which the instrument should be drafted in order to achieve the desired halakhic and legal result.
  I. The Propriety of a Hetter Iska
  Biblical law forbids both exaction of interest by a creditor and payment of interest by a debtor. The underlying considerations are two-fold in nature: 1) The obligation to perform charitable deeds includes a responsibility to help the indigent by means of non-interest bearing provident loans. 2) Interest accepted in return for extending a loan involving neither risk of capital nor personal service is perceived as a form of unjust enrichment.
  Although in biblical days the economic structure of society was predominantly agrarian and loans were sought primarily for personal purposes, the strictures against usury apply with equal force to loans extended for commercial purposes. During the Middle Ages, Jews, particularly when they were forbidden to own land, turned to commercial pursuits. Frequently, they became merchants engaged in the buying and selling of goods and produce. Such pursuits required capital investment of sums which few individual Jews possessed. Loans were generally not available unless they were also of financial benefit to the lender. Since interest was not permitted, ways and means had to be found to facilitate commercial endeavors without violation of either the letter or the spirit of Jewish law.
  The earliest attempt to overcome this difficulty was that of the sixteenth-century Polish scholar known as R. Mendel Avigdors of Cracow, who composed the earliest version of the <i>hetter iska</i>. The <i>hetter iska</i> is a document which specifies the terms and conditions under which money is advanced by one individual to another. Its legal purpose is to create a partnership arrangement as distinct from a debtor-creditor relationship.
  The issue with which rabbinic thinkers were forced to grapple was the propriety of seizing upon lacunae in the law and fashioning contrivances in order to circumvent the violation. A subterfuge may avoid technical violation of the law but nevertheless thwart the spirit of the law.
  Virtually every system of law employs legal fictions for reasons of expediency. A corporation is a legal person and is accorded such status in order to achieve perfectly cogent and moral results. Such "myths" are essentially definitional. The law determines how it wishes to treat certain arrangements and coins a term for use as a type of verbal shorthand in referring to such arrangements. At times, the law, particularly in areas of taxation, provides for options. Determination by a businessman of the beginning and end of his fiscal year is an obvious but relatively trivial example. In such matters, as Judge Learned Hand famously remarked, "Anyone may arrange his affairs so that his taxes shall be as low as possible…. Over and over again the Courts have said that there is nothing sinister in so arranging affairs as to keep taxes as low as possible." 2<i class="footnote"><i>Gregory</i> v. <i>Helvering</i> 69 F.2d 809, 810 (2d Cir. 1934), aff’d, 293 U.S. 465, 55 S. Ct. 266, 79 L.Ed. 596 (1935).</i> An individual acting in such a manner is not guilty of moral turpitude.
  Jewish law is quite familiar with arrangements designed to avoid the onus of applying the full force of the law. In rabbinic literature there is a term for such an arrangement; it is known as a "<i>ha'aramah</i>," a term that probably may best be translated literally as "a cleverness." The term "device" probably best captures the flavor of the term without connotation of a moral judgment. Such devices fall into three distinct categories: (a) those that are disdained as inconsistent with the spirit of the law; (b) those that, for one reason or another, are regarded as commendable and actively encouraged; and (c) those with regard to which there is an attitude of complete neutrality.
  Perhaps the earliest example of a <i>ha'aramah</i> involves regulations pertaining to tithing. Talmudic exegesis establishes that biblical law requires tithing only of produce brought into an abode through the gates of the courtyard. The Gemara, <i>Berakhot</i> 35b, bemoans the fact that, "while earlier generations were wont to bring in their produce by way of the front door in order to make it liable to tithing, later generations bring in [the produce] by way of roofs, yards, or enclosures in order to exempt [the produce] from tithing." Clearly, that statement reflects a negative attitude toward the moral legitimacy of the "later generations" who avail themselves of such expedients. Indeed, according to some early-day authorities, the subterfuge in question was formally banned by rabbinic law.
  The reason for the negative view <i>vis-á-vis</i> utilization of such expedients is self-evident. The purpose of the tithe was to provide a livelihood for the Levites who were required to serve in the Temple and who constituted a leisure class charged with providing religious instruction on behalf of the entire populace. Were tithes not to be forthcoming, the Levites would have been compelled to seek employment as artisans or laborers, thereby making it impossible for them to discharge their sacred mission. The spirit of the law would certainly have been thwarted. Thus, this is an example of a disdained <i>ha</i>'<i>aramah</i>.
  The second category includes devices that are designed actively to promote conformity with the manifold provisions of law. The first-born male offspring of a cow or a sheep belonging to a Jew is sanctified from birth to be brought to the Temple as a sacrificial offering. With the destruction of the Temple, sacrifices are no longer feasible. However the sanctity of the first-born animal is in no way mitigated. The animal may not be used as a beast of burden; it is forbidden to drink its milk or use its wool. The animal is literally a "sacred cow" that roams virtually at will—as is the case even today in some Hindu communities with regard to animals they regard as sacred—until the animal develops a blemish rendering it unfit as a sacrifice. Unless and until that occurs, the animal is a halakhic nuisance in the sense that it is of no benefit to either God or man but poses an ongoing potential for inadvertent sin, "a tort waiting to happen."
  The sole remedy is to prevent the animal from achieving sanctified status by transferring ownership of the pregnant animal to a non-Jew before it gives birth. Subsequent to parturition both animals may be reconveyed to the original Jewish owner. Commandments regarding the first-born commemorate the sparing of Jewish first-born in Egypt while the first-born of the Egyptians perished in the last of the ten plagues. Hence, the first-born of an animal belonging to a non-Jew is not sanctified.3<i class="footnote">The transfer of ownership, however, must be carried out in accordance with the provisions of Jewish law. Such transfer entails a formal act of <i>kinyan</i>, or conveyance. Rules regarding <i>kinyan</i> are quite complex. Modes of <i>kinyan</i> for transfer of chattel between a Jew and a non-Jew differ from the modes of <i>kinyan</i> governing transfer of title between a Jew and a fellow Jew. As a result, one can open virtually any volume of 18th- or 19th- century responsa and find a discussion of at least one case in which the efficacy of an act of transfer performed in such a situation is questioned. Those discussions invariably focus upon technicalities, not upon the propriety of the procedure itself.</i>
  No one has ever suggested that it is improper to seek such transfer of title on the grounds that it is a subterfuge designed to frustrate the purpose of the law. Such circumvention of the law represents an instance in which fulfillment of the underlying purpose has become impossible, i.e., the animal cannot be brought as a sacrificial offering because the Temple no longer stands. Despite that factor, the law is not a dead letter; the restrictions governing sanctified animals remain in full force and effect. The only way to avoid transgression and violation of both the letter and spirit of the law lies in obviating its provisions by causing title to become vested in a non-Jew. That is so even if the calf or lamb and its mother are both subsequently repurchased from the non-Jew. Under such circumstances, employment of a "device" to accomplish that end is worthy of approbation.
  The final category is best exemplified by the practice of selling <i>ḥamez</i>, i.e., food products not suitable for consumption during Passover, to a non-Jew. The sale is effected before the advent of the holiday and, typically, the same foodstuffs are repurchased from the non-Jew upon conclusion of the holiday. To some, the transaction has the appearance of a charade since the parties are fully aware <i>ab initio</i> that the effect of the sale will be rescinded at the earliest permissible opportunity.
  In point of fact, this arrangement is entirely innocuous. Scripture forbids a Jew to retain any economic interest in <i>ḥamez</i> during the course of the Passover week. A Jew in possession of such foodstuffs on the eve of Passover may avoid the onus of transgression in one of three ways: (1) He may consume such foods in their entirety, a feat beyond human capability if the quantity is gargantuan, or inadvisable in the extreme in the case of a copious quantity of alcoholic beverages made from grain. (2) He may totally destroy such items by burning or the like, a procedure that may prove to be cumbersome and is wasteful but, nevertheless, entirely permissible. (3) He may transfer title to a non-Jew who, since the 613 biblical commandments are binding only upon Jews, is in no way subject to the prohibition. Jewish law does not accord preference to any one of these options over the others. Accordingly, the expedient of sale to a non-Jew is regarded as both halakhically and morally neutral; it is neither encouraged nor discouraged. Choice of that option is generally dictated by considerations of practicality and conservation of wealth. There are no restrictions upon acquiring <i>ḥamez</i> after the holiday, even <i>ḥamez</i> owned by a Jew before Passover, provided that the sale to a non-Jew prior to the advent of the holiday has been properly consummated.4<i class="footnote">See R. Moshe Sofer, <i>Teshuvot Ḥatam Sofer</i>, <i>Oraḥ Ḥayyim</i>, no. 102. See also R. Shlomoh Yosef Zevin, <i>Ha</i>-<i>Mo</i>’<i>adim be</i>-<i>Halakhah</i>, 7th ed. (Tel Aviv, 1960), pp. 245-255.</i>
  To the extent that there are intimations of criticism in rabbinic literature regarding such sales, the concern is entirely with regard to legal niceties.5<i class="footnote">Cf., a variant reading of the Tosefta, <i>Pesaḥim</i> 1:6, that is cited and dismissed by <i>Bet Yosef</i>, <i>Oraḥ Ḥayyim</i> 448.The objection voiced by R. Alexander Schorr, <i>Tevu</i>’<i>ot Shor</i>, <i>Pesaḥim</i> 21a, was limited to selling animals for the purpose of feeding them <i>ḥameẓ</i> during the course of the holiday. That objection was emphatically dismissed by a plethora of rabbinic scholars. See <i>Ha</i>-<i>Mo</i>’<i>adim be</i>-<i>Halakhah</i>, p. 250 and p. 250, note 22.</i> Proper modes of formal conveyance, or <i>kinyan</i>, must be employed in transferring title. More significantly, there can be no valid transfer without a meeting of minds. Both parties must grasp the final and irrevocable nature of the transaction. Neither party may regard it is a <i>pro forma</i> ritual or liturgical act devoid of actionable effect. Small wonder, then, that rabbinic eyebrows were raised on occasion, particularly when the purchaser was an untutored and legally unsophisticated peasant.
  In the modern world loans occur in situations varied and sundry. In antiquity, loans were almost always provident in nature. A farmer's crop failed and as a result he was forced to resort to a loan in order to purchase seed for planting the next year's crop. A person became temporarily incapacitated and was reduced to seeking a loan to put bread on the table. A loan of that nature is described by Scripture as an act of charity: "that your brother may live with you" (Leviticus 25:36). Charity assumes many guises. A loan represents a higher form of charity than a gift because the impoverished recipient receives full benefit of the funds advanced without being deprived of dignity. Such loans represent fulfillment of a divine command for which a person dare not demand compensation at the hands of his fellow. Moreover, unlike the majority of commandments with regard to which Scripture does not reveal the underlying rationale, in promulgating the requirement that a loan be interest-free, Scripture is quite explicit with regard to the reasoning: the borrower is in need of funds to sustain himself. The Bible regards profiting from human misery as ignominious; to demand interest is to compound the plight of an already destitute person. In earlier centuries, the Church regarded the taking of interest to be prohibited by natural law; the odiousness of interest-taking was not based upon dogmatic revelation but was regarded as readily apprehensible by the unaided light of moral reason.
  In antiquity, all loans were provident in nature. At some point in human history society found it expeditious to rely upon middlemen to transfer commodities from producer to consumer. Tradesmen require at least a modest amount of capital in order to function in that capacity. Only with the advent of the industrial revolution and the construction of factories for the production of consumer goods did the need for vast amounts of capital arise; only in the modern era does capital represent the most prevalent form of generating more capital.
  A person is under divine command to come to the assistance of his fellow who is in need. However, he is under no obligation to assist his fellow in amassing wealth. The rationale underlying the prohibition against interest-taking is alleviation of human misery. Yet the law acquires a life of its own. In general, Jewish law regards implementation of the law to be independent of the ends the law is designed to promote. That is the case even when those ends have been revealed to man. For that reason, in the Jewish legal system, policy considerations do not figure prominently in judicial decision-making. Hence, the prohibition against giving or taking interest is fully applicable, not only in situations in which societal goals are not advanced thereby, but even in situations in which societal goals may actually be thwarted.
  A device that would enable a lender to reap profit from a provident loan would certainly offend the spirit of the law. But a device that restructures the transaction and renders it an investment makes it possible for the individual providing the financing to participate in the profits generated by the investment. Properly crafted so that it does not violate technical strictures, such a device is morally neutral when used in the context of a commercial enterprise.
  Many such "devices," although entirely permissible under biblical law, were interdicted by rabbinic decree. Those restrictions were instituted either as a means of constructing a "fence" around the biblical law because of fear that if a particular legitimate practice were to be sanctioned it would rapidly result in other practices violative of biblical law or because the arrangement might be regarded as interest-taking in the eye of an unsophisticated beholder. Practices that have come to be known as Islamic banking include a variety of devices forbidden to Jews by rabbinic decree rather than by biblical law. A repurchase agreement is a case in point: instead of lending money at interest, a "lender" sells a piece of property and agrees to repurchase it at a specified future date for a higher sum. The transaction is formally structured as a sale coupled with a repurchase agreement, but is actually a subterfuge. The funds advanced for the "purchase" of the parcel of real property are really a disguised loan since recovery is guaranteed. The "seller's" sole motivation is use of the funds in the interim; the "purchaser's" sole motivation is the interest he receives in the form of an enhanced repurchase price.
  The primary device recognized by Jewish law as an acceptable means of avoiding the prohibition against extending or receiving an interest-bearing loan is known as a <i>hetter iska</i> or "permissible venture."6<i class="footnote">R. Baruch ha-Levi Epstein, <i>Torah Temimah</i>, Leviticus 25:36, sec. 192, defends the introduction of the <i>hetter iska</i> during the medieval period:</i> Its function is to substitute some form of profit-sharing representing a return on invested funds for what would otherwise have been structured as an interest-bearing loan.
  II. The Nature of a Hetter Iska
  The formula of the original <i>hetter iska</i> was based upon a ruling recorded in <i>Shulḥan Arukh</i>, <i>Yoreh De'ah</i> 167:1, which provides that funds may be advanced on the condition that the recipient use the funds to buy and sell merchandise on behalf of the person advancing the funds until such time as a stipulated profit has been earned; profit over and above that amount then accrues without limit to the recipient. During the initial period, the recipient acts solely as a commercial agent on behalf of the principal who advanced the funds; only after the originally stipulated profit has been earned does the deposit revert to the status of a loan and the recipient, who was heretofore considered a bailee, becomes a debtor. Subsequently, no profit whatsoever accrues to the lender and hence there is no question of forbidden payment of interest. The text incorporating this formula, which was drawn up by R. Mendel Avigdors for use in lieu of a promissory note, appears in <i>Naḥalat Shiv'ah</i>, no. 40, and is known as the "<i>Hetter Iska</i> of Maharam."
  The Ga'on of Vilna, (known as the Gra), <i>Bi'ur ha-Gra</i>, <i>Yoreh De'ah</i> 167:1, expresses amazement at <i>Shulḥan Arukh</i>'s ruling that money may be lent in this manner. In the situation which is outlined, the "agent" undertakes to engage in the purchase and sale of merchandise on behalf of the "principal." However, his sole motivation for doing so is the stipulated agreement between the parties to the effect that the sum advanced will subsequently be available for the recipient's own personal use as a loan. Labor or personal services performed on behalf of a creditor as an inducement to extend a loan is a form of interest forbidden by Jewish law. Performance of such services without compensation is, in fact, a disguised interest premium exacted as a precondition demanded for conversion of the funds to a loan. It is for this reason that <i>Shulḥan Arukh</i> stipulates that the "agent" must be paid for his services, although the payment may be nominal in nature. Such nominal payment to avoid the stricture against usury is analogous to the peppercorn recognized in common law as consideration validating a contract. The Gra quite evidently regarded token payment as inadequate; forgoing usual compensation was regarded by him as a form of interest-taking.7<i class="footnote">One way to avoid the problem is to structure the half-loan, half-deposit so that, although profits are shared equally, the recipient is to bear maximum loss of only one third of the capital. In effect, the difference between one half and one third is “insured” by the financier in return for the services performed by the recipient. That arrangement renders performance of personal services as payment in lieu of a “premium” that might ordinarily be required to insure against loss of the difference between one third and one half of the investment. See the <i>hetter iska</i> of R. Abraham Brode published in <i>Noheg ke-Ẓon Yosef, masa u-matan</i>, sec. 4. Cf., R. Ya’akov Yeshaye Blau, <i>Brit Yehudah</i> 37:4, 40:6, note 12 and 40:1, note 9. This expedient is employed in the <i>hetter iska</i> used by Israeli banks.</i>
  In terms of making capital available for commercial purposes, an obvious drawback with regard to this arrangement is that the "interest" received by the lender is not tied to the period of time his capital is used by the borrower and does not provide for accrual of interest or sharing of additional profit in the event that repayment is delayed. Subsequently, a modified form of the <i>hetter iska</i> which had the effect of eliminating this difficulty was developed by the 18th-century authority, R. Abraham Danzig of Vilna, renowned as the author of <i>Hayyei Adam</i> and <i>Hokhmat Adam.</i>8<i class="footnote">See <i>Ginat Veradim</i>, <i>Yoreh De’ah</i>, <i>klal</i> 7, no. 4.</i> Under the terms of this modified <i>hetter iska</i>, the recipient accepts fifty percent of the funds as a deposit and fifty percent as a loan. The sum advanced in this manner is termed "half-loan, half-deposit" in rabbinic literature. Any expenditure of a commercial or investment nature is deemed to be a joint venture in which both parties share equally. Thus, technically, half the profits accrue to the "principal" and half to the borrower. The principal may, of course, agree to accept no more than a certain stipulated amount of the profits and the agent may demand that the agreement specify that profits exceeding that amount accrue to the agent as compensation for his services.
  Transformation of the <i>hetter iska</i> into a transaction in the form of half loan, half deposit is to the advantage of the investor in the sense that only half his funds are at even theoretical risk. Another advantage is that it makes accrual of presumed periodic profits possible with the result that procrastination in a repayment results in greater profit to the investor.
  It may well be the case that this form of <i>hetter iska</i> was also developed, in part, to overcome the objection raised by the Gra, <i>viz</i>., that mere token consideration in return for personal services is not sufficient to overcome the taint of usury. That objection is grounded in the concern that the recipient expends time and effort on behalf of the financier during the initial period of the arrangement in which the funds are invested solely on behalf of the financier and, clearly, that he is motivated to do so only because he desires that the sum advanced together with the profits generated during that period convert to an interest-free loan as rapidly as possible. However, it may be argued that this is not the case with regard to funds received as a half loan, half deposit. In the latter case, the recipient would expend no less time and effort were he to seek investment opportunities only for the funds he receives as an interest-free loan. No additional time or effort is expended because the investor's funds are commingled with the recipient's borrowed funds. Quite to the contrary, it is more likely than not that any entrepreneur would welcome the opportunity to double the quantity of merchandise he seeks to buy with his own already available funds by acting at the same time as an agent for another individual desirous of acquiring the same type of merchandise or in making a similar investment. Doubling the quantity purchased or the amount of money invested affords the opportunity for taking advantage of economy of scale.
  To offer a simple example: Jumbo certificates of deposit require a much higher minimum investment than ordinary certificates of deposit, but yield a higher premium. The amount of paper work involved in securing a jumbo certificate of deposit is no different from the amount of paper work required to secure an ordinary certificate of deposit. Assume that A has only $50,000 to invest but that the minimum requirement for a jumbo certificate of deposit is $100,000. Assume also that B has $50,000 available to invest. Under such circumstances prudence and good business sense would dictate that one of the parties should offer to take custody of the other party's $50,000, commingle the funds, and invest them in a joint certificate of deposit as tenants in common. The motive for doing so is not altruism in helping a friend but self-interest in becoming able to qualify for a higher yield on one's own investment. Similarly, in purchasing merchandise, it is highly likely that, the greater the quantity bargained for, the lower the cost per unit. Hence, it is certainly arguable that in accepting funds in the form of half loan, half deposit, the recipient is motivated solely by self-interest and performs no additional services as compensation for receipt of the loan.9<i class="footnote">Cf., <i>Tosafot</i>, <i>Bava Meẓi’a</i> 104b, s.v. <i>hai iska</i>. The arrangement described by <i>Tosafot</i> does not seem to involve a potential for economy of scale.</i>
  In neither of these formulations is the <i>hetter iska</i> a subterfuge or legal fiction. The determining characteristic which distinguishes a loan from a deposit or bailment is that a debtor must always repay the full amount borrowed, regardless of any losses he may suffer, while the laws of bailment provide that the bailee need not indemnify the bailor in the event of certain types of losses. Since the <i>hetter iska</i> generates a principal-agent relationship, no payment need be made if <i>bona fide</i> losses occur while the borrower functions as a bailee. Under the terms of Maharam's formula there is no obligation whatsoever on the part of the recipient if the loss is sustained during the initial period; according to the modified version, such losses are borne equally by the "principal" and the "borrower." The "principal" may, however, set forth certain stipulations to protect himself against loss. He may, for example, stipulate that the agent must bear all losses unless the merchandise purchased is at all times under the latter's personal supervision or unless the agent produces two trustworthy witnesses who testify that he had indeed properly discharged all aspects of his trust.
  Conditions such as these serve to protect the capital of the creditor, but do not at all guarantee the payment of "interest" in the event that the debtor denies that he has earned a profit. This drawback was overcome by means of a further stipulation providing that the "principal" may assume that a profit has been earned and that this presumption may be nullified only upon the solemn oath of the "agent" to the contrary. That stipulation has meaningful implications because of a singular socio-religious factor unique to Jews, Quakers, and few others. Pious Jews abjure oaths, even when called upon to swear to matters which are entirely true. Accordingly, the net effect of a stipulation requiring an oath to substantiate an absence of profit is to assure that the creditor will receive "interest" in the form of a guaranteed "profit" even if, in fact, no profit has been earned in any commercial endeavor. Thus, even if no profit has been realized, the stipulated payment is legitimately proffered as a fee for exoneration from the oath that would otherwise be imposed in order to substantiate the absence of profits. The payment is in the nature of an accord and satisfaction.
  Whether or not such an oath may be demanded when the "principal" has personal knowledge of the truth of the "agent's" claim that no profit has been earned is a matter of some dispute. Some authorities assert that an oath attesting to facts already known by the party demanding the oath is forbidden as an oath taken "in vain," i.e., to no purpose. Of course, if an oath cannot be exacted, payment in lieu of the oath cannot be demanded.10<i class="footnote"><i>Teshuvot Shai</i>, I, no. 3; <i>Teshuvot Panim Me’irot</i>, II, no. 3; <i>Teshuvot Divrei Ḥayyim</i>, II, <i>hashmattot</i>, no. 16; and R. Moshe Feinstein, <i>Iggerot Mosheh</i>, <i>Yoreh De’ah</i>, II, nos. 62 and 63, maintain that such an oath cannot be demanded. However, R. Shimon Grunwald, <i>Teshuvot Maharshag</i>, <i>Yoreh De’ah</i>, no. 4, disagrees.</i>
  In an article that originally appeared in <i>Torah she-be-al Peh</i>, XIX (5727), 100-105, and was reprinted in his posthumously published <i>Mishkan Shiloh</i> (Jerusalem, 5755), pp. 281-288, Rabbi Shiloh Rafael notes that, in our day, an aversion to oath-taking is no longer to be taken for granted. Understandably, there may therefore be reluctance on the part of some persons to utilize the customary <i>hetter iska</i>. The lender may well anticipate a contingency in which the borrower will indeed swear quite truthfully that he has earned no profit. Rabbi Rafael suggests that the requirement for an oath on the part of the debtor be replaced with a stipulation giving the creditor the right to examine the financial records of the debtor and the right to participate in all decisions with regard to the expenditure of the sum advanced. The agreement between the parties would provide that this right be abrogated upon payment of a specified sum or percentage of the capital at the discretion of the debtor.
  Nevertheless, in recent years, perhaps because of concern for the qualms expressed by the Ga'on of Vilna, a new form of <i>hetter iska</i> has developed among members of the stringently observant community. That agreement is colloquially referred to as the "<i>hetter iska</i> of Bnei Brak" because of the locale in which it gained currency.11<i class="footnote">See R. Moshe Sternbuch, <i>Mo’adim u-Zemanim,</i> VI, no. 41, secs. 1(3) and 1(7). This form of <i>hetter iska</i> seems to have been introduced by R. Ya’akov Breisch in Switzerland. See R. Ya’akov Breisch, <i>Teshuvot Ḥelkat Ya’akov</i>, III, no. 188, sec. 2, no. 189, sec. 1 and no. 194, sec. 8, s.v. <i>akhen</i>.</i> The arrangement is designed to eliminate any possible halakhic pitfall by providing that the funds are advanced in their entirety solely as an investment. Provision is made for payment of a stipulated periodic rate of return as profit upon the investment with any balance in excess of that rate of return to be retained by the recipient. Since there is no loan whatsoever, there can be no taint of interest. All other provisions of such an agreement are identical to those of a standard <i>hetter iska</i>. A return, even when no profit has been realized, is assured by stipulating a presumption of profit rebuttable only by a solemn oath. Capital is protected by stipulating that only the testimony of two halakhically qualified witnesses is acceptable as proof of loss. The drawback of that arrangement is that, unlike with regard to other forms of <i>hetter iska</i>, since no portion is in the form of a loan, the funds advanced are at risk in their entirety.
  The <i>hetter iska</i> was originally designed to facilitate the lending of capital for commercial purposes. Since the capital was, in fact, invested in mercantile enterprises, and since the creditor assumed a measure of risk, the relationship which was formed was—factually as well as technically—that of partners in a business venture. However, with the passage of time, use of the <i>hetter iska</i> became more and more widespread. Greater and greater numbers of Jews earned their livelihood as craftsmen or as hired laborers. The use of a <i>hetter iska</i> by a person who is not in any way engaged in a business enterprise appears to be nothing more than a subterfuge. A prominent twentieth-century decisor, Rabbi Moshe Feinstein, <i>Iggerot Mosheh</i>, <i>Yoreh De'ah</i>, II, no. 62, totally ignoring earlier responsa dealing with this problem, strongly implies that a <i>hetter iska</i> may by employed only when the borrower is engaged in business ventures. Moreover, declares Rabbi Feinstein, both parties must know the nature of the commercial venture for which the money is advanced for "[the <i>hetter iska</i>] is not an incantation or charm."12<i class="footnote">Similarly, <i>Mo’adim u-Zemanim</i>, VI, no. 41, sec. 1(1) writes: “Many err with regard to the intention of Maharam and imagine that [the <i>hetter iska</i>] is a chanson (<i>pizmon</i>) to render the prohibition against interest inoperative.”</i> Rabbi Feinstein adds that the written agreement should state that the money advanced must be used for ventures in which it may reasonably be anticipated that the total profit will be at least twice the amount of the stipulated return. Thus, according to <i>Iggerot Mosheh</i>, the <i>hetter iska</i> is reserved for use in situations in which the interest paid is, in reality, no more than the actual anticipated profit realized on the portion of the merchandise acquired on behalf of the creditor.13<i class="footnote">Conflicting opinions with regard to the efficacy of a <i>hetter iska</i> in conjunction with non-commercial loans and theories upon which they may be justified are discussed in this writer’s <i>Contemporary Halakhic Problems</i>, II, 379-381.</i>
  The question of the efficacy of a <i>hetter iska</i> in conjunction with non-personal loans was raised earlier, at approximately the turn of the twentieth century, and discussed in detail by Rabbi Shalom Mordecai Schwadron, <i>Teshuvot Maharsham</i>, II, no. 216. Advancing a view contrary to that later espoused by <i>Iggerot Mosheh</i>, Rabbi Schwadron justifies the use of a <i>hetter iska</i> in conjunction with loans that are entirely of a personal nature. He concedes that when the funds have not been used for commercial purposes, and no profits have been earned, no interest need be paid. The borrower, however, certainly has the legal and halakhic option of using the money advanced for such purposes; indeed, that is the <i>prima facie</i> purpose of the loan as stated in the written agreement. Under the terms of the agreement the financier may demand a solemn oath that the money has not, in fact, been used to generate profit and that, in fact, no profit has been earned. The amount stipulated as "accord and satisfaction" is then paid by the recipient solely as a means of avoiding the oath which he is otherwise bound to swear.
  In the nineteenth century, R. Joseph Saul Nathanson, <i>Teshuvot Sho'el u-Meshiv</i>, <i>Mahadurah Kamma</i>, III, no. 160, justified an <i>iska</i> arrangement in conjunction with personal loans, for at least some purposes, on entirely different grounds. According to this authority, a loan which makes it possible for an individual to continue practicing his profession or to retain his income-generating employment is to be deemed a commercial enterprise. Therefore, according to <i>Sho'el u-Meshiv</i>, a teacher who is burdened by debt or by personal expenses may borrow funds under an <i>iska</i> arrangement if failure to receive the loan would force him to abandon his profession and to seek a livelihood in some potentially more lucrative endeavor. Since the loan enables him to continue in his occupation, the loan constitutes a form of <i>iska</i> in which the financier becomes a partner. The application of this line of reasoning to student loans designed to enable the student to pursue studies which will qualify him to enter a profession or to make him eligible for higher salaried employment is obvious.
  In another responsum, <i>Sho'el u-Meshiv</i>, <i>Mahadurah Telita'ah</i>, I, no. 133, Rabbi Nathanson sanctioned an <i>iska</i> arrangement in order to enable the recipient to repay a debt and thereby to avert the forced sale of his home. This is categorized as an <i>iska</i> by <i>Sho'el u-Meshiv</i> by virtue of its service in preventing loss of capital resulting from the otherwise economically forced sale of the house and/or the opportunity to realize the contemplated appreciated value of the property.
  The permissive views of <i>Sho'el u-Meshiv</i> were sharply disputed in the early part of the twentieth century by R. Meir Arak, <i>Teshuvot Imrei Yosher</i>, I, no. 108. <i>Imrei Yosher</i> maintained that the concept of <i>iska</i> is limited to profits realized from the investment of capital advanced or from sale of merchandise or goods acquired with such capital. Profits realized from avoidance of loss or from income derived from personal employment do not constitute the fruits of a joint venture and hence, according to <i>Imrei Yosher</i>, do not justify use of a <i>hetter iska</i>. This latter view is consistent with the position of most authorities, including <i>Ginat Veradim</i>, <i>Yoreh De'ah</i>, <i>klal</i> 6, no. 4; <i>Shulḥan Arukh ha-Rav</i>, <i>Hilkhot Ribbit</i>, sec. 42; <i>Kizur Shulḥan Arukh</i> 66:10; and <i>Erekh Shai</i> 177:7.
  A contemporary scholar, Rabbi Moshe Sternbuch, <i>Mo'adim u-Zemanim</i>, VI, no. 41, sec. 2(7), s.v. <i>ve-hineh,</i> points out that with widespread investment in the stock market on the part of small investors there can be no question with regard to the creditor's lack of personal knowledge that no profit has been earned. Indeed, it is not uncommon for individuals who borrow money for personal purposes to use such funds in an attempt to reap short-term profits in the securities market. Since the creditor cannot know that this has not, in fact, occurred, he may, according to all authorities, stipulate that an oath be sworn to that effect.
  III. Must a Hetter Iska Be Enforceable in a Civil Court?
  Whether or not the terms of a <i>hetter iska</i> are enforceable in a secular court, and whether a civil court would treat the instrument as a loan agreement or as establishing a joint venture would appear to be of little import in determining its validity in Jewish law. The principle "<i>dina de-malkhuta dina</i>—the law of the land is the law" is irrelevant with regard to financial and commercial dealings between members of the Jewish faith.14<i class="footnote">See <i>Shakh</i>, <i>Ḥoshen Mishpat</i> 73:39 and <i>Ḥazon Ish</i>, <i>Ḥoshen Mishpat</i>, <i>Likkutim</i> 16:1.</i> The law of the land does not supersede the rules of Jewish jurisprudence, most particularly in situations in which the state has no interest in thwarting private arrangements between the parties.
  Nevertheless, the matter becomes more complex with regard to the validity of contracts. Contracts are generally not enforceable unless there is a meeting of minds, i.e., seriousness of intent and reliance by both parties. Thus, even though transfer of real property can be consummated by delivery of consideration, the Gemara, <i>Kiddushin</i> 26a, declares that in a locale in which a deed is customarily delivered to the purchaser the sale is not valid unless such a deed is executed. The reason, as explained by Rashi, <i>ad locum</i>, is that in the absence of a deed, the purchaser does not rely upon the conveyance. The question, then, is whether there is seriousness of intent with regard to assumption of a contractual obligation that cannot be enforced in a secular court.15<i class="footnote">Cf., however, <i>Ḥazon Ish</i>, <i>Ḥoshen Mishpat</i>, <i>Likkutim</i> 16:5, who asserts that a biblical conveyance is valid regardless of local practice. The rule formulated in <i>Kiddushin</i> 26a, asserts <i>Ḥazon Ish</i>, is a rabbinic decree limited to transfer of real property without a deed. However, later, <i>Bava Kamma</i> 10:9, <i>Ḥazon Ish</i> reversed himself and ruled that as a matter of intrinsic law title remains vested in a Jewish seller until such time as the conveyance is consummated in accordance with applicable civil law. This is also the position of R. Shlomoh Zalman Auerbach, <i>Ma’adanei Ereẓ</i>, no. 18 and no. 19, sec. 9.</i> Ordinarily, that is not a problem because Jewish law requires its adherents to abjure recourse to secular courts and to submit all disputes to a <i>bet din</i> for adjudication. It also categorizes any recovery obtained in a civil court that would not have been allowed by a <i>bet din</i> as extortion.
  The issue becomes germane primarily in instances involving a Jew and a non-Jew, in which the contract is not enforceable in civil law but is designed to satisfy a requirement of Jewish law.16<i class="footnote">Brief surveys of this issue are presented by R. Shlomoh Yosef Zevin, <i>Ha-Mo’adim be-Halakhah</i>, 7th ed. (Tel Aviv, 5720), p. 249 and <i>idem</i>, <i>Le-Or ha-Halakhah</i>, 2nd ed. (Tel Aviv, 5717), pp. 119-122.</i> The earliest discussion of this issue arose with regard to the sale of <i>ḥamez</i> before Passover to a non-Jew. In the 19th century, the Austro-Hungarian Empire imposed a tax upon conveyance of chattel in the form of payment for a tax stamp that they required be affixed to the bill of sale. Jews were not in the practice of purchasing stamps for use in conjunction with the sale of <i>ḥamez</i> and were denounced to government officials as tax-evaders. The matter reached the ears of the Emperor Franz Josef, who had a justly-deserved reputation as a philosemite. His reaction was that it is common knowledge that sale of <i>ḥamez</i> is not a mercantile transaction designed for profit but entirely a matter of religious scruple. As such, the Emperor had no desire to burden religious practice by taxation.17<i class="footnote">A similar rationale is invoked by R. Moshe Sternbuch, <i>Teshuvot ve-Hanhagot</i>, I, no. 295, in sanctioning sale of <i>ḥameẓ</i> in South Africa to a black person during the period in which apartheid laws were in effect. Apparently, the concern was with regard to the sale or lease of real property in which the <i>ḥameẓ</i> was located. See his <i>Mo’adim u-Zemanim</i>, III, no. 269, note 1, s.v. <i>amnam</i>, where Rabbi Sternbuch makes a similar point regarding subleasing the places in which <i>ḥameẓ</i> is stored in situations in which the lessee is not entitled to sublet. The identical argument appears in <i>Yerushat Pleitah</i>, no. 12, with regard to sale of <i>ḥameẓ</i> in which the government has established a legally enforceable monopoly on the purchase of grain.<i>Ḥazon Ish</i>, <i>Ḥoshen Mishpat, Likkukim</i> 16:7, states that conveyances between Jews in reliance upon Jewish law are effective in jurisdictions in which the authorities raise no objection to such <i>ultra vires</i> arrangements and in which Jews are permitted recourse to a <i>bet din</i>. See also R. Yitzchak Ya’akov Weisz, <i>Minḥat Yiẓḥak</i>, VI, no. 170, sec. 22.</i>
  Although the Emperor was satisfied, R. Baruch Frankel, author of <i>Barukh Ta'am,</i> reportedly was not. R. Baruch Frankel inferred from the Emperor's response that failure to affix tax stamps did indeed render the sale void and unenforceable. If so, he was hesitant with regard to its validity in Jewish law. Presumably his concern was that, even if in Jewish law the contract is valid in form, the non-Jew would not rely upon the conveyance. R. Moshe Sofer, <i>Teshuvot Hatam Sofer</i>, <i>Oraḥ Hayyim</i>, no. 113, disagreed, but only because he understood Austro-Hungarian law differently (and probably more correctly). <i>Hatam Sofer</i> asserted that failure to abide by the tax statute did not serve to void the sale; it compelled only payment of the tax prior to institution of legal proceedings based upon the transaction.18<i class="footnote">See also R. Menachem Krochmal, <i>Teshuvot Ẓemah Ẓedek</i>, no. 61; <i>Teshuvot Shev Ya’akov, Oraḥ Ḥayyim</i>, no. 21 as well as R. Samuel Eliezer Stern, <i>Mekhirat Ḥameẓ ke-Hilkhato</i> (Bnei Bark, 5747) 16:6. Cf., the bill of sale drafted by R. Shenuer Zalman of Liadi, addendum to <i>Shulḥan Arukh ha-Rav</i> (Brooklyn, 5745), III-IV, 100 [1406], authorizing the non-Jew to translate the document and to pay the applicable taxes in order to render the contract enforceable in civil courts.</i>
  A somewhat different version of the problem appears in a later responsum authored by the son-in-law of R. Baruch Frankel, R. Chaim Halberstam, <i>Teshuvot Divrei Hayyim</i>, II, <i>Oraḥ Hayyim</i>, no. 36. For reasons of Jewish law, conveyance of <i>ḥamez</i> is usually accompanied by sale or lease of real property so that the <i>ḥamez</i> may be transferred by <i>kinyan</i> in the form of <i>ḥazer</i>, i.e., placement in the beneficiary's domain, and by accession or <i>agav karka</i>. The Austro-Hungarian Empire apparently had a law similar to England's Statute of Frauds requiring that transfer of real property be in writing. <i>Divrei Hayyim</i>'s interlocutor apparently had concern with regard to the validity in Austro-Hungarian law of an instrument drafted in Hebrew.19<i class="footnote">Other authorities preferred use of the vernacular in order to assure that the non-Jew understands the nature of the transaction and that he enter into it with the requisite seriousness. Cf., <i>Mekhirat Ḥameẓ ke-Hilkhato</i> 10:10 and chap. 16, note 10.</i> Apparently, in that jurisdiction, instruments written in a foreign language were of no legal effect.20<i class="footnote">See <i>Ḥelkat Ya’akov</i>, III, no. 192. Cf., however, <i>Teshuvot Maharsham</i>, II, no. 223, who cites statutes indicating that drafting an instrument in a foreign language did not void what would otherwise have been a valid oral contract. However, <i>Maharsham</i> contended that, in any event, a contract for the sale of <i>ḥameẓ</i> was not enforceable under applicable civil statutes because it is in the nature of a <i>Scheinvertrag</i>, i.e., a sham or fictitious contract.</i> <i>Divrei Hayyim</i> responds that, under the applicable civil law, absent a valid deed, the sale is not void but merely voidable 21<i class="footnote">A number of subsequent authorities adopt the view that the sale is efficacious for purposes of Jewish law so long as the civil authorities do not interfere with transfers of property not in conformity with civil statutes. See <i>Ḥazon Ish</i>, <i>Ḥoshen Mishpat</i>, <i>Likkutim</i> 16:5 and 16:8; R. Moshe Sternbuch, <i>Teshuvot ve</i>-<i>Hanhagot</i>, I, no. 295; <i>Yerushat Pleitah</i> no. 12; and R. Moshe Nachum Spira, <i>Mishnat Kesef</i>, I, no. 41. As noted <i>supra</i>, note 15, <i>Ḥazon Ish</i>, <i>Bava Kamma</i> 10:9, later reversed himself. See also R. Moshe Sternbuch, <i>Shemitah ke-Hilkhatah</i>, rev. ed. (Jerusalem, 5753), pp. 124-125.</i> and, moreover, that even when accompanied by a written instrument, the sale is voidable unless registered in a land registry office.22<i class="footnote">For an extensive discussion of sources addressing the validity of transfer of real property absent registration in a registry office even when the sale is not effected in order to avoid transgression of religious law, see R. Ya’akov Yeshaye Blau, <i>Pitḥei Ḥoshen</i>, VII (Jerusalem, 5754) 2:11, note 14. For a discussion of the applicability in such circumstances of <i>dina de-bar meẓra</i>, i.e., the law giving the owner of an abutting property priority with regard to purchase, see R. David Horowitz, <i>Teshuvot Kinyan Torah</i>, V, <i>Ḥoshen Mishpat</i>, no. 146, sec. 4. See also <i>supra</i>, note 21. Cf., R. Yechiel Michel Epstein, <i>Arukh ha-Shulḥan</i>, <i>Ḥoshen Mishpat</i> 190:25, who regards registration of the deed as a condition that upon fulfillment retroactively confers validity upon the transfer.</i> Nevertheless, argues <i>Divrei Hayyim</i>, the validity of both an oral transfer and an unregistered transfer, even if void under the law of the land, is determined solely by Jewish law. <i>Divrei Hayyim</i> dismisses the contention that the non-Jew does not rely upon such a conveyance with the comment that a non-Jew "knows that the Jew will not renege without acquiescence of the non-Jew in order that his <i>ḥamez</i> not become forbidden." In other words, the non-Jew recognizes the critical nature of the transaction in avoiding transgression of a biblical command and, since he is confident that the Jew will not wish to sin, he relies upon the conveyance because he is convinced that the Jew will make no attempt to seek to rescind the sale by suing in a civil court.
  A rather similar argument was advanced by R. Yitzchak Elchanan Spektor, <i>Be'er Yizḥak</i>, <i>Oraḥ Hayyim</i>, no. 7. As earlier noted, Jewish law provides that in a locale in which sale of real estate is generally memorialized by a deed, a sale for cash alone is not valid. <i>Be'er Yizḥak</i> contends that if, however, it is patently clear that the seller will not renege, e.g., he seeks to divest himself of land because it is not productive (<i>mipnei ra'atah</i>), the sale is valid even if consummated only by delivery of consideration. <i>Be'er Yizḥak</i> argues that a Jew will not renege on a sale of <i>ḥamez</i> on the eve of Passover because invalidating the sale would leave him with <i>ḥamez</i> from which he may derive no benefit. Hence the non-Jew has every reason to rely upon even an unconventional mode of transfer.23<i class="footnote">Cf., the opposing view earlier advanced by R. Isaac Trani, <i>Teshuvot Maharit</i>, II<i>, Ḥoshen Mishpat</i>, no. 65, who limits the rule to situations in which the deed serves as <i>kinyan</i>. In such cases, in a sale <i>mipnei ra’atah</i>, there is reliance upon the deed alone. <i>Maharit</i> contends that in a locale in which a deed is customarily executed there is no reliance upon payment of cash alone since there is no proof of sale. The issue discussed by <i>Be’er Yiẓḥak</i> is the sale of <i>ḥameẓ</i> to a non-Jew on the eve of Passover without a valid written conveyance. The question is, since the sale is <i>maḥmat ra’atah</i>, can payment of cash be relied upon as a mode of conveyance? <i>Be’er Yiẓḥak</i> rebuts <i>Maharit</i>’s view on the basis of statements of <i>Teshuvot ha-Rashba</i>, I, no. 1, 226 and <i>Teshuvot R. Beẓalel Ashkenazi</i>, no. 8. See also <i>Netivot ha-Mishpat</i> 191:2.</i>
  A similar problem presented itself with regard to sale of farmland in the Land of Israel in anticipation of the sabbatical year in order to avoid difficulties arising from observance of the laws of the sabbatical year. Palestine was part of the Ottoman Empire from 1517 until after World War I. Ottoman law did not recognize the validity of transfer of real property unless the transfer was recorded in a land-registry office. That procedure was not followed in the transfer of land in advance of the sabbatical year. R. Abraham I. Kook, <i>Shabbat ha-Arez</i>, introduction, sec. 13, arrives at a conclusion similar to that earlier recorded by <i>Divrei Hayyim</i> in the context of the sale of <i>ḥamez</i>, <i>viz</i>., "when the essence of the transfer is to obviate a prohibition, the grantor firmly intends to convey [by means of] such a transfer and the grantee [firmly intends] to acquire, there is proper reliance."24<i class="footnote">See the similar position of R. Isaac Elchanan Spektor expressed in a letter appended to R. Jonathan Abelman, <i>Torat Yehonatan</i> (Vilna, 5649), affirming the validity of sale of land in <i>Ereẓ Yisra’el</i> in a manner valid only according to Jewish law in order to avoid <i>shmittah</i> prohibitions. See also R. Shlomoh Zalman Auerbach, <i>Ma’adanei Ereẓ</i>, no. 18 and no. 19, sec. 9 and R. Shlomoh Yosef Zevin, <i>Le-Or ha-Halakhah</i>, 2nd ed. (Tel Aviv, 5717), pp. 119-121. See the contradictory view of <i>Divrei Emet</i> and its rejection by <i>Ma’adanei Ereẓ</i>, <i>loc. cit</i>.</i>
  In a similar manner, R. Shalom Mordecai Schwadron, <i>Teshuvot Maharsham</i>, I, no. 20, permitted use of a <i>hetter iska</i> by a bank even though, contrary to applicable civil law, no provision for a <i>hetter iska</i> was incorporated in the by-laws of the bank. That view was endorsed by R. Meir Arak, <i>Teshuvot Imrei Yosher</i>, I, no. 189.25<i class="footnote">R. Ya’akov Yeshaye Blau, <i>Brit Yehudah</i> 40:11, note 21, cites R. Yitzchak Schmelkes, <i>Teshuvot Bet Yiẓḥak</i>, <i>Yoreh De’ah</i>, II, <i>Kuntres Aḥaron</i>, no. 32, as expressing an opposing view. However, it seems to this writer that <i>Bet Yiẓḥak</i> is in agreement with <i>Maharsham</i> since <i>Bet Yiẓḥak</i> writes that the <i>hetter iska</i> is effective because, since it is written to satisfy provisions of Jewish law, it may be presumed that the creditor will abide by the decision of a <i>bet din</i>.</i> This also appears to be the view of R. Joseph Saul Nathanson, <i>Teshuvot Sho'el u-Meshiv</i>, <i>Mahadura Kamma</i>, II, no. 136, s.v. <i>u-mah she-katavta</i>.
  As stated earlier, both the Jew and the non-Jew rely upon Jewish law as governing the sale of <i>ḥamez</i> because the non-Jew recognizes that the Jewish party will abide by the agreement rather than commit a transgression. The non-Jew will not renege because the terms of the sale guarantee him a profit. But is this also the case with regard to a <i>hetter iska</i> in which one of the parties is a non-observant Jew who is unconcerned with the prohibition against usury and who may find it to his financial advantage to challenge the validity of the agreement in a civil court?
  R. Moshe Schick, <i>Teshuvot Maharam Shik, Yoreh De'ah</i>, no. 161, s.v. <i>mihu</i>, observes that, even if an attempt is made by the financier to bring an action in a civil court to demand that the document be construed as an ordinary interest-bearing loan, the recipient of the funds incurs no transgression. Nor, from the vantage point of Jewish law, has the financier transgressed the prohibition against accepting interest; the financier is an extortionist but not a recipient of interest.26<i class="footnote">See also <i>Teshuvot Ḥelkat Ya’akov</i>, III, no. 195, sec. 9.</i>
  R. Yitzchak Ya'akov Weisz, <i>Teshuvot Minḥat Yizḥak</i>, IV, no. 16, secs. 4, 7-8 and no. 18, sec. 4, questions the validity of a <i>hetter iska</i> whose provisions would not be enforced by a civil court in situations in which one or both of the parties would have no compunction with regard to having recourse to a civil court rather than to a <i>bet din</i>. His concern is that some individuals do not really intend to be bound by the provisions of the <i>hetter iska</i>,27<i class="footnote">Although oral contracts are valid in Jewish law, failure to reduce the agreement to writing may create a situation in which the requisite seriousness of intent does not exist. For that reason, R. Menachem Mendel Schneersohn, <i>Teshuvot Ẓemaḥ Ẓedek</i>, <i>Yoreh De’ah</i>, no. 88, and R. Shalom Mordecai Schwadron, <i>Teshuvot Maharsham</i>, II, no. 123, insist that the <i>hetter iska</i> must be executed in writing. See also R. Chaim Yehudah Litvin, <i>Sha’arei De’ah</i>, no. 5. <i>Maharsham</i> also makes the same point with regard to sale of <i>ḥameẓ</i>. Cf., <i>Shitah Mekubbeẓet</i>, <i>Ketubot</i> 56a, s.v. <i>ve-zeh leshon ha-Ritva</i>.</i> and hence a <i>hetter iska</i> that is not enforceable in a civil court does not constitute a valid agreement.28<i class="footnote">Cf., however, <i>Teshuvot Ḥelkat Ya’akov</i>, III, no. 194, sec. 8, who argues that, fundamentally, as stated by Rambam, <i>Hilkhot Geirushin</i> 2:20, every Jew desires to abide by the laws of the Torah and hence the requisite state of mind exists at the time the transaction is consummated. Cf., however, <i>idem</i>, no. 197, sec. 23.</i> He does, however, suggest that this may not be a concern if it is only the recipient of the funds who is unconcerned with regard to the prohibition against bringing suit in a civil court. Since the provisions of the <i>hetter iska</i> are to the advantage of the recipient in providing for release from financial obligations upon adducing the stipulated proof of loss of capital and release from payment of a share of profits upon his oath that there was no profit, it is to his advantage to rely upon the <i>hetter iska</i> and hence there is no reason why he should not reach the requisite determination to be bound by its provisions. However, <i>Minḥat Yizḥak'</i>s argument fails in a situation in which the stipulated return would constitute usury in civil law. In such an event, the recipient would certainly have a motive to seek to have a civil court invalidate the <i>hetter iska</i> because, in light of the usurious return stipulated by the parties, the debt would not be actionable in a civil court.
  IV. The Hetter Iska in American Law
  As has been explained earlier, the <i>hetter iska</i> constitutes a device that is in full conformity with Jewish law. The <i>hetter iska</i> has also been recognized by the civil courts as an instrument creating a <i>bona fide</i> partnership interest providing for a limited return upon an investment rather than as a loan instrument requiring payment of interest on borrowed funds. In <i>Leibovici</i> v. <i>Rawicki</i>, 57 Misc.2d 141, 290 N.Y.S. 2d 997 (Civ. Ct. 1968), the Court upheld the legality of a <i>hetter iska</i> which provided a return greater than was permitted by usury laws in effect at that time. In this decision, the Court specifically denied the plea that the agreement was merely a disguised loan and hence constituted a subterfuge to avoid the taint of usury. In a number of cases, the courts have ruled explicitly that payment of a portion of profits in lieu of interest with no guarantee of profit is not usury.29<i class="footnote">See <i>Clift</i> v. <i>Barrow</i>, 108 N.Y. 187, 192-194, 15 N.E. 327, 328-330 (1888); and <i>Mueller</i> v. <i>Brennan</i>, 68 N.Y.S. 2d 517 (Sup. Ct., 1947).</i> In <i>Leibovici</i> v. <i>Rawicki</i> the Court recognized that a <i>hetter iska</i> constitutes an agreement of precisely this nature.
  There are, nevertheless, cases in which American courts, primarily New York State courts, have refused to recognize a <i>hetter iska</i> as creating a partnership or joint venture. However, in each of those cases, either (a) the <i>hetter iska</i> issue was irrelevant; (b) the testimony presented to the court with regard to the nature of a <i>hetter iska</i> was patently incorrect; or (c) the <i>hetter iska</i> was improperly drafted. Those cases underscore the need to draft a <i>hetter iska</i> properly and, when testimony is required, to have knowledgeable scholars serve as expert witnesses.
  One decision in particular has great relevance with regard to a type of <i>hetter iska</i> that has unjustifiably gained currency because of its simplicity. There appears to be a fairly widespread practice within some business circles to use a standard promissory note and to write the words "<i>al pi hetter iska</i>" ("in accordance with <i>hetter iska"</i>), or to use a rubber stamp bearing those words, above the signatures of the parties. The intent of that legend is to nullify explicit references in the instrument to a loan and interest by declaring actual intent to be a joint venture and shared profit. Read literally and in its entirety, the result is an ambiguity creating doubt with regard to the nature of the instrument and hence the expedient is of dubious efficacy in obviating the prohibition against interest.30<i class="footnote">See <i>Brit Yehudah</i> 40:8. Cf., <i>Teshuvot Maharsham</i>, II, no. 252.</i> Moreover, the term <i>hetter iska</i> is generic in nature and does not indicate which of the available forumulae or variations thereof is intended. In addition, a significant problem arises in the event that no profits are realized from investment of the funds received. Since there is no specific provision for an oath supporting a claim of the absence of profit, there is no basis for a fee in the form of accord and satisfaction in lieu of such an oath. Accordingly, if the transaction is governed by such a <i>hetter iska</i> and no profits have been realized, any sum paid the creditor in excess of principal in the absence of profit is, at best, a "gift" in return for the loan, which is itself a proscribed form of interest.
  It is quite likely that the practice of using the formula "<i>al pi hetter iska</i>" evolved from an earlier-employed expedient that was not open to these challenges. In earlier times, printed promissory notes were not used. Instead the recipient wrote out a simple declaration in the nature of "I acknowledge receipt of the sum of _______ from _______ in accordance with the <i>hetter iska</i> instituted by Maharam Avigdors of Cracow."31<i class="footnote">Cf., the comments of R. Ephraim Zalman Margolies published by R. Aaron Walden in <i>Shem ha-Gedolim ha-Ḥadash</i> (Warsaw, 5625), <i>Ma’arekhet Gedolim, ot mem</i>, sec. 51.</i> The instrument (a) contained no contradictory reference to interest; and (b) provided both for profit-sharing and for payment in lieu of an oath by means of incorporation of the provisions of a particular well-known and published <i>hetter iska</i>, <i>viz</i>., the <i>hetter iska</i> authored by Maharam Avigdors. At times, the slightly shortened formula "<i>al pi hetter iska shel Maharam</i>" was used on the strength of the common recognition that the only "Maharam" associated with a <i>hetter iska</i> instrument is Maharam Avigdors. Use of the formula "<i>al pi hetter iska shel Maharam"</i> when the intention is half loan, half deposit is clearly inappropriate.32<i class="footnote">Cf., <i>Mo’adim u-Zemanim</i>, VI, no. 41, sec. 1(2). Cf. also, the comments of R. Ephraim Zalman Margolies, cited <i>supra</i>, note 31, in which, although he decries the practice, Rabbi Margolies endeavors to justify its acceptance on the basis of the contention that because the error is so widespread it may be assumed that the party’s intention is half loan, half deposit.</i>
  A New York trial court found the abbreviated formula to be ambiguous and ordered a trial to determine, <i>inter alia</i>, the nature of the instrument. In <i>Arnav Industries</i>, <i>Inc.</i> v. <i>Westside Realty Assoc</i>., 180 A.D.2d 463, 579 N.Y.S.2d 382 (1st Dep't 1992), the Appellate Division found that inclusion of the phrase did not generate a threshold level of ambiguity sufficient to warrant denial of summary judgment on the promissory note. The promissory note that was the subject of dispute in <i>Arnav</i> contained a clause specifically stating that "[n]othing herein or in Mortgage is intended to create a joint venture, partnership, tenancy-in-common, or joint tenancy relationship between Borrower and Lender."33<i class="footnote"><i>Arnav Industries, Inc.</i> v. <i>Westside Realty Assoc</i>. at 383.</i> That clause constitutes an explicit rejection of the relationship a <i>hetter iska</i> is designed to create. Absent that clause, the issue of whether references to a loan and to payment of interest are rendered ambiguous by insertion of the phrase "<i>al pi hetter iska</i>" remains an open legal question. Nor, if an ambiguity is found to exist, is there any way to predict how a trial court might resolve that ambiguity.
  The decision in <i>Arnav</i> should serve as a caution against use of an abbreviated <i>hetter iska</i> consisting of but a vague and ambiguous phrase. Since that procedure is in any event of dubious halakhic validity, the decision in <i>Arnav</i> places no significant burden upon utilization of a <i>hetter iska</i>.
  <i>Barclay Commerce Corp.</i> v. <i>Abraham Finkelstein</i>, 11 A.D.2d 327, 205 N.Y.S.2d 551 (1st Dep't 1960) involved a case in which the plaintiff and the corporate defendant entered into a factoring agreement guaranteeing payment of monies which might become due. The suit was based upon the plaintiff's contention that the accounts receivable that had been assigned were fraudulent. The Appellate Court found that since the alleged fraud was not denied, the question of whether or not the <i>hetter iska</i> created a partnership was a "phantom" issue.34<i class="footnote"><i>Barclay Commerce Corp</i>. v. <i>Finkelstein</i> at 553.</i>
  In <i>Bollag</i> v. <i>Dresdner</i>, 130 Misc.2d 221, 495 N.Y.S.2d 560 (N.Y. City Civ. Ct. 1985), the parties entered into a properly drafted <i>hetter iska</i>. The defendant maintained that (a) the contract was void as usurious; and (b) the transaction constituted an investment by the plaintiff subject to any profits earned or losses sustained. The plaintiff relied upon the <i>hetter iska</i> in countering the usury defense but also incongruously alleged that he was entitled to repayment of principal and interest regardless of any profit or loss. Further muddying the waters, the plaintiff's complaint and bill of particulars categorized the instrument as a note and presented an expert witness who testified imprecisely that such forms are employed "when one party is lending money and the other party is borrowing money."35<i class="footnote"><i>Bollag</i> v. <i>Dresdner</i> at 562.</i> The Court also noted discrepancies between the <i>hetter iska</i> and testimony regarding liability in the event of absence of profit or of loss of capital. Most damaging was the unequivocal, but patently false, testimony of the expert who "testified emphatically…that the agreement did not create a joint venture or partnership."36<i class="footnote"><i>Id.</i> at 562.</i> Not only is that statement false, it is absurd, since, if true, it would render the <i>hetter iska</i> devoid of purpose. The Court took note of the inherent contradiction in the plaintiff's position in observing, "Yet despite his repeated use of the above-quoted categorization of the transaction, it is 'done,' he said, 'specifically not to pay interest.'"37<i class="footnote"><i>Id.</i> at 562.</i>
  The Court, in this decision, expressly affirmed in principle the validity of a properly drafted <i>hetter iska</i> in finding that the <i>hetter iska</i> presented to the court "as interpreted by the parties and their witnesses at trial reveals that the transaction was a loan (not an 'investment') and that plaintiff's clear intent was to exact a higher rate of interest than is permitted by law."38<i class="footnote"><i>Id.</i> at 225.</i> The Court distinguished the case before it from <i>Leibovici</i> v. <i>Rawicki</i> on the grounds that in <i>Leibovici</i> "the funds were clearly treated as a joint investment."39<i class="footnote"><i>Id.</i> at 225.</i>
  In <i>Bollag</i>, the Court found that the defendant requested the loan that was the subject of litigation for business purposes rather than for personal needs. Accordingly, it found that the policy and reasoning behind the general obligation law §5-521 serve to disallow a usury defense in such circumstances. In doing so, the Court affirmed the <i>hetter iska</i> as a valid business contract. However, incongruously, and perhaps cynically, its disallowance of the usury defense notwithstanding, the Court refused to allow "interest." The Court found that the plaintiff was prevented from collecting interest by his own pleading. The plaintiff had maintained that the <i>hetter iska</i> was entered into specifically to avoid payment of interest. Yet, the plaintiff's own expert witness incorrectly categorized the claim as one to recover "interest." Giving equal credence to both the plaintiff's own statement and the testimony of plaintiff's expert witness the Court declared, "[P]laintiff has made it plain he neither wishes to nor can collect interest. Accordingly, none shall be awarded."40<i class="footnote"><i>Id.</i> at 564.</i> The defendant, in effect, was hung from his own petard.
  Although recovery of accrued interest was denied, the Court allowed interest on the judgment at the statutory rate from the date of its entry on the grounds that there was "no testimony which would bar plaintiff from receiving interest on a judgment (as distinct from interest on a loan)." That finding reflects an erroneous assessment of Jewish law. Recovery of interest on a judgment is indeed precluded by Jewish law. Absence of testimony to that effect is but one more example of the comedy of errors and incompetence on the part of counsel and witnesses reflected in this case.
  In an unreported case, <i>Berger</i> v. <i>Moskowitz</i>, N.Y.L.J 25, (col. 3), vol. 206, no. 85 (October 30, 1991), 25 the Court found that a perfunctory <i>hetter iska</i> did not succeed in establishing a partnership agreement because it failed "to identify the nature of the venture let alone the specifics relating to the terms of such venture." The <i>hetter iska</i> in question was titled "Business Agreement" and did recite that the sum in question was advanced "for business and the profit of which agreed on will be ten percent, and the duration [of] the business will be three years from the signing of this agreement." Although inelegantly drafted, the agreement itself would seem to this writer to reflect the intent of the parties that the funds be applied to a business venture and that the 10% be paid out of profits, the implication being that such profit-sharing was predicated upon actual realization of profits. However, in this case also, the defendant was found liable by virtue of his own admission. The Court declared, "Nor can defendants label this transaction a business venture since MOSKOWITZ wrote a letter to the plaintiffs' attorney stating in part: 'I acknowledge that the unpaid balance of the loan with interest through November 30, 1990, is $268,200.00.'"
  The decision of a U.S. Bankruptcy Court in <i>In re Stephen Douglas</i>, 174 B.R. 16 (Bankr. E.D.N.Y. 1994), does not involve any substantive matter pertaining to the legal validity of a <i>hetter iska.</i> The issue was whether a certain payment constituted a fraudulent conveyance under applicable statutes. Section 273 of the New York Debtor and Creditor Law denominates as fraudulent any conveyance made by an insolvent for inadequate consideration, without regard to actual intent. The books and records of the debtors reflected no consideration or economic benefit conferred upon Douglas by the defendant. It was also alleged that a promissory note reflecting a loan was executed but was lost. A <i>hetter iska</i> was introduced in evidence in support of existence of a loan. That evidence was dismissed on the grounds that it was written on the stationery of a corporation not involved in the proceedings. Since the <i>hetter iska</i> was dismissed as irrelevant, the Court found that there was no need to address the legal import of a <i>hetter iska.</i>
  <i>Heimbinder</i> v. <i>Berkovitz</i>, 175 Misc.2d 808, 670 N.Y.S.2d 301 (N.Y.Sup. 1998), involved an action to invalidate various transfers to the defendant corporation which would have otherwise been available to satisfy a judgment against the defendant. The plaintiff sought to hold defendant directors personally liable for breach of fiduciary duty in effecting such transfers. Since a <i>hetter iska</i> was employed, the plaintiff also contended that the defendant, in signing the document, personally guaranteed payment of the obligation. The court cited with approval an earlier unpublished decision in <i>Burger</i> v. <i>Baruch Ha'Levi Moskowitz</i>, Kings County Sup.Ct., Index NO. 15600/91, in which a <i>hetter iska</i> constituted the sole contract between two individual parties and was found to establish a valid guarantee. In <i>Heimbinder</i>, however, the evidence indicated that plaintiff's attorney had previously requested a personal guarantee and that the request had been refused by the defendant's attorney. Moreover the "security agreement" executed by the parties contained a provision for a "guarantee" next to which the word "omit" was entered in handwriting; the defendant's name had been typed beneath the signature line for the guarantor but was crossed out. The plaintiff conceded that there had been a discussion of guarantees but that no agreement was reached; that he had been unaware of the existence of a guarantee provision in the form that was used and that the <i>hetter iska</i> was produced by the defendant only after discussions in which he refused a personal guarantee had been concluded; that he could not read Hebrew and relied upon the defendant's categorization of the document; and that the defendant did not portray it as a personal guarantee. Accordingly, the Court found that, in the case before it, the <i>hetter iska</i> was not intended to generate a personal guarantee of corporate debt. Nothing in the <i>Heimbinder</i> decision casts doubt upon the enforceability of the provisions of the <i>hetter iska</i> itself.
  <i>Heimbinder</i>, however, serves to establish a significant point of law. A standard <i>hetter iska</i> signed by an officer of a corporation in circumstances other that those of <i>Heimbinder</i> may well establish a personal obligation even absent explicit indication of such intention. If this is not the desired outcome, the remedy is to indicate explicitly in the document that it is being executed on behalf of the corporation and not in an individual capacity.
  <i>Wiesel</i> v<i>. Rubinstein</i>, 12 Misc.3d 1168(A) (N.Y.Sup. 2006), represents a foregone opportunity to seek application of a collateral provision of a <i>hetter iska</i>. Kalman Rubinstein signed a series of promissory notes in order to obtain funds to open a kosher pizza shop. The rate of return on each of the notes, most of which were short term in nature, was 14%. Despite an agreement to submit the dispute to a <i>bet din</i> for resolution, Rubinstein failed to appear before the <i>bet din</i> to which the parties agreed. The <i>bet din</i> declined to issue a default judgment. Thereupon, the estates of Sol and Harriet Wiesel sued in New York State Supreme Court. Apparently, the plaintiff made no attempt to have the Supreme Court enforce compliance with the arbitration agreement or to issue a default judgment upon failure to submit to arbitration despite the express agreement between the parties to do so that was incorporated in the <i>hetter iska</i>.
  Rubinstein seems to have been under the impression that he would prevail in a civil court both because the plaintiffs' claim was barred by the statute of limitations and because the rate of interest was usurious. Nevertheless, he asserted an additional affirmative defense, <i>viz</i>., since "each of the notes contains a notation in Hebrew indicating the notes are subject to <i>hetter iska</i> failure of the business absolved him of the obligation to repay the amounts due on the notes."41<i class="footnote"><i>Wiesel</i> v. <i>Rubinstein</i> at 1.</i>
  The Court denied summary judgment based on the statute of limitations on the grounds that the alleged partial payment of the debt had the effect of extending the period within which a suit might be initiated. The Court also interpreted the note as prescribing 14% per annum rather than a usurious 14% for a much briefer period. The <i>hetter iska</i> defense was ignored in the Court's rejection of the motion for summary judgment, perhaps because it was self-evident that whether or not the business did fail, and whether or not a <i>hetter iska</i> provides that such failure cancels the obligation, are triable issues.
  In point of fact, a standard <i>hetter iska</i> would require a solemn oath to substantiate a claim of non-realization of profit and testimony of two halakhically qualified witnesses to establish loss of capital. In this case, an ordinary promissory note seems to have been employed but modified by inclusion of a "notation in Hebrew indicating that the notes are subject to the <i>hetter iska.</i>" That "notation" was the incorporation of the earlier-discussed phrase "<i>al pi hetter iska.</i>" As noted earlier, that phrase may signify only that the term "loan" should be understood as "investment" and that the term "interest" should not be construed as such but as the return of a share of the profits earned by the investment. If so, as discussed earlier, entirely different and much less onerous standards of proof might apply to establishing absence of profit and loss of capital. Whether or not "failure of the business" did indeed absolve the defendant "of the obligation to repay the amounts due on the notes" is a matter to be resolved by a <i>bet din</i>.
  The above notwithstanding, in a Report and Recommendation on Motion to Dismiss in the case of <i>Edelkind</i> v<i>. Fairmont Funding,</i> 539 F.Supp.2d 449 (D. Mass. 2008), a federal magistrate asserted that in civil courts "<i>hetter iska</i> agreements have been interpreted as 'merely a compliance in form with Hebraic law'42<i class="footnote">See <i>Barclay Commerce Corp</i>. v. <i>Finkelstein</i> at 329.</i> that does not create a partnership between the parties."43<i class="footnote"><i>Edelkind</i> v. <i>Fairmont Funding</i> at 454.</i> In support of that assertion, the Magistrate cited <i>Arnav Indus. v. Westside Realty Assoc</i>., <i>Barclay Commerce Corp</i>. and <i>Heimbinder</i>. As has been shown, those cases do not serve to establish the Magistrate's conclusion. Quite to the contrary, each of those cases reflects acceptance of the validity of a properly drafted <i>hetter iska</i> in creating a partnership between the parties but finds the particular <i>hetter iska</i> instruments submitted in those proceedings to be faulty. The <i>hetter iska</i> employed in <i>Edelkind</i> may also have been defective but that issue was not addressed. These cases serve to underscore the necessity for careful drafting of a halakhically valid <i>hetter iska</i> agreement. A sample text of an English-language <i>hetter iska</i> agreement may be found in this writer's <i>Contemporary Halakhic Problems</i>, II (New York, 1983), 386-388.
  The claim advanced in <i>Edelkind</i> was dismissed because Jamie Edelkind was found to have no standing to bring a suit as an alleged third party beneficiary of a mortgage agreement entered into by his wife. The findings with regard to the <i>hetter iska</i>, also apparently signed by the wife, constitute dicta having no bearing upon the merits of the case. Nevertheless, the facts presented in <i>Edelkind</i> underscore a serious pitfall based, not upon the validity of the <i>hetter iska</i> itself, but upon the way it was employed.
  It is quite common for a lender to execute a <i>hetter iska</i> and immediately thereafter to execute a second, standard mortgage agreement. It is commonly assumed that the lender is better protected legally if he is in possession of a mortgage instrument. Whether or not that is true, there is no gainsaying the fact that the lender feels more secure with a mortgage in his possession. R. Moshe Feinstein, <i>Iggerot Mosheh</i>, <i>Hoshen Mishpat</i>, no. 62, sanctions the practice from the perspective of Jewish law in describing the mortgage instrument as devoid of halakhic validity provided that the <i>iska</i> agreement states that it is the governing instrument and the mortgage is designed merely as "<i>bitḥonot</i>," best translated as "reliances" or "assurances."44<i class="footnote">Cf., <i>Teshuvot Ḥatam Sofer</i>, <i>Ḥoshen Mishpat</i>, no. 48, who writes that the formula “<i>al pi hetter Maharam</i> as was spoken between us” rather than the full text of a <i>hetter iska</i> was utilized in order to give full force to the written agreement in civil law as an ordinary loan agreement. The concept seems to be identical to that formulated by <i>Iggerot Mosheh</i>. See also the discussion of R. Shlomoh Kluger, <i>Teshuvot Tuv Ta’am va-Da’at</i>, I, no. 208. The basic principle is also accepted by R. Yitzchak Ya’akov Weisz, <i>Teshuvot Minḥat Yiẓḥak</i>, IV, no. 16, sec. 2. <i>Ḥelkat Ya’akov</i>, III, no. 195, sec. 6, regards the practice as permissible even if the mortgage is executed prior to the <i>hetter iska</i>.</i> Nevertheless, one legal ramification of such practice is evident in <i>Edelkind</i>.
  In instances such as those described in <i>Edelkind</i>, the mortgage agreement is crucial to the enterprise. Fairmont Funding is a mortgage broker. The principals are observant Jews. The firm enters into mortgage agreements in conjunction with the purchase of real property and then sells the mortgage loans to banks. Therein lies the dilemma: The mortgage broker is the lender, both halakhically and legally; the mortgage broker contracts for interest to be paid to himself. This, as an observant Jew, he may not do. Therefore, he avails himself of a <i>hetter iska</i>. The mortgage bankers are non-Jews; they do not understand, and would not purchase, a <i>hetter iska</i>. Therefore, the mortgage broker, in addition to a <i>hetter iska</i>, insists upon a mortgage note which he then sells to the mortgage bank, presumably without informing the bank of the existence of the <i>hetter iska</i>.
  That was precisely the chain of events reported in <i>Edelkind v. Fairmont</i>. The contemplated procedure was recited in detail in the preamble to the <i>hetter iska</i> agreement. Jamie Edelkind argued that the <i>hetter iska</i> created a partnership in the mortgaged property and, accordingly, Fairmont Funding was not empowered to sell the purported loan without permission or notification of his wife who was thereby denied her option to repurchase Fairmont's investment. Edelkind categorized the mortgage document as a "fiction" and accordingly contended that neither Fairmont nor any subsequent holder of the instrument had any legal rights in the mortgage. Hence, he argued, Fairmont and his wife still own the property jointly as partners and neither Fairmont nor any subsequent holder had any foreclosure rights.
  There is merit in Jamie Edelkind's contention but, if so, both he and Fairmont perpetrated a fraud upon subsequent holders of the mortgage. The obvious remedy is to insert a clause in the <i>hetter iska</i> agreement authorizing the mortgage broker to mortgage the entire property.45<i class="footnote">The problem may also be remedied by inserting a clause similar to the one described in the following section.</i> Indeed, a careful reading of the preamble to the <i>hetter iska</i> agreement between Edelkind and Fairmont Funding reveals that this was agreed to by the parties. Clearer language would have prevented Jamie Edelkind from arguing otherwise. In any event, suppression of the <i>hetter iska</i> agreement gives rise to an aura of fraud and should not be countenanced. In addition, authorization to assign a mortgage against the entire property should be carefully spelled out in the body of the <i>hetter iska</i> agreement.46<i class="footnote">There is an additional vague reference to <i>hetter iska</i> in a decision of a federal appeals court, <i>Barclays Discount Bank, Ltd</i>. v. <i>Bogharian Bros.,</i> 743 F.2d 722, 724 (9th Cir. Cal. 1984). In that case a district court denied a motion for reconsideration on the grounds that the <i>hetter iska</i> issue had not been advanced in a timely fashion and further observed that the argument based upon the <i>hetter iska</i> contradicted the defendant’s own description of the relationship between the parties. In a footnote, the Appellate Court stated, “A <i>hetter iska</i> appears to be a religious document purporting to characterize the bank and those to whom the bank charges interest as a ‘venture’ to avoid violation of religious law.” <i>Id.</i> at note 2.</i>
  V. Assuring The Validity of a Hetter Iska
  It seems to this writer that, even according to the authorities who maintain that a <i>hetter iska</i> must be enforceable in a civil court, a contract containing an enforceable clause requiring arbitration before a <i>bet din</i> is sufficient. The position of those authorities is based on the contention that a contract requires a meeting of minds. Hence, if one or the other of the parties regards the agreement as a charade, the <i>hetter iska</i> is of no effect. Awareness that the provisions of that instrument are not civilly enforceable, they contend, has the effect of thwarting such a meeting of minds. However, whether the provisions are directly enforceable under applicable civil law or are enforceable only by virtue of an arbitration clause is of no material significance in this regard. So long as both parties recognize that the provisions of the <i>hetter iska</i> will be enforced by a court of competent jurisdiction, whether directly or by means of confirmation of an arbitration award, there must be seriousness of intent to abide by the provisions of a <i>hetter iska</i>.
  There is, of course, the possibility that, for one reason or another, one of the parties may prove to be recalcitrant and refuse to appear before a <i>bet din</i> or that, as occurred in <i>Wiesel</i>, the <i>bet din</i> may decline to issue an award and grant leave to have recourse to a secular court. In such instances, it may be to the advantage of the person bringing a cause of action to have the court treat the instrument as an ordinary loan.
  Since the parties have bound themselves to enforceable arbitration before a <i>bet din</i>, and hence at the time of the execution of the <i>hetter iska</i> there is a proper meeting of minds, the manner in which the instrument is construed by a civil court in cases of default is of no halakhic significance. The validity of that conclusion is evidenced both by <i>Hatam Sofer</i>'s ruling that an interest-bearing loan agreement may be presented to a civil court if there is default on the underlying <i>hetter iska</i>, and by the ruling of <i>Iggerot Mosheh</i> that a conventional mortgage may be executed for purposes of "<i>bitḥonot</i>."
  Accordingly, this writer recommends that a <i>hetter iska</i> agreement incorporate two separate clauses: (a) a clause providing for binding arbitration before a <i>bet din</i>; and (b) a clause providing that if, for any reason, the transaction governed by the <i>hetter iska</i> becomes the subject of litigation before a civil court, it is the intent of the parties that the court construe the transaction as an ordinary loan and enforce it as such. In this writer's opinion, the coupling of those two clauses renders the arrangement acceptable according to all authorities. The following text may be used to achieve that effect:47<i class="footnote">A full text of a <i>hetter iska</i> agreement can found in this writer’s <i>Contemporary Halakhic Problems</i>, IV (New York, 1995), 380-383. The manifold provisions contained in that form may be incorporated by reference and made an integral part of any loan agreement. A standard loan form may be used provided that the words “loan” and “interest” are eliminated wherever they may occur and the words “funds” and “premium” are substituted in their stead and provided that the following clause is inserted:</i>
  In the event of any controversy arising out of, or related to, this agreement the same shall be submitted to binding arbitration in accordance with Jewish law by a tripartite panel of qualified rabbis versed in such law.48<i class="footnote">The arbitration clause should clearly state that arbitration before a <i>bet din</i> is a condition precedent to any action before a civil court. In the most recent case involving a <i>hetter iska</i> agreement, <i>Koenig v. Middlebury Land Associates</i>, LLC, 16 Conn. L. Rptr. 90 (Conn. Sup. Ct. 2008), the issue was not whether the transaction was a loan or a joint business venture, but whether an arbitration clause in the <i>hetter iska</i> directing that disputes be heard by a <i>bet din</i> deprived the court of subject matter jurisdiction. The Court held that the language of the arbitration clause of the <i>hetter iska</i> did not expressly provide or necessarily imply that arbitration before a <i>bet din</i> was a condition precedent to any court action and denied the defendant’s motion to dismiss.</i> Each party shall be entitled to appoint one member of this panel; the two members appointed in this manner shall appoint the third member of the panel. The award by a majority of such panel shall be confirmed in any court of competent jurisdiction pursuant to the CPLR without any right of appeal therefrom. An action shall be brought before a civil court only in the event of failure of one of the parties to appear before a rabbinic tribunal and only upon leave from a rabbinic tribunal.
  This undertaking shall be governed solely by the stipulations contained in this agreement. Any further documents pertaining to this transaction bearing the signatures of the undersigned is hereby declared null and void insofar as Jewish law is concerned. Any such document is to be construed solely as an expedient designed to provide relief in a civil court in accordance with usual judicial procedures in the event of the undersigned's failure to appear before a rabbinic tribunal or failure to abide by the decision of that tribunal. Accordingly, it is expressly acknowledged that, in the event of recourse to a civil court by the undersigned or by any other party, the claims and privileges of the undersigned arising from any other document or from any other source are in no way to be diminished or compromised by virtue of this agreement.49<i class="footnote">One reason for inserting such a clause is the possible effect upon third parties who are not themselves parties to the <i>hetter iska</i>. A problem of that nature arose some time ago in <i>First International Bank, Ltd.</i> v. <i>L. Blankstein &amp; Son, Inc.</i>, 88 A.D.2d, 501, 449 N.Y.S.2d 737 (1st Dep’t 1983), aff’d, 59 N.Y.2d 436 (N.Y. 1983). That case involved an Israeli diamond merchant sued by an Israeli bank in a New York court because he failed to honor a promissory note. In a suit filed in New York State Supreme Court it was alleged that the practice in Israeli diamond circles permits return of gems to the wholesaler for any reason or for no reason. A similar issue was presented in <i>Israel Discount Bank Ltd</i>. v. <i>Rappaport</i>, 90 A.D.2d 740, 456 N.Y.S.2d 988 (1st Dep’t 1982). In Israel, diamonds are typically purchased on the strength of a promissory note. That practice is described in another case, <i>Barclays Discount Bank Ltd</i>. v. <i>Levy</i>, 743 F.2d 722 (9th Cir. Cal. 1984), involving Siegman as a diamond dealer but in which he was not a party. When the market for diamonds fell dramatically in the early 1980’s, Leo Siegman, recognizing that he could not realize a profit from the resale of diamonds that he had purchased, returned the diamonds to the seller. He contended that in doing so he voided the promissory note. However, in the interim, the seller had discounted the promissory note by presenting it to the Israeli bank. Executory agreements that allow for cancellation of a debt are not binding on an assignee unless the assignee has prior notice of such an agreement. The bank argued that, as an assignee, it was not bound by any executory agreement between Siegman and the seller and, accordingly, sued to collect the debt represented by the promissory note. The facts in <i>First International Bank</i> v. <i>Siegman</i> were similar to those before the Court in <i>Barclays</i> v. <i>Levy</i>. In <i>First International Bank</i> v. <i>Siegman</i>, Siegman was in a position to counter that all arrangements between Israeli banks and their clients are governed by a <i>hetter iska</i> and that the <i>hetter iska</i> creates a partnership between the bank and the party to whom it advances funds in return for assignment of the debt represented by the <i>hetter iska</i>. The Uniform Partnership Act § 12 provides that notice to one partner is tantamount to notice to all partners. See Alan R. Bromberg, <i>Crane and Bromberg on Partnership</i> (St. Paul, 1968), p. 322 and p. 322, note 73. Since the bank had effective notice of the agreement permitting cancellation of the debt upon return of the diamonds through notice to its partner, i.e., the seller, Siegman might have argued he was relieved of liability to the bank as well.Although the <i>hetter iska</i> was not raised as a consideration in <i>First International Bank</i>, nevertheless, that case serves to underscore the fact that complications affecting third parties may arise as the result of a <i>hetter iska</i> arrangement. A clause in the nature here proposed would result in the court considering the <i>hetter iska</i> as a conventional loan agreement with regard to any suit brought by a third party.</i>
  At the time the Torah was given, the basis of livelihood and commerce of Jews was agrarian; agriculture was the source and support of their wealth and status for he who had copious produce was considered a rich man. Money was not fundamental or the basis of their activities; and when money was loaned it was not for the purpose of trade and acquisition, but in order to acquire bread and clothing by a person who could not afford such from the yield of his land and work. Accordingly, it is obvious that this poor debtor could not afford to pay interest, while the wealthy lender would suffer no perceivable loss in lending without interest….However, in the Middle Ages, by which time Israel had lost its share and inheritance in working the land and become a nation that sustained itself solely on commerce, the prohibition became onerous, for by then money had become the basis of [the Jews'] life and commerce; it became a business commodity to the lender and to the borrower equally. "That your brother may live with you" applied to both equally. Therefore, our sages saw fit to find a permissive manner to avoid this prohibition considering that, for such purpose, the Torah did not entirely forbid this matter.
  The parties to this agreement desire to comply fully with the provisions of Jewish law prohibiting payment and acceptance of interest and for this purpose agree that the terms of this agreement shall be made subject to the terms of <i>Hetter Iska</i> as provided in the form set forth in <i>Contemporary Halakhic Problems</i>, IV (New York, 1995), pp. 380-383. All provisions of said <i>Hetter Iska</i> form shall be incorporated and made part of this agreement as is fully set forth herein.
  
  Chapter 5 Medical Malpractice and Jewish Law
  
  <i>My God, heal me and I shall be healed.Let not Your anger be kindled that I be consumed.My drugs and potions are Yours, whether good or bad,whether strong or weak.It is You who shall choose and not I;Of Your will is the harmful and the effective.Not upon my healing do I rely;Only for Your healing do I watch.</i>
  JUDAH HALEVI1<i class="footnote"><i>Kol Shirei Rabbi Yehudah ha-Levi</i>, ed. Israel Zemorah, 2nd ed. (Tel Aviv, 5730), II, 227.</i>
  I. Introduction
  Common law categorizes efficacious treatment carried out without consent of the patient, not as malpractice, but as simple battery. Unlike common law which regards any intentional non-consensual tactile contact with another person as a tort, Halakhah does not regard therapeutic "wounding" (<i>ḥavalah</i>) as a tortious battery. Jewish law does not accept the notion that lack of consent renders medical intervention tortious and certainly provides no basis for monetary recovery unless there is resultant physical harm. But, when the physician does cause damage, it would seem at least as a first impression, that the usual rules governing battery are applicable. Thus, in principle, Jewish law regards malpractice, not as an independent tort, but as a form of <i>ḥavalah</i> or battery.
  In general, Jewish law assigns to the tortfeasor absolute liability for harm to person or property arising from his or her physical act. The connotation of the talmudic principle formulated by the Gemara, <i>Bava Kamma</i> 26b, "<i>Adam mu'ad le-olam</i>" ("Man is always forewarned") is that a person is always liable for damage resulting from an act committed by his body. It would then follow that a physician whose treatment results in harm to the patient should be liable for compensation even if the harm is the result of misadventure rather than negligence.2<i class="footnote">It is indeed the case that <i>Tosafot</i>, <i>Bava Kamma</i> 27b, <i>Bava Meẓi’a</i> 82b and <i>Bava Batra</i> 93b, maintain that there is no liability in instances of <i>ones gamur</i>, i.e., in situations in which the harm is entirely unforeseeable. For a detailed discussion of the position of <i>Tosafot</i> and concurring authorities see <i>Oẓar Mefarashei ha-Tal-mud</i>, <i>Bava Kamma</i>, II (Jerusalem, 5748), pp. 31-36 and accompanying notes. That exclusion is not applicable to medical ministration. <i>Tosafot’s</i> position is disputed by Ramban, <i>Milḥamot ha-Shem</i>, <i>Bava Kamma</i> 29a, as well as in his commentary on <i>Bava Meẓi’a</i> 82b and cited by <i>Shitah Mekubbeẓet</i>, <i>Bava Meẓi’a</i> 82b. For sources concurring in and elucidating Ramban’s opinion see <i>Oẓar Mefarashei ha-Talmud</i>, <i>Bava Kamma</i>, II, p. 33, note 123. For an analysis of Rambam’s view see <i>Maggid Mishneh</i>, <i>Hilkhot Ḥovel u-Mazik</i> 1:11, 1:16, 6:1 and 6:3, as well as <i>Shakh</i>, <i>Ḥoshen Mishpat</i> 378:1. See also <i>Sema</i>, <i>Ḥoshen Mishpat</i> 378:2 and 421:8, as well as <i>Maḥaneh Efrayim</i>, <i>Hilkhot Nizkei Mamon</i>, no. 5.</i> Nevertheless, in point of fact, Jewish law does not hold the physician liable in any sense unless he has been negligent in his treatment. The reason or reasons for treating malpractice differently from ordinary forms of <i>adam ha-mazik</i> (lit.: man as tortfeasor) require elucidation. Moreover, the definition or delineation of culpable negligence as applied to medical malpractice is problematic.
  The task of determining with precision the circumstances in which a physician will be held culpable is made difficult by the fact that the talmudic sources are sparse and cryptic. Moreover, early-day commentaries and codifiers offer little further clarification; quite to the contrary, their comments present further difficulties. In addition, at least until the contemporary period, the topic has, relatively speaking, received but scant attention in the writings of latter-day authorities.
  One can but speculate with regard to the relative paucity of material devoted to this issue. It may well have been the case that during the medieval period, when in some countries Jewish medical practitioners were common, modalities of treatment were quite limited with the result that demonstrable malpractice was much rarer than at present. In later times, with the exception of observant physicians of German extraction, the religious commitment of most Jews who entered the medical profession was generally quite tenuous rendering it unlikely that such a physician would heed the summons of a <i>bet din</i> or feel himself bound by the provisions of Jewish law.
  At present, although the Jewish community is graced by a sizeable and growing number of observant physicians, adjudication of a malpractice suit by a <i>bet din</i> is a rarity for the simple reason that, at present, all physicians are covered by malpractice insurance. Insurance companies are publicly held corporations and are hardly likely to avail themselves of the services of a <i>bet din</i>. Elsewhere,3<i class="footnote">J. David Bleich, <i>Contemporary Halakhic Problems</i>, V (Southfield, Michigan, 2006), 33-35.</i> this writer has expressed the opinion that a Jew may properly sue a fellow Jew in a secular court provided that he does not accept a recovery greater than the insurance coverage. Although, technically, it is the insured who is the named defendant, the actual party interest is, and is commonly known to be, the insurance company. Hence, since the real party in interest is not Jewish, the prohibition against recourse to civil courts is not operative.4<i class="footnote">See <i>ibid</i>., pp. 35-37.</i>
  Recovery from an insurance company, in this writer's opinion, is appropriate even in instances in which such recovery would not be available in a <i>bet din</i> or in which the sum awarded is greater than would be allowed by a <i>bet din</i>. That is so even if, as a matter of Jewish law, the Jewish plaintiff cannot directly claim that, in an action against a non-Jew, the appropriate measure of damages is the enhanced standard adopted by the civil legal system.5<i class="footnote">Rambam, <i>Hilkhot Melakhim</i> 10:22, codifies the rule that provides that in a controversy involving a non-Jew a Jewish litigant may claim any advantage accruing to him under applicable civil law. However, R. Akiva Eger, in his glosses to the Mishnah, <i>Bava Kamma</i> 1:3, sec. 11, and <i>Ḥazon Ish al ha-Rambam</i>, <i>Hilkhot Melakhim</i> 10:12, declare that this rule applies only to a defendant but not to a plaintiff seeking recovery. See also Rosh, <i>Bava Kamma</i> 1:19.</i> The insurance company is liable in contract rather than in tort. The claimant is a third party beneficiary of that contract. The contractual obligation is to indemnify the policyholder or the victim to the extent that a civil court would award damages against the insured. Thus, whether or not Jewish law would make a similar award for tort damages is rendered irrelevant. The insurance company has contracted to indemnify its insured in that amount simply because, in fact, it has been, or would have been, awarded by the court.
  Parenthetically, the question of whether a contract of that nature is invalid by reason of <i>asmakhta</i>, i.e., lack of anticipation that the circumstances necessitating performance would actually arise, is a moot point.6<i class="footnote">See the dispute between R. Isaac ben Sheshet, <i>Teshuvot Rivash</i>, no. 308, and R. David ibn Zimra, <i>Teshuvot Radvaz</i>, I, no. 797, regarding insurance of merchandise against loss at sea. The salient portions of those responsa are translated by Stephen M. Passamaneck, <i>Insurance in Rabbinic Law</i> (Chicago, 1974), pp. 33-41. Sources dealing with the issue of <i>asmakhta</i> as it relates to insurance contracts are also cited by R. Menachem Slae, <i>Ha-Bituaḥ be-Halakhah</i> (Tel Aviv, 5740), pp. 79-82. That work also contains a valuable historical survey of the treatment of insurance in rabbinic literature as well as a discussion of a number of halakhic theories for enforcement of insurance contracts. See in particular pp. 72-78. In the English translation of that work by Bracha and Menachem Slae, <i>Insurance in the Halakhah</i> (Tel Aviv, 1982), this material is presented on pp. 98-113.</i> Any insurance company that might choose to deny an otherwise valid claim on the plea that the insurance policy serving as the basis of the claim is unenforceable by reason of <i>asmakhta</i> would not long remain in business. No rational policyholder would continue to pay a premium for coverage once he has discovered that the insurer will not honor any legitimate claim.7<i class="footnote">Cf., R. Zalman Nechemiah Goldberg, <i>Teḥumin</i>, XIX (5759), 322.</i>
  II. Early Sources
  1. The Tosefta
  The principles governing financial liability for malpractice are closely tied to the principles governing punishment for inadvertent manslaughter applicable in situations in which malpractice leads to death. Jewish law denies recovery of damages for wrongful death regardless of whether the death results from an act of homicide, negligent manslaughter or misadventure. Under certain carefully defined circumstances Jewish law prescribes exile to one of the designated cities of refuge as punishment for negligent manslaughter. Exile was imposed only during the historical period in which there was an incumbent High Priest and in which capital punishment might be imposed for willful acts.8<i class="footnote">See <i>Encyclopedia Talmudit</i>, VI (Jerusalem, 5714), 135.</i>
  In limiting the circumstances in which exile was required the Mishnah, <i>Makkot</i> 8a, states:
  If [a person] throws a stone into his courtyard and slays: if the victim had permission to enter therein [the slayer] goes into exile; but if not, he does not go into exile, as it is written "and when a person goes with his friend into a forest" (Deuteronomy 19:5)—just as a forest [both] the victim and the slayer are permitted therein, similarly [the law applies] in every domain in which [both] the victim and the slayer are entitled to enter, to the exclusion of the courtyard of the householder since the victim does not have the right to enter therein. Abba Saul says: Just as the cutting of wood is a discretionary act, similarly [the law applies] in all instances of discretionary acts, to the exclusion of a father who beats his son, a teacher who strikes his pupil and a messenger of the court [who administers lashes].9<i class="footnote">There is a significant disagreement among early-day authorities with regard to the nature of the activity undertaken by the agent of the <i>bet din</i> that, when it results in unforeseen death, is not punishable by exile. Rambam, <i>Hilkhot Roẓeaḥ</i> 5:16, codifies the rule formulated in the Mishnah as applicable to a bailiff who seeks to compel a litigant to appear before the court. Ra’avad, <i>ad locum</i>, disagrees and asserts that the reference in the Mishnah is to an agent of the court who administers the punishment of flogging. Rabbenu Yonatan of Lunel limits the agent’s immunity from exile to situations in which the judges erred in ordering more than the appropriate number of lashes and the agent simply carried out their instruction. R. Jacob Reischer, <i>Teshuvot Shevut Ya’akov</i>, III, no. 140, understands Ra’avad as adopting that position as well. However, Ramban, <i>Makkot</i> 8a, apparently understands the reference to be to an agent of the <i>bet din</i> who administers the proper number of lashes as determined by the <i>bet din</i> in accordance with the transgressor’s physical condition but who nevertheless causes the death of the transgressor. See also R. Jacob Ettlinger, <i>Arukh la-Ner, Makkot</i> 22b; R. Zevi Alexander Halperin, <i>Imrei ha-Ẓevi</i>, <i>Bava Kamma</i> 32b; and R. Yitzchak Ya’akov Weisz, <i>Teshuvot Minḥat Yiẓḥak</i>, III, no. 10, sec. 2.</i>
  Abba Saul's distinction is quite easy to grasp. The paradigm presented in Scripture involves accidental death resulting from an attempt to fell a tree in a forest. Such activity is entirely discretionary; the slayer was perfectly free to refrain from wood-chopping. Had he done so, no misadventure would have occurred. Since he was under no legal or moral duty to engage in that activity, he is held to a relatively high standard of prudence. A father who punishes his son, a teacher who chastises a pupil and the court official who administers punishment to a miscreant are all engaged in discharging a duty. Since they are duty-bound to perform such acts they are not punished even in cases of avoidable misadventure.
  The Tosefta, <i>Makkot</i>, addenda 2:5, presents a rather different rule:
  A messenger of the court who administers lashes with permission of the court goes into exile [if the victim dies]. A skilled physician who heals with license of the court and kills [his patient] goes into exile. One who performs an embryotomy [upon a fetus] in the woman's womb with license of the court and kills [the mother] goes into exile.10<i class="footnote">The identical rule with regard to both a messenger of the court and a physician is also formulated in the Tosefta, <i>Bava Kamma</i> 9:3.</i>
  The rule formulated by the Tosefta seems to be at variance from the rule presented in the Mishnah in the name of Abba Saul. Abba Saul formulates the principle that a person who causes the death of another in the course of performing a duty is exempt from exile. The examples presented by the Tosefta, <i>viz.</i>, a messenger of the court and a physician, seem similarly to be instances of persons engaged in discharging a duty and, in accordance with the rule formulated by Abba Saul, such persons should not be subject to exile.
  R. Meir Simchah of Dvinsk, <i>Or Sameaḥ</i>, <i>Hilkhot Rozeaḥ</i> 5:6, suggests that the Tosefta expresses a view that is in disagreement with Abba Saul. Indeed, as recorded by the Gemara, <i>Makkot</i> 8a, in presenting his principle, Abba Saul appears to be disagreeing with the anonymous, and hence presumably majority, view recorded in the immediately prior sentence of the same Mishnah.11<i class="footnote">See also R. Samuel Schoppen, <i>Kos ha-Yeshu’ot</i>, <i>Makkot</i> 8a, who writes that there appears to be a controversy between Abba Saul and the anonymous author of the earlier statement.</i> The principle presented in the earlier statement of the Mishnah is predicated upon the identical biblical verse adduced by Abba Saul in support of his rule and appears to be an alternative, and hence conflicting, exegetical interpretation of the cited phrase. Assuming that there is a controversy between Abba Saul and the exponents of the first principle, the statement recorded in the Tosefta may be understood as an expression of the view of those who, as recorded in the Mishnah, disagree with Abba Saul.12<i class="footnote">Cf., R. Jacob Ettlinger, <i>Teshuvot Binyan Ẓion</i>, no. 111, who questions whether the physician’s act is in the nature of a permissible <i>davar she’eino mitkhaven</i> (unintended effect) for which there can be no culpability and suggests that the Tosefta must either maintain that, by its nature, the physician’s act is not a <i>davar she’eino mitkhaven</i> or that the Tosefta rejects the permissibilty of any <i>davar she-eino mitkhaven</i>.</i>
  <i>Or Sameaḥ</i>, however, recognizes a difficulty inherent in his position. Rambam, <i>Hilkhot Rozeaḥ</i> 5:6, rules in accordance with the position of Abba Saul in exempting a father and a messenger of the <i>bet din</i> from exile.13<i class="footnote">R. Judah Ayash, <i>Matteh Yehudah, Yoreh De’ah</i> 336, advances the curious view that, in actuality, the culpability of a physician is no greater than that of a father or teacher and hence the exile referred to by the Tosefta and <i>Shulḥan Arukh</i> is not statutory in nature but a voluntary undertaking “in order to satisfy the requirements of Heaven.”</i> Hence, according to <i>Or Sameaḥ</i>, it logically follows that a physician is also exempt from exile. <i>Or Sameaḥ</i> expresses amazement that Ramban, in his <i>Torat ha-Adam</i>,14<i class="footnote">See <i>Kol Kitvei ha-Ramban</i>, ed. R. Bernard Chavel (Jerusalem, 5724), II, 41.</i> cites the Tosefta's ruling with regard to the culpability of the physician without noting that the Tosefta's statement is at variance with the position of Abba Saul. <i>Or Sameaḥ</i> further observes that the Tosefta is cited as normative by both <i>Tur</i> and <i>Shulḥan Arukh</i>, <i>Yoreh De'ah</i> 336:1.
  Moreover, it is far from clear that there exists a controversy between Abba Saul and the authors of the first principle recorded in the Mishnah. R. Menachem ha-Me'iri, <i>Hiddushei ha-Me'iri</i>, <i>ad locum</i>, expressly declares that there is no controversy and that the two principles are not mutually exclusive.15<i class="footnote">See also <i>Oẓar Mefarashei ha-Talmud</i>, <i>Makkot</i>, p. 350, note 12. R. David Pardo, <i>Shoshanim le-David</i>, <i>Makkot</i> 8a, points out that the first rule enumerated in the Mishnah and Abba Saul’s rule are derived from different words in the verse. The first rule is derived from the word “<i>ya’ar</i>” while, as explained in the Gemara, <i>Makkot</i> 8a, Abba Saul derives his rule from the word “<i>asher</i>” that, in this context, is rendered as “if” and thus, according to Abba Saul, the verse should be rendered “and if a person goes with his friend into a forest.” The term “if” in this context is understood as having the connotation that the slayer’s presence is discretionary. <i>Shoshanim le-David</i> further points out that Abba Saul’s rule is subsumed in the Mishnah’s earlier statement: Culpability is assessed only if the slayer had no right to be in the victim’s domain. The individual seeking to fulfill a <i>miẓvah</i> has an obligation to enter the domain of the victim. On the basis of that argument it should follow that Abba Saul may indeed disagree in rejecting the broader exclusion of the author of the first statement of the Mishnah but that Abba Saul’s rule is accepted by all: If permission of the householder to be present in his courtyard is sufficient to exonerate the slayer from the penalty of exile, <i>a fortiori</i>, divine dispensation to engage in an act that results in misadventure must be sufficient to exonerate a person who does so from the penalty of exile.</i> Me'iri regards the diverse exegetical interpretations of the phrase "in a forest" presented in the Mishnah as complementary rather than contradictory. For Me'iri, the term "<i>ya'ar</i>" is paradigmatic in serving to exclude arenas that are not comparable to a "forest" in any salient respect. Moreover, Rambam, <i>Hilkhot Rozeaḥ</i> 6:11, codifies the principle formulated in the earlier clause of the Mishnah. Thus, Rambam who, as previously noted, rules in accordance with Abba Saul, apparently saw no contradiction between Abba Saul's view and the principle earlier formulated in the Mishnah.
  In stating that the Tosefta expresses a view at variance from that of Abba Saul, <i>Or Sameaḥ</i> does not expressly assert that the Tosefta follows the position expressed in the initial statement of the Mishnah. <i>Or Sameaḥ</i> may not have intended to posit a controversy between the Tanna'im cited in the Mishnah but rather to have regarded the Tosefta as expressing the novel, extra-canonical view of a Tanna whose position was rejected by the Mishnah.16<i class="footnote">This possible reading of <i>Or Sameaḥ</i> seems to have eluded the editors of <i>Oẓar Mefarashei ha-Talmud</i>, <i>Makkot</i>, p. 351 and p. 351, note 19, who cite <i>Or Sameaḥ</i> as postulating a controversy within the Mishnah.</i> If so, the ruling of <i>Shulḥan Arukh</i>, <i>Yoreh De'ah</i> 336:1, in accordance with the Tosefta is all the more astonishing.
  The apparent contradiction between the principle formulated by Abba Saul and the ruling of the Tosefta was first commented upon by R. Simon ben Zemaḥ Duran, <i>Tashbaz</i>, III, no. 82. <i>Tashbaz</i> raises the objection that "the physician is also engaged in a <i>mizvah</i>" and hence questions why the Tosefta regards the physician as culpable for misadventure. <i>Tashbaz</i> cryptically comments that the case of the physician can be distinguished from that of a father or a teacher but fails to indicate the substantive nature of the distinction. That lacuna is filled by <i>Teshuvot Besamim Rosh</i>, no. 386, a work of uncertain provenance commonly, but probably spuriously, attributed to Rabbenu Asher.17<i class="footnote">See particularly, <i>Teshuvot Ḥatam Sofer, Oraḥ Ḥayyim</i>, no. 154 and R. Shlomoh Yosef Zevin, <i>Ha-Mo’adim be-Halakhah</i> (Tel Aviv, 5720), p. 247. R. Chaim Joseph David Azulai, <i>Shem ha-Gedolim</i>, II, <i>Ma’arekhet Sefarim</i>, sec. 127, opines that the work was authored by Rabbenu Asher but that portions of the text reflect tampering in the form of interpolations and modifications. Cf., R. Joseph Saul Nathanson, <i>Divrei Sha’ul</i>, <i>Yoreh De’ah</i> 336:1 and R. Yechezkel Abramsky, <i>Ḥazon Yeḥezkel</i>, <i>Bava Kamma</i> 9:3. In a posthumously published article that appeared in No’am, II (5719), 317-324, R. Yerucham Fishel Perla presents a critical survey of all prior discussions of the authenticity of <i>Besamim Rosh</i>, including discussions that appeared in the periodical literature, and provides his own evidence of the spurious nature of <i>Besamim Rosh</i>.</i> <i>Besamim Rosh</i> observes that, whatever the result, in striking a child or a student, the father and the teacher are engaged in chastising the youth. Thus, even if the act had an untoward outcome it was nevertheless designed for an entirely different effect and, indeed, despite the subsequent tragic consequence, may well have served to cause the child to resolve to correct his behavior.18<i class="footnote">Cf., however, Teshuvot <i>Ḥatam Sofer, Oraḥ Ḥayyim</i>, no. 177, who remarks that “neither the father nor the teacher nor the agent of the <i>bet din</i> has fulfilled his <i>miẓvah</i>.”</i> Not so a battery committed by a physician that fails in the therapeutic goal. The physician's obligation is to heal; an act that causes the death of a patient is not at all an act of healing. Hence the act, albeit well-intended, does not serve to fulfill any <i>mizvah</i>.19<i class="footnote">Students of Plato will hear in these words an echo of Plato’s comments in his <i>Republic</i>, I, 340. Plato remarks that at the moment at which a craftsman’s knowledge fails him he is no longer a craftsman. By way of example Plato comments that, at the moment that a physician makes a mistake in treating his patient, he cannot, properly speaking, be called a physician.</i> Accordingly, since it turns out that a physician who causes the death of his patient has not fulfilled any <i>mizvah</i> or discharged a duty, he is not excused from the penalty of exile. A similar distinction is drawn by <i>Yad Avraham</i>, <i>Yoreh De'ah</i> 336:1, in the name of <i>Ma'aseh Rokeaḥ</i>.
  R. Chaim Joseph David Azulai, Birkei Yosef, <i>Yoreh De'ah</i> 336:6, also presents the same solution in the name of the author of <i>Ma'aseh Rokeaḥ</i>, R. Eli'ezer of Cracow, but with a slight variation. That authority asserts that the Mishnah's exoneration of the father and the teacher is limited to usual situations in which the misadventure occurred despite the fact that the father or the teacher did not employ excessive force, i.e., they acted in a manner that was incumbent upon them.20<i class="footnote">See R. Gershon Koblentz, <i>Kiryat Ḥannah</i>, no. 22, who distinguishes between chastisement that may legitimately be administrated by a father and force used by a teacher to correct a student. The latter may utilize only “a small strap” in order to impose discipline whereas the father, whose corrections go beyond assuring that the child is attentive, may engage in corporal punishment. Hence, the Mishnah speaks of a father who “beats” his son in contradistinction to a teacher who may only “strike” a pupil. According to <i>Kiryat Ḥannah</i>, a teacher who employs excessive force does not therein fulfill a <i>miẓvah</i> and hence, if the child dies as a result, the teacher is exiled. That position is consistent with the comment of <i>Besamim Rosh</i>; cf., however, R. Joshua of Kutna, <i>Teshuvot Yeshu’ot Yisra’el</i>, <i>Ḥoshen Mishpat</i> 25:3, cited <i>infra</i>, note 22.For a fuller discussion of appropriate versus inappropriate forms of chastisement see R. Ya’akov Meir Stern, <i>Imrei Ya’akov</i> (Bnei Brak, 5756), <i>Bi’urim</i>, pp. 39-43.</i> However, the physician who causes the death of a patient has erred either in his assessment of the nature of the malady or of the appropriateness of the therapy he has administered.21<i class="footnote">See <i>infra</i>, notes 49 and 50 and accompanying text.</i> Since the physician's act was performed in error,22<i class="footnote">Teshuvot <i>Yeshu’ot Yisra’el</i>, <i>Ḥoshen Mishpat</i> 25:3, formulates the distinction in a somewhat different manner: The father and the teacher commenced their act in an entirely legitimate manner; only later, in not curtailing the chastisement, did they apply excessive force. Hence, they are not punished by exile. In effect, asserts <i>Yeshu’ot Yisra’el</i>, since the battery is privileged, they are immune to punishment despite their abuse of the privilege. The physician, on the other hand, erred in the first instance in attempting to amputate a limb or in making an incision. From its very inception, his act served to fulfill no <i>miẓvah</i> and hence was not privileged. It would seem to follow that, according to <i>Yeshu’ot Yisra’el</i>, the case of a surgeon who correctly commences a procedure but, for example, later negligently nicks an artery and thereby causes the death of his patient is analogous to the case of the father and the teacher and, in such instances, the physician is not punished by exile. R. Shlomoh Zalman Auerbach, as cited by Abraham S. Abraham, <i>Nishmat Avraham</i>, <i>Yoreh De’ah</i> 336:1, note 7, makes much the same point in suggesting that the physician is liable to exile only if his act was not at all appropriate, e.g., he made a surgical incision in the wrong site, in which case he was not at all engaged in a <i>miẓvah</i>. If, however, he commences a procedure that is indeed therapeutic but errs in assessing its execution his situation is comparable to that of a father or teacher who errs in assessing the proper measure of chastisement.</i> the physician, despite his noble intentions, cannot be described as having been engaged in performing an act endowed with the nature of a <i>mizvah</i>.23<i class="footnote">The third example of an individual who is forced to go into exile offered by the Tosefta, i.e., a messenger of the <i>bet din</i> who causes death, remains somewhat problematic since the act of the agent of the <i>bet din</i> seems to be analogous to that of a father or a teacher and indeed Abba Saul, as recorded in the Mishnah, explicitly exempts such an official from exile. Nevertheless, the Tosefta may be understood as referring to an agent of the <i>bet din</i> who erroneously administers more than the prescribed number of lashes. In no way does the additional stroke of the lash constitute fulfillment of a duty. Accordingly, Rambam, <i>Hilkhot Sanhedrin</i> 16:2, rules that under such circumstances the messenger is liable to the punishment of exile. See <i>Oẓar Mefarashei ha-Talmud</i>, p. 351, note 20. Rambam, <i>Hilkhot Roẓeaḥ</i> 5:5, understands Abba Saul’s exemption of the agent of the <i>bet din</i> from exile as applying to the case of a messenger who uses physical force to compel the appearance of a person summoned by the <i>bet din</i>. That situation is comparable to the case of the father or teacher in that the messenger is indeed properly discharging a duty incumbent upon him.</i> It would follow that, according to <i>Birkei Yosef</i>, a physician who properly assesses his patient's condition and administers appropriate treatment is not subject to exile if the patient unpredictably dies for an unknown reason.
  As will be shown subsequently, a number of contemporary authorities also understand the Tosefta's statement with regard to imposition of exile to be limited to situations involving particular forms of negligence on the part of the physician. If so, there is no contradiction between the Tosefta and the general rule formulated in the Mishnah.
  2. Ramban
  The Gemara, <i>Bava Kamma</i> 85a, declares "'… and he shall cause him to be thoroughly healed' (Exodus 21:19.)—From here [it is derived] that the physician has been given authority to heal." In Jewish law, as in all coherent legal systems, all activities are permissible unless they are expressly prohibited or contraindicated on the basis of some legal or moral consideration. If so, in the absence of any countervailing consideration, why should a physician require explicit permission to practice the healing arts? Absent the scriptural dispensation provided by this verse, why should a physician shrink from using his skills in order to cure a patient? In their respective commentaries on this talmudic passage, Rashi, <i>Tosafot</i> and Rashba explain that the constraint is theological in nature. To paraphrase Rashi's formulation: "If God afflicts, how dare man attempt to cure?" Or, as expressed by <i>Tosafot</i>, in curing the patient, the medical practitioner "appears to thwart the divine decree." Such would be the physician's concern in the absence of specific dispensation; once permission is given, practice of the healing arts becomes intrinsic to God's providential guardianship of man and hence medical ministration is not only permissible or even commendable but is mandatory.24<i class="footnote">See this writer’s “The Obligation to Heal in the Judaic Tradition,” <i>Jewish Bioethics</i>, ed. Fred Rosner and J. David Bleich, 2nd edition (New York, 2000), pp. 22-30.</i>
  In his <i>Torat ha-Adam, Shaar ha-Sakanah</i>,25<i class="footnote"><i>Kol Kitvei ha-Ramban</i>, ed. R. Bernard Chavel (Jerusalem, 5724), II, 42.</i> Ramban presents the foregoing rationale for the need of specific authorization to practice medicine but prefaces that comment with an entirely different explanation. The verse in question, declares Ramban, is necessary in order to teach that the practice of medicine is not forbidden because of the possibility of a disastrous outcome. The physician is given authority to heal "lest the physician say: 'Why should I [seek] this vexation? Perhaps I will err with the result that I become an inadvertent slayer of human souls.'"26<i class="footnote"><i>Ibid</i>., p. 41.</i> Elsewhere in his <i>Torat ha-Adam</i>,27<i class="footnote"><i>Ibid</i>., p. 43.</i> Ramban writes, "… since license was given to the physician to heal and, moreover, it is a <i>mizvah</i> that is incumbent upon him, he need have no concern; for if he conducts himself appropriately in accordance with his opinion he has naught but a <i>mizvah</i> in his medical ministrations, for God commands him to heal and his intellect coerced him to err."
  Ramban is however troubled by the fact that, despite scriptural reassurance that he should have no moral qualms, the physician, found qualified and licensed to practice medicine by the <i>bet din</i>, who errs and causes the death of his patient is nevertheless held culpable of manslaughter and, if his error is non-lethal, is liable—at least according to the "laws of Heaven"—to pay compensation to his victim. If the physician is commanded to treat his patient and to ignore the potential for inadvertent error, how can he be held liable for doing what he is duty-bound to do?
  Ramban's resolution of the problem is startling, to say the least. His solution is novel but seemingly paradoxical. Ramban responds by stating that the situation of the physician is analogous to that of a judge. The <i>dayyan</i> is commanded to sit in judgment and, declares the Gemara, <i>Sanhedrin</i> 6b, "Lest the judge say, 'Why should I seek this vexation?' the verse states '[God is] with you in the matter of judgment' (II Chronicles 19:6)—the <i>dayyan</i> has only what his eyes see." Nevertheless, in some circumstances, the <i>dayyan</i> will be held liable for judicial malpractice. Ramban adds the remarkable comment that the judge is liable "if he errs and it becomes known to the <i>bet din</i> that he erred … similarly [in the case of the physician], according to the laws of man he is not liable to payment but according to the laws of Heaven he is not quit until he pays for the damage and goes into exile for the death <i>since it has become known that he erred</i> [emphasis added] and has caused damage or caused death by means of a direct act."
  Ramban formulates the curious position that the judge and the physician are liable in the eyes of Heaven—but only if they are found out. It is of course readily understood that neither the judge nor the physician can be required, even by Heaven, to make restitution if he remains unaware of his error and hence of his liability. But certainly Heaven is aware of the error and hence the person should not be guiltless in the eyes of Heaven even if, as a practical matter, he cannot be called upon to redress the wrong. A layman who commits an ordinary tort without ever becoming aware of the damage he caused, e.g., a person who throws a rock and unknowingly shatters a valuable vase, and whose act was unobserved, will never be called upon to make restitution simply because he is not identifiable as the tortfeasor but, assuredly, he is not guiltless in the eyes of Heaven. Yet the <i>dayyan</i> and the physician, declares Ramban, are guilty "at the hands of Heaven" only if it becomes known to them that they have caused harm.
  Furthermore, Ramban's thesis is formulated in an attempt to resolve a difficulty but apparently falls short of doing so. Absent scriptural reassurance, the prudent physician and judge are pictured as abjuring their callings for fear of potential punishment. Assurance that liability is contingent upon actual awareness of resultant harm hardly seems to assuage such a concern.28<i class="footnote">Cf., R. Zalman Nechemiah Goldberg, <i>Teḥumin</i>, XIX, 318-320. Rabbi Goldberg accepts the notion that liability “at the hands of Heaven” is triggered by awareness. In this writer’s opinion that thesis is unsubstantiated and counterintuitive. Moreover, it fails to explain how the concerns of the judge or the physician are thereby assuaged.</i>
  It seems to this writer that Ramban's comment must be understood in light of the dual nature of the transgression inherent in misappropriation of property, commission of a tort and manslaughter. A person who deprives another of his property has committed an act of theft and in doing so has sinned against his fellow. In addition, he has violated the divine commandment prohibiting theft and has thereby sinned against God. The <i>dayyan's</i> putative concern that he may err in rendering judgment is not born of a concern for the welfare of the financially harmed litigant but of the <i>dayyan's</i> fear of sullying his own immortal soul by sinning against God. The verse "God is with you in the matter of judgment" serves to assure the judge that God joins in the judgment even if it is in error, i.e., insofar as God is concerned, the qualified judge who errs commits no sin. Indeed, theologically speaking, his error may, in a certain sense, be providentially ordained.29<i class="footnote">In explaining why false witnesses whose testimony leads not only to conviction but to actual execution are not put to death, Ramban, <i>Commentary on the Bible</i>, Deuteronomy 19:19, asserts that God does not abandon the totally guiltless and, moreover, God Himself participates in the judgment of the court.</i> In concurring in the judgment of the <i>bet din</i>, The Heavenly Court renders judgment not simply upon the pleadings of the litigants but upon a totality of considerations, many of which are likely to be unknown to mortals.
  That consideration, however, is relevant only from the divine perspective; insofar as the obligations of a man to his fellow man are concerned, "[the Torah] is not in Heaven" (Deuteronomy 30:12). Hence the <i>dayyan</i> may be required to compensate the victim of his mistake if, in terms of terrestrial considerations, the <i>dayyan</i> has committed a culpable error. But, since in the eyes of Heaven he is entirely guiltless and a person who is unaware of any harm that he may have caused cannot compensate the victim, Heaven must hold him guiltless in every sense unless he becomes aware of his error. Thus assured, the <i>dayyan</i> has no reason to shirk the duties of his office. The <i>dayyan</i> is dedicated to his sacred calling and, if competent, dare not be dissuaded by fear of potential financial loss.30<i class="footnote">Regarding the obligation of a qualified scholar not to demur when requested to serve as a <i>dayyan</i> see R. Moses Sofer, <i>Teshuvot Ḥatam Sofer</i>, <i>Ḥoshen Mishpat</i>, no. 169 and this writer’s elucidation of that obligation in <i>Be-Netivot ha-Halakhah</i>, IV (New York, 5771), 180-181.</i> If the <i>dayyan</i> errs, he will dutifully make restitution. His real fear is the fear of inadvertent sin and on that score the biblical verse serves to put his mind at ease.
  Ramban, in this comment, formulates a remarkable thesis: Every trespass against one's fellow man constitutes a dual infraction: 1) an offense against one's fellow (<i>bein adam le-ḥavero</i>); and 2) an offense against God. Halakhah is replete with examples in which a <i>bet din</i> will not issue a monetary reward but the individual remains liable "at the hands of Heaven." In such cases, Heaven will not grant forgiveness despite the fact that the aggrieved party is non-suited by the <i>bet din</i>. The dual nature of the infractions explains why there may be an obligation "at the hands of Heaven" even when there is no obligation at the hands of man.
  Herein, in effect, Ramban posits the converse, i.e., an obligation "at the hands of man" despite the absence of liability in the eyes of Heaven. The <i>dayyan</i> who, in his error, unwittingly carried out a divine decree has committed no transgression against Heaven, but since the Torah "is not in Heaven" and since in terms of human considerations he has ruled improperly, he may be required to make restitution entirely as a matter of <i>bein adam le-ḥavero</i>. But such an obligation, since it is entirely between man and his fellow man, can be meaningful only when and if the existence of such an obligation becomes known to the guilty party. If it fails to become known, Heaven will impose no penalty because in the eyes of Heaven the individual in question is guiltless.
  Having established this principle with regard to the <i>dayyan</i>, Ramban applies it as well to the physician who errs in treating his patient. A person who commits a battery commits a transgression <i>vis-à-vis</i> his fellow man as evidenced by the requirement to compensate for the harm done. In causing physical harm to another, he also transgresses the divine commandment "he shall not exceed" (Deuteronomy 25:3).31<i class="footnote">The literal import of the verse is an admonition not to impose more than the prescribed number of lashes in punishing a miscreant. According to the Oral Law tradition the prohibition applies <i>a fortiori</i> to beating or “wounding” an innocent person. See Rashi, Deuteronomy 25:3; <i>Sanhedrin</i> 85a and Rashi, <i>ad locum</i>, s.v. <i>ve-aḥer</i>; and Rambam, <i>Hilkhot Ḥoval u-Mazik</i> 5:1.</i> The physician who errs in treating his patient is not engaged in therapeutic "wounding" excluded from that prohibition; he has committed a simple battery and has reason to fear that he has committed an offence against the Deity. Fear of committing such a transgression might well serve to discourage the physician from engaging in his profession. The verse "and he shall cause him to be thoroughly healed," according to the understanding of the passage advanced by Ramban, serves completely to exonerate the physician from any transgression. In drawing a parallel between the <i>dayyan</i> and the physician Ramban undoubtedly intends to assert that any harm caused by the physician is the result of Heavenly decree. That concept is eloquently captured in the medical context in an aphorism coined by R. Moshe Hagiz: "The unintentional act of the physician is the intent of the Creator."32<i class="footnote">This aphorism appears in R. Moshe Ḥagiz’ letter of approbation to R. Shlomoh Zalman Henau’s work on grammar, <i>Sha’arei Torah</i> (Hamburg, 5478).</i> Since the erring physician is, in effect, a divine messenger, Heaven can hardly hold him responsible for the untoward results of his ministration.
  But, again, the Torah "is not in Heaven." Hence, in terms of obligation to his fellow man, the physician remains responsible for making his victim whole. But that is an obligation the physician incurs entirely <i>vis-à-vis</i> his fellow man and is devoid of any "religious" implications.33<i class="footnote">A similar notion with regard to the financial liability of a <i>get</i> for obligations incurred while he was a non-Jew and for an adult for obligations incurred while yet a minor is formulated by <i>Teshuvot Bet Ya’akov</i> on the basis of his understanding of <i>Tosafot, Sanhedrin</i> 71a. Cf., however, R. Jacob Reischer, <i>Teshuvot Shevut Ya’akov</i>, no. 177.</i> In other cases failure to satisfy a financial obligation constitutes a "religious" infraction as well. Depending upon the circumstances, the "religious" obligation with regard to monetary compensation is born of an obligation to restore stolen or lost property, of contractual liability or of an obligation in tort. The physician's "religious" obligation could only be of the latter category but since he acts at the divine behest no such "religious" obligation exists. Accordingly, his obligation is solely in the nature of a duty owed a fellow man with no parallel duty to God to fulfill that duty. Hence, according to Ramban, if the physician remains unaware of his error he cannot be held culpable in any sense of the term.
  However, it should then follow that when the error becomes known, the physician, no less so than the <i>dayyan</i>, should be held liable by the <i>bet din</i>. That is indeed the case according to biblical law. However, the Tosefta, <i>Gittin</i> 3:13, declares that the Sages found it necessary to grant the physician immunity "for the welfare of society" (<i>mipnei tikkun ha-olam</i>). In effect, they recognized that failure to hold a physician guiltless in instances of misadventure would result in many physicians declining to practice medicine for fear of financial liability. Nevertheless, the obligation "<i>bein adam le-ḥavero</i>" remains "at the hands of Heaven" even though in the eyes of Heaven the physician has done no wrong. The guilt recognized by Heaven is not for a sin against the Deity but for an offense against the victim. Apparently no similar conferral of blanket immunity was regarded as necessary in order to entice a qualified scholar to serve as a <i>dayyan</i>.34<i class="footnote">For a detailed survey of the circumstances in which a <i>dayyan</i> is liable for judicial error see <i>Encyclopedia Talmudit</i>, XX (Jerusalem, 5751), 620-637.</i>
  The statement that the physician is liable "at the hands of Heaven" even though in the eyes of Heaven he has committed no harm means simply that a moral obligation <i>bein adam le-ḥavero</i> does exist. Moreover, the statement has a very practical effect, <i>viz.</i>, the victim is entitled to exercise self-help (<i>tefisah</i>), i.e., seizure of property or funds, without authorization of the <i>bet din</i>. According to many authorities self-help is a remedy available in all cases in which Halakhah posits an obligation "at the hands of Heaven"35<i class="footnote">See <i>Nimukei Yosef</i>, <i>Bava Meẓi’a</i> 67b and <i>Pitḥei Teshuvah</i>, <i>Ḥoshen Mishpat</i> 28:6. For a survey of the various opinions regarding this question see <i>Encyclopedia Talmudit</i>, VII (Jerusalem, 5716), 395-396. It is indeed the case that Ramban, as cited by <i>Teshuvot Rivash</i>, no. 392, rejects self-help as a remedy with regard to an obligation “at the hands of Heaven.” However, Ramban’s comment is limited to situations in which the plaintiff asserts a claim certain and the defendant professes ignorance. In such a situation the nature of the obligation “at the hands of Heaven” is to abjure tainted funds. But if the claim is false there can be no obligation even “at the hands of Heaven.” Since in such a case no evidence is offered in support of the claim, Ramban maintains that self-help is not available. It cannot be deduced from that position that self-help is not a legitimate remedy in situations in which the obligation in the eyes of Heaven is a certainty.It is further arguable that the opinion of the authorities who maintain that self-help is of no avail is limited to situations in which the obligation, by virtue of its nature, is only in the eyes of Heaven but not to the situation addressed by Ramban in which, according to the analysis presented herein, the claim is fully actionable according to biblical law but the victim is non-suited by rabbinic decree “for the benefit of the universe.” In such instances it might well be maintained that the rabbinic decree merely constrains the <i>bet din</i> but does not extinguish underlying liability and hence self-help may be warranted.</i> particularly in instances in which the obligation is biblical in nature.36<i class="footnote">See R. Jacob Reischer, <i>Teshuvot Shevut Ya’akov</i>, I, no. 146, cited by <i>Pitḥei Teshuvah</i>, <i>Ḥoshen Mishpat</i> 28:6.</i>
  Ramban's comment with regard to culpability for exile may be understood in a similar fashion. The nature of exile in instances of inadvertent manslaughter is complex. In a number of instances37<i class="footnote">See <i>Makkot</i> 2b, 8b and 11b and <i>Ketubot</i> 36b and 38a. See also the early-day commentators cited in <i>Encyclopedia Talmudit</i>, VI, 123, note 16.</i> the Gemara describes exile as a form of expiation, i.e., as atonement for inadvertent transgression of the prohibition against homicide. However, since Ramban asserts that the physician who "conducts himself appropriately … has naught but a <i>mizvah</i>," such a physician is clearly not in need of expiation.
  Exile serves not only as expiation but as a punishment as well.38<i class="footnote">See <i>Ner Aharon</i>, no. 6.</i> Evidence that exile is also a form of punishment is found in the discussion of the Gemara, <i>Makkot</i> 2b, regarding the punishment of witnesses who by accusing a person of manslaughter seek to have him banished to one of the cities of refuge and whose testimony is impeached (<i>edim zomemim</i>). The general rule is that such witnesses receive punishment in kind, i.e., they are accorded the selfsame punishment they sought to have meted out to the defendant against whom they testified. Witnesses who testify with regard to unintentional manslaughter but who are impeached are not themselves subjected to exile. However, that is so only because they are expressly excluded from exile on the basis of exegetical interpretation of a biblical verse. It is readily established that the punishment of impeached witnesses is not designed to expiate their transgression as evidenced by the fact that the Gemara, <i>Makkot</i> 2b, declares that witnesses are not liable to monetary payment in the form of <i>kofer</i>, or "atonement," if their testimony concerning an ox that gores a person is impeached.
  The master of an ox that has previously committed such acts and who has been properly admonished regarding the irascible nature of his animal is liable to payment of a sum of money in an amount equal to the amount the victim might have commanded were he to be sold as a slave. That payment is designated as "<i>kofer</i>" or "atonement," i.e., it is designed, not as restitution, but as expiation of the transgression inherent in the master's negligence in not exercising proper supervision over his ox. However, in contradistinction to other cases of false testimony, witnesses who seek to subject the master of an ox to such punishment but whose testimony is impeached are not subject to monetary punishment in the form of <i>kofer</i>. The principle reflected in this provision is that impeached witnesses are subject to punishment but their punishment does not serve to expiate their transgression.39<i class="footnote">See Ritva, <i>Makkot</i> 2b.</i> Accordingly, since the essence and purpose of <i>kofer</i> is expiation and atonement, whereas the punishment of impeached witnesses is not designed to serve as expiation, impeached witnesses are not called upon to make such payment. Thus, since an explicit biblical verse is necessary to exclude impeached witnesses from the penalty of exile, it is evident that the exile to which impeached witnesses would have been subjected had Scripture not excluded them from that penalty would have been in the nature of punishment rather than of expiation. But, assuredly, the physician who "has naught but a <i>mizvah</i>" does not deserve to be punished.
  However, exile serves a third purpose as well, <i>viz.</i>, it serves to preserve the perpetrator from vengeance at the hands of the blood-avenger.40<i class="footnote">Indeed, Maharsha, <i>Ḥiddushei Aggadot</i>, <i>Makkot</i> 10b, asserts that this is the sole purpose of exile. For sources raising obvious objections to the Maharsha’s position see <i>Encyclopedia Talmudit</i>, VII, 123, note 15. See also <i>Or Sameaḥ</i>, <i>Hilkhot Roẓeaḥ</i> 6:12.</i>
  Ramban's comment to the effect that the verse "and he shall cause him to be thoroughly healed" serves to assure the physician that he need not be concerned over possible unintentional manslaughter "unless he becomes aware that he has erred" can be understood on the basis of the multifaceted purpose of exile. In this case as well, the physician whom Scripture seeks to reassure is not a physician whose concern is his own safety or well-being but the physician whose concern is the possibility of transgression. The verse serves to establish that a physician who errs inadvertently has committed no transgression whatsoever. Hence, he incurs no Heavenly punishment and requires no expiation. Accordingly, if he never discovers his error, nothing is lost. He is, however, subject to the wrath of the blood-avenger.41<i class="footnote">The blood-avenger lacks licence to execute a father or teacher who inadvertently causes the death of a child because, in committing the ill-fated act, they were actually engaged in fulfilling a <i>miẓvah</i>. See <i>supra</i>, note 18, and accompanying text. The physician who inadvertently causes harm intends to perform a <i>miẓvah</i> but does not actually do so. See, however, <i>infra</i>, section III.</i> That, however, is a contingency that he need not fear unless his error becomes known. But even if his error becomes known he faces no temporal danger because of the safety afforded him through exile in a city of refuge.42<i class="footnote">This analysis assumes that the blood-avenger has license to seek vengeance but is not engaged in a form of extra-judicial punishment. Cf., however, <i>Or Sameaḥ</i>, <i>Hilkhot Melakhim</i> 3:10.An interesting question presents itself in the case of a physician who inadvertently causes the death of a Canaanite slave. Canaanite slaves have no halakhically recognized biological relatives; hence a Canaanite slave can have no blood-avenger. If, as Ramban is herein understood, the physician is not subject to punishment and requires no expiation, a physician who causes the death of a Canaanite slave should be exempt from exile since he is not at risk of vengeance at the hands of the blood-avenger. Cf., <i>Or Sameaḥ</i>, <i>Hilkhot Roẓeaḥ</i> 6:12.It cannot be countered that the regulations pertaining to exile are rules of general application that do not admit exceptions. Although it is the case that they do not admit of <i>ad hoc</i> exceptions, e.g., in the case of a particular individual, such as a proselyte who does not happen to have a blood relative, they do admit of exceptions in instances in which a particular concern cannot possibly be manifest. For example, a resident-alien is not subject to exile because he is not thereby granted immunity from the blood-avenger. Nevertheless, as recorded by Rambam, <i>Hilkhot Roẓeaḥ</i> 5:3, a resident-alien who causes the death of a Canaanite slave is exiled even though the Canaanite slave cannot possibly have a blood-avenger; in that instance, exile is required for purposes of expiation. See <i>Or Sameaḥ</i>, <i>Hilkhot Roẓeaḥ</i> 6:12. In light of the herein presented understanding of Ramban, it stands to reason that a physician who causes the death of a Canaanite slave should be exempt from exile since he requires neither expiation nor refuge from the blood-avenger.</i>
  III. Exclusion from Culpability for Exile
  As noted earlier, <i>Shulḥan Arukh</i>, <i>Yoreh De'ah</i> 336:1, in accordance with the statement recorded in the Tosefta, rules that a physician who causes the death of his patient is liable to exile. Ostensibly, the sources recording the physician's culpability for exile declare that he is liable to banishment for causing the death of the patient regardless of whether he has been negligent. Ofttimes a surgeon performs an operation knowing quite well that there is a statistical probability that the patient will not survive the surgery. Nevertheless, both the surgeon and the patient assume that risk because, absent surgical intervention, death is either a certainty or more likely to occur. However, on the basis of the distinction drawn by the <i>Birkei Yosef</i> between a father or a teacher on the one hand and a physician on the other, there would appear to be no reason to require exile in such circumstances. In such an instance the physician commits no error. He is engaged in an attempt to perform a <i>mizvah</i> no less so than the father who engages in an attempt to chastise a son or a teacher who endeavors to correct a pupil. However, that conclusion does not flow from the earlier-cited distinction formulated by <i>Besamim Rosh</i>.43<i class="footnote">Nor does it flow necessarily from the distinction drawn by <i>Yeshu’ot Yisra’el</i> and R. Shlomoh Zalman Auerbach, cited <i>supra</i>, note 21. Nevertheless, Rabbi Auerbach asserts that a physician is liable to exile only if the patient’s death results from an error in judgment. See <i>Nishmat Avraham</i>, <i>Yoreh De’ah</i> 376:1, note 1.</i>
  The question of the culpability of a physician who causes the death of his patient but who has committed no error is also addressed by a number of contemporary authorities. R. Zalman Nechemiah Goldberg, <i>Teḥumin</i>, XIX (5759), 320, asserts that, since there is no element of "error," the situation is comparable to a case of <i>ones</i>, or unavoidable death, which does not entail exile. That analysis is imprecise in the sense that in cases of statistical danger the physician knowingly and willingly assumes the attendant risk.44<i class="footnote">See R. Chaim Ozer Grodzinski, <i>Teshuvot Aḥi’ezer</i>, II, no. 16, sec. 6, with regard to whether or not there must be a fifty percent chance of survival in order to justify the risk.</i> Assuming that negligence is not a necessary element in imposition of the penalty of exile, the physician should indeed be culpable since the situation is not akin to a case of unavoidable or unpreventable death and certainly not to a case of <i>force majeure</i>.
  Other scholars clearly go beyond Rabbi Goldberg's position and maintain that a physician who causes the death of a patient through no fault of his own but as the result of a <i>bona fide</i> error of judgment is not liable to exile. One authority, R. Yechiel Michel Epstein, <i>Arukh ha-Shulḥan</i>, <i>Yoreh De'ah</i> 336:2, asserts that a physician is exiled only if the misadventure he has caused was the result of "laziness" (<i>hitrashlut</i>) or failure to exercise due diligence (<i>lo iyein yafeh</i>). From a parenthetical comment incorporated in that ruling it appears that <i>Arukh ha-Shulḥan</i> understands that the distinction between a father or a teacher and a physician as reflected in the Tosefta lies in the consideration that the physician to whom the Tosefta refers is a physician who has been negligent in discharging his duties whereas the father and the teacher of whom the Mishnah speaks have not been negligent.
  R. Shlomoh Zalman Auerbach, as quoted by Abraham S. Abraham, <i>Nishmat Avraham</i>, <i>Yoreh De'ah</i> 376:1, note 7, suggests that the penalty of exile is not imposed in instances of an error in judgment. That penalty, he suggests, is imposed only if the physician performs an unintended act, e.g., he reaches for the wrong medicine or erroneously takes hold of a non-sterilized scalpel.45<i class="footnote">See also R. Shlomoh Zalman Auerbach, <i>Minḥat Shlomoh</i>, I, no. 32, note 1, where that thesis is introduced with the words “<i>akh efshar</i> (but perhaps).”</i>
  A similar but markedly different view is reflected in the comments of R. Moses Feinstein, <i>Iggerot Mosheh, Even ha-Ezer</i>, IV, no. 31, s.v. <i>u-mah she-nizkar</i>. <i>Iggerot Mosheh</i> declares that as long as there is no better-qualified physician available and the physician has acted with due deliberation in accordance with his knowledge and skill his act is regarded as tantamount to that of an <i>anus</i> i.e., a person whose act is unavoidable, and he is not liable to exile. Action on the part of the physician in accordance with the dictates of his intellect resulting in misadventure is thus equated with <i>force majeure</i>. Thus, <i>Iggerot Mosheh</i> is in agreement with <i>Arukh ha-Shulḥan</i> and Rabbi Auerbach in ruling that non-negligent error does not render the physician culpable for exile. However, unlike <i>Arukh ha-Shulḥan</i>, <i>Iggerot Mosheh</i> asserts that if the physician acted "in haste" (<i>be-behilut</i>), even if it seemed to him that no further reflection was necessary, the physician is also exempt from exile. In that case, however, the physician is exempt from exile, not because he is without guilt, but because the physician's negligence is of a magnitude for which exile is an insufficient punishment. Exile, opines <i>Iggerot Mosheh</i>, is reserved solely for situations in which
  the matter was not urgent and it would have been possible to wait for a more highly qualified physician or in situations in which the physician was the most highly qualified but he did not reflect more than his wont, then if [he acted] in accordance with the manner of physicians [in which case] there is no negligence, he is liable to exile and [exile] serves as expiation [for his transgression].
  Presumably, reaching in haste for the wrong medicine or for an unsterilized scalpel, according to <i>Iggerot Mosheh</i>, would not result in exile because such an act constitutes gross negligence for which exile is an insufficient expiation.
  Failure to administer a diagnostic test that would have led to proper treatment represents an act of nonfeasance rather than malfeasance. Accordingly, it would follow that such forms of malpractice should not result either in subjecting the physician to exile or to liability to tort damages. However, if failure to administer such a test leads to overt intervention resulting in the death of the patient, the physician, according to <i>Arukh ha-Shulḥan</i>, would seem to be culpable for such intervention since failure to administer such a test represents failure to exercise due diligence (<i>lo iyein yafeh</i>). Similarly, according to <i>Iggerot Mosheh</i>, the physician must be judged to have acted "in haste" (<i>be-behilut</i>). However, according to Rabbi Auerbach, if the physician did not order the test because he erroneously regarded it to be unnecessary, and proceeded with treatment that results in harm he has committed an error of judgment for which he does not incur the penalty of exile.
  Similarly, an act of omission cannot result in liability for tort damages, although if the physician is regarded as a bailee, as will be discussed later, he may be liable for resultant damages by virtue of having failed to perform his duties as a bailee.46<i class="footnote">See <i>infra</i>, section VI.</i> Tort liability for harm caused by active intervention that would have been avoided had the diagnostic test been ordered must be examined in the context of the general provisions governing a physician's liability and, as will be discussed subsequently,47<i class="footnote">See <i>infra</i>, note 56 and accompanying text.</i> may be contingent upon a possible distinction between ordinary negligence and gross negligence in the physician's failure to administer the diagnostic test.
  R. Moses Sofer, <i>Teshuvot Hatam Sofer, Oraḥ Hayyim</i>, no. 177, presents a position that appears not only to qualify the ruling of <i>Shulḥan Arukh</i> but also to be at variance from that normative ruling. In that responsum <i>Hatam Sofer</i> addresses a tragic case of inadvertent poisoning. A servant girl fainted from sudden fright. Her mistress sought to revive her by giving her whiskey to drink. However, in reaching for the flask of whiskey she inadvertently took hold of a jar of "petrol" (probably kerosene). The liquid poured into the young lady's throat "went down into her innards and the lass was burned." Her mistress, understandably consumed by feelings of guilt, sought advice with regard to a proper form of expiation. In his responsum, <i>Hatam Sofer</i> opines that the woman who administered the poisonous substance was far less culpable than a father or a teacher who chastises a child. Hence, asserts <i>Hatam Sofer</i>, were such a penalty still imposed in our day, she would not be subject to the punishment of exile. <i>Rav Pe'alim</i>, III, <i>Oraḥ Hayyim</i>, no. 36, s.v. <i>ve-atah</i>, objects to <i>Hatam Sofer's</i> analysis on the grounds that the case of the woman is comparable to that of a physician who, according to the Tosefta and the ruling of <i>Shulḥan Arukh</i>, <i>Yoreh De'ah</i> 336:1, is liable to exile. The same objection was later raised by R. Yitzchak Ya'akov Weisz, <i>Teshuvot Minḥat Yizḥak</i>, III, no. 104, sec. 5.48<i class="footnote">Cf., the view of <i>Matteh Yehudah, Yoreh De’ah</i> 336, discussed <i>supra</i>, note 13.</i>
  R. Samson Aaron Polonski, <i>Divrei Aharon</i>, no. 34, sec. 2, attempts to defend <i>Hatam Sofer's</i> response by arguing that it is predicated upon an earlier ruling of R. Simon ben Ẓemaḥ Duran, <i>Tashbaz</i>, III, no. 82. As will later be more fully discussed, <i>Tashbaz</i> asserts that a physician is culpable only for death resulting from an act performed by his own hand or by an instrument wielded by him, but that a physician who treats patients by means of "liquids or medicaments" is not similarly liable even "at the hands of Heaven." <i>Tashbaz</i>' distinction is presumably based upon the consideration that culpability is contingent upon proximate cause which, in Jewish law, is narrowly defined.49<i class="footnote">See <i>Teshuvot Minḥat Yiẓḥak</i>, III, no. 104, sec. 1, who understands <i>Tashbaẓ</i>’ distinction as reflecting this consideration. However, R. Yitzchak Zilberstein, <i>Halakhah u-Refu’ah</i>, ed. R. Moshe Hershler, II (Jerusalem, 5741), 288, [reprinted with minor additions in <i>Emek Halakhah: Assia</i>, ed. R. Mordecai Halperin (Jerusalem, 5746), p. 131], understands the distinction as reflecting an entirely different concept. Rabbi Zilberstein asserts that <i>Tashbaẓ</i> is distinguishing between <i>bona fide</i> error and negligence: the internist diagnoses and provides treatment in accordance with his assessment of the malady; the surgeon of whom <i>Tashbaẓ</i> speaks harms the patient as a result of negligent cutting. Hence, he asserts, a surgeon cannot be held culpable for honest misdiagnosis that leads him to perform an unnecessary surgical procedure. If so, <i>Tashbaẓ</i>’ comments are simply the harbinger of the position later advanced by <i>Arukh ha-Shulḥan</i>, Rabbi Auerbach and <i>Iggerot Mosheh</i>. Needless to say, if such a distinction were indeed intended by <i>Tashbaẓ</i> he would have been expected expressly to indicate that the surgeon is not liable in any and all circumstances.</i> Harm done by a hand is direct; poison, however, must first be absorbed by the body and hence the resultant harm is indirect, i.e., it is effected through a form of <i>gerama</i>.50<i class="footnote">R. Chaim Joseph David Azulai, <i>Birkei Yosef</i>, <i>Yoreh De’ah</i> 336:8, cites <i>Tashbaẓ’</i> ruling but notes that neither Ramban nor Tur nor <i>Shulḥan Arukh</i> makes such a distinction and suggests that <i>Tashbaẓ’</i> position is open to rebuttal. A point similar to that of <i>Tashbaẓ</i> is made by R. Jacob Schorr in a responsum published in <i>Teshuvot Ge’onei Batra’i</i> (Prague, 5576), no. 6, a compendium edited by <i>Sha’agat Aryeh</i>. R. Judah Ayash, <i>Teshuvot Bet Yehudah</i>, <i>Even ha-Ezer</i>, no. 14, maintains that abortion induced by chemical potions is rabbinically forbidden whereas direct destruction of the fetus is biblically proscribed. See also R. Ovadiah Yosef, <i>Yabi’a Omer</i>, IV, <i>Even ha-Ezer</i>, no. 1, sec. 5. Cf., R. Eliezer Waldenberg, <i>Ẓiẓ Eli’ezer</i>, VIII, no. 36, who fails to make this distinction but who indicates elsewhere, <i>Ẓiẓ Eli’ezer</i>, IX, no. 51, <i>sha’ar</i> 3, chap. 3, sec. 11 and <i>sikum</i>, sec. 16, that, when termination of pregnancy is permissible, it is preferable to induce abortion by chemical means.</i>
  That analysis of <i>Hatam Sofer's</i> position seems implausible. <i>Hatam Sofer's</i> choice of words in describing the effect of the poison as "burning" internal organs was probably expressly intended to distinguish the case under discussion from the type of act for which <i>Tashbaz</i> declined to assign culpability.51<i class="footnote">Cf., <i>Teshuvot Minḥat Yiẓḥak</i>, III, no. 104, sec. 1, who suggests that <i>Ḥatam Sofer</i> sought to exclude the possibility of some other supervening cause of death. <i>Minḥat Yiẓḥak’s</i> further suggestion that an act of <i>garmi</i> constitutes capital homicide is rather novel.</i> Pouring acid on a person's body is no less the proximate cause of resultant harm than is incision with a scalpel. The effect of acid is far different and far more immediate than poison that, to do harm, must first enter the bloodstream. <i>Hatam Sofer's</i> description of the "burning" effect of "petrol" clearly indicates that he presumed the effect to be comparable to that of acid.
  Even more problematic is the fact that, in addressing an entirely different incident in <i>Teshuvot Hatam Sofer</i>, <i>Hoshen Mishpat</i>, no. 184, <i>Hatam Sofer</i> remarks, "for indeed the Sages, <i>ipso facto</i>, exempted from exile a physician who causes the death of a patient." That categorical statement seems to be erroneous since the only explicit mention of a physician occurs in the Tosefta and the Tosefta declares the physician to be subject to exile.52<i class="footnote">See <i>Teshuvot Minḥat Yiẓḥak</i>, III, no. 104, sec. 4.<i>Ḥatam Sofer</i> may perhaps be understood as accepting the view postulated by <i>Arukh ha-Shulḥan</i> to the effect that the physician is subject to exile only in cases of negligence. See Rabbi Zilberstein, <i>Halakhah u-Refu’ah</i>, p. 293 and <i>Emek Halakhah,</i> p. 135. However, the terminology employed by <i>Ḥatam Sofer</i>, <i>viz.</i>, “for indeed the Sages exempted a physician who causes the death of a patient” would seem to indicate an explicit, rather than an inferential, exemption.R. Zevi Spitz, <i>Mishpetei ha-Torah</i>, I, no.12, note 3, seems to follow <i>Arukh ha-Shulḥan</i> in assuming that a physician is liable to exile only if he is negligent. He then proceeds to argue that a physician is held to a higher standard of care than is a layman, <i>viz.</i>, while treating his patient the physician must remain calm and collected. Hence a physician who panics or acts precipitously is liable to exile whereas the woman in question was exonerated by <i>Ḥatam Sofer</i>, claims Rabbi Spitz, because her state of excitement rendered her act non-negligent in nature. It should be noted that <i>Rav Pe’alim</i> pointedly states that the woman, who was surely aware of the presence of the poisonous substance in proximity to the whiskey, was negligent in not inspecting the flask. Moreover, Halakhah does not follow community standards in assessing negligence. If panic vitiates negligence, it does so for a physician no less so than for a layman. In addition, Rabbi Spitz overlooks the most significant point, <i>viz.</i>, in <i>Teshuvot Ḥatam Sofer</i>, <i>Ḥoshen Mishpat</i>, no. 184, <i>Ḥatam Sofer</i> declares that even a physician is exempt from exile.</i>
  IV. Financial Liability
  1. The Problem
  Unlike exile, which has lapsed in our era, a suit for actual physical harm continues to be actionable before a <i>bet din</i>. The rule with regard to financial liability in instances of malpractice is recorded in <i>Shulḥan Arukh</i>, <i>Yoreh De'ah</i> 336:1. <i>Shulḥan Arukh</i> declares:
  [A physician] should not engage in the practice of medicine unless he is proficient and [only if] there is no one greater than he in that [locale] for otherwise he is a shedder of blood. If [the physician] practices medicine without license of a <i>bet din</i> [and causes harm] he is liable for payment even if he is proficient. However, if he practices medicine with license of a <i>bet din</i> but errs and causes harm he is not liable according to the laws of man but he is liable according to the laws of Heaven.
  That ruling is based upon two separate statements of the Tosefta. The Tosefta, <i>Bava Kamma</i>, 6:6, states that a physician who practices with license of a <i>bet din</i> and causes harm "is exempt according to the laws of man but his judgment is turned over to Heaven" and posits the same rule with regard to a physician who performs an embryotomy. In antiquity, since a Caesarian section almost always resulted in the death of the mother, an embryotomy was the sole available means of preserving the life of a woman carrying a hydrocephalic fetus whose head was too large to pass through the birth canal. The rule with regard to medical malpractice is repeated in somewhat different language by the Tosefta, <i>Bava Kamma</i> 9:3. In the latter formulation the rule pertaining to the physician is coupled with identical rules applicable to a father, a teacher and an agent of the <i>bet din</i>. With regard to each of those individuals the rule presented by the Tosefta provides that when the individual acts with permission of the <i>bet din</i> he is not liable but includes the caveat that "if he wounds more than is proper he is liable."
  <i>Magen Avraham</i>, in his commentary on the earlier statement of the Tosefta, <i>Bava Kamma</i> 6:6, explains that the provision indicating that the judgment of the physician who causes harm "is turned over to Heaven" connotes nothing more than a declaration of the physician's liability for intentional infliction of harm.53<i class="footnote">R. Menachem Azariah Meir Castelenuovo, <i>Misgeret ha-Shulḥan</i>, <i>Yoreh De’ah</i> 336:1, declares that only if the physician “errs” is he exonerated but if he has been negligent in any manner he is culpable because “negligence is close to intent.”</i> <i>Magen Avraham</i> understands the Tosefta as declaring simply that, since God knows the physician's intention, He will hold the physician liable for willful battery; however, a physician who had no intention of causing harm is guiltless even in the eyes of Heaven. <i>Magen Avraham</i>'s analysis of the Tosefta is clearly contradicted by <i>Shulḥan Arukh</i>. Following Ramban's earlier-cited statement, <i>Shulḥan Arukh</i> declares unequivocally that the physician "is liable according to the laws of Heaven." Unlike the phrase "turned over to Heaven," which may connote that the matter is left to Heaven for adjudication in light of the attendant circumstances, the term "is liable according to the law of Heaven" is a declaration of absolute liability that is quite independent of intent. <i>Shulḥan Arukh</i> and Ramban apparently understood the phrase "but his judgment is turned over to Heaven" as an unqualified statement indicating liability, at least in the eyes of Heaven, for even inadvertent harm.
  The notion of exoneration by a human court but culpability in the eyes of Heaven is not immediately recognizable as an application of general principles of tort liability with regard to battery. As previously noted, the Gemara, <i>Bava Kamma</i> 26b, establishes the principle that, unlike harm caused by a person's animal or property, a person is always liable for damage caused by his body (<i>adam ha-mazik</i>). If so, it would follow that a physician should be held responsible for any harm that he causes, regardless of whether the damage is the result of negligence or of misadventure. Although Ramban, <i>Bava Mezi'a</i> 82b, and as cited by <i>Shitah Mekubbezet</i>, <i>Bava Mezi'a</i> 82b, s.v. <i>ve-atah be-shem</i>, disagrees, <i>Tosafot</i>, <i>Bava Mezi'a</i> 82b, do carve out an exception to that rule in instances of an entirely unavoidable harm (<i>ones gamur</i>) as, for example, in the case of a person who rolls over in his sleep and in the course of doing so breaks utensils that have been placed next to him.
  Presumably, the harm for which a physician is not liable by a <i>bet din</i> but for which he is held accountable in the eyes of Heaven is not of the nature of an entirely unavoidable misadventure for which, according to <i>Tosafot</i>, there is no liability. This assessment is based upon two factors: 1) according to <i>Tosafot</i>, if the harm was entirely unavoidable, there is no liability even in the eyes of Heaven; 2) <i>Shulḥan Arukh</i> rules that the same act, if it leads to the death of the patient, results in exile of the physician. As shown earlier, many latter-day authorities maintain that a physician is subject to exile only if he could have avoided the death of the patient by modifying the therapeutic procedure or if the procedure was performed in error. It is certainly the case that, according to <i>Tosafot</i>, a physician who properly performs a needed procedure that inadvertently results in harm as a result of sheer misadventure incurs no liability even in the eyes of Heaven. Thus, according to <i>Tosafot</i>, the liability "according to the laws of Heaven" of which the Tosefta and <i>Shulḥan Arukh</i> speak is limited to instances of harm resulting from some form of negligence. According to <i>Tosafot</i>, there are no exemptions from tort liability with regard to harm caused by a person; hence, according to <i>Tosafot</i>, financial liability should exist not only "according to the laws of Heaven" but should be imposed by a terrestrial <i>bet din</i> as well.
  2. Mipnei Tikkun ha-Olam
  a) Financial Immunity
  The concept of liability "at the hands of Heaven" must be understood on the basis of yet a third statement recorded in the Tosefta. The Tosefta, <i>Gittin</i> 3:13, declares: "A physician who practices medicine with the license of a <i>bet din</i> and causes harm: if inadvertently, he is not liable; if intentionally, he is liable—because of the welfare of society (<i>mipnei tikkun ha-olam</i>)." The phrase "<i>mipnei tikkun ha-olam</i>" must be understood as explaining the first portion of the Tosefta's compound statement, i.e., as explaining why the physician is exonerated in cases of inadvertent harm, rather than as explaining why he is liable for intentional harm as recorded in the second portion of that statement.54<i class="footnote">The contention of R. Moshe Ze’ev Zorger, <i>Va-Yashev Mosheh</i>, II, no. 8 sec. 2, that the concept “<i>mipnei tikkun ha-olam</i>” is adduced by the Tosefta, not to explain conferral of immunity upon the physician in instances of negligence, but to explain why there is actionable liability in cases of purposeful malpractice is untenable. <i>Va-Yashev Mosheh</i> suggests that the harm caused by the physician is a <i>hezek she-eino nikkar</i>, i.e., the harm is not physically perceivable as, for example, is the case with regard to damage inflicted in defiling a foodstuff. Harms of that nature are not actionable other than on the basis of rabbinic decree. That principle cannot be applied in instances of malpractice since, assuredly, physical harm resulting from malpractice is recognizable. Cf., R. Mordecai Elon, <i>Torah she-be-al Peh</i>, XVIII (Jerusalem, 5737), 75.</i> Thus the Tosefta, in effect, informs us that, the physician in principle should be fully liable even for harm caused inadvertently but that he is exonerated because of concern for the welfare of society, i.e., he is exonerated from tort liability by rabbinic decree designed to promote the welfare of society. The Sages were concerned that a person possessing medical skills might be fearful of malpractice liability and therefore renounce the practice of medicine. Accordingly, the Sages conferred qualified immunity upon the physician in order that he not be discouraged from practicing his profession. The Sages decreed only that the <i>bet din</i> not render judgment against the physician; they did not seek to abrogate the fundamental biblical law. Accordingly, the physician remains liable "at the hands of Heaven."
  Thus the basic problem is resolved: In terms of biblical law the physician is fully liable in accordance with the principle <i>adam mu'ad le-olam</i>; he is exonerated solely by reason of rabbinic enactment <i>mipnei tikkun haolam</i>. The remaining question is a determination of the parameters of the immunity conferred upon the physician by the Sages.
  b) Exclusions from Immunity
  (1) Improper Conduct
  As noted earlier, the Tosefta, <i>Bava Kamma</i> 9:3, declares that a physician who "wounds more than is proper" is liable for damages. The clear implication is that such damages are to be awarded by the <i>bet din</i>. In order to make that statement compatible with the statements of the Tosefta, <i>Gittin</i> 3:83 and <i>Bava Kamma</i> 6:6, exonerating the physician in cases of error, R. Simon ben Ẓemaḥ Duran, <i>Tashbaz</i>, III, no. 82, explains that the phrase "more than is proper" refers to intentional infliction of harm for which the physician is, of course, liable as is expressly stated by the Tosefta, <i>Gittin</i> 3:13. A similar act resulting in the death of the victim, adds <i>Tashbaz</i>, would not lead to exile because a person who willfully takes a life is not subject to exile even though he cannot be executed because of lack of prior admonition. R. Yechezkel Abramsky, <i>Hazon Yeḥezkel</i>, <i>Bava Kamma</i> 9:3, <i>Hiddushim</i>, depicts wounding "more than is proper" as an act that is "<i>kemezid</i>," i.e., as an act that is akin to an intentional act. In his <i>Bi'urim</i>, <i>loc. cit.</i>, Rabbi Abramsky adds the comment "for to that extent it is within his power to safeguard himself that he not wound."
  There appears to be a significant difference between the comments of <i>Tashbaz</i> and those of Rabbi Abramsky with regard to the proper understanding of the Tosefta. According to <i>Tashbaz</i>, a physician can never be held liable for negligence; his liability, as announced by the Tosefta, is limited to wanton infliction of harm that is tantamount to mayhem. Rabbi Abramsky, however, understands the Tosefta as extending the physician's liability beyond intentional infliction of harm.55<i class="footnote"><i>Misgeret ha-Shulḥan</i>, <i>Yoreh De’ah</i> 336:1, similarly declares that only if the physician “errs” is he exonerated but if he has been “negligent in any manner” he is liable because “negligence is close to intent.” See <i>supra</i>, note 53. <i>Misgeret ha-Shulḥan</i> seems to assert that <i>Tashbaẓ</i> would concede that, if the physician made a factual error because of failure to master the appropriate medical information, the physician is liable. It would follow that <i>Misgeret ha-Shulḥan</i> understands gross negligence to be included in <i>Tashbaẓ’</i> concept of intentional harm but has a broader view of the definition of gross negligence than is found in other sources. According to <i>Misgeret ha-Shulḥan</i> the concept includes not only carelessness in performing an act but also undertaking an act one is not qualified to perform.</i> According to Rabbi Abramsky, the phrase "wounding more than is proper" connotes extending an incision or incising the wrong area and is paradigmatic of an act that a prudent physician would not perform even inadvertently. As noted earlier, according to <i>Tosafot</i>, a person is never liable, even in the eyes of Heaven, for a harm that is entirely unforeseen and hence completely unavoidable.
  In what case, then, is there liability in the eyes of Heaven but no award of damages by a <i>bet din</i>? The Tosefta, according to Rabbi Abramsky's understanding, seems to refer to an act that in other legal systems would be categorized as gross negligence as opposed to ordinary negligence. Constant vigilance requires uninterrupted attention and concentration and a person, although endeavoring to be prudent, may momentarily relax his vigilance and commit a negligent act that might have been avoided, but only with a greater degree of attentiveness. Gross negligence results, not from a momentary lapse of concentration, but from a type of thoughtlessness or inattention that is readily avoidable. The Tosefta presumes that cutting tissue more extensively than is necessary is an example of gross negligence. In exempting the physician from liability lest he be deterred from practicing medicine, the Sages found it necessary to exempt the physician only from liability resulting from ordinary negligence. Since no person is always unflaggingly vigilant, sooner or later every physician will commit an act that must technically be categorized as a negligent act. Failure to protect a physician from liability that he must anticipate as a virtual certainty may prompt the physician to seek another means of earning a livelihood. Therefore, the Sages conferred immunity from financial liability upon him "<i>mipnei tikkun ha-olam</i>." Gross negligence, however, is not at all inevitable. A competent physician does not anticipate that in the course of his career he will be guilty of gross negligence. Hence the specter of liability in instances of gross negligence will not induce him to forsake medicine. Moreover, it may well be assumed that the Sages would have indeed preferred that a person who, knowing his own nature, recognizes that he will be prone to gross negligence desist from the practice of medicine.
  Curiously, R. Samuel ha-Levi Woszner, <i>Teshuvot Shevet ha-Levi</i>, <i>Yoreh De'ah</i>, no. 151, cites <i>Tashbaz'</i> interpretation of the phrase "more than is proper" verbatim but proceeds to rule in a manner consistent with Rabbi Abramsky's understanding of the Tosefta. The question that came before Rabbi Woszner involved a dentist who, in the course of drilling a diseased tooth, caused the drill to penetrate a healthy tooth. The issue confronting Rabbi Woszner was the dentist's liability for expenses incurred in the repair of the healthy tooth that the dentist had damaged. In his responsum <i>Shevet ha-Levi</i> cites the comments of <i>Tashbaz</i> and without further elaboration adds "but if … he drilled in a site in which he did not need [to drill], even if it was in the nature of an inadvertent act, with regard to the law of compensation, it is [regarded as] purposeful."56<i class="footnote"><i>Shevet ha-Levi’s</i> understanding of <i>Tashbaẓ</i> is probably also shared by R. Chaim Joseph David Azulai, <i>Tov Ayin</i>, no. 9, sec. 8. <i>Tov Ayin</i> quotes <i>Tashbaẓ</i> and comments that, according to all authorities, it is possible that, if the physician did not properly diagnose the illness and therefore prescribed a drug that caused the patient’s death, he is liable “because it was possible for him to reflect and to be accurate.” <i>Tov Ayin</i> apparently understands the “willfulness” to which <i>Tashbaẓ</i> refers as reflective of a rather broad concept. Conceptually, however, it is readily understandable that the Sages did not seek to excuse the physician from liability in instances of gross negligence.</i>
  As <i>Shevet ha-Levi</i> himself notes, there are situations in which cutting or drilling healthy tissue is not necessarily indicative of negligence. Nevertheless, in the overwhelming majority of cases, it is likely that incision of healthy tissue occurs because of negligence. Somewhat surprisingly, <i>Shevet ha-Levi</i> concludes that the physician or dentist cannot be exonerated unless the practitioner "had no control over his hand whatsoever." That conclusion is surprising since <i>Shevet ha-Levi</i> presumably accepts a distinction between what has earlier been categorized as "gross negligence" versus ordinary negligence.57<i class="footnote">Cf., <i>Va-Yashev Mosheh</i>, II, no. 8, sec. 3, who takes issue with <i>Shevet ha-Levi</i> and asserts that the phrase “more than is proper” connotes <i>hitrashlut</i>, i.e., laziness or inattention.</i> If so, there are undoubtedly some situations in which a practitioner has not lost control of his hand but, nevertheless, has not been grossly negligent.58<i class="footnote">In all cases in which the physician denies negligence there is a significant question with regard to which party bears the burden of proof. Does the patient have the burden in conformity with the general rule that burden of proof is on the plaintiff or, since the act has certainly been committed, is absence of negligence merely an affirmative defense with the burden of proof upon the putative tortfeasor? See <i>Olat Shmu’el</i>, no. 73; R. Shlomoh Heyman, <i>Ḥiddushei Rabbi Shlomoh</i>, II, nos. 14-15; <i>Ḥazon Ish</i>, <i>Bava Kamma</i> 7:18 and R. Mordecai Elon, <i>Torah she-be-al Peh</i>, XVIII, 71-73 and 75.</i>
  (2) Absence of Licensure
  In conferring financial immunity upon the physician the Sages limited this enactment to a physician who ministers to patients "with permission of the <i>bet din</i>." Since, in our day, the practice of <i>bet din</i> licensure of physicians has lapsed, the applicability of the rule providing for such immunity is subject to question. Citing <i>Maharil, Birkei Yosef, Shiyurei Berakhah, Oraḥ Hayyim</i> 328:1, declares that when the civil authorities regulate the practice of medicine and grant licenses to practitioners no further permission from a <i>bet din</i> is needed. <i>Arukh ha-Shulḥan</i>, <i>Yoreh De'ah</i> 336:2, also seems to regard government licensure as tantamount to authority conveyed by a <i>bet din</i>. In his earlier-cited responsum, <i>Shevet ha-Levi</i> is even more explicit in stating that, in our day, the <i>bet din</i>, in effect, constructively acquiesces to licensure by civil authorities.
  <i>Divrei Sha'ul</i>, <i>Yoreh De'ah</i> 336:1, cites <i>Bet Hillel</i> who defines a "proficient" physician (<i>mumḥeh</i>) as one who possesses a "document [issued] by the great physicians in the places where he studied the science [testifying] that he studied and is proficient."
  <i>Bet Hillel</i>, <i>Yoreh De'ah</i> 336:1, declares that the status of any physician accepted by the populace of the city is identical to that of a physician who has been licensed by a <i>bet din</i>.59<i class="footnote">See also <i>Ramat Raḥel</i>, no. 22.</i> <i>Bet Hillel's</i> ruling is predicated upon the consideration that, in licensing the physicians, the <i>bet din</i> is not performing a judicial function. Rather, the <i>bet din</i> is acting in an administrative capacity in order to preserve the health and welfare of the members of their society. Preservation of social welfare is an inherent right and duty of society. Thus, in this matter, as in other matters, the <i>bet din</i> acts in a representative capacity in exercising the regulatory power of the society it represents. Nevertheless, in a final comment, <i>Divrei Sha'ul</i> seems to bemoan the fact that licensure by the <i>bet din</i> has lapsed.
  <i>Shulḥan Arukh</i>, <i>Yoreh De'ah</i> 336:1, rules that, despite having received permission of the <i>bet din</i> to practice medicine, a physician should not treat a patient if a more qualified practitioner is available. That ruling apparently applies even to a physician who has been granted permission to practice medicine by a <i>bet din</i>. <i>Hazon Yeḥezkel</i>, <i>Bava Kamma</i> 9:3, explicitly declares that a physician who fails to defer to a more qualified practitioner "has no license from the <i>bet din</i>" and, in case of misadventure, is to be held liable as would be the case were he an unlicensed practitioner. In effect, the license granted a physician by the <i>bet din</i> is qualified in nature and is contingent upon the unavailability of a more proficient practitioner.
  <i>Shevet ha-Levi</i>, IV, <i>Yoreh De'ah</i>, no. 151, expresses doubt with regard to whether, in contemporary society, the obligation to defer to a more qualified practitioner remains in effect since, in our day, physicians who are both "more and less qualified practice this profession and all of them are licensed." Although R. Eliezer Waldenberg, <i>Ramat Raḥel</i>, no. 22, sec. 5, declares that it is not necessary to refer a patient to a more competent specialist when both the complaint and the treatment are well-known and routine, <i>Shevet ha-Levi</i> is certainly in error with regard to any non-routine matter requiring informed medical judgment and sophisticated decision-making. In such cases the patient must be referred to the more competent and more experienced specialist provided that time permits and that the specialist is willing and able to treat the patient. Similarly, when the sheer number of patients is so great that the more proficient specialist cannot adequately treat all who are in need of his services, a less competent physician is not only permitted to treat, but must treat, the patient as long as he is the best qualified of the physicians available to the patient.60<i class="footnote">See <i>ibid</i>., secs. 2 and 5.</i>
  (3) Mipnei Tikkun ha-Olam in Light of Ramban's Analysis of the Liability of an Artisan
  According to Ramban, who, as noted earlier, maintains that an individual is liable for even unforeseen and unavoidable harm caused by his person, it would seem that a rabbinic decree conferring immunity from tort liability was necessary to provide immunity not only in instances of ordinary negligence but in cases of absolutely unavoidable harm as well. However, such a conclusion may not be correct. Indeed, according to Ramban, <i>Bava Mezi'a</i> 82b and as cited by <i>Shita Mekubbezet</i>, <i>ibid</i>., s.v. <i>ve-atah be-shem</i>, the principle <i>adam mu'ad le-olam</i> should not apply to a physician acting with the consent of his patient and not compensated for his services. The Gemara, <i>Bava Kamma</i> 99b, declares that an uncompensated ritual slaughterer who, in the process of slaughtering the animal, inadvertently renders it non-kosher is not liable for damages; if, however, he receives a fee for his services he is liable. Ramban apparently maintains that an artisan is not liable for damages in tort since he acts with the permission of the owner of the property. In effect, Ramban concedes that although, in principle, a person who causes unavoidable harm is liable for his act, nevertheless, one who performs the act with the permission of the person harmed thereby is not liable in tort for unavoidable damage. Hence an artisan is liable as a bailee rather than as a tortfeasor. There is, however, a difference between liability as a tortfeasor and liability as a bailee. Were the artisan to be regarded as a tortfeasor, his liability would be absolute; whereas, as a gratuitous bailee, he is liable only if he is negligent. If the artisan receives compensation his status is that of a bailee for hire who is liable for all damages other than those incurred as a result of <i>ones</i>, i.e., damages totally beyond his control.
  The statement of the Tosefta indicating that a physician is immune from liability only by virtue of rabbinic decree is problematic when examined from the vantage point of Ramban's position. It would stand to reason that the status of the physician should be identical to that of the slaughterer, i.e., the physician who acts with the consent of the patient should not be liable as a tortfeasor just as the slaughterer is not regarded as a tortfeasor. Thus a physician who receives no fee should not be liable for damages. Even assuming that the physician is regarded in Halakhah as a bailee,61<i class="footnote">Cf., however, <i>infra</i>, section VI.</i> the physician who has faithfully discharged his duties cannot be held liable even in the eyes of Heaven unless he is compensated for his services. However, in assigning liability to the physician according to biblical law, the Tosefta makes no distinction between a physician who charges a fee and a physician who renders his services <i>pro bono</i>. The problem, then, is that, even if he is regarded as a bailee, a physician who is not compensated should, according to Ramban, have the status of a gratuitous bailee who, even absent a rabbinic decree, is not liable other than for damage that occurs as a result of negligence.62<i class="footnote">See Rabbi Zilberstein, <i>Halakhah u-Refu’ah</i>, II, 125.Rabbi Mordecai Elon, <i>Torah she-be-al Peh</i>, XVIII, 73f., develops a theory according to which Ramban would regard an artisan who is compensated for his services as exempt only from liability for damage to property but not from harm to another individual’s person. However, Rabbi Elon does not support that theory with citation of precedent or evidence from earlier sources.</i>
  R. Yechezkel Abramsky, <i>Hazon Yeḥezkel</i>, <i>Bava Kamma</i> 9:3, in drawing a distinction between a physician and a father or a teacher identical to the distinction cited earlier in the name of <i>Besamim Rosh</i>, asserts that Ramban's rationale in exempting an artisan from tort liability is applicable only if the damage is caused by the artisan in the course of fulfilling his function as an artisan. However, argues Rabbi Abramsky, a physician who harms rather than heals is not engaged in medical ministration and hence is not functioning in his capacity as an artisan.63<i class="footnote">See <i>supra</i>, note 19.</i> Hence, according to Ramban, absent a rabbinic decree, even an uncompensated physician would be liable for damages.
  However, even conceding that a physician who commits an error is not functioning in his capacity as a physician, that consideration does not seem to explain the physician's liability according to Ramban. There is no reason to assume that an artisan is not liable for damages simply because of the fact that he is an artisan as Rabbi Abramsky seems to imply. Rather, as explained earlier, his immunity from liability is based upon the consideration that he has been granted license to perform the act that caused harm. In effect, his act becomes a licensed tort for which there is no liability as <i>adam ha-mazik</i>. The artisan remains liable only for such loss as a bailee is liable by virtue of his duty of care as a bailee.
  The problem is resolved somewhat differently and, in this writer's opinion, more cogently by R. Joshua of Kutna, <i>Yeshu'ot Yisra'el</i>, <i>Hoshen Mishpat</i> 25:3. <i>Yeshu'ot Yisra'el</i> asserts that an artisan is immune from liability according to Ramban only because he was granted permission to ply his craft by the owner of the property. However, a physician who performs an unnecessary procedure does not act with the permission of the patient since the patient does not give permission for unnecessary procedures and hence the physician is exonerated only by virtue of rabbinic enactment "for the welfare of society." <i>Yeshu'ot Yisra'el</i> apparently maintains that this is also the case in situations in which the patient did not survive the surgery because of "weakness;" in effect, such a procedure is unwarranted and the patient did not give permission for an unwarranted procedure. That analysis is similar to the earlier-cited distinction between a father or teacher, who are not culpable because they were engaged in discharging a duty, and a physician who, if acting in error, fulfills no obligation and would not apply to a case in which the procedure was appropriate but a mishap occurred in the course of its execution. Accordingly, if the surgery was appropriate but the patient succumbs as a result of inadvertent error committed by the surgeon, Ramban, according to <i>Yeshu'ot Yisra'el</i>, would maintain that the surgeon is not to be held liable even absent the consideration of "the welfare of society."
  V. Divergent Positions
  1. Rabbenu Nissim
  Rabbenu Nissim (Ran), in his commentary on <i>Sanhedrin</i> 84b, espouses a view entirely at variance with that of Ramban and <i>Shulḥan Arukh</i> and apparently also in contradiction to the rule formulated by the Tosefta. Rabbenu Nissim maintains that a physician is in no way liable for inadvertent error.64<i class="footnote">R. Menachem ben Zeraḥ, <i>Ẓeidah la-Derekh</i>, <i>ma’amar</i> 4, <i>klal</i> 2, chapter 2, interprets the Tosefta in a manner that dispels any contradiction to the position of Rabbenu Nissim. <i>Ẓeidah la-Derekh</i> understands the comment of the Tosefta, <i>Bava Kamma</i> 6:6, declaring that the judgment of the physician is “delivered to Heaven” as restricted to a physician who, although licensed by the <i>bet din</i>, is, in reality, unqualified to practice medicine. Presumably, that individual was licensed by the <i>bet din</i> because the <i>bet din</i> erroneously believed him to be a qualified practitioner. The same position is espoused by R. Moses Mat of Premishla (Przemysl) in his <i>Matteh Mosheh</i>, part V, chapter 3. According to this view, it is only an unqualified physician who, although exonerated by the <i>bet din</i> because of his licensure, remains guilty in the eyes of Heaven whereas a qualified and properly licensed physician is absolved from liability even in the eyes of Heaven if he causes damage inadvertently. <i>Ramat Raḥel</i>, no. 23, sec. 3, rather implausibly suggests that this position can be read into the words of <i>Shulḥan Arukh</i> as well.</i> Rabbenu Nissim explains that "when a proficient physician errs in his medical ministrations he is not acting inadvertently (<i>shogeg</i>) but out of compulsion (<i>ones</i>) for he heals with [divine] authority… and he has only that which his eyes see as we say with regard to a judge who has erred, [i.e.], that his heart coerced him."65<i class="footnote"><i>Tosafot</i> cannot be understood as accepting the notion that “his heart coerced him” as constituting, not simply an <i>ones</i>, but an <i>ones gamur</i> for which there is no liability since, were that so, there would be no liability even “at the hands of Heaven” as posited by the Tosefta.</i>
  Despite the rejection of the view by both Ramban66<i class="footnote"><i>Divrei Sha’ul</i>, <i>Yoreh De’ah</i> 336:1, attempts to show that Ramban, as cited in Ran’s commentary on <i>Rif</i>, contradicts the position Ramban espouses in his <i>Torat ha-Adam</i> and is in agreement with Rabbenu Nissim’s view. However, as shown by <i>Ẓiẓ Eli’ezer</i>, IV, no. 13, sec. 3, <i>Divrei Sha’ul’s</i> analysis is not compelling.</i> and <i>Shulḥan Arukh</i>, Ran's position is cited with approbation by the nineteenth-century authority, R. Joseph Saul Nathanson, <i>Divrei Sha'ul</i>, <i>Yoreh De'ah</i> 336:1. <i>Divrei Sha'ul</i> notes, as did <i>Besamim Rosh</i> before him, that a person who inadvertently commits a transgression in the course of attempting to fulfill a <i>mizvah</i> is exempt from the sacrificial offering required for expiation of inadvertent sin. By the same token, asserts <i>Divrei Sha'ul</i>, it would be reasonable to assume that if a person commits a tort in the course of fulfilling a <i>mizvah</i> there should be no financial liability in the eyes of Heaven. More significantly, the Gemara, <i>Sanhedrin</i> 84b, establishes that therapeutic "wounding" is not proscribed by the biblical prohibition against battery. The exclusion of therapeutic "wounding" is predicated upon a principle of rabbinic exegesis knows as a <i>hekesh</i>, i.e., the juxtaposition of two different legal categories, which has the effect of transposing one or more halakhic provisions already established in one of those categories to the other category. In this instance the <i>hekesh</i> is based upon the juxtaposition of references to the smiting of an animal and the smiting of a man in Leviticus 24:17-18. The Gemara declares, "Just as a person who smites an animal for therapeutic purposes is not liable so also a person who smites a man for therapeutic purposes is not liable." The inference to be drawn from that statement, argues <i>Divrei Sha'ul</i>, is that a person who inadvertently causes harm in the course of attempting to perform a therapeutic procedure is totally exonerated even in the eyes of Heaven, as is the case with regard to a person who causes harm to an animal in the course of a failed attempt to cure the animal. <i>Divrei Sha'ul</i> apparently regards the Gemara's formulation of this rule to be at variance with the rule posited by the Tosefta holding the physician accountable at the hands of Heaven and, applying usual canons of halakhic decision-making, <i>Divrei Sha'ul</i> asserts that the rule formulated by the Gemara should be given preference over that recorded in the Tosefta.67<i class="footnote">See <i>Ẓiẓ Eli’ezer</i>, IV, no. 13, sec. 3.</i>
  As noted earlier, <i>Arukh ha-Shulḥan</i>, <i>Yoreh De'ah</i> 336:2, rules that a physician is culpable at the hands of Heaven only if he has been remiss in some manner in his treatment of the patient or if he has not been sufficiently diligent in determining the proper treatment. Although his view appears to reflect the position of Rabbenu Nissim, <i>Arukh ha-Shulḥan</i> surprisingly asserts that his view reflects his understanding of the Tosefta as well as of the rulings of Ramban and <i>Shulḥan Arukh</i>.
  Despite the fact that Rabbenu Nissim is widely regarded as ruling that the physician is immune to suit for usual forms of medical malpractice, thoughtful application of his position to many common instances of malpractice results in an entirely different conclusion. Rabbenu Nissim's position exonerating a physician from malpractice liability is limited to genuine, albeit avoidable, errors of judgment as reflected in his appeal to the concept of "his heart compelled him." A surgeon who removes the wrong organ because he failed to consult the patient's medical chart or who fails to read the label on a vial of medicine and administers a toxic drug has not committed an error of judgment compelled by his intellect.68<i class="footnote">See R. Eliezer Waldenberg, <i>Ramat Raḥel</i>, no. 23, sec. 2.</i> The same is true in the situation of a physician who simply fails to examine a patient properly because of the pressure of time or because of sheer laziness.69<i class="footnote">See <i>Tov Ayin</i>, no. 9, sec. 8, cited <i>supra</i>, note 56.</i> Those situations represent examples of negligence for which immunity has not been conferred upon the physician. Moreover, according to Rabbenu Nissim, the physician is liable for such forms of negligent malpractice not only "at the hands of Heaven" but will be held liable by the <i>bet din</i> as well. Relief from liability by operation of rabbinic decree is recorded in the Tosefta solely in conjunction with the Tosefta's formulation of a biblical law doctrine of strict liability applicable to a physician. If the Tosefta's position is rejected in favor of the authoritative ruling of the Babylonian Talmud, there is no other evidence pointing to the existence of a rabbinic decree circumscribing the power of a <i>bet din</i> to order compensation.70<i class="footnote">Of course, as shown earlier, according to <i>Ḥazon Yeḥezkel’s</i> understanding of the Tosefta, the physician is liable for gross negligence even according to the normative view of the Tosefta and <i>Shulḥan Arukh</i>.</i>
  2. Tashbaz
  (a) Gerama vs. Garmi
  R. Zalman Nechemiah Goldberg, <i>Teḥumin</i>, XIX (5759), 321, asserts that a physician who causes harm by offering poor advice, recommending a harmful procedure or prescribing a medication that is deleterious is liable not simply because of <i>gerama</i> but because his act is in the nature of <i>garmi</i>. In his initial discussion Rabbi Goldberg expresses some doubt with regard to whether, even according to the ruling of the Tosefta, a physician who incorrectly prescribes a medication, but does not personally administer the drug, bears liability even in the eyes of Heaven. He similarly expresses some doubt with regard to liability at the hands of Heaven with regard to a physician who orders an injection or counsels surgery but does not himself perform the procedure. Nevertheless, he concludes that, at the minimum, such acts constitute <i>gerama</i>, or an indirect cause, for which there is liability in the eyes of Heaven. Moreover, Rabbi Goldberg further declares the matter to be analogous to <i>mesirah</i>, i.e., "informing" an evildoer of the location of money or property and thereby enabling the malfeasor to seize the property. Such an act is categorized as <i>garmi</i> for which there is complete liability, i.e, for which there is liability not only in the eyes of Heaven but which is actionable before a <i>bet din</i> as well.
  Jewish law recognizes two distinct forms of indirect causation: <i>gerama</i> and <i>garmi</i>. Damage caused by <i>gerama</i> results in liability only at the hands of Heaven; damage as a result of <i>garmi</i> is actionable before a <i>bet din</i> despite the absence of proximate cause. The precise nature of the distinction between the categories of <i>gerama</i> and <i>garmi</i> is the subject of considerable controversy among early-day authorities.71<i class="footnote">For a survey of the various opinions regarding this question see <i>Encyclopedia Talmudit</i>, VI, 461ff.</i> In defining the concept of <i>gerama</i> in contradistinction to that of <i>garmi</i>, <i>Tosafot</i>, <i>Bava Batra</i> 26b, assert that, when the harm is a necessary and inescapable result of the tortfeasor's act, the tortfeasor is liable even if the resultant damage is caused by the tortfeasor only indirectly.72<i class="footnote">The Tosefta, <i>Bava Kamma</i> 6:5, declares that a person who pours poison down the throat of an animal is not held liable for damages by a <i>bet din</i>. <i>Ḥasdei David</i>, in his commentary <i>ad locum</i>, explains that the Tosefta, contrary to the position of <i>Tosafot</i>, maintains that there is no liability for indirect damages even in cases of <i>garmi</i>. The inference is that direct injection of a deleterious substance by a physician is in the category of <i>garmi</i> and, according to <i>Tosafot</i>, the physician would be culpable. See <i>Minḥat Yiẓḥak</i>, III, no. 104, sec. 1.</i>
  If the physician's referral to the surgeon or his order to the nurse is deemed to be in the nature of <i>garmi</i>, the physician would be fully liable but for the immunity conferred upon him by rabbinic edict. In comparing dispensing medical advice with "informing" and categorizing such conduct as <i>garmi</i>, Rabbi Goldberg's theory is based upon the presumption that the specialist's advice will certainly be heeded, i.e., the nurse will obey his orders and the patient will fill the prescription, and is further predicated upon the premise that even an indirect act that will inevitably result in pecuniary harm constitutes actionable <i>garmi</i>. There are, however, grounds to question whether erroneously prescribing a medication, ordering an injection or counseling surgery generate liability even at the hands of Heaven. In point of fact, other than perhaps in the case of an order issued to a nurse, it is far from certain that there is a necessary causal relationship between the practitioner's advice and the resultant harm since patients frequently seek second opinions or ignore medical advice.
  Quite apart from the foregoing consideration, it would appear that the harm caused by a physician is not in the nature of <i>garmi</i>. The relevant paradigm is the case of a person who consults a money-changer (<i>shulḥani</i>) with regard to the value of a coin proffered to him. A proficient <i>shulḥani</i>, i.e., a master money-changer requiring no further training, who renders an erroneous opinion regarding the value of a coin is not liable for any loss incurred in acceptance of the coin since he has not been negligent in any way. There is a dispute between early-day authorities with regard to whether the money-changer incurs no liability even if the recipient of the coin expressly indicates that he is relying upon the money-changer's opinion.73<i class="footnote">See <i>Hagahot Asheri</i>, <i>Bava Kamma</i> 9:18.</i> The consensus of opinion is that there is no obligation even under such circumstances.74<i class="footnote">See <i>Shiltei Gibborim</i>, <i>Bava Kamma</i> 99b. However, a <i>shulḥani</i> who receives a fee for his services is liable as a bailee for hire. See <i>Shulḥan Arukh</i>, <i>Ḥoshen Mishpat</i> 306:6.</i> The same rule would apply to a proficient physician who makes a <i>bona fide</i> error in prescribing a medication or the like.
  Most significantly, as explained by Rosh, <i>Bava Kamma</i> 9:13, liability by virtue of <i>garmi</i> is limited to situations in which the harm is immediately consequent upon performance of the culpable act. Since the money-changer's liability in terms of <i>garmi</i> is predicated upon the fact that the resultant damage is not only certain to occur but is also immediate, i.e., the proferred coin is immediately accepted in reliance upon the money-changer and the person who presents the coin has no further liability. Accordingly, no such liability would result from the writing of a prescription which must be taken to a pharmacy or from an order directed to a nurse in any situation in which the order is not carried out immediately.
  Of course the issue is entirely theoretical since, even if the internist's referral to the surgeon or the physician's order to the nurse constitutes a form of <i>garmi</i>, any resulting liability is cancelled by virtue of the rabbinic decree promulgated <i>mipnei tikkun ha-olam</i>.
  The liability of a nurse who administers an injection at the direction of a physician or of a surgeon who heeds the internist's diagnosis and performs an operation is a separate issue. It would certainly seem that according to the ruling of <i>Shulḥan Arukh</i>, the nurse and the surgeon are culpable in the eyes of Heaven. Thus it would follow that both the physician and the nurse, as well as the internist and the surgeon, are equally liable, at least in the eyes of Heaven. Rabbi Goldberg, however, seems to imply that liability cannot be shared jointly by the physician and nurse or by both the internist and the surgeon.
  It should also be noted that, according to Rabbenu Nissim, it would appear that the nurse and the surgeon are not culpable since they are "coerced" by the advice they have received and upon which they have every right to rely. Thus, according to Rabbenu Nissim, under such circumstances the nurse or surgeon cannot be deemed liable. Rabbi Goldberg makes a different and more novel point in asserting that the actions of individuals such as the nurse or the surgeon who act upon the advice of a qualified professional are in the nature of an <i>ones gamur</i> for which, according to the opinion of <i>Tosafot</i>, there is no liability.
  b) Tashbaz' Ruling
  In his earlier-cited responsum, <i>Tashbaz</i> draws a distinction between physical intervention in the form of a surgical procedure or the like and the administration of medication, just as he drew such a distinction with regard to exile in instances of inadvertent manslaughter. In support of that distinction <i>Tashbaz</i> asserts that the term "<i>uman</i>" employed by the Tosefta denotes an artisan who employs a sharp instrument and hence all references in the Tosefta are to a surgeon (<i>rofeh uman</i>) who heals by means of "work of the hand" but that a physician who cures the sick by means of potions, laxatives or medicaments is not referred to by the appellation "<i>rofeh uman</i>." Thus the doctrine of liability formulated by the Tosefta is limited solely to a surgeon whose intervention results in misadventure but does not apply to a physician since the latter's ministration "does not enter into the realm of wounding that he be liable for damages. [Therefore] whether [he acts] unintentionally or intentionally and causes death or adds suffering to the sickness, [since] he intended to cure and did not intend to do harm, he is not liable even according to the laws of Heaven."
  As explained earlier, the distinction drawn by <i>Tashbaz</i> reflects the halakhic notion of proximate cause.75<i class="footnote">See <i>supra</i> note 49 and accompanying text. Cf., R. Yitzchak Zilberstein’s analysis of <i>Tashbaẓ</i> discussed in that note.</i> A tortfeasor is liable only for damages that are a direct result of his act; he is not liable for damages he causes only indirectly. Damage to a limb or organ arising from "wounding," i.e., surgical incision or excision is clearly direct; harm caused by drugs or medications, even if administered directly by inoculation or the like, is regarded by <i>Tashbaz</i> as indirect.76<i class="footnote">See <i>supra</i>, note 49 and accompanying text. The incredulity voiced by <i>Ẓiẓ Eli’ezer</i>, IV, no. 13, sec. 3, in the form of a rhetorical question, “If someone administers a poisonous drink to his fellow and causes him to die thereby, shall he not be culpable for murder?” is understandable but misplaced when analyzed in terms of the distinction between proximate cause and <i>gerama</i> as formulated by many halakhic authorities. Cf., however, <i>supra</i> note 72.</i> Even those authorities who regard the harm caused by injection or ingestion of a drug to be the direct effect of that cause must concede that merely prescribing, or even handing a patient, a drug can be no more than a <i>gerama</i>. That conclusion is evident from the statement of the Gemara, <i>Bava Kamma</i> 47b, describing the act of placing a poison before an animal as a mere <i>gerama</i> with the result that, although a person causing such harm is liable "according to the laws of Heaven," a <i>bet din</i> cannot require compensation.
  <i>Tashbaz'</i> distinction between a surgeon and a practitioner of internal medicine presents a number of difficulties. As noted earlier, the salient distinction between mere <i>gerama</i> and <i>garmi</i> is that in instances of <i>garmi</i>, despite the absence of proximate cause as defined by Halakhah, the resultant harm is certain to occur. Erroneous injection of a toxic substance, for example, should, in light of the previously cited comments of <i>Tosafot</i>, <i>Bava Batra</i> 26b, be treated no differently than an error committed in the course of surgery. <i>Tashbaz</i> may well have accepted the position of the early-day authorities who, because of a lack of proximate cause, regard even such harm to be in the category of unactionable <i>gerama</i>.77<i class="footnote">See <i>Encyclopedia Talmudit</i>, VI, 464-467.</i>
  A close reading of <i>Tasbaz'</i> comment "since [the physician] intended to heal and did not intend to harm," reveals that <i>Tashbaz</i> regards the physician to be free of liability for harm resulting from administration of medication only if the improper medication was administered in error. The implication of that statement is that if the physician intentionally administers a harmful medication he is indeed liable. However, as Rabbi Mordecai Elon, <i>Torah she-be-al Peh</i>, XVIII, 74, points out in questioning <i>Tashbaz</i> ́ comment, a tortfeasor is not liable even for intentional damage arising only indirectly from his act.
  An even greater difficulty lies in the fact that a tortfeasor is liable in the eyes of Heaven even when the damages result from an act in the nature of <i>gerama</i>. Indeed, as recorded by Rema, <i>Hoshen Mishpat</i> 346:3, although the <i>bet din</i> is not empowered to order compensation directly and hence may not seize the tortfeasor's property, the <i>bet din</i> may nevertheless apply sanctions against the tortfeasor in the form of excommunication in order to prompt him to discharge the obligation that exists in the eyes of Heaven. <i>Tashbaz</i> does not at all explain why a physician should not be held accountable in the eyes of Heaven for malpractice in the form of <i>gerama</i>. This difficulty is noted by <i>Minḥat Yizḥak</i>, III, no. 104, sec. 1.
  It seems to this writer that <i>Tashbaz</i>' position should be understood in light of the view expressed by Me'iri, <i>Bava Kamma</i> 56a. Me'iri maintains that liability "at the hands of Heaven" in instances of <i>gerama</i> is limited to situations in which the tortfeasor "intends to do harm." Thus, the physician, even if he has erred in his ministrations, certainly did not intend to cause harm and hence he incurs no liability even "at the hands of Heaven." Indeed, <i>Tashbaz</i> qualifies the scope of his ruling by describing the salutary intention of the physician to heal and adds the phrase "and he did not intend to do harm." If <i>Tashbaz</i> maintains, as did Me'iri before him, that liability for <i>gerama</i> "in the eyes of Heaven" is limited to situations in which there is intention to do harm, the incorporation of that phrase seems to explain why the physician's act, unlike other forms of <i>gerama</i>, does not result in culpability in the eyes of Heaven.
  R. Chaim Joseph David Azulai, <i>Birkei Yosef</i>, <i>Yoreh De'ah</i> 336:7, notes that, in affirming the physician's liability in the eyes of Heaven, both Ramban and <i>Shulḥan Arukh</i> are apparently in disagreement with <i>Tashbaz</i>, since they do not in any way qualify or limit their assertion that the physician is liable in the eyes of Heaven. In light of the foregoing discussion it is not difficult to explain why those authorities maintain that the physician is liable in the eyes of Heaven even in instances of <i>gerama</i>. Moreover, there are numerous authorities who maintain that harm caused by administration of medication must be regarded as direct rather than as indirect.78<i class="footnote">See <i>supra</i>, note 50 and accompanying text as well as <i>Ẓiẓ Eli’ezer</i>, IV, no. 13, sec. 2. See also <i>Imrei Binah</i>, II, <i>Dinei Dayyanim</i>, no. 30, who remarks that he sees no difference between “wounding by means of iron or by means of a compress or some other remedy.”</i> Nevertheless, <i>Shevet ha-Levi</i>, IV, <i>Yoreh De'ah</i>, no. 151, comments that "it is difficult" to dispute <i>Tashbaz'</i> opinion or to rule contrary to the position of <i>Tashbaz</i>.79<i class="footnote"><i>Shevet ha-Levi</i> cites <i>Tashbaẓ’</i> employment of the phrase “for [the physician] has only that which his eyes see” in support of a caveat to the effect that <i>Tashbaẓ</i> ruling does not apply in situations in which the error would have been avoided had an x-ray examination been undertaken. However, that conclusion is simply unwarranted. Cf., <i>Va-Yashev Mosheh</i>, II, no. 8, sec. 3. <i>Shevet ha-Levi’s</i> comment might be cogent in elucidating Rabbenu Nissim’s position but for the fact that a <i>bona fide</i> error in deciding that x-rays are superfluous (and, if so, to be avoided for sound medical reasons) is also within the category of “his heart compelled him.”</i>
  VI. Liability as a Bailee
  R. Meir Auerbach, <i>Imrei Binah</i>, II, <i>Dinei Dayyanim</i>, no. 30, accepts <i>Tashbaz'</i> distinction insofar as tort liability is concerned in principle but does not agree that either the Tosefta or Ramban intended to draw such a distinction. Nevertheless, argues <i>Imrei Binah</i>, although in the absence of proximate cause there may be no tort liability, the physician is also a bailee and is liable for violation of his duties as a bailee. The rule with regard to an artisan is that an artisan who receives no compensation is a gratuitous bailee with regard to any property entrusted to him in the practice of his craft. However, as stated in the Mishnah, <i>Bava Mezi'a</i> 80b, if the artisan is compensated for his services, his status is that of a bailee for hire. A gratuitous bailee is liable only if he is negligent; a bailee for hire is liable for any loss except that which occurs through <i>ones</i>. <i>Imrei Binah</i> notes that physicians are generally compensated for their services. When compensated, writes <i>Imrei Binah</i>, a physician has the status of a bailee for hire.80<i class="footnote">It may be argued that even a physician who receives no remuneration for his services has the status of a bailee for hire. The Gemara (<i>Bava Kamma</i> 56b; <i>Bava Meẓi’a</i> 29a and 82a; <i>Shevu’ot</i> 44a; and <i>Nedarim</i> 33b) records a controversy between Rabbah and Rav Yosef with regard to whether a person engaged in restoring lost property has the status of a gratuitous bailee or of a bailee for hire. Rav Yosef maintains that since the finder is engaged in fulfilling a <i>miẓvah</i> he is exempt from discharging other obligations and hence enjoys the tangible benefit of not being required to bestow even minimal alms upon a mendicant. That dispute is mirrored in a controversy recorded by <i>Shulḥan Arukh</i> and Rema, <i>Ḥoshen Mishpat</i> 267:16.The Gemara, <i>Sanhedrin</i> 73a, declares that the obligation to restore lost property entails, a <i>fortiori</i>, an obligation to prevent loss of life. Rambam, <i>Commentary on the Mishnah</i>, <i>Nedarim</i> 4:4, states that a physician is obligated to treat a patient by virtue of that commandment. Hence, it should follow that a physician, assuming he is a bailee, is similarly exempt from the <i>miẓvah</i> of charity while treating patients. Therefore, according to the earlier-cited opinion with regard to the status of the finder of lost property, he should have the status of a bailee for hire even if he is not compensated for his services.Similarly, although <i>Imrei Binah</i> presents a detailed analysis of the category of bailment ascribed to a <i>dayyan</i>, he fails to analyze the possible status of every physician as a bailee for hire. One argument made by <i>Imrei Binah</i> with regard to a <i>dayyan</i>, <i>viz.</i>, that while committing an error he performs no <i>miẓvah</i> and hence is not exempt from the <i>miẓvah</i> of charity may be applicable to the physician as well. Cf., <i>supra</i>, note 18.</i> Thus, even if he has not been negligent, the physician should be liable for any harm that he causes, even if he causes such harm only indirectly, unless the harm is in the nature of <i>ones</i>. In exempting the physician from financial liability, the Sages, <i>ipso facto</i>, exempted him not only from tort liability but also from liability as a bailee. However, since that exemption is only from liability imposed by a <i>bet din</i>, the physician remains liable in the eyes of Heaven even for damage caused indirectly, albeit not as a tortfeasor, but as a bailee.
  <i>Imrei Binah's</i> thesis presents a number of difficulties:
  1) A bailee for hire, as <i>Imrei Binah</i> notes in a different context, becomes liable only upon receiving payment in advance or upon taking custody of the bailment by means of formal <i>kinyan</i>, e.g., by means of lifting or moving the object. A physician enters into no such <i>kinyan</i> and is generally not paid for his services before they are rendered. That objection may be resolved on the basis of a comment of Ramban cited by <i>Nimukei Yosef</i>, <i>Bava Mezi'a</i> 94b. Ramban states that a <i>kinyan</i> is necessary solely in order to demonstrate that the bailed object has been accepted as a bailment by the bailee and that he has assumed the obligations of a bailee. An artisan who acquires the status of a bailee does so by commencing his labor; for the artisan, commencement of his work is, in effect, commencement of the bailment. Accordingly, the physician, if he has the status of a bailee, acquires that status immediately upon commencement of medical ministration.
  2) As recorded in Exodus 22:14, a bailee is not liable in situations in which the bailor accompanies his property at the inception of the bailment and is himself in a position to participate in the safeguarding of his property. Granted that the patient has the status of a bailed item, the patient, who is also the bailor, is physically present during treatment81<i class="footnote">As stated by Ra’avad in a gloss to a ruling of Rambam, <i>Hilkhot Sekhirut</i> 2:1, presence of the owner does not serve to exonerate the bailee from liability in tort. This is true despite the ostensibly contradictory ruling of Ra’avad, <i>Hilkhot Ishut</i> 21:9, declaring that, by virtue of the husband’s presence in the home, a housewife is exempt from liability for household utensils that she may break. The housewife, however, is exempt from tort liability because she has the status of an artisan and an artisan who labors with permission of his or her client is not liable in tort for unintentional damage. See <i>Teshuvot Rabbi Eli’ezer</i>, no. 2. Cf., <i>Maḥaneh Efrayim</i>, <i>Hilkhot Shomrim</i>, no. 39.</i> and indeed is present at the time that the harm occurs.82<i class="footnote">See R. Mordecai Elon, <i>Torah she-be-al Peh</i>, XVIII, 76.</i> That factor would serve to exonerate the physician from liability as a bailee.
  3) The most complex problem is <i>Imrei Binah's</i> assumption that a human being can be bailed and that the obligation of a bailee can extend to a physician who assumes a duty of care <i>vis-à-vis</i> a patient. The general rule, recorded by Rambam, <i>Hilkhot Sekhirut</i> 2:1, is that a bailee incurs no disability with regard to real property, slaves or promissory notes. As will be shown, there are authorities who regard not only slaves but all human beings as not subject to bailment.
  Although Ra'avad disagrees, Rambam, <i>Hilkhot Sekhirut</i> 2:3, maintains that a bailee, although exempt from liability in the event of damage resulting from other causes, nevertheless remains liable for negligence even with regard to real property, slaves and legal instruments. However, even according to Rambam, physicians might be held accountable according to the "laws of Heaven" only if the harm suffered by the patient results from the physician's negligence, whereas Ramban and <i>Shulḥan Arukh</i> apparently assign such liability even in the absence of negligence.
  Moreover, the status of a human being as a bailed object for purposes of liability under the laws of bailment is far from clear.83<i class="footnote">See the discussion of this issue by R. Moshe Bleich, “The Halakha Corner: A School’s Liability for a Student’s Injury,” <i>Ten Da’at</i>, vol. X, no. 1 (Spring, 1997), p. 79, note 1.</i> Among early-day authorities, Rashi, <i>Kiddushin</i> 7a, s.v. <i>sheyesh lahem aḥarayut</i>, states clearly that all human beings have the halakhic status of real property in this regard.84<i class="footnote">For additional halakhic ramifications of such classification see <i>Encyclopedia Talmudit</i>, I, 2nd ed. (Jerusalem, 5733), 160-161.</i> However, <i>Tosafot</i>, <i>ad. locum</i>, s.v. <i>im ken</i>, followed by Ramban, Rashba and Ritva, <i>ad locum</i>, take issue with Rashi85<i class="footnote">See also Rashba, <i>Shevu’ot</i> 42a and <i>Maggid Mishnah</i>, <i>Hilkhot To’en ve-Nit’an</i> 5:12.</i> and declare that the principle is limited to Canaanite slaves.86<i class="footnote">For further analysis of the positions of Rashi and <i>Tosafot</i> see <i>Sha’ar ha-Melekh</i>, <i>Hilkhot To’en ve-Nit’an</i> 5:2. <i>Shakh</i>, <i>Ḥoshen Mishpat</i> 95:18, marshals sources in support of each of these contradictory views but concludes by affirming Rashi’s position that all human beings have the status of real property. An opposing conclusion is reached by R. Jonathan Eibeschutz, <i>Urim ve-Tumim, Tumim</i> 95:7.It should be noted that R. Akiva Eger, in his glosses to the Palestinian Talmud at the beginning of the seventh chapter of <i>Bava Kamma</i>, accepts the view of <i>Tosafot</i> to the effect that a Canaanite slave has the status of real property. Cf., however, <i>Or Sameaḥ</i>, <i>Hilkhot To’en ve-Nit’an</i> 5:2. It should also be noted that R. Akiva Eger in his glosses to <i>Yoreh De’ah</i> 6:2 and in his responsa, <i>Teshuvot R. Akiva Eger</i>, no. 51, upon analyzing the ruling of <i>Shulḥan Arukh</i>, arrives at a conclusion that is at variance with that of <i>Shakh</i> and maintains that only a Canaanite slave has the status of real property. Cf., <i>Or Sameaḥ</i>, <i>Hilkhot Gerushin</i> 1:6, who resolves the difficulties identified by R. Akiva Eger in a different manner. For an analysis of the parameters and limitation of the parallels (<i>hekesh</i>) between an <i>eved</i> and real property see R. Chaim Soloveichik, <i>Ḥiddushei ha-Graḥ al ha-Rambam, Hilkhot Geneivah ve-Aveidah</i> 9:1 and Rabbi Leib Mallen, <i>Ḥiddushei Rav Aryeh Leib</i>, I, no. 62.Regarding the status of an <i>eved ivri</i> and, by extension, that of a worker as realty with regard to the appropriate <i>kinyanim</i> see <i>Shakh</i>, <i>Ḥoshen Mishpat</i> 333:25 and 95:18; <i>Netivot ha-Mishpat</i> 333:1; <i>Ḥazon Ish</i>, <i>Bava Kamma</i> 23:7; R. Isaac Elchanan Spektor, <i>Naḥal Yiẓḥak</i>, no. 39, sec. 13 and no. 74, sec. 3; and <i>Maḥaneh Efrayim, Hilkhot Sekhirat Po’alim</i>, no. 1.</i>
  Among latter-day authorities, R. Jonathan Eibeschutz, <i>Urim ve-Tumim</i>, <i>Tumim</i> 95:7, maintains that only slaves have the status of real property. Adopting a somewhat different position, <i>Shulḥan Arukh</i>, <i>Hoshen Mishpat</i>, 227:19, implies that a Hebrew slave and an employee contracted for his services also have that status but that a freeman does not. However, <i>Shakh</i>, <i>Hoshen Mishpat</i> 95:17, rules in accordance with the view of Rashi in extending that status to all human beings. Hence, whether or not a patient has the status of real property and, consequently, whether a patient can or cannot be the subject of a bailee's liability is a matter of considerable controversy among both early-day and latter-day authorities.87<i class="footnote">See, R. Ya’akov Y. Blau, <i>Pitḥei Ḥoshen</i>, II, <i>Hilkhot Pikadon u-She’elah</i> 1:21, note 49. See also R. Moshe Feinstein, <i>Iggerot Mosheh</i>, <i>Even ha-Ezer</i>, I, no. 106, s.v. <i>ve-im ken nimẓa</i>, who suggests that a mother may be liable for harm caused to her children as a result of inadequate nutrition “if we say (<i>im naima</i>) that a free man is not [compared] to slaves and real property and that the laws of bailment pertain to him.”</i>
  VI. A Concluding Comment
  The physician, far more so than the practitioner of any other profession, must maintain constant and uninterrupted vigilance. Any momentary lapse on his part can lead to disastrous consequences. The physician's responsibilities are awesome and indeed they strain the limits of human capability. He or she cannot possibly maintain ongoing vigilance without a firm and dedicated resolve to do so and even then the physician must rely upon divine assistance. As expressed in the final section of the "Daily Prayer of a Physician," attributed (probably spuriously) to Rambam, the physician must constantly pray:
  Illumine my mind that it recognize what presents itself and that it may comprehend what is absent or hidden. Let it not fail to see what is visible but do not permit it to arrogate to itself the power to see what cannot be seen, for delicate and indefinite are the bounds of the great art of caring for the lives and health of Your creatures. Let me never be absent-minded. May no strange thoughts divert my attention at the bedside of the sick or disturb my mind in its silent labors, for great and sacred are the thoughtful deliberations required to preserve the lives and health of Your creatures.88<i class="footnote">Translated by Harry Friedenwald, <i>The Jews and Medicine</i> (Baltimore, 1944), I, 29.</i>
  
  Chapter 6 Medical Questions
  
  <i>When afflictions are inflicted upon a person they are sworn…not to depart other than on a specific day, at a specific hour, through the intermediacy of a specific person and by means of a specific medicament.</i>
  AVODAH ZARAH 55a
  Choosing Between Therapies: A Painful Dilemma
  The widely-acclaimed Jerusalem Institute of Technology, of which the late R. Shlomoh Zalman Auerbach was a guiding spirit, has, for the past number of years, been compiling and publishing an annual volume of essays devoted to topical halakhic issues pertaining to technology and medicine. Those volumes, bearing the title <i>Ateret Shlomoh</i>, are published in memory of Rabbi Auerbach.
  Volume IV (Adar 5759) of that series features an article by the highly regarded head of the Institute, R. Levi Yitzchak Halperin, devoted to a heart-wrenching medical problem. The hypothetical case involves a child afflicted with a malignant brain tumor that would almost certainly be fatal if left untreated. The tumor in question can be treated in one of two ways: 1) Chemotherapy can be used in treating this form of cancer and will be successful in approximately 60% of all cases. 2) The tumor can also be treated by whole-brain radiation which is effective in virtually all cases. The problem with the latter form of treatment is that, in a pediatric patient whose brain is not fully developed, there is a strong likelihood that the patient will suffer loss of neural function so severe that it will render the child a <i>shoteh</i>; i.e., the child will suffer extreme mental retardation. For purposes of analyzing the problem it must be assumed that radiation cannot be held in abeyance until it is determined whether or not chemotherapy will be successful.
  Assuming that there is no cogent reason to fear that radiation poses a risk of foreshortening life, there is little question that, were radiation the sole therapy available, it would be obligatory to treat the patient in that manner even were resultant severe mental retardation to be a certainty. The issue in the case described is whether a statistically less efficacious therapy may be employed in the hope of not compromising quality of life.
  This writer has consulted a number of medical specialists in the fields of pediatrics, oncology, neurology and radiology in an attempt to identify the type of tumor for which these possible modes of therapy are accompanied by the statistical risks described. That effort did not meet with success. It must be presumed that the discussion refers to a hypothetical situation. The fact that, as presented, the dilemma is not actual in no way diminishes the importance of the issue involved. The halakhic principles to be applied are applicable in many other situations as well.
  Rabbi Halperin frames the question in terms of the consideration that it will become impossible for the mentally incompetent patient to perform <i>mizvot</i>. Framed in those terms, the issue is whether or not a person may accept a significant degree of risk, not in order to effect a cure, but to be assured of the ability to perform <i>mizvot</i>. The broader question is whether a person may accept a significant degree of risk in order to improve, or prevent a deterioration of, quality of life. The dilemma as posed by Rabbi Halperin is whether severe mental retardation must be accepted as the cost of improving chances of survival. The broader question is whether severe physical incapacity such as, for example, paralysis or blindness, must be accepted as the cost of improving chances of survival.
  Rabbi Halperin tentatively suggests that there may be no obligation of rescue in situations in which the person whose life is saved will remain mentally incompetent. The Gemara, <i>Sanhedrin</i> 73a, bases the obligation to preserve the life of one's fellow upon the obligation to return lost property: Since a person is obligated to return lost property, <i>a fortiori</i>, he must be obligated to restore a person's life no less so than he is obligated to restore that person's property. However, as recorded in <i>Shulḥan Arukh</i>, <i>Hoshen Mishpat</i> 261:4, there is no obligation to restore the property of a person who willfully abandons or attempts to destroy his property. The fact that the Gemara derives the obligation of rescue from the obligation to restore lost property prompted <i>Minḥat Hinnukh</i>, <i>Komez Minḥah</i>, no. 237, and R. Yerucham Perla, in <i>Mahari Perla</i>, his commentary on R. Sa'adia Ga'on's <i>Sefer ha-Mizvot</i>, <i>mizvot aseh</i>, no. 28, s.v. <i>u-milevad zeh</i>, to reach the astonishing conclusion that there is no obligation to intervene in order to prevent a would-be suicide.
  Surprisingly, but on the basis of the same line of reasoning, R. Shlomoh Kluger, <i>Hokhmat Shlomoh</i>, <i>Hoshen Mishpat</i> 426:1, asserts that there is no obligation to save one's fellow from imminent death if the act of rescue would involve humiliation or embarrassment to the rescuer. <i>Hokhmat Shlomoh</i> argues that the exemption from taking custody of and returning lost property in such circumstances extends to preservation of life.1<i class="footnote">R. Meir Don Plocki, <i>Klei Ḥemdah</i>, <i>Parashat Ki Teẓe</i>, sec. 6 (1), rebuts that contention on the basis of the categorization by the Gemara, <i>Sotah</i> 21b, of a person who declined to rescue a drowning woman as a “stupid pietist” (<i>ḥasid shoteh</i>). <i>Klei Ḥemdah</i>, <i>ibid.</i> 6(2), also notes that prevention of suicide is mandated not only by virtue of the obligation to preserve life but also by virtue of the obligation to prevent a fellow Jew from transgression. The latter obligation admits of no exception because of shame or embarrassment.Moreover, a person is exempt from restoring lost property if rescuing such property would cause him humiliation or embarrassment only if he would be willing to suffer the loss of such property were it his own in order to spare himself similar humiliation or embarrassment. It is highly unlikely that a person would decline to rescue a close relative whose life is endangered simply because rescue would entail a measure of humiliation or embarrassment. See R. Joseph Shalom Eliashiv, <i>Sefer ha-Zikaron le-Maran ha-Grib Zolti</i>, ed. R. Joseph Buchsbaum (Jerusalem, 5747), p. 404.</i>
  Rabbi Halperin applies a similar parallelism in the medical context. The Gemara, <i>Bava Mezi'a</i> 30b, establishes that a person must trouble himself to return another person's property only in situations in which he would devote similar time and expend similar effort in preserving his own property. Apparently assuming that the same exemption exists in a situation in which the owner of the property would not deem it worth his while to preserve his own property, argues Rabbi Halperin, there is no obligation to provide life-prolonging treatment for a person who does not wish to live. Such an individual would not seek to preserve his own life; hence, argues Rabbi Halperin, according to the earlier-cited authorities, it is not incumbent upon others to do so on his behalf.2<i class="footnote">The application of this argument to the case under discussion is far from clear. It may well be the case that a mentally competent, fully rational individual would deem a life of mental incompetence worse than no life at all. An infant, however, never having experienced life as a person endowed with reason, has no basis for an aversion to a life bereft of reason. Moreover, lacking mental development, an infant does not have the cognitive capacity to formulate a conscious desire either to survive or not survive.</i>
  That argument fails for a number of reasons. A person has no obligation to prevent his property from becoming destroyed. The prohibition of <i>bal tashḥit</i>, i.e., wasteful or wanton destruction of property, prohibits only wanton acts of destruction but does not mandate conservation of resources. Hence, a person may legitimately decline to preserve property facing imminent destruction, particularly when intervention would involve expenditure of time and effort not commensurate with the value of the endangered property. Not so with regard to life. According to Rabbenu Nissim, a person is under obligation to preserve his own life even if he prefers death because failure to do so is tantamount to suicide.3<i class="footnote">See Rabbenu Nissim, <i>Shevu’ot</i> 25a. See also this author’s <i>Bioethical Dilemmas</i>, II (Southfield, Michigan, 2006), 168.</i> It then follows that, since a person is obligated to preserve his own life under any and all circumstances, others are similarly required to engage in acts of rescue on his behalf regardless of his desires.
  Moreover, as Rabbi Halperin himself concedes, the positions of <i>Minḥat Hinnukh</i>, <i>Mahari Perla</i> and <i>Hokhmat Shlomoh</i> are rejected by the consensus of rabbinic authorities. The Gemara, <i>Sanhedrin</i> 73a, posits a further obligation to preserve the life of one's fellow based upon the verse "Nor shall you stand idly by the blood of your fellow" (Leviticus 19:16). That negative commandment seems to be quite independent of the obligation regarding restoration of property—and the <i>a fortiori</i> obligation to rescue life—and, accordingly, does not admit of any of the exclusions attendant upon the obligation to restore lost property.4<i class="footnote">The notion that the negative commandment is not at all contingent upon the obligation of restoring lost property is developed by Rabbi Eliashiv, <i>Sefer ha-Zikaron</i>, p. 404, and invoked by him in explaining why a person involved in fulfilling a <i>miẓvah</i> is not thereby exempt from the obligation of rescuing a person whose life is endangered.</i>
  Even if the earlier-cited authorities are correct in assuming that the prohibition "Nor shall you stand idly by the blood of your fellow" is limited to situations in which intervention is mandated by "and you shall restore it to him" (Deuteronomy 22:2) and does not establish an entirely independent obligation, the conclusion reached by those authorities is not compelled. A person's life, unlike his property, does not belong to himself but to the Creator.5<i class="footnote">See Rambam <i>Hilkhot Roẓeaḥ</i> 1:4; Radvaz, <i>Hilkhot Sanhedrin</i> 18:6; <i>Shulḥan Arukh ha-Rav</i>, V, <i>Hilkhot Nizkei Guf va-Nefesh</i>, sec. 4; and R. Shlomoh Yosef Zevin, <i>Le-Or ha-Halakhah</i> (Tel Aviv, 5717), pp. 222f. Cf., R. Shiloh Rafael, <i>Torah she-be-al Peh</i>, XXXIII (5752), 74-81, reprinted in <i>idem, Mishkan Shiloh</i> (Jerusalem, 5756), pp. 212-221.</i> Accordingly, argues R. Yo'av Yehoshu'a Weingarten of Kintzk (Konskie), in a responsum included in R. Issachar Berish Graubert's <i>Teshuvot Divrei Yissakhar</i>, no. 168, the obligation to preserve life is not a duty owed to the potential victim but to the Deity and that duty is shared equally by the rescuer and the person rescued.6<i class="footnote">Indeed, <i>Tur Shulḥan Arukh, Ḥoshen Mishpat</i> 261, maintains that willfully abandoned property or property that a person attempts to destroy becomes <i>res nullius</i>. However, Rambam, <i>Hilkhot Gezelah ve-Avedah</i> 11:11, maintains that an attempt to destroy one’s property does not render it <i>res nullius</i> but that, nevertheless, there is no obligation to preserve such property. R. Yo’av Yeshoshu’a Weingarten’s line of reasoning is cogent according to both positions. See Rabbi Eliashiv, <i>Sefer ha-Zikaron</i>, p. 404.</i> In a responsum of which the earlier-cited authorities were apparently unaware, R. Meir of Rothenberg, <i>Teshuvot Maharam ben Barukh</i> (Prague, 5368), no. 39, rules unequivocally that an individual must be rescued from potential death even if he protests and vehemently demands that there be no intervention. A similar rebuttal is offered by R. Meir Dan Plocki, <i>Klei Hemdah, Parashat Ki Teze</i>, sec. 6.7<i class="footnote">See also R. Chaim Joseph David Azulai, <i>Birkei Yosef, Oraḥ Ḥayyim</i>, 301:6, who rules that Sabbath restrictions are suspended for purposes of saving the life of a would-be suicide. Similar rulings are recorded in <i>Teshuvot Maharil Diskin, Kuntres Aḥaron</i>, no. 5, sec. 34, as well as in <i>Iggerot Mosheh, Yoreh De’ah</i>, II, no. 174, <i>anaf</i> 3 and <i>Yoreh De’ah</i>, III, no. 90.</i>
  By far the sharpest language employed in rejecting those views is that of R. Moshe Feinstein, <i>Iggerot Mosheh, Yoreh De'ah</i>, II, no. 174, <i>anaf</i> 3:
  … with the forgiveness of these illustrious scholars, it is clear that it is absolute error, may their Master forgive them! Heaven forfend that the words of <i>Hokhmat Shlomoh</i> be uttered, for if the honor of Heaven is set aside … for the sake of saving the life of even the most inferior person … certainly the honor of a mortal, even of the greatest of the great, must be set aside.
  <i>Iggerot Mosheh</i> argues that although preservation of personal dignity may be a consideration sufficient to negate the obligation to restore lost property, it cannot be invoked to excuse oneself from the obligation of rescuing life: God, in permitting violation of His commandments for purposes of <i>pikuaḥ nefesh</i> has waived His own honor; <i>a fortiori</i>, the dignity of a mortal cannot represent a value that might serve to abrogate the duty of rescue.
  <i>Iggerot Mosheh</i> further argues that one must conduct oneself <i>vis-à-vis</i> another person's lost property as one would conduct oneself <i>vis-à-vis</i> one's own and hence one must act the same way with regard to preservation of the life of another. <i>Iggerot Mosheh</i> argues that since a person would seek to save his own life under any and all circumstances he cannot plead humiliation as an exception from the obligation to save the life of another. <i>Iggerot Mosheh</i> adds that even if the individual in question
  is a person to whom dignity is preferable to life, behold it is forbidden to die because of [avoidance of] demeaned dignity. Therefore, that the person would be [willing to be] wicked is meaningless in terms of exempting him from rescuing others because of his wickedness. Besides, no person has credibility to declare that his dignity is preferable to him over his life.
  Rabbi Halperin advances another consideration relevant solely to the particular situation he addresses, <i>viz.</i>, potential mental incompetence. As a rationale for the principle that Sabbath restrictions are set aside for preservation of life, the Gemara, <i>Shabbat</i> 151b and <i>Yoma</i> 85b, advances the argument: "Better that they desecrate one Sabbath on his behalf so that he may observe many Sabbaths." Rabbenu Nissim, <i>Yoma</i> 82a, offers that rationale, <i>inter alia</i>, in ruling that Sabbath restrictions must be set aside in order to preserve the life of a fetus even in the earliest stages of development. Rabbenu Nissim clearly maintains that this consideration is both normative and independent of other considerations in terms of setting aside halakhic restrictions.
  Rashba, cited by <i>Bet Yosef, Oraḥ Hayyim</i> 306, apparently disagrees. Rashba addressed a question involving a person whose daughter was abducted on <i>Shabbat</i> and faced forced apostasy. The interlocutor sought advice with regard to the propriety of engaging in acts ordinarily forbidden on <i>Shabbat</i> in an attempt to rescue his daughter. Rashba replied that the matter required further study but that he was inclined to the negative. Since an apostate will surely desecrate the Sabbath, acts of rescue could readily be justified on the basis of the rationale "Better that they desecrate one Sabbath in order to observe many Sabbaths," were that consideration to be considered normative. Rashba's failure to issue a positive ruling indicates that he did not regard that rationale to be applicable.8<i class="footnote">The parameters of an entirely different principle that might serve as a basis for intervention, <i>viz.</i>, “A scholar would prefer to commit a lesser transgression so that an ignoramus not commit a graver transgression” (<i>Eruvin</i> 32b), are discussed by <i>Tosafot, Shabbat</i> 4a. Cf., <i>Magen Avraham, Oraḥ Ḥayyim</i> 306:29.</i>
  Nevertheless, both <i>Taz, Oraḥ Hayyim</i> 306:5, and <i>Magen Avraham, Oraḥ Hayyim</i> 306:29, understood Rema, <i>Hilkhot Shabbat</i> 328:10, as ruling that Sabbath prohibitions may be ignored in order to save a Jew from apostasy for precisely that reason, i.e., so that he may continue "to observe many Sabbaths." If so, argues Rabbi Halperin, it would follow that Sabbath restrictions may also be ignored in order to cure a person of lunacy or mental incompetence "so that he may observe many Sabbaths." Actually, that point was made much earlier by R. Iser Yehudah Unterman, <i>Shevet me-Yehudah</i>, I (Jerusalem, 5715), 49 and 64. Rabbi Unterman, however, declares that the Gemara posits that rationale only in situations in which the undertaking is assured of success so that the observance of "many Sabbaths" is a certainty.
  Moreover, the argument is inherently flawed. The notion that a single Sabbath may be set aside in order to gain the observance of many Sabbaths reflects the notion that a certain degree of risk-taking <i>vis-à-vis</i> Sabbath observance, and even compromise thereof, is warranted in order to gain maximum Sabbath observance. It does not at all follow that even <i>ḥayyei sha'ah</i>, i.e., ephemeral longevity, may be ventured solely for the sake of maximizing Sabbath observance.
  Rabbi Halperin also notes that mental incompetence is regarded by Halakhah as itself constituting a life-threatening condition. A person afflicted in that manner poses a danger to himself as well as to others and certainly lacks the mental capacity to avoid situations of danger. In support of that contention, Rabbi Halperin cites a statement of <i>Bet Yosef, Yoreh De'ah</i> 228, in the name of Rashba. That responsum is published in <i>Teshuvot ha-Rashba ha-Meyuḥasot le-ha-Ramban</i>, no. 281.
  The earliest reference to mental disease sufficiently grave to imperil the life of the afflicted occurs in <i>Issur ve-Hetter he-Arukh</i>,9<i class="footnote"><i>Issur ve-Hetter he-Arukh</i> (Vilna, 5751), no. 59, sec. 35. Cf. also <i>Hagahot Maimuniyot, Hilkhot Ma’akhalot Assurot</i> 14:15.</i> attributed to Rabbenu Yonah of Gerondi. <i>Issur ve-Hetter he-Arukh</i> cites a specific query addressed to an earlier authority, Maharam, concerning an epileptic who sought advice concerning the permissibility of partaking of a forbidden food reported to possess medicinal properties capable of curing his illness. The decision, in which Ramban acquiesces, is in the affirmative, provided that the efficacy of the remedy has been established. That decision is predicated upon a determination that epilepsy constitutes a danger to life since, at times, an epileptic may endanger himself by "falling into fire or water." R. Israel Meir Mizrachi,10<i class="footnote">Cf., <i>Piskei Teshuvah</i>, ed. R. Abraham Pieterkovsky (Pietrokow, 5693), II, no. 261.</i> <i>Teshuvot Pri ha-Arez</i>, III, <i>Yoreh De'ah</i> (Jerusalem, 5665), no. 21, relies upon the decision of Ramban in ruling that insanity also constitutes a danger to life and accordingly permits an abortion when it is feared that the mother may otherwise become mentally deranged. Rashba, cited by <i>Bet Yosef, Yoreh De'ah</i> 228, also regards mental incompetence as life-threatening in nature.11<i class="footnote">Cf., <i>Teshuvot ha-Rashba ha-Meyuḥasot le-ha-Ramban</i>, no. 281; <i>Magen Avraham, Oraḥ Ḥayyim</i> 554:8 and <i>Pri Megadim, ad loc</i>.</i> This position is also adopted by R. Mordecai Winkler, <i>Teshuvot Levushei Mordekhai, Hoshen Mishpat</i>, no. 39, who is cited by R. Waldenberg, <i>Ẓiz Eli'ezer</i>, IX, no. 51, <i>sha'ar</i> 3, chap. 3, sec. 9. Other authorities, including <i>Teshuvot Admat Kodesh</i>, I, <i>Yoreh De'ah</i>, no. 6; <i>Teshuvot Pri ha-Arez</i>, II, <i>Yoreh De'ah</i>, no. 2; <i>Birkei Yosef, Shiyurei Berakhah, Yoreh De'ah</i> 155; <i>Teshuvot Nezer Mata'ai</i>, I, no. 8; <i>Iggerot Mosheh, Even ha-Ezer</i>, I, no. 67; <i>Ẓiz Eli'ezer</i>, IV, no. 13, sec. 3; and R. Yosef Shalom Eliashiv, as cited by R. Yitzchak Zilberstein, <i>Assia</i>, no. 42-43 (Nissan 5746), pp. 26f., also maintain that mental incompetence constitutes a threat to life.12<i class="footnote">Cf., however, R. Iser Yehudah Unterman, <i>Ha-Torah ve-ha-Medinah</i>, IV, 27, who argues that the instinct for self-preservation is so deeply ingrained and suicide so rare that a suicide complex cannot be considered to be within the category of illnesses that endanger life. That assumption is quite evidently regarded as contrafactual by the many authorities who adopt an opposing view. See also this writer’s <i>Contemporary Halakhic Problems</i>, I (New York, 1977), 363.</i>
  Blood-Sugar Tests
  1. The Shabbat Problem
  Drawing blood on <i>Shabbat</i> is ordinarily forbidden. According to Rashi, <i>Shabbat</i> 107a, and most other early-day authorities, the prohibition is classified as a form of "slaughter," one of the thirty-nine paradigmatic categories of labor prohibited on the Sabbath. Scripture declares, "for the blood is the life (<i>nefesh</i>)" (Deuteronomy 12:23), and hence removal of any quantity of blood is, halakhically speaking, tantamount to extinguishing a measure of "life." According to Rambam, <i>Hilkhot Shabbat</i> 8:7, drawing blood is categorized as a derivative form of "threshing," another of the prohibited categories of labor. Threshing involves extracting a kernel from its husk. Removal of blood, according to Rambam, is similar in that it involves expressing blood from the tissues in which it is imbedded.
  Accordingly, drawing blood on <i>Shabbat</i> is sanctioned only if necessary for the treatment of someone suffering, or possibly suffering, from a life-threatening illness. Untreated or insufficiently treated diabetes is certainly an illness of that nature. Crucial in the treatment of diabetics is regulation of blood sugar levels. Since a diabetic's body either does not produce insulin, as in Type I diabetes, or cannot process the insulin that is produced, as is the case in Type II diabetes, the diabetic's blood sugar levels are likely to fluctuate much more widely than those of a healthy person. Current medical practice calls for taking blood samples periodically during the course of the day, particularly before meals, in order to titrate the amount of insulin or other medication required to achieve optimum blood sugar levels. Patients are trained to draw small quantities of blood and to perform a simple blood analysis themselves.
  Diabetics whose insulin is insufficient for their caloric intake are at risk for developing life-threatening effects of hyperglycemia, including diabetic coma. Those whose dose of insulin is too high for the quantity of calories they have consumed are in danger of potentially life-threatening hypoglycemia that can result in insulin shock. Indeed, the halakhic discussions of glucose testing on <i>Shabbat</i> focus upon the immediate dangers. However, from the halakhic perspective, the danger must be defined far more broadly.
  It is certainly true that lack of precision in determining dosages, particularly over a short period of time, poses no imminent danger to the patient. Deleterious effects of minor increases of blood sugar are unlikely to become manifest until after the passage of time. However, the nature of diabetes is such that, barring a supervening illness or accident, the diabetic's longevity anticipation will be shortened as a result of complications of that disease. The effects of higher than normal blood sugar are cumulative. Medical science has not established a threshold of safety for fluctuation of blood sugar or a time span within which such fluctuation is innocuous.
  Accordingly, it seems to this writer that any act designed to achieve an optimal blood sugar level for even a minimal period of time is in the category of possible <i>pikuaḥ nefesh</i> and hence is not only permitted but is mandatory on <i>Shabbat</i> as well as on weekdays.
  In practical terms, the result is that a diabetic is required to perform the necessary blood test as often as recommended by his or her physician. Of course, a rabbinic authority knowledgeable in the technical aspects of the testing procedure should be consulted for advice with regard to minimizing the nature of the Sabbath infractions involved.13<i class="footnote">A symposium devoted to the proper methods to be employed in performing blood tests on <i>Shabbat</i> in order to minimize the severity of the infraction was published in the Tevet, 5763 issue of <i>Or Yisra’el</i> (vol. 8, no. 2). That symposium also addresses the issue of whether intake of carbohydrates should be minimized in order to reduce the number of blood tests required during the course of the Sabbath.R. Shalom Elyakim Berkowitz, <i>Or Yisra’el, ibid.</i>, p. 36, notes that many authorities maintain that rabbinic prohibitions are <i>hutrah</i>, i.e., entirely suspended, when permitted for a sick person. See R. Meir Arak, <i>Teshuvot Imrei Yosher</i>, I, no. 185; <i>Teshuvot Emunat Shmu’el</i>, no. 47, p. 47, s.v. <i>ve-hineh</i> and <i>ibid.</i>, appendix, <i>kelalim</i> 21:8; R. Yo’av Yehoshu’a Weingarten, <i>Teshuvot Ḥelkat Yo’av, Oraḥ Ḥayyim</i>, no. 14, s.v. <i>akh</i>; <i>Derekh ha-Melekh, Hilkhot Shabbat</i> 6:9, sec. 1; <i>Teshuvot Avnei Nezer, Oraḥ Ḥayyim</i>, I, no. 118, sec. 5; and R. Eli’ezer David Grunwald, <i>Teshuvot Keren le-David</i>, no. 70, s.v. <i>me-attah</i>. See also <i>Teshuvot Ḥesed le-Avraham, Mahadura Tinyana, Oraḥ Ḥayyim</i>, no. 67. Since blood should be drawn with a <i>shinuy</i>, i.e., in an “unusual” manner, thereby rendering the infraction rabbinic rather than biblical, the number of times blood must be drawn need not be minimized according to this view.</i> The sole remaining halakhic issue is whether, in drawing blood, the patient may extract only the minimum quantity of blood necessary for analysis or whether the amount of blood removed in the performance of a single act is irrelevant. There are multiple self-use kits available for this purpose. Some draw more blood than others. Assuming, as is likely the case, that all are equally efficacious, is the patient obligated to use a device, or method, that results in removal of the least possible quantity of blood? That issue is addressed in a number of contributions to <i>Assia</i>, no. 73-74, Nisan 5764 (vol. 19, no. 1-2).
  The most commonly used method of obtaining a blood sample is by means of a finger prick. The blood that is released is placed on a testing strip. The strip is then placed into a processing device that measures and announces the blood sugar level.
  Since receiving F.D.A. approval late in the year 2000, a device has been available that accomplishes lancing, blood collection and glucose testing with a single press of a button. The device employs a vacuum mechanism that holds the skin in place while an integrated apparatus lances the skin and automatically transfers the blood to a bio-sensor strip that tests the blood and records the result. This device has the advantage of causing significantly less pain than traditional finger-stick testing and also requires only a minute quantity of blood. A survey of various self-testing devices of this material presently marketed in the United States indicates that the amount of blood drawn varies between .6 microliters and four microliters. Standard type lances are much cheaper but draw more blood than necessary.
  In his article in <i>Assia</i>, Rabbi Menasheh Klein astutely draws attention to the fact that an issue exists with regard to the proper procedure to be followed on weekdays as well as on <i>Shabbat</i>. Drawing blood constitutes "wounding" and is forbidden other than for therapeutic purposes. The Gemara, <i>Sanhedrin</i> 84b, indicates that, even in a medical context, "wounding" beyond that which is therapeutically necessary constitutes a transgression. Accordingly, Rabbi Klein asserts that drawing more blood than is necessary for diagnostic testing is forbidden.14<i class="footnote">Rabbi Klein states that he had earlier expressed this view in his <i>Mal ve-lo Para</i> (Brooklyn, 5744). The reference is presumably to <i>netiv</i> 24 of that work in which Rabbi Klein demonstrates that, as is readily apparent from <i>Sanhedrin</i> 84b, removal of more tissue than necessary, or extending an incision more than medically necessary, constitutes illicit “wounding.” Rabbi Klein regards drawing additional blood in the same light.</i>
  R. Meir Simchah ha-Kohen of Dvinsk, <i>Or Sameaḥ, Hilkhot Shabbat</i> 18:1, establishes a fundamental point with regard to Sabbath regulations. The general rule is that even when an otherwise forbidden act is sanctioned, e.g., for purposes of saving a life, it is forbidden to add to the quantity of a substance that is the subject of that act even if no additional physical act is required. Thus, on the Sabbath, if a limited quantity of soup is required for a patient, it is not permitted to add additional water to the pot even before the pot is placed on the stove. This is so despite the fact that only a single act of "cooking" is required regardless of whether the pot contains a large or small quantity of water. The concern is that, if adding food before the pot is placed on the fire is permitted, the cook may err in adding additional food in the pot after the pot has been placed on the fire. Placing a second quantity of food in a pot already on the fire constitutes an additional discrete act and, since not necessary for the purpose of preserving life, constitutes a biblical infraction. Placing more food in the pot than is required for the patient constitutes <i>ribbuy be-shi'urin</i> i.e., increasing the quantity of the substance upon which the forbidden act is performed.
  Among the various categories of "labor" prohibited on <i>Shabbat</i>, many forms of "labor" result in culpability at the hands of the <i>bet din</i> only if the prohibited act is performed upon a minimum quantity of a substance. Accordingly, although cooking even a minimum quantity of a foodstuff is proscribed, there is no culpability unless a minimum quantity equal in weight to a <i>grogeret</i>, or dried fig, is cooked. According to <i>Tosafot, Menaḥot</i> 64a, s.v. <i>shetayim</i>, and Rashba, <i>Hullin</i> 15b, such acts are prohibited only by virtue of rabbinic decree. According to Rabbenu Nissim, <i>Beizah</i> 17a, the prohibition against <i>ribbuy be-shi'urin</i> is biblical in nature.
  With regard to some forbidden forms of activity no minimum quantity or size is prescribed. Accordingly, "slaughter" or killing a live creature is cause for punishment by the <i>bet din</i> even if the creature is a mere insect. <i>Or Sameaḥ</i> asserts that there is no concept of <i>ribbuy be-shi'urin</i>, i.e., forbidden enhancement of quantity, with regard to forbidden forms of labor that require no minimum quantity or size for culpability. Accordingly, he asserts that when an animal must be slaughtered on behalf of a patient on <i>Shabbat</i> and two animals are available, one large and one small, there is no reason to slaughter the smaller, leaner animal rather than the larger.
  <i>Or Sameaḥ</i>'s position is in contradiction to the earlier published view of R. Joseph Babad, <i>Minḥat Hinnukh</i>, no. 32, <i>Mosekh ha-Shabbat</i> 39:2. When two animals are available, <i>Minḥat Hinnukh</i> regards the slaughter of an animal larger than necessary for the needs of the patient to be rabbinically forbidden even according to Rashba.15<i class="footnote"><i>Minḥat Ḥinnukh</i>’s position is also espoused by R. Joseph Chaim ben Eliyahu, <i>Teshuvot Rav Pe’alim</i>, I, <i>Oraḥ Ḥayyim</i>, no. 21; <i>Arukh ha-Shulḥan, Oraḥ Ḥayyim</i> 318:16; and <i>Kaf ha-Ḥayyim, Oraḥ Ḥayyim</i> 328:87.</i> A similar rule exists with regard to cutting the branch of a tree bearing fruit when fruit is necessary for a patient on <i>Shabbat</i>. Plucking fruit, cutting a branch or otherwise severing vegetative matter from the ground to which it is attached constitutes a category of labor that is forbidden on <i>Shabbat</i>. In a situation in which a patient is in need of no more than two fruits and two branches are available, one bearing two fruits and the second three fruits, only the branch bearing two fruits may be severed despite the fact that the physical "labor" involved in severing the branch bearing a larger number of fruits is absolutely identical to the "labor" involved in severing the branch bearing a smaller quantity of fruit. The reason, asserts <i>Kesef Mishneh, Hilkhot Shabbat</i> 2:8, is that <i>ribbuy be-shi'iurin</i> is forbidden. In effect, each fruit is discrete and each is the subject of the prohibition against plucking; hence more fruit than needed may not be "plucked" even by means of severing the branch through a single act.
  R. Zevi Pesach Frank, <i>Teshuvot Har Ẓevi, Oraḥ Hayyim</i>, no. 177, cites R. Meir Auerbach, <i>Imrei Binah, Dinei Shabbat</i>, no. 17, in distinguishing between severing a branch bearing two fruits rather than a branch bearing three fruits and slaughtering a heavier rather than a lighter animal and asserts that the rule applicable to the branches does not apply to the slaughter of an animal. Since an animal is an integral whole, argues <i>Har Ẓevi</i>, slaughter is always in the nature of a single act performed upon a single animal having the effect of extinguishing a single life regardless of the size of the animal.16<i class="footnote"><i>Har Ẓevi</i> disputes and rejects the ruling of <i>Teshuvot Rav Pe’alim</i>, I, <i>Oraḥ Ḥayyim</i>, no. 21, who asserts that even with regard to a single animal “the limbs are separate and stand separately” and therefore invokes the rule applicable to branches of a tree in forbidding the slaughter of the larger of the two animals.</i> Consequently, <i>Har Ẓevi</i> permits the slaughter of the larger animal.17<i class="footnote">This is also the position of R. Shlomoh Zalman Margulies, <i>Matteh Efrayim, Keẓeh ha-Matteh</i> 618:14 and addenda and, as reported by R. Joshua Neuwirth, <i>Shemirat Shabbat ke-Hilkhatah</i>, 2nd edition (Jerusalem, 5739), I, 32:29, note 90, was also endorsed by R. Shlomoh Zalman Auerbach. See also R. Eli’ezer Waldenberg, <i>Ẓiẓ Eli’ezer</i>, VIII, no. 15, sec. 6.</i> <i>Har Ẓevi</i> similarly suggests that lighting a lamp containing more fuel than is necessary to satisfy the needs of the patient may also be permissible. The situation involving the lamp is more complex and involves additional considerations that are not relevant to this discussion.18<i class="footnote">Applying this principle, <i>Ẓiẓ Eli’ezer</i>, VIII, no. 15, sec. 6, permits a physician who must write a prescription for 10 pills, representing the dose to be administered on <i>Shabbat</i>, to write a prescription for 20 pills. His ruling is based upon the consideration that no additional letter or number need be written; the “labor” involved in writing a “2” is identical to that entailed in writing a “1.” <i>Ẓiẓ Eli’ezer</i> compares that situation to the slaughter of a larger animal rather than a smaller one.</i>
  Applying that controversy to the issue of drawing a greater than necessary quantity of blood on <i>Shabbat</i>, Rabbi Israel Rosen, writing in the same issue of <i>Assia</i>, concludes that, although according to <i>Minḥat Hinnukh</i> the permissibility of the procedure is questionable, nevertheless, according to <i>Or Sameaḥ</i> and <i>Har Ẓevi</i>, there is no reason to employ a procedure that would express a smaller quantity of blood. Rabbi Klein, however, dismisses that conclusion. Rabbi Klein understands <i>Or Sameaḥ</i>'s thesis as applicable only to matters such as slaughter in which there is only a single life to be taken regardless of the size of the animal. "Wounding," he argues, even though it is a sub-category of "slaughter,"19<i class="footnote">See <i>Magen Avraham, Oraḥ Ḥayyim</i>, 316:15.</i> can be performed repeatedly upon the same animal and hence, he argues, each drop of blood expressed represents a discrete transgression.
  Rabbi Klein's argument seems to be flawed in the sense that it must have been impliedly rejected by both <i>Or Sameaḥ</i> and <i>Har Ẓevi</i>. Every act of slaughter also involves blood-letting. The blood of a large animal is certainly more copious than that of a small animal. If so, even though each animal, large or small, has but a single life, slaughter of the larger animal should be forbidden because, <i>ipso facto</i>, it also involves "wounding" of a quantitatively greater nature, i.e., the extraction of a larger quantity of blood, each drop of which constitutes a separate infraction. In failing to reach that conclusion <i>Or Sameaḥ</i> and <i>Har Ẓevi</i> must have reasoned that since "wounding" is a sub-category of "slaughter" there can be no <i>ribbuy be-shi'urin</i> with regard to "wounding" just as there is no <i>ribbuy be-shi'urin</i> with regard to "slaughter."
  Putting that matter aside, <i>Or Sameaḥ</i>'s thesis is advanced only with regard to forms of labor that demand no minimum quantity for purposes of liability. If, as Rambam maintains, "wounding" is prohibited as a form of "threshing,"20<i class="footnote">Rambam’s position is understood in this manner by <i>Leḥem Mishneh, Hilkhot Shabbat</i> 8:9; <i>Minḥat Ḥinnukh</i>, no. 32, <i>Mosekh ha-Shabbat</i> 5:3; and R. Meir Dan Plocki, <i>Ḥemdat Yisra’el, Kuntres Ner Miẓvah</i>, pp. 47b-48a.</i> causing blood to flow in the course of slaughter should also be a form of "threshing." Since culpability for "threshing" requires a minimum quantity of a <i>grogeret</i> it should follow that <i>ribbuy be-shi'urin</i> with regard to "threshing" is prohibited21<i class="footnote">Rashi, <i>Shabbat</i> 107a, cites an opinion to the effect that “wounding” constitutes a form of “dyeing,” which also requires a minimum quantity for culpability.</i> with the result that slaughter of a large animal on behalf of a sick person should not be warranted when slaughter of a smaller animal would suffice.22<i class="footnote">Rabbi Rosen’s comments in this regard are somewhat confusing. He seems to ignore the fact that <i>Or Sameaḥ</i>’s thesis is limited to forms of labor having no minimum <i>shi’ur</i>.</i>
  Rabbi Rosen does, however, raise an additional point that is both empirical and intriguing. The amount of blood expressed in utilization of any of the available methods is much lower than the minimum quantity of <i>grogeret</i>. Even assuming that an act involving a <i>ḥazi shi'ur</i>, i.e., a quantity insufficient to generate culpability, is biblically proscribed, the issue that seems heretofore not to have been addressed is whether there is a prohibition of <i>ribbuy be-shi'urin</i> when, in aggregate, the total quantity will remain below the <i>shi'ur</i>, or threshold, that gives rise to culpability.
  The issue raised by Rabbi Rosen is germane only with regard to the Sabbath prohibition associated with drawing blood. It is not at all relevant with regard to the issue of "wounding" since punishment for violation of that prohibition is not contingent upon causing a fixed minimum amount of bleeding. In a brief communication, published in the same issue of <i>Assia</i>, R. Joshua Neuwirth, author of <i>Shemirat Shabbat ke-Hilkhatah</i>, dismisses the question of <i>ribbuy be-shi'urin</i>. He reasons that in applying a lance, the patient is, in effect, performing an act comparable to opening a faucet. In the latter case, other than the first gush of water, the resultant flow is automatic and only an indirect result of the act of turning on the spigot. The same is the case, asserts Rabbi Neuwirth, with regard to the flow of blood following puncture of the skin: pricking the skin allows pent up blood to flow of its own accord.23<i class="footnote">See, however, <i>Magen Avraham, Oraḥ Ḥayyim</i> 328:53, and R. Meir Auerbach, <i>Imrei Binah, Dinei Shabbat</i>, who maintains that use of a leech in bloodletting constitutes a direct act rather than <i>gerama</i>. Cf., R. Moshe Schick, <i>Teshuvot Maharam Shik, Oraḥ Ḥayyim</i>, no. 115. Rabbi Berkowitz, <i>Or Yisra’el</i>, <i>ibid.</i>, p. 35, argues that, when a lance is utilized, the resultant flow of blood should be regarded in the same light as leeching. However, it may be the case that <i>Magen Avraham</i> maintains that it is only the blood drawn immediately by the bite of the leech that is the product of a direct act and hence only the first gush of blood following lancing would be regarded in that manner.</i> Rabbi Rosen responds that, in point of fact, blood (often) does not flow of its own but requires application of pressure to express the required quantity of blood; hence, the issue of <i>ribbuy be-shi'urin</i> is germane.
  Interesting, but apparently ignored, is a point that arises from Rabbi Neuwirth's analogy to opening a faucet. In that case, the initial gush of water as a result of built-up pressure is deemed a human act for all purposes of Halakhah. The ongoing flow is regarded to be in the nature of <i>gerama</i>, i.e., an act caused only indirectly. In the absence of proximate cause, such indirect acts are generally proscribed by rabbinic decree in all situations in which a direct act is biblically forbidden. The issue not addressed is whether there is a prohibition of <i>ribbuy be-shi'urin</i> with regard to acts in the nature of <i>gerama</i>.
  2. Unnecessary "Wounding"
  As previously noted, Rabbi Klein is adamant in his view that drawing more blood than necessary for diagnostic purposes is forbidden and accordingly draws attention to the common medical practice of filling relatively large vials with blood despite the fact that only a fraction of the quantity drawn is actually needed for blood tests.24<i class="footnote">Dr. Jocelyn Hicks, “Excessive Blood Drawing for Laboratory Tests,” <i>New England Journal of Medicine</i>, vol. 340, no. 21 (May 27, 1999), p. 1690, presents the results of a survey showing that community and university hospitals typically draw 2 1/2 to 7 times the amount of blood as do children’s hospitals for identical tests. Dr. Hicks theorizes that unnecessary quantities of blood are drawn because in years gone by when the practice was developed most instruments required large quantities of serum for analysis. Earlier, J. C. Dale and S. K. Pruett, “Phlebotomy: A Minimalist Approach,” <i>Mayo Clinic Proceedings</i>, vol. 68, no. 3 (March,1993), pp. 249-255, reported that a range of 2 to 102 times and a mean of 45 times the required volume required of diagnostic tests was obtained for a sample of 113 hospital patients.The prohibition against “wounding” is not necessarily predicated upon concern for the health of a particular victim or patient. However, unnecessary drawing of more than minimal quantities of blood is not without risk. Phlebotomy is highly associated with changes in hemoglobin and hematocrit levels and can contribute to anemia. See Paaladinesh Thavendiranathan, Akshay Bagai, Albert Ebidia <i>et al.</i>, “Do Blood Tests Cause Anemia in Hospitalized Patients: The Effect of Diagnostic Phlebotomy on Hemoglobin and Hematocrit Levels,” <i>Journal of General Internal Medicine</i>, vol. 20, no. 6 (June, 2005), pp. 520-524. Although blood loss from laboratory testing is not likely to pose a problem for most patients, blood loss is greater in intensive care patients and after cardiovascular surgery. Other risk groups include low-birth weight newborns, patients with already existing anemia and the elderly. See Dirk Wisser, Klaus van Ackern, Ernst Knoll <i>et al.</i>, “Blood Loss from Laboratory Tests,” <i>Clinical Chemistry</i>, vol. 49, no. 10 (October, 2003), pp. 1651-1655. Obviously, internal bleeding and dehydration are additional risk factors. The proposed remedy is simply to use smaller tubes for drawing blood. See Jeannette O. Andrews, “A QI (Quality Improvement) Project to Reduce Nosocomial Blood Loss,” <i>Dimensions of Critical Care Nursing</i>, vol. 17, no. 2 (March-April, 1998), pp. 92-99 and Terry Rudisill, “Smaller Tubes Reduce the Volume of Blood Drawn,” <i>Critical Care Nurse</i>, vol. 22, no. 3 (June, 2002), pp. 96-95.</i> The practice, he emphasizes, is forbidden on weekdays as well as on <i>Shabbat</i>. As noted earlier, Rabbi Neuwirth clearly disagrees because, in his opinion, the additional blood flows of its own accord. Although, again as noted, that is frequently not the case with the minute quantity of capillary blood obtained by means of a needle prick or by lancing, it is certainly true with regard to venous blood drawn for more extensive diagnostic purposes. Again, as stated earlier, it seems to this writer that causing flow of blood in that manner is in the nature of <i>gerama</i> that is rabbinically forbidden. This writer is unaware of any source that specifically addresses the issue of whether there is a prohibition of <i>ribbuy be-shi'urin</i> with regard to an act in the nature of a <i>gerama</i>. Even if <i>ribbuy be-shi'urin</i> represented a rabbinic rather than a biblical prohibition, such an act should nevertheless be prohibited even when performed indirectly. However, it would seem to be the case that it is permissible to direct a non-Jew to perform such an act. An act that would be permissible but for the cumulative effect of three separate rabbinic prohibitions remains permissible. In this case, the act would be forbidden only because of a conjunction of three distinct rabbinic prohibitions: 1) <i>gerama</i> is rabbinically forbidden; 2) <i>ribbuy beshi'urin</i> is rabbinically forbidden; and 3) instructing a non-Jew to perform a proscribed act is rabbinically forbidden. Hence, since rabbinic legislation does not forbid acts involving a configuration of three rabbinic decrees,25<i class="footnote">See <i>Pri Megadim</i>, introduction to <i>Oraḥ Ḥayyim, Hanhagot ha-Sho’el veha-Nish’al</i> 1:74. See also R. Mordecai Brisk, <i>Teshuvot Maharam Brisk</i>, I, no. 23. That position seems to be reflected in the comments of R. Ezekiel Landau, <i>Dagul me-Revavah, Oraḥ Ḥayyim, Eshel Avraham</i> 340:5, who advances this principle only tentatively. The principle is rejected by R. Henoch Pak, <i>Zikhron Yosef</i>, secs. 8 and 167.</i> the act is not forbidden.
  However, the prohibition against drawing excess blood on a weekday is not as clear-cut as it might seem. R. Moshe Feinstein, <i>Iggerot Mosheh, Hoshen Mishpat</i>, I, no. 103 and <i>Oraḥ Hayyim</i>, IV, no. 102, permits donations of blood to a blood bank on various grounds. <i>Inter alia</i>, he argues, the Sages regarded blood-letting not only as a therapy for various maladies but also regarded it as having prophylactic value.26<i class="footnote">As early as the fifteenth century B.C.E., Hippocrates asserted that bloodletting restored the harmony of the body’s humors. There are reports of bloodletting by English monks in 674 C.E. Bloodletting reached the height of its popularity in Europe between 1825 and 1835. Venesection, or bloodletting, does serve to reduce hypertension but any benefit is transient. Modern medicine has come to recognize the value of “therapeutic phlebotomy” in the treatment of a few relatively rare diseases. Among those are: polycythemia, a hematologic disorder characterized by an increased concentration of red blood cells; hemochromatosis, a hereditary disorder that causes the small intestines to absorb excessive amounts of iron; and porphyia cartenea tarfa (PCT), one of several conditions caused by an enzyme deficiency that are collectively known as porphyrias. PCT is caused by deficiency of an enzyme called uroporphyrinogen decarboxylase and exacerbated by exposure to toxic chemicals including Agent Orange and certain herbicides, alcohol and drugs such as estrogen and iron supplements. See Salena M. Wright and Jean Finical, “Beyond Leeches: Therapeutic Phlebotomy Today,” <i>American Journal of Nursing</i>, vol. 100, no. 7 (July, 2000) pp. 55-56, 58-59 and 61-63.Use of leeches has also had a limited comeback in modern medicine. In the early years of the 20th century it was discovered that the mouth and throat of the leech contain a substance that prevents blood from coagulating. See R. T. Sawyer, “Leeches—New Role for an Old Medicine,” <i>Ward’s Bulletin</i>, Spring, 1986, pp. 1-4. The compound was isolated and named hirudin. However, with the advent of heparin, there is no longer significant reason to use leeches for that purpose. Beginning in the 1970’s medicinal leech therapy was introduced primarily for use in plastic and reconstructive surgery, postoperative venous congestion and graft rejection. Leech application often has a positive effect in treatment of superficial phlebitis and may improve the cosmetic appearance of spider-burst veins. The anti-inflammatory and circulation-enhancing effects of chemicals in leech saliva combined with blood and lymph drainage are of benefit in arthrosis pain management. The most valuable use of leech therapy is in preventing thrombosis or insufficiency of anastomosed arteries that may occur in corrective microsurgery. In such procedures, leeches serve to drain replanted or transplanted tissue. The most comprehensive study of leech therapy is Andreas Michalsen, Manfred Roth and Gustav Dobos, <i>Medicinal Leech Therapy</i> (New York, 2007). See also Michelle M. O’Hara, “Leeching: A Modern Use for an Ancient Remedy,” <i>American Journal of Nursing</i>, December, 1988, pp. 1656-1658.</i>
  Accordingly, <i>Iggerot Mosheh</i> concludes that blood donations are permitted because of their medical benefit to the donor. According to <i>Iggerot Mosheh</i>, filling a larger than necessary vial of blood when drawing blood for diagnostic purposes should be regarded simply as a form of bloodletting permitted by the Sages because of its potential health benefit. Rabbi Klein, consistent with his view regarding drawing excess blood for diagnostic purposes, regards donation to blood banks as prohibited although, presumably, he, too, would permit such donations in a situation of a <i>ḥoleh le-faneinu</i>, i.e., an imminent danger as that concept is expansively defined by Halakhah.27<i class="footnote">See J. David Bleich, <i>Contemporary Halakhic Problems</i>, IV(New York, 1995) 188-192.</i>
  Permissibility of drawing more blood than necessary would thus be subject to the dispute between Rabbi Klein and <i>Iggerot Mosheh</i>. Assuming that drawing more blood than is necessary constitutes illicit wounding, there is certainly cause for a Jewish physician, nurse or phlebotomist to be concerned with drawing blood in excess of the necessary quantity. However, it seems to this writer, that a Jewish patient need not protest if a non-Jewish health-care practitioner does so. The excess blood is neither drawn at the behest of the patient nor for his benefit. The benefit, if any, is convenience for the laboratory and for the person drawing the blood who wishes to avoid difficulties with his or her employer. Although the patient must cooperate and provide assistance in placement of the needle, there is no <i>mesaye'ah</i> or "assisting" in drawing the quantity that is in excess of medical need. At that point, the patient merely holds his or her hand rigid and provides no further active assistance.
  Use of Viagra on Shabbat
  The Sages prohibited many acts on <i>Shabbat</i> despite the fact that, in and of themselves, those acts involve no aspect of forbidden labor and are in no way inimical to the Sabbath spirit. They feared that engaging in such acts might, through excitement, confusion, overpowering desire etc., lead a person to desecrate the Sabbath, whether inadvertentently or knowingly, by performing a biblically proscribed act of labor. Thus, for example, the Sages severely limited the type and quantity of possessions that may be rescued from a burning building on <i>Shabbat</i> by removing them into a courtyard. Their concern was that, once rescue of property is countenanced, the prospect of loss of a significant portion of a person's fortune might engender emotional confusion that would prompt him to transport valuables into a public thoroughfare as well.
  The prohibition against taking medication or engaging in therapeutic activity on <i>Shabbat</i> or <i>Yom Tov</i> is among rabbinic prohibitions of that nature. According to Rif, <i>Shabbat</i> 53b, and Rosh, <i>Shabbat</i> 5:3, the Sages feared that a person in pain or discomfort, if permitted to take medicine, might be led to pulverize the components used in compounding such medications. <i>Hayyei Adam</i> 69:1 explains that the prohibition is rooted in a concern that, if any type of medical treatment is permitted, an observer might infer that Sabbath restrictions, including the biblical prohibition involved in pulverizing medicaments, are suspended for all medical purposes even in the absence of danger to life.
  Since even biblical restrictions are suspended in situations involving danger to life, no restriction against availing oneself of medication on <i>Shabbat</i> exists with regard to a patient afflicted with a potentially life-threatening illness or even of a malady that might cause permanent impairment. Nor did the Sages see fit to restrict medication or treatment of an even temporarily bedridden or incapacitated person. Essentially, the restriction is limited to individuals suffering mild discomfort or pain that is not incapacitating in nature.
  Contemporary reality makes it difficult to appreciate the concern that prompted a rabbinic enactment limiting therapeutic measures on <i>Shabbat</i>. Although, on occasion, a pill may be crushed and mixed with applesauce or the like, in our day, virtually no consumer of pharmaceuticals has ever compounded a medication and consequently there is scant reason to fear that a patient may be prompted to crush medicinal ingredients. Even pharmacists are no longer apothecaries. Pharmacists dispense medications that have been produced in bulk quantities by pharmaceutical companies. As a result, neither the patient, the care-giver nor the health-care professional in personal contact with the patient has occasion inappropriately to crush or pulverize medicines on <i>Shabbat</i>. The concern prompting promulgation of this rabbinic decree has largely lapsed.28<i class="footnote">Cf., R. Abraham Chaim Noe, <i>Keẓot ha-Shulḥan</i>, VII, no. 134; <i>Badei ha-Shulḥan</i>, sec. 7; and R. Noah Siegel, <i>Teshuvot Olat Noaḥ</i>, no. 3; and <i>Ẓiẓ Eli’ezer</i>, VIII, no. 15, chap. 15, secs. 2-4.</i> Nevertheless, the decree remains in force and is fully binding until rescinded by a court "larger in number and wisdom" than the court that enacted the decree originally.
  Sildendfil, marketed by Pfizer Pharmaceuticals under the name Viagra, is the first Food and Drug Administration approved oral pill for treatment of impotence. Use of Viagra on <i>Shabbat</i> is addressed by Rabbi Joel Catane in the earlier cited issue of <i>Assia</i>, Nisan 5764. Rabbi Catane points to a series of considerations that serve to render use of erectile dysfunction remedies permissible on <i>Shabbat</i>. Those considerations merit careful examination.
  Viagra does not cause penile erections directly. Rather, Viagra acts by enhancing the effect of nitric oxide that is normally released in response to sexual stimulation. Viagra removes impediments to the smooth muscle relaxant effects of nitric oxide which, in turn, allows increased blood flow into the areas necessary to cause and sustain an erection.29<i class="footnote">Side effects may include headache, flushing, indigestion and changes in vision, principally altered vision perception and blurred vision caused by a decrease in blood flow to the optic nerve and possibly also a sudden decrease or loss of hearing. In combination with organic materials, Viagra may cause a sudden, unsafe drop in blood pressure that can lead to fainting, stroke or a heart attack and hence is contraindicated for any patient taking certain medications including nitroglycerine.</i> Sildenafil is also marketed under the brand name Revatio. Tadalafil, marketed as Cialis, has much the same effect and is accompanied by the same side effects. An erection is reversed when a chemical known as phosphodiesterase type 5 (PDE5) breaks down the chemicals that caused the muscles to relax in order to accommodate an increased flow of blood. Cialis operates by flushing PDE5 and hence prevents contraction of the muscle tissue with the result that blood remains in the tissue for a longer period of time and prolongs the erection. A third FDA approved drug, Levitra, is similar to Cialis.
  On first analysis, it might appear that erectile dysfunction remedies such as Viagra, Cialis and Levitra should not be categorized as medications within the ambit of the edict prohibiting use of medications on <i>Shabbat</i> by virtue of the fact that the person taking the medication suffers from no illness. The problem confronting a person seeking to avail himself of one of these pharmaceutical products is that he either cannot achieve an erection or cannot sustain an erection long enough to engage in intercourse. Yet, as codified by <i>Mishnah Berurah</i> 328:120, medications are forbidden, not only when they are designed to alleviate an underlying condition, but also when they are designed <i>le-ḥazek ha-mezeg</i>, i.e., to "strengthen," enhance or promote ongoing "pharmacological balance" or good health. Of course, even ordinary food is designed to achieve that goal. The rule of thumb is that whatever constitutes a <i>ma'akhal bari</i>, or "food of a healthy person," i.e., a substance that would be consumed by a normal person for its nutritional value in the absence of a particular health concern, is permitted whereas a substance that would not ordinarily be consumed by a healthy person is forbidden. A food that would be eaten by a healthy person may be eaten by a sickly individual even though the latter's purpose in consuming such food is therapeutic in nature.30<i class="footnote">Cf., however, <i>Magen Avraham</i> 328:43 who permits use of such substances only if imbibed by the patient to satisfy hunger, to slake thirst or to satisfy desire. Cf., <i>Mishnah Berurah</i> 328:120 and 328:121. <i>Magen Avraham</i>’s view is rebutted by <i>Eglei Tal, Melekhet Toḥen</i>, sec. 47; <i>Torat Shlomoh</i> 328:49; and <i>Bi’ur Halakhah</i> 328:36. See also <i>Ẓiẓ Eli’ezer</i>, XIV, no. 50, sec. 2.</i> Understood in this manner, use of erectile dysfunction remedies is certainly problematic: Young, virile men have no need for a pharmaceutical stimulant. The medication is designed to increase blood flow to the penis, a goal which, even if it is not to be characterized as the cure of a malady, is certainly in the nature of a measure intended <i>le-ḥazek ha-mezeg</i>, i.e., to restore physical prowess and performance. Thus, even if Viagra or the like is not a "medicine," that serves to cure a physiological disorder, it is nevertheless not a <i>ma'akhal bari</i> and hence may well be prohibited on <i>Shabbat</i>.
  There may, however, be other grounds that would serve to exclude use of erectile dysfunction medications from the ambit of the prohibition against use of medicine on <i>Shabbat</i>. There are a host of authorities who rule that a medicine that must be taken over a period of several days may be utilized on <i>Shabbat</i> as well. The earliest authority to permit such medications is R. Shlomoh Kluger, <i>Sefer ha-Hayyim</i> 328:1,31<i class="footnote">This was also the view of <i>Ḥazon Ish</i> and R. Ya’akov Yisra’el Kanievsky. See sources cited by R. Joshua Neuwirth, <i>Shemirat Shabbat ke-Hilkhatah</i>, 2nd edition, I, 34:17, note 79 and R. Abraham ha-Levi Hurwitz, <i>Orḥot Rabbenu ha-Kehillot Ya’akov</i>, I, no. 214, pp. 155f. That position is also accepted by <i>Iggerot Mosheh, Oraḥ Ḥayyim</i>, III, no. 53.<i>Sefer ha-Ḥayyim</i>’s view is disputed by R. Shalom Mordecai Schwadron, <i>Da’at Torah, Oraḥ Ḥayyim</i> 328:36. See also <i>idem, Hagahot Maharsham, Orḥot Ḥayyim</i> 328:27. R. Moshe Stern, <i>Teshuvot Be’er Mosheh</i>, I, no. 33, sec. 7, follows the position of <i>Da’at Torah</i>. See also <i>Ẓiẓ Eli’ezer</i>, VIII, no. 15, chap. 15, sec. 15. In light of Rambam’s ruling cited herein, it would appear that the controversy between <i>Sefer ha-Ḥayyim</i> and <i>Da’at Torah</i> is limited to situations in which the earlier illness will not again became manifest if the medication is interrupted on <i>Shabbat</i>.It should be noted that in a later work, <i>Shenot Ḥayyim</i>, I, no. 152, sec. 5, R. Shlomoh Kluger reiterates his earlier recorded “commonly known leniency” but declares that it applies only if the illness has not dissipated (<i>lo pasak</i>) before <i>Shabbat</i>; apparently, then, if symptoms of the illness are no longer present, the medication may not be continued on <i>Shabbat</i>. However, as evident from Rambam’s rulings, if discontinuation of the medication will cause a new outbreak of the illness, the medication may be continued. If so, there may be no controversy between <i>Da’at Torah</i> and <i>Sefer ha-Ḥayyim</i>. Cf., however, <i>Ẓiẓ Eli’ezer</i>, VIII, no. 15, chap. 15, sec. 16, who apparently assumes that <i>Da’at Torah</i> forbids continuation of medication on <i>Shabbat</i> even in such circumstances.Moreover, in <i>Kinat Soferim</i>, II, <i>Hashmatot le-Shnot Ḥayyim</i>, no. 125, R. Shlomoh Kluger clarifies his position in stating that he had permitted only medicinal potions that can be eaten or imbibed but would not ordinarily be consumed by healthy persons. A person already availing himself of such a therapy, since he is aware of his need before <i>Shabbat</i>, will prepare his medications in advance; thus there is no fear that he will pulverize the medicaments on <i>Shabbat</i>. Other forms of therapy, he rules, are forbidden on the Sabbath under all circumstances because of a separate consideration, <i>viz.</i>, “<i>shema yimraḥ</i>,” lest he reduce a medicinal substance by smearing. When such a consideration pertains, it cannot be obviated by advance preparation. That comment is also included in Rabbi Kluger’s <i>Shnot Ḥayyim</i>, no. 152, addendum to sec. 4.</i> who states that this is a commonly known leniency (<i>margela be-puma de-insha</i>). R. Joshua Neuwirth cites Rambam's rulings, <i>Hilkhot Shabbat</i> 21:22 and 22:7, to the effect that one who drinks a potion known as <i>haltit</i> for medicinal purposes several days before <i>Shabbat</i> may continue to do so on <i>Shabbat</i>. Indeed, as reported by R. Abraham ha-Levi Hurwitz, <i>Orḥot Rabbenu ha-Kehillot Ya'akov</i>, I, no. 214, pp.155f., <i>Hazon Ish</i> goes beyond the position of other authorities in ruling that, in such circumstances, even the initial dose may be taken on <i>Shabbat</i>.32<i class="footnote">The source of this ruling, the Gemara, <i>Shabbat</i> 140a, permits use of medication on <i>Shabbat</i>, at least under some circumstances, when the medication had previously been used on Thursday and Friday. Rambam, <i>Hilkhot Shabbat</i> 22:7, codifies the rule in that manner. However, in <i>Hilkhot Shabbat</i> 21:22, Rambam states only that “If he drank <i>haltit</i> before <i>Shabbat</i> he may continue drinking it on <i>Shabbat</i>.” Cf., <i>Ẓiẓ Eli’ezer</i>, ibid., sec. 16. There seems to be no explicit source in the writings of early-day authorities permitting commencement of therapy on <i>Shabbat</i>.</i> In <i>Hilkhot Shabbat</i> 22:7 Rambam adds the qualifying phrase "so that he not become ill if he interrupts drinking." Apparently, the reason for this exclusion from the prohibition is that the Sages sought only to forbid use of medication by a person suffering mild pain or discomfort but did not seek to include in their edict those who are ill, even if not severely so. Requirement of medication over a period of days was probably assessed as being an indication of illness rather than mere discomfort (<i>meiḥush</i>).33<i class="footnote">See <i>Orḥot Rabbenu</i>, I, no. 214, pp.155f. Cf., R. Shlomoh Kluger, <i>Sefer ha-Ḥayyim</i> 328:6; who asserts that a person requiring medication for a period of days prepares a sufficient quantity in advance and hence has no reason to pulverize ingredients on <i>Shabbat</i>. <i>Ẓiẓ Eli’ezer</i>, VIII, no. 15, chap. 15, sec. 15, refutes that view and <i>ibid.</i>, sec. 17, offers the unlikely explanation that only a person who believes that medication will cure him the same day will be likely, in his agitation or excitement, to pulverize medicaments.</i>
  Ingestion of erectile dysfunction medication is not necessary on a daily or periodic basis. Nevertheless, Rabbi Catane suggests that the need for such medication in conjunction with every occasion of coitus is similarly an indication of sickness rather than mere discomfort. That argument is not persuasive for two reasons: 1) Rambam's phraseology in <i>Hilkhot Shabbat</i> 22:7 seems to indicate that the crucial factor is that interruption of the medication on <i>Shabbat</i> would frustrate the benefit of weekday administration of the medicine by causing the earlier illness to express itself anew. That explication of this rule reflects the incident reported by the Gemara, <i>Shabbat</i> 140a, which serves at the basis of the ruling.34<i class="footnote">See <i>Ẓiẓ Eli’ezer</i>, VIII, no. 15, chap. 15, sec. 16 and <i>Teshuvot Be’er Mosheh</i>, I, no. 33, sec. 7.</i> As recorded in that discussion, R. Hiyya ben Avin was permitted to continue taking medication on <i>Shabbat</i> with the explanation that failure on his part to do so would make him ill. Erectile dysfunction medication does not "cure" an underlying malady and hence its interruption does not negate the benefit of earlier doses. 2) The leniency formulated by <i>Sefer ha-Hayyim</i> does not seem to apply to a person who suffers from (mild) headaches intermittently. Typically, such a person takes an aspirin when he experiences a headache but does not do so prophylactically or to prevent a recurrence. Such an individual would not be permitted to take aspirin on <i>Shabbat</i> because each headache is a new episode.35<i class="footnote">Cf., however, the controversial ruling of R. Yonatan Steif regarding use of aspirin on <i>Shabbat</i>, cited by <i>Be’er Mosheh</i>, I, no. 33, sec. 5, II, no. 32 and VI, no. 39. See R. Abraham Chaim Noe, <i>Keẓot ha-Shulḥan</i> 138:31; R. Jacob Breisch, <i>Teshuvot Ḥelkat Ya’akov</i>, IV, no. 41; and R. Yitzchak Ya’akov Weisz, <i>Teshuvot Minḥat Yiẓḥak</i>, II, no. 35, who rule that the usual regulations governing use of medications on <i>Shabbat</i> apply to aspirin as well.</i> Viagra and other erectile dysfunction medications are similar in that they are used to cause or to sustain an erection whenever such physical prowess is desired. That effect is certainly in the nature of <i>le-ḥazek ha-mezeg</i>, i.e., achievement of more optimal physical function. A patient experiencing sexual dysfunction does not become sexually rejuvenated as a result of taking the medication over a period of time, which might render taking the medication on <i>Shabbat</i> permissible. Rather, the medication is used on an <i>ad hoc</i> basis to facilitate intercourse just as aspirin is used to alleviate an intermittent headache.
  Rabbi Catane finds other grounds to permit Viagra and the like in the rulings of <i>Ẓiz Eli'ezer</i>, XIV, no. 50, sec. 5, regarding tranquilizers and <i>Teshuvot Be'er Mosheh</i>, I, no. 33, sec. 19, and VI, no 39, with regard to stimulants used to prevent drowsiness. In both cases there is neither an underlying physical malady nor is the drug designed to "strengthen the constitution." As recorded by <i>Shulḥan Arukh, Oraḥ Hayyim</i> 328:37, a person suffering from no illness may avail himself on <i>Shabbat</i> of a substance not normally consumed by a healthy person. Rabbi Catane argues that Viagra should similarly be permitted since it does not function as a cure or as a means of "strengthening the constitution." However, the comparison does not seem apt since the purpose of an erectile dysfunction remedy is clearly to "strengthen the constitution," i.e., to sustain penile erection in a person otherwise incapable of doing so. It is of interest to note that Rabbi Catane cites a personal communication to him from Rabbi Neuwirth in which the latter apparently takes issue with the permissive rulings of <i>Ẓiz Eli'ezer</i> and <i>Be'er Mosheh</i> regarding tranquilizers and stimulants. It should also be noted that in sec. 13 of the summary appended to <i>Ẓiz Eli'ezer</i>, VIII, no. 15, Rabbi Waldenberg without explanation forbids use of stimulants to prevent sleep.
  A much stronger argument cited by Rabbi Catane was earlier advanced by Dr. Abraham S. Abraham with regard to penile injections on <i>Shabbat</i> prior to intercourse. Although painful and cumbersome, the purpose of such injections is the same as that of Viagra, <i>viz.</i>, to cause the blood vessels to become engorged in order to make erection possible. Dr. Abraham's argument permitting penile injections is based upon another exclusion from the prohibition against medical treatment on <i>Shabbat</i>. Even in situations in which there is no threat to life, treatment is permissible if failure to treat will result in loss or dysfunction of an organ or limb (<i>sakkanat ever</i>). <i>Kezot ha-Hoshen, Badei ha-Shulḥan</i> 138:18, and <i>Shemirat Shabbat ke-Hilkhatah</i> 33:1, in a gloss appended to sec. 5, rule that this exclusion applies even if foregoing treatment will result in only partial dysfunction of the organ, e.g., failure to set a fracture will not render the person lame but will result in a limp that will compromise function of the foot. Thus, Rabbi Neuwirth is cited in <i>Nishmat Avraham</i>, IV, introduction to <i>Oraḥ Hayyim</i> 328, sec. 4, as permitting fertility treatments on <i>Shabbat</i> on the grounds that inability to conceive represents dysfunction of the reproductive organs and hence constitutes <i>sakkanat ever</i>.36<i class="footnote">See also <i>Teshuvot Ḥelkat Ya’akov</i>, III, no. 23. <i>Be’er Mosheh</i>, I, no. 33, sec. 8, permits fertility medication on <i>Shabbat</i> for a completely opposite reason. Although his view is certainly subject to dispute, <i>Be’er Mosheh</i> asserts that failure to conceive does not constitute a malady within the ambit of the prohibition against therapeutic treatment on <i>Shabbat</i>; hence, as noted earlier, treatment is permitted since medications are permitted to an entirely healthy person.</i>
  However, at least in cases in which the patient is not completely impotent, the application of this thesis to penile injections or to ingestion of Viagra and the like, is not as strong as it might appear. The terminology employed by the codifiers of this rule is "<i>sakkanat ever</i>" or danger to an organ. The concern reflected in this exclusion seems to be the threat of permanent loss or dysfunction resulting from non-treatment. Thus loss of a foot, lameness and inability to bear children each represents an irreversible loss. Reproductive organs are designed for procreation. Although fertility treatment does not restore those organs to ongoing normalcy, such treatment serves to bring the <i>raison d'être</i> of those organs to actualization by making pregnancy possible.
  However, there is no evidence indicating that transient dysfunction constitutes <i>sakkanat ever</i> or that enhancement of the function of a limb or organ is permitted. Thus, for example, medical treatment for a sprain that results only in temporary lameness does not seem to be encompassed within this category. Similarly, it is not at all apparent that a person rendered lame by a sprain but who, hypothetically, can walk for a brief period of time after being injected with steroids, might avail himself of such treatment on <i>Shabbat</i>. No "danger" to the limb is forestalled by such treatment. Temporary dysfunction of an organ certainly does not constitute <i>sakkanat ever</i>; hence, restoration of such function does not seem to constitute a remedy for <i>sakkanat ever</i>. A medication that would entirely reverse impotence would certainly be permissible on <i>Shabbat</i>. However, Viagra does not do so; it only makes sustained erection possible for a limited period of time.37<i class="footnote">Use of medication for temporary reversal of male infertility would be comparable to female fertility treatment designed to restore the function of reproductive organs and thus make childbearing possible, at least while the hormone regimen is followed.</i> Nor does failure to use Viagra for any period of time compromise its future efficacy as a treatment for impotence.
  The strongest argument advanced by Rabbi Catane in support of the permissibility of use of Viagra and similar medications on <i>Shabbat</i> is based upon a responsum of R. Yitzchak Ya'akov Weisz, <i>Teshuvot Minḥat Yizḥak</i>, I, no. 108. <i>Minḥat Yizḥak</i> permitted a husband to take medication before and after intercourse "so that he not be harmed" by intercourse. Presumably, in the case brought to his attention, nitroglycerin was prescribed for a patient suffering from angina in order to dilate the cardiac vessels so that they might accommodate without danger the more rapid blood flow that results from exertion during intercourse.
  <i>Minḥat Yizḥak</i>'s ruling is based on the principle that a non-Jew may be instructed to perform a rabbinically forbidden act on <i>Shabbat</i> if such is necessary in order to fulfill a <i>mizvah</i>. As recorded in <i>Shulḥan Arukh, Oraḥ Hayyim</i> 140:11, intercourse is not permitted in an illuminated room. On <i>Shabbat</i>, when a lamp cannot be extinguished, <i>Hayyei Adam, Hilkhot Shabbat</i> 29:4, rules that it is also forbidden to interpose a "wall" between the lamp and the couple in order to block the light. This is forbidden even though construction of a "temporary wall" is only rabbinically proscribed. Nevertheless, a non-Jew may be requested to put a temporary wall in place on the basis of the principle that it is permitted to instruct a non-Jew to perform a rabbinically proscribed act in order to facilitate fulfillment of a <i>mizvah</i>, in this case, fulfillment of conjugal obligations. <i>Minḥat Yizḥak</i> cites <i>Teshuvot Radvaz</i>, III, no. 640, in demonstrating that availing oneself of medication on <i>Shabbat</i> is an infraction of lesser severity than commanding a non-Jew to perform a rabbinically proscribed act of labor. Hence, he argues, if ordering a non-Jew to perform such an act in order to fulfill a <i>mizvah</i> is permitted, <i>a fortiori</i>, taking medication must also be permissible for the purpose of fulfilling a <i>mizvah</i>. <i>Minḥat Yizḥak</i>'s ruling is endorsed by <i>Ẓiz Eli'ezer</i>, VIII, no. 15, chap. 15, sec. 14, and is cited by <i>Nishmat Avraham</i>, I, <i>Oraḥ Hayyim</i> 321:11, note 3. A similar ruling based on the same source was issued by <i>Be'er Mosheh</i>, I, no. 33, sec. 9 and reiterated in volume VI, no. 39 of the same work.
  However, Rabbi Neuwirth, <i>Shemirat Shabbat ke-Hilkhatah</i>, 2nd edition (Jerusalem, 5739), I, 33:20, note 81, apparently regards any act performed by a person himself, including taking medications on <i>Shabbat</i>, as a more severe infraction than the infraction involved in directing a non-Jew to perform a forbidden act.38<i class="footnote">See also <i>Nishmat Avraham</i>, I, <i>Oraḥ Ḥayyim</i> 321:2.</i> Nevertheless, on the basis of the clear rulings of <i>Minḥat Yizḥak, Ẓiz Eli'ezer</i> and <i>Be'er Mosheh</i>, and in light of the somewhat tentative nature of Rabbi Neuwirth's comment, the weight of opinion supports the conclusion that use of erectile dysfunction medications on <i>Shabbat</i> is permitted.
  
  Chapter 7 New York City Water
  
  <i>"You shall not make yourselves detestable with any swarming thing that swarms and you shall not defile yourselves with them…" (Leviticus 11:43). [The Torah] included in this negative commandment things that swarm on the ground, winged swarming things and things that swarm in the water. What are things that swarm in the water? Those small creatures such as worms and leeches that are in the water and exceedingly large creatures that are animals of the sea.</i>
  RAMBAM, HILKHOT MA'AKHALOT ASSUROT 2:12
  I. The Problem
  The saga of what one magazine has termed the "Jewish Watergate" began in July, 2003. For a number of years an Israeli firm had been marketing Romaine lettuce and other produce certified as free of insect infestation. In the course of demonstrating proper methods of vegetable inspection an instructor at the Mashgiach Training Institute of Yeshiva Birkas Reuven discovered insects in the water used for washing lettuce.1<i class="footnote">See “Jewish Watergate: Does Drinking Water Need to be Filtered?” <i>Kashrus Magazine</i>, July, 2004, p. 28 and <i>New York Times</i>, June 1, 2004, p. B2.</i> Samples were subsequently sent to the firm marketing the produce. In due course a response was received indicating that the insect that had been identified was known to spawn in water rather than in vegetables. Hence, the problem was in the water used for cleansing the produce rather than in the vegetables themselves! That finding was rapidly confirmed by examining tap water in the locale in which the lettuce had been rinsed.2<i class="footnote">See <i>Kashrus Magazine</i>, July, 2004, p. 36 and <i>New York Times, loc. cit.</i> A somewhat different version of the events leading to discovery of the problem is recounted in <i>Yated Ne’eman</i>, 28 Iyar 5764, p. 107.</i>
  The insects that were detected are tiny aquatic creatures, crustaceans called copepods which are millimeter-long zooplankton common both in the ocean and in ground water. These creatures have a thick body, long antennae and numerous legs.3<i class="footnote">The dominant species is Diacyclops thomasi, commonly known as Cyclops. Cyclops have five pairs of legs and divided tail-like appendages called furca as well as two sets of antennae. As the name suggests, Cyclops have one black or red eye in the middle of the head. Adults of the species range in size from 0.8 millimeter to 1.4 millimeters. Other species of copepods are present in the water supply as well.</i> Since the water supply is treated with chlorine very few, if any, of the insects found in the water have been found to be alive. The color of these insects is whitish grey which renders them translucent and accounts for why they are not easily spotted. They are, however, visible to the naked eye without magnification.
  New York City's Department of Environmental Protection has long been aware of the existence of these copepods. A manual published by that agency specifically states that "several aquatic species have been recognized as nuisance organisms in potable water supplies. Although these organisms are not considered public health risks, they are often the source of consumer complaints."4<i class="footnote">See <i>Yated Ne’eman</i>, 23 Iyar 5764, p. 107. The New York State Attorney General’s office has indeed filed a number of actions over a period of years in repeated attempts to force the city to filter water that is vulnerable to contamination. Despite a hue and cry raised in some quarters regarding a possible health hazard the Attorney General does not appear to have based his complaints upon the presence of copepods. Cf., <i>Yated Ne’eman</i>, 22 Sivan 5764, pp. 89 and 98.</i> The gauge of most filters routinely affixed to kitchen faucets is probably too large to prevent the escape of such tiny organisms. However, filters with impressively low gauges, some as low as .05 microns, are readily available. Such low-gauge filters effectively eliminate the targeted organisms.
  A number of prominent halakhic authorities have affixed their signatures to statements varying in terms of stridency but unequivocal in admonishing the public to filter water prior to use. The most significant of those is a ruling authored by R. David Feinstein bearing the endorsements of R. Joseph Shalom Eliashiv and R. Chaim Pinchas Scheinberg.5<i class="footnote">See <i>Yated Ne’eman</i>, 29 Elul 5746, p. 19.</i> That statement containing a succinct explanation of the grounds for the prohibition was reproduced and widely disseminated. Prominent among the signators of other statements are R. Feivel Cohen, author of <i>Badei ha-Shulḥan</i>, and R. Yechezkel Roth, formerly the <i>dayyan</i> of the Satmar community in Borough Park.6<i class="footnote">See <i>The Jewish Press</i>, June 4, 2004, p. 72 and June 18, 2004, p. 82. See also the statement of R. Hershel Schachter, <i>Kashrus Magazine</i>, October, 2004, p. 199.</i> A comprehensive article authored by R. Chaim Oberlander discussing all factors involved in resolution of this issue, including a number of possible leniencies, is featured in vol. IX, no. 4 (Tammuz 5764) of <i>Or Yisra'el</i>, published in Monsey, N.Y. Additional articles by R. Shlomoh Gross, <i>dayyan</i> of the Belz community in Borough Park, and R. Yitzchak Bistritsky, a <i>dayyan</i> of the Satmar community in Borough Park, containing valuable information also appear in the same issue of <i>Or Yisra'el</i>.7<i class="footnote">In addition, a pamphlet authored by Rabbi Yitzchak Raitport, <i>Teshuvah Halakhah le-Ma’aseh be-Din ha-Mayim she-be-Brooklyn</i>, New York, dated Elul 5764, permitting use of New York City water has been circulated within the rabbinic community. An article authored by R. Israel Belsky permitting New York water appears in <i>Sha’ashu’ei Oraita</i>, no. 3 (Kislev 5765). This article has been reprinted in the 5765 issue of <i>Ha-Metivta</i>, published by Mesivta Torah va-Daath. The reprinted version includes an additional section in which the author rules against use of a filter on <i>Shabbat</i>. The Yiddish-language publication, <i>Der Yiddisher Flam</i>, Av 5764, features a survey of the problem by Yitzchak Shlomoh Dresner as well as two halakhic articles, one by R. Elchanan Breisch forbidding use of unfiltered water and a second by R. Naphtali Oshry permitting use of such water.</i>
  The general rule with regard to examination for the presence of insects is that examination is required in all instances in which infestation is common but that examination is not required in situations in which there are no grounds to suspect that infestation has occurred. Thus, for example, <i>Hokhmat Adam</i> 38:13, rules that, in Germany and France, all cherries, save for one species known as "sour cherries," require examination but reports that in Danzig insect infestation of cherries is not common and hence examination is not required unless the presence of worms is indicated by pock marks on the skin. Plums, he reports, must be presumed to be infested in all countries with the exception of Lithuania where the plums are puny and insect-free. Certainly, the situation has changed significantly since the time of <i>Hokhmat Adam</i>, undoubtedly as a result of widespread spraying. Whether or not particular species of fruits and vegetables require examination depends upon the species and the locale.
  In some areas of Brooklyn, particularly in Borough Park and Flatbush, as well as in Washington Heights, copepods have been found in the drinking water. Where infestation has been demonstrated, filtration of the water is required. Where filtration is required, it is necessary not only for drinking-water but also water used in cooking, rinsing food, gargling and brushing teeth.8<i class="footnote">Cf., <i>Ḥokhmat Adam</i> 38:6.</i> However, even within those neighborhoods, in many locations no insects have been detected despite repeated examination.9<i class="footnote">It has been asserted that there can be no halakhic problem associated with drinking New York water because innumerable glasses of water containing copepods have been consumed by pious and learned Jews over the course of decades. In light of the talmudic dictum “The Holy One, blessed be He, does not permit the beasts of the righteous to sin in error, how much less the righteous themselves” (<i>Ḥullin</i> 5b), it is argued, it is unthinkable that such individuals should have been guilty of even inadvertent transgression. See David Berger, “On the Prohibition of Water: An Appeal to <i>Poskim</i>,” <i>The Jewish Press</i>, October 22, 2004, p. 7. Quite apart from the fact that the cited talmudic generalization has never been recognized as a dispositive consideration in ruling a questionable foodstuff to be permissible, considering the far from ubiquitous presence of copepods in New York water, there is no evidence that any of the luminaries named by Professor Berger did actually imbibe such creatures. The talmudic citation is not apropos by reason of the qualifying statement found in <i>Bereshit Rabbah</i> 60:10 declaring, “If the ancients…were humans we are as animals and not even as the donkey of R. Pinchas ben Ya’ir.” A similar statement is recorded in <i>Shabbat</i> 12b. See also <i>Tosafot</i>, s.v. <i>R. Natan omer</i>.</i> The Environmental Protection Agency is reported to have determined that infestation does not occur in the reservoir but at some point close to the point of discovery.10<i class="footnote">See <i>The Jewish Press</i>, June 18, 2004, p. 82.</i> There is, however, strong scientific evidence suggesting that the species of copepods found in New York water originate in the reservoirs rather than in localized water mains.11<i class="footnote">See Yaakov D. Lach’s unpublished report commissioned by the Orthodox Union, “Copepods in NYC Tap Water” (July 19, 2004), p. 9 and pp. 30-31.</i> Nevertheless, in some buildings, copepods may fall to the bottom of the water tower or, for some unexplained reason, their movement may be impeded in some manner by the plumbing system. If so, it seems to this writer that, if upon multiple examinations it is evident that the water in any particular building is insect-free, further examination is unnecessary. Examination is best carried out against a black background or, less optimally, by placing a cloth at the tap in order to catch the copepods as they emerge. Care should be taken that the tap be only partially opened in order to prevent dismemberment of the insects by the pressure of the water running through the cloth.
  II. Minuscule Insects
  The almost reflexive reaction of many people to reports of infestation of New York water by copepods was that the issue is hardly novel. Probably every person who has had the benefit of a freshman biology course has had the experience of placing a droplet of water on a slide and examining it under a microscope. The experience of viewing countless numbers of microorganisms is both thrilling and disconcerting: Thrilling because the microscope becomes a window opening upon the hidden mysteries of divine creation through which a tiny cross-section of the myriad complexities of the natural order is revealed; disconcerting, at least to a Jew, because, mindful as he is of the biblical admonition "of all that creeps in the water and of all the living creatures that creep in the waters … you shall not eat of their flesh" (Leviticus 11:10-11), he becomes jarringly aware that with every sip of water he imbibes copious quantities of such creeping things.
  In a nutshell, the problem and its resolution requires a determination of whether, in seeking to abide by the commandments of the Torah, a Jew must be concerned with subvisual phenomena or only with that which is detectable upon gross observation. The question is by no means limited to establishment of the parameters of the dietary code. A quite similar issue arises, <i>inter alia</i>, with regard to the validity of a Torah scroll. A severed or disconnected letter disqualifies the entire scroll. But even with, relatively speaking, moderate magnification, what appear to be vast chasms of open space can be seen in the structure of the letters of every Torah scroll. The discharge of even a minute quantity of menstrual blood renders a woman a <i>niddah</i>. What is the status of a woman if the discharge is too slight to be seen by the naked eye but can be readily perceived upon magnification? More recently, the selfsame issue has arisen with regard to the status of a fertilized ovum that is as yet invisible to the naked eye.12<i class="footnote">For other areas of Halakhah in which this issue arises and further discussions of the issue see R. Shlomoh Kluger, <i>Teshuvot Tuv Ta’am va-Da’at, Mahadura Tinyana, hashmatot</i>, no. 35; R. Shalom Mordecai Schwadron, <i>Teshuvot Maharsham</i>, III, no. 357; R. Benjamin Aryeh Weiss, <i>Teshuvot Even Yekarah, Mahadura Tinyana</i>, no. 33; R. Ovadiah Yosef, <i>Yabi’a Omer</i>, IV, <i>Yoreh De’ah</i>, no. 20, sec. 8 and no. 21 sec. 7; R. Ze’ev Wolf Leiter, <i>Teshuvot Bet David</i>, no. 8; R. Moshe Feinstein, <i>Iggerot Mosheh, Yoreh De’ah</i>, II, no. 146; R. Betzalel Zolti, <i>Mishnat Ya’aveẓ</i>, no. 66; R. Moshe Stern, <i>Teshuvot Be’er Mosheh</i>, V, no. 16; R. Eliezer Waldenberg, <i>Ẓiẓ Eli’ezer</i>, VIII, no. 15, chap. 14, sec. 10; R. Menasheh Klein, <i>Mishneh Halakhot</i>, IV, no. 129; <i>Darkei Teshuvah</i> 18:18; and <i>Mishnah Berurah</i> 648:46. Cf., <i>Teshuvot Divrei Yosher</i>, no. 30. See also R. Moshe Sternbuch, <i>Mo’adim u-Zemanim</i>, VIII, addenda to vol. II, no. 124.</i> The resolution of that question may impact upon the propriety of utilizing the "morning-after pill" for termination of pregnancy and upon the permissibility of destroying a fertilized ovum at the very earliest stages of cell division.13<i class="footnote">For a further discussion of this question see this writer’s <i>Bioethical Dilemmas: A Jewish Perspective</i>, vol. II (Southfield, Michigan, 2006), chap. 7, “Stem Cell Research,” pp. 199-238.</i>
  The earliest discussion of halakhic implications of subvisual phenomena is that of R. Israel Lipschutz in his commentary on the Mishnah, <i>Tiferet Yisra'el, Avodah Zarah, Bo'az</i> 2:3. <i>Tiferet Yisra'el</i> reports that some fifty or sixty years earlier a prominent rabbinic authority ruled that a fish, theretofore presumed to be non-kosher because it lacked scales, was indeed kosher. That authority issued his ruling subsequent to examining the skin of the fish with a "microscope" (<i>sic</i>: probably a magnifying glass) and discovering scales. In that instance, magnification was utilized to permit that which otherwise would be regarded as forbidden rather than to forbid the otherwise permissible. <i>Tiferet Yisra'el</i> peremptorily dismisses that view on the grounds that the Gemara, <i>Niddah</i> 51b, categorizes the scales that are the hallmark of kosher species as being in the nature of a "garment" (<i>levush</i>). <i>Tiferet Yisra'el</i> asserts that each of the minuscule scales of the species in question is "no more than a tiny dot which the eye cannot perceive." Accordingly, he concludes that such scales "certainly cannot be categorized as a garment" and hence the species must be regarded as non-kosher. <i>Tiferet Yisra'el</i> bases his position entirely upon the idiosyncratic halakhic definition of a "scale" rather than upon a global assertion that Halakhah takes no cognizance of subclinical phenomena. 14<i class="footnote">See, however, R. Yechiel Michel Epstein, <i>Arukh ha-Shulḥan, Yoreh De’ah</i> 83:15, who stipulates that the scales must be visible to the naked eye and also readily removable from the skin. See also R. Moshe Chalfon, <i>Sho’el ve-Nish’al</i>, V, <i>Yoreh De’ah</i>, no. 64. Cf., however, R. Moshe Strug, <i>Teshuvot Yashiv Mosheh</i>, no. 214, who rules that such fish are permissible.</i>
  Later, in discussing a somewhat different matter, R. Shlomoh Kluger, <i>Teshuvot Tuv Ta'am va-Da'at, Mahadura Tinyana, kuntres aḥaron</i>, no. 53, comments upon the method commonly employed in examining the lungs of an animal subsequent to slaughter. Whenever an adhesion is found, a possibility exists that it occurred because of a perforation in the underlying pulmonary tissue. Therefore, the adhesion is peeled away and the lung is blown up in order to examine for the presence of a lesion. The practice, as described by R. Shlomoh Kluger, involves placing water or saliva upon the exposed site of the adhesion and then looking for bubbling caused by escaping air which, in turn, is evidence of the presence of a minuscule hole. That examination is not unlike the procedure commonly employed by an automobile repairman seeking to locate the site of a puncture that has caused a flat tire. <i>Tuv Ta'am va-Da'at</i> dismisses as absurd his own suggestion that a magnifying glass might be used to search for a perforation visually and to substitute that procedure for the usual mode of examination. In context, <i>Tuv Ta'am va-Da'at</i> dismisses magnification as an unacceptable means of examination. More to the point is the fact that magnification of a sufficiently high power will always disclose the presence of holes with the result that, if cognizance must be taken of that phenomenon, no animal is kosher!
  Elsewhere, in his <i>Hokhmat Shlomoh, Yoreh De'ah</i> 84:1, R. Shlomoh Kluger remarks that with proper magnification minuscule creatures can always be found in water, but "the Torah was not given to ministering angels" with the result that creatures that cannot be perceived visually are not prohibited.15<i class="footnote">See also <i>Darkei Teshuvah</i> 84:9 and 84:28.</i>
  The classic and most frequently quoted source with regard to the issue of microscopic organisms is the comment of R. Yechiel Michel Epstein, <i>Arukh ha-Shulḥan, Yoreh De'ah</i> 84:36. <i>Arukh ha-Shulḥan</i> writes:
  I have heard that every drop of water, and particularly of rainwater, is full of minuscule creatures that the eye cannot see. In my youth, I heard from a person who had been in distant places and saw in water, by means of a lens of exceedingly great magnification, [i.e., that magnifies] tens of thousands of times, many species of creatures.
  In light of this, how may we drink water … ? Indeed, the truth is that the Torah did not forbid that which the eye cannot perceive, for the Torah was not given to angels. For, if not so, many scientists have written that the entire atmosphere is also full of extremely minuscule creatures and that when a person opens his mouth he swallows a number of them … . Even if this is so, since the eye cannot perceive them, it is of no significance. However, that which the eye can see, even [if only] against the sun and even if it is the tiniest of the tiny, is a veritable insect.16<i class="footnote">See also <i>Ḥokhmat Adam, Sha’ar Issur ve-Hetter, Binat Adam</i>, no. 34. <i>Ḥokhmat Adam</i> 38:49 negates the opinion of <i>Sefer ha-Brit</i> who forbade use of vinegar because it contains minuscule organisms. See also R. Shlomoh Kluger, <i>Ḥokhmat Shlomoh, Yoreh De’ah</i> 84:1; <i>Darkei Teshuvah</i> 84:66 and 84:94; <i>Ḥazon Ish, Yoreh De’ah</i> 14:6, s.v. <i>u-be-ma’aseh</i>; R. Shlomoh Zalman Auerbach, <i>Minḥat Shlomoh</i>, <i>Tinyana</i>, no. 63, sec. 2; R. Ovadiah Yosef, <i>Yeḥaveh Da’at</i>, II, no. 47; R. Samuel ha-Levi Woszner, <i>Teshuvot Shevet ha-Levi</i>, VII, no. 2, sec. 10 and no. 123, sec. 2; R. Moshe Feinstein, <i>Iggerot Mosheh, Yoreh De’ah</i>, II, no. 46 and <i>Even ha-Ezer</i>, III, no. 33; R. Chaim Kanievsky, <i>Derekh Emunah</i>, IV, <i>Hilkhot Shmittah ve-Yovel</i> 4:1, <i>Bi’ur ha-Halakhah</i>, s.v. <i>bein min ha-asabin</i>; R. Nathan Gestetner, <i>Teshuvot Le-Horot Natan</i>, IX, no. 25, sec. 13; and the far-reaching discussion of R. Pesach Eliyahu Falk, <i>Teshuvot Maḥazeh Eliyahu</i>, no. 91. Cf., however, R. Jacob Emden, <i>She’ilat Ya’aveẓ</i>, II, no. 124, who advises that rice be examined for ants “in the sun or by means of a magnifying glass.” Cf., also, Rabbi A. Y. Zeleznick, <i>Halikhot Sadeh</i>, no. 51 (5747), p. 33.In stark contradiction, both Rabbi Raitport, p. 11, and Rabbi Oshry, p. 33, cite the position of <i>Halakhot u-Minhagei Rabbeinu Shalom me-Neustadt</i>, ed. Shlomoh Spitzer (Jerusalem, 5737), sec. 464, who ostensively maintains that insects that are visible only in bright sunlight are not prohibited. Cf., <i>Magen Avraham</i> 32:15. Cf., however, the note appended by the editor in which he assumes that authority’s position to be consistent with the accepted view.See also R. Elyakim Dworkes, <i>Be-Shevilei ha-Halakhah</i>, II (Jerusalem, 5752), 52, who cites unnamed sources who understand <i>She’ilat Ya’aveẓ</i> as merely commending use of a magnifying glass in order to ease the difficulty of close scrutiny with the naked eye. See also <i>Mo’adim u-Zemanim</i>, VIII, addenda to vol. II, no. 124. <i>Maḥazeh Eliyahu</i>, no. 91, sec. 8, cites <i>She’ilat Ya’aveẓ</i>, no. 124 and <i>Minhagei Ḥatam Sofer</i> (Tel Aviv, 5758) 10:18 in support of that practice.</i>
  Without explicitly citing the earlier comments of <i>Tiferet Yisra'el, Arukh ha-Shulḥan</i> declares the converse to be true as well, i.e., when visual perception is required to satisfy any halakhic requirement, artificial magnification is of no avail. Thus, <i>Arukh ha-Shulḥan, Yoreh De'ah</i> 83:15, writes:
  And know that it is clear that, with regard to a matter that requires visual perception, it is of no avail if one sees by means of a lens that magnifies visual perception; rather, one must see by means of the eye itself. Such is the case with regard to matters in the whole Torah in its entirety with regard to that which requires visual perception.
  As noted earlier, the requirement that scales must be visually perceivable for an aquatic species to be regarded as kosher is, according to <i>Tiferet Yisra'el</i>, quite independent of the general question of the status of subvisual phenomena. However, the general rule formulated by <i>Arukh ha-Shulḥan</i> seems to be a matter of dispute. The rule with regard to Torah scrolls, <i>tefillin</i> and <i>mezuzot</i> is that no letter may extend to the edge of the parchment; some blank space must remain as a margin on each side of the letter. Is the letter validly written if examination with a magnifying glass reveals that there is indeed empty space in the margin even though the blank space is not manifestly evident to the naked eye? R. Moshe Sternbuch, <i>Mo'adim u-Zemanim</i>, II, no. 124, cites the late <i>Tchebiner Rav</i>, R. Dov Berish Weidenfeld, as ruling that a scroll written in such a manner is invalid.17<i class="footnote">See also the discussions of R. Israel Veltz, <i>Divrei Yisra’el</i>, III, nos. 110 and 111; R. Samuel ha-Levi Woszner, <i>Teshuvot Shevet ha-Levi</i>, I, no. 7, sec. 8; and R. Shammai Kehot Gross, <i>Teshuvot Shevet ha-Kehati</i>, I, no. 36.</i> That ruling is entirely consistent with the position of <i>Arukh ha-Shulḥan</i>. However, <i>Mo'adim u-Zemanim</i> also cites R. Ze'ev Mintzberg as disputing that view in ruling that it is sufficient that a blank margin be discernible upon magnification. Rabbi Mintzberg's view is recorded in his responsa collection, <i>She'erit Yisra'el</i>, nos. 11 and 12.18<i class="footnote">See also R. Moshe Stern, <i>Teshuvot Be’er Mosheh</i>, V, no. 16 and R. Menasheh Klein, <i>Mishneh Halakhot</i>, IV, no. 128 and VII, no. 9.</i>
  Somewhat surprisingly, <i>Mo'adim u-Zemanim</i> proceeds to discuss the status of an <i>etrog</i> in which a very small piece of the flesh is missing and to relate that issue to this dispute. As recorded in <i>Shulḥan Arukh, Oraḥ Hayyim</i> 648:2, at least optimally, the <i>etrog</i> should be whole. What is the status of an <i>etrog</i> in which the hole can be seen only with a magnifying glass? <i>Mo'adim u-Zemanim</i> asserts that the status of such an <i>etrog</i> is contingent upon resolution of the controversy between the <i>Tchebiner Rav</i> and Rabbi Mintzberg.
  To the mind of this writer, that conclusion is unwarranted. The position of <i>Arukh ha-Shulḥan</i> and the <i>Tchebiner Rav</i> is readily grasped: The requirements of the Torah are to be understood as determinable under ordinary circumstances. The "creeping" organisms interdicted by Scripture are defined as creatures that would have been perceived as "creeping" at the time of revelation at Mount Sinai. "<i>Ein ha-Kadosh barukh Hu ba be-teruniyah im beriyotav</i>—The Holy One, blessed be He, does not act despotically with His creatures." (<i>Avodah Zarah</i> 3a), i.e., at Sinai, the Deity did not establish requirements that they could not possibly meet. The Torah was given to humans, not to angels (<i>Berakhot</i> 25b; <i>Yoma</i> 30a; <i>Kiddushin</i> 54a; and <i>Me'ilah</i> 14b). Hence, perforce, the connotation of a "creeping" organism, for example, must be limited to creatures that could have been identified at that time by use of conventional modes of observation. Conversely, according to <i>Arukh ha-Shulḥan</i>, if God stipulated blank margins for Torah scrolls, <i>tefillin</i> and <i>mezuzot</i>, the term "<i>mukaf gevil</i>" (surrounded by a margin) must have been understood as a visually perceivable margin and hence that must have been—and must remain—the meaning of the term.
  Rabbi Mintzberg's position is equally comprehensible. As has been noted earlier, the Gemara declares that "the Torah was not given to ministering angels." Hence microscopic organisms could not have been banned at Mount Sinai. The Torah also requires that scrolls have margins. Of course, at Sinai, a scribe upon completing the writing of such a scroll could not have been certain that a margin existed unless he could have seen the empty space with his naked eye. For much the same reason, a person intent upon fulfilling the <i>mizvah</i> of eating <i>mazah</i>, which requires him to eat a quantity of <i>mazah</i> equal to a <i>ke-zayit</i> (an olive) would perforce have to eat a bit more than a <i>ke-zayit</i> since absolutely precise measurements are beyond human capacity. That, however, does not imply that the Torah insists that sufficient space be left so that it may be perceived by the naked eye any more so than that the Torah insists that a person eat a bit more than a <i>ke-zayit</i> of <i>mazah</i>. That which is added for reasons of doubt is not intrinsic to the <i>mizvah</i> itself. A person informed by the Holy Spirit that he has consumed a <i>ke-zayit</i> of <i>mazah</i> need eat no more. Similarly, according to Rabbi Mintzberg, a scribe informed by the Holy Spirit—or by a magnifying glass—that a margin does exist may declare the scroll to be valid.
  The question of the suitability of an <i>etrog</i> in which a tiny portion has been eroded involves an entirely different issue. Resolution of that question depends upon the definition of the term "missing" (<i>ḥaser</i>). Presumably, all would agree that, if it is only angels who realize that a tiny fragment is missing, the <i>etrog</i> may be used without hesitation for purposes of fulfilling the <i>mizvah</i>. Hence all would agree that "<i>ḥaser</i>" must be defined as that which normal human beings, using methods of examination naturally available to them, would perceive as being <i>ḥaser</i>. If so, a microscopic erosion, by virtue of halakhic definition, does not render the <i>etrog</i> a <i>ḥaser</i>.19<i class="footnote">Cf., however, R. Joseph Lieberman’s comment in one of a series of notes appended in unnumbered pages to R. Elyakim Dworkes’ <i>Be-Shevilei ha- Parashah</i> (Jerusalem, 5762). Basing himself upon a responsum of R. Akiva Eger published in <i>Drush ve-Ḥiddush R. Akiva Eger</i> (New York, 5709), I, 176, Rabbi Lieberman asserts that an <i>etrog</i> missing a portion of its flesh must be disqualified even if that fact can be ascertained only upon magnification. R. Akiva Eger cites <i>Magen Avraham, Oraḥ Ḥayyim</i> 648:16, who rules that a <i>ḥazazit</i>, i.e., lichen or a scab on the surface of an <i>etrog</i>, disqualifies the <i>etrog</i> only if it is perceived by the eye (<i>nireh le-ayin</i>). R. Akiva Eger, however, asserts that <i>Magen Avraham</i>’s comment is applicable only to a <i>ḥazazit</i> which disqualifies an <i>etrog</i> because the fruit is no longer <i>hadar</i> or “beautiful.” Since “beauty” in the first instance is literally determined by the eye of the perceiver, a blemish that cannot be perceived cannot mar the beauty of the fruit. However, asserts R. Akiva Eger, despite the terminology employed by <i>Magen Avraham, Oraḥ Ḥayyim</i> 648:9, when the disqualification is not because of the requirement of <i>hadar</i> but because of the removal or the absence (<i>ḥaser</i>) of a part of the <i>etrog</i>, the <i>etrog</i> is disqualified even if the affected area is much smaller. Based upon that comment, Rabbi Lieberman concludes that, according to R. Akiva Eger, halakhic determinations must take cognizance of perceptions that may be experienced only upon magnification.However, a close reading of R. Akiva Eger’s comments does not support that conclusion. R. Akiva Eger carefully states: “ … whatever is actually separated and is perceived by precise placing of the eye (<i>nireh be-simat ayin be-diyuk</i>) is disqualified because it is <i>ḥaser</i>.” R. Akiva Eger carefully speaks of “precise placing of the eye”—not of artificial magnification—as opposed to mere “perception by the eye”, i.e., that which can be perceived only by close scrutiny as distinct from that which strikes the eye immediately. Only a <i>ḥazazit</i> that “strikes the eye” disqualifies an <i>etrog</i>; one that can be detected only upon “precise placing of the eye” does not. In contradistinction, <i>ḥaser</i> is a disqualification even if the absence of a very small portion of the <i>etrog</i> can be determined only upon close visual scrutiny. There is, however, no reason to conclude that R. Akiva Eger would disqualify an <i>etrog</i> missing a portion of its flesh so minute that the imperfection can be determined only upon artificial magnification. See also R. Joseph Mashash, <i>Teshuvot Mayim Ḥayyim</i>, no. 259, who rules unhesitatingly that such an <i>etrog</i> is kosher.</i>
  The principle enunciated by <i>Arukh ha-Shulḥan</i> is readily traceable to the Mishnah, <i>Nega'im</i> 4:4. The various forms of <i>zara'at</i> described in Leviticus 13 cause defilement only if the manifestation covers the area of a <i>gris</i>, usually assumed to be approximately the size of a dime. The Mishnah, <i>Nega'im</i> 4:4, states that an afflicted area the size of a <i>gris</i> causes defilement even if a hair is encompassed in that area. The hair certainly occupies space and hence the tiny opening through which the hair invades the <i>zara'at</i> and displaces the skin cannot itself constitute a site manifesting a symptom of <i>zara'at</i>. Nevertheless, the Mishnah declares, "We are not concerned that the space of the black hair may have diminished the <i>baheret</i> (one of the forms of <i>zara'at</i>) for [the hair] is devoid of substance." Rambam, in his <i>Commentary on the Mishnah, ad locum</i>, interprets that comment as meaning that "the size of the hole from which the hair emerges is extremely minuscule and lacks <i>perceivable</i> [emphasis added] size that would diminish the size [of the <i>baheret</i>]."
  However, the ruling of the Mishnah with regard to <i>zara'at</i> does not really serve to establish a precedent for <i>Arukh ha-Shulḥan</i>'s thesis. Determining the criteria of what constitutes <i>zara'at</i> is distinguishable from other aspects of Jewish law in that Scripture explicitly provides that the determination is to be made "entirely in accordance with the sight of the eyes of the priest" (Leviticus13:12). The determination of impurity by reason of <i>zara'at</i> is thus made expressly contingent upon subjective appearance "in the eyes of the priest" rather than upon objective reality.
  The basic principle of Jewish law formulated by <i>Arukh ha-Shulḥan</i> has a parallel in the Roman law maxim "<i>De minimis non curat lex</i>—The law takes no cognizance of a trifle." However, the trifle that Jewish law declines to recognize is considerably more trivial than that to which Roman law refers. In fact, it is so trivial as to be, literally, non-perceivable.
  One caveat must be added. Assuming that terms employed by Scripture in describing physical entities must be understood as entities perceivable at the time of Revelation, it does not follow that any particular present-day subclinical phenomenon was also subclinical at the time of the giving of the Torah at Mount Sinai. In a note published in R. Yechiel Michel Tucatzinsky's <i>Bein ha-Shemashot</i> (Jerusalem, 5689), p. 153, and frequently cited in rabbinic discussions of this topic, R. Iser Zalman Meltzer suggests that human eyesight was far more keen in ancient times than at present. The Mishnah, <i>Bekhorot</i> 54b, states that even in open places animals may be regarded as members of a single group for purposes of tithing so long as the animals are all present within a limited, specified area. As Rashi explains the Mishnah, that area is defined as the area in which the animals can pasture under the supervision of a single shepherd. That, in turn, is defined by the Mishnah as an area having a radius of sixteen <i>mil</i>. The Gemara explains that the key to establishing the determination of that radius is the ability of a single shepherd to see, and hence to safeguard, the entire group of animals. As expressed by the Gemara, <i>ad locum</i>: "The Sages determined that the eye of a shepherd can master sixteen <i>mil</i>." There are various conflicting opinions among rabbinic decisors with regard to the definition of a <i>mil</i>.20<i class="footnote">For a comprehensive survey of the bases and ramifications of those opinions see R. Jacob Gershon Weiss, <i>Middot u-Mishkalot shel Torah</i> (Jerusalem, 5745), pp. 198-264.</i> According to the most widely accepted views, calculations of the distance of a <i>mil</i> vary from between 960 meters (3,149.60 feet)21<i class="footnote">R. Abraham Chaim Noe, <i>Shi’urei Torah</i> (Jerusalem, 5707), p. 249, calculates the <i>amah</i>, or cubit, as 48 centimeters (18.897 inches). The identical calculation was earlier advanced by R. Shalom Mordecai Schwadron, <i>Da’at Torah</i> 35:116, and <i>Darkei Teshuvah</i> 19:27. Accordingly, the <i>mil</i>, which equals 2,000 cubits, is equal to a distance of 960 meters (3,149.60 feet). Rabbi Noe gives alternative calculations of the <i>amah</i> as equal to 47 centimeters (18.5 inches) or 49 centimeters (19.29 inches). The <i>mil</i> would then be the equivalent of 940 (3,083.98 feet) or 980 meters (3,215.22 feet).</i> to approximately 1,160 meters (3,805.77 feet).22<i class="footnote"><i>Ḥazon Ish, Kuntres ha-Shi’urin, Oraḥ Ḥayyim</i>: <i>Mo’ed</i> 39:5-6 and 39:9, calculates the <i>amah</i> as 58 centimeters (22.83 inches) and, accordingly, a <i>mil</i> would equal 1,160 meters (3,805.77 feet). <i>Ḥazon Ish</i> also cites a report to the effect that R. Shmu’el Salant accepted calculations that would render the <i>amah</i> 60 centimeters or 23.62 inches and the <i>mil</i> 1200 meters or 3937.01 feet. Nevertheless, for purposes of stringency, R. Shmu’el Salant is reported to have accepted calculation that would yield 48 centimeters or 18.90 inches for the <i>amah</i> and 960 meters or 3149.61 feet for the <i>mil</i>.Cf., R. Ya’akov Yisra’el Kanievsky, <i>Shi’urin shel Torah</i> (Bnei Brak, 5729), who, in a note on p. 63, calculates the <i>amah</i> as 57 2/3 centimeters (22.70 inches) and possibly, according to Rashba, as 59.66 centimeters (23.49 inches). The <i>mil</i> would then be the equivalent of either a bit more than 1,153.2 meters (3,783.46 feet) or a bit more than 1,193.2 meters (3,914.6 feet). Actually, according to Rabbi Kanievsky’s own premises, the calculation of the <i>amah</i> as 57 2/3 centimeters is somewhat imprecise. Rabbi Kanievsky reports that “according to that which has been received by us from <i>gedolei hora’ah</i>, that the <i>amah</i> equals 13 <i>vershak</i> …” The <i>vershak</i> is a Russian measuring length equal to 1.75 inches or 4.445 centimeters. If so an <i>amah</i> is the equivalent of 57.78 centimeters (22.73 inches) and thus a <i>mil</i> is equal to 1,155.7 meters (3,791.66 feet). See <i>Encylopedia Britannica</i>, 15th ed. (Chicago, 1995), XIX, 735 (where the term is transliterated as <i>verchok</i>).A table compiled by Moshe, Rosa and Shimon Bodenheimer, <i>Shi’urin u-Middot Ḥazal</i> (Jerusalem, 5725), gives the measurement of the <i>amah</i> as 57.6 centimeters (22.67 inches). Cf., the table published by Y. Avi-Zevi, <i>Shanah be-Shanah</i>, 5730, p. 125, that gives the measurement of the <i>amah</i> as 56 centimeters and that of the <i>mil</i> as 1120.37 meters. Those measurements presumably reflect the view of R. Chaim Jacob Sheftel, <i>Erekh Milin</i> (Berditchev, 5667), <i>erekh eẓba</i>, according to whom the <i>amah</i> measures approximately 56 centimeters. See <i>Encyclopedia Talmudit</i>, II (Jerusalem, 5760), 29, note 56.Other less frequently cited opinions with regard to the length of an <i>amah</i> include those of R. Naphtali Zevi Judah Berlin, <i>Meshiv Davar</i>, I, no. 24; <i>Arukh ha-Shulḥan, Yoreh De’ah</i> 201:3; R. Moshe Sofer, <i>Teshuvot Ḥatam Sofer, Oraḥ Ḥayyim</i>, no. 181; and the earlier noted view of <i>Erekh Milin, erekh eẓba</i>.<i>Meshiv Davar</i> rejects what he terms “the customarily accepted calculations” according to which the length of the <i>amah</i> would equal 60 centimeters and advances considerations that would yield the measurement of an <i>amah</i> as 48 centimeters. Nevertheless, in his concluding remarks he retracts that view and advances a calculation that would result in an <i>amah</i> measured “in excess of” 48 centimeters and endorses the practice of Frankfurt am Main according to which the length of the <i>amah</i> is calculated as 57 centimeters or 22.44 inches.<i>Arukh ha-Shulḥan</i> states that the length of an <i>amah</i> is 12 “<i>veierskes</i>,” which is apparently the Yiddish term for the Russian <i>vershak</i>. Thus, according to <i>Arukh ha-Shulḥan</i>, the length of an <i>amah</i> equals 53.34 centimeters or 21.0 inches. Cf., <i>Encylopedia Talmudit</i>, II, 29, where the length of an <i>amah</i> is given as 54 centimeters according to <i>Arukh ha-Shulḥan</i>.<i>Teshuvot Ḥatam Sofer</i> states that the length of a fingerwidth is a “<i>zoll</i>.” Since a biblical <i>amah</i> is equal to 24 fingerwidths, the length of an <i>amah</i> according to <i>Ḥatam Sofer</i> equals 24 <i>zoll</i>. <i>Da’at Torah, Yoreh De’ah, Kuntres Aḥaron</i> 35:116 states that the <i>zoll</i> is equal to 2.6 centimeters. That calculation is apparently also assumed by <i>Darkei Teshuvah</i> and <i>Encyclopedia Talmudit</i>. [Cf., however, <i>Encyclopedia Britannica, loc. cit</i>., which describes the <i>zoll</i> as a Swiss unit of measure equal to 3 centimeters. The German <i>zoll</i> is usually 1/12 of a foot although in some places it was 1/11 or 1/10 of a foot. The German foot varied from a minimum of 23.51 centimeters in Wesel and a maximum of 40.83 centimeters in Freier. The <i>zoll</i> measured 2.36 centimeters in Saxonia, 2.43 centimeters in Bavaria and 2.62 centimeters in Prussia. The <i>zoll</i> in Austria measured 2.65 centimeters, approximately the measurement assumed by <i>Da’at Torah</i> and <i>Darkei Teshuvah</i>.] <i>Encyclopedia Talmudit, ad locum</i>, indicates that, according to <i>Ḥatam Sofer</i>, the length of an <i>amah</i> is 62.4 centimeters or 24.567 inches. Thus a <i>mil</i>, according to <i>Ḥatam Sofer</i>, would equal 1248.0 meters or 4094.49 feet. <i>Darkei Teshuvah</i> 19:27 notes that the <i>amah</i> was marginally enlarged by rabbinic decree and, accordingly, should be calculated as 24 1/2 <i>zoll</i>. <i>Darkei Teshuvah</i> endorses the position of <i>Ḥatam Sofer</i> and states that the length of an <i>amah</i> according to <i>Ḥatam Sofer</i> is 63.8 centimeters. Thus, according to <i>Darkei Teshuvah</i>, an <i>amah</i> equals 25.12 inches and, accordingly, a <i>mil</i> equals 1276 meters or 4186.35 feet. <i>Ḥatam Sofer</i>’s calculations are thus significantly greater than those of the earlier-cited authorities.<i>Darkei Teshuvah</i> also cites a certain “Rabbi of Yasi” who calculated the <i>amah</i> as “no more than” 52 or 53 centimeters and an anonymous scholar who differed and calculated the <i>amah</i> as approximately 58 1/2 centimeters. <i>Darkei Teshuvah</i> also quotes <i>She’erit Yisra’el, Parashat Meẓora</i>, who calculates the <i>amah</i> as 21 1/6 <i>zoll</i> or 56 centimeters and further opines that one who calculates the <i>amah</i> as 22 1/8 <i>zoll</i> or 57 centimeters is to be commended. <i>Darkei Teshuvah</i> cites an additional source for the latter calculation as well.<i>Da’at Torah, Yoreh De’ah, Kuntres Aḥaron</i> 35:116, cites authorities who variously posit the length of an <i>amah</i> as 47 centimeters, 55.25 centimeters, a bit more than 57 centimeters and 58.5 centimeters, as well as one authority who asserts that according to Rashi the <i>amah</i> equals 43.8 centimeters, according to <i>Shakh</i> 45.02 centimeters, according to <i>Taz</i> 47.34 centimeters and according to Rambam 50 centimeters. See also the measurements ascribed to earlier authorities by Rabbi Weiss, <i>Middot u-Mishkalot shel Torah</i>, pp. 198-264 as well as the table presented <i>ibid.</i>, introduction, p. 8. See also R. Moshe Shimshon Bacharach, <i>Ḥut ha-Shani</i>, no. 97.For a table correlating many of these measurements of the <i>amah</i> with the volume of the liquid <i>revi’it</i> see <i>Middot u-Mishkalot shel Torah</i>, introd., p. 8. Rabbi Kanievsky, <i>Shi’urin shel Torah</i>, p. 51, cites a discussion in the Palestinian Talmud, <i>Terumot</i> 10:7, that is understood by some authorities, including Riva, <i>Commentary on the Pentateuch, Parashat Mishpatim</i>; <i>Ma’adanei Yom Tov, Berakhot</i> 3:30; and <i>Tosafot Yom Tov, Pe’ah</i> 6:6, as indicating that a <i>revi’it</i> equals 240 grams. If so, an <i>amah</i> would appear to equal 73.368 centimeters or 28.885 inches. Rabbi Kanievsky categorizes that calculation of the <i>revi’it</i> as “extremely large” and as “not imagined by early-day authorities.”It may be of interest, albeit of no halakhic import, to note that, based upon Josephus’ description of the circumference of the pillars of the Temple, the renowned physicist Sir Isaac Newton postulated the length of the “sacred” cubit to be between 24.90 and 25.02 inches as distinct from the “profane” cubit of 20.63 inches that he believed was the measurement employed by the Egyptians in construction of the Great Pyramid. Newton’s calculations are the subject of his “De magnitudine cubiti sacri.” An English translation of that essay titled “A Dissertation Upon the Sacred Cubit of the Jews and the Cubit of Several Nations: in which, from the Dimensions of the Greatest Pyramid, as taken by Mr. John Greaves, the ancient Cubit of Memphis is determined” was published in the <i>Miscellaneous Works of John Greaves, Professor of Astronomy in the University of Oxford</i>, ed. Thomas Birch, II (London, 1737), 405-433.Perhaps of even greater interest is the fact that the long or royal Egyptian cubit was 20.67 inches or 52.5 centimeters in length. Specimens of that cubit are on display in the Egyptian Gallery of the British Museum as well as in the Egyptian Museum of Torino. That cubit was in use in Egypt in antiquity and was widely employed throughout the ancient world. A second unit of measure, known as the short cubit, equaling 45 centimeters or approximately 18 inches, was also used at that time. For an interesting discussion of those measurements and archaeological evidence that the Temple Mount measured 500 x 500 long cubits see Leen Ritmeyer, <i>The Quest: Revealing the Temple Mount in Jerusalem</i> (Jerusalem, 2006), p. 170.</i> Rabbi Meltzer observes that it is quite impossible for a person to see objects at such a great distance.23<i class="footnote">In absolute terms, the distance a person possessing normal vision is capable of seeing with the naked eye is limited primarily by the curvature of the earth. Thus, when vision is not obstructed, a five-foot-tall person standing at sea level can perceive objects, at least faintly, over a distance of 4.5 kilometers or 2.8 miles. See <i>Encyclopedia Britannica</i>, VI, 60. According to all calculations, the distance subject to a shepherd’s surveillance is much less. Presumably, then, contemporary inability to see objects at such a distance is due to atmospheric conditions, including, particularly in inhabited areas, pollution, rather than diminished visual prowess.</i> Accordingly, he concludes that, over the ages, our sense of sight has become dulled.24<i class="footnote">See also <i>Mo’adim u-Zemanim</i>, II, no. 124 and VIII, addenda to vol. II, no. 124.</i> Rabbi Meltzer attributes that phenomenon to widespread reliance upon artificial illumination which, he opines, has caused natural eyesight partially to atrophy. In support of that opinion he comments, "We know that, even today, the Bedouin who do not use our lamps can see very far."
  Thus it is certainly arguable that the organisms denoted by Leviticus 11:10-11 are those that were perceivable at the time the commandment was announced and that those organisms remain forbidden even if, in our age, they are no longer visible to the naked eye. However, this is not to say that microscopic organisms are forbidden. Even the shepherd of antiquity did not enjoy omnivision; after all, he could see only a distance of sixteen <i>mil</i>. Since that time there has been, at worst, only limited optical degeneration. Perhaps in antiquity the average man possessing eyesight of the quality common among his peers could see with his naked eye even that which is visible to us only upon two-three- or even ten-power magnification. Consequently, organisms that are but marginally subvisual may well be forbidden but those that can be seen only with a microscope rather than with a magnifying glass could not conceivably be within the ambit of the prohibition.
  <i>Arukh ha-Shulḥan</i>'s position with regard to minute organisms is amplified by R. Samuel ha-Levi Woszner, <i>Teshuvot Shevet ha-Levi</i>, VII, no. 122. <i>Shevet ha-Levi</i> addresses the problem of insects that are visible to the naked eye but which, upon gross examination, can be seen only as specks or black dots and hence are not recognizable as insects. Since they cannot be recognized as insects by the naked eye due to lack of perceivable limbs, antennae or other discernible body parts, <i>Shevet ha-Levi</i> regards them as permissible.25<i class="footnote">See also R. Yecheskel Roth, <i>Emek ha-Teshuvah</i>, IV, no. 57, s.v. <i>ve-od</i>.</i>
  An obvious objection to that view is the ruling of Rema, <i>Yoreh De'ah</i> 84:6, with regard to fruit in which a "black dot" is evident. Rema rules that it is forbidden to eat such fruit without first removing the infested portion because the black dot "is the place from which the worm begins to develop … and is forbidden just as the worm itself [is forbidden]. The "black dot" is certainly not perceivable as an organism but it is nevertheless prohibited by Rema. <i>Shevet ha-Levi</i> counters that objection by asserting that the organisms or larvae described by Rema will eventually develop into macroorganisms and for that reason they are forbidden even in earlier stages of development. Thus the "black dots" described by Rema are distinguishable from microorganisms that will never become visibly discernible.26<i class="footnote">This is also the view of R. Moshe Sternbuch in a letter included by R. Yisra’el Dov Halberstam in his <i>Kuntres Ve-Anokhi Tola’at</i>, appended to his anonymously published monograph <i>Yad le-Mikveh</i> (n.p., 5754), p. 20.</i> Basing himself upon <i>Teshuvot ha-Rashba</i>, no. 275, <i>Shevet ha-Levi</i> also acknowledges that, if the organism can be perceived, or will at any time be perceived, as propelling itself, such locomotion, in and of itself, serves to establish its identity as a forbidden creature.
  However, R. Moshe Viya, <i>Bedikat ha-Mazon ke-Halakhah</i>, I (Jerusalem, 5758) 2:4, reports that the late R. Shlomoh Zalman Auerbach ruled that organisms that are visible but which are not perceivable as "creeping things" are forbidden. A somewhat different version of Rabbi Auerbach's position is reported by R. Joshua Neuwirth, <i>Shemirat Shabbat ke-Hilkhatah</i>, I, 2nd edition (Jerusalem, 5739) 3:37, note 105. Rabbi Neuwirth relates that Rabbi Auerbach originally opined that an organism that can never be perceived as living or mobile cannot be regarded as a "creeping thing that creeps upon the earth" (Leviticus 11:41). However, Rabbi Auerbach later recounted that he heard from persons close to <i>Hazon Ish</i> that the latter considered such organisms to be forbidden. In point of fact, <i>Hazon Ish, Yoreh De'ah</i> 14:6, s.v. <i>ve-shi'ur</i>, writes explicitly: "[Even] if the eye does not recognize it because of its small size, if it is yet whole it does not become nullified by virtue of rabbinic decree because of the law of <i>biryah</i>."27<i class="footnote">Cf., the rather cryptic and tentative statement of R. Moshe Feinstein, <i>Iggerot Mosheh, Yoreh De’ah</i>, IV, no. 2, “<i>she-efshar she-davar she-lo nireh le-ma’aseh le-einayim eino assur</i>—it is possible that something that cannot actually be seen by the eye is not forbidden,” that continues to declare “<i>u-le-khol ha-paḥot eino be-ḥashivut biryah</i>—and at the very minimum does not have the status of a <i>biryah</i>.” See <i>infra</i>, notes 50 and 51, and accompanying text.</i> <i>Hazon Ish</i> clearly maintains that even organisms that cannot be identified as insects are forbidden.28<i class="footnote">As noted <i>supra</i>, note 16, <i>Ḥazon Ish</i> does indeed endorse the view that microscopic organisms that could not have been visually perceived were not banned at Mount Sinai. Accordingly, he as well as the other authorities who rule that even creatures that cannot be readily identified as insects are prohibited, must be understood as maintaining that visual perception of such creatures as “black dots” creates a <i>safek</i>, or doubt, and as such are subject to the rules governing doubtful situations. Moreover, since the “doubt” is perceived and can now readily be resolved by means of magnification, this “doubt” can no longer be willfully ignored. That consideration is sufficient to prohibit ingesting such creatures. See R. Chaim Chizkiyahu Medini, <i>Sedei Ḥemed</i>, V, <i>Kellalim, ma’arekhet ha-samakh, klal</i> 51.</i> R. Joseph Shalom Eliashiv is also quoted in <i>Bedikat ha-Mazon ke-Halakhah</i>, I, 2:4, note 4, as ruling that all such creatures are forbidden even if they cannot immediately be identified as insects.29<i class="footnote"><i>Arukh ha-Shulḥan, Yoreh De’ah</i> 84:36, rules that insects are forbidden even if they are visible only when scrutinized in direct sunlight. R. Shlomoh Zalman Auerbach, <i>Halikhot Sadeh</i>, no. 51, indicates that this is the case even if the insects can be seen only by employing an electric light.</i>
  In point of fact this issue has no bearing upon the problem of New York City water for the simple reason that many of the copepods in question are clearly recognizable as insects.30<i class="footnote">See <i>Kashrus Magazine</i>, July, 2004, p. 376. See also R. Ephraim Zalman Margulies, <i>Teshuvot Bet Efrayim ha-Ḥadashot</i>, no. 62, cited in <i>Darkei Teshuvah</i> 84:28, who suggests that minuscule creatures having the appearance of “thin, moving kernels of wheat” are permissible because, in reality, they are not insects but nascent fishlings. There is no hint in that source that inability visually to identify those creatures as insects should in any way be a consideration. Cf., also, R. Chaim Halberstam, <i>Teshuvot Divrei Ḥayyim</i>, II, <i>Yoreh De’ah</i>, no. 54, who rejects the position of <i>Bet Efrayim</i>.</i>
  III. Permitted Insects
  A. Insects Spawned in Stagnant Water
  As formulated in Leviticus 11:10, the prohibition against forbidden aquatic creatures reads: "And all that do not have fins and scales in the seas and in the rivers, of all that creep in the waters and of all living creatures that are in the water, they are detestable unto you." The Gemara, <i>Hullin</i> 66b, declares that the requirement that aquatic creatures have fins and scales as a condition of dietary acceptability is limited to those that spawn in "the seas and in the rivers" and hence creatures that spawn in stagnant pits or utensils are permissible. The Gemara, <i>Hullin</i> 67a, qualifies that exclusion in indicating that creatures that spawn in utensils are permissible only so long as they remain in their aquatic element. Upon emerging, or upon being removed, from water they acquire the forbidden status of "creeping things that creep upon the earth" (Leviticus 11:29) and remain prohibited even if they return to the water from which they emerged. Thus, insects that spawn in stagnant water collected in stationary cisterns or the like are not forbidden unless they become separated from the water that is their natural habitat.
  Although the status of reservoirs themselves in this regard is a matter of some controversy,31<i class="footnote">See, for example, <i>Darkei Teshuvah</i> 84:26-27. For the status of the water remaining in a reservoir that overflows its banks see <i>Darkei Teshuvah</i> 84:6.Most significantly, since the water flows out of the reservoir into the distribution system, the reservoir cannot be deemed to be a pit of stagnant water. See Ra’avad, <i>Sifri</i>, <i>Parashat Shemini</i> 3:2; <i>Teshuvot Ḥatam Sofer</i>, <i>Even ha-Ezer</i>, II, no. 30; and <i>Darkei Teshuvah</i> 84:6. There appears to be no distinction in this regard between a situation in which the water flows naturally into a canal and a situation in which the flow is controlled by man, i.e., by means of outflow valves, as in the case of a reservoir. Thus the statement published in the <i>Yated Ne’eman, supra</i>, note 5, indicating that the reservoirs in question have the status of rivers is entirely correct. In that ruling, Rabbi Feinstein, in accordance with the view cited, <i>infra</i>, note 33, states that the outflow system is sufficient to guarantee that status and adds that, since the reservoir is fed by flowing water, it must be regarded as having both an inflow and an outflow and hence must be deemed a “river” according to all authorities even though the water appears to be stationary. Cf., <i>Sha’ashu’ei Oraita</i>, p. 156.</i> bugs found in water that is collected in a pit or the like, but in situations in which the water originates in a river or stream, cannot <i>ipso facto</i> be presumed to be permissible. The status of such organisms is perforce the subject of doubt: Did the insects originate in the free-flowing source and hence they are forbidden or did they spawn in stagnant water subsequent to entering the reservoir? <i>Darkei Teshuvah</i> 84:5 rules that such insects are forbidden because of their doubtful status.32<i class="footnote">A publication of the New York City Department of Environmental Protection, “Copepods in New York City Drinking Water” (July, 2004), p. 3, states that copepods “are found almost everywhere where water is available” and that “various species can be found in all kinds of freshwater habitats from small creeks to glacial lakes.” Cyclops and other copepods that are predominant in New York water are limnetic, i.e., they live in open areas of still bodies of water. These species are not endemic to fast moving mountain streams or rivers and hence must breed in the reservoirs or at some point beyond the reservoir in the distribution system. See “Copepods in NYC Tap Water,” p. 9.</i>
  Moreover, as recorded by Rambam, <i>Hilkhot Ma'akhalot Assurot</i> 2:18, and <i>Shulḥan Arukh, Yoreh De'ah</i> 84:2, this exclusion from the prohibition is limited to creatures that spawn in stationary bodies of water. According to those authorities, creatures that spawn in water that flows from a stationary collection of water are forbidden because all flowing water is encompassed within the category of "rivers."33<i class="footnote">See <i>Shakh, Yoreh De’ah</i> 84:8; <i>Pri Megadim, Yoreh De’ah, Siftei Da’at</i> 84:8; and <i>Teshuvot Ḥatam Sofer</i>, <i>Even ha-Ezer</i>, II, no. 30. Cf., however, <i>Tosafot, Ḥullin</i> 67a, s.v. <i>ba-mayim</i>; Rosh, <i>Ḥullin</i> 3:68; and <i>Teshuvot Mishkenot Ya’akov, Yoreh De’ah</i>, no. 27, who rule that such insects are permitted.Rabbi Raitport’s major argument for permitting Brooklyn water in which such insects are found is that Brooklyn water is derived from the Catskill/Delaware System which receives its water from reservoirs in the Catskill Mountains. Rabbi Raitport received a letter from the Assistant Director of the New York State Bureau of Water Supply Protection informing him that those reservoirs are “recharged by rainfall and snowmelt from the Catskill Mountain” area but have “no river intakes.” See <i>Teshuvah Halakhah le-Ma’aseh</i>, p. 37. Rabbi Raitport develops the novel thesis that, for the insects to be prohibited, the stagnant water must not only flow out but water must also flow into the stagnant area. See <i>ibid</i>., pp. 18-19. That thesis, however, is contradicted by numerous sources, including Ra’avad, <i>Sifri, Parashat Shemini</i> 3:2; <i>Teshuvot Ḥatam Sofer</i>, <i>Even ha-Ezer</i>, II, no. 30; as well as <i>Darkei Teshuvah</i> 84:6.Moreover, the empirical premise upon which this argument is based is incorrect. The reservoirs in question are indeed “recharged by rainfall and snowmelt.” However, the rainwater and melting snow are first absorbed by the ground and form aquifiers which, in turn, form springs that flow into the reservoirs. See “Copepods in NYC Tap Water,” p. 15. Such springs clearly do not have the status of rainwater. See R. Menachem Mendel Schneersohn, <i>Teshuvot Ẓemaḥ Ẓedek, Yoreh De’ah</i>, no. 176 and R. Naphtali Zevi Judah Berlin, <i>Teshuvot Meshiv Davar</i>, II, no. 41.</i>
  To be sure, as indicated by <i>Darkei Teshuvah</i> 84:30, bugs found in water flowing from a cistern through a pipe are indeed permissible since such insects have not emerged from a utensil. Accordingly, insects that spawn in the pipes that form the distribution system would appear to be permissible. However, such insects may be permissible only if the pipe is not attached to the ground. <i>Darkei Teshuvah</i>, <i>loc. cit.</i>, suggests that a pipe attached to the ground loses its status as a utensil and becomes an integral part of the earth to which it is attached or in which it is implanted. Accordingly, insects that spawn in such pipes, or in an aqueduct, are in the category, not of "creeping things that creep in the water," but of the entirely distinct category of "creeping things that creep on the ground." Creatures that do not spawn in flowing water or in utensils are creatures that "creep on the ground" and hence are forbidden despite the fact that they spawn in a liquid medium.
  Furthermore, and perhaps most significantly, upon recording the permissibility of ingesting insects found in stagnant water, <i>Shulḥan Arukh, Yoreh De'ah</i> 84:1, carefully adds, "therefore one may <i>bend</i> and drink from them." Rema appends the explicit comment that it is forbidden to draw water in a utensil in order to drink the water drawn in such manner. As explained by <i>Shakh</i> and <i>Taz, ad locum</i>, when a utensil is used for drawing water there is a distinct possibility that the insect may attach itself to the wall of the utensil. Insects that spawn in utensils are permissible only so long as they do not emerge from their aquatic habitat. Upon attaching themselves to the wall of the utensil (for some authorities only if the insect precedes the flow of water but for other authorities even if the insect rises from the water in the utensil to the dry portion of the utensil) the insect becomes prohibited and remains so even upon its return to an aquatic environment.
  <i>Kisei Eliyahu, Yoreh De'ah</i>, 84:1, cited by <i>Darkei Teshuvah</i> 84:9, suggests that, in actuality, the problem associated with water drawn in a utensil is a matter of practical concern only with regard to relatively large insects. <i>Kisei Eliyahu</i> suggests that extremely small, barely perceivable insects, once they have emerged from the water, are not capable of returning. Nevertheless <i>Kisei Eliyahu</i> urges that one not drink water infested even with minuscule organisms without prior straining.
  In contradistinction to the understanding of <i>Shakh</i> and <i>Taz, Darkei Teshuvah</i> 84:13 cites <i>Pri Tohar</i> 84:1 who, basing himself upon <i>Issur ve-Hetter</i>, rules that it is permissible only to bend and drink water but that otherwise the insects, once they are removed from their original habitat even if only by means of a utensil, are forbidden. Thus, according to <i>Darkei Teshuvah</i>, if insects are present in the water, they become forbidden immediately upon being drawn into a glass or a pot.
  B. Insects Produced by Spontaneous Generation
  <i>Hazon Ish, Yoreh De'ah</i> 14:10, advances a further consideration that would serve effectively to prohibit imbibing any organism that spawns in reservoirs. <i>Hazon Ish</i> assumes that the exclusion of organisms spawned in stagnant water from the prohibition concerning "creeping things" is based upon the rationale that they are not the product of sexual reproduction but are generated by the "hylic power" of the water. Otherwise, contends <i>Hazon Ish</i>, they should be forbidden, if not by virtue of their own identity, then by virtue of the fact that they are the "<i>yozei</i>" or emission of a non-kosher entity and the applicable principle is that any food product that "emerges" from a non-kosher organism is, <i>ipso facto</i>, non-kosher regardless of its own characteristics.34<i class="footnote">That argument has no validity if the highly novel thesis of the <i>Brisker Rav</i>, R. Yitzchak Ze’ev Soloveichik, is accepted. The <i>Brisker Rav</i> observes that, insofar as other members of the animal kingdom are concerned, the presence or absence of the criteria recorded in Scripture as distinguishing kosher species from non-kosher species is irrelevant with regard to determining the status of any individual contemporary animal. That point is evidenced by the rule that pronounces a pig born of a sheep to be a kosher animal while a lamb born of a pig to be non-kosher. See <i>Bekhorot</i> 5b. The principle of <i>yoẓei</i> defines membership in a species on the basis of the identity of the progenitor, or of the progenitors, of the animal in question. The anatomical criteria spelled out in Scripture are significant only in establishing the identity of the early progenitors of present-day animals. Not so with regard to “creeping things that creep on the ground,” asserts the <i>Brisker Rav</i>. In that context, he observes, Scripture fails to provide anatomical criteria and omits the phrase “according to their species.” Hence the <i>kashrut</i> of those creatures is determined by classification of each individual member of those species on the basis of its appearance (<i>to’ar</i>) as a “creeping thing.” The <i>Brisker Rav</i> makes the same assertion with regard to “creeping things that creep on the ground.” In support of that position he cites Rambam’s formulation of the prohibition in <i>Hilkhot Ma’akhalot Assurot</i> 2:23 and his ruling in <i>Hilkhot Ma’akhalot Assurot</i> 3:8 as well as the midrashic statement indicating that it was necessary for God to demonstrate a “creeping thing” to Moses in order for him properly to apprehend the characteristics of the prohibited species. This principle also seems to be reflected in Rambam, <i>Hilkhot Ma’akhalot Assurot</i> 2:12. See <i>Ḥiddushei ha-Griz al ha-Shas</i>, ed. Y. Cohen (Jerusalem, 5729), p. 259. Cf., the terminology employed by Rambam, <i>Guide of the Perplexed</i>, III, chap. 48, which does not necessarily contradict this thesis. For a fuller discussion see this writer’s <i>Be-Netivot ha-Halakhah</i>, I (New York, 5756), 132-133. Since “creeping things” do not acquire their status from their progenitors but are classified as kosher or non-kosher solely on the basis of their individual criteria, it should logically follow that organisms spawned in stagnant water are always permitted since 1) they are excluded from the requirement of fins and scales and 2) since there is no reference to identification “according to their species,”the identity of their progenitors is irrelevant.</i> By way of comparison, <i>Hazon Ish</i> points out that a fish hatched in stagnant water from an egg laid by a non-kosher mother is certainly non-kosher.35<i class="footnote">See the novel thesis advanced by the <i>Brisker Rav</i> cited <i>supra</i>, note 34. The <i>Brisker Rav</i> would certainly concede <i>Ḥazon Ish</i>’s halakhic conclusion since <i>Ḥazon Ish</i> explicitly limits his thesis to species having the appearance of “creeping things” but not to fish whose status Scripture predicates upon the criteria of fins and scales. See also <i>Teshuvot Ḥatam Sofer, Yoreh De’ah</i>, no. 75 and <i>Be-Netivot ha-Halakhah</i>, p. 132. According to the <i>Brisker Rav</i>, the argument would be that the exclusion from the requirement for fins and scales is intrinsically limited to organisms that spawn in stagnant water and does not extend to organisms generated by sexual reproduction.</i>
  <i>Hazon Ish</i>'s position has implications beyond his actual statement. <i>Hazon Ish</i> appears to address only the issue of the status of an organism produced by progenitors who were themselves spawned in naturally flowing bodies of water. However, if his underlying position is taken literally, <i>viz.</i>, that only aquatic organisms arising in stagnant water <i>sui generis</i> are exempt from the requirement of fins and scales, organisms that may be so categorized may well constitute a null class.
  The Gemara, <i>Shabbat</i> 107b, declares that <i>kinim</i>, generally identified as lice, may be exterminated on <i>Shabbat</i> with impunity since they do not reproduce sexually. The implication is that lice are the product of spontaneous generation.36<i class="footnote">See also Rambam, <i>Hilkhot Ma’akhalot Assurot</i> 2:17 and <i>idem, Commentary on the Mishnah, Shabbat</i> 126b, as well as <i>idem, Sefer ha-Miẓvot, lo ta’aseh</i>, no. 179. In his <i>Sefer ha-Miẓvot</i> Rambam describes those who deny spontaneous generation as “fools who have no knowledge of science.”</i> However, since the time of Pasteur, science has rejected the notion of spontaneous generation and insists that no living organisms are reproduced asexually. That perception obviously has implications with regard to the halakhic status of <i>kinim</i>.37<i class="footnote">For a survey of the issues surrounding <i>kinim</i> see R. Nosson Slifkin, <i>Mysterious Creatures</i> (Jerusalem, 2003), pp. 191-204.In a non-halakhic context R. Avraham ben ha-Rambam writes: “The great stature and quality of the sages of the Talmud in perfection of their talent in explication of the Torah and its minutiae … does not necessitate that we argue on their behalf and substantiate their opinion in all of their dicta regarding medical care, natural science and astronomy.” See “Ma’amar al odot Derashot Ḥazal Le-R. Avaraham ben ha-Rambam Zal,” <i>Koveẓ Teshuvot ha-Rambam</i> (Leipzig, 5619), p. 41. That treatise was also published in <i>Kerem Ḥemed</i>, II (5676), 7-16 and in <i>Rabbeinu Avraham ben ha-Rambam: Milḥamot ha-Shem</i>, ed. R. Reuven Margolies (Jerusalem, 5713), pp. 81-98 and is also included in the prefatory material published in the Vilna edition of <i>Ein Ya’akov</i> (Vilna, 5637) and its reprints as well as in the introduction to the Vilna edtion of <i>Midrash Rabbah</i> (Vilna, 5603).The identical point was made much earlier by R. Sherira Ga’on with regard to the various medical remedies recorded in the Gemara. See <i>Oẓar ha-Geonim</i>, ed. M. B. Levine (Jerusalem, 5744), X, <i>Gittin</i> 68b, <i>Oẓar ha-Teshuvot</i>, no. 376. Cf., Abraham S. Abraham, <i>Lev Avraham</i>, II (Jerusalem, 5738), chap. 14, sec. 8 and the comment of R. Shlomoh Zalman Auerbach, <i>ibid.</i>, p. 19. Alternative explanations for the lack of efficacy of the medical remedies recommended by the Gemara are presented by <i>Tosafot, Mo’ed Katan</i> 11a, <i>Avodah Zarah</i> 24b and <i>Ḥullin</i> 47a; R. Shlomoh Luria, <i>Yam shel Shlomoh, Ḥullin</i> 8:12; R. Chaim Joseph David Azulai, <i>Brit Olam</i> (<i>commentary on Sefer Ḥasidim</i>), sec. 477; R. Israel Lipschutz, <i>Tiferet Yisra’el, Bo’az, Shabbat</i> 19:1; <i>Magen Avraham</i> 173:1 and 179:8; <i>Ḥazon Ish, Even ha-Ezer: Nashim</i> 27:3 and <i>Yoreh De’ah</i> 5:3; and R. Yair Chaim Bacharach, <i>Teshuvot Ḥavvot Ya’ir</i>, no. 234. Cf., the gloss of R. Akiva Eger to <i>Yoreh De’ah</i> 336:1 and <i>Kaf ha-Ḥayyim, Yoreh De’ah</i> 24:50.</i> Some scholars have taken the position that scientific reports based upon the clinical observations of researchers are unreliable and hence the Halakhah does not change.38<i class="footnote">See, for example, the opinion of R. Yehudah Brill, cited by R. Isaac Lampronti, <i>Paḥad Yizhak, erekh ẓeidah; Sefer ha-Brit</i>, I, chap. 14, note 8; and R. Jonathan Eibeschutz, <i>Kereti u-Peleti, Peleti</i> 40:4. See also the comments of R. Chaim Joseph David Azulai, <i>Shem ha-Gedolim, Ma’arekhet Sefarim, ot heh</i>, sec. 82, who staunchly affirms the inerrancy of the Sages of the Talmud with regard to matters of science and attributes their knowledge to the influence of the Holy Spirit and revelation by the prophet Elijah. See also <i>Sefer Ha-Brit, Ma’amar Yod</i>, chap. 8; <i>Tosefet Yom Tov, Kila’im</i> 5:5, s.v. <i>u-me-atah</i> and R. Yecheskel Roth, <i>Emek ha-Teshuvah</i>, IV, no. 57.In another comment, Rabbi Brill suggests that the premise of the Sages is not contradicted by contemporary science. If my reading of that passage is correct, Rabbi Brill maintains that the original <i>kinim</i> were generated from “sweat” but now reproduce sexually. If so, he ascribes to the Sages the novel position that even sexually reproduced progeny of spontaneously-generated creatures are excluded from the <i>Shabbat</i> prohibition. On that analysis the Sages posited spontaneous generation only of primordial creatures at the time of creation.For statements in entirely different contexts rejecting scientific findings at variance with halakhic presumptions see <i>Teshuvot Rivash</i>, no. 447; R. Jacob Reischer, <i>Teshuvot Shevut Ya’akov</i>, III, no. 20; and <i>Arukh ha-Shulḥan, Even ha-Ezer</i> 13:30. See also <i>Teshuvot ha-Rashba</i>, I, no. 98. <i>Arukh ha-Shulḥan</i> adds the comment that “he who questions their words testifies with regard to himself that he does not believe in the Oral Law although he is ashamed to say this explicitly.” See also R. Chaim Oberlander, <i>Or Yisra’el</i>, Tevet 5757, pp. 210-214.</i> Alternatively, it is possible to acknowledge the reliability of the scientific evidence but to recognize that the categorical proposition that no organisms can arise <i>sui generis</i> is an inductive generalization which may admit of exceptions and hence may not be universally correct. It may also be maintained that presently extant lice are not identical with those categorized in this manner by the Gemara and that the <i>kinim</i> described by the Gemara no longer exist.39<i class="footnote">See R. Nissim Karelitz, <i>Ḥut Shani</i>, vol. I, chap. 15, sec. 1, p. 125; <i>Orḥot Shabbat</i>, I, 14:30; and <i>Teshuvot Shevet ha-Kehati</i>, III, no. 126. See also Herman Branover, “Torah and Science: Basic Principles,” <i>Encounter: Essays on Torah and Modern Life</i>, ed. H. Chaim Schimmel and Aryeh Carmel (Jerusalem, 1989), p. 239.Neither the thesis that there are exceptional species in which spontaneous generation is present nor the thesis that the <i>kinim</i> described by the Gemara are no longer extant resolves the broader problem posed by Rambam’s comments in <i>Hilkhot Ma’akhalot Assurot</i> 2:13:Rambam describes two dichotomous classes of “creeping things,” each subject to its own separate prohibition: those that procreate sexually and those generated by putrefaction. According to Rambam, <i>kinim</i> are not to be identified as members of an idiosyncratic and perhaps exotic species but are merely paradigmatic of the multitude of tiny organisms that do not reproduce sexually but arise out of putrid substances.</i>
  At least one scholar, R. Isaac Lampronti, <i>Paḥad Yizḥak, erekh zeidah</i>, accepts the scientific finding negating spontaneous generation and pursues it to its logical conclusion. <i>Paḥad Yizḥak</i> asserts that there is no specific Oral Law tradition with regard to <i>kinim</i> as a particular species. <i>Paḥad Yizḥak</i> insists that the principle announced by the Sages is entirely correct in theory but that, empirically, the class of organisms that reproduce spontaneously may be a null class. He points to another talmudic discussion recorded in the Gemara, <i>Pesaḥim</i> 94b, involving a scientific disagreement between the Sages and the "wise men of the nations of the world" in which the Gemara reports that R. Judah the Prince conceded that the "wise men" were correct and that the Sages had erred.40<i class="footnote">Regarding the nature of that retraction see also the comments of R. Yehudah Brill quoted by <i>Paḥad Yiẓḥak</i> and <i>Shitah Mekubbeẓet, Ketubot</i> 13b, as well as the discussion of R. Shimon Schwab, <i>Kol Torah</i>, Tishri 5757, pp. 217-218.</i> Accordingly, he concludes that, in light of present-day scientific knowledge, killing lice on <i>Shabbat</i> must be regarded as a capital offense.41<i class="footnote">See also <i>Torat Shabbat</i> 316:15 and R. Joseph Shalom Eliashiv, cited in <i>Orḥot Shabbat</i>, I, 14:30, note 47.</i>
  If any one of these methods of reconciling the Gemara's position with regard to <i>kinim</i> and scientific theory is accepted and is coupled with the view expressed by <i>Hazon Ish</i> regarding the rationale underlying the exclusion from the requirement of fins and scales of organisms spawned in stagnant water, it follows that the provision of law formulated in <i>Hullin</i> 66b permitting such insects is a dead letter and that no such organism is permitted.
  This writer has long contended that the halakhic principle enunciated by the Gemara with regard to lice is not intended as a literal statement with regard to reproduction. Rather, it is but another manifestation of the principle that Halakhah is predicated upon gross phenomena. As noted earlier, halakhic authorities have uniformly recognized that dietary restrictions do not apply to the imbibing of subvisual creatures. It is similarly arguable that references to regulations applicable solely to species that engage in sexual reproduction are intended as references to <i>perceivable</i> sexual reproduction42<i class="footnote">See R. Chaim Kanievsky, <i>Derekh Emunah</i>, IV, <i>Hilkhot Shmittah ve-Yovel</i> 4:1, <i>Bi’ur ha-Halakhah</i>, s.v. <i>bein min ha-asabin</i>, who employs the same principle also with regard to the Gemara’s categorization of mushrooms as drawing nutrients from the air and hence not classified as vegetables and to Rambam’s reference to plants that appear during the <i>shmittah</i> year “of their own accord.” In each case, the classification, contends Rabbi Kanievsky, is predicated upon human perception rather than upon objective reality. See also Leo Levi, <i>The Science of Torah: The Scientific Knowledge of the Talmudic Sages</i> (New York, 2004), pp. 49-50.</i>; hence sexual reproduction that is not perceived as such43<i class="footnote">The Gemara, <i>Shabbat</i> 107b, questions the statement that <i>kinim</i> do not reproduce sexually on the basis of the dictum “The Holy One, blessed be He, sits and sustains [all creatures] from the horns of the eggs of <i>kinim</i>” or, according to Rashi, “tiny <i>kinim</i> as they emerge from the eggs.” Ostensibly, the Gemara finds evidence that <i>kinim</i> reproduce sexually in the employment of the term “eggs of <i>kinim</i>” in the dictum. If so, it might be argued that, if the sexual nature of reproduction was acknowledged by the Gemara in its original contention but purported to be subvisual, the term “eggs of <i>kinim</i>” might be understood as referring to microscopic eggs and should not have been regarded as anomalous.It may be countered that, even if the thesis of spontaneous generation is understood literally, there is no reason to presume that <i>kinim</i> arise spontaneously as mature creatures. Certainly, divine providence would perforce necessarily extend even to spontaneously generated <i>kinim</i>. If so, God’s providence would indeed be necessary regardless of whether or not eggs of <i>kinim</i> are visually perceivable and regardless of whether or not they can be perceived as reproducing sexually. How then, does the cited dictum negate the assertion that <i>kinim</i> are the product of spontaneous generation?The Gemara’s objection is indeed based upon employment of the phrase “eggs of <i>kinim</i>” but the objection need not be understood as based upon the literal description of <i>kinim</i> as arising from eggs and hence the conclusion that they are sexually reproduced; rather, the objection is that the dictum indicates that divine providence is perceived by man as being exercised over development of those minuscule eggs, thereby indicating that the reproductive process is entirely usual, i.e., sexual in nature and perceived as such.</i> is treated by Halakhah as if it occurred spontaneously.44<i class="footnote">If this analysis is correct, the <i>kinim</i> described by the Gemara cannot be identical to present-day head lice since eggs of the latter species are clearly visible to the naked eye.</i> Put somewhat differently, Halakhah is not based upon ontological reality but upon phenomenological perception.45<i class="footnote">Cf., the comments attributed to R. Shlomoh Zalman Auerbach in <i>Halikhot Sadeh</i>, no. 51 and by R. Joshua Neuwirth, <i>Shemirat Shabbat ke-Hilkhatah</i> 3:105 as well as the comments attributed to R. Eliyahu Dessler, <i>Mikhtav me-Eliyahu</i>, ed. Aryeh Carmell, IV (Jerusalem, 5743), 355-356, note 4 and R. Shimon Schwab, <i>Kol Torah</i>, p. 217. See also R. Pesach Eliyahu Falk, <i>Madrikh le-Bedikat Tola’im: Guide to the Inspection of Fruits and Vegetables for Insects</i> (Gateshead, 5744), p. 92.</i>
  R. Meir Simchah ha-Kohen of Dvinsk, <i>Meshekh Hokhmah, Parashat Bo</i>, cites R. Elijah of Vilna who maintains that the reasons or considerations advanced by the Sages in explanation of the various ordinances that they promulgated are not exhaustive in nature; rather, for reasons best known to themselves, they frequently reserved other considerations <i>in pectore</i>. In actuality, that position was formulated much earlier during the Ge'onic period in a responsum of Rav Ha'i Ga'on as recorded in <i>Teshuvot ha-Ge'onim</i>, ed. <i>Mekizei Nirdamim</i> (Lych 5683-5684), no. 1.
  In a comment published in R. Eliyahu Dessler's <i>Mikhtav me-Eliyahu</i>, IV, 355, note 4, the editor reports that Rabbi Dessler resolved the problem concerning <i>kinim</i> in a similar fashion: The halakhic ruling was transmitted to the Sages as a tradition received at Sinai and, although they endeavored to present a reasoned basis for that tradition, the rationale they advanced was neither exhaustive nor necessarily accurate. In his words:
  The explanation does not determine the law. Rather the opposite: the law determines the explanation and the reason mentioned in the Gemara is not the sole possible reason with regard to the matter and if, at times, they presented reasons that are in accordance with the knowledge of nature of their day it is our obligation to seek other explanations in accordance with the knowledge of nature of our day by means of which the law will remain steadfast in its place.
  … But even if a proper reason is not found, we believe with perfect faith that the law is a true law and we hope to God that He will enlighten our eyes to find an appropriate explanation.46<i class="footnote">See also R. Moshe Schick, <i>Teshuvot Maharam Shik, Yoreh De’ah</i>, no. 244 and R. Chaim Oberlander, <i>Or Yisra’el</i>, Tevet 5757, pp. 212-214. Cf., also, <i>Ḥazon Ish, Yoreh De’ah</i> 5:3 and R. Moshe Feinstein, <i>Iggerot Mosheh, Even ha-Ezer</i>, II, no. 3, <i>anaf</i> 2.</i>
  If either of these theses is correct—or if the stance that science is simply wrong is adopted—the Halakhah with regard to exterminating <i>kinim</i> on <i>Shabbat</i> as well as the Halakhah regarding creatures spawned in stagnant water remains unchanged.
  IV. Bittul (Nullification)
  Infested water might nevertheless be permissible were the principle of <i>bittul</i> applicable, i.e., nullification by means of adulteration in a permitted substance sixty times as great. However, despite the minuscule size of the copepods that may be present in tap water, they do not become nullified by virtue of the relatively copious quantity of water in which they are found. The most obvious consideration rendering the principle of nullification inoperative is the fact that even a tiny insect is, by definition, a "creature" or <i>beriyah</i> (but among talmudists commonly vocalized as "<i>biryah</i>" or "<i>beryah</i>") that is not subject to nullification.47<i class="footnote">R. Chaim Oberlander, <i>Or Yisra’el</i>, Tammuz 5764, pp. 188-189, notes that if the copepods in question spawn in stagnant water, and hence become forbidden only upon emerging from their natural aquatic habitat, they may not have the status of a <i>biryah</i>. Many early-day authorities maintain that a <i>biryah</i> that is not subject to nullification is, by definition, an entity that is prohibited as food from the moment of its creation. See <i>Encylopedia Talmudit</i>, IV, 209. R. Jonathan Eibeschutz, <i>Kereti u-Peleti, Peleti</i> 100:4, rules that, since insects indigenous to fruit become prohibited only upon emerging from the fruit in which they develop, they do not have the status of a <i>biryah</i> and may become nullified. See also R. Shlomoh Kluger, <i>Teshuvot Tuv Ta’am va-Da’at, Mahadura Telita’a</i>, I, no. 160. Nevertheless, <i>Kereti</i> cautions that this ruling should not be relied upon in practice. A similar view was earlier formulated and rejected by R. Chaim Joseph David Azulai, <i>Shiyurei Berakhah, Yoreh De’ah</i> 100:4. See also <i>idem, Maḥazik Berakhah, Yoreh De’ah</i> 84:10. See also R. Jacob Meshulam Orenstein, <i>Yeshu’ot Ya’akov, Yoreh De’ah</i> 84:1; <i>Teshuvot Tuv Ta’am va-Da’at, Mahadura Telita’a</i>, I, no. 162; <i>Teshuvot Bet David, Yoreh De’ah</i>, no. 22; and R. Shlomoh Drimer, <i>Teshuvot Bet Shlomoh, Yoreh De’ah</i>, no. 155. Cf., the gloss authored by R. Betzalel ha-Kohen of Vilna cited by <i>Matteh Yehonatan, Yoreh De’ah</i> 100:1. See also R. Jacob of Lissa, <i>Ḥavvat Da’at</i> 100:5 and <i>Pri Megadim, Siftei Da’at</i> 84:31, who reject the position of <i>Kereti</i>. See also R. Ezekiel Landau, <i>Dagul me-Revavah, Tinyana, Yoreh De’ah</i> 84, and R. Raphael Joseph Hazan, <i>Teshuvot Ḥikrei Lev, Yoreh De’ah</i>, I, no. 63, who demonstrate that <i>Kereti</i>’s position is contradicted by the comments of Rabbenu Nissim, <i>Ḥullin</i> 58b. See also <i>Teshuvot Kol Eliyahu, Yoreh De’ah</i>, no. 18; <i>Erekh ha-Shulḥan, Yoreh De’ah</i> 84:2; and <i>Teshuvot Afarkasta de-Anya</i>, no. 20, as well as sources cited by <i>Darkei Teshuvah, Yoreh De’ah</i> 100:4 and <i>Yeḥaveh Da’at</i>, VI, no. 47.It should be noted that <i>Arukh ha-Shulḥan, Yoreh De’ah</i> 100:17, declares that repulsive creatures (<i>devarim ma’usim</i>) such as “flies and ants” are nullified when mixed with foods sixty times greater in quantity even if a whole insect is present in the mixture. However, Rabbi Gross, <i>Or Yisra’el</i>, p. 198, correctly points out that <i>Shulḥan Arukh, Yoreh De’ah</i> 100:1, declares, “A <i>biryah</i>, that is, an ant etc., is not nullified even [in food] a thousand times [as great].” That statement is in direct contradiction to the ruling of <i>Arukh ha-Shulḥan</i>.Nevertheless, it should further be noted that R. Abraham Benjamin Samuel Sofer<i>, Teshuvot Ketav Sofer, Yoreh De’ah</i>, no. 63, cites early-day authorities who maintain that a <i>biryah</i> is nullified when adulterated in 960 parts of a permissible foodstuff and rules that, although that opinion is not normative, all authorities would agree that a <i>biryah</i> that is itself disdainful (<i>pegumah</i>) becomes nullified in such a quantity of permissible food.</i> A second consideration rendering nullification inoperative is the fact that nullification occurs when the identity of the prohibited substance is lost by virtue of the fact that its identity becomes submerged in that of the overwhelmingly larger substance.48<i class="footnote">See the terminology employed by Rambam, <i>Hilkhot Ma’akhalot Assurot</i> 15:4.</i> Hence, in circumstances in which the forbidden substance remains recognizable despite adulteration, it has certainly not lost its identity and hence is not subject to nullification.49<i class="footnote">See <i>Tosafot, Ḥullin</i> 95a, s.v. <i>sefeko</i> and <i>Sukkah</i> 9b, s.v. <i>ha</i>; Rosh, <i>Ḥullin</i> 7:37; <i>Taz</i>, <i>Oraḥ Ḥayyim</i> 632:3 and <i>Yoreh De’ah</i> 104:1; <i>Pri Ḥadash, Yoreh De’ah</i> 104:3; and <i>Ḥokhmat Adam</i> 51:1.</i> R. Moshe Viya, <i>Bedikat ha-Mazon ke-Halakhah</i>, I (Jerusalem, 5758), p. 142, note 12, cites R. Joseph Shalom Eliashiv, R. Nissim Karelitz, R. Samuel ha-Levi Woszner and R. Ben-Zion Abba Shaul as maintaining that a forbidden substance that can be discerned only with "great effort" (<i>toraḥ gadol</i>) is not regarded as "perceivable" (<i>nikar</i>) and hence is subject to nullification provided that it is not a <i>biryah</i>.50<i class="footnote">See also <i>Bedikat ha-Mazon ke-Halakhah</i>, I, 7: introd., note 1. Cf., however, <i>Iggerot Mosheh, Yoreh De’ah</i>, IV, no. 2, who suggests that a creature that is not “readily perceivable” does not have the enhanced status of a <i>biryah</i>. See <i>supra</i>, note 27.</i> That point is also made by R. Shlomoh Zalman Auerbach in a responsum published in that volume, pp. 178-179.51<i class="footnote">In his contribution to <i>Halikhot Sadeh</i>, no. 51, mentioned earlier, R. Shlomoh Zalman Auerbach states that an extremely small insect, although biblically prohibited, is not deemed to be a <i>biryah</i> and hence is subject to nullification. A <i>biryah</i> is not nullified because of its significance, but such minuscule creatures, he contends, are insignificant. See <i>Rosh Yosef, Ḥullin</i> 96b. However, Rabbenu Nissim asserts that a <i>biryah</i> is not subject to nullification because its significance stems from the fact that consumption of a <i>biryah</i> is subject to statutory punishment despite its small size. Accordingly, concludes R. Schneur Zalman Revach, <i>Tola’at Shani</i> (Bet Uzi’el, 5767), I, no. 4, even the smallest prohibited insect cannot be nullified.</i>
  R. Chaim Oberlander seems to suggest that even if it is assumed that extremely tiny insects are subject to nullification because they can be recognized only with "great effort" they may nevertheless not be subject to nullification for an entirely different reason. Rema, <i>Yoreh De'ah</i> 98:4, rules that if a small quantity of milk falls into a large pot containing a meat product the milk is not automatically nullified. Rather, cold water must be poured into the pot. The cold water will cause the milk to rise so that it may be skimmed from the top. The reason nullification of the milk does not occur is that the food is a "<i>davar she-yesh lo matirin</i>," i.e., the principle of nullification applies only if the food cannot otherwise be eaten in a permissible manner. Since cold water will cause the milk to rise, the remaining food may be eaten in an entirely permissible manner. However, rules R. Jonathan Eibeschutz, <i>Kereti u-Peleti, Peleti</i> 98:61, in a situation in which food would become spoiled in the process of being mixed with cold water and thereby become inedible, the usual principle of nullification would apply. <i>Kereti</i>, however, apparently accepts the basic principle that prohibited substances that can be recognized only upon exertion of effort do become nullified.
  However, <i>Pri Megadim, Mishbezot Zahav</i> 98:7, advances an entirely different explanation for Rema's ruling. <i>Pri Megadim</i> explains that, in the circumstances described, the milk is regarded as "recognizable" since it can be rendered visually perceivable by adding cold water. In effect, <i>Pri Megadim</i> posits that even a merely potentially recognizable entity is considered to be "recognizable."52<i class="footnote">Rashba, <i>Torat ha-Bayit, bayit dalet, sha’ar dalet</i>, rules that a non-kosher pot that becomes mingled with a larger number of kosher pots becomes nullified. Even though the pot can be rendered permissible by means of a <i>kashering</i> process, that procedure is not required because of the expense and labor entailed in such a procedure. R. Aaron ha-Levi, <i>Bedek ha-Bayit, ad locum</i>, disagrees and rules that, since the prohibited substance absorbed in the walls of the utensil can be purged by <i>kashering</i>, that substance is regarded as recognizable and hence not subject to nullification. That dispute is mirrored in a controversy between <i>Shakh, Yoreh De’ah</i> 102:8 and <i>Ḥavvat Da’at, Yoreh De’ah</i> 102:3. A similar dispute exists between <i>Pri Ḥadash, Yoreh De’ah</i> 102:8, who regards nullification to be efficacious even when the taste of a forbidden substance might be detected by a professional taster, and <i>Shakh, Yoreh De’ah</i> 109:8 and <i>Pri Megadim, Yoreh De’ah, Siftei Da’at</i> 102:3, who reject that position.The most obvious analysis of the controversy between <i>Torat ha-Bayit</i> and <i>Bedek ha-Bayit</i> is that it centers upon the issue of whether potential separation and removability of a prohibited substance renders it “recognizable” for purposes of Halakhah. If so, the dispute between <i>Kereti u-Peleti</i> and <i>Pri Megadim</i> simply mirrors the earlier controversy of <i>Torat ha-Bayit</i> and <i>Bedek ha-Bayit</i>. See R. Shimon Shkop, <i>Sha’arei Yosher, sha’ar gimmel</i>, chap. 19. Cf., however, R. Chaim Oberlander, <i>Or Yisra’el</i>, p. 182.Rabbi Oberlander cites rulings in other areas of Halakhah that also seem to hinge upon the resolution of this issue. <i>Teshuvot Yad Eliyahu</i>, no. 88, discusses the case of a Torah scroll that is unfit for use because it contains an error that becomes mingled with other kosher Torah scrolls and rules that the invalid scroll becomes nullified among the valid scrolls. <i>Teshuvot Ḥatam Sofer, Yoreh De’ah</i>, no. 277, disagrees and asserts that, since by careful examination of the full text of each of the scrolls the invalid scroll may be identified, that scroll does not become nullified. Rabbi Oberlander suggests that the controversy between <i>Yad Eliyahu</i> and <i>Ḥatam Sofer</i> centers upon precisely this issue, i.e., whether potential identification of a defect, albeit only upon expenditure of significant effort, renders the defect presently “recognizable” for purposes of Halakhah. He also draws attention to the statement of <i>Arukh ha-Shulḥan, Yoreh De’ah</i> 100:7, to the effect that, were it not a <i>biryah</i>, a fertilized egg that could be recognized as such upon meticulous external examination, that becomes intermingled with a large quantity of unfertilized eggs, would become nullified despite the fact that the fertilized egg is potentially identifiable.</i>
  It is, however, not at all clear that <i>Kereti</i> accepts the notion that nullification occurs when the prohibited substance can be recognized only with "great effort." The situation involving milk falling into a pot of meat is readily distinguishable from instances in which small particles of a forbidden substance became mingled with a much greater quantity of a permitted foodstuff. In the former case the milk is presently not at all recognizable; unless and until cold water is added to the mixture it is impossible to detect the presence of milk. In the latter case, the exceedingly small forbidden entities can presently be detected, albeit only with great effort, upon meticulous sifting and scrutiny. The milk may be subject to nullification since it is presently entirely undetectable whereas the minute but detectable particles may not be subject to nullification.53<i class="footnote">See <i>Bedikat ha-Mazon ke-Halakhah</i>, I, 2: introd., note 1.</i> Indeed, R. Jonathan Eibeschutz himself, <i>Peleti</i> 84:19, rules that a forbidden substance, when recognizable, is not subject to nullification even if it can be recognized only upon exertion of great effort. That is also the view of <i>Pri Megadim, Yoreh De'ah, Siftei Da'at</i> 84:35 and <i>Hazon Ish, Yoreh De'ah</i> 14:6.54<i class="footnote">See also <i>Ḥokhmat Adam, Binat Adam, sha’ar rov ve-ḥazakah</i>, no. 9.</i>
  However, R. Menachem Mendel Schneersohn, <i>Teshuvot Ẓemaḥ Ẓedek, Yoreh De'ah</i>, no. 70, asserts that a forbidden substance that can be recognized only upon "great effort" does become nullified. That is apparently also the position of <i>Arukh ha-Shulḥan, Yoreh De'ah</i> 100:7. Rabbi Viya, <i>Bedikat ha-Mazon ke-Hilkhatah</i>, I, 7: introd., note 1, states that a similar conclusion can be inferred from the discussions of <i>Teshuvot ha-Rashba</i>, I, no. 259 and R. Chaim Halberstam, <i>Teshuvot Divrei Hayyim</i>, II, <i>Yoreh De'ah</i>, no. 54.
  Even more akin to the case at hand is the ruling of <i>Issur ve-Hetter</i> 32:9, cited by <i>Darkei Mosheh, Yoreh De'ah</i> 104:1, to the effect that a prohibited substance that falls into a liquid but which can be removed by straining is not subject to nullification. Copepods can readily be eliminated from the tap water by means of filtration.
  V. Boiled Water and Cooked Food
  There are, however, circumstances in which <i>bittul</i>, or nullification, may be applicable. With regard to fruit that requires examination for possible insect infestation, <i>Shulḥan Arukh, Yoreh De'ah</i> 84:9, rules that, if the unexamined fruit has been cooked in a manner that renders subsequent examination impossible, <i>post factum</i>, the cooked fruit is permissible. As explained by <i>Shakh, Yoreh De'ah</i> 84:29, that ruling is predicated upon the principle of <i>sefek sefeika</i> or "double doubt," i.e. first, the presence of an insect is doubtful and secondly, even if present, it is possible that the insect has been crushed and hence has become nullified in the larger quantity of permissible foodstuff. Citing earlier sources, <i>Shakh</i> concludes that, in a locale in which infestation by insects has been established "<i>huḥzeku</i>," the presence of insects in the fruit must be regarded as tantamount to a certainty and hence the food is forbidden.55<i class="footnote">See <i>Kereti u-Peleti, Kereti</i> 84:16 and <i>Pri Ḥadash, Yoreh De’ah</i> 84:31. Cf., however, <i>Ẓemaḥ Ẓedek, Piskei Dinim, Yoreh De’ah</i> 84:9, who asserts that, even according to <i>Shakh</i>, the food is prohibited only if insects had been found in the fruit prior to cooking. For further discussions of the status of cooked foods that may be insect-infested see <i>Darkei Teshuvah</i> 84:131.</i>
  However, <i>Taz, Yoreh De'ah</i> 84:17, disagrees with <i>Shakh</i> in maintaining that, upon cooking, the foodstuff is permissible even if it is known that such fruit is commonly infested. According to <i>Taz</i>, even if there is but a "remote possibility" (<i>zad raḥok</i>) that no insect was present, the food is permissible on the basis of the principle of <i>sefek sefeika</i>.56<i class="footnote"><i>Ḥokhmat Adam</i> 38:5 explains <i>Taz</i>’ position on the basis of the fact that the rule that a <i>biryah</i> cannot be nullified is the product of a rabbinic edict. Hence, in principle, if there is a question with regard to whether the prohibited organism is still a <i>biryah</i> or whether it has lost that status by virtue of having become crushed, the food may be regarded as permissible on the basis of the rule that rabbinic edicts do not extend to situations of doubt. That principle, explains <i>Ḥokhmat Adam</i>, does not apply in cases of <i>itḥazek issura</i>, i.e., in situations in which the object was known with certainty to have been forbidden and the sole question is whether its status has changed. Hence, explains <i>Ḥokhmat Adam, Taz</i> maintains that, so long as there exists even a “remote possibility” that there never was a forbidden insect in the fruit, the foodstuff was never known to have been forbidden (<i>itḥazek issura</i>) and hence, despite the fact that the majority of such fruit is infested, the cooked fruit is permitted because of the possibility that any organism that was present may have been crushed. Since it is certain that the insect represents but a tiny fraction of the entire quantity of cooked food, the resultant doubt is only with regard to whether the rabbinic edict canceling nullification in cases of <i>biryah</i> is applicable. Since the doubt is with regard to the applicability of a rabbinic prohibition, the food is permissible.</i>
  Rabbi Oberlander seeks to relate that controversy to the situation of unexamined water used for cooking. Applying the principle of <i>sefek sefeika</i>, he asserts that the cooked food is permissible. Even in situations in which the presence of copepods is widespread, the cooked food or boiled water, he argues, is permissible according to <i>Taz</i>. However, in the opinion of this writer that conclusion is not warranted. It is surely the case that insects typically found in fruit quite often become crushed or dismembered in the process of cooking; hence the status of such an organism in food that has been cooked is indeed a matter of doubt. There is, however, to the best of this writer's knowledge, no evidence that such crushing takes place when extremely minuscule, barely visible organisms (such as copepods) find their way into a pot. Such tiny creatures, it would seem, will remain intact unless a concerted effort is made finely to grind the food in question. Accordingly, insofar as use of boiled water for tea, coffee and the like is concerned, there is no reason to assume that any organism becomes crushed in the boiling process. Moreover, as noted by <i>Teshuvot Maharam Lublin</i>, no. 27, as well as by R. Samuel Landau, <i>Teshuvot Shivat Ẓion</i>, no. 28, and <i>Teshuvot Rav Pe'alim</i>, IV, <i>Yoreh De'ah</i>, no. 8, although in cooking, a process in which food is often rinsed by hand and stirred by means of a cooking utensil, there is a realistic likelihood that any organism present may become crushed, such crushing is not likely to occur when water is simply boiled in a kettle or a pot.
  Indeed, R. Yitzchak Bistritsky, <i>Or Yisra'el</i>, p. 203, reports that, after boiling water for three quarters of an hour, he found copepods that were completely intact.57<i class="footnote">Cf., however, Rabbi Gross, <i>Or Torah</i>, p. 198, who assumes that this organism loses its status as a <i>biryah</i> upon cooking. <i>Ḥazon Ish, Yoreh De’ah</i> 14:6, rules that an insect does not retain the status of a <i>biryah</i> if its shape becomes distorted in cooking. It is not clear that that phenomenon is likely to occur with an extremely tiny organism.</i> Hence, empirically, there is no <i>sefek sefeika</i> that would render such cooked food permissible. Moreover, as Rabbi Oberlander himself notes, tea or coffee can readily be strained even after it has been fully prepared. Obviously, although it is not permitted to cook forbidden objects in reliance upon nullification, coffee that has already been brewed by a method in which a filter is employed is permissible.
  VI. Use of Filters on Shabbat
  Use on <i>Shabbat</i> of a faucet to which a filter has been attached presents an entirely different problem. Removal of an insect from a food product or from a beverage on <i>Shabbat</i> presents a problem of <i>borer</i>, i.e., the prohibited "selection" or removal of a waste product or undesired object, e.g., from food to be eaten or from other objects to be used for some other purpose. Ordinarily, an insect might be removed together with a small quantity of the food or beverage in which it is found rather than in isolation. Since a portion of the perfectly desirable foodstuff is removed together with the disdained entity, the separation remains incomplete and hence permissible. However, a filter will remove only the insect as well as any other fragments of undesired dross. Hence the problem of <i>borer</i>, or impermissible separation, arises.
  The problem, however, exists only if there is a desire and need to achieve a separation. A person who does not suspect the presence of a forbidden organism and who would be perfectly willing to drink unfiltered water may open a faucet to which a filter has been attached. By the same token, there is no problem with using filtered water for washing one's hands on <i>Shabbat</i> since there is no reason to be concerned with the presence of insects in water used for that purpose. A similar problem is identified by R. Joshua Neuwirth, <i>Shemirat Shabbat ke-Hilkhatah</i>, I, 3:56, note 163, who reports that R. Shlomoh Zalman Auerbach expressed astonishment at the use of a filter on the first and last days of <i>Pesaḥ</i> by individuals concerned with regard to possible contamination of the water supply by <i>ḥamez</i>.
  Comprehensive discussions of the issues involved in straining water on <i>Shabbat</i> are presented by R. Chaim Oberlander and R. Joel ha-Levi Rosner in the Tishri 5765 issue of <i>Or Yisra'el</i>.58<i class="footnote">Rabbi Oberlander, <i>Or Yisra’el</i>, Tishri 5765, pp. 111-115, endeavors, <i>inter alia</i>, to demonstrate that: 1) under these circumstances the act of <i>borer</i> is only rabbinically proscribed; 2) since it is unlikely that copepods are immediately present in the first gush of water released when the faucet is opened, the act of <i>borer</i> is in the nature of <i>gerama</i>, i.e. opening the faucet is not the proximate cause of any resulting separation; and 3) <i>gerama</i> is permitted with regard to rabbinic prohibitions. Of course, each of those points merits careful analysis and substantiation. However, Rabbi Oberlander’s discussion is predicated upon a fundamentally flawed presumption. It is quite clear that it is extremely unlikely that copepods will be present in the first gush of water following attachment of a filter. However, if any such creatures are present in the water they will certainly be trapped by the filter when they do appear and will remain in close proximity to the filter at the time of every future use of the filter. Accordingly, if, for example, a filter is installed in an infested building early in the week, opening the faucet some days later on <i>Shabbat</i> will result in an immediate act of <i>borer</i>.In sharp contrast, Rabbi Rosner, <i>Or Yisra’el</i>, p. 126, cites the thesis propounded by <i>Ḥazon Ish, Oraḥ Ḥayyim: Mo’ed</i> 36:1, with regard to culpability for use of a water mill on <i>Shabbat</i>. <i>Ḥazon Ish</i> asserts that, with regard to activities proscribed on <i>Shabbat</i>, there is no distinction between proximate cause and <i>gerama</i> with regard to acts that are customarily performed by way of <i>gerama</i>. <i>Ḥazon Ish, ibid</i>., 156: <i>le-siman</i> 252, makes the same point with regard to agricultural irrigation, i.e., he regards opening a dam in order to facilitate the flow of water for the purpose of irrigating vegetables to be a biblically culpable act even in the absence of proximate cause and also adopts the same position with regard to setting electrical machines into operation on <i>Shabbat</i>. The paradigm upon which the principle is predicated is, according to <i>Ḥazon Ish</i>’s analysis, acts of trapping that are forbidden on <i>Shabbat</i>, i.e., a trap is set by the hunter but the animal is caught only later by means of its own act of triggering the trap. <i>Ḥazon Ish</i> reasons that proximate cause is lacking but that the hunter is nevertheless liable because such is the normal manner of carrying out the act of trapping. Rabbi Rosner applies the same principle to use of a filter: since “removing the impediment” to the free flow of water through a filter is the usual form of “separation” of foreign objects from the water, such an act, he contends, constitutes a biblical transgression according to <i>Ḥazon Ish</i>.</i>
  The issue of straining water on <i>Shabbat</i> for a comparable purpose was first addressed by <i>Kaf ha-Hayyim, Oraḥ Hayyim</i> 319:120. <i>Kaf ha-Hayyim</i> ruled that, in Jerusalem, water might be strained on <i>Shabbat</i>. Apparently, during the period in which that work was written, there was concern that insects might be present in the Jerusalem water supply. However, even if present, those insects were not forbidden aquatic creatures since they spawned in cisterns. The concern was solely due to the fact that consumption of insects is revolting to most people and hence is forbidden by virtue of the prohibition "You shall not make yourselves repulsive" (Leviticus 11:43). However, that prohibition is operative only if the presence of insects is known with certainty; if their presence is merely doubtful there is no revulsion attendant upon drinking the water. Basically, <i>Kaf ha-Hayyim</i>'s position is that, if straining is required as a matter of Halakhah, it is forbidden on <i>Shabbat</i> but, if straining is not normatively required, it is permissible. However, in a subsequent comment, <i>Kaf ha-Hayyim, Oraḥ Hayyim</i> 319: 223, cites earlier authorities who ruled that straining is indeed required. Nevertheless, <i>Kaf ha-Hayyim</i> permits straining on <i>Shabbat</i> with the stipulation that the straining be performed by two persons acting simultaneously and that the straining be performed in an unusual manner.
  R. Abraham Chaim Noe, <i>Kezot ha-Shulḥan</i>, IV, 125:14, <i>Badei ha-Shulḥan</i>, no. 37, cites <i>Hayyei Adam, Hilkhot Shabbat, Nishmat Adam</i> 16:5, who suggests that it may be permissible to strain insects that are not repulsive on <i>Shabbat</i>. Rabbi Noe asserts that, since the individual has no personal desire to effect the separation but rather the sole consideration motivating a person not to imbibe such insects is the prohibition attendant upon them, there is no prohibition of <i>borer</i>. The argument is that the prohibition is attendant only upon separation of something that is not desired from something that is desired. The insects in question are neither objectively nor subjectively undesirable for any mundane reason; they are eschewed only because of concern lest a halakhic prohibition be violated. Selection and separation on the basis of a consideration that is best described as metaphysical, he argues, does not entail a forbidden act of <i>borer</i>. However, <i>Nishmat Adam</i> regards that distinction as questionable.59<i class="footnote">Cf., however, <i>Teshuvot Maharam Shik</i>, <i>Oraḥ Ḥayyim</i>, no. 134, who regards such selection as involving a prohibited act of <i>borer</i>. Rabbi Joel ha-Levi Rosner, <i>Or Yisra’el</i>, p. 118, finds support for that position in the comments of Ramban, <i>Shabbat</i> 144b, who remarks that inability to partake of a particular food because of an ancillary Sabbath prohibition does not lead to a problem of <i>borer</i>. The implication of that statement is that an intrinsic prohibition attendant upon the foodstuff <i>qua</i> foodstuff does give rise to such a problem.</i>
  <i>Kezot ha-Shulḥan</i> observes that whether or not insects are present in the water is itself a matter of empirical doubt. Consequently, <i>Kezot ha-Shulḥan</i> concludes that, according to <i>Nishmat Adam</i>, straining on <i>Shabbat</i> would be permissible on the basis of the principle of <i>sefek sefeika</i> or "double doubt." The identical consideration would render filtering New York City water on <i>Shabbat</i> permissible since at any given time the presence of insects is doubtful. The "double doubt" lies in the combination of the doubt with regard to the fact, i.e., the presence of copepods, and the halakhic doubt regarding the permissibility of straining such copepods on <i>Shabbat</i>.
  <i>Kezot ha-Shulḥan</i>, however, draws an inference from a ruling of <i>Shulḥan Arukh ha-Rav, Oraḥ Hayyim</i> 319:23, that serves to establish that straining on <i>Shabbat</i> is forbidden even if the purpose is to remove insects that are not repulsive. On the basis of that inference, <i>Kezot ha-Shulḥan</i> rules contrary to the position of <i>Nishmat Adam</i>. The source for the ruling that a foodstuff not intrinsically disdained, but rejected only because of its nature as a halakhically prohibited substance, is not subject to the prohibition of <i>borer</i> is a statement of <i>Magen Avraham, Oraḥ Hayyim</i> 500:12, with regard to separation of prohibited fat (<i>ḥelev</i>) from meat on <i>Shabbat</i>.60<i class="footnote">See also <i>Pri Ḥadash</i>, <i>Oraḥ Ḥayyim</i> 500:6; <i>Shulḥan Arukh ha-Rav</i>, <i>Oraḥ Ḥayyim</i> 500:18; <i>Mishnah Berurah</i>, <i>Sha’ar ha-Ẓiyyun</i> 500:49; R. Abraham Borenstein, <i>Eglei Tal</i>, <i>mel’ekhet borer</i>, sec. 11; <i>Minḥat Ḥinnukh</i>, <i>Mosekh ha-Shabbat</i>, <i>mel’ekhet borer</i>, sec. 6; R. Shalom Mordecai Schwadron, <i>Da’at Torah</i>, <i>Oraḥ Ḥayyim</i> 500:6; no. 3; and R. Shimon Grunfeld, <i>Teshuvot Maharshag</i>, I, no. 47.</i> A similar statement made in a different context appears in <i>Magen Avraham</i> 466:8. <i>Kezot ha-Shulḥan</i> observes that <i>Magen Avraham</i>'s ruling is limited to situations in which both the separated portion and the consumed portion are of the same species, e.g., both are meat, and hence fungible other than because of the prohibition.61<i class="footnote">A similar distinction is drawn by <i>Eglei Tal</i>, <i>mel’ekhet borer</i>, sec. 20 and <i>Tehilah le-David</i> 319:25. Cf., <i>Magen Avraham</i> 466:8; <i>Pri Megadim</i>, <i>Oraḥ Ḥayyim</i>, <i>Eshel Avraham</i> 319:3; <i>Shulḥan Arukh ha-Rav</i>, <i>Oraḥ Ḥayyim</i> 466:8; <i>Da’at Torah</i>, <i>Oraḥ Ḥayyim</i> 500:6; <i>Shevitat ha-Shabbat, mel’ekhet borer</i>, sec. 10; and <i>Tehilah le-David</i> 319:25, who apparently maintain that separation of such a forbidden substance constitutes a rabbinic transgression. However, <i>Eglei Tal</i>, <i>mel’ekhet dash</i> 17:54, maintains that such separation constitutes a biblical prohibition.</i> However, insects and water are intrinsically different substances and hence, asserts <i>Kezot ha-Shulḥan</i>, even if insects are eschewed only because of an attendant prohibition, the restriction of <i>borer</i> applies.
  <i>Hazon Ish, Oraḥ Hayyim: Mo'ed</i> 53, suggests that a filter may be used on <i>Shabbat</i> to assure that insects are not present in water flowing from the faucet on condition that the glass or container in which the water is collected be removed from the faucet before the spigot is turned off so that a portion of the water filtered in that manner is wasted in the act of filtration.62<i class="footnote">See also <i>Tehilah le-David</i> 319:25. Rabbi Rosner, <i>Or Yisra’el</i>, pp. 121-122, takes sharp issue with <i>Ḥazon Ish</i>’s position. <i>Ḥazon Ish</i>’s ruling is based on the consideration that a beverage may be gently poured from a cup of liquid into which an inedible object has been introduced. Similarly, according to <i>Ḥazon Ish</i>, there is no problem of <i>borer</i> other than with regard to the last portion of water flowing from the faucet; hence if that water is wasted there is no act of <i>borer</i>. In the case of liquid into which an inedible object has fallen, argues Rabbi Rosner, the liquid is spilled in a normal manner and does not “appear as separation.” However, use of a filter, he contends, clearly represents “separation” of any and all water in the faucet from the creatures left behind.<i>Minḥat Yiẓḥak</i> also notes that the faucet always remains filled with water when it is turned off and, moreover, there is no separation between that water and the source of the water supply. Therefore, since some water always remains in the system, allowing some water to go to waste would be superfluous. Rabbi Rosner challenges that statement and indeed it would seem that in conventional sinks turning off the faucet impedes the flow of water and leaves the faucet empty. Hence there is “separation” of the filtered entities from a clearly demarcated body of water.<i>Minḥat Yiẓḥak</i> further notes that <i>Ḥazon Ish</i> permits use on <i>Shabbat</i> of a teapot with a built-in strainer. <i>Minḥat Yiẓḥak</i> apparently understands <i>Ḥazon Ish</i> as maintaining that, since the strainer remains in place even when straining is not required, i.e., when there are no tea leaves in the teapot, there is no prohibition of <i>borer</i> provided that the tea is poured for immediate consumption. See <i>infra</i>, note 63. Similarly, argues <i>Minḥat Yiẓḥak</i>, since the filter remains in place even when water is drawn for washing, there is no prohibition of <i>borer</i>.Rabbi Rosner, however, argues that <i>Ḥazon Ish</i> permits use of such a teapot only because the strainer is an integral and inseparable part of the teapot. A filter, however, is readily removable and hence does not become part of the faucet. See also Rabbi Oberlander, <i>Or Yisra’el</i>, Tishri 5765, p. 110. Cf., <i>Shemirat Shabbat ke-Hilkhatah</i>, I, 3:57, note 166, who explains <i>Ḥazon Ish</i>’s ruling on the basis of R. Shlomoh Zalman Auerbach’s view that a utensil that is used only for food that will be consumed immediately is not regarded as a “utensil” for purposes of separation. Separation other than by means of a utensil for immediate consumption does not constitute a forbidden act of <i>borer</i>.It should also be noted that <i>Ḥazon Ish</i> advances his opinion with regard to use of the teapot only tentatively (<i>mihu efshar</i>). Cf., however, R. Chaim Kanievsky, <i>Ta’ama de-Kra: Hanhagot ha-Ḥazon Ish</i>, sec. 40, who reports that <i>Ḥazon Ish</i> permitted use of such a teapot in his household without care being taken to leave some residual liquid in the utensil with the tea leaves. See, however, R. Samuel ha-Levi Woszner, <i>Teshuvot Shevet ha-Levi</i>, I, no. 84, who refuses to permit use of a teapot with a built-in strainer on <i>Shabbat</i>.</i> However, <i>Hazon Ish</i> concludes that the expedient is of no avail if the water is infested by "many" insects.63<i class="footnote">For an analysis of <i>Ḥazon Ish</i>’s position see R. Chaim Oberlander, <i>Or Yisra’el</i>, Tishri 5765, pp. 107-110.</i> Nevertheless, R. Yitzchak Ya'akov Weisz, <i>Teshuvot Minḥat Yizḥak</i>, VII, no. 23, advises that the expedient suggested by <i>Hazon Ish</i> be employed on <i>Shabbat</i>.64<i class="footnote">Rabbi Oberlander, <i>ibid</i>., p. 110, observes that, according to <i>Minḥat Yiẓḥak</i>’s understanding of <i>Ḥazon Ish</i>, use of a filter is permissible provided that some liquid is left behind and only if the water is obtained immediately prior to its consumption but not if the water is to be consumed only at a later time.The requirement that the water be used immediately would require that filtration take place at the point at which the water leaves the faucet rather than, for example, at the site of the intake valve since water filtered at a remote site will not necessarily emerge immediately. However, Rabbi Rosner, <i>Or Yisra’el</i>, pp. 134-135, finds grounds to permit even such filters. As has been stated earlier, there is no problem with regard to filtration of water to be used for washing since there is no reason to prefer insect-free water for such purposes. Filtration of water that remains in the plumbing system is an effect that is not at all intended by the person presently draining water from the faucet. An act producing an unintentional result is permitted unless the unintended result is a necessary outcome of the otherwise permissible act (<i>pesik reisha</i>). Since the next use of water remaining in the system may well be for purposes of washing, and since insect-free water is not required for that purpose, “separation” of insects from water to be used for such purposes is permitted. Accordingly, there is no certainty that a forbidden form of “separation” will indeed take place. Since forbidden “separation” is not a necessary outcome and, if it does result, is unintended, argues Rabbi Rosner, use of such filters on <i>Shabbat</i> is permissible.</i>
  VII. Concluding Comments
  During the period in which the attention of some sectors of the Jewish community in New York City was focused upon possible halakhic problems regarding drinking water the parasitic disease section of the Centers for Disease Control and Prevention engaged in an active campaign to inform the public of potential health hazards associated with swimming in any body of water. The concern of the Centers for Disease Control focused upon swimming pools because of the prevalence of a parasite called cryptosporidium that may be responsible for eye, skin and respiratory infections as well as for gastrointestinal disorders.65<i class="footnote">See <i>New York Times</i>, June 8, 2004, p. F5 and www.healthyswimming.org.</i> Clearly, swimming pools are not risk free.
  Nor is New York City drinking water free of contaminations that pose risks to human health. In the <i>New York City 2003 Drinking Water Supply Quality Report</i> published by the Department of Environmental Protection that city agency advised that a significant majority of water samples taken from the Kensico Reservoir and the New Croton Reservoir were positive for giardia and cyptosporidium, microscopic pathogens that may cause intestinal illness.66<i class="footnote"><i>New York City 2003 Drinking Water Supply and Quality Report</i>, p. 11.</i> The DEP advises that immuno-compromised persons, such as patients undergoing chemotherapy, transplant recipients, persons suffering from Crohn's disease or HIV, as well as the elderly and infants "should seek advice from their health care providers about their drinking water."67<i class="footnote"><i>Loc. cit</i>.</i>
  Elsewhere in the same document, the DEP reports that treatment of water with chlorine to destroy microorganisms results in the forming of haloacetic acids and concludes with the following observation:
  New York State requires the following statement: <i>Some studies of people who drank chlorinated drinking water for 20 to 30 years show that long-term exposure to disinfection by-products (possibly including haloacetic acids) is associated with an increased risk for certain types of cancer. However, how long and how frequently people actually drank the water as well as how much haloacetic acids the water contained is not known for certain. Therefore, we do not know for sure if the observed increased risk for cancer is due to halaoacetic acids, other disinfection by-products, or some other factors. The individual haloacetic acids dichloracetic acid and trichloroacetic acid cause cancer in laboratory animals exposed to high levels over their lifetimes. Dichloroacetic acid and trichloroacetic acid are also known to cause other effects in laboratory animals after high levels of exposure, primarily on the liver, kidney and nervous system and on their ability to bear healthy offspring. Chemicals that cause effects in animals after high levels of exposure may pose a risk to humans exposed to similar or lower levels over long periods of time.</i>68<i class="footnote"><i>Ibid</i>., p. 9.</i>
  Certainly the authors of that material do not anticipate, nor do they desire, that all readers immediately desist from utilizing New York water for drinking purposes. The physicians and epidemiologists employed by the Centers for Disease Control find it their duty to make the public mindful of potential harm so that individual members of society may make their own determinations with regard to the risks they are willing to assume.
  The United States has difficulty exporting its grain to European countries because of a perceived danger associated with genetically engineered food products. One need only enter any supermarket to observe organically grown produce and non-organically grown produce displayed side by side. The price discrepancy between organic and non-organic fruits and vegetables is justified solely on the basis of the fact that organic produce is free of carcinogens associated with pesticides. Obviously, some consumers are willing to pay a premium for avoidance of even a slight health risk. Dentists routinely seek a full set of x-rays of a patient's teeth annually. Some patients just as routinely decline the potential benefit of dental x-rays despite the low dose of the radiation involved because the effects of radiation are cumulative over a lifetime and no one has definitively established an absolutely safe threshold.
  There is little question that in each of these cases the risk involved is well within the category of "<i>shomer petai'im Ha-Shem</i>—the Lord pre-serves the simple" (Psalms 116:6)69<i class="footnote"><i>Ketubot</i> 39a, <i>Yevamot</i> 12b and 72a and <i>Niddah</i> 45b.</i> and hence is halakhically acceptable. But to say that a specific form of conduct is halakhically acceptable is not to say that it is sagacious or even prudent. Some people are more prudent than others; some are more open to risk-taking than others. Such variations of temperament serve to explain why two equally intelligent individuals may make different choices with regard to assumption of minimal, albeit real, risk of life or well-being. Where real risks exist, or when the absence of risk cannot reasonably be excluded, a person who seeks to avoid the risk in question is neither neurotic nor irrational and should not be regarded in that light by a person willing to assume the risk, just as a person who is willing to accept a degree of risk associated with an unsecured financial venture in return for the anticipation of a higher return on his investment should not be regarded as irrational by a person who willingly forgoes a higher return in favor of guaranteed preservation of capital.
  The same is true with regard to potential transgression and its attendant spiritual harm. Halakhah generally forbids even acts that are doubtfully forbidden. However, again generally speaking, when the chance of transgression is remote, Halakhah does not forbid such acts. Nevertheless, concerned individuals to whom the onus of transgression is as serious as is the fear of bodily harm will refrain from acts that harbor even the possibility of transgression.
  It seems to this writer that this is the thrust of the remarks of R. Moshe Chaim Luzzato, <i>Mesilat Yesharim</i>, chap. 14. The topic addressed by <i>Mesilat Yesharim</i> is "<i>perishut</i>," literally, asceticism. Asceticism is certainly not an ideal in Judaism. Yet certain forms of self-denial are regarded as salutary. One of those is identified by <i>Mesilat Yesharim</i> as "<i>perishut be-dinim</i>" or "asceticism with regard to laws." As <i>Mesilat Yesharim</i> explains, persons who eschew certain pleasures or benefits because of fear of transgression, even when Halakhah does not mandate such a high state of vigilance, act in a commendable manner.
  In small towns in Europe inhabitants not only raised chickens for food but often maintained cows in their backyards in order to obtain milk. R. Israel Meir ha-Kohen, the <i>Hafez Hayyim</i>, also kept a cow for that purpose. Members of <i>Hafez Hayyim</i>'s family relate that one day he suddenly stopped drinking milk or eating any dairy products prepared from milk given by that cow. When questioned about the sudden change in his eating practices, <i>Hafez Hayyim</i> turned a deaf ear and refused to respond. Repeated questioning elicited only adamant refusal to answer. Eventually, the cow stopped giving milk. Since there was no longer any reason to maintain the cow it was slaughtered. Lo and behold, upon examination by the <i>shoḥet</i>, adhesions were found on the lungs and the cow was pronounced non-kosher. Suddenly, <i>Hafez Hayyim</i>'s earlier demurral to drink milk derived from the cow became cogent: the milk of a non-kosher animal is non-kosher.
  How did <i>Hafez Hayyim</i> become aware of the fact that his cow had developed adhesions? Certainly not on the basis of some prophetic inspiration. Had such information been revealed to him, presumably he would have been duty-bound to inform his family so that they not inadvertently fall into transgression.70<i class="footnote">Although the principle “it is not in heaven”(Deuteronomy 30:11) establishes that subsequent to revelation of the Torah at Sinai there will never be a supplemental revelation of halakhic information, that principle does not preclude prophetic revelation of factual information having halakhic ramifications. See R. Zevi Hirsch Chajes, <i>Torat Nevi’im: Eleh ha-Miẓvot</i>, chap. 2 and addendum to chap. 2, as well as R. Elchanan Wasserman, <i>Kuntres Divrei Soferim</i> (Pietrokow, 5684) no. 5, secs. 3-7. Republished in his <i>Koveẓ Shi’urim</i>, II (5720).</i> There is a quite simple explanation for <i>Hafez Hayyim</i>'s behavior. As animals age, they, no less so than humans, become prone to illness. <i>Hafez Hayyim</i> realized that his cow had reached a stage in bovine life at which disease and illness were more than remote possibilities. Concern for his spiritual welfare caused him to eschew milk that might be tainted with even the suspicion of spiritual harm. Was he halakhically required to act as he did? Certainly not. Unless empirically rebutted, the halakhic presumption that the majority of animals are not <i>tereifot</i> applies even to aging animals. <i>Hafez Hayyim</i>'s avoidance of the milk of that particular cow reflected a finely-honed sensitivity to the possibility of transgression.
  The concern with regard to New York water is far more profound. It is quite conceivable that, at least in some neighborhoods, the incidence of infestation does not rise to a level that serves to generate a halakhic concern. The general rule, subject to limited exceptions, is <i>lo maḥazikin rei'uta</i>, i.e., one is not required to suspect halakhic contamination of one type or another unless there is basis for such suspicion. In the absence of halakhically recognized grounds for suspicion, subjective concern may be grounds for personal stringency but not for a normative prohibition. However, when cogent grounds for fear rising to the halakhic threshold of suspicion do exist, the obligation not to risk transgression becomes a matter of normative law. The fundamental issue with regard to New York water is whether the grounds for suspicion rise to that level.
  An isolated sighting of a forbidden insect would not result in a mandatory requirement for examination of all water. Such an event might be dismissed as an isolated occurrence. Animals are not routinely examined upon slaughter for every possible anomaly. Since the majority of all animals are not <i>tereifot</i>, there is no halakhic requirement to suspect that any given animal is a member of the minority class of <i>tereifot</i>. However, examination of the lungs for the presence of adhesions is required despite the fact that the majority of animals do not develop disqualifying adhesions. Examination of the lungs is required because, unlike other <i>tereifot</i>, adhesions are present in a <i>mi'ut ha-mazui</i>, i.e., in a significant minority of animals.71<i class="footnote">A determination of the percentage of animals presently declared non-kosher on the basis of <i>sirkhot</i>, or adhesions on the lungs, would not establish the definition of <i>mi’ut ha-maẓui</i> for three reasons: 1) Determining the percentage of animals having <i>sirkhot</i> would establish only the threshold level of <i>mi’ut ha-maẓui</i>, i.e., a percentage that certainly constitutes a <i>mi’ut ha-maẓui</i>, but leave open the possibility that a lower frequency may also constitute a <i>mi’ut ha-maẓui</i>. 2) Many types of <i>sirkhot</i> are declared non-kosher because of disputes among halakhic authorities with regard to their status. Thus, in talmudic times the number of animals actually declared non-kosher for that reason may well have been less since at least some of those <i>sirkhot</i> may have been accepted as kosher. 3) The incidence of <i>sirkhot</i> is subject to great fluctuation depending upon the age of the animal, the fodder it is provided and various other factors.</i> <i>Shakh, Yoreh De'ah</i> 84:28, indicates that examination of food products for the presence of insects is required as a matter of normative law if insect infestation represents a <i>mi'ut ha-mazui</i>.72<i class="footnote">Cf., <i>Teshuvot Dvar Shmu’el</i>, no. 260. <i>Mishkenot Ya’akov, Yoreh De’ah</i>, no. 17, asserts that 10% constitutes a <i>mi’ut ha-maẓui</i>. <i>Bedikat ha-Mazon ke-Halakhah</i> reports that this position was also endorsed by R. Shlomoh Zalman Auerbach. However, the same author reports that R. Joseph Shalom Eliashiv, R. Nissim Karelitz and R. Chaim Kanievsky maintain that <i>mi’ut ha-maẓui</i> should be defined as 5%. In the second edition of that work (Jerusalem, 5765) Rabbi Eliashiv is quoted as maintaining 4% is a <i>mi’ut ha-maẓui</i>.R. Samuel ha-Levi Woszner, <i>Teshuvot Shevet ha-Levi</i>, IV, no. 81, challenges the relevance of the sources upon which that definition is based and asserts that examination may be required even when the percentage is much lower. However, <i>Shevet ha-Levi</i> fails to present a precise definition of <i>mi’ut ha-maẓui</i> and implies that a decision to declare examination mandatory is to be made “in accordance with the judgment of the decisor.” Cf. also, <i>Teshuvot ha-Rivash</i> no. 191, who asserts that <i>a mi’ut ha-maẓui</i> is to be defined as a frequency close to one half (<i>karov le-meḥeẓah</i>). See also <i>Teshuvot Rashi</i>, no. 62, who ascribes a similar view to the Ge’onim.</i>
  Thus, the crucial question with regard to whether or not a halakhic problem exists with regard to the New York water supply is whether the incidence of infestation rises to the level of a <i>mi'ut ha-mazui</i>.73<i class="footnote">Assuming that <i>mi’ut ha-maẓui</i> is defined as 10%, for purposes of establishing a <i>mi’ut ha-maẓui</i> it would be necessary to determine that there is a 10% chance that a forbidden creature will be present in the quantity of water drawn on any given occasion. For other opinions regarding the determination of a <i>mi’ut hamaẓui</i> see <i>Bedikat ha-Mazon ke-Halakhah</i>, I, 4:2, note 4.Moreover, the concept of <i>rov</i> and the exclusion of <i>mi’ut ha-maẓui</i> applies only when the presence of a forbidden entity has not been established with certainty, e.g., an animal or fruit. However, when the presence of a forbidden entity has been established in any particular place with certainty (<i>kavu’a</i>), but the person is in doubt as to whether the entity he has selected is permitted or forbidden, the principle of <i>rov</i> does not apply (<i>kol kavu’a ke-meḥeẓah al meḥeẓah dami</i>). It is arguable that since all the water in the faucet and plumbing as well as in the distribution system and the reservoir is contiguous and it is known with certainty that there are at least some copepods in the system, turning on the faucet is tantamount to removing the water from its source and hence the principle of <i>rov</i> does not apply.</i> The answer is not necessarily uniform for all areas throughout the city. Some areas may be <i>muḥzak be-tola'im</i>, i.e., they may be known to be plagued by repeated instances of infestation; other areas, and even some buildings within infested areas, may have repeatedly been shown to be free of infestation. Although copepods are known to be present in the reservoir, water in buildings shown to be free of infestation has, in effect, undergone a self-filtration process.74<i class="footnote">The situation in such buildings is comparable to that which results from the procedure commonly employed to avoid the tedious and time-consuming examination of vegetables in which insect infestation is likely, <i>viz.</i>, the vegetables are soaked in a solution containing vinegar or some other agent that serves to dislodge insects that may be present. When and if the efficacy of the method used has been demonstrated, the foodstuff is no longer deemed to be <i>muḥzak be-tola’im</i> and hence further examination is unnecessary.</i> In this writer's opinion, where copepods have been found, filtration of water is certainly to be advised; in buildings shown to be free of infestation, filtration is not required.
  The species which originate in dust heaps and carcasses, such as maggots, worms and their like and which are born, not through intercourse between male and female, but out of putrefaction of dung and the like are called "<i>creeping things that move upon the earth</i>" (Leviticus 11:44). … On the other hand, "<i>all creeping things that creep upon the earth</i>" (Leviticus 11:42) refers to such that procreate by intercourse between male and female.
  
  Chapter 8 Piscatorial Parasites
  
  <i>"…you shall not defile yourselves with them, that you should be defiled by them" (Leviticus 11:43). A person must guard with much caution and alacrity against anything that possibly enters into the category of detestable [creatures], especially in these times when the atmosphere and terrestrial areas have all become polluted.</i>
  OR HA-HAYYIM, AD LOCUM
  I. The Anisakis Problem and its Precursors
  Fish has always been the food of choice for those concerned with matters of <i>kashrut</i>.1<i class="footnote">The Gemara, <i>Bava Batra</i> 75a, declares that in the eschatological era God will prepare a feast for the righteous at which the Leviathan will be served. R. Elijah Levita, known as R. Elijah Baḥur, <i>Tishbi,</i> s.v. <i>Yukhnah</i>, records a tradition to the effect that both the Leviathan and the <i>shor ha-bar</i>, a legendary wild ox, will be featured at this banquet<i>.</i> A folk witticism has it that the Leviathan will be available to those who will request fish since even then they will be unwilling to rely upon the <i>kashrut</i> of the meat.</i> Unlike meat, that not only must be slaughtered, porged, soaked and salted in accordance with the myriad provisions of Halakhah, any fish having fins and scales may be consumed without further ado. Questions concerning the permissibility of fish which have become a staple in the diet of observant Jews are quite unsettling.
  A vigorous debate has erupted among rabbinic authorities both in Israel and in the Diaspora regarding the permissibility of consuming fish known to be infested by a parasitic nematode known as Anisakis. Among the suspect species are wild salmon,2<i class="footnote">R. Moshe Mordecai Karp, <i>Or Yisra’el,</i> no. 54 (Tevet 5769), p. 63 and no. 56 (Tammuz 5769), p. 49, reports that wild salmon imported from Norway and Chile have not been found to be infested. Those salmon do, however, require examination for the presence of lice on the skin.</i> cod, flounder, Alaskan sole,3<i class="footnote">Rabbi Karp, <i>loc. cit</i>., also reports that sole imported from Holland are free of infestation.</i> halibut, pollack, red snapper, perch, Pacific sable, mackerel, blue whiting, herring and sardines imported from Norway and Scotland. The problem is limited to fish harvested in their natural habitats in which, in many cases, the Anisakis also thrives. Farmed fish present no problem since, due to the absence of the marine species in which Anisakis lay their eggs, Anisakis larvae have no opportunity to infest the waters in which farmed fish are raised. Farmed salmon is much less expensive, and hence more common, than wild salmon. The same is true of smoked and cured farmed salmon. However, for a number of reasons, canned salmon is usually of a wild salmon variety and have been found to be infested.4<i class="footnote">See R. Schneur Zalman Revach, <i>Or Yisra’el</i>, no. 56, p. 48.</i>
  By no means is this the first controversy of such nature in modern times. In recent decades there have been a number of contretemps centering upon parasites now commonly found in fish. The first such problem arose in 1978 with regard to the permissibility of sable5<i class="footnote">A similar problem was reported thereafter by R. Pesach Eliyahu Falk, <i>Madrikh le-Bedikat Tola’im</i>: <i>Guide to the Inspection of Fruits and Vegetables for Insects</i> (Gateshead, 5744), p. 92, with regard to cod, whiting and a number of other fish. As reported in “Advisory Pamphlet No. 29,” published by the British Ministry of Agriculture, fertilized eggs of parasites living in the stomach of seal pass into the sea with excreta of the seal. The eggs hatch into microscopic worms that are swallowed by a shrimp-like aquatic animal. Cod become infested by eating the shrimp-like organism or small fish containing the tiny worm. The life cycle of the organism is completed when the seal eats the infested cod.</i> and a subsequent issue with regard to worms found in other fish, including Canadian whitefish, was debated in the late 1990s.6<i class="footnote">That problem involved a worm called <i>Triaenophorus crassus</i> and was quite similar to the problem involving the Anisakis. <i>Triaenophorus</i> thrive in the intestines of pike. The adult worm releases its eggs in the spring when the pike spawns and then dies. The microscopic eggs are released into the water by the pike. The eggs hatch into larvae known as coracidia and are covered by hair-like filaments that enable them to move in the water. Coracidia are microscopic and will perish within approximately two weeks unless consumed by another creature. Coracidia are eaten by zooplankton and evolve into a second larval stage called a procercoid. A procercoid can grow to a length of over 300 microns at which stage it is visible to a diligent observer. The procercoid will die in approximately one month unless the zooplankton is consumed by a fish.When the zooplankton is eaten and digested by a whitefish the procercoid is released into the stomach of its host. Within hours it develops into a tiny microscopic worm. The infinitesimal worm releases enzymes that enable it to penetrate the stomach wall and embed itself in the muscle tissue of the whitefish. The whitefish reacts to the alien organism by encapsulating it within a small cyst within which the <i>Triaenophorus crassus</i> develops into a thin organism between four and fifteen inches in length. The worm can survive in that habitat for some three or four years but as the cyst gradually shrivels it will die without progeny. However, if the whitefish is eaten by a pike before the encysted worm dies, it will escape from the cyst and live to maturity within the viscera of the pike where it will spawn the following spring and restart the cycle. The problem was encountered when the cysts containing the worms were discovered in infested whitefish.</i> Extensive analyses of the questions posed at those times are presented in various contributions to <i>Or Yisra'el</i>, no. 6 (Tevet 5757), "Kuntres ve-Yirdu be-Degat ha-Yam," pp. 137-250.
  The facts surrounding the first of that series of controversies concerning parasites found in sable are described in detail by R. Samuel ha-Levi Woszner, <i>Teshuot Shevet ha-Levi</i>, IV, <i>Yoreh De'ah</i>, no. 83. As reported in the name of ichthyologists, the source of those parasites are sailfish. The parasites lay eggs in the sailfish which in turn excrete them into the ocean. The eggs (or probably larvae) are ingested by shrimp in whose intestines the parasites hatch. The shrimp, in turn, are eaten and digested by cod and sable. The parasites survive and infest the flesh of the sable.
  In the earlier instances the matter was dismissed by most rabbinic scholars because of lack of empirical evidence that the parasites in question were of a category that is halakhicly forbidden.7<i class="footnote">See R. Moshe Viya, <i>Bedikat ha-Mazon ke-Halakhah</i>, 2nd ed. (Jerusalem, 5765), I, s<i>ha’ar sheni</i>, chap. 3, note 22. See also <i>idem, Or Yisra’el</i>, no. 6 (Tevet 5757), p. 18 and R. Moshe Feinstein, cited by R. Shlomoh Gross, <i>Mishneh Shlomoh</i>, no. 31. Cf., however, R. Samuel ha-Levi Woszner, <i>Teshuvot Shevet ha-Levi</i>, IV, no. 83, who at that time wrote that “it is impossible to resolve [the question] definitively until we know if there is support for that which these non-Jewish scientists are saying” and therefore tentatively forbade consumption of such fish. A succinct and much less tentative restrictive ruling was later published in <i>Teshuvot Shevet ha-Levi</i>, VII, no. 127, sec.3. Restrictive rulings were also issued by R. Yisra’el Dov Halberstam of Antwerp in an essay titled <i>Kuntres Ve-Anokhi Tola’at,</i> appended to his anonymously published monograph <i>Yad le-Mikveh</i> (n.p., 5754), as well as by R. Nathan Gestetner, <i>Teshuvot Le-Horot Natan</i>, IX, nos. 23-25, and R. Joseph David Weiss, <i>dayyan</i> of the Satmar community in Antwerp, <i>Teshuvot Va-Ya’an David</i>, I, no. 114. Rabbi Viya, <i>Bedikat ha-Mazon</i>, note 22, reports that in 1997 R. Joseph Shalom Eliashiv ruled against eating whitefish on the basis of scientific testimony demonstrating that the parasites originate outside the fish. A letter from a son of <i>Shevet ha-Levi</i> affirming that his father ruled against eating suspect fish is affixed to <i>Ve-Anokhi Tola’at</i>, p. 21.</i> However, with regard to the current controversy concerning the Anisakis worm there is a plethora of evidence, both empirical and scientific, giving substance to the problem.8<i class="footnote">The statement signed by R. Israel Belsky, published in <i>Hamodia</i>, 28 Iyar 5771, p. D14, stating that the facts (<i>meẓ’iut</i>) have not changed is, at best, misleading. The facts may be the same but the evidence on the basis of which facts are established is indisputably different.</i> R. Schneur Zalman Revach, head of the Machon le-Mizvot ha-Teluy'ot ba-Arez, was apparently the first to bring the Anisakis issue to the attention of rabbinic decisors. Rabbi Revach conducted a thorough investigation of the problem. He personally performed numerous examinations of infested fish and has observed the Anisakis in various stages of development. Detailed information derived from those investigations is presented by Rabbi Revach in his <i>Tola'at Shani</i> (Bet Uzi'el, 5767), II, <i>Birurei Halakhah,</i> chap. 5, reprinted in <i>Or Yisra'el</i>, no. 54 (Tevet 5769), pp. 34-41, and in subsequent contributions to that journal. An excellent and widely circulated, but regrettably unpublished, analysis of the problem has been authored by Rabbi Gershon Bess of Los Angeles. Many of the pertinent halakhic issues were earlier discussed by R. Yisra'el Dov Halberstam of Antwerp in <i>Kuntres Ve-Anokhi Tola'at</i>, included as an addendum to his anonymously published <i>Yad Mikveh</i> (n.p., 5754).
  Fish infested by Anisakis parasites have been banned by virtually all recognized Israeli authorities, including R. Joseph Shalom Eliashiv,9<i class="footnote">See <i>Or Yisra’el</i>, no. 61 (Tishri 5771), p. 60.</i> R. Nissim Karelitz,10<i class="footnote">Letter dated 7 Ḥeshvan 5770, published in <i>Or Yisra’el</i>, no. 61, p. 60.</i> R. Samuel ha-Levi Woszner,11<i class="footnote">Letter dated 9 Iyar 5770, published in <i>Or Yisra’el, ibid</i>., p. 63, note 19. See also <i>Or Yisra’el</i>, no. 56, pp. 51 and 55.</i> R. Chaim Kanievsky,12<i class="footnote">See <i>Or Yisra’el</i>, no. 60 (Tammuz 5770), p. 50.</i> R. Moshe Sternbuch,13<i class="footnote"><i>Loc. cit.</i></i> R. Chaim Greineman14<i class="footnote">See <i>Or Yisra’el</i>, no. 54, pp. 40 and 54 as well as <i>Or Yisra’el</i>, no. 56, pp. 43 and 55. See also R. Schneur Zalman Revach, <i>Tola’at Shani</i> (Bet Uzi’el, 5767), II, 197 and 210.</i> and R. Shlomoh Amar.15<i class="footnote">Letter of approbation published as the preface to <i>Tola’at Shani</i>, II, reprinted in <i>Or Yisra’el</i>, no. 54, p. 33.</i> The most notable Israeli exception to this consensus is R. Moshe Landau of Bnei Brak.16<i class="footnote">See <i>Pa’amei Ya’akov</i>, Elul 5770.</i> The ruling forbidding such parasites was affirmed in a statement dated 18 Iyar 5771 signed by Rabbis Eliashiv and Woszner as well as by R. Feivel Cohen, R. David Feinstein and R. Aaron Schechter and published in <i>Hamodia</i>, 28 Iyar 5771, p. D5 and in <i>Yated Ne'eman</i>, 5 Sivan 5771, p. 41. A number of American rabbinic personalities, primarily among those identified with the hasidic sector, including R. Yechezkel Roth, formerly a <i>dayyan</i> of the Satmar community, have ruled permissively.17<i class="footnote">See R. Yechezkel Roth, <i>Or Yisra’el,</i> no. 6<i>,</i> pp<i>.</i> 153-158, reprinted in <i>idem, Emek ha-Teshuvah</i>, IV, no. 57. That responsum was authored in 5757. However, his son, R. Moshe Roth, <i>Or Yisra’el</i>, no. 57 (Tishri 5770), pp. 148-154, reports that his father maintains that the arguments and conclusions pertain to the present problem as well. See also R. Asher Anshel Eckstein, a <i>dayyan</i> of the Belz community in the United States, <i>Or Yisra’el,</i> no. 54, pp. 60-62.</i> Numerous articles representing both sides of the controversy were published in <i>Or Yisra'el</i>, no. 54 (Tevet 5769), pp. 18-56; no. 56 (Tammuz 5769), pp. 43-56; no. 57 (Tishri 5770), pp. 148-154; no. 58 (Tevet 5770), 146-152; no. 60 (Tammuz 5770), pp. 32-54; and no. 61 (Tishri 5771), pp. 15-72.
  The life cycle of the Anisakis worm is quite remarkable. The cycle begins with mature worms present in marine mammals such as whales or dolphins. The worm deposits its eggs in the mammal. The marine mammal excretes unembryonated eggs which then slowly fall to the ocean floor. The Anisakis worms become embryonated in the water and larvae develop from the eggs. The free-swimming larvae are ingested by crustaceans, most significantly by krill, and mature within their host. The host is then consumed by a predator, e.g., a marine mammal, salmon or flounder, etc., that digests the host but not the Anisakis. The Anisakis then proceeds to encyst itself in the viscera of the host fish. Some of the parasites succeed in piercing through the walls of the visceral organs and migrate into the flesh of the fish.18<i class="footnote">See Leo M.L. Nollet and Fidel Toldra, <i>Handbook of Seafood and Seafood Products Analysis</i> (Boca Raton, 2010), p. 591.</i> This occurs most frequently after the death of the host fish even though the fish is refrigerated. The parasites die only when the fish is frozen.19<i class="footnote">See <i>Tola’at Shani</i>, II, 192 and 198 and <i>Ve-Anokhi Tola’at</i>, p. 5. Parasites are killed when fish are frozen at -20° centigrade for 24 hours. See Robert E. Olson, “Marine Fish Parasites of Public Health Importance,” <i>Seafood Quality Determination</i>, ed. D.E. Kramer and J. Liston (Amsterdam, 1986), p. 339. See also <i>Modern Food Microbiology</i>, 7th edition., ed. by James M. Jay, Martin J. Loessner and David A. Golden (New York, 2005), p. 704.In his unpublished essay, Rabbi Gershon Bess reports that worms have been found only in the intestines of a certain species of hake processed in Argentina whereas the flesh of the same fish processed in China is highly infested. In China, the fish are only partially eviscerated after being caught and are not kept in a cold state with the result that, as the fish warms, the worms migrate to the flesh.Rabbi Bess further reports that an experiment was conducted by <i>kashrut</i> supervisors in Norway. Thousands of herring were frozen at -2° centigrade. Upon examination, no parasites were found in their flesh. Ten fish were left unfrozen and allowed to warm for a day after their capture. The flesh of nine of those ten herring were found to be infested with parasites.</i>
  Creatures lacking fins and scales "of all that swarms in the waters" (Leviticus 11:10) are biblically proscribed. Included in the prohibition against consuming such marine creatures are not only fish and crustaceans but also insects and the like whose natural habitat are "the seas and the rivers." Such creatures are forbidden even after having been ingested by a kosher fish but, as explained by the Gemara, <i>Hullin</i> 67b,20<i class="footnote">Actually, the Gemara discusses rules applying to animals but does not explicitly refer to fish. <i>Tosafot, ad locum</i>, state that the underlying principle applies equally to fish.</i> similar organisms generated within an animal or the flesh of the fish are permitted. Accordingly, as recorded in <i>Shulḥan Arukh, Yoreh De'ah</i> 84:16, worms found within21<i class="footnote">R. Ephraim Zalman Margulies, <i>Teshuvot Bet Efrayim,Yoreh De’ah</i>, no. 25, writes that, even when found on the external surface of the entrails, such organisms are forbidden. See, <i>infra</i>, note 50 and accompanying text.</i> the entrails22<i class="footnote">As understood by Rashi and <i>Tosafot</i>, the Gemara, <i>Ḥullin</i> 67b, indicates that such creatures are also forbidden when found in the liver or lungs of the animal or of the fish on the suspicion that they entered through the “nostrils.” <i>Ḥiddushei R. Akiva Eger, ad locum</i>, adds the phrase “or in the brain” reflecting the possibility that the worms might have entered through the nose and penetrated the cranial cavity.</i> of a fish are forbidden because it is to be presumed23<i class="footnote"><i>Pri Megadim, Siftei Da’at</i> 84:43, indicates that such creatures are prohibited, not with certainty, but as a matter of doubt since their presence within the digestive organs strongly suggests that they originated outside the fish. This view was expressed much earlier by <i>Issur ve-Hetter he-Arukh, sha’ar</i> 41, sec. 15. The Gemara found no reason to assume that organisms found elsewhere within the fish originated outside the fish and hence, if found under the skin or in the flesh of the fish, they are permitted. The matter can best be categorized as an application of the principal “<i>ka’an nimẓa ka’an ḥayah</i>—what is found here was always here [and not elsewhere]” other than in situations in which there exists a <i>rei’uta</i>, or unusual factor, giving rise to suspicion of the contrary.Consistent with that principle, <i>Knesset ha-Gedolah</i>, followed by <i>Baḥ</i> and <i>Yam shel Shlomoh</i> in his commentaries on <i>Sefer ha-Terumot</i> and on <i>Ḥullin</i> 3:104, concludes that worms are permitted only if found in a whole fish and upon examination it is ascertained that there is no lesion through which the worms might have entered. However, once the fish has been cut, any worms that are found are forbidden because of the fear that, in cutting the fish, they may have penetrated the flesh from the viscera.</i> that they were earlier present in the water and were ingested by the fish, whereas such creatures, when found between the skin and flesh, or, according to most authorities,24<i class="footnote">See, however, <i>Baḥ, Yoreh De’ah</i> 84, <i>Knesset ha-Gedolah, Yoreh De’ah</i> 84:101 and <i>Yam shel Shlomoh, Ḥullin</i> 3:106, who permit such creatures only when found between the skin and the flesh. See <i>infra</i>, note 25.</i> within the flesh of the fish itself,25<i class="footnote">Rashi identifies the “<i>darni</i>” explicitly permitted by the Gemara, <i>Ḥullin</i> 67b, as “worms found between the skin and the flesh.” Rashba, <i>Torat ha-Bayit ha-Kaẓer, bayit slishi, sha’ar shlishi</i>, adds the phrase “or in the flesh” and that emendation is incorporated by virtually all subsequent authorities.</i> are permitted.26<i class="footnote">Nevertheless, when detached or separated from the fish, such worms are forbidden because of <i>ma’arit ayin</i>, i.e., because they appear to be forbidden creatures. Contrary to other authorities, <i>Shakh, Yoreh De’ah</i> 84:46, permits such creatures as long as they remain in the plate in which they are served but forbids them once they leave the plate for another surface. Cf., <i>Pri Ḥadash, Yoreh De’ah</i> 84:9 and <i>Ḥokhmat Adam</i> 38:29.</i> Even if they develop within the fish while the fish is still alive, such creatures are regarded as having originated in the fish itself.27<i class="footnote">R. Hai Ga’on, <i>She’iltot</i>, <i>Parashat Shemini</i>, <i>she’ilta</i> 4, explains that, since such creatures are generated by the flesh of the fish or animal, they have the halakhic status of their progenitor.It is generally presumed that the Gemara asserts that such creatures arise in the flesh of their host as a result of spontaneous generation. However, Rashi, <i>Ḥullin</i> 67b, defines the term “<i>gavli</i>” in the phrase <i>mineih gavli</i> as meaning “<i>gadli,</i>” i.e., “grow” or develop.” That understanding is entirely consistent with the notion that the parasites enter the flesh while yet microscopic in nature—and hence are accorded no halakhic cognizance—and later develop within the flesh. Since their existence is recognized only when they become visually perceivable within the flesh of the host, their halakhic identity is that of the host in which they “grow” or “develop.” See <i>infra</i>, note 57.</i> As enunciated by the Gemara, the applicable principle is that creatures that grow from the flesh of a kosher fish or of an animal that has been slaughtered28<i class="footnote">Only worms generated in the flesh of an animal after it has been slaughtered are permitted; a worm generated in the flesh while the animal is yet alive is forbidden as “an organ torn from a living creature.” Since the worm enjoys independent animation, it is not rendered permissible by the slaughter of its host. Fish do not require slaughter and are not subject to the prohibition against an organ torn from a living creature. Hence, worms generated by the flesh of a fish, even when the fish is yet alive, are permitted.</i> are entirely permissible.
  Despite the relative clarity—in application, if not in theory–-of the principles governing the status of insects found in fish, a number of controversies arose in previous generations. Insects found on the surface of a fish are, quite understandably, presumed to have flourished in water rather than in the fish itself and hence are forbidden. R. Abraham Danzig, <i>Hokhmat Adam</i> 38:28, reports that, beginning in the spring and until the onset of winter, he "many times" observed the presence of worms in the mouths and ears of various species of fish, particularly hecht. During that season, rules <i>Hokhmat Adam</i>, fish must be examined for the presence of such worms. <i>Hokhmat Adam</i> further reports that, although they were not to be found in his own city of Vilna, in many locales, "fish lice" were also found to be present in the proximity of the fins and in the mouths of fish. The fish lice are described as "round as barley" with tiny black eyes placed close together. In Prague "everyone recognized them" and a well-known ban existed against consuming those fish without vigorous scraping in the area of the fins and the tail and examination behind the ears and mouth.
  Nevertheless, R. Meir Arak, <i>Teshuvot Imrei Yosher</i>, II, no. 11, reports that, at least in some areas, many people ate carp regularly without performing such an examination despite the known presence of the lice described by <i>Hokhmat Adam</i>. In defending that practice, <i>Imrei Yosher</i> entertains the possibility that only worms found on the surface of the fish must be presumed to originate in the water that is the fish's habitat but that perhaps those found "under the scales" are more likely to be generated within the fish itself. <i>Imrei Yosher</i> concludes that there are grounds to permit eating such fish without examination but only if scientists will testify that the source of the worm is the flesh of the fish. Not surprisingly, there is no indication that such substantiation was forthcoming. R. Israel Jacob Klapholz, <i>Admorei Belz</i>, IV (Bnei Brak, 5736),118, reports that R. Aaron Rokeach refused to accept a lenient ruling reported in the name of R. Shalom Mordecai Schwadron (Maharsham) with regard to this matter and also rejected the expedient of removing the scales of the fish to examine for the presence of parasites. Moreover, when told that a respected personality, R. Saul Brock, had issued a permissive ruling, he responded to his informant by saying, "I appoint you my agent to ask him to retract his ruling."
  II. The Anisakis in Light of Rambam's Stringent Ruling
  Contradicting the majority of early-day authorities, Rambam, <i>Hilkhot Ma'akhalot Assurot</i> 2:17, rules that only worms generated in the rotting flesh of dead fish are permitted but those generated while the fish are still alive, even if the worms are found embedded in the flesh, are forbidden. As explained by <i>Maggid Mishneh, ad locum</i>, such worms are forbidden as a matter of doubt because of the possibility that they may have been ingested together with water and swallowed by the fish and later migrated into the flesh, whereas it is the nature of decaying tissue to generate worms.29<i class="footnote">Rambam clearly affirms the spontaneous generation of such worms. In his <i>Sefer ha-Miẓvot, lo ta’aseh</i>, no. 179, Rambam describes those who deny that phenomenon as “fools who have no knowledge of science.” See also Rambam, <i>Commentary on the Mishnah, Shabbat</i> 126b.</i> <i>Maggid Mishneh</i> explains that the word "<i>darni</i>"30<i class="footnote">In explaining the similar position of <i>Sefer ha-Terumot, Knesset ha-Gedolah, Yoreh De’ah</i> 84:101, suggests that “<i>darna</i>” (plural, <i>darni</i>) is the name of a particular species that was known to the Sages to be generated by the flesh of an animal or fish but expresses uncertainty with regard to whether that statement is to be accepted paradigmatically as permitting other species regarding which that fact is not known with certainty. <i>Migdal Oz, Hilkhot Ma’akhalot Assurot</i> 2:17, declares that only the species known as <i>darni</i> are permitted. The identity of that species is unknown to us.<i>Teshuvot Shevet ha-Levi</i>, IV, <i>Yoreh De’ah</i>, no. 83 and VII, no. 127, sec. 3, also asserts that “<i>darna</i>” is the name of a specific species but, unlike <i>Migdal Oz</i>, he regards it as a paradigm for all such species. Nevertheless, <i>Shevet ha-Levi</i> agrees that only creatures known with certainty to generate within the fish are permitted. See also, <i>infra</i>, note 57 and accompanying text.Cf., an unpublished responsum authored by R. Pesach Eliyahu Falk, quoted by R. Levi Baum, <i>Or Yisra’el</i>, no. 60, p. 59, who cites <i>Shabbat</i> 54b, 70a and 102b in arguing that “<i>darni</i>” is a generic term for numerous species of worms. There is however, a variant reading in which the term “<i>murani</i>” (rather than “<i>darni</i>”) appears in the texts of <i>Ḥullin</i> 67b<i>. Shulḥan Arukh, Yoreh De’ah</i> 84:1, speaks of “worms” and hence certainly did not accept the notion that the provision is limited to a particular species.</i> employed by the Gemara to denote permitted worms specifically refers to creatures generated in decaying flesh.31<i class="footnote">R. David Arama, in a gloss on Rambam, <i>ad locum</i>, published in the Shabbetai Frankel edition of the <i>Mishneh Torah</i> in the section titled <i>Likkutim</i>, explains Rambam’s position in a manner that renders forbidden even insects that are known with certainty to have been generated by the flesh of a live fish. Worms that develop in a fruit after it has been plucked are not deemed to be creatures that “creep on the earth” until they emerge from the fruit. However, if they develop while the fruit is still attached to the tree, which itself is rooted in the ground, the worms are prohibited. Similarly, asserts this authority, a fish while alive in water is “attached” to the water with the result that all insects that develop in a live fish, even if they are generated in the flesh of the fish, are forbidden because they “creep in the water,” i.e., in the fish that is regarded as attached to the water. (Cf., <i>Yoreh De’ah</i> 201:33 regarding ritual immersion.) Hence, only insects that develop in the flesh of the fish, i.e., after it has died, are permissible. That analysis of Rambam’s position is developed at length by R. Shlomoh Amar in his letter of approbation to <i>Tola’at Shani</i>, reprinted in <i>Or Yisra’el</i>, no. 54, pp. 30-32.</i> According to Rambam, only creatures found in the decaying flesh of a dead organism are presumed to have been generated by that organism.32<i class="footnote">See the somewhat different explanation of Rambam offered by <i>Teshuvot Shevet ha-Levi</i>, IV, no. 83<i>.</i></i>
  As understood by <i>Maggid Mishneh</i>, the controversy between Rambam and opposing authorities is limited in nature and involves no dispute with regard to any fundamental halakhic or scientific principle. All agree that organisms generated outside the animal or fish are forbidden and all accept both the principle of spontaneous generation or its equivalent and the principle that, when known to have been generated within an animal or fish, such organisms are permitted. The sole dispute is with regard to the presumption attendant upon organisms found embedded in the flesh of the animal or fish. Rambam maintains that the origin of such organisms is a matter of doubt and hence they are forbidden unless found in flesh that has begun to rot since only in that case may they be presumed to have been generated by the decaying flesh. Rambam's opponents rule that, when found in the flesh of the creature rather than in the creature's internal organs, it may be assumed that the organisms were generated within the flesh of the creature and hence are permitted.33<i class="footnote">See <i>supra</i>, note 23.</i>
  R. Ephraim Zalman Margulies, <i>Teshuvot Bet Efrayim, Yoreh De'ah</i>, no. 25, asserts that numerous early-day authorities, including <i>Sefer ha-Terumot</i>, no. 36; <i>Sefer Mizvot Gadol, lo ta'aseh</i>, no. 132; <i>Sha'arei Dura</i>, no. 54; and <i>Rokeaḥ</i>, concur in Rambam's opinion, and, for that reason, <i>Bet Efrayim</i> strongly leans toward ruling in accordance with Rambam's stringent view. Although <i>Shulḥan Arukh</i> accepts the permissive view of the many early-day authorities who disagree with Rambam, <i>Shulḥan Arukh</i> concludes his codification of that ruling by declaring, "There are those who forbid worms that are generated after slaughter." <i>Be'er ha-Golah</i> and <i>Arukh ha-Shulḥan, ad locum</i>, identify the anonymous authority cited by <i>Shulḥan Arukh</i> as Rambam. R. Moshe Mordechai Karp, in a brief monograph titled <i>Le-Afrushei me-Issura</i>: <i>Be-Inyan Kashrut ha-Dagim</i> (5770), cites <i>Shakh, Yoreh De'ah</i> 243: <i>kizur klalei issur ve-hetter</i>, sec. 5, and <i>Kaf ha-Hayyim</i> 243:164 who declare that whenever an authority is cited in that manner by <i>Shulḥan Arukh</i>, even though the normative rule is permissive, nevertheless, "a pious person should be stringent for himself." According to that view, all fish infested with parasites should be abjured even in the absence of positive evidence that the parasites are commonly imbibed by the fish from the water in which it swims. Thus, concludes Rabbi Karp, "a pious person" should abstain from eating Anisakis-infested fish regardless of how the controversy is resolved.
  III. Anisakis and the Talmudic Presumption of Permissibility
  The authorities who are unconcerned with regard to possible Anisakis infestation of commonly consumed fish do not accept Rambam's position. For the most part, they regard the principle formulated by the Gemara regarding spontaneous generation of insects in the flesh of fish as empirically correct, universal in nature and an irrebuttable presumption not subject to scientific challenge. They regard any challenge to that position as either bordering upon, or as actual, heresy. The many scholars who regard the problem in a serious light either maintain that the Gemara refers to a specifically named organism or do not regard the statement of the Gemara as universally applicable to all organisms and hence distinguish the Anisakis on either halakhic or empirical grounds. Some authorities declare that the principle of "<i>nishtaneh ha-teva</i>—nature has changed"34<i class="footnote">See, for example, Rabbi Halberstam, addenda to <i>Ve-Anokhi Tola’at</i>, p. 40 and Rabbi Revach, <i>Or Yisra’el</i>, no. 54, p. 39.</i> is applicable and assert that, although spontaneous generation occurred in ancient times, it is no longer a reality or, alternatively, state simply that, although parasites such as Anisakis were rare in earlier ages, they have become much more common in our era.35<i class="footnote">Cf., R. Chaim ibn Attar, <i>Or ha-Ḥayyim,</i> Leviticus 11:43, who asserts that the incidence of “creeping things” has increased “in these times when the atmosphere and terrestrial areas have all become polluted.”</i> Indeed, <i>Shakh, Yoreh De'ah</i> 84:22, states that frequency of infestation varies from place to place and from era to era.36<i class="footnote">Rabbi Halberstam, addenda to <i>Ve-Anokhi Tola’at</i>, p. 39, draws attention to two studies conducted in a fishing area north of Scotland, one in 1960 and one in 1972. The earlier study found an infestation rate of one and one-half percent, while the second revealed infestation in sixty percent of the fish examined.</i> Rabbi Halberstam, addenda to <i>Ve-Anokhi Tola'at</i>, p. 39, attributes increased infestation in our day to present use of refrigeration enabling fish to be shipped long distances without being eviscerated.37<i class="footnote">See Allan Roepstorff, Horst Karl, Bouke Bloemsma <i>et al</i>., “Catch Handling and the Possible Migration of <i>Anisakis</i> Larvae in Herring, <i>Clupea harengus</i>,” <i>Journal of Food Protection</i>, vol. 56, no. 9 (Sept., 1993), p. 783.</i> As noted earlier, migration of Anisakis from the intestine takes place after the death of the host.38<i class="footnote">See John W. Smith and R. Wootten, “Experimental Studies on the Migration of <i>Anisakis</i> Sp. Larvae (Nematoda: ascaradida) into the Flesh of Herring <i>Clupea Harengus</i> L.,” <i>International Journal for Parasitology</i>, vol. 5, no. 2 (April, 1975), pp. 133-136 and John W. Smith, “The Abundance of <i>Anisakis simplex</i> L.3 in the Body-cavity and Flesh of Marine Teleosts,” <i>International Journal for Parasitology</i>, vol. 14, no. 5 (Oct. 1984), pp. 491-495.</i> Some scientists attribute proliferation of parasites to the warming of sea waters.39<i class="footnote">See Patricia Noguera, “Red Vent Syndrome in Wild Atlantic Salmon <i>Salmo Salar</i> in Scotland is Associated with <i>Anisakis Simplex</i> Sensu Stricto (Nematoda: Anisakidae),” <i>Diseases of Aquatic Organisms</i>, vol. 87, no. 3 (Dec. 3, 2009), pp. 199-215. Cf., sources cited by Harford Williams and Arlene Jones, <i>Parasitic Worms of Fish</i>, (London, 1994), p. 417.</i> Accordingly, it is not surprising that changed empirical reality results in halakhic rulings differing from those applicable in circumstances that prevailed in earlier times.40<i class="footnote">For a fuller discussion of similar controversies in other contexts see chapter seven of this volume, pp. 210-217.</i>
  Rabbi Karp, <i>Or Yisra'el</i> no. 58, p. 147, and R. Moshe Eliezer Bloom, <i>Or Yisra'el</i>, no. 54, p. 20, cite numerous authorities, including Rabbenu Nissim, Rashba, Ran, Re'ah, Me'iri and <i>Hagahot Asheri</i>, who employ language indicating that the worms permitted by the Gemara are sanctioned only because, in the assumed circumstances, it is impossible that they might have spawned in water and subsequently have been swallowed by the fish. The Gemara declares that, were that the case, worms of the same species would also be found in the excretory organs. Thus, for example, Ran, <i>Hullin 66b,</i> writes<i>,</i> "Worms found in meat that could not possibly come from outside, but are generated from the meat, are permitted." Re'ah, <i>loc. cit.</i>, similarly states that, "Worms found in meat in a place in which it is not possible41<i class="footnote">The notion of certainty voiced by these authorities must assuredly be understood not in the sense of logical certainty but as connoting the absence of any cogent reason for suspecting otherwise. See <i>supra</i>, note 23. The crucial point is that the cited statements unequivocally indicate that the rules established by the Gemara are not universal and do not apply when “certainty” is dispelled by contradictory evidence.</i> that they came from outside" are permitted.42<i class="footnote">Cf., R. Moshe Roth, <i>Or Yisra’el</i>, no. 57 (Tishri 5770), pp. 153f., and R. Asher Anshel Eckstein, <i>Or Yisra’el,</i> no. 54, p. 61, who unconvincingly strain to interpret those authorities in a manner that renders their comments compatible with the position of other decisors. Cf., also, an unpublished responsum of Rabbi Falk cited in addenda to <i>Ve-Anokhi Tola’at</i>, p. 41, in which Rabbi Falk argues that these sources establish the opposite point, <i>viz.</i>, that they declare that under no circumstances can worms found in the flesh of the fish be prohibited. His understanding leads to an argument quite similar to the argument made by other writers with regard to the terminology employed by <i>Shulḥan Arukh</i> 84:16. See <i>infra</i>, note 51 and accompanying text.</i>
  In their discussions, both Rabbi Karp and Rabbi Bloom (<i>ibid</i>., pp. 22-23) assert that parasites found in the flesh are permitted only because there is no evidence, and no reason to assume, that they migrated from the digestive tract. However, they cogently contend, when there are grounds to suspect otherwise, such organisms are prohibited. That view was previously formulated by R. Shlomoh Zalman Ehrenreich, <i>Teshuvot Leḥem Shlomoh, Yoreh De'ah</i>, no. 71, and by Rabbi Halberstam, <i>Ve-Anokhi Tola'at</i>, p. 7. Rabbi Karp notes that <i>Baḥ, Yoreh De'ah</i> 84; <i>Knesset ha-Gedolah, Yoreh De'ah</i> 84:101; and R. Shlomoh Luria, both in his glosses to <i>Sha'arei Dura</i> 47:3 and in <i>Yam shel Shlomoh, Hullin</i> 3:106, declare that only worms found in a previously uncut and uneviscerated fish are permitted because it is virtually impossible that the worms could have come from the water and penetrated that deeply into the fish whereas the provenance of worms found in an already opened or cut fish is indeterminable and hence such worms are forbidden. R. Samuel ha-Levi Woszner, <i>Teshuvot Shevet ha-Levi</i>, IV, no. 83 and VII, no. 127, sec. 3, and R. Nathan Gestetner, <i>Teshuvot Le-Horot Natan</i>, IX, no. 25, similarly maintain that piscatorial organisms are permissible only when it can be determined with certainty that they were not ingested by the fish.43<i class="footnote">See <i>infra</i>, note 57 and accompanying text.</i>
  The identical argument is also advanced by R. Moshe Saul Klein, an associate of Rabbi Woszner, in a contribution to the symposium published in <i>Or Yisra'el</i>, no. 61, pp. 33-35. Rabbi Klein distinguishes the Anisakis from the worms permitted by the Sages. The permitted worms, he asserts, were sanctioned only because they are "certainly" generated within the flesh of the fish itself. Citing <i>Teshuvot Bet Efrayim, Yoreh De'ah</i>, no. 25, Rabbi Klein argues that such a conclusion is warranted only if evidenced by the absence of a perforation between the skeletal flesh and the internal portion of the fish through which a worm generated externally and swallowed by the fish might have migrated. According to <i>Bet Efrayim</i>, absence of such a perforation is required to confirm that the worm could not have bored its way through the intestinal wall. However, when Anisakis is found in the flesh, argues Rabbi Klein, absence of such a perforation is meaningless because that nematode is so tiny that any hole it created would not be of sufficient size to be perceived. As will be noted subsequently, actual presence of such holes is additional evidence of the worm's origin in the digestive tract.
  IV. Parasites Found Both in the Flesh and in the Digestive Tract
  Rabbi Karp makes one additional point. The Gemara speaks only of worms found in the digestive tract or in the flesh. However, all early-day authorities discuss the status of worms found either in the digestive tract, in the flesh or between the flesh and the skin. They do not directly address the situation in which the same species of worm or insect is found both in the digestive tract and in the flesh. The latter is virtually always the case with regard to Anisakis. As clearly expressed in numerous sources, worms found in the flesh are permitted because there is no reason to suspect that the worms originated outside the fish. Worms found in the digestive tract are always deemed to have come from outside the fish and to have been ingested by the fish. According to Rabbi Karp, if forbidden worms are indeed found in the digestive tract and other worms of the same species are also found in the flesh, there is certainly at least as much reason to assume that they penetrated the viscera and entered the flesh as there is to assume that they were generated within the flesh of the fish.44<i class="footnote">A direct association has been found between the number of parasites found in the viscera and the number found in the flesh, a fact that certainly suggests a common origin. See Smith and Wooten, “Experimental Studies” cited <i>supra</i>, note 38.</i> <i>Teshuvot Le-Horot Natan,</i> IX, no. 24, sec. 12 and no. 25, reaches a similar conclusion.45<i class="footnote">That contention, however, is contradicted by one source. <i>Hagahot Sha’arei Dura, Hilkhot Tola’im</i> 47:2, describes certain worms that migrate from the flesh of the fish into the intestine and later return to the flesh of the fish. <i>Hagahot Sha’arei Dura</i> regards those worms as permissible “because it is obvious and known that such worms are generated in the flesh.” The implication is that, when found in the intestines as well as in the flesh, even the worms found in the intestine are permitted. Rabbi Halberstam, addenda to <i>Ve-Anokhi Tola’at</i>, p. 43, understands the reference to be to worms generally found to be in the flesh but only rarely in the intestines. That, he asserts, was accepted by <i>Sha’arei Dura</i> as evidence that even the worms found in the intestine originated in the flesh of the fish. Moreover, that statement is missing in many editions of <i>Sha’arei Dura</i> and is not quoted by later authorities who cite <i>Sha’arei Dura</i>.</i> That is certainly the case in situations in which the worms are found in tissue immediately adjacent to the internal organs. Such a conclusion is even more compelling when, as is the case with regard to Anisakis, tiny holes in the intestinal wall can be observed upon examination.46<i class="footnote">See <i>Tola’at Shani,</i> II, 191. On page 96 of that work, Rabbi Revach reproduces a photograph clearly picturing such holes in the flesh of infested salmon. See also the photograph published in <i>Parasitology International,</i> described <i>infra,</i> note 53. Somewhat astonishingly, Rabbi Roth, cited <i>supra</i>, note 42, denies that, even if such perforations exist, the parasites are capable of passing through them.</i>
  In advancing an opposing position, R. Moshe Roth, <i>Or Yisra'el</i>, no. 57, pp. 151-153, contends that it is no more likely that worms penetrate from the digestive tract into the flesh than vice versa. Rabbi Roth asserts that the Sages assumed that such creatures, at least in a mature state, are not capable of piercing the walls of an organ in migrating from one site to another.47<i class="footnote">R. Joseph Mordecai Silver, <i>Or Yisra’el</i>, no. 54, pp.62ff. and no. 56, pp. 67f., concedes that parasites found in the flesh may have had their origin in the digestive tract but asserts that they migrate while yet subvisual eggs and hence are deemed to be the product of the flesh. That suggestion must be rejected in light of the evidence demonstrating that migration takes place after the death of the fish which would not allow sufficient time for the development of an egg into a mature parasite.</i> Consequently, even if mature organisms are present in the viscera, any parasites found in the flesh of the fish, even if they are of the same species as those found in the viscera, must have developed within the flesh and hence, he declares, such creatures are permissible.48<i class="footnote">Rabbi Roth, <i>Or Yisra’el</i>, no. 57, p. 153, further asserts that any creature entering through the mouth of the fish will be digested by the fish. Accordingly, he understands the Gemara as postulating that worms found in the intestines either originate in the tissue of the intestine itself or, as stated by the Gemara, are imbibed “through the nostrils while the fish is asleep.” As will be shown later, that presumption is contrafactual and, moreover, there is no compelling reason to interpret the Gemara’s discussion in that manner.</i> The authorities who rule permissively fail to address the status of an Anisakis found partially within the intestine and partially within the flesh.49<i class="footnote">See the distinction made in another context by R. Shalom Mordecai Schwadron, <i>Da’at Torah, Yoreh De’ah</i> 31:20, based upon the direction which the worm’s head faces.</i>
  Recognition of the ability of parasites to pierce the walls of the digestive tract is reflected in the fact that numerous authorities prohibit worms that are found between the digestive tract and surrounding muscle or skeletal tissue.50<i class="footnote">Those authorities clearly contradict the assertion of R. Israel Belsky concerning the status of worms found in the abdominal cavity as presented in his essay posted on  in_fish.doc. Rabbi Belsky’s intemperate remarks concerning those who prohibit Anasakis contain the canard that the many eminent authorities who ruled permissively with regard to the same issue raised in the 1990s but more recently prohibited Anasakis did so because of misinformation now presented to them. As has been documented herein that is patently incorrect. The substantive points in that article have been discussed in the earlier published literature. The online article has now been published in <i>Mesorah</i>, no. 25 (Ḥeshvan, 5772), and in Rabbi Belsky’s <i>Shulḥan ha-Levi</i> (Modi’in Illit, 5771), <i>Ḥelek Birurei Halakhah</i>, no. 23.</i> Among those are <i>Teshuvot Bet Efrayim, Yoreh De'ah</i>, no. 25, who forbids organisms found on the external surface of visceral organs; R. Yehudah Ayash, <i>Leḥem Yehudah, Hilkhot Ma'akhalot Assurot</i> 2:17, R. Chaim Pelaggi, <i>Mo'ed le-Khol Hai</i> 12:20 and <i>Teshuvot Matteh Re'uven</i>, no. 167, who declare that the entire abdominal cavity is included in the category of viscera; and R. Shlomoh Ganzfried, <i>Kizur Shulḥan Arukh</i> 46:43, who prohibits worms found on the internal fatty tissue of herring.
  As noted earlier, the controversy actually hinges upon a contingency not addressed by the Gemara. The Gemara speaks of worms found within the digestive tract and other internal organs and of worms found in the flesh. The Gemara does not address the case of worms found simultaneously in the flesh and in internal organs. Assuming that organisms can penetrate the intestinal wall, their presence in the digestive tract compromises the presumption that identical organisms found in the flesh of the fish could not have been spawned outside the fish. Thus, as has been discussed earlier, the many authorities who refuse to sanction consumption of Anisakis-infested fish arrive at that position because they assume, as a matter of course, that all organisms present simultaneously in the viscera and the flesh are forbidden. Their opponents begin with a conflicting premise: they assume that organisms found in the flesh are permitted even when similar organisms are also present in the internal organs.
  The assumption that organisms found in the flesh are always permitted and the antecedent premise that parasites cannot migrate from the viscera to the flesh are based upon a textual inference. Writing in 1957, R. Yechezkel Roth, <i>Emek ha-Teshuvah</i>, IV, no. 57, dismissed the possibility that parasites may penetrate the flesh of a fish on the basis of the blanket talmudic ruling declaring that all insects found in the flesh are permitted. In the immediately following responsum, no. 58, his son, R. Yitzchak Roth, presents a wide-ranging discussion of when such global talmudic statements may be regarded simply as generalizations admitting of at least rare exceptions and when they must be understood literally. Rabbi Roth asserts that there can be no universal statement of such nature that admits of an exception if erroneous application of the generalization would lead to transgression. Consequently, he concludes, the generalization could have been enunciated only if it is to be applied literally without exception.
  That argument is misplaced for the simple reason that the alleged universal proposition is not talmudic in origin but is first formulated by <i>Shulḥan Arukh, Yoreh De'ah</i> 84:16. Moreover, as pointed out by R. Moshe Eliezer Bloom, <i>Or Yisra'el</i>, no. 56, p. 47, the citation of <i>Shulḥan Arukh</i>'s statement is misconstrued.51<i class="footnote">In an earlier article, <i>Or Yisra’el</i>, no. 54, p. 25, Rabbi Bloom dismisses the argument based upon <i>Shulḥan Arukh’s</i> codification in the form of a universal rule with the comment that <i>Shulḥan Arukh</i> had no need to record the obvious point that the codified rule applies only in circumstances in which the underlying rationale pertains.</i> <i>Shulḥan Arukh</i> rules, "All worms found in an animal, whether between the hide and the flesh or in its viscera, are forbidden. Those found in fish: in the viscera, they are forbidden; between the skin and the flesh or in the flesh, they are permitted." <i>Shulḥan Arukh</i>'s global statement is made only with regard to animals. The word "<i>kol</i>," meaning "all," is noticeably absent in the portion of the ruling dealing with fish. It is certainly possible that the same generalization was intended; it is equally possible that it was not. The phraseology is simply ambiguous.
  <i>Shevet ha-Levi</i> explains the terminology employed by <i>Shulḥan Arukh</i> in a somewhat different manner. <i>Shevet ha-Levi</i> perceives no contradiction whatsoever between scientific findings and the discussion of the Gemara, <i>Hullin</i> 67b. He cites Me'iri in asserting that "<i>darna</i>" permitted by the Gemara is the name of a particular species known to the Sages.52<i class="footnote">See <i>supra</i>, note 30.</i> Since only that species is capable of generating worms in its flesh while yet alive, and since we cannot identify the <i>darna</i>, we must regard all fish infested by parasites as forbidden. Consequently, asserts <i>Shevet ha-Levi</i>, there is no controversy between Rambam and other early-day authorities, nor does Rambam contradict the Gemara in any way. Rambam, he explains, rules that worms found in a living fish are forbidden simply because we cannot identify the <i>darna</i>. Rambam does permit worms found in a dead fish but is careful to describe them as permissible when they "sprout before us" (<i>hitli'ah le-faneinu</i>) because only under those circumstances can worms be assumed to be generated by decaying tissue.
  <i>Shevet ha-Levi</i> fully conceded that <i>Tur</i> and <i>Shulḥan Arukh</i>, in positing a general rule with regard to all marine species, disagree with regard to this point. Those authorities maintain that the rule is not limited to a specific species but is general in nature. However, <i>Shevet ha-Levi</i> asserts that the controversy is only with regard to the assumption that pertains in the generality of cases "but all agree that the opposite also exists," <i>viz</i>., species whose flesh certainly does not generate parasites and, consequently, any parasite found in such fish must have originated elsewhere. In his original discussion in 5751, <i>Shevet ha-Levi</i> ruled that, if the scientific evidence is corroborated, infested sable may not be consumed.
  It seems to this writer that the assertion that parasites are capable of migrating from the internal organs to the flesh of the fish is supported by the discussion recorded by the Gemara, <i>Hullin</i> 67b. The Gemara begins its discussion by citing a dictum of Rav Shisha the son of Rav Idi to the effect that "<i>kuki'ani</i>" are forbidden because they originate outside the organism in which they are found. Rav Ashi rejects that view because, if that were the case, similar creatures should be found in the excretory organs as well. The Gemara later concludes that such creatures are forbidden because they might have entered through the nostrils while the animal was asleep. Hence, as both Rashi and <i>Tosafot</i> indicate, they are forbidden even if found in the liver or in the lungs and their absence in the excretory organs is of no significance. However, the Gemara, at the initial stage of its discussion, had no problem accepting the proposition that worms are forbidden—apparently at whatever site they may be found—provided they are also present in the viscera; an explanation of how such creatures found their way into the fish from outside the animal becomes necessary only if they are not also present in the excretory organs. That premise is not at all rejected. In a second version of that exchange recorded by the Gemara, Rav Shisha the son of Rav Idi declared "<i>kuki'ani"</i> to be permissible because they are generated by their host. To that assertion, Rav Ashi responded by saying, "Obviously, for, if they came from outside, they should be found in the excretory organs." Again, the clear inference is that, if the same species of insect is indeed found in the digestive system in addition to being found in the flesh, the creatures found in the flesh are also forbidden because of the strong possibility that they migrated to the flesh from the viscera.
  It is at least theoretically possible to construct a crucial experiment having the potential to demonstrate that the hypothesis Rabbi Roth, <i>Or Yisra'el</i>, no. 57, p. 151, ascribes to the Sages, <i>viz</i>., that parasites cannot migrate from the viscera to the flesh, is fallacious. A quantity of farm-grown fish might be introduced into a tank of parasite-free water. It is probably possible to modify parasites genetically so that they acquire a distinctive color as has been done with various species of fish. Those genetically modified parasites would then be introduced into the tank. If, upon harvesting the fish, only such genetically modified parasites are found in the flesh of the fish, it would defy credulity to believe that those parasites arose from the flesh rather than from the water.53<i class="footnote">A less convincing form of this experiment has already been conducted. Rabbi Halberstam, <i>Ve-Anokhi Tola’at</i>, p. 18, reports that fish raised in water that did not harbor “shrimp,” as is generally the case with regard to fish raised in aqua culture, were found to be free of infestation. Those fish were then transferred to water in which shrimp made their habitat. A plethora of worms were subsequently found in those fish. See also <i>Tola’at Shani</i>, II, 192. Those who are unconvinced by that demonstration would presumably respond that, while parasites found in the digestive tract did indeed originate in the organisms upon which the fish feed, those found in the flesh were coincidentally produced by spontaneous generation. It would, however, be preposterous to attribute an identical genetic modification to mere coincidence.A report of a similar scientific experiment appeared in <i>Parasitology International</i>. A quantity of hatchery-bred rainbow trout were placed in a tank. Anisakis larvae were embedded in feed pellets and fed to the trout orally immediately after preparation of the pellets. Other trout were fed larvae-free pellets as a control. The fish were maintained in the experimental tank for 35 days. During the balance of that period all trout were fed uncontaminated pellets. Fish were removed from the tank at 7, 14, 21, 28, and 35 days post-infestation. Worms were recovered both from the body cavity and muscle tissue at each of those times. Sections of body muscle were thinly sliced, pressed between two glass plates placed under a stereo-microscope and examined for the presence of larvae migrating into muscle tissue. Photographs were published showing a larva penetrating the hypaxial muscle in which the anterior end of the larva is wedged into the body muscle and the posterior end remains in the body cavity of the host fish. Also published is a picture of a histological section of infested muscles showing the presence of broken muscle tissue surrounding the Anisakis. No signs of infestation was found in the control group. See Karl Marx, A. Quiazon, Tomoyoshi Yoshinaga and Kazuo Ogawa, “Experimental Challenge of <i>Anisakis simplex</i> sensu strict and <i>Anisakis pegreffi</i> (Nemotada: Anisakidae) in Rainbow Trout and Olive Flounder,” <i>Parasitology International</i> (2011).</i> Less dramatic, but far simpler and only slightly less convincing, would be an experiment requiring no more than a DNA analysis of the parasites introduced into the tank containing the fish and a subsequent DNA analysis of the parasites later found in the flesh of those fish. DNA evidence would serve to establish whether the parasites found in the flesh are genetically descended from the parasites originally placed in the water or whether they are indeed unrelated to those parasites.
  V. Subvisual Organisms
  As will subsequently be explained, subvisual organisms, i.e., creatures that cannot be perceived by the naked eye, are not prohibited.54<i class="footnote">See also this volume, pp. 203-217. Rejection of this basic premise is the central thesis of Rabbi Halberstam’s <i>Kuntres Ayin Lo Ra’atah</i>, also appended to his <i>Yad le-Mikveh</i>. Rabbi Halberstam’s view was earlier enunciated by Rabbi Abraham Jacob Zeleznick, Rosh Yeshiva of Yeshivat Eitz Chaim in Jerusalem, and originally published in a little-known Israeli journal. The letter is reprinted as an appendix to <i>Ayin Lo Ra’atah</i>, p. 36. In support of his position, Rabbi Halberstam, <i>ibid,</i> pp. 27-28, 31 and 44, cites terminology employed by <i>Me’orei Or</i>, cited by <i>Darkei Teshuvah, Yoreh De’ah</i> 84:59, and two works of R. Shlomoh Kluger, <i>Teshuvot Tuv Ta’am va-Da’at, Mahadura Tinyana, Kuntres Aḥaron</i>, no.53, and <i>Ḥokhmat Shlomoh, Yoreh De’ah</i> 84. [This writer has been unable to locate the source cited in <i>Ḥokhmat Shlomoh</i>.] He also draws a supporting inference from R. Jonathan Eibeschutz, <i>Kereti u-Peleti, Peleti, Yoreh De’ah</i> 100:4. Rabbi Halberstam, <i>ibid</i>., pp. 27f., distinguishes subvisual insects and the like from microscopic creatures found in air and water in declaring that the latter are incapable of independent locomotion. That assumption, however, is factually incorrect. Many microbes are capable of movement in their environment. Many possess a long, flexible, spiral-shaped structure known as a flagellum that pushes the microbe through its surrounding medium. See Varun Shastri, <i>Microbes</i> (Delhi, 2006), p. 2.</i> It may be argued that parasites that enter the intestinal tract of the fish when they have not yet reached a state of development at which they are visible should be permitted even if they migrate to the flesh of the fish while yet in that early stage of development and grow in size. R. Pesach Eliyahu Falk, <i>Madrikh le-Bedikat Tola'im: Guide to the Inspection of Fruits and Vegetables for Insects</i> (Gateshead, 5744), p. 92, contends that, if no cognizance is taken of them in a subvisual state, their juridical identity is established at the time at which they do become perceivable. If so, their development within the fish should, arguably, give them the status of worms generated by the fish itself.55<i class="footnote">In an unpublished responsum, Rabbi Falk employs this thesis as a means of reconciling the statements of the Gemara with scientific rejection of spontaneous generation. According to Rabbi Falk, the Gemara, in speaking of worms being generated by the flesh of an animal, is actually referring to the genesis of halakhic recognition of their existence. See <i>Ve-Anokhi Tola’at</i>, p. 7. That analysis finds support in a comment of Rashi, <i>Ḥullin</i> 67b. See <i>supra</i>, note 27. Cf., however, <i>infra</i>, note 57.</i> Accordingly, parasites known to have spawned outside of the fish are permitted when found in the flesh of the fish because they acquire the halakhic identity of the host fish at the time at which they become visible.56<i class="footnote">R. Moshe Viya, <i>Or Yisra’el</i>, no. 6, p. 180, reports that this view was endorsed by the late R. Shlomoh Zalman Auerbach and R. Ben-Zion Abba Sha’ul as well as by R. Nissim Karelitz and R. Ya’akov Fisher. Cf., the diametrically opposing view discussed <i>infra</i>, note 57.If it is true that an organism acquires a halakhic identity only upon becoming visible and that this is the concept expressed in the principle <i>mineih gavli</i>, it should then be the case that such organisms are permitted even if found within the digestive organs. If so, the thesis might seem to be contradicted by the Gemara. <i>Pri Megadim, Siftei Da’at</i> 84:43, states that the Gemara forbids organisms found in the digestive tract, not because it assumes with certainty that they were generated externally and subsequently imbibed by the fish, but because, since that is a distinct possibility, they are forbidden because their status is doubtful. If so, the doubt is really a “double doubt” or <i>sefek sefeika,</i> and hence the organism should be permitted. The first doubt is whether the organisms were spawned outside the fish but imbibed while yet microscopic; the second doubt is whether those organisms were generated by the host itself and hence are entirely permissible. Since the Gemara explicitly prohibits such worms, it should then be concluded that the doubt predicated upon the possibility that external organisms reaching visual size are permitted was not considered precisely because such organisms are forbidden.However, a <i>sefek sefeika</i> is not really present. The reason that a microscopic organism entering the fish is permissible is because it acquires halakhic identity only when becoming visible and hence is halakhically regarded as the product of its host. That is precisely the reason that creatures generated by the host itself are permitted. See <i>supra</i>, note 55. If so, the two doubts are <i>shem ehad</i>, i.e., really a single halakhic doubt that does not qualify as a <i>sefek sefeika.</i></i>
  Nevertheless, it is not at all clear that this is so if the parasites develop in the intestinal tract before migrating to other parts of the fish. Worms found in the flesh are permitted because, since they feed upon the flesh, they are deemed to have been produced by the fish itself. Worms or parasites that develop in the intestinal tract presumably are nourished, not by the flesh of the fish, but by nutrients ingested by the fish and hence should have the same prohibited status as worms that originate outside the body of the fish.
  More fundamentally, R. Nathan Gestetner, <i>Le-Horot Natan</i>, IX, no. 25, sec. 14, disputes the basic contention that the status of the parasite is determined by its location at the time it becomes visible. <i>Le-Horot Natan</i> asserts that, even if the parasite is subvisual at the time it migrates to the flesh of the fish, upon becoming visual it acquires the status of a prohibited creature. <i>Le-Horot Natan</i> argues that the reason certain worms are permitted is because, as stated by the Gemara, they are produced by the flesh of the fish. In effect, the worm and the fish are considered to be a single entity. Accordingly, although microscopic larvae are themselves not forbidden, the development of parasites from such larvae cannot be attributed to the flesh of the fish and it is only a worm whose genesis can be ascribed to the fish that may be deemed to have the status of the host. Hence, the genesis of the parasite must be ascribed to its actual progenitor even if in the earliest stage of its development it is infinitesimally small and hence not halakhically cognized.
  In support of that thesis, <i>Le-Horot Natan</i> cites <i>Hazon Ish, Yoreh De'ah</i> 14:1, secs. 10 and 12, with regard to a parallel rule regarding the permissibility of creatures that spawn in "vessels." <i>Hazon Ish</i> comments that the rule applies "only when the hylic power of the water generates them." However, if an adult creature excreted eggs in the utensil, the organism that develops from such eggs is forbidden. That is so despite the fact that such individual eggs deposited in the utensil are certainly not visible to the naked eye.
  A similar argument is advanced by Rabbi Klein, <i>Or Yisra'el</i>, no. 61, p. 35. Rabbi Klein asserts that the principle with regard to subvisual creatures is simply an expression of the principle "The Torah was not given to ministering angels" (<i>Berakhot</i> 28b) and means only that such an organism cannot be prohibited if it cannot be perceived, but does not imply that such existence is to be ignored. Accordingly, if such creatures are known to have "swarmed," even if such swarming took place while they were as yet not perceivable, those organisms are prohibited when they do become visible. Rabbi Klein reports that both R. Joseph Shalom Eliashiv and R. Samuel Woszner concur in that conclusion.
  As evidence in support of that position, Rabbi Klein cites R. Jonathan Eibeschutz, <i>Kereti u-Peleti, Peleti, Yoreh De'ah</i> 84:5. <i>Shulḥan Arukh, Yoreh De'ah</i> 84:6, rules that a worm found in a fruit that has not been plucked, and hence is still attached to the ground, is forbidden provided that the worm has "crept" within the fruit. Since the fruit is still attached to the ground, the worm is regarded as having "swarmed" on the ground. Nevertheless, <i>Shulḥan Arukh, loc. cit</i>., rules that similar worms found under the skin of "beans and peas" are permitted because, due to the confining nature of the space in which they thrive, there is no room for them to "swarm."
  <i>Kereti u-Peleti</i> explains that when such a worm comes into existence it is so infinitesimally small that "no eye can perceive it." At that early stage of its development the worm is capable of at least minimal movement within the host fruit. Hence, according to <i>Kereti u-Peleti</i>, locomotion by a "swarming creature" while it is yet subvisual renders the organism a prohibited creature when it does mature and becomes visible.57<i class="footnote">Rabbi Halberstam, <i>Ayin Lo Ra’atah</i>, pp. 23 and 35, maintains that a minuscule creature destined to mature to a recognizable state is forbidden while yet subvisual even according to the authorities who maintain that Halakhah does not take cognizance of subclinical entities or phenomena. Those authorities, he contends, would concede that subvisual parasites that are ingested are forbidden since they will eventually grow to a state in which they are readily perceivable. Hence, he argues, parasites spawned in water are always forbidden regardless of their size at the time of ingestion. Cf., however, <i>Teshuvot Shevet ha-Levi</i>, VII, no. 122. That is also the view of Rabbi Falk, <i>Teshuvot Maḥazeh Eliyahu</i>, no. 91, sec. 6, and of R. Moshe Sternbuch as recorded in a letter appended to <i>Ve-Anokhi Tola’at</i>, p. 20.Rabbi Falk’s position, as expressed in <i>Maḥazeh Eliyahu</i>, can be understood as compatible with his earlier-cited explanation of why worms found in flesh are permissible only by positing the notion that, although the Torah does take cognizance of subvisual creatures that are destined to become visible, their halakhic status is determined by the nature of their subsequent visual state. If so, it does not necessarily follow that subvisual organisms spawned in water but which develop in the flesh of the fish are always forbidden as is contended by Rabbi Halberstam. Rather, if the parasite succeeds in migrating into the flesh while still in a subvisual state, its subsequent development is as a permitted creature nurtured by the flesh of a fish; hence, it was never destined to develop as a forbidden creature. Cf., Rabbi Silver, <i>Or Yisra’el</i>, no. 56, pp. 57f. However, if it is capable of migrating only after maturation, as is probably the case, it is forbidden as having already become a perceivable, forbidden creature in the intestines of the fish.</i>
  At issue is the nature of the principle that subclinical phenomena are to be disregarded. The concept can be understood in two distinct ways: 1) The Torah simply ignores any and all subclinical entities and phenomena and regards them as non-existent. That notion mirrors the reasoning of Resh Lakish, <i>Yoma</i> 73b, who maintains that a quantity of forbidden food less than the minimum for which statutory punishment is prescribed, is entirely permissible according to biblical law because, in effect, the Torah completely disregards its existence.58<i class="footnote">Rabbi Halberstam, <i>Ayin Lo Ra’atah</i>, pp. 27f., cites the comments of the <i>Brisker Rav</i>, R. Yitzchak Ze’ev Soloveitchik, <i>Ḥiddushei Rabbeinu ha-Griz al ha-Torah</i> (n.d.) and <i>idem, Ḥiddushei Maran Riz ha-Levi al ha-Torah</i> (Jerusalem, 5723), Exodus 12:2, in support of his position regarding recognition of the sub-visual state. <i>Mekhilta</i> 1:6, cited by Rashi, Exodus 12:2, reports that Moses was perplexed with regard to the precise time at which the new moon might be sanctified. God responded by showing him the nascent moon and declaring, “This is what you shall see and sanctify.” The <i>Brisker Rav</i> explains that, in replying to Moses, God announced a <i>shi’ur</i>, i.e., he specified the minimum arc that must be visible as a requirement for announcing the new month. That exchange, argues Rabbi Halberstam, presumes that Moses knew that the Torah generally does take cognizance of subvisual phenomena but that he was in doubt only with regard to the <i>shi’ur</i> for sanctification of the new moon.That deduction is not at all warranted. As the <i>Brisker Rav</i> himself remarks, it would have been impossible to sanctify the new moon on the basis of sighting the moon before it becomes visible. Moses’ doubt was with regard to whether sanctification is contingent solely upon the astronomical phenomenon of the emergence of the new moon, in which case “sighting” is merely confirmatory evidence of the moon’s return to a particular point in its orbit, or whether the new month does not commence until a visible portion of the moon has emerged. When the new moon is sanctified on the basis of actual sighting, resolution of that question is of no consequence. However, when the new moon is sanctified on the basis of astronomical calculations—as was the method utilized by Hillel II in establishing a perpetual calendar—the question assumes profound significance: Is it the exposure of even a minuscule sliver of the nascent moon that should be calculated or is it the appearance, some six hours later, of a portion of the moon of sufficient size to be perceived on earth that must be calculated? God’s reply affirmed the latter to be the case. The <i>Peirush</i> (anonymous marginal commentary published in standard editions of the <i>Mishneh Torah</i>), Rambam, <i>Hilkhot Kiddush ha-Ḥodesh</i> 7:1, cites a commentary on <i>Rosh ha-Shanah</i> attributed to Rambam indicating that it is the time of possible visual appearance of the moon that must be calculated. That issue is in no way related to the general question of the halakhic status of subclinical phenomena or entities.Nor can <i>Mekhilta</i> be cited as proof that events taking place in a subclinical state are not given cognizance even when their results later become perceivable. The fact that the non-perceived emergence of the nascent moon is of no import even though the moon subsequently becomes visible does not constitute evidence for that thesis. The response, “This is what you shall see and sanctify” establishes a <i>shi’ur</i>, namely, that the new month does not begin with the earliest emergence of the moon but commences only when a portion of the moon sufficient for it to be visible to the human eye actually emerges.</i> 2) The phenomena are indeed recognized as real but, since the Torah was given to human beings rather than to angels, no mandate or stricture can be attendant upon such entities or phenomena59<i class="footnote">Cf., R. Ephraim Fishel Siegal, <i>Or Yisra’el</i>, no. 61, p. 69. There is actually a third possibility, <i>viz.</i>, that “the Torah was not given to angels” means only that humans cannot be held accountable for what they cannot know, just as man cannot be held culpable in instances of <i>force majeure</i>, but, objectively speaking, such phenomena are regarded as existent. That formulation of the principle is quite correctly dismissed by R. Zevi Fried, <i>Or Yisra’el</i>, no. 61, p. 49 and by Rabbi Siegal, <i>ibid</i>., p. 68.</i> unless and until there is perceivable evidence of such entity or occurrence.60<i class="footnote">It has been argued that genetic engineering involving the introduction of one or more genes from another citrus fruit into the seed of an <i>etrog</i> does not result in a halakhic <i>murkav</i> or hybrid because the foreign gene is microscopic at the time that genetic manipulation is undertaken. <i>Kereti</i>’s thesis raises the possibility that the <i>etrog</i> may indeed be <i>pasul</i>, or disqualified, if characteristics of the alien fruit become recognizable in the mature <i>etrog</i>. See this writer’s <i>Bioethical Dilemmas</i> (Southfield, Michigan, 2006), II, 213f.</i> If all subvisual events are to be entirely ignored, the "creeping" of minuscule Anisakis is of no moment. If, however, subclinical phenomena are ignored only until there is clinical evidence of their occurrence, the "creeping" of the Anisakis, while yet microscopic, renders it a "swarming creature" at least at such time as it becomes visually perceivable.61<i class="footnote">A difficulty arises from the fact that fermentation of wine as detected by taste buds is evidence of the presence of yeast in the wine. There is no source indicating that wine is permitted only because it is explicitly sanctioned by Scripture. Cf., <i>Ḥokhmat Adam, Binat Adam</i>, no. 35. Yeast, however, are incapable of independent locomotion and hence are presumably not members of the prohibited class of “swarming creatures.” See Claudio Delfini and Joseph V. Formica, <i>Wine Microbiology: Science and Technology</i> (New York, 2001), p. 49.</i>
  An ostensively even stronger argument is formulated by Rabbi Halberstam, <i>Ve-Anokhi Tola'at</i>, p. 11, and by R. Ephraim Fishel Siegal, <i>Or Yisra'el</i>, no. 61, pp. 71-72. They contend that, even if minuscule creatures are not halakhically cognized, and, as Rabbi Falk argues, the species identity of such creatures is established at the time it becomes discernible to the naked eye, the Anisakis should nevertheless be forbidden. An Anisakis that reaches such a stage of development within crustaceans should be regarded either as a <i>yozei</i> of a forbidden fish62<i class="footnote">Cf., however, <i>Ḥavvat Da’at</i> 81:1, who maintains that there is no prohibition of <i>yoẓei</i> with regard to forbidden fish. See, however, <i>Tosafot</i>, <i>Bekhorot</i> 7b, s.v. <i>rov dagim,</i> who apparently espouse a contradictory view. See also Malbim on Leviticus 11:11.</i> or as having acquired the identity of the species of fish of which its host is a member.63<i class="footnote">Rabbi Viya, <i>Bedikat ha-Mazon,</i> I, <i>sha’ar sheni</i>, chap. 3, note 22, rebuts that argument by citing <i>Ḥavvat Da’at</i> 81:2 who asserts that a fully-developed entity is not prohibited as <i>yoẓei</i>. However, <i>Ḥavvat Da’at</i> agrees that, by virtue of an entirely different scriptural derivation, identity as a member of a species is acquired by virtue of the species identity of the progenitor. If so, an organism juridically regarded as having been generated by a non-kosher creature should also acquire identity as a member of the species of its progenitor.</i> Consistent with that assessment, <i>Minḥat Hinnukh</i>, no. 163, declares that, although worms that develop in the flesh of a dead animal are not prohibited as the <i>yozei</i> of a limb of a living animal, nevertheless, if the animal is a member of a non-kosher species, such worms are forbidden by virtue of the prohibition against eating the flesh of a non-kosher animal.64<i class="footnote">Cf., <i>Ḥazon Ish, Yoreh De’ah</i> 17:13. <i>Ḥazon Ish</i> asserts that worms that are not <i>piresh</i> are permissible because they are generated by a foodstuff whose identity they acquire. That view is entirely consistent with the position of R. Hai Ga’on cited <i>supra</i>, note 27. <i>Ḥazon Ish</i>, however, further declares that a worm that is not <i>piresh</i> is permissible only if it acquires the identity of a permissible foodstuff as evidenced by the fact that worms found in a living animal are not forbidden as <i>ever min ha-ḥai</i>. Hence, according to <i>Ḥazon Ish</i>, a worm generated by a non-kosher organism would always be treated as <i>piresh</i>.</i> That ruling is recorded by <i>Darkei Teshuvah</i> 84:177.65<i class="footnote">See also the commentary of Malbim on Leviticus 11:11.</i>
  However, as stated earlier, worms that develop within flesh of the fish are regarded as the product of the fish because they are generated by, and feed upon, the flesh in which they are embedded. Parasites developing in the digestive tract of a non-kosher fish might be regarded as the product of partially digested food found within those organs, but not of the non-kosher fish or crustacean.
  Moreover, and most significantly, it does not appear to be the case that the Anisakis are imbibed by krill only as subvisual larvae. The larvae are reported to be no smaller than 200 microns long and 14 microns wide.66<i class="footnote">See R. Zevi Fried, <i>Or Yisra’el</i>, no. 61, p. 54. See Jadwiga Grabda, “Studies on the Life Cycle and Morphogenesis of <i>Anisakis simplex</i> (Rudolphi, 1809) (Nematoda: Anisakidae) Cultured in vitro,” <i>Acta Ichthyologica et Piscatoria</i>, vol. 6, no. 1 (1976), pp. 133f., reports that Anisakis larvae when measured without their sheath are between 281 and 293 microns in length. The smallest reported measurements are 144 to 215 microns in length. See F. J. Adroher, D. Malagón, A. Valero <i>et al.</i>, “<i>In vitro</i> Development of the Fish Parasite <i>Hysterothylacium aduncum</i> from the Third Larval Stage Recovered from a Host to the Third Larval Stage Hatched from the Egg,” <i>Diseases of Aquatic Organisms</i>, vol. 58 (Jan. 28, 2004), no. 1, p. 44.</i> R. Moshe Viya, <i>Bedikat Mazon ke-Halakhah,</i> I, <i>sha'ar sheni</i>, chap. 2, note 6, reports that researchers in ophthalmology have established that an object five microns long and one-half micron wide can be seen at a distance of twenty-five centimeters by a person with normal vision.67<i class="footnote">See also the discussion of visibility to the naked eye, <i>supra</i>, pp. 210-217. It should also be noted that Anisakis as well as other parasites coil themselves within their host and thereby enhance perception of their width.</i> If krill ingest already perceivable Anisakis,68<i class="footnote">Some researchers claim that smaller larvae perish within their hosts with the result that “only third-stage larvae longer than 4-5 mm would moult and survive in the final host.” See Marianne Kфie and H.P. Fagerholm “The Life Cycle of <i>Contracaecum osculatum</i> (Rudolphi, 1802) sensu stricto (Nematoda, Ascaridoidea, Anisakidae) in View of Experimental Infections,” <i>Parasitology Research</i> vol. 81, no. 6, (1995) p. 488.</i> the argument for declaring such parasites permissible is entirely undermined.
  VI. Scientific Evidence
  The reasoning advanced by the Gemara, <i>Hullin</i> 67b, in permitting worms found between the skin and the flesh, <i>viz</i>., "they are generated from it," certainly appears to reflect reliance upon a notion of spontaneous generation. Whether that statement is to be understood literally and, if so, whether rejection of that concept by modern science has any bearing upon Halakhah, or whether the Gemara's statement should be understood as expressing a concept that is compatible with contemporary scientific theory are intriguing questions.69<i class="footnote">As noted earlier, scientific testimony not in contradiction to talmudic sources regarding the origin of various aquatic creatures was accepted by R. Meir Arak, <i>Teshuvot Imrei Yosher</i>, II, no. 11 and <i>Ḥazon Ish</i>, <i>Yoreh De’ah</i> 14:8<i>.</i></i> Resolution of those questions is, however, irrelevant to the points that have been made herein.70<i class="footnote">The same issues arise concerning the statement of the Gemara, <i>Shabbat</i> 107b, with regard to lice. For a discussion of those questions see chapter seven of this volume, pp. 222-228. In addition to possible resolutions of the issue of spontaneous generation discussed in that article R. David Sinzheim, noted for his role as president of the Sanhedrin convened by Napoleon, <i>Minḥat Ani, ma’amar sheva ḥakirot,</i> p. 111a, offers a rather implausible explanation of the concept “<i>heinu reviteih”</i> or <i>“mineih gavli</i>” employed by the Gemara with regard to organisms generated in stagnant water or in the flesh of animals and fish. <i>Minḥat Ani</i> suggests that those organisms are regarded as integral parts of their hosts because they cannot be nurtured in any other medium and hence members of those particular species are never found elsewhere. For that reason, he claims, worms found in cheese and fruit, for example, are readily recognized as different species.</i> Those points are entirely compatible with even the most literal reading of the text and do not at all entail the slightest negation of any empirical presumption made by the Sages. Moreover, if the notion of spontaneous generation is rejected and the various theories advanced to reconcile the apparently contradictory talmudic statements with contemporary science are rejected, the resulting conclusion that, contra unequivocal dicta and precedents spanning more than two millennia, all worms and piscatorial parasites found in the flesh of fish are forbidden is compelled. To date, no rabbinic scholar has espoused such a conclusion with regard to piscatorial parasites.71<i class="footnote">See, however, R. Isaac Lampronti, <i>Paḥad Yiẓḥak, erekh ẓeidah</i>, with regard to the issue of <i>kinim</i>.</i> To be sure, some protagonists in the modern-day controversies have been accused of harboring such predilections and certainly of rejecting talmudic teachings in favor of fidelity to science.72<i class="footnote">See, for example, R. Yekuti’el Yehudah Halberstam, the <i>Klausenberger Rebbe, Or Yisra’el</i>, no. 6, p. 136; R. Menasheh Klein<i>, ibid.,</i> pp. 147ff.; R. Yechezkel Roth, <i>ibid</i>., p. 155, and <i>idem, Emek ha-Teshuvah</i>, IV, no. 57, s.v. <i>ve-amrina;</i> and R. Shlomoh Gross, <i>Or Yisra’el</i>, no. 6, p. 203.</i> Those charges have been vigorously and vehemently denied. A fair reading of the published material exposes such charges as spurious. The authorities who prohibit consumption of fish infested with Anisakis assert simply that the Sages never claimed that all "creeping things" found in the flesh of animals or fish are spontaneously generated. Hence, in situations in which there is evidence to the contrary, such organisms must be regarded as forbidden.
  Rabbi Revach strongly insists that his conclusions are in no way predicated upon reports of scientists who reject the possibility of spontaneous generation73<i class="footnote">See, in particular, <i>Or Yisra’el</i>, no. 54, p. 44, where Rabbi Revach describes parasites found encased in cysts within the flesh of the fish that are never found within internal organs. Rabbi Revach suggests that those organisms are the <i>darni</i> described by the Sages and are permissible. Indeed, Rabbi Revach, <i>ibid</i>., no. 56, p. 52, permits white organisms resembling grains of rice found in the flesh of “all species of sole” as conforming to the criteria for assuming that they are spontaneously generated by the fish.</i> but are based upon his own observations of the Anisakis at various stages of development, many of which he has recorded photographically. Similarly, Rabbi Halberstam, <i>Ve-Anokhi Tola'at</i>, pp. 5-7, emphasizes that the available scientific information regarding Anisakis is not based upon denial of the possibility of spontaneous generation on the part of scientists but upon empirical observation. Unlike those of many parasites, the eggs of Anisakis do not survive within a predator; Anisakis survive within the body of their host only if imbibed as larvae. Hence it is not necessary to deny the phenomenon of spontaneous generation of other aquatic organisms while negating such a possibility with regard to Anisakis.
  More significantly, Anisakis have never been observed in the flesh of a fish other than when Anisakis have also been found to be present in its digestive system. Since the Anisakis found in the digestive tract certainly originate outside of the fish, it is logical to conclude that creatures of the same species found in the flesh of the fish share a common origin and were also spawned outside the fish. Photographs have been taken showing individual Anisakis partially within the intestine and partially within the flesh of the host. That phenomenon establishes that, at least after the death of its host, the Anisakis is capable of migrating from the digestive tract into the flesh of the fish. Since that is a distinct possibility, even were Anisakis to be found solely in the flesh, it cannot be claimed with certainty that those organisms did not also migrate from the digestive tract.
  Rabbi Roth, <i>Or Yisra'el</i>, no. 57, p. 151, counters that, on the basis of the statements of the Gemara, one must conclude that Anisakis never migrate from viscera to flesh and, accordingly, those found within internal organs are forbidden, whereas those found in the flesh of the same fish are permitted because they must have been generated by the flesh itself.
  As recorded in <i>Shevet ha-Levi</i>, IV, <i>Yoreh De'ah</i>, no. 83, Rabbi Woszner's interlocutor, R. Shmu'el Shmelka Friedman, <i>dayyan</i> of the Pupa community and one of the halakhic authorities of the Central Congress of American Rabbis (Hitachdut ha-Rabbanim), advanced an interesting argument designed to render the scientific evidence moot. Rabbi Friedman assumed that larvae do not emerge from the eggs of the parasite until they are swallowed by shrimp. Since shrimp are swallowed whole by sable, the parasites never become organisms that "creep in the waters;" rather, their life cycle unfolds wholly within marine organisms.
  <i>Shevet ha-Levi</i> regards that argument as unfounded. The same basic principles apply with regard to insect-infested fruit, i.e., an insect that develops wholly within a fruit is permissible but becomes forbidden when it emerges and "creeps upon the earth." The Gemara questions the permissibility of an insect that bores its way "from one date to another date," i.e., out of one plucked fruit and into another immediately touching fruit. In effect, the question is whether the insect is forbidden only if, at the minimum it becomes exposed to the atmosphere or whether it lacks the status of a creature that "creeps upon the earth" as long as it remains inside a fruit even if that particular fruit is not the organism's progenitor. The situation with regard to the parasite found in sable is the same: the parasite travels from one fish, <i>viz</i>., the shrimp, to another, <i>viz</i>., the sable, without exiting the body of a marine species.74<i class="footnote"><i>Ḥazon Ish, Yoreh De’ah</i> 14:3, asserts that the doubt expressed by the Gemara is only with regard to a worm that passes from one date to a second date that is in close proximity to the first. If, however, the second date is moved, the second date can no longer be regarded as the original, and hence natural, habitat of the worm. Rabbi Viya, <i>op. cit.,</i> note 22, seems to suggest that, since the parasite ingested with krill will never return to its original <i>situ</i>, it must certainly be regarded as <i>piresh</i>, or as having removed itself from its place of origin. Presumably, that argument is based upon the fact that the krill is digested by its host and hence the parasite is “removed” from its original <i>situ</i>. However, that phenomenon certainly does not seem to constitute <i>piresh</i> on the part of the parasite. See <i>infra</i>, note 75.</i> The question with regard to infested fruit is left unresolved by the Gemara. Since the status of such creatures remains doubtful, they are forbidden. The same principle, concludes <i>Shevet ha-Levi</i>, applies to marine parasites as well.75<i class="footnote">That argument is also advanced by Rabbi Halberstam, <i>Ve-Anokhi Tola’at</i>, pp. 10f. Rabbi Viya, <i>loc. cit.</i>, finds reason to distinguish this case from that of a worm passing from one fruit to another. Unlike the worm which leaves its original habitat, the parasite does not initiate the change of <i>situ</i> but is merely within a marine creature swallowed by a larger fish. Hence the parasite has not separated itself from its natural habitat.</i> Moreover, argues <i>Shevet ha-Levi</i>, the Gemara expresses doubt only with regard to a creature that passes (<i>piresh</i>) "from one date to another date" in which the host fruits share a common identity as members of a particular species. There is no indication of a similar doubt in a situation in which an already recognizable creature passes from one species to another,76<i class="footnote">Although <i>Shevet ha-Levi</i> does not cite a source, the position that an organism that passes from one species of fruit to another is prohibited as a matter of certainty was formulated by an early-day authority, R. Chaim ben Yitzchak, <i>Teshuvot Or Zaru’a,</i> no. 26.</i> e.g., from a date to a fig or from a shrimp to a sable.77<i class="footnote">R. Ben-Zion Woszner, <i>Or Yisra’el</i>, no. 6, p. 170, suggests that the opposite is the case, i.e., that an organism that has been swallowed together with water by a fish is not to be considered as separated from its original <i>situ</i>. Rabbi Woszner compares that situation to that of an aquatic organism that comes to rest upon a foodstuff within the same water. An organism generated in stagnant water is permitted because it is deemed to have been generated by the water. That organism is regarded by <i>Pri Megadim, Mishbeẓot Zahav</i> 84:5, as possibly permissible because it may be deemed not to have been separated from its original habitat. <i>Ḥazon Ish, Yoreh De’ah</i> 14:3, similarly regards an aquatic organism arising in stagnant water that comes to rest upon a fish within the same body of water as not having been “separated” from its original aquatic habitat.This writer finds the comparison elusive. The situations considered by <i>Pri Megadim</i> and <i>Ḥazon Ish</i> are ones in which the aquatic organism remains immersed in its original habitat, <i>viz.,</i> in water; a parasite migrating from one fish to another is not in its original habitat just as a worm that passes from one date to another is no longer in its original habitat. Indeed, <i>Ḥazon Ish</i> explicitly states that an organism coming to rest upon a fish becomes forbidden once the fish is removed from the water.</i>
  Moreover, since the eggs hatch in the water to release free-swimming larvae,78<i class="footnote">See Ronald J. Roberts, <i>Fish Pathology</i>, 3rd ed. (London 2001), p. 264.</i> the larvae consumed by the shrimp acquire the status of organisms that "swarm in the water" before being ingested by the shrimp. This factual consideration renders the exchange between Rabbi Friedman and <i>Shevet ha-Levi</i> entirely moot.
  Rabbi Friedman further argued that, although the fertilized egg of a land creature is forbidden, the fertilized egg of an aquatic creature is not forbidden until the hatchling is exposed to the natural elements. <i>Shevet ha-Levi</i> counters that such considerations are germane only with regard to the status of the fertilized egg itself but that once larvae emerge from the egg their status is that of the species of which they are members.
  VII. Eggs of Parasites
  By far the most engaging—and most novel—argument in support of ignoring the newly-presented realia regarding the life cycle of the problematic Anisakis parasites is formulated by R. Moshe Landau, Chief Rabbi of Bnei Brak, in a contribution to the Elul 5770 issue of the Israeli Torah journal <i>Pa'amei Ya'akov</i>. Insofar as factual issues are concerned, Rabbi Landau dismisses the contention that delay in processing the fish experienced in contemporary circumstances allows the ingested larvae to mature, to migrate from the stomach and to penetrate the flesh of the fish whereas in earlier times fish were immediately eviscerated and the larvae disposed of together with the entrails. Rabbi Landau asserts that herring, one of the species subject to parasite infestation, were always salted for extended periods of time and that the fish were placed in salt without prior removal of internal organs because those organs enhance the taste of the herring.
  Rabbi Landau's halakhic argument is based upon one salient point. The survival of the parasites in question is attributed to the manner in which they are transferred from one aquatic species to another. Larger fish prey upon smaller fish infested with live parasites. The parasites deposit eggs in the intestines of the predator. Those eggs are excreted by the larger fish together with a glob of nutrients needed to sustain the larvae upon hatching. The vast majority of those eggs fall to the floor of the ocean and are destroyed. A small percent of the eggs are consumed by krill and hatch within the krill. The krill are then consumed by larger kosher fish and the parasites make their way into the flesh of the fish brought to market. Rabbi Landau focuses upon the fact that the eggs, prior to hatching, do not have the power of locomotion and hence cannot be categorized as creatures that "creep in the water." His novel point is that the eggs of the parasite are not prohibited. Rabbi Landau notes that <i>Sifra</i> 4:12, as understood by Ra'avad, <i>ad locum</i>, establishes that secretions, skin and eggs of forbidden birds are prohibited on the basis of a pleonasm in the biblical text that records the biblical prohibition. But in a parallel statement, <i>Sifra</i> 4:8, according to Ra'avad, prohibits secretions of forbidden marine creatures lacking fins and scales but fails to include eggs within the ambit of that prohibition.
  However, the statement of <i>Sifra</i> appears to be problematic. There is a general rule to the effect that any substance secreted by a forbidden creature (<i>ha-yozei min ha-tame</i>) is itself forbidden. If so, even absent a specific pleonasm, the eggs of all forbidden creatures, including eggs of marine parasites, would be forbidden. Rabbi Landau explains the need for a pleonasm as recorded in <i>Sifra</i> by citing <i>Pri Hadash, Yoreh De'ah</i> 103:1, and <i>Teshuvot Ẓemaḥ Ẓedek, Yoreh De'ah</i>, no. 63, sec. 3, who state that since insects and the like are repulsive they would not qualify as a "food" and hence would not be included in the category of forbidden species but for the fact that they are explicitly proscribed as a foodstuff despite their repugnance. That biblical prohibition, argues Rabbi Landau, applies only to the creeping creatures themselves. Since such creatures are inherently repugnant, the independent general prohibition against secretions emitted by forbidden creatures is not attendant upon creeping organisms. Hence, contends Rabbi Landau, since eggs are not enumerated by <i>Sifra</i> 4:8 as being explicitly prohibited, there is no basis to deem the eggs of insects and the like to be forbidden as the secretion of a forbidden creature.
  Accordingly, argues Rabbi Landau, there is no reason to regard the parasites as forbidden until the eggs hatch as larvae and become organisms that "creep in the water." But, asserts Rabbi Landau, Anisakis larvae emerge only within the body of the host fish. Hence, concludes Rabbi Landau, since at no time in the course of their development did the parasites "creep in the water," there is no basis to prohibit them when found in kosher fish. The presence in the water of eggs from which the parasites develop is regarded by Rabbi Landau as inconsequential since the eggs are "disgusting" and hence are not a foodstuff.
  R. Shmu'el Shmelka Friedman, <i>Or Yisra'el,</i> no. 61, p. 19, similarly asserts that a creature that has not actually "swarmed" in water is not forbidden. Rabbi Friedman regards the eggs of "creeping creatures" as permissible for a different reason. As explained in the Gemara, the eggs even of kosher fowl would be forbidden as <i>yozei</i>, i.e., as the product of a forbidden organ of a living creature, but for the fact that the Torah specifically permits consumption of eggs. As explained by <i>Tosafot, Hullin</i> 64a, the eggs of a member of a forbidden species would also have been included in that dispensation and are forbidden solely on the basis of a pleonasm. Since there is no similar pleonasm governing the status of eggs of insects, argues Rabbi Friedman, the prohibition of <i>yozei</i> does not apply; rather, the eggs of insects are included in the general exclusion of eggs from the prohibition of <i>yozei.</i>
  However, the facts as presented in the scientific literature are somewhat at variance from the scenario described by Rabbi Landau. Rabbi Landau speaks of the deposit of "eggs" in the sea and the ingestion of the "eggs" of the parasite by krill. It appears that, as has been explained earlier, at least at the time of ingestion by krill, the nascent creatures are not eggs but have already hatched as larvae. Presumably, Rabbi Landau would counter that since larvae do not have the power of locomotion and do not "swarm in the water" their status is indistinguishable from the status of eggs. But that, too, is contrafactual. As noted earlier, larvae are animated and do propel themselves in the water. Indeed, some scientists theorize that it is the wriggling in the water that attracts the krill.
  The novel points in Rabbi Landau's thesis are: 1) eggs of "swarming" creatures are not prohibited;79<i class="footnote">It should however be noted that Rambam enumerates the various species of forbidden creatures and the statutory punishment for consuming them in the first two chapters of <i>Hilkhot Ma’akhalot Assurot</i>. Included also, in <i>Hilkhot Ma’akhalot Assurot</i> 2:12, are swarming aquatic creatures. Thereupon, in <i>Hilkhot Ma’akhalot Assurot</i> 3:1, Rambam records the rule that “all food which is the product of <i>any</i> (emphasis added) of the forbidden species whose consumption is punished by flogging is likewise forbidden to be eaten on the basis of biblical law.”</i> and 2) organisms not yet capable of independent locomotion are not "creeping things" and hence are no different from "eggs" for halakhic purposes.
  There are yet other grounds to argue that larvae and "swarming creatures" hatched from eggs are not included in the category of <i>yozei</i>. The Gemara, <i>Temurah</i> 31a, declares that the prohibition of <i>yozei</i> is not attendant upon a bird hatched from an egg because the egg turns to mere "dust" before the embryo develops.80<i class="footnote">For an analysis of the apparently contradictory position of <i>Tosafot, Ḥullin</i> 62b, see this writer’s “The Problem of Identity in Rashi, Rambam and the Tosafists,” <i>Tradition</i>, vol. 41, no. 2 (Summer, 2008) pp. 30-42. The comment of <i>Tosafot</i> is germane only with regard to the question of species identity but not to the concept of <i>yoẓei</i> as the product of a forbidden substance. Rabbi Friedman’s discussion is somewhat puzzling.</i> The same principle would apply to fish hatched from eggs.
  Arguably, it is possible to accept Rabbi Landau's second assertion maintaining that the Anisakis is not forbidden because it does not "swarm in the water" and to apply it in practice without also accepting his primary thesis establishing that the restrictions regarding <i>yozei</i> do not apply to eggs or larvae of such species. Developing that argument, however, requires reformulating Rabbi Landau's thesis by predicating it upon an entirely different premise.
  In comments published in <i>Hiddushei ha-Graḥ al ha-Shas</i>, ed. Y. Cohen (Jerusalem, 5729), <i>Hullin</i> 67a, the <i>Brisker Rav</i>, R. Yitzchak Ze'ev Soloveitchik, is quoted as espousing the position that, although identity as a member of a forbidden animal species is established by progenital relationship without regard to the presence or absence in the progeny of the anatomical criteria enumerated by Scripture, that is not the case with regard to <i>sherazim</i> or "swarming things." The <i>Brisker Rav</i> observes that Scripture neither names species forbidden as "creeping things" nor does it spell out identifying criteria. The prohibition is couched simply in the phrase "all swarming things that swarm upon the earth you shall not eat" (Leviticus 11:42). Accordingly, he argues, the fact that the organism "swarms" is the determining factor in establishing the prohibited nature of such a creature.81<i class="footnote">For further discussion of the <i>Brisker Rav</i>’s view see chapter seven of this volume, p. 221, note 34.</i>
  In establishing that thesis, the <i>Brisker Rav</i> cites an intriguing rule recorded by Rambam, <i>Hilkhot Ma'akhalot Assurot</i> 3:8. A single creature may be thrice prohibited, namely as 1) a swarming land creature, 2) a swarming aquatic creature and 3) a swarming flying creature. Since consumption of creatures that creep on land, swarm in water and those that fly are each prohibited by a separate negative commandment, Rambam, <i>Hilkhot Ma'akhalot Assurot</i> 2:23, rules that a person who consumes a land creature that reproduces in bodies of water and also sprouts wings is punished by having the statutory number of stripes administered three times. Rambam, <i>Hilkhot Ma'akhalot Assurot</i> 3:8, further rules that a person who consumes an unhatched bird incurs punishment for violation of the prohibition against consuming "winged swarming things that go upon all fours" (Leviticus 11:20). Even though, if eaten only upon hatching, punishment incurred in eating such a creature would be for consuming an "unclean bird," nevertheless, while yet in the egg, its appearance is that of a "swarming" creature and punishment is for consuming a "winged swarming thing."
  Thus, although R. Chaim Soloveitchik is quoted as having famously declared, "A horse is a horse, not because it is a horse, but because its mother was a horse," nevertheless, a "swarming thing" is a "swarming thing" because it itself creeps and hence, arguably, only if it actually creeps. If so, as Rabbi Landau has argued, a creature acquires the status of one that "swarms in the water" only if it actually "swarms" in the water— but not if it "swarms" only within a host fish.
  It is possible that the <i>Brisker Rav</i> would concede that the secretions— and even the eggs of—a creature that "swarms in the water" are forbidden as <i>yozei</i>, but would maintain that only when the egg metamorphasizes into an independently animated creature is its status dependent upon whether or not it "swarms." Alternatively, it is conceivable that the <i>Brisker Rav</i> might concur in the thesis that, in banning only a creature having the appearance of a "swarming thing," the Torah has obviated the stricture regarding <i>yozei</i> with regard to "swarming things." Of course, the objection that the larvae themselves, as distinct from eggs, are "swarming things" remains cogent.
  Nevertheless, this conclusion, as well as that of Rabbi Landau, appears to be incorrect. In the above-cited ruling of Rambam, <i>Hilkhot Ma'akhalot Assurot</i> 3:8, regarding an unhatched embryo endowed with wings, Rambam rules that the unhatched embryo of a forbidden bird possessing wings is a <i>sherez ha-of</i>, i.e., a swarming bird-creature. That prohibition attaches to the unhatched creature despite the fact that its wings were never used and it never engaged in flight.82<i class="footnote">Rabbi Friedman does take note of Rambam’s ruling and somewhat implausibly distinguishes the two cases: In the case of the unhatched bird, the embryo, if left undisturbed, will hatch and actually fly; the Anisakis however, is destined to be ingested by a larger fish and will never actually “creep” in water.</i> It might be countered that, despite the grammatical identity of the two phrases, the phrase "<i>mi-kol sherez hamayim</i>"(Leviticus 11:10) translates literally as "of all that swarms in water" while <i>kol sherez ha-of</i> (Leviticus 10:20) should be translated as "all of the swarming birds," i.e., creatures having <i>capacity</i> for flight but not necessarily creatures that have flown. More likely, in both the case of a <i>sherez ha-of</i> and in the case of a <i>sherez ha-mayim</i>, it is the "<i>to'ar</i>" (to use the terminology of the <i>Brisker Rav</i>), i.e., the appearance or physical characteristics of the creature that are determinative rather than the actual performance of the activity common to such creatures. Hence, any organism having the appearance of a creature that "swarms in water" would be prohibited even if that creature has not actually swarmed in water.
  Moreover, the Gemara, in telling us that worms found between the skin and the flesh are permissible because they are generated by the fish itself, seems to negate the notion that only creatures that actually swarm in the water are forbidden. Were that the case, creatures originating within the fish itself would be permitted because of that reason—a reason more fundamental and more inclusive than the reason actually advanced by the Gemara, i.e., that they are generated from the flesh of the fish.83<i class="footnote">Cf., R. Ephraim Fishel Siegal, <i>Or Yisra’el</i>, no. 61, pp. 70-71.</i>
  VIII. Examination for Parasites
  Despite his strong insistence that the creatures found in many species of fish are forbidden, Rabbi Karp, <i>Or Yisra'el</i>, no. 61, pp. 38-48, rules permissively with regard to consumption of herring without examination. Rabbi Karp reports that initial examination of sample quantities of herring yielded no parasites but that a second examination found thirty percent of whole herring to be infested. He quite plausibly attributes the disparity to the fact that the second examination was conducted by use of an electric light-box or blue light. That type of examination is not possible for the ordinary consumer.
  The crucial consideration is that a clear majority of the fish in question are free of infestation. Nevertheless, presence of insects in a significant minority of fish, a <i>mi'ut ha-mazui</i>, does necessitate examination by virtue of rabbinic decree.84<i class="footnote">For a definition of “<i>mi’ut ha-maẓui</i>” see <i>Bedikat ha-Mazon ke-Halakhah</i>, I, 4:2, note 4 and this volume, <i>supra</i>, pp. 244-245, notes 70-72. In calculating the rate of infestation in food that constitutes the threshold level of <i>mi’ut ha-maẓui, Teshuvot Ḥatam Sofer, Yoreh De’ah</i>, no. 77, rules that the controlling factor is the number of fruit infested compared to the total of number of fruit rather than the number of infested receptacles versus the number of non-infested receptacles. Rabbi Karp, <i>Or Yisra’el</i>, no. 61, p. 47, reports that with regard to fish such as herring that are cut into pieces before serving, Rabbi Eliashiv stated that it is the number of pieces likely to be infested, rather than the number of fish that must be calculated.</i> Rashba, <i>Hullin</i> 9b and <i>Torat ha-Bayit he-Arukh, bayit sheni, sha'ar shlishi</i>, followed by <i>Pri Megadim, Yoreh De'ah</i> 39:introduction, and <i>idem, Siftei Da'at</i> 39:2, explain that the Sages required such examination because failure to do so is comparable "to averting one's eyes from that which is forbidden." Despite the clear presence of infestation in a significant minority of fish, Rabbi Karp finds examination of the fish in question to be unnecessary for a number of reasons:
  1. As has been shown earlier, insects found even in the intestines of a fish are forbidden only because of doubt with regard to their origin. Although this view is disputed by <i>Mishkenot Ya'akov, Yoreh De'ah</i>, no. 17, <i>Teshuvot Bet Efrayim, Yoreh De'ah</i>, no. 6, rules that there is no requirement to examine for the presence of something that is only doubtfully prohibited. R. Shalom Mordecai Schwadron, <i>Da'at Torah, Yoreh De'ah</i> 39:introduction, sec. 1 and <i>Kanfei Yonah</i> 31 concur in that opinion.85<i class="footnote">See additional sources cited by <i>Darkei Teshuvah</i> 39:3.</i>2. In line with Rashba's explanation that examination is required because failure to examine is tantamount to "averting one's eyes," Rabbi Karp contends that the concept of "a significant minority" applies only to a minority that, if actually present, is likely to be discerned in the normal course of eating the possibly infested foodstuff. The parasites within the fish in question, even if present, will never be perceived by a person eating the fish.3. R. Shlomoh Kluger, <i>Teshuvot Tuv Ta'am va-Da'at, Mahadura Tinyana, Kuntres Aḥaron,</i> no. 53, maintains that since, as Rashba declares, the obligation to conduct an examination is because failure to do so is tantamount "to averting one's eyes," there is no obligation to search for such creatures with a magnifying glass even though, when discerned, they can be perceived with the naked eye.86<i class="footnote">See <i>Darkei Teshuvah</i> 39:24.</i> His position is that, although averting one's eyes is forbidden, it is not necessary to actively enhance the possibility of discerning the presence of such creatures.4. When examination for the presence of a <i>mi'ut ha-mazui</i> is not possible, no examination is required. <i>Pri Megadim, Yoreh De'ah</i> 39:introduction and <i>Tevu'ot Shor</i> 1:3, rule that examination that requires inordinate travail or inconvenience is similarly not required. Examination by means of a light-box or the like, argues Rabbi Karp, falls into that category.
  Of course, none of those considerations apply in situations in which the majority of the suspected fish are known to be infested. Rabbi Karp, <i>Or Yisra'el</i>, no. 61, p. 47, considers such to be the case with regard to salmon and sole in particular.
  R. Moshe Chaim Padwa of London, <i>Or Yisra'el</i>, no. 61, p. 15, finds grounds to permit eating doubtfully infested fish after they have been cooked. <i>Taz, Yoreh De'ah</i> 84:17, permitted consumption of baked foods containing raisins that might have been contaminated by ants on the basis of a <i>sefek sefeika</i> or "double doubt": 1) the presence of ants in the raisins was doubtful; 2) even if present, the ants may have been crushed in preparing the dough and hence been nullified in the much larger quantity of flour and other ingredients. However, <i>Nekudat ha-Kesef, loc. cit.,</i> disagrees and maintains that a <i>safek ha-ragil</i>, i.e., doubt with regard to a forbidden substance the presence of which is "usual," does not qualify as a "doubt" for this purpose.87<i class="footnote">Cf., <i>Ḥokhmat Adam, Binat Adam</i>, secs. 33 and 35 and <i>Teshuvot Ẓemaḥ Ẓedek</i>, as cited by <i>Darkei Teshuvah</i> 84:123.</i> As stated by <i>Shakh, Yoreh De'ah</i> 84:29, that is the case when infestation is known to have occurred in such environs. <i>Pri Megadim, Mishbezot Zahav</i> 84:16, comments that one should not protest if a lenient ruling has been issued in such a case in an instance of "great loss and enjoyment of <i>Yom Tov</i>."
  IX. Concluding Observations
  The Anisakis controversy is born of a confrontation between the principles governing nature accepted by the Sages and the conclusions of contemporary science. In other instances of such conflict, e.g., spontaneous generation of "<i>kinim</i>" versus sexual reproduction, methodology for resolving the confrontation requires that we either a) reject the findings of science as false;88<i class="footnote">See <i>Teshuvot Rivash</i>, no. 447; <i>Sefer ha-Brit</i>, I, <i>ma’amar</i> 14, chap. 8; and <i>Kereti u-Peleti, Peleti,Yoreh De’ah</i> 40:4.</i> b) assert that Halakhah as transmitted by Moses at Sinai is predicated upon real or perceived reality at that time;89<i class="footnote">See <i>Ḥazon Ish</i>, <i>Even ha-Ezer: Nashim</i> 27:3.</i> c) redefine the relevant talmudic statements as references to extinct or otherwise unobservable entities or phenomena; or d) redefine the talmudic concepts in a manner that renders them non-literal.
  Application of these diverse methods to resolution of the Anisakis problem yields diverse results. The Gemara declares organisms found within the flesh of fish to be permissible because they are not generated outside the muscle tissue. That claim is difficult to maintain when identical organisms are also present in the digestive organs, particularly when the walls of those organs are punctured (presumably by the organisms present in the digestive tract) and the punctures are large enough to accommodate migration of those organisms. Nor is it possible to assert that the talmudic statements reflect the reality of antiquity since the talmudic discussion focuses on empirical evidence employed to determine whether the worms in question were generated outside the fish. Hence those who permit the Anisakis found in the flesh of fish are forced to ascribe the genesis of those organisms to spontaneous generation no matter how implausible that conclusion may be.
  Assertion that nature has changed and that either present-day products of spontaneous generation constitute a null class or that they are unidentifiable by us would yield a diametrically opposed position. The result would be acceptance of the conclusion that all aquatic organisms presently found in any area of the fish are forbidden.
  Those who forbid Anisakis and similar parasites but not other organisms assess the talmudic statements in one of two ways. Some limit the relevant talmudic presumption to organisms with regard to which there is no compelling evidence disproving the possibility of spontaneous generation and prohibit only species found both in the viscera and flesh but accept the talmudic statements as presumptive and hence controlling only where there is no compelling evidence to the contrary. They do not regard the relevant talmudic statements as universal and irrebuttable and, consequently, their conclusion prohibiting Anisakis and permitting other species is not inconsistent.
  Other authorities redefine the underlying talmudic principle, not as enunciating a principle of spontaneous generation, but as a declaration that subvisual phenomena are to be disregarded. Upon redefinition of the principle, the application involves no inconsistency or difficulty whatsoever. Any and all organisms imbibed at a stage at which they can be perceived by the naked eye are forbidden. Those imbibed while not yet visible and found in the flesh of the fish, since they have no prior juridical identity, are treated as having been generated by the fish itself. Those found in the digestive tract, even if imbibed in a subvisual state, do not develop by parasitic feeding upon the flesh of the host but appropriate to themselves nutrients eaten by the fish that are found in the digestive organs and hence are treated no differently from organisms that feed in the water outside the fish.
  In the opinion of this writer, the halakhic arguments of those who prohibit Anisakis-infested fish are compelling. Moreover, their resolution of the underlying confrontation serves to affirm the veracity of talmudic teaching without ignoring empirical reality and does so in a manner that is intellectually consistent.
  
  Chapter 9 Is the Milk We Drink Kosher
  
  <i>He who commanded and admonished regarding treifah commanded us to rely on rov.</i>
  TESHUVOT HATAM SOFER, ORAH HAYYIM, no. 83
  I. The Problem
  The Gemara, <i>Bekhorot</i> 6b, establishes that it is forbidden to drink the milk of a non-kosher animal. That prohibition applies not only to the milk of members of non-kosher species but also to the milk of an animal of a kosher species that has become a <i>treifah</i>, i.e., an animal that has either suffered a perforation of a particular organ or that manifests any one of a number of anatomical anomalies. The perforation and anomalies that render an animal a <i>treifah</i> are enumerated by the Gemara, <i>Hullin</i> 42a, and catalogued by Rambam, <i>Hilkhot Sheḥitah</i> 10:9.
  Determining with perfect certainty that any particular animal is not a <i>treifah</i> would require a laborious and time-consuming post-mortem examination. Indeed, since perforation of the major portion of the trachea or even a minute hole in the esophagus renders an animal a <i>treifah</i>, it is impossible to make an absolute determination that the animal is not a <i>treifah</i> because, in an otherwise kosher animal, the perforation may be masked by the slaughterer's incision. The Gemara, <i>Hullin</i> 11a, determines that no such anatomical examination is necessary. Instead, the Gemara declares that the rabbinic rendition of the verse "according to the majority to determine" (Exodus 23:2) serves to establish the principle of <i>rov</i>, or "majority." The principle of <i>rov</i> establishes that, given the existence of a major set and of a minor set, each set containing entities of a single species but of differing halakhic status, one may assume that any unexamined entity within that species is a member of the major set. Applied to the <i>kashrut</i> status of any unexamined animal, the principle of <i>rov</i> serves to establish that the animal in question is not a <i>treifah</i>: The majority of animals are not <i>treifot</i> and hence constitute the major set. Therefore, despite the existence of a minor set comprised of animals that are <i>treifot</i>, any slaughtered animal must be assumed to belong to the major set of non-<i>treifah</i> animals. Hence the meat of an unexamined animal—and its milk while the animal is yet alive—may be consumed by reason of reliance upon the principle of <i>rov</i>.
  Nevertheless, despite the principle of <i>rov</i>, by virtue of rabbinic decree, one is required to examine for the presence of any factor that might render the animal or foodstuff impermissible if the magnitude of the minor set rises to the level of a <i>mi'ut ha-mazui</i>, i.e., "a frequently encountered minority." Hence, since adhesions symptomatic of an underlying perforation (or a degeneration of tissue that would culminate in a perforation) are quite common, rabbinic law decrees that the lungs of slaughtered animals must be examined in order to confirm the absence of disqualifying adhesions.1<i class="footnote">See Rashi, <i>Ḥullin</i> 12a, s.v. <i>Pesaḥ</i>. Cf. however, Ran, <i>Ḥullin</i> 11a, and <i>Bi’ur ha-Gra, Yoreh De’ah</i> 1:4, who assert that examination is required in the presence of a <i>rov</i> in all instances in which such examination is possible without inordinate difficulty, e.g., examination of the lungs. Cf. also, Rashba, <i>Ḥullin</i> 9a, who cites an earlier authority who advances entirely different considerations in explanation of the requirement for examination of the lungs.</i> Similarly, flour, rice and fruit or the like need not ordinarily be examined for insect infestation since the majority of such foods are free of insects. However, if the percentage of such foodstuff known to be infested constitutes a <i>mi'ut ha-mazui</i> all samples of that species must be examined for insects.
  The definition of a <i>mi'ut ha-mazui</i> is a matter of significant controversy.2<i class="footnote">See also the discussion of <i>mi’at ha-maẓui</i> in chapter seven of this volume, pp. 245-246.</i> An early-day authority, <i>Teshuvot Rivash</i>, no. 131, defines <i>mi'ut hamazui</i> as "close" to fifty percent.3<i class="footnote">That position is also reflected in <i>Teshuvot Rashi</i>, no. 62, and is ascribed by Rashi to Geonic sources. For other opinions regarding the determination of a <i>mi’ut ha-maẓui</i> see R. Moshe Viya, <i>Bedikat ha-Mazon ke-Halakhah</i>, 2nd edition (Jerusalem, 5765), I, <i>sha’ar bet</i> 4:2, note 3. See also R. Shmu’el ha-Levi Woszner, <i>Teshuvot Shevet ha-Levi</i>, IV, no. 81.</i> At the other extreme, some contemporary authorities regard even five percent,4<i class="footnote">R. Chaim Kanievski and R. Nisim Karelitz, quoted in <i>Bedikat ha-Mazon ke-Halakhah</i>, 2nd edition, I, <i>sha’ar bet</i> 4:2, note 3. Cf., <i>Bedikat ha-Mazon ke-Halakhah, loc. cit.</i>, who suggests even greater stringencies in defining a <i>mi’ut ha-maẓui</i>.</i> or possibly four percent,5<i class="footnote">R. Joseph Eliashiv, quoted in <i>Bedikat ha-Mazon ke-Halakhah, loc. cit</i>.</i> as "a <i>mi'ut ha-mazui</i>." The most widely cited position is that of R. Jacob of Karlin, <i>Teshuvot Mishkenot Ya'akov, Yoreh De'ah</i>, no. 17. <i>Mishkenot Ya'akov</i> adduces talmudic evidence suggesting that examination is required in circumstances in which the incidence of concern is higher than ten percent.6<i class="footnote">The Mishnah, <i>Gittin</i> 31a, indicates that, at certain times of the year, wine that is to be designated as <i>terumah</i> must be examined to determine that it has not become vinegar and therefore disqualified for designation as <i>terumah</i> for wine. The Gemara, <i>Bava Batra</i> 93b, declares that a person who buys a quantity of casks of wine must accept among those casks wine that has spoiled and become vinegar provided that the vinegar is not more than ten percent of the entire quantity. Accordingly, asserts <i>Mishkenot Ya’akov</i>, examination before designating wine as <i>terumah</i> is necessary only when the incidence of spoilage rises above the acceptable level of ten percent.</i> Since there is no evidence that a lower incidence was also regarded as a <i>mi'ut ha-mazui</i> triggering a requirement for examination, <i>Mishkenot Ya'akov</i> regards ten percent as the threshold level of <i>mi'ut ha-mazui</i>.7<i class="footnote">Nevertheless, in the same responsum, s.v. <i>u-le-inyan she’elatenu, Mishkenot Ya’akov</i> writes that “everything that is perceived as always likely (<i>maẓui tadir</i>)” must be considered to be a <i>mi’ut ha-maẓui</i>, apparently even if the incidence is less than ten percent.</i>
  Thus, were it to be established that a certain anomaly rendering an animal a <i>treifah</i> is prevalent in a particular locale to a degree greater than the threshold level of a <i>mi'ut ha-mazui</i>, it would be necessary to examine every suspect animal in order to establish the absence of that anomaly. For example, some years ago, in Israel, Newcastle's disease became widespread among poultry with the result that many rabbinic authorities required examination of fowl for the presence of anomalies caused by that disease.8<i class="footnote">See R. Samuel ha-Levi Woszner, <i>Teshuvot Shevet ha-Levi</i>, II, <i>Yoreh De’ah</i>, nos. 14-15; R. Yehudah Adas, <i>Birur Halakhah: Be-Inyan Ḥoli ha-Maẓui be-Zmaneinu be-Rei’ot shel Of</i> (Jerusalem, 5732); as well as authorities cited by R. Benjamin Adler, <i>Kashrut u-Treifut be-Of</i> (Jerusalem, 5739) 39:2 and <i>ibid</i>., 37:52. Cf., however, R. Yisra’el Ya’akov Fisher, <i>Even Yisra’el</i>, III, <i>Hilkhot Sheḥitah</i> 7:9. Cf. also, the exchange between R. Joseph Saul Nathanson and R. Moses Jerusalimski, <i>Minḥat Mosheh, Yoreh De’ah</i>, no. 8.</i>
  The rule requiring examination for a <i>mi'ut ha-mazui</i> is limited to situations in which such examination is feasible. Thus, if the lungs are lost or misplaced before the requisite examination has been carried out the meat of the animal may nevertheless be eaten, at least in situations in which declaring the animal a <i>treifah</i> would result in a significant financial loss, in reliance upon the principle that the animal may be judged to be a member of the major set of non-<i>treifot</i>. It is precisely because examination of the lungs of a living animal is impossible that it is permissible to drink the milk of an unexamined animal of a kosher species.9<i class="footnote">Even with presently available technology most adhesions would not be detected by means of an MRI or CAT scan. Furthermore, to the best of available information, an apparatus large enough to accommodate a steer is simply not available. Moreover, the inordinate cost and difficulty that would be entailed serve to render the requirement for such examination nugatory. Cf., R. Menasheh Klein, <i>Mishneh Halakhot</i>, XI, no. 114 and XII, no. 165, as well as Rabbi Klein’s later-cited article published in <i>Ha-Be’er</i> (no. 23), vol. 5, no. 1 (Tishri 5764).</i>
  In days gone by, when milk for home consumption was obtained from a single cow or from a handful of cows, the applicability of the principle of <i>rov</i> was obvious. Even though, statistically, some animals certainly were <i>treifot</i>, there was no reason not to apply the principle of <i>rov</i> to the animal milked and hence no reason to challenge the <i>kashrut</i> of the milk produced by the cow. In the modern age, commercial dairies collect milk from, literally, hundreds of cows. Milk from all those cows is combined, pasteurized and then bottled. Statistically, since a <i>mi'ut ha-mazui</i> of dairy cows are indeed <i>treifot</i>, it is virtually certain that milk bottled in a dairy contains an admixture of non-kosher milk. The percentage of non-kosher milk processed in a dairy is proportionate to the percentage of non-kosher cows in the dairy herd or herds from which the milk is collected. Thus, if, for example, milk in a specific dairy is collected from one hundred cows and if, statistically, the minority of cows known to be <i>treifot</i> is ten percent, ten of those cows may be presumed to be <i>treifot</i> and the milk produced by those cows is likewise non-kosher.10<i class="footnote">The probability that at least ten cows are <i>treifot</i> is determined by the formula 1 – P(0) – P(1) – P(2) – P(3) – P(4) – P(5) – P(6) – P(7) – P(8) – P(9)where the notation P(k) is used to express the probability that <i>exactly</i> k cows are <i>treifot</i>. That probability can be computed by<img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAA7AVwDASIA
  AhEBAxEB/8QAHQABAAMAAwEBAQAAAAAAAAAAAAYHCAMEBQIBCf/EAE0QAAEDBAAEAwUCBwoMBwAA
  AAECAwQABQYRBxIhMQgTFBUiMkFRFmEJFyNCcYGFMzdGUmJ2gqG0xBgkJVZng5GWoqXT4zhDcpOx
  tdH/xAAVAQEBAAAAAAAAAAAAAAAAAAAAAf/EABcRAQEBAQAAAAAAAAAAAAAAAAABMUH/2gAMAwEA
  AhEDEQA/ANl0rK3iezDivwpudm9kcRpc2DdUPFCZVrheY2poo5tqSyAQQ4n5DsavOViWZCL/AIjx
  TvYlBPxSrZAdaKvvShhCtfdzfroJtSqQwzi5kFq4rnhbxMhQWro8R7OukAKSxLChtHMhRPKVaI2D
  rmHLr51ZvETMrLguNO3y9Or8sKDTDDSeZ6S6r4Wm0/nKP/6T0FBIqVAIUfijecVkznbxasbu81SX
  I0IwPVogtBKtNqVzp53SSkqX1SOUgJPeqv8ABXnOWZxNzSXlV8lXNxn0PkpcIS21zeo5uRCQEp3y
  p3oDeh9KDR9KVX3EPKb7gV0GR3FKLlhbqm2poaZ1ItROkh4a/dWifiGuZJOxsdAFg0rihSo02GzM
  hvtyIz7aXGnW1BSVoUNhQI7gg73XLQKUpQKUrguEyLb4L8+dIajRY7anXnnFBKW0JGyok9gBQfs+
  JFnwZEGdGZlRJLSmX2Hmwtt1ChpSFJPRSSCQQehBrmqvuHmU3/Pbl9o7e0m14Y2paIQea3JupG0+
  ad/uTQPwj4lEbOh0Ng0ClR2HnGIzMouOMxb/AAnrta45kz2EL2IzYOiXF/CkgkbSTsd9ariwPiBh
  2dquQxG+sXcWx4MylsoWEIUd60ogBYOj1SSPvoJPSq5yHMrrgeWst5e6xIxS7yvKhXVDYbVbnlbK
  WJAHQtn810a1rSv41WMOo2KBSlKASACSQAO5NRu4Z5htvx9rIJ+SW6LannVNMynXQlDy0kghBPx/
  CdFO9gbGxUa45XB15eLYSw6tpOUXZMWYpCilRhtpLj6AR2KkgI/Qo10/E5K9m8FpdjtbTbUi8PRr
  NBZQkBILqwnlAHYciVaAoLDg361TsabySJJL1rcjerQ+lpfvNcvNzBOuY9OoGtn5V27dNiXGAxPg
  SWpUWQgOMvNLCkOJI2CCOhFQrFrveWr0zjGP45bpWL2plqJ7Y9qqTrkSEKbS15JClpKSCErKfkVB
  W0jz8IliwcZ8mwNr3bfKgtZBAZHwseY4W5CU/RKnAF67AqV9aCzKUrzMqv8AacXx+Zfr5MRDt8Nv
  zHnV/L5AAfMk6AA6kkCg9OlQrhrdsqydKsou0ZFnsstr/JdqU2DILZIKX31/mqUOzaeiQrqSe3Hx
  czq5YVYZs61Yjcr87EiKlvLQUtRmGxvaluKPUjlJKUBR11OgQaCc0qpfC1n9/wCJGDXbIMhVGD6L
  07HZbjtciGmgyyoIHcnqtR2ST179qtqgUpSgUpXDObeehPsxpKory21JbeSgKLSiNBQB6HR66PTp
  Qc1Kr7C83nIy93h/mzceLkjbXnwpLAKY91j9fyjYJJSsaPM3s60SCR24+Pl4lwsatNgt8hyLKye8
  xbL57Z0tlp1R81aT8jyJUAe4KgRQTizXa23mO7ItctuWy08thTrfVHOk6UArsrR6EjY2CO4Nd2oP
  xPyqHwo4aKvUPHnJlutoajpixXENJYQSEIJKuyeYpT7oUfeHTWyBzubD4j2bEb1jxgIvsd922SUz
  A6pSmUhbjbqAkBCgk791Sx99BOK8ax5Tj18vF1tFou0ebNtC0t3BpklXp1qKwEqPbm22sEA7Guuq
  9mqLsl8OPcauM86PHZlzSbIiHDW6psynjEXptJShauY9eyT22dDZAXPebtbrPHakXSW3EZdeQwlx
  zYRzrOkgnsnZ6bOhsgdyK7tR+yJumRYu/HzTG4NtVMSpp63onesSWlJAIWry0Dm6qBA2OgPMd9Ir
  wOyRb2MXKxXicXZeN3iVZS+8ra322VDy1k/M+WpAJ+ZBNBS/4Q3+A37Q/u1apuVxt9sirlXKdFhR
  0DanZDqW0JH1JUQBWVvwhv8AAb9of3atFR+HPD2O6HY+B4sy4OoUi0MJI/WEUXjPj0SRxk8VNqyv
  G47zmLY0Y6XLqUFLL6mHFO6QT8RK1cvT5DfavUnXNXEXxmQrI+rzbNhzLjzbJ6pVISlPM4R9Q6tA
  /wBWPrWlGWmmWktMtobbQNJQhIASPoAKynwDbXD8Z3EJmWCHHU3FTe/oqW0tP/BQavrI3gCmRLfB
  z+dOkNRozCIC3XXFBKUJHqdkk1rmv51cJMQyjKOHeay8VuU1qXZ37fNXCjrKfVhPqCCNdStBHMkf
  p6E8uhGubFZuKl/v8u6DiHd7JjC1bgRpFphGc6n5qVtkBpP8UKBXr4gDVlzLQxcMbdsV2dcuLEiI
  Yspx5KQt9KkcqlKCAEgq6k8oA2egHaq38M/FdniXhoRPcQjIralLdwaGh5o7JeSPorXUDsrfyI3b
  NEZ28F+TTUxMj4bXV8vP43LUIqlHr5RWpK0foStO/wCnr5VMPx2/6IuLP+7n/cqqvC+2t7xTcTJj
  O/Shc9JI7bVOSU/1JVVq/wCEfwY/zy/5ZL/6VFPx2/6IuLP+7n/cqQ4DxE+114dtv2FziweXHU/6
  m92n0zC9KSnkSrmO1nm2B9Eq+lR7/CP4Mf55f8sl/wDSqQ4DxZ4f55eHbRil/wDaM1mOqS416N9r
  TYUlJVtxCR3Wka3vrQTis5eNDIpsgYvwytj6mXMimIMsp7lvzEobT94KyT/qxWjayh4oG1x/FPw0
  nvgiIVQEgnttE5RV/UpNCNS2iBEtVqiWuAylmJEZQww2nshCQEpH+wCu1SlEZ84Y2aDlHHDipksx
  mMjHLfPj2tLHIA269Fb27zjsUpWorKTsFRST8PX1fCLLjXTEr1lj0lj2jlt7m3dLBWA6iMHfJbHL
  35E8hAP31aFlw/G7LZrnaLZbER4V0ffkTmw4tRfde/dVqUSVbV+n9Gq4cDwTD8EtqrfiOPwrSwvX
  mFlG3HNdudxW1r/pE0HDxex1jK+GWQ2F9sL9TBc8rY+F1I5m1fqWlJ/VVY+FbOb/AJXwRciQxDm5
  BY3RBb9e+tttxv3S2pa0pUrogqT0B2WxvW91ekpaG4zrjuuRKCVb+mutZZ/B6sPptuZSVBXkLeho
  QflzJS8Vf1KTRXe4Y8WuLmXcZL7hjsfHmHrfHksrbS2r08Vxp9Danidlx3XvJCQpIJWCdAdOfi1l
  vG3hBcYORXS/2vLcZkPhl5pNtRF8pR2eT3dqTsA8quZQ2Oo7A+F4bf8AxhcSv2r/APYNVaPjKXHT
  4fb4HuXzFvRQzv8Aj+obJ1/RCqD44iXOFfbdw14rW1SlWmBcm35KlDqxGko8pa1fTkUU8306n5Vz
  casZm51xHwjGXIlzTYIypNzuM2OFoQ24lHKwkOp6JXzFR1vejuu/4arMtjw84zarxGS6iRCdW4y8
  naVtPOOLSkg9wULHT6VP8ctEexWdi0xH5TsaPtLPqHS4tCNkpRzHqUpBCRvZ0Bsmgg/BaRkNqsUL
  B73ZJolWMOQ3bipktxnmG+kdxtRGnCtBTsA7SUq5tHQVwYzFN58RWTZQx1h2ezMWAOD4XHlOGQ6k
  H6oBQD9CrX1qyZzBlQX4yZD0cvNqbDzJAcb2NcySQQFDuOh611ces1usFpZtVqjCPFZ2QnmKlKUT
  tSlKPVSlEklRJJJJNEehWafEHdF5p4gsI4UcxVaWZDU65Mjs8r3l8ih9A0g/+4fpWlqyhdG1xPwg
  8N6SCG5CUqZ38wbcpA1/TBosavAAAAGgOwqJ8Z/3ns0/m/P/ALOupZUT4z/vPZp/N+f/AGddEVL4
  Bv3nrt/OB7+zx60NWefAN+89dv5wPf2ePVy599t/Y7X2C+z3tP1CfN9t+d5Hk8qt8vle9z83JrfT
  XN91FupDWbPFFxZ4pcOp8SJAj49BgXMv+ilNc8mRytlIJWFpShKtLSdcqh11s662LZfx/wDtiF7Z
  /Fj7M9Q36z0nrvP8nmHP5fN7vPy71vpvW6pv8Ib/AAG/aH92oRa2c2/jtPsrl5xvJLJZZaEea3Y2
  4CXlKGt8ipDmwXPl0QlO+m9da6/hj4wSeJdqn22/RWomQ2op9QG0lKH2ySAsJPVJBGlDtsgjvoXJ
  WU/CnCXI8R/Eu8QU/wCS2nZbCVJ+Hbkzmb/4W1UE18aEF+Lgdoze1uKjXfHLo09HkJ+JCFnlI/QV
  hs/q1867vES6DLeFOC8TYjJ8q03SBfJjSAVFDCdpkAfXk5lEn6INd3xiLQnw9ZEF62pcQI/T6lo/
  /ANdvwqxXEeHnF48xHN5jD6ilY2Chb7qkgj6cpFBx+KtiZd+BF4ttlt1wu0u4riojNW+K5JUvT7b
  pVpsHSeVCjzHpvQ3sjfU4gSJM/jlwmu8SyX96BFanuypCbTI5IwlMJbZDp5PyauYHmSrRQOqtDrV
  m4xY4eO2sWu3LkeibWosMuuFYYQezaCeoQPkk70Og6AAepREe+x1o9se1fWZD6j1HqOT7Qz/ACOf
  m5teT53l8m/zOXl101rpVEXrF7leOMHE3KIELIrNeLeYU3HbmbTJLT640csvtDSNOoWSE8qdlQ95
  IUAd6YpQQrhRms7LcX9bf8Zu+M3OKhPro9xhOx296PvtLcACke6T32npvuCY3wKsLNxsN9yiYysN
  ZHkE26wgfdPplqCWiR/KSgKH3KFWLk9jhZFaza7kp8wlrCn2WnCgSEDu2sjqUH5pBGwNHoSD6TTb
  bTSGmkJbbQkJSlI0EgdgB8hQZw4+8I+LHFm42x6Q5hlqi21DiWGUXGS6olwp5ipRjjfwJ+Q7Vd+C
  qzQW7yc0i2JuU0hCUP2uW66l86PMVJcaRydhrRVvZ7a6yKlArPfF+yucPuO9g4yR21exJakwMgUk
  bDHOnykvK1+brk/W2P4wrQlcFxhxLjAfgT4zUqLIbLbzLqQpDiCNFJB7gig83KnsiNk58QYtEqe5
  8CrhKW0ylJSff222srO+X3egIJ94a0aT8N/CXiRwnu1x9SvFLlbrsqOmV5c+Qh1kNlfvIBY0o6cV
  7pI2QPeFXDgeKnEIb9qi3eXLs6VD2dFkgKVBR120lz4lo7coVspA1sjWpJQZnuPBLiXa+M8ziBw/
  l4nYG3ZClphuTZC0OpV8YWkMAALI5igH3Seh6A1deS5ZIw/hpMyjMGrfGmQo6lusw5CnWXHdkNob
  UtKFErPKNFI0SR1A2ZbUUu2Ew75l7F9yCY7co0BSXLZbFpCY0Z0DReUn/wA1ze9KV0SD0G+pKgXh
  MwG4YnhczIMgaU3fskeEyShadLab6lCVD5KPMpRHy5gD1FXRSlEKUpQKprxX8P5+YYVFvePtKXkG
  OvGZEQgbU6joXEJHzV7qVAfMp186uWlBH+HOUwc1wq15Lb1pLc1gKcQD1acHRbZ+9KgR+qpBUTtO
  ExbHmUm/2Cc9botwUpy6WxKAqNJeI0Hkju05vXMU9FAdRvrUsoKY4g+30eIvAbRbszv4h3FUufcL
  WlxpEZpiO0nkACG0rUFuEAhalb69ugq2rfd7TcZcyJb7nClyILgbltMPpWuOs9kuAHaT9x1WbOKO
  T3GD4gsjyZhlabJjlutdgu1wbe5FwmZkgOuuN9D+UAUgFX5qdnYOq0PimLY3i0RyNjdkt9raeUFv
  elZSgvK18a1DqtX8pRJP1oIv4g8qTivC66LY5nLpcmzbrZHbHM49IeBQkJT3JAJVr+TXF4c8AVw6
  4YQrPLSkXOSozLgU9dPLA9zfz5UhKfoSCfnXtw8JhqzRzLr3MdvFzbK0W0PJCWbcyr81pA6c5GuZ
  w7UfuHSvfvUuTAtj0uHa5V1fbA5IkZbSHHNkDoXVoQNb2dqHQHWzoEMh8FGL874ueILuPS4bMpiR
  dHFtS0KLUlv1yEltSk9UdVBQUAdFI6EE1d+Y8O8m4lXi2N57JtUDGbc+JPsi1vuPqmOgaBdeWhvS
  QCRpKfzj17EVpwhxDihiPHLJc+uvDic7CvnrPyEa6QVOs+dIS8PieSFa5ddx3rUlFr5abbaaQ00h
  LbaEhKUpGgkDsAPkK+qUohSlKBVC+KDFp8C/Y3xgsUVcmXjL7ZuTDY2p2IlfMVD/ANO1g/cvfZNX
  1X4pKVpKVJCkkaII2CKDq2W5QbzaIl2tkhEmFMZS8w6g9FoUNg/7DUP4xWvO8hxm543ikbHURrnA
  XFfmXGa8hxvzApCwltDKgfdPRRX3J93p19HDMLj4hcpqbHPfYsUnmcRZ1JCmYrylbUplXdCD1231
  Ts7GuxlVBR/h14fcSuFtoex+a1idytkq4erdeZuUhDzPMlCFlKTHIX7qAQklPX5jexeFKUCsm/hD
  f4DftD+7VrKsx+LLCeI/E67WeNjmCzBEsypSfVP3CGgSfMLYCkJ87YTprY5tH3uoGqLFt3OFxacs
  CrHCn4wiQpryfbynHg8BrXmCKEFIc11/deXfy10r0eEfD6zcN8TRYrSVvuLWXpct0flJLp7rV9B0
  0B8h9Tsn28WuVyulrEm7Y9MsMoK5VRpL7LpPQHmSpla0lOyR1IPQ9B0335yJDsJ9qJITHkLbUlp5
  TfOG1kdFFOxzaOjrY3RFGeKcTM2n45whx9XPcLnKTOuK0jYiRG9jnc+gJOx9SjXzG7rx+1Q7HYoF
  lt7flxIMduMwn6IQkJH69CvIwfDLXiolyWXH593uCw5cbpLIVIlLHbmIACUjslCQEpHYVJaBSlKB
  SlKBSlKBSlKBSlKBSlKBSlKBSlKBSlKBSlKBSlKCEt8McZ9k5jbZHrJbWYSHn7op9xJVtxAQEtkJ
  HKlAA5e5Gt7Ne3g+L23D8cj2O1rmOssgcz0uQp995WgOZa1dSdAD5AAAAAACvbpQKVGeE90n3zhZ
  iV7uj/qJ9wskKVKd5Ep8x1xhClq0kADaiToAD6VJqBSlKBSlKBSlKBSlKBSlKBSlKBSlKBSlKBSl
  KBSlKBSlKBSlKD//2Q==
  "> [<img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAApADEDASIA
  AhEBAxEB/8QAGwAAAgMBAQEAAAAAAAAAAAAAAAgFBgcEAwL/xAA4EAABAwMDAgMEBQ0AAAAAAAAB
  AgMEBQYRAAchEhMIMUEiI3GhFBg3VKIWQlFSYWJzdIGRk7O0/8QAFQEBAQAAAAAAAAAAAAAAAAAA
  AAH/xAAWEQEBAQAAAAAAAAAAAAAAAAAAARH/2gAMAwEAAhEDEQA/AN83WvypR7ppe21lLZN11gFa
  5LiOtumRgCVPrT5KVgHpSeCQM+YBu1nW7EtijCnxpEuW4tZeky5bxdekvEAKcWo+pwOBgAAAAAaX
  bwlz1XpvbuLfctXceV0MsFXPQ044roSPghhA+A00Wi1BX9cjdo2jUriep02oNwI631sRUpKylCSp
  RJUQlIASSST6cZOAczuffk0faGLfkay5tQdXCjzZsRExDbUFD/T20uPqTytQWghCEKVhQJABBMp4
  sahLh7FV6DTkLcqFY7VKjNpBJWp9xKFDj9wrP9NUTxDQqXado7XbboZmPUyRXoyp5jxXJDshiNhb
  uUNgqUpa1JPAPPpoje6jTotzWwYFZhuNtTWEF5gOqStpXCsBacEKSoDBGDkZ1ntjXfU7a3FXtVek
  5c2S4z9Jt+rO4C58fn3buOO8jpUM/nBOTg46tNpMw1CmR5qocqEXkBfYlICXW8+ikgnB/ZnjS8eO
  hD9Lo9nXjT1lio0qrlDDyfNJUjuD5sj56EMjo0v31nLe+4j/ACaNBVvBBFXb1+7i2rLBRKjOMoUk
  +fuXHkE/jH99NVrD9yLekbf7uxt5KNGdepMlr6HdEdhJK0MnAEkJHJCelBUBz7GfUkbTT5kWoQWJ
  0GQ1JiyGw4y80oKQtJGQQR5gjRa9lKCUlSiAAMkk8DXJTKtSqoqQmm1OFNVGX23xHfS4WlYz0q6S
  ek45wdfVWgU+p02RAq0OLNgvoKX2JLSXGnE+oUlQII+Ol78PtyW9Y+z0262aHOd/Kq4KhUabRqHT
  y9JWyHe2hDTKMYQhDY5OEgKHPtDJDHaXHx9yUjbSg08cvP1pLiEgZJCGXQfm4NMSy+lcRElaFMJU
  2FqS7gFAxnCvQEeusMdpJ3n3lptyFsrsW0lKEJ5Q9ipzOoFSm/1mkqSgFXkejAyCcFjGvq1XX92V
  89Gnd0aGhQCgUqAIPBB9dRtu0Gj27Ddh0SA1AiuvqfUyzkNhasdRSnySDjOE4GcnHJ1JaNEUjfq4
  Ba2zF21zudtxilvJZVnydcT22/xrTrK9mn07QXjFsW8qogoqlu092jyHWwkh5v3T8NvAyo9ag4Ej
  JPWo8knV58U/2NTv5+B/1ta6L/8At020/hVb/SzoL7XaTT65SX6VVYyZUKQAHmVKIS4AQcHBGRxy
  PIjIOQSNdMVhiLGbjRWW2GGkhDbbaQlKEgYAAHAA/Rr00aA0aNGg/9k=
  "> is the binomial coefficient defined as <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAA3AGwDASIA
  AhEBAxEB/8QAHAABAAMAAwEBAAAAAAAAAAAAAAQGBwIFCAED/8QAOxAAAQMDAwEGAwUECwAAAAAA
  AQIDBAAFBgcREiETFBcxQZYIWNMVIjJRYRYjQnEmMzZDYmdzgYKh4//EABQBAQAAAAAAAAAAAAAA
  AAAAAAD/xAAUEQEAAAAAAAAAAAAAAAAAAAAA/9oADAMBAAIRAxEAPwD2XSlKBSlKBVW1Pz3H9OsW
  fyHIlyu7NDo3FYU64s7gdAOgG5A5KISCQCRuKtNYv8XapFxwWxYTCcSmVleRwbcOSeQS2HO1Usj1
  Sns07j9aCx5prHiuKyLRGmRL1MkXOTHjdnChdoIjj+xbQ+skIbUQd+HIqI6gEdai60ouFxzbTKwQ
  X3WkPZF9oSwhZSlbMRlbhC9vNPMt9D05caomtdwxmyauaUYbcbizAtFukyMgmLkubqeebSoR1H1c
  dW6Vn1Uok7bmtOj3rEr5rSxBjX9DuQ2OyPKXau7rSppqSuMrtVKIACtkNjj5gOddt6C+UpSgUpSg
  UpSgUrGJemutrsp5xj4h3mGlrUpDX7JRVcEk9E78+uw6b1+Xhlrj8xr3tCL9Sg2ylYn4Za4/Ma97
  Qi/Up4Za4/Ma97Qi/UoNsqLMttumS4suZAiSJMNRXFddZStbCiNiUKI3SSPUVjvhlrj8xr3tCL9S
  nhlrj8xr3tCL9Sg156zWh68s3p61QXLoy12TU1cdBfbRuTxSsjkE7knYHbrX1i0Wpi7yLwxbITVy
  lIS3ImIYSl51CfwpUsDkoD0BPSsg8MtcfmNe9oRfqU8MtcfmNe9oRfqUG2UrE/DLXH5jXvaEX6lP
  DLXH5jXvaEX6lBtlKxPwy1x+Y172hF+pTwy1x+Y172hF+pQbZSsT8MtcfmNe9oRfqVe9PcczOx2V
  6JlWoC8pmrkKcRLVamonBspSA3wQog7EKO/n97b0oO6zH7WGPSnLNeINoktp7Qy5kJUpttCeqiWw
  43udh58un5Gql8OF+yXKtHbJkuWSkSbncw7IJQwlpKWi6oNgJT6cAk/n18zXD4m72qwaC5fNaKu2
  et6oTIT1UVyCGU7fru5/1VS0pxuwZ5j8S15FFuLTeCzGbXGsq5CmUsOxkNlEh1Lah2q1jioct0gH
  iATyUoNAzrMpdhzvCMWhQWn3MkmSG3XXFH9yywyXVqAHmT0G5Ow39ak6lZenFLZDbiRUz75dpSYN
  ogc+PeH1eqj/AAtoAK1q9Eg+pAMa4WZ+461Wq8PxXDDstjkJjvKQeHbyXmwdj6qCGDuPQLH5iqdK
  K7v8ZMONK3VGx/DVy4aD/C/Ik9mtY/m2OP8AtQazZGJ8a0xmLpOE6clsd4kJaDaXF+ZKUj8Kd+gH
  U7AbkncmZWY/FNNRbdBMruCp82EWYSg2uLIUypTi/wB22kqT1481pJAI322O4JBx/WLEZLuhGBYp
  dH57dwusy12O029EhbaIiSE83nUJIDrpQhXLnulHPZI3BUoPV1K4tIS00ltA2ShISkfoKxvx9/yV
  1m9r/wDrQbNWSa/X7O8fuGKpxbIrbCZvd8h2juqrX2sglxSi46HVOFISEJ8uz/5dRVo0yz/9uftD
  +hWZ4z3Lsv7Q2vufeOfP+q+8rlx4fe8tuSfzrKPiYyxyy6uYhIYtki6DFrRcMhkx2k8kpBSGG3V9
  R91BLivPc7bDqaD0VVPm5LJtmrlvxic4kwL5anXraeIBTJjLBeRv5nk262ob+XZq/OvmC4Vj1quk
  jMYkqXdb1eIyBJusiUtfeGzspIQjfg2jy2ShIAG3nVL1ZW5L+JLR+3w9y9GN2mySn+7Z7sEAn9FE
  lP8AOguOomZS8cybDLDBgtSXsjupiLU4o/uWUNqccWAPM7J289hvud/KrpVLvdmfuesmN3N2K4YV
  ktU5xDxQeHeX1MNpAPluG0PdP8Qq6UFR1PwpOcw7JBfuaocS33qNc5DSWefe0sKKgyTuOKSriSev
  4fKobGnSY2rc/O4WQ3GGxco7CZ9qYCUtS32QpLbq1/i2CCBxG2/EbkjpSlBNxnFbzas8yPI5+ZXO
  6wbr2Qh2l4bMW8IGx4feIJUfUBP68j1rrMgsqrfrhjmZsJCk3C2yMfmDcbpG/emV/rsWnUn/AFB6
  bmlKCfq3gzeoWOw7DJuSoUNu5R5kpKWQ53ltpXPsepHEKUE9evl5V1mrGmcjO8oxK9M5ZNsP7OPv
  vpTEjNuOOrcShIUFObpQUhKtiUK/F6bUpQXy2REwLdHhJfkPpYaS2HZDpcdXsNuSlHqpR9SakUpQ
  KpKNPIb2oOT5XdZpuDd9tTNpEJTPFMeMkK5o5bkq5qWonoNvLr50pQcdMcSVpnght1yyq532Pb4+
  yH5aQkR4zQPBtCEDySnoSd1Hbz2CUiPp3j0qblVy1Lv7Qbul1joiW6NyCu4W5J5IbJBILjij2i9i
  QCQkEhO5UoOzsWK3mBqRfsql5lc7hbrkw01Esrg2jQeAAUpP3iCpRBO4CfxHfl0IttKUH//Z
  ">]If the incidence of <i>treifot</i> to non-<i>treifot</i> is 10% , p=.1. For a sample of size 100, the value of n is 100.The probability that exactly 0 of n cows is a <i>treifah</i>, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAArATgDASIA
  AhEBAxEB/8QAHQAAAgIDAQEBAAAAAAAAAAAAAAcGCAMEBQIJAf/EAEkQAAEDAwMCAwIIBhEFAQAA
  AAECAwQFBhEABxIhMQgTURRBFSIyUmFxgZEJFhdCkqEjKDM3Q0ZXYmd2haWmsbTE0iRTcoSVwf/E
  ABYBAQEBAAAAAAAAAAAAAAAAAAABAv/EABgRAQEBAQEAAAAAAAAAAAAAAAABQTER/9oADAMBAAIR
  AxEAPwCz/wCU3bb+UK0v/sx/+evbG4+3khwNsX5azqz0CUVdhRP2Beq0/hA6VTolTtGpRobDMuYi
  Y3IdQgJU6EFkp5Ed8c1d/XVsqjSKTUoJgVCmQpkQp4eQ+wlbfHtjiRjGitqO8zIYQ/HdbeaWMoWh
  QUlQ9QR3171VB9+Rs14qaTatrSHkWxcqo6naUXCppkvuKaykH5JStPIH0OO2m3vhf9So9UotgWet
  s3dcbgbYdWnkmCxkhUhSfeQAogfzVH3YI8M5uVGclPRG5DK5DASp1pKwVthWeJUO4BwcZ74OtOu1
  +hUFlD9drVNpTSzhK5spDKVH6CsjOo5Ep9A2o27qlSQHXxDjOTqhLfXykz3UpJK3Fnqpaj0HuGQB
  gaW3hOhSLui1fdi6uM6u1SatiI44OQiR0Y+I0D8gciodPckep0Q8qPVqXWYYm0epQ6jFJwHor6XU
  E+nJJI1mEuKZpgiSz7UGw6WOY8wIJICuPfjkEZ7ZGq9eIKado9yra3JoLfs0KqPKhXBEZHFuWkYU
  lZSOhc4leFd8pT6nLY3HtBm76OzUqNMFPuKGjz6NVmDhbSyMhJP5zS+yknIIPbONBM9Gl9sVuKnc
  K1nnJsdMGv0t4w6vC/7TycjkB34qwcehCh1xnTB0Bo0aNAaNGjQGvLi0NNqccWlCEAqUpRwAB3JO
  vRIAyTgaRtBqqt8r5qjXmr/J3bz4Y8hCsJrModcuEd2U9Dw7KyknOcAHdGfZkx2pMZ5t5h1AW242
  oKStJGQoEdCCOudZNeWm22mkNNIS22hIShCRgJA7AD3DS/8AEXd1UsXaCu3PSH48eZEZCWXHUBeH
  FrS2jCT0J5KB69MA9PQGFo0mrxui84N5bRW1BrRbmVt1a6w37K0RJYYYSt5SspJb5Htx44yR9Tl0
  GGFLizorcuFJZkx3BlDrKwtCh6gjodZtJTc193Zqux77ojK/xTqMtLNxUtofsbK1/JmMp7IXnooD
  AUSn3nIcsGVGnQWJ0N5D8aQ2l1l1BylaFDKVA+hBB0GbRrSRWKS49MZRVIKnIKQqWgSElUcHOCsZ
  +KDxVjOOx9Na1Due2q6+6xRLhpFUdaGXEQ5rbykD6QknGg5u493tWlTInkxDUKvU5SIVLgpXxMh9
  fbKuvFCRlSlY6AeuBrlUi6Lhp24kKzbsNIfdqlPcmQZNPZcZSFtKAcZUla1lWAoKCwU5wfijUcqr
  blb8W1IjSMmJbtsOz2EnsH33S0o/oY/R15lrcuDxZwW4oKo1qW+tcpY7IfknCWz9JQQr7DopxaNG
  l1vxuN+T+24yKbGTOuOsPiHR4Z7OOkgc1D5qSpPT3kpHTOQRPzKjCamEZDIlKbLoZ5jmUAgFXHvg
  EgZ7ZI1m1ENs7LRa1PXMqUpdWuWoALqtVe6uPufMT81pPZKBgADtnOuJ4g6RXZ23tw1Cn3dUaLGp
  9IflCPBQlC33G0LWQt05VwIATxTx95JOcAGHEmw5in0xJbEhUZ0svhpwKLTgAJQrHyVYUk4PXBHr
  rPqvXgIJVtBV1KJJNwPEk+//AKePqwugNGjWrValTqTDVMqk+LAjI+U9JeS2hP1qUQNBtawzZcWF
  HVJmSWYzCSApx1YQkEkADJ6dSQB9JGudXLntqhPNs1u4aRS3XOqETJrbKlfUFEZ1tzYtNrdIciy2
  YtQp0xritCwHGnkKH3EHQbmjSWti4JW2e6sbbCty3pNu1hvzbYmSFlS46s4MNaj1UAeiCckAoHXP
  Td8UNaq1JtWiRqDWarTqlVa0xT2kU8oDjqXM8u6SroB04lPVQydA3NGlDXpNeXv5Y9Fj1me04IEy
  o1mK1KWY3s/ENso8v5KsOEjmocievToAaBV/hDf4jf2h/ttWAlXXdZaIgbX19T2Pi+11CntN5+lS
  JDhA+pJ1Xz8IY4gvWQyFAuJTPUpOeoB9nAP6j92rZsuIeZQ80sLbWkKSoHIIPUHRcJazdpK7Ut2F
  bp7kToL1Vax8HUyAVKYiAAhGVqAKikEnAGORKs+7UD2tmLunxv3ZUJRKzSI8pmOD2bDKm43T0+Uo
  /Wo+urT6qztpCXafjguinygUCtRpT8Yns4HVIkdPXHBY+tJ0DT8Wa3UeHu6iySFeXHBx80yWgr9W
  da3g9ShPh4tspxlSpZV9ftTo/wAgNT3cW3GrusWtW06sNiow3GELPZCyPiK+xWD9mlV4N570Kxql
  YNXbMSuW3UXW34jnRYbcPNKx6pKivqOnY+8aGOX4+kp/JJR1H5YrzQH1ezv5/wAhpxbTqeXtZaS5
  GfOVRIZcz35eQjP69J/xawZF9XJZO2FIJcmzJip0zh19ljpHDzV+gwpzHqU4941YCFGZhQmIcZAb
  YYbS02kfmpSMAfcNBV+wJq7Z8ct0UCOeMatpcLjY7FamEygr688/0jq02qs2JCXcnjpuauxwVRqK
  hYdcHUBaY6IvHPrkq/ROrTaFQe/ra3ArNYalWpuZ+K0JMdLbkT4CYmc3ApRLnNwgjIKRx7fFz79R
  78RN5/5ef8IxP+WpDf35W/hhr8QvxH+DPZ0+b8N+1ef53JWePlfF4ceGM9c8vo1Hv2yX9E394aA/
  ETef+Xn/AAjE/wCWmzpTftkv6Jv7w02dEL/xGVt+3tkbqqcVZbeEL2dC0nBSXlpayPpHmZ1GPBfA
  ah7BUqQ2kBU6VKkOHHdQdU3n7mx92pT4h6FIuTZS6aTFQXH1QvPbQkZK1NKS6Ej6Twx9uor4Lag3
  M2DpkZCgVQJcmO4PQl0u4+5waLh0aXu+Vh1DcOl27Ro8uIxTotfi1CqoeKsvxmuRU0kAEEqJT3wO
  mmFoPQZOiFpX7EuOo76xb6j1Snx6bEt5VMipUhTj7D63ubjqUEcDlsBIJPc5KSBgsvSOYvC872sK
  6Nx7auQUOj00yzQoqYTLzdQbihXJ2QpxJVxcWhaQGygpAByo6aO21xi79v6BdHkCOapT2ZSmgcht
  S0AqSD7wDkZ0HD8Q1PbqeyF4RnUhSUUp6QAfVoeaD96BpceGCNC3H8OUegXCqY6xTZr0IFiY6wog
  JC0AltSSoBLwAScp+Knp0GmT4gp7VN2RvGQ8oJSukvxwSfznU+Wkfesah3gsob9H2NivyEFCqpNe
  nJSRg8TxbSftDYI+gjRcJzwNWrRbodu4V+L8IRI3sShCeJVGdWrz8Lcb+Ssp4njyyByJxnBHS8V1
  jQdr6tb+5O3zaaBI9t8h5qKODQd4laFJSOgBSlaVJ+SRjp1Ocv4PSSwmXesRTyBIdRCcQ2T8ZSUl
  8KIHoCtOf/IalfjNL91uWjtjQkiVW6jUhMLKevktpQtAWv5qfjrOfRCtF0yY9Fk3U7b25tu1Rmi1
  mZQ22nS/EMpl6M8EvBCkBaDySo5SoK95yCDqQWHaEG1IkwtyHp1SqUgyqlUJAHmynj7zjolIHRKR
  0SPtJ6luUtmiW9TaLGUVMwIjUVsnuUtoCR+oa39GRqrF8TF3J467bokklUajIb8pB7BaY65XLHrk
  p6/zR6atPqrN/Ql2145rYr0gFMatpbDbh7FamFReP154fpDRYtNqJ7z/ALz16f1fn/6depZqI71K
  SjZ29CtQSPgCcMk+8sLA/XohT+Ab956rf1ge/wBPH1YbVefAMR+R+rDIyK+8cf8Arx9OW/rJti/K
  O1SLrpnwjCZkJkttee41hwJUkKy2pJ7LUMZx10W9SHVM/HlbNKo1btysQRLEqrGaqWXpbrySUFoj
  iFqIQP2RQ4pwAMADpp/UXYLaWjViFV6bafkTYMhuTGd+EZSuDiFBSVYU6QcEA4II0m/whv8AEb+0
  P9toTp2V7ZXbyt2/IptRobUiZIQfMqznx56nMfupePxirPXBPH3Yx00n/BXXazSrwuvaypylyWKU
  XXY4USQypp4NOhOeyVFSTjsCCfedWZfq1MYoi647OYRTER/aVSiseWGuPLnn0x1zqvPhBt+ZU7uv
  PdeVGcjxa7LfRTgtOC4hb5dcV9WQhIPqFemg9+PWOpiy7XuCOstTIFY8tl1PRSCttS8g/Wyk/Zpm
  R7f/AB9qG3m4UqoBuPToJnogBjIdeksJwsr5dOGcgcT19+lp49pBesW2aEwlTkubWQ4y0nqpfBpa
  MAe/q6kfbp82ZSlUKz6LRFEFVPp7EUkdiW20p/8AzQR1e36nN0597O1t0tzILEMw0s8VIS0vnxDo
  VngpWCpPHrjGcEgmpxo0Qs6zsPtdWpntlZoE2pSccfOl1uc6vHpyU8TqW2XZ9Bs6GuHQGprEZYSk
  Mv1GRJQ2E5wEJdWoI7nPHGemc4GO/o0BpW75WDUK3Mo19WklsXdbboeioWeKZrIOVR1H3ZyrB/nK
  HTlkNLRoOBYl20q8KKKjTVLbdbPlzIbw4yIbw+U06g9UqB9e/cZGvFxWTbNeqbVVqFNKakyjg3Oi
  yHYslKfm+aypK+P0ZxruNRIjMt+W1FYbkSAkPOpbAW4E548j3OMnGe2TrNoOJbNp29bbsl+kU1DM
  mUQZEpxxb0h7Hbm64VLVj6Sdczcu80WvTkRKbFVVblnpKKVSmerj7nzlfNaT3Us4AA75xqXawiLF
  E1U0RmRKU2Gi/wAB5hQCSE8u+ASTjtknQQHYfbk7f2zINSkInXFV3zMq8wfwjqsngk+9Kcn6yVHp
  nAYmjRoDRo0aA0aNGgNJCjUobI7gVOUEFG31yPh1TyR8SjzO37J81lecBfZOEg4AyXfry8028ytl
  5tDjTiSlaFpylQPcEHuNANOIdaQ60tK21pCkqSchQPYg+8a424CpabDuBUAKVLFLklgJ7lzylccf
  TnGuvGYZjR240ZltllpAQ222kJShIGAAB0AA6Y1k0Fb6VPYt7wGQWoA82TUrf9ghtNdVuyphLYQg
  Durm6Tgeh9NO/bSgKtXby3bbWQpymUyPFcUD0UtDaUqP2kE60aNtpZFIqzNSp9EDTsd5ciMyqS8u
  NFdXnk4ywpZaaUcnqhKT1PrqX6BKbpMyN5KsxYVAeWLWhS0u3FV2/wBzcUg5ERlXZa89VEdEEJz1
  HEuSnQ4tOp8enwWER4sZpLLLSBhKEJGEpH0AADX7CixYMVuJCjMxo7Q4ttMoCEIHoAOg0nPGNd1x
  2btbDn2zVHKbKlVZuI862hJUWlMvKKQVA8TlCTkYPTvoE/4FLbo1wi8TVIinHYxgmPIZeWw+yVe0
  BXB1tSVozgZwRnAz21ae17Jte2Z0moUelJbny+kia+85Ikuj0U66pSyOg6Zx01Qbw43fcdtbo0Kn
  0OqOQ4lZq0KJUGghCkvtF4J4nkDjotQyMEZODr6N6LRo0aNEGl7vrt0ncK12UQpCYVwUt4TKRMP8
  E8nB4k/NVgZ9CEnrjBYWjQRDbG80XTTFRKlHNMuangNVelu9HGHfepI/OaV3SsZBB75zrNfG39rX
  thNyxZ81nyw2Y6KpKZYUASQS024lBOT8ojPbr0GJH7JE9uE/2Vj2sNFoP+WPMCCclPLvxyAcdsjW
  bQQK09n7AtOYmVbdLqFLcS6l0pj1qalDiknI5o87isfQoEEdCCNT3Ro0Bqpv4Q3+I39of7bVstfO
  PxG3dcdy7pV6BW6o5Mi0arTYlPaKEpSw0HlJCQEgZOEJBJyTgZOixeiZtZYsxvyJFFcXC5+Z8Hid
  ITC5Zzn2YL8nv1+RqTyX6XQaMt+Q5EptMhM5UpRS00w2kfYEgDSj8Hd33Hee102oXPVHKlLjVZyI
  064hCVBpLLKgk8QM9VqOTk9e+nHMixZscx5kZmSyopUW3UBaSUkKBwenQgEfSBohN0C3Ze5+68Pc
  qtw3ots0RHC24clspclLzkzFoPVKScFAOCeKD0x1dWjRoDRo0aD/2Q==
  ">.Substituting p = .1, n = 100, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0a
  HBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIy
  MjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAAkAiYDASIA
  AhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQA
  AAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3
  ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWm
  p6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEA
  AwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSEx
  BhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElK
  U1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3
  uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwDrvFHx
  G8R+E9asdNvdC06U3xAhlivH2nLBTkFMjBYV1dxe+LrWAyjSNLvNvJigvHVz9N6AV5z8bP8Akb/B
  3/XU/wDo6GvZpp4baB5p5UiiQEs8jBQB6kmgDnvCnjfTfFguIYEmtb+1bbc2VwuJIjnHbgjPGRV/
  XfENnoMEPnCSa6uH8u1tIBulnf0UfzJ4A615d4FjbWvi14h8V2YZNFAkRZyMJKSFGQe4+Ut+Nanw
  5u38Y+Mte8WXYLJbsLLT0bkQpyWx7kbef9o0Ad3fa9FoXh1tW13ZaCNA0saPv2seiqf4j2rKt9e8
  S3ukrq1voFuts6ebHaS3RW5dOo4C7QxHOCa4v40XbXeteF/D5J8m5uleRc8Nl1QZ+m4/nXr4UIoV
  RgKMAewoAwtK8TReI/DbaroUQuJRlfs0z+WyyA/NG3909fbpVvQdfs9etJJbffHNC5iubaYbZYJB
  1Vh/XoRyK8z+Hd0dM+Lni7QYzi2kkedU7KysMn8Q/wClXvHF63gr4iaJ4jgytrqf+h6gg6OARtb6
  gHr7UAeqUVzuu+N9C8N3UVtqdxcJLLH5qCK0llBXJGcopHasr/hbPhD/AJ+77/wWXH/xFAHb1HJI
  kMbyyOqIgLMzHAAHUk1ydl8TPC+o6hb2VtdXbT3EixxhtPnUFicDJKAD6mud+KWqzahq+h+CbWRk
  GqzKbxkOCYd33eOmQCfwoA7XQ/EH/CQSy3FlauNJXKw3shx9oYHkovXZ1+Y9e3HNbtRW9vFa28dv
  BGscMShERRgKoGABUtAHmfxZsZYfCuo3g1PUWmu5IbW2tluDHFGzsE4VMFu5O4kVoeNtUu/DHg7T
  NK0iTy9SvpoNKtJByY2b5S/4AH8SKwfiP4o0a88R+GdJ+1rJDbasJtQZQSsIjyMOcYHzH9K1/HCp
  fy+EvEtrm60zT9RW4neBS+ImGPMwOSFIBPtzQBT8aaPYeDtJ0fU9Hh8jU4tStoTdhj51yrthxK3W
  TcM53Z56Vb07Z448ba9/aMQuNF0WQWVvZycxST9ZHdejEcAZyAD0zzS6ts8eeJtCh08GfRNMuPt1
  1dgERySKMRxIf4jnk46YHrVXwpqFt4KuPE1hrRlivJ9VmvbdBEzNdxuF2mPA+Y8EYHIPWgC/4Dla
  18VeL9Dg3Lpmn3cRtYtxKw74wWRc9FzyB0Hauz1Oa7t7CSWytRdTpgrCZNm/nkA9jjp71zXw/wBE
  vdNsdQ1PVIzFqWsXb3k8RIJiB4SM47hcZ96689KAMzQtdstfsTdWbMNrGOWGRdskLjqjr2IrUryr
  xLfHwT8VtK1SM7NP11RbXyfwlwQFk+oyMn0zXqgoADWDeeJEGuJoumwG9vxte5CtiO1jJ+9I3Yns
  o5P05pvjjxEPC3hG/wBVUAzRpthU9DIxwv1wTn6A1lfC3SzZeCbW9nLSX+pk3t1M5y0jOeCT/u4/
  WgDtR0qG6uoLK2kuLmVIoYxlnc4AFT0x0DghlDA9iOKAOFsfifpureObbw3ptrPL5nmGW5lUxhdq
  FhtUjJzjqcV2d9JdRWcslnbpPcKMpE77A59M9q8cH/J0Lcf8sf8A20Fe20AY+ha/aa/aySW4eKeB
  zFc20w2ywSDqrj9QehHIrI8aatdRy6T4f06V4L7WbjyfPT70MKjdK6+jbeB9faua8eXp8F+P9D8T
  W+VttQ/0LUUHSQDlWx6gE8/7Irc8Qolj480HX7ttumxW08Dz4ykTtgqzegIyM+v1oAia3tfD3xJ0
  DT9KgW1ivrK5F1HHx5nl7Srt6sCT8x5OTzXe1xeiWM2t+NrvxVPFJFZxQCy01JFKsy5zJLg8gMeB
  6gZrtD0oAo6xq1jommS3+oXCwW8Q5Y8knsABySTwAOTTNGvbvUbH7VdWD2PmNmKGVgZNnYuBwrHn
  jtXnTXp8Y/Ggaa536X4eQzeX/C8/A3H1wTj2xXqoFAA3SuQtfGU2v6re2Phqziu47Jtk99cSlId/
  91doJY+4GK6LWo7mXQ7+Ozz9pa2kWLB53lTt/XFeY/AO6iHhrVNPbi7gvTJKpGCAyqBx9UagDrl8
  YSad4gttE8Q2a2M95n7JdRSb4J2/uZIBVvYitDWfEI0C8hfULZk0qXCG+VsrBITgCQdlPHzdAeuK
  4j48og8F2VyOJ4r+Py3H3lyrZx+QrurBV8Q+DbQajEHW/sYzOhHB3oC386ANpWDKGBBB5BFcB8Rr
  KHUdU8Naf+9Fxd6gqkrM4URJ875UHBzgDJHeqfwl1y4MOq+FdQkL3GizGOJ3OS0OSACe+MY+hFPt
  tc0zXvi7bPDdo8Fjp7JbPg7ZZpG+YKSMEhVHT1oA2PF2o3E2r6N4WsZngl1N2e5ljba0dsgy+D2L
  fdB+vesua5sfCXxBFvZwrbac2izXd1DCMLuiYbXx03EEjPU8ZzV3WFj0n4l2Wvagxj05tKktFnIJ
  WObzQ3JHTKkgeuMVj33h7UvE1p4s13yJIpr+yNlpkEi4fyV53EHoXPQemM0AR3Nsb/4Yan4v1Zc6
  tcWkl9aynrZqBmJYj/DwATjBJJzmvRtHuJLzRbG6m/1s1vHI/HcqCf1NcJqTr4t8M6X4W0gSCOVY
  V1FyhAtIUALI2f4yQF2/U9K9FhiSCFIY12oihVUdgOAKAJKK5rXfHGg+G9QFlqdxcJO0YlAis5pR
  tJIHKKR/CeKzf+Fs+EP+fy+/8Flx/wDEUAdvRXJad8SPDOrajb2Fnc3bXE77I1ewnQE4z95kAHTu
  a6taAHVDc3EVpbSXFxIscMSl3djgKo5JNTV5t8cNUk074eyRRMVN5cJAxH93Bcj8QuKANjSfFOre
  J7WTUNB0uD+zQ7LDNfTNG1xj+JVVThfQmr/hvxXbeIXvLVoHs9TsJPLu7OUgtGexBHDKexFT+ELO
  Ow8H6PbRDCpZxfmVBP6mvNdbvj4d/aDtpouE1CxUTKOjna4Gf+/YoA7q+8XtJ4mbw5oloL3UYk8y
  5d5NkNsvbewBJJz0ApP+Etm0zX7PR/ENpFaSX2RaXUEpeCVx/AcgFW9M8GuM+BbHUbfxFrUxL3F3
  egM7dSNu/wD9nrR+OUAbwNFeISs9rexNE46qWO3IoA7fxD4g0/wxo02qalIUgj4AUZZ2PRVHcmsg
  a74mOkf2sNAg+zmPzvsZuj9p2Yz/AHdu7H8OfavNfGmtP4o1b4eadKT5N6sF1KueGaRguT9MH8zX
  u3b9aAMvw/r9h4m0aHVNNlLwS8YPDIw6qw7EVj/8JfNqmv3mjeHrOO7lssC7u55SkETdkBAJZvoO
  K838G623hjWPiJpsRPkWS3F3CvZTGxXj65X8q6b4GW4XwPPeMS09zeyNI55LFcDJoA6ay8WtH4kX
  w7rdotlqMsfm2zxyb4ble+xiAQw7qRUni7xnp3g+yimu1lmnnbZb2sAzJMfYeg9a4j45O2n2vh3W
  oSUuLS+IRx1A27//AGSu/s9HtrzWE8RzAS3ElrHHArKMQIfmbb7knk+wFAHmifHW4tr5I9W8L3Fp
  bsePnYSY9QrKAfwNeu6bqNrq2nW9/ZTLNa3CB43XoQa4/wCLltYzfDrUpbuNDLEqtbMw+YS5AGPr
  yPeo/CryeBPhBFdamjeZa28ly0THBBZiyp7HkCgDovEPizSfDSRfb52NxOdsFrCpeaY9MKo5/Gl0
  2/12/eOabSYbC2YjMdxcbpseuFBUH2zXlHwvuI9V1XUvGWum6vtTaQxQrBbSTGEY5ICg7R2HsK9X
  0XxdoWu3UtpYX6tdxZ8y2lRopVx1+RgD+lAGrd3UVjaTXVw+yGFC7t6KBk148/xm8R+a/leBrkx7
  jtLCcHHYkeXXXfE7UdcstMsF0TRpdTL3IknRI2dQifMA23nBbH/fNcvpnx1jjvBaeItDuLBg2Gki
  Ytt9yhAOPoTQB3vgrV9X17Qv7U1izWxeeRjDahWBjjHAyW5JPJ5A60mreNdO0/VV0ezjn1PWH/5c
  rQBmT3dj8qD6msb4ieO4tE8CJqWj3KSzahiO0mQ5ABGS4+g/Uj0rG+GraV4X8IDVLpb64vb/ADPd
  3cdnLMAOyl1U9ByfcmgD0XTptXmYvqNpaW0ZXKpHO0jg+hO0D8qKNE8QaT4itDc6TfQ3cSnDFDyp
  9GB5B+tFAHkvxwRZPFfhFHGVd2UjOMgywivTR4K8PFgX0yObB+7NI8g/JmIriPGHw+8W+MNXsdQm
  1DR7U2P+ojjSVsHcGySevKjsOlekaQNW+ykax9i+0BuGtC+1hgckN0Oc+tAEerWoi8OX1vaRLGBa
  yrGkahQDsOMAe9ec/AAgeDtSQ/6xdQO7/v2n9Qa9ZPIrzHwxYHwF8QdR0mUbNJ1xvO0+U/dEoyTF
  n1wTj1wKAMv4y2ptfEXhPXCD5EF2scjdlxIrjP5GvYcgjrwazNe0Gw8TaPPpWpReZbzrzg4Knsyn
  sRWTaaZ4ssdOTTI9T0+dI18tL+eJ/OCgYBZB8rMB3yM0AcT8P7Q3/wAY/GGsoCbeF2hD9izFeP8A
  yH+tL8fwW8O6Mq/61r1tnr/q2H8yK9F8MeGrLwrpAsbTc7M5lnnk5eeQ8s7e5/SuK1+x/wCE8+JN
  hYxDfpGgN5t5KPutOSCIge5AAz6ZNAHS33jXw/4WhsbPXNTFrdSWqSBGjkYkYxn5VPcGqf8Awtrw
  N/0H0/78Tf8AxFdsBRj2oA5Cy+Jvg7Ur+3srTW0lubiRYooxFKNzE4A5XH51w3iVXT9onw9JL/qm
  jj2Z/wB2QH9SK9nIrz74m+G7u7TTfEmlQmXUtGmE3lKOZYgQWUe/GfzoA9DHSiqWlaraazpNtqVl
  IJLe4jEiMPfsfQg8Eeoq7QAzbj1/OlC49fzp1FADcVz+p+JTb60uiaZZtqGqmLz5IhKI44Y84DSO
  QcZPQAEnr0roT0rzz4abr/WPGetykmW51hrYFuojiUBB+G4igDqPDviKPXDfW728lpf6fN5F3bOw
  by2IyCGHDKQcg1tmvPvAH+m+MfHGrJu8ibUUt427MYkCt+vFegnpQB4z8fVeSLw3HFnzmuZQmOuS
  Fx+teypwoz6V5je2B8d/FO1lQb9F8On95L1WW5znYp77SBn6V6eKAPM/jrHK/wAPQ0edqXsTP/u4
  YfzIrsfBjK3gjQivT+z4P/Ra5qfxJokHiPw9e6RcnEdzEUDf3W6q34EA/hXM/DC9uLfQn8Namvla
  rozeTKhP3oySUceqkcA+1AHd5oyK5fW9A1vUPFuj6nY67JZ6faf8fNmoOJ+c844ORxz07Vv3n2sW
  khslha5x+7ExITPuQCcfSgDxwf8AJ0Lf9cf/AG0Fe15GK8qTwB4tHxDPjFr/AEZro8eRslCbdmzG
  ev3e9emWZujaRG9WFbnH7wQklAfYkA4+tAHln7QHzeGNJRf9Yb87fX/Vv/XFeoaWjrpNmsmd4gjD
  fXaM1554jsf+E9+IunabEN+kaC3m30o+60xwRED/AHgAM+gJr08CgAAxS0Uh6UAeL/C9Hi+LXjOO
  bIl3OefTziR+hFe0V5jr2nnwh8TLbxeqkaVqC/ZdRYDiFzgLIf8AZOFBPavQtRt577Sbu3s7o21x
  NCyRXCjJjYjhh9DzQBcNc1eeCNFudZbV4UnsdSkz5lxZTtC0n+9jg/iKseEdI1TRfDlvY6vqj6le
  IWLXDZJwTwMtycepqneaN4m1cNDc65Dp1oxwyabCfNZfTzHPy/UDNAHm/jPTYfGHjXT/AAlpMlxd
  vbP5mo30spk8le656AgZ7dSB2r22GJLe3SGJQkcahEUdAAMAVl+HvC+leF7E2ulWqxKx3SSE7nlb
  1ZjyTU+uava6Bo1zqV44SGBC3J+8eyj1JOBigDynwCrt8dPF7pnyQJg2OmTImP5NXs2Pr+dcD8Lf
  DF3pOmXus6pGY9U1iX7RLG33okySqn35JP1r0CgBMGjHtS0UAJiloooA5rXPHXhrw3qC2Or6qttc
  tGJRGY5GO0kgH5VI6g1m/wDC2vA3/QfT/vxN/wDEV23OecUY9qAOT074k+ENW1G30+x1lJrq4fZF
  GIpRuOCcZKgdjXVrS4oAoAU15p8ctOkv/h400QLfY7hJmA/ukMhP4bs10njTQNa1+ytItF12TSZY
  rgSSOmfnX0454646HvW/cWkV3ZyWlzGs0MsZjkRxw4IwQRQBneFbmO78JaTPE2UaziwfogH9K8z8
  QWB179oKxgi5WzsA0rdduFkIz/32PzrtNI8Oa34VtDpujXdndaYrE28d9vElsCc7Qy53r6ZxV7w7
  4Uj0a7v9TuZzeatqD77m6ZdvA6Io7KBgAewoA4X4EobHT/EGkygrcWt8N6HqPkCf+yVo/HKYL4BW
  BfmlnvYVjQdWIOcCt+78JTWviaTxHoNxFbXtwgjvLeZSYboDoTjlWHqM/SiTwtd65rNjqXiKa2kj
  09jJbWNsGMQk/wCejs3LEdhgAUAeaeL9Gbw1rvw4vZuIbRbe1lY8BWR1bB/76P5V7tkdPwrK8R+H
  LHxRos2l6ijGJ8MrocPG46Mp7EVlx6b4tj04ab/amnsAnlDUTE/nbem7Z93djvnGaAPNPCeiyeI9
  d+JF7CMxXS3FrE3Zmdi3H/fI/MV1HwNnVvAT254lgvZRIh6qSQcGu28N+HLLwvokWmWCsI1Jd5HO
  Xlc/edj6mshfCV1omvXmreHLiCJb8h7uxuQfKd/+eileUb14INAHJ/HSNr/T/D+kwgvPdahhVHU/
  IV/9nr05Gg0rSlM8yxwWsIDyOcAKowST+FYVt4VnvPEkHiDXp4ri7tYzHZ28CkQ2+fvMCeWc8cnG
  MdKyvH/hPxV4raG003WLTT9NiKyFMP5kjg5BJHYdh68+lAGhb6dP4s1ODVtUgeLS7V/M0+xlXBkf
  tPKvr/dU9Op56RfEuMap4A1/T7WRZLqGBJ3iRssqq4fke4Q4riP+FXfEM9fHk3/gVc//ABVd94D8
  J3XhbRZ4dRvft2oXc7S3FwSzb+yjLcnAA6+9AFD4QtA/w00oQ7cqHWXH9/ec598EVx/xcgk0vx74
  V1jTQV1Kd9hKdWKMgGfXIcj3AxXd2HhK88MXd0/hm4t0sbqQyvp10G8uNz1aNl5UH+7gin2/g+S9
  8UQ+I9enjuby2TZZ28KkQ23qwzyzc9Tj9BQB1YxjntxXM+OvDGmeJPDd4l9DH50ULPDcEDfEwBII
  Pp6jpWXq3hXxePFcms6H4q8uCTGbC8RnhUAY2hQcY9xg89a2b3RdV8QaebHV7uG2tZBtuIrDdumX
  uu9uVB74GcdxQB85z2WoXHwhsL+RHNpb6rII89leNc/hvVvx+tfUti0Emn2r2u0W7RKYtvA2kDGP
  wqvL4f0ubQTob2UX9mGLyfs4GFC+3oe+eueaxdL0PX/DtimmabeWd5YQ5W3N8HEsKdlJXhwO3Tig
  Dx/xDNqnhP4ta3H4XVleaFZHjjHAD7WPA9GP6n1or2Xw54Oh0W/vtWup/tusag2bi6dNoA7Ii/wq
  MD3OKKBnUUUUUCEPSqOpaTY61pz2V/brNA5zg8FWHIZSOQQeQRzRRQBLY25tLSG3M80+xdvmzNud
  sdycDJ96tUUUAVdQtvtljJbefNAJQFMkDbXUZ5we31pmmaZZaRZR2NhbpBbxj5UUdz1JPcnuTyaK
  KALi9KWiigApnWT8KKKAMyx0Gy0rUbq6sfMgW6y8tujfuTJnlwvZj3xjPetYdBRRQAUUUUAIeleL
  eKNbvfA/jqXRNEdYrXXpftU7OoZ4JXwrNEegzgHDBhmiigD1fQdEsvD2kwabp8bLBCOrNuZ2PJZj
  3Ynkmp9Usxf2D2rTzwpKQrPA+x8Z5AbqM9MjmiigB1hp9ppdlFZWNvHb20S7UjjGABVqiigBrHn8
  KytS0Kzv7y21BvMhvrQ/urmBtr7SRlCf4lPcHiiigDXooooAKqX9ubuzltxPNB5g2mSBtrqM84PY
  +9FFADdM0uy0eySy0+3WC3jzhV7k9ST1JPcnk1doooAKD0oooAhnt4bq3kguI0lhlUq8bqCrAjkE
  elVNG0mDRbI2VrJO1ujHykmkL+UuB8ik87R2BziiigDRooooAKyrvQ7PUNXgv7wPO1qAYIXbMUb8
  /OF6F+2TnHbFFFAGqOKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiig
  AooooAKKKKACiiigAooooAKKKKAP/9k=
  ">.The probability that exactly 1 of n cows is a <i>treifah,</i> <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAAuAWgDASIA
  AhEBAxEB/8QAGwAAAwEBAQEBAAAAAAAAAAAAAAYHCAUEAwH/xABQEAABAwMDAQMHAwoUBwAAAAAB
  AgMEAAURBgcSIQgTMRQVIjJBUWEWgbQJFzc4QlJxdoOEGCMlM0ZWZ3JzgoWRlqGipcTR0+MkSVWS
  k/Dx/8QAFQEBAQAAAAAAAAAAAAAAAAAAAAH/xAAWEQEBAQAAAAAAAAAAAAAAAAAAAUH/2gAMAwEA
  AhEDEQA/ANl0VlbtPaw3X2pudm80bjS5sG6oeKEyrXC7xtTRRyypLIBBDifYPA1c5WktZCL/AMDu
  nexKCfWlWyA60VfFKGEKx8OXz0DtRUQ0Zu5qC1brna3cyFBaujxHm66QApLEsKGUckKJ4lWCMg45
  Djj21TdxNZWXQumnb5enV92FBphhpPJ6S6r1Wm0/dKP+ZPQUDFRSzoBerpcB266uTFhPzOK49qjp
  5CA31wlbni44cjkeiQRgDxJUN+N0r1t/p+dPs2jZ1yEQtoeuEjDUNlTnEJ9vN3qpIPEAAnBUCCKC
  q0UgdnnVF31ntBZdTX15t24zVSS6ptsIThMl1CQAPABKUj39OuT1r1bmXfVOl0J1TamEXayw2s3W
  1JbAkd0CSp9hftUkeLauigOhB8QdaK52mb5a9SWGHfbLMbmW+Y2HGXUeBHuI9hByCD1BBBro0BRR
  RQFFFfC4TItvgvz50hqNFjtqdeecUEpbQkZKiT4ACg+9FT7bzVN/17cvlHb2k2vRjalohB5rMm6k
  ZT3pz+tNA+qPWURk4HQ0Gg+M6XEgxlypspiKwgZW684EISPiT0FcLSWu9H6tuVyt+mNQwbw/bA2Z
  hhr7xtvvOXD9MHoEngroCcY610L/AKesWoExU32zwbmiI938dEthLqW3MEBYSoEcgCcH2ZqPdklL
  Vwi7g64wlDd91XLVHUBgeSs4S18wysfNQVLU+tdN6bnM2+5znVT32y63ChxHpklTYOCvuWULXwB6
  csY+Ne7S2obLqizt3ewXFmfCcUpIcbJ9FSThSVA4KVA9CkgEe0VJOyS89qmzao3RuIK52qL08WFq
  8W4TB7tloe4JIX/9qV3jWdz0xpDfyfYHFtOzdXC124tHHGS7+lvqT7l8QTkdcgGg0dO3V0DClusS
  L9hpiR5M/NTEfXCZezju3JSUFlCsnBClgg9DTqCCAQQQfAipLuZYrXonsoag04000Ilu0u9ET6PR
  ThZKQv8AfFw8s+85pp2PMtWzGilT1LVJNghFwr9YnuEePx99A40UlbmXfVOl0J1TamEXayw2s3W1
  JbAkd0CSp9hftUkeLauigOhB8WTTN8tepLDDvtlmNzLfMbDjLqPAj3Eewg5BB6ggg0HRooooCiii
  g4+rNUae0pbhcNSXiHa4qlcEuSHAnmrGeKR4k4BOBXqsF3t9+ssS82mR5TAmNh1h3gpHNB8DhQBw
  fwUgdp65vwNnLrBhHM68ratMVOfXW+sIUn/s50/6dtjFksFus0UYjwIrUZrpj0UJCR/UKD3UUUob
  m67g6KgxEeSu3O9XN7ya1WthQDst046ZPqoGRyWegHxwKBvorjaOZ1G1ZgvVU2JJubyy64iI1wZj
  ggYaQT6SgnHrK6kk+AwBJe0TvPqXQNpLli0dKDbklUNF2uKQhjvQFHDbYPNfRKiFHik4yOQoLnRS
  9tncpl6240xeLi6Hps60RJMhwJCebi2UKUcDAGST0HSmGgKKKKAoopZ3AXq6Jb2rrpHyaY/CKnH7
  U+jAnt9MoQ54tuDB4nqCTgj2gGail7b3WFm1zphi/wBkdUplwlDrLg4ux3R6zTifuVD/ACIyCDXG
  1JuK3ZNyrXot2xyl+Xx3JPl3fN8EoSPBKEkrUor4oAITkq6E0D1RSXozXEi/681RpOTY/IXbAmMp
  T6ZQeS4H0FaUqASOCwB1AKh49fe6UCzqjX+jdM3WNab3qCHEuMpSEsxMlbyytXFOEJBV1PQHFFJW
  636v72bb6SHpsw337/MT973CeLCv/IVCiglH1Q39g38of4atU3K42+2RVyrlOiwo6BlTsh1LaEj3
  kqIArK31Q39g38of4atFR9udvY7odj6D0sy4OoUi0MJI+cIouM+PRJG8naptWq9Nx3nNLaaMdLl1
  KCll9TDincIJ9YlauPT2DPhXUnXNW4vbMhWR9Xe2bRzLjzbJ6pVISlPJwj3h1aB+THvrSjLTTLSW
  mW0NtoGEoQkAJHuAFZT2DbXD7Z24TMsEOOpuKm8+5Utpaf7FBq+pN2v/ALXbVH5p9LZqs1Ju1/8A
  a7ao/NPpbNEg7IH2u2l/zv6W9VZICgQQCD0INSbsgfa7aX/O/pb1VmhWa+zrdl6O311vtIVlNr8p
  dnWtonoz1CuCfgW1pP5P4mtKVlG1trkfVB5rsbJbYQpT2PDHm5KDn+MRWrqLSPr7cT5I3hq2/IXX
  F/7yOl/ymyWnylhGVKTwUrkMLHHJHuUn30vfXt/ci3Z/o5/uUw6+3Z2/0HeGrRqu/wDm6a9HTJba
  8jfdy2VKSFZbQoeKFDGc9KXv0R+zH7cv7sl/6VAfXt/ci3Z/o5/uUl9tDUU2QNL7ZWx9TLmopiDL
  KfEt94lDafiCsk/kxTp+iP2Y/bl/dkv/AEqlPagbXH7U+2k98ERCqAkE+GUTlFX9Sk0GpbRAiWq1
  RLXAZSzEiMoYYbT4IQkBKR/MBXqooohM3b1c7pfS07zfZ9QXW7vwn/IGbXaJEvLwThAUtpBS36RH
  rEe3GcUjdklAg7K2jRFxseobTdIkZ9U1E+zSoqMuPLUeLrjYQpWFjoFE49mBVsooINszMv8AtftM
  3t5J0bf7jqSzuymYIjQHFQ7h3j7jjTolY7ptB7wcuakqTg9M9K5uodlr5bezvHssIIvGqo98a1Lc
  UIUEifL7zk6hJOPuDxSTjPAeGa0XRQRjdHznvDYYmhbLZb/a7TPksuagn3O3Owe4jNrCyy2HQkuO
  rUlIyjkgAHJqxxWGYsZqNHbS0y0gIbQkYCUgYAHwAqU7n73w9G2aRcYmjdS3ZtqaICHlR/I47r5U
  UhKVvYUsEg4U2haemc4qspyUgqABx1AOcUH6QFAggEHoQazX2dbsvR2+ut9pCsptflLs61tE9Geo
  VwT8C2tJ/J/E1pSso2ttcj6oPNdjZLbCFKex4Y83JQc/xiKLGrqlnaM1lrrQmjpOpNMW+wvW+Kls
  SXprrqnkKccDY4tJASQCpHUr9p9Hp1qdSbtf/a7ao/NPpbNEjg7U3reHcrae3XqNqSz6ffeL+Jy7
  cmS7KUl5YGEZS20hIAR1C1HiT09vM2l3c1tbd23tqd1UxXrktZRDuLDaWwtfHkkEJASUrT6pABB6
  Edejj2QPtdtL/nf0t6pZvVCXdO2toWNbE8n2WoT8nj4gNPOurz+TSPmop+3bu7l83U0PbE6e1U9Y
  7JdnJ9zmN2CYtkPtIIYCSG/0wc8+knKcEEEirdRRRBWaNlbmrcrtQat1jLV30LT0dUK0oPVLSVLK
  ErHuKkpdJ/hD7q0vWUOwM2uHfNfwJQIltqiJWD45QqQFf1kUVq+s89vn7D1p/GBn6PIrQ1Z57fP2
  HrT+MDP0eRQitbMfYe0X+L8D6OimylPZj7D2i/xfgfR0V49ffXb88NfIL5D+bPJ097578q7/AL7k
  rPHuvR4ceGM9c8vhRDlOVKTCfVBbZdlBtRYQ84UNqXj0QpQCilJOMkAkD2HwrNI3d3dV2io+3MiB
  p1Ljbi0LixlK7l0KjKdQtbywV8UgpWeKUk8eOOtWTQP12/PDvy9+Q/mzydXdeZPKu/77knHLvfR4
  ceecdc8fjUF/5if/AL/0iiw2bzXPfjby3fLSPquzX20R1pM63t2hLKGUqIAIyVOKRkgE8wRkHGMk
  VraTW8HcPQVv1RCaLHlAKH45VyLLqThaM+0Z6g+0EHpXm34XHRstrIyuPdmyygM/fltQR/aKanvY
  at8uFskp+SlSW511fkR8+1sJbbyPhybXQcKz3VW3fbHuenGld3ZdYNokln7lElSCQsfEuJcT+U+A
  p80/p+beu0jqPV92tsxmFZLdHtlncfZUht8rBcdcbJGFcSVJyMj0qlO/ja5XbO29ai5LjaLepzj4
  4TLdWr+zWrqCQbGWy8vK1hMvlluNuev15mPS3H+8iuttdG2G2lDCjhAUrvEkYyMEnwoVqsdr0uxM
  nR378+kNFTqZNzm3FXFIJ9Btxbh5fBA5HoOvSu7RREW26nyL52htT6kn6f1Lb467bHtllenWWSwh
  xlPJx4la0BLfpgYCiCcj29KKtNFBm7f3aPdjdm42x6Q5oy1RbahxLDKLjJdUS4U8ipRjjPqJ9g8K
  t+hVa0Fu7nWkWxNymkISh+1y3XUvnB5FSXGkcPAYwVZyfDHViooCs97v2Vzb7fewbyR21eZJakwN
  QKSMhjmnukvKx9zjh87Y++FaEr4XGHEuMB+BPjNSoshstvMupCkOIIwUkHxBFB9m1ocQlxtSVoUA
  UqScgg+0VKd/9Kbia+0vcNIWFrS8S0y1slcuZPf79YQpLmO7SwUo9NIGeSsgezOA56D0qdIQ37VF
  u8uXZ0qHm6LJAUqCjrlpLnrLR4cQrJSBjJGMMlBK9gtK7g6C0pbdH32PpiVa4aniJkO4P9+kLWtz
  HdKYCVemrGeacA+0jBpV4uMK0WqVdblIRGhxGlPPurOEoQkZJPzV6qVtZaLi6uuUQX2c/Iscbi4q
  zpSEsyXgrIW8fFxI6Yb6JyMnl0ACV9l/TE66aj1NvHe4q4z+pJDnmtl1OFtxCvlyP77CAPgjPUKF
  X2vxCUoQlCEhKUjCUgYAHur9oCiiigKjXav2/n6w0VFven2lL1Bp14zIiEDKnUdC4hI9qvRSoD2l
  OPbVlooF/bnVMHWuirXqW3rSW5rAU4gHq04Oi2z8UqBHzUwUp2nRMWx6yk3+wTnrdFuClOXS2JQF
  RpLxGA8keLTmccinooDqM9abKAooooCiivxwrCFFtKVLweIUcAn4nrigiHaT/VvcXaLRA9NM3Uhu
  rzf3zcNvmQfgQtX83wq4VB73oneK6b4WLcl2BoZLNlguxI9sF7lHJcCwpwu+SeOFAer9yKucNUlU
  RlUxppqSW0l5tpwuIQvHpBKilJUAc4JSM+4eFB8rxcYVotUq63KQiNDiNKefdWcJQhIySfmqGdl/
  TE66aj1NvHe4q4z+pJDnmtl1OFtxCvlyP77CAPgjPUKFVTWWi4urrlEF9nPyLHG4uKs6UhLMl4Ky
  FvHxcSOmG+icjJ5dAGhCUoQlCEhKUjCUgYAHuoP2pN2v/tdtUfmn0tmqzWb+21uRbLZpSXtoqDMc
  uV3ix5YkDiGW2kyc+OeRVlgjGMYIOfZRY93ZXTr2JsTZHbKmwXOFIMlUdqfIdiriESHUkckNuB1J
  Ukq8EEFRGSMU9bZbZuWHVV211qi5NXnV129F2Q20UMRWugDTKSScAJSOR6kJHh1zN+xNuPa7npOH
  toiDMbudoiyZa5BCSw42qTnoc8grL4GMY6Hr7K0hQooooogrPLdoO0/adXfFp7vS2uAqOX8ehHnK
  UFhKj7OSweP8IfvTWhq5up7DaNTWKVZL5BamwJSOLrTg8fcQfEEHqCOoIyKDpVDO0Vt5ubulbWbD
  CTpK22mJPMppx24SFvvcUrQgqAY4o9FZJSCrqfWOMmtaNs82w2VNrmXyVeUsrIjvykp75LOBxQtQ
  /XFDr6eAT0z16ns0CPtFbdb2DTNr03qiHp4x7Xb24jMu3T3nFud2lKEBTS2UgeiDlQWeo8OvR4oo
  oCseajj3WT2+3WrJOZhXAKQtl15ouN+ja0qKFJBB4qAKSQcgKyOorWWp7vG0/pq6X+Y265GtsN6Y
  8hoArUhtBWoJBIGcJOMkfhrCSd4LV+ih+uwq1TfNneY8lBT33DyTyfPjxz91jPwz7aLGqdwdGa73
  HgM6c1DMstg04p1LlwTa5LsmTMCSCEBS2m0tpyAfBRyB4+BpFnt1tsNkjWu3MNQ7fBYS0y2OiW20
  jA6n4DxP4TXz0xd42oNNWu/w23W41yhszGUOgBaUOIC0hQBIzhQzgn8Nc3XWlvldEYtcy6y4tnUV
  ecIkb0FTk9MNqcHpJb8eQTgqBxkDOSI5tJZ17ib/AF/3gebPmKAo2+wrUnpIKE90p5PvTjmc+9z3
  pNaGrz26FDtsBiBb4rMWJHbDbLLSAlDaQMAADoBXooCiiigKKKKD/9k=
  ">.Substituting p = .1, n = 100, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0a
  HBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIy
  MjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAApAiYDASIA
  AhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQA
  AAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3
  ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWm
  p6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEA
  AwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSEx
  BhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElK
  U1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3
  uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD0fUPi
  n4f0q9Sy1CLVbW5fGyKWwkBbJwMcc8mtKfxrZWcZmvdP1e1hH3pZbB9q/UjOK84+NoA8V+ED3MhB
  /wC/0Ne1YBUgjg54oAp6VrGna3ZpeaZew3du3SSFgw/+sasXl5bWFnLdXc8cFvEu55JG2qo9Sa8b
  8Gg2Px28Q2GkjGlkO00afcVsKenQfOWFbmm33/Cw/iDepLiTw/oDARw/wXFzk4dvULg4H0NAHoth
  eQ6hZRXkG8wyjchdChI7HB5/+tVDU/E+j6TqFpp93fRJeXcyQw24+Z2ZiAPlHIHPU1qyIZI2QlgG
  BBKnB/A9q8I8ZaNYaL8avCMWn24hWaW2kl+Yku/nkbiTyTgCgD3aR/LiaQgkKCSFGSceg9araVqt
  lrFmLuwuFmiLFTjgqw6qwPIYdwaunoK828b3D+BdfsfFtmpFldyraatAvSQEfLJj+8ACM9+BQB6V
  RUMVzDJBHMsqGORQyPnhgRkEfhTvPi/56J/30KAJKKj8+L/nqn/fQqSgArO1PWLHSRD9rn2vPII4
  YlBZ5W9FUcn+lLrOq22iaRd6ndtiC2iMje+OgHuTgfjXH/DeC41y3l8aauN+oagzJbKelrbBsBEH
  bJGSe/FAHoA6Vm6sdW8iMaQtmJmkxI92W2omDyAvLHOOMj61pVT1O8TT9Ku712CpBC8pJ7YUmgDk
  /h7rOs65Jr82p3iXMFvqLWlsUgES7UGGIGSeT6k9Kj0bU9U8dNqGoWGqz6ZpENw9tYtbRxs9wU4a
  Vi6sChPAUAHg5NVPh3bXMXwdSeEs17ewXF3lepkkLEY/HFJ4D1Kx8OfA7TtReVVht7OSVj6yFmJX
  3O44x60ARW/xFvm8D2lx5VvN4gu9QbSoFwRHJMHK+YR1CgfMQPpV/WNQ1fwfqnh+S41qfUrfUr5b
  G6inhjUKzKSHj2AFcEfdJbg9a43QtCu9I1f4aQamnlvLLe3kqsOk8iFwp9wMflXXeLEOvfEPwtol
  uA6adK2qXpHPlKo2xg+7EnH0oA9A5xVGx1ixv7y7s4Jh9ptX2zQsCrp6HB/hPYjg1eA4964n4jWN
  xb6SPFGlHytX0geajD/lrD/HE3qpHOOx5oA7iisnw1rtv4k0C01a14juIwxXPKN0ZT9DkVqnigBk
  siRI0jsERRlmY4AHqTVXStUtNYsvtdjIZbcsVSXaQr47rnqPfvXAatqDeNfiKvhBG/4k2nJ9o1MK
  f+PhgRtiP+zkjI74NelxosaKiKFVRgKBgAegoAdVHVNW0/RbRrvUryG1gXq8rYH0HqfYVeryb426
  Hp6+FJdXMG6/+1wqJncsVUnBVcnAHsKAPTtL1G21fSrXUbNy9tcxLLExUglSMg4PSoYtYsptVn0t
  Zgt9CA7QuCrFT/EufvDtkVl/D3/knXh3/sHw/wDoAqLx3osmo6FJfWEhg1fTgbmyuF+8jKMlfdWA
  II6GgDqq4rxZqOs2niXw9Y6dqQhj1C6EckIt1Y+WgLuxcnjIGOnfrWj4H8TxeLvC1tqqqI5jmO4i
  H8Eo+8P5EexFZcf/ABNfjBI/WLRtNCAjkebM2T+IVR+dAHcUUVg+L/EkPhPwzd6tMA7RLiKMn78h
  4Vf89gaAL13rFjZ31tYSzj7XdHEUKDc5HdsDoo9TxV4ZrkPh7pU0WiLrmqMZta1ZRcXMzdVU8pGP
  RVGOB3zXYGgCrqOpWWk2Ul7f3MVtbR8vJK2AKxH8b6XFbi7uItQgsTgi8ls3WLB6Etjge5rgfide
  NdfFLwhot1/yDfMimdCfldmkKnP0Cj/vqvX5Io54XimRXjkUq6sMhgeCCKAIG1G3/s06hHJ59sI/
  NDwDzNy4zlcdfwp2nX9pqdlHeWNxHcW8oykkZyD/APX9q8s+Emoy2PiHxL4PaRnttPuZHtQxzsUS
  FGA9s4P5+taGv33/AAr7xzZahGdmh65J5V7EPuxTjpMPTIIz64JoA6vxbrlxpMNha2AjOo6ndLaW
  zSDKxkglnYdwFBOO5xVFL+/0XxhpWkXOpS6hBqcMpBnRA8ciAHIKADaRngg4I61d8W6PFqFpZ3x1
  BNPn0ucXkV1Ku6NSAQQ4yMqQT3Fc14Qt9S8T+LpfGGo8WFvC1rpa+UY/NUn5pQpJIBxxkk4NAFuz
  1TV/EmhX/iO01R7C0QynT4Y40ZHSPI3y7gSdxU8ArgV1HhnV/wC3/DOmat5flm8tkmKf3SwyRXm+
  sWOqaVZ/8IB4a1FLg3juQqwYewtnYl/MkBxjk7eAfrXqOk6dDpGkWmnW4xDbQrEn0AxQBcoprSIn
  3nVfqcU3zov+eif99CgCSimedETjzE/76FPoAKD0orE8Yay3h/wjqmqJjzbeAmPPTeeF/UigBl34
  s0y21KTTojPe30QDS29nC0zRDtuxwv0Jq1o/iDTdcEwsbjdLA2yeF1KSRN6Mp5Fcf8HLDyfAkeoy
  EyXWpTSXM0rcs53EDJ/A/nWP46vG8J/Ffw5rkBMceop9lvFHSRQwGT6kBlx/uigD0rWdd03QreOb
  UblYhI/lxpgs8jHoqqOSaoweLtNkv4LG5FzY3Nx/qI72BofO9lJ4J9utcFoV0fFXx01S5nO+30SF
  obVD0Vs7SwHrndXT/FXTUv8A4d6m+MTWii7icdUZCDkHtxkfjQB2RYDJ7fWue/4TbSpWm+xLeX8U
  DFJZ7O2aWNCOo3Dg49s15x4l8fXU/wAErK8WYrqGof6JM6kg/LnzCD6kAf8AfRr1DwnpUWieEtK0
  +FAohtk3YHViMsfqSSaAL+l6pY6zYR3unXUdzbSZ2yRnI+nsfY1neIvF+h+Fo1bVr9IHcZSIAtI/
  0Uc4rgdBvD4b+OOr6BGdthqqC5WIHhZSu7IHvhs10Vj8PNN1G8utZ8UWUWoaneSMxjnPmR28ecJG
  o6cLjJ9c0AX/AA98RPDPie6+y6dqOboglYJo2jdgO4B610lzcwWdtJcXMqQwxqWeR2wqgdya8C+K
  Pg208Ka9oeoeGozazXdxtjt4ycJKpBUr6ZJAxXR/HPULiO28P6UWK2l3cF7kDgPtKDafb5ifwoA7
  tfG+lvbfbIodQlsMZ+2R2btFj1zjke+K29P1Gz1WyjvbC5iubeUZSWJsg1LBHFFBHHCFESKFQDoF
  A4x+FeS+D70+HfjL4h8LQcafck3MMQ+7G+xZDgdhhiMew9KAPT9a1my0DSbjU9Rl8q1gALsFLHk4
  AAHJOSK4pvjb4NCFhPfHAzgWT8/pVPx34z8MW3imDQvEkT3FlBCJ5IRCJEaVvubxn+FckdeW9qu+
  HtP+FviRgdG03RZp1+byjbBJBjvtYA/jQB3OnXb3+nW120D25njWTynILJkZwcd6o33ijR9P1e00
  me+iF/dyCOK3U7nJPPIHQcdTWnLAsts8B3KjoUOxipAIxwRyD714ZrmkWGi/H/w9b6fbrBE/kyso
  JO5j5oLEnqeBQB7uWCpuJAUckngAetcXffEbwlpdzNP5ksrNhZbq1spJUIXOAZFXBxk9+Mmuo1PT
  YtVsVtbhn+zsymWNTjzVH8B/2ScZ9QMdDSX1zp2k6Y8l7LbWllGuG83CxgemOn4UAQaB4k0jxNZt
  daRfR3ManDheGQ+hHUUV538LvDVzF4m1vxLBC1lol8WWwtzwXQuGDbewABxn+8aKAMz44hj4n8JB
  W2sXcBsZwfNhwcd69Jl0PXrqNopvFEscbcMbWzjif8GJbFeYfE8ax4n8Q6Jc6V4a1uW304lmdrNk
  3kujYAPPRO/rXsekaourWhnFne2hBw0d3A0TA4B4z1HPUcUAY9h4X0vwfoGof2RbskzQySSTyOXl
  kYKSCzHk81xX7P8Ah/DOrTt/rZL4bvX/AFan+ZNetyIskbI4yjAhge4NeV/C+1fwr4v8SeFLr5SX
  F5ak8ebGeMj8NtAHq9eJ/Ef/AJLf4M+tt/6UGvaZZFjidyGYKpJCqSTj0A614f4zGra18TdB16x8
  Na5JYacYfMZrJlZtspc7VPPQ9/SgD3IdK4L4zRJJ8MNSZgDskhYf9/VH8ia7LTdQTUbbz0guoBkr
  suYGifI/2WAOPeuC+M08l14csfDtmpkvtXvEijiHUqp3E/QHbQBN4U0HS/GPw08OjxBYR3oigygl
  3fKQSoPBHYCro+E3gX/oW7X/AMf/APiquv4E0a+8OaXo2pQPPDp8ahDHPJF823Bb5CM556+tUR8J
  PBv/AEDbn/wYXH/xdACn4TeBf+hbtfzf/wCKrtEUJGqqMBRgD0FcSfhJ4NII/s25IIx/yEbj/wCL
  rto1CRqgGAowOaAPO/jbPLD8O51jJCy3MMb47qSTj9BXReAEWP4e+HgvQ6fC34lQT+ppfHPh8+Jv
  B2o6ZGB58ke6En++vI/PGPxrL+FGpfbvAFlbyDZdaeWs54m4MbIeAR/u4oA7esfXPDOleI4kh1W2
  e4iQMPL86RFYHGchWAboOua2KKAMbQ/C+k+G0ZNKtnt42UL5fnyOqgHPCsxA6npiok8G+H4743ia
  ZEJPN87ZljGJP7/l52bvfGa3qKAM/VNHsNatPsuo2qXEQYOobIKsOjKRypHqDmsvT7jwt4f1BNJs
  7qzhvruTBhE2+aVgOrEksTgdWroJ4Y7m3kglUNHIpRlPcEYIrzWPSrFfjXp1jp1nb21ro+kST7IY
  woDyts7d8AGgD07OazfECLL4e1KNwCrWsoOf9w1o9K5n4gaxHovgbVbpmw7QGGJe7O/ygD35oA4/
  4A3Msvgi6hcnZDeEJ9Cisf1Jr1YmuN+FvhyTwz4Gs7W4TZdTZuJl7qW6A/QYFdkaAPF/gxI934y8
  Y3c+TO0i5J68ySZ/9BFe0ZHrXk/h2yPg/wCNOqWUo2WeuwtPaP2Lg7in1GX/AMmvSNbur6z0W8ud
  MtBd3sUTNDATgSN2FAGj3rzf43/8k7k/6/Lf/wBDrr/Ct/q2p+HLS81uwFhqEiky24yNvJwcHJGR
  g4PTNcT8XZL3WfD76Hpei6reXIuYpGkitGMYC/Nw3RuuOKAOq+Hv/JOvDv8A2D4f/QRXROquhVhl
  WGCPUGuR+Hd5KnhbStIutM1Gzu7OzSOQXNqyISoC8ORg59M56+lbviHVodE8P6hqU0gRLaB3yfXH
  H5nFAHlXwJZprfxPYtu+zrcIQASOW8xTjHThV6V6Xpfg/RNG1F7+wsmhuZP9ZJ58jGT3bLEN+Ncv
  8GvD0+jeD3vbuMx3OqS/aSjDlU6KD9eT/wACr0agArx39oK4lj8N6XApPlyXTlx7qmR/OvYq4T4s
  +GpfEngi4S1j33VowuI1A5YAHco+o/lQB2VkixWNvGnCJEqqB6YFWMisDwXq8eu+DdKv42BZ7dVk
  HdZFG1gfxBqtdaz4ji8eWmlwaIr6FJCXmv8AJyjYPGc4HIAwRk59qAGeNvAtl4ztrcyTSWl9aNvt
  rqMAlDwcEdxkD8qDJ4l0vSJZNW1PSVit4i0l6sDhsAfeKE7c/j1ra1XVTpiR7NOvr2STO1LWLd09
  WJCr+JFcdqvhzxH48kjt9b26LoKuHexglEk9zg8b2HyqPYZ/kaAMD4KabPeX2veK7jeRfTNHDI4w
  ZPmLu2Prjp3z6Vf+PManwJbyEfMt/GAfTKv/AICvSrGxttMsobKzhSC2hQJHGg4UCvNPi9HL4gu/
  D/hGyO65vbrz3A/5ZxqMbz7DLflQB2Wj2NrrvhbQ7rU7dLiQ2UMhWXLLuKKclTwTn1BxW68SvE0R
  HyMpUhSRx06jp+FJa28dpaQ20K7YoUWNB6ADAqagCnY6ZZabG6WdtHCHO5yo+Zz6serH3JJq5RRQ
  Bg6/4Q0DxPLA+taZFetAGWIybvlBIJ6EegrI/wCFTeBf+hbtPzf/AOKrW8QeDtE8USwSavayzNAr
  LGY7mSLAYgn7jDPQdax/+FS+Df8AoHXP/gwuP/i6AJYPhb4JtriOeHw9bJLG4dGBfIYHIP3vUV2V
  cZB8K/CFvcRTxafcCSJ1dCb+c4IORwX9RXZ9qAEJHrXGfFeB7n4Za0kYJKpHIQPRZFY/oDWt4u1L
  WdK8O3F3oOnLqF+hXZAQTkE8nAIJwOwNWrVJdY8OxR6vZrDLd2oW6tc5Cll+Zc/iRQBz3wmZT8Md
  DwckRMD9fMauP+N0TXuveDrOHmaS4kUAdeWiA/ka6HwnZ6j4AgudDvbK7vtKWZpLG8tIjMQrfwOi
  /MCPXGDU8Ph678TeP7fxPqVq9pYadD5dhbTY8yRySTK6/wAIGeAeeATjFAHK/C+NrX4r+NYJeHMj
  sB7GZm/kwr0H4hOqfDvxCWPBsJV59SpA/nWNrHhq+0bx3F4x0W2N0s0fkajZoQHdccSJnALDA49u
  Kd4s/tDxrox0DS7C9tYrp1F3eXkDQiGIEEhQ2C7HAGBx70AeM61ptxH8FNBumVvKOoXDg9sOCB+o
  NfTdlIsljbup+VolIx6YFc/q/grTtT8E/wDCMAeVbJCqQvjJjZfut7nPX6mqPh3VNU0PQ7fSNd0i
  /kvbNBAs9nAZo7hV4Vgw+6SMZDYoA46/ge6/aWsvK5+z2ySPj0ETg/8AoQr167u7ews5bq7mSG3h
  QvJI5wFA6k1yHhHwxeQ+ItW8V6xGsWo6i22K2DBvs0IxhSRwWOATjgVyHi/xN41v/ESf2X4NvLjS
  rN8xR3dpIVmkB4kIBHT+EH6nnoAdjpujy+JfEcHinVoGigtVK6VZyjDID1mcdmbsOwx36afjDwhp
  /jPRjp99vQq3mQzp96JumR6jsRXmknj74sPGyr4MCEjhlsJSR9MuRXptlJc+HvDFil9HfalehAJj
  BGZXaUjcx9AM59hxQBT0qx8VaTpyWd1qel3MMCbVu5InSQIBwWGdpIHfiuA+GtpN4i+J+ueKmma5
  tYd0MVyUCeaxAXgDgDaucdgRmut1nTvFXjeI6dJGPD2iucTkyLLdTr/dAXKoD3ySf5V1Wk6JaeHN
  Ej03SLZI4YEPloT95vVj6k9TQAzU/DGh6yztqOk2Vy7ja0kkKlyP97r+teL/ABK+HkHgyODxN4bn
  ltUjnUPEHJMLE4VkPXGcDBrvtD8Z+K01CSz8T+D7yEbiEutPiaWMexGST9R+Qq34n0O98dxW+lyQ
  yWGiLKs1zJLxNcbeQiL1UZ6lsH0FAG94V1OXWfC2l6lOu2W6tklcehI5ryzxhz+0R4a/65wfzmr2
  JY4NN07ZFCwgt4sLHEhY7VHRVHJPHQcmvFvEQ1bUfi9pXiS38M642nWXlK7GyYMwXeSQp5/j/SgD
  3EEbRXJeIvhx4d8S+ZJfwXH2hiSsy3LkoT/dBJA+gAqfXdb1tPDC6j4a0lrq7EgDWt7E8LlBncQp
  wc8DH1zzS6T4ruNRsVa48OaxZ3pHNtLb4GfaQ4XHuSKAPE9M1zW/hV4s1TQkka9tkAKxN905wyyA
  diQecetFeu6F4KLeItS8Ta+kMuo3wEaW6/MltEMYXP8AE3AyenHFFAztdo9KQLg06igQVg+IfDSa
  y1reW05s9WsmL2l4q5KE9VYfxIehH8q3qKAKtj9saxhN+kKXW396sLFkz7Z5x35qxsHoKdRQBFIH
  EbmNVZ8HaCcAntzWBpHhqSPV31/WZUu9YkTy0KjEVrH/AM84wefqx5P04rpKKAEAI60tFFABRRRQ
  AmK5q68OXFl4gfXNBeKGe52rf2suRFcqOj8fdkGTz36H1rpqKACiiigAooooAr3tw9rZyTpbTXLI
  MiGALvf2G4gfmRXnPhdddh+IniHWtS8M6lBDqjQRWzloW8qNBg78SZHY8Z6mvTj0pn8X40AO68Vz
  U3hubWdej1DXJI5LWykLWNjHnYG7SyZ+8/oOi+55rpP4x+NOoAQDFBpaKAMfxD4dtPEVisE7PDPC
  4mtbqLiS3lHR1P8AToas6SupiwVNXNu92hKtJb5CyDs20/dJ9OcetX6KAEAwKMA9qWigBu0Zrm9V
  8Nz+I9VQavJGdGtpFkiso8/6Q45DSn0B6KOO5z0rpqKAGqu0AAAAdAKdRRQAUhGRilooA5m08OXG
  h69JdaLJEmnXsvmXljJkKjnrLER0J7r0PtXSbe/enUUAJjnkUoGKKKAILv7QLWU2qRvcBT5ayMVU
  t2yR0FY2g+GRpt7c6tfTi91q7AWe5K4VVHSOMfwoPzPU10FFACDgUtFFABRRRQAUUUUAFFFFACY4
  oxS0UAIVyaAMUtFACGk206igBMUbR6UtFACEUBRjpS0UAN2c9BRtp1FACYpaKKAGlQTnApcUtFAA
  RkU3YPQU6igBoXFG0DoBTqKAEAopaKAP/9k=
  ">.The probability that exactly 2 of n cows are <i>treifot</i>, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAAoAVYDASIA
  AhEBAxEB/8QAHAAAAgMBAQEBAAAAAAAAAAAAAAgFBgcEAwEC/8QATRAAAQMEAQIDBQEKCAsJAAAA
  AQIDBAAFBhEhBxIIMUETFCJRYRUJFhcyQnGBgqG0GDdWV3aRldIjJTNGUoOFoqXE0yRDU2Nyc5Ox
  0f/EABUBAQEAAAAAAAAAAAAAAAAAAAAB/8QAGBEBAQEBAQAAAAAAAAAAAAAAAAERQTH/2gAMAwEA
  AhEDEQA/ANP6tde8o6Y3OJCyTprEUJiFLjPxcgK23AkgKA3HBBGx5geYrRpV76lxovvBwCySj27L
  EXJVF38w9pGQnf62vrWB/dDf8xv9of8ALU2VFZ7056t45mN9lY0uNPsWSw9h+03JsId48yggkLA8
  +Odc61zV6uM2HbYD8+4SmYsSO2XHnnVhKG0gbJJPAFKt1PT9r+N/F2sZ+OXCEQ3JbPPb2KWt3uI/
  8kpSd/QfSrV1ZvK+oviAsXR9pxX2Bb1Jn35CTxJUhHtUtK+aNdg183PmkUMbHgmVjL4b91hWqXFs
  xUBAmSR2Kmp524ls/Elvy7SrRVvegNbxzqL4rMTx28vWvHrPIyRTCyh2QmSGGCocHsV2qK/z6APo
  SOa2rOLPNvOF3Gw2iWi2vTI/uqXwNexbVpKykDyUEFXb9deVfcVxLHMXsDVjsloiRILbYQUJbG3O
  OSs+ayfUne6Izrop4gMW6lXP7E90kWW9KQVtRX3AtD4A2fZrGtkDZIIB1yN6OrvmeaRsQucI32C+
  xY5Xa0q8JUFMxnlK0lDw80JPGnOU7Oj2+ZWrNsHhWjxo4tBw2KiI2+uNdJTEdPa2wErWXgAOEgob
  3ry2vQ8wKbO92yDerRLtNzjokwpjKmX2ljhaFDRFFrrQpK0hSVBSSNgg7BFfawXwvZVPg3zJOj99
  lrkzMYfcTbn3D8TsRK+0A/8Ap2gj6LA8k1p34Tem384WJf2zH/v0RbKKqf4Tem384WJf2zH/AL9S
  FgzDEcgmLhWDKbHdpSGy6tmFcGn1pQCAVFKFEgbUBv6j50E5RRWLeLTP7jimHQsex51Td+yR4xI6
  0K0tpvgLUk+iiVJSD6dxI5FBfbRnEK+ZhIsGPxHrnHgKUi53JCgmNFdA2GUq/wC8c3raU8JB5O+D
  bKgOnmLW/C8MtmNW1CQzCYShSwNF1zzW4fqpWyfz1P0HLMuduhzIkOZcIkeTNWURWXXkoW+oAqKU
  JJ2ogAnQ3wK8YN8ss+6TbVBu9vlXCB2++RWZKFvR+78X2iAdo3o62BusbytpGQeMPFbZFbBGN2KV
  dpbuye1cjtjhI+R7UJ4413FQ5r38PCot26i9Uc27mUG6X/7KhJ2AXGYDQb70D1BKudfKg1TMrzMs
  FlVdItkl3hDKwZLERSfbJZ0e5aEn/KEcfACCRvWzwffGL7aMlscW92Oc1OgSkdzTzZ4PzBHmCDwQ
  eQRo1JUvKrv+CfxOosLavZ4tnATISx5IjTlKKCpI9O5YG/8A3B/oigYaiiqr1Kzm24FZDdrpbb1O
  YSha1fZ0Fb4bSnWytQ0hscjlahvnXkdBaqKxnA+uqs6sM6biuB3m5z48pTKISHm0JDYQlSXHXl9r
  bfcVKAQCtXwE615fOn3XyBe88VguWYxOxC/lz2bDEp4OocXrYR3dqdKUOU8EK9DyNjGi5tirGXMM
  226TZaLNpRlw4zy2TLPHalbiCFBA+IlII7iU7Oho5x0Wx77zer+Y4rjcmW5iMaJGeEd55TiYcxfJ
  aQpXPKPiProp36Vbuo/UGz4/cWMaRkVktd5mN+09tcZbbTUNnyLqgtQ7leiUDlR89JCiJDppKxBy
  0yIeIXaPd2Y7xVMmsvB/28hfxLWt1PwrcOwTo8bA0BoUFrr4tSUIUtaglKRtSidAD519rAfE/k86
  6ZPjPRyyS3Iz2Rvtm6PtHS0RFL7e0H6hKyfogDyUaDVMMzWLl9ymfYMF+RY4vc2LwohLEl4KAKGR
  5uJHO3OE7Gh3cke2e53ieC2z3/KL1GgIIJbaUe5176IQPiV+gaHrqpmzW2DZ7TFtVtjNxoURpLLD
  SBpKEJGgBVU62W63u9K80muwYq5X3uzEe3U0kudqWXFJHdregokgehJNB+ejHUi39T8euF9tdvkw
  ocW5LhNCQoFboS22vvIHCd+01rZ8vPnQvFLz4Bv4nrt/SB793j0w1CiiisRyvxH41YcxgY07jmRN
  vSJLLbz0+GuIlppawkuhCwXFgDZ12DetA0G3VwZBLnQLNJmW21rukppIUiIh1LaneRsJUrjetkA6
  BIA2N7rJ+oPWnI8QipvMnpLfjjpUAq4Py2m1pBOgpTKQtSAeNd5T5gHR4rRuneZWLPMVjZHj0hTs
  R/aVJWO1xlwfjNrHoof1cgjYINB64TldmzCzfadmfWpKHFMyGHkFD8Z5PCmnUHlCwfMH842NGpyl
  66z3U9I+tuP57DJasmS/9hyBhP4ilI12v6/0wlW9/Jsj8o0wqSFAKSQQeQR60HjPlxYEJ+dNkNRo
  sdtTrzzqwlDaEjalKJ4AABO6q56lYR96z2UN31t+zsrWhclhh10fBrvUAlJUpKdjagCkfOuzqbfx
  i/T2/wCQdwSuDAeda36uBJ7B+lXaP01g1ysErFfCzYMUW923vLpMO3FavNr3l0uqRz6BJX3E/lKU
  fLQAMpb5ka4QI8+E8l+LJaS8y4nyWhQBSofQgg1E5ljoyeC3a5VwlxrYtRMxqK6ppySnXDZcSQpK
  N8q7dE6A2BsHiQzfJNqt/wB5d3s1utTUcNNIn2d2SpaUHtSpBTIa0gpA1wdjRB0a4eoOfW3E0QbN
  LvljjX+e3/glzpCY0dpI4W+sLXsIB3pAUVKPwg8KUkKH09xZWCdd71jeDOyDja7E3MmQZEhbrMOW
  p7tQApRJClNpWrROyPPjt0VofTGZiDkedDxi/wAS/wApK0ybpcGHkvF95zY7luI+Du0jQQD8KQkA
  AaooF++6G/5jf7Q/5amAlYTdZbRZl9ScycaUNKS2YTBP67UZCh+gilz8YK7/ANRrjjzGK4NmcuPa
  0SC7IXj8ppK1OlvQSFICuA36gedM5guVNZTbveBZ77aZDaEF+PdLY9FUlSgeElxISvRB32k643rY
  ovHhgfT3EcIS+rHrShiTJO5Mt1anpD5J2e9xZKjs863rfpS89AVrk+M3qE7K2XG03FLe/kmW0lP+
  7TX0tV2taunfjHt2ROpLdlzFtccPfkokKSAUE/MuJbV/rPoaEMrUBnOUxMVtKZLrDs2dJcDFvt7H
  L0x8j4W0D9pUeEgEngV3ZNdk2OwzLsqBPuHuzfeIsGOp994+QShCQSSSR9B5nQBNKSvqV16Tms7J
  2ukU1+Q6CzDE2wznTCj/APhNlJSBvQKla2o+Z0EgAxPS7BX7LcblmGTOMzMwvZCprzfLcVsa7IzO
  /wAhICQT5qI2fSr9SlOdUvEpkhbsCOnz1j+0HExjcGrBMbMYLISV97iilIG99xHGt8U1lsit261x
  oKHHFtxWUNJW6sqUoJSBtSj5njkmhStWpa4n3QaazGOm30KS9r1BtyVn/fApgvwZdNv5vcS/saP/
  AHKxbw9WtWZ9fs46r9pVam5DsG2unyeV8KO9P0DSE/8AyD5VfP4OHRj+Rv8AxOX/ANWgtv4Mum38
  3uJf2NH/ALlSFgw/EcfmLm2DFrHaZS2y0t6Fb2mFqQSCUlSEgkbSDr6D5VQv4OHRj+Rv/E5f/Vqw
  4D0m6f4HeHbvilg+zpr0dUZx33x93bZUlRTpxah5oSd63xQXilQ8T61u+KnppEe37qFW9QB8tqnK
  Cv2JTTX0uvjPxqaiNjfUq1MKefxuWkykpHPsitKkL/Mladfr79KEMVRXJZrjEu9oh3WA6HYkxhD7
  Cx+UhaQpJ/qNddEQjGO47a8kumYIhsx7pOjobnTVuq+JpofCD3HtSkDngD5mq70vb6UW9yRa+n1y
  x2RISpx15qFckSnkd6u5ez3qUlJVyRwN1cb1arfercu3XWKiXDcUlTjLm+xfaoKAUPyk7A2k8EcE
  EHVZL1XtiL71t6b2rHozbd0skpV1uMxlASYlvCSgNKUPR5W0hPr2qOtA0GzUqPj3WuJfsAnxTqWh
  ctSCPPaVRyn9pNNdS1dZrYrqV4osTxCKPbQsdjpnXVY5S0FLCyg/VSUtD9cfKiwytVLrRz0ezTf8
  n537uurbVT6z/wAT2af0fn/u66IyXwDAfgfux1z98D37vHqoeOuJ7jm+C322I7Ls6HW0rR+MSy40
  pr9IU4rVd3glyZyx9NbszNsd4fty7w4tubAhOS/8L7FkKbU20FOJ0AghXb2nuI2NVcHMPvfVTrRb
  M0yKzS7NimOJH2XEnI7JM14K7vaKbPLaO4JOlckISNcnRetVu2CYPd7g7cbthuO3Ca8QXZEq2Muu
  L0AB3KUkk8ADn5VJ2Ky2awwTBsVpgWqIVlZYhxkMt9x1tXakAbOhz9K76KIKVC5LXI+6ERGpOy2y
  lKWd/IW1Shr9Ymmvpa/ETal4b13wjq0EEWsyWoN0cA4ZPxJ71fnbWof6v60WGUqp9Z/4ns0/o/P/
  AHddWwEKAIIIPIIrOevd/VFwHIMdhWHIrvc7paH48du3WiRIbBdQtsFTiEFCdHZKSru1rjkbIz/w
  DfxPXb+kD37vHrZc+yn70bO1cvvcyG/+0kJY92skL3l9G0qV3qTsaQO3RPzUn51h3g3lXHD8PmYt
  k2KZZbJsu8F9hx6wyvYqS4202O5YbIRooOyrQAO9+emSot9ZjZesH2neIVt/Bd1Ohe9yG2PeZdg9
  mwz3qCe9xXee1A3sn0ANY14kgD4wumwIGj9lb+v+MHabOlD8VL0uP4rcAkW+J77MaYty48fvCPbO
  Cc8Uo7jwNnQ2eBuhDXZDbol4sM+1T20uRJkZxh5KhsFCkkH9hpZvue70tVmzCOsq90RIiraHp7RS
  XAvX10lH7K1jqBm17uOKS7PhuIZM9kM9lUZsS7a5GYhlY0XHH3AGz2gkjsUrZA9OakOgvTmP0ywB
  ixB1Eie8sybhIQPhceUACE757UgBI+et6G6DM/H22g9KbK6de0TfG0p/MWHt/wD0K2fpY+/K6Y4r
  JlEl96yw3HSfPuLKCf21ivi9jyc1ynBul1oJXNny1TZOufYNAdgcUPkEl4/q/WmHt0RiBb48GKjs
  YjNJZaT8kpAAH9QoPxd7bbrvbnbddoEW4QntB2PKZS60vRBHclQIOiAefUCvO62e0XaOzHutqgz2
  WHEutNyY6HUtrT+KpIUCARs6I5Fd1FEFQF6wjDL5PVcL1iNgucxYCVSJdtZecIA0AVKSToCp+igj
  cfsFix6M5FsFlttpYcX7RbUGKhhClaA7iEAAnQA39KKkqKAooooCq91Cw+zZzjD9hvbSiyshbLzZ
  7XY7o/FdbV6KH/6DsEiiig8un6cujW9215eI0qRCKW2LpHWALgjR0tTfm24NDuHIJO0n0FmoooCq
  X1JtOU5OhOL2mQiz2aW1/jO6pcBkezJIUwwj0WoebiuEhXAJ8iigsOK2C04vj8Ow2OGiHb4bfs2W
  k+nqST5kk7JJ5JJNSdFFAUUUUBXjNixp0N6HMYbkRn21NutOJCkrQoaKSD5gg6oooKJ08xe/YFcz
  jdvKLlhbqnHYKnHtSLUTtRZO/wDKtE/ikHuTvR2Oa0GiigqfVfLZmGYg9drbjd5yK4KUGo0K2QXJ
  K1LO/iUED4UDRJJI35A7IrOumHUmMzdGLX+Crq01cbzMQbhebnjYZbW6rSfaurDh9m0kaAA4QkAD
  yoooNYzB/ImbOUYtBiSbm8sNNrlu9jMcEHbq9fEoJ1+KnkkgcDZEV0zwSBhcKY5707c71c3vebrd
  H0gOy3TvnQ4SgbPagcAfXZoooLdWe+IXJLDYelGSRLvdocKTcrPNjwWHXQHJDimSntQnzVytO9eW
  xuiiixk3gNyOwsYJcMZfu8Nq9P3l6QxCW6Euut+7tfEhJ5Vr2a963oJJNM3RRQooooogqOyax2rJ
  bDMsV7htzLfMbLbzS/Ij5g+YIOiCOQQCKKKCt9M7TlWMIVit2dRdrLCaAtd2LgD/ALIEBLD6PVaR
  5OJ4UByAfO60UUBRRRQFJd4gMzxa5+KXCL7bb7Bm2q1m3CZMjuhxpvsmLcX8Sdg6SoE63RRRYcWy
  3W2Xu2M3Sz3CLcIL+/ZSIzqXG16JSdKSdHRBB+RBFfjIJFyi2eS/Z7ci4z0pAYjreDSVqJA+JZ3p
  I3s8E6B0CdCiiiKt07wM2K7XDK8gmou+W3UATJqUdrbLY12x2EnlLSdDz5URs/IXiiigKKKKAooo
  oCiiig//2Q==
  ">.Substituting p = .1, n = 100, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0a
  HBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIy
  MjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAAgAiYDASIA
  AhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQA
  AAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3
  ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWm
  p6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEA
  AwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSEx
  BhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElK
  U1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3
  uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwDsNb8a
  +KtE8caV4blg0aT+0ivlXKpKNoJIOV3diOma6PVtR8V6JYyX/wBj03VYIVLyxW5kgl2jqV3FgfpX
  C/Ee4itPjN4NuJ5FjhjVWd26KPMbmu51jxlZf2VcRaPHNq1+8ZSG3tIWfLEYG44woHU5PSgDT8M+
  JNP8V6JDqunOxhkJVkcYaNh1VvcVDpviI65q00elwiTTLZmjnvmbCySD+CIfxYPVunYZ7eW3dnqX
  wx+ClxazSCPVNSuNjbGz5O5QCAR3CJ27mvWfDekRaR4U07TIR5aQ2qodnHzEfMfqSSaAON134ja0
  19c23hHwxcavFayGOa8Knyi46quMZx0zn8Kn8A/E6PxbfzaTf2B0/VYlLeXklXA64zyGHoa7iwsL
  bS7CCxtIhHBAgVFHb3+uec153oGirrHxh1bxVbKF021T7LHKowLifbtkI9QOQT6j60AdiPEDWXiF
  dI1WFbc3TE2FyrZjnx1Q/wB2QDt0Pb0reFcf8UNOF98P9TkUlbizUXcEgODG8Z3ZHvjP51FovivV
  tU8CaRrGn6M2qXdwuy4iS5jh2MuQzZfg/MOnvQB21Ga40eJPGH/Qg3H/AINbb/GrWm654lutRhhv
  vB81jbOxEly2oQSCPg/wqcnnA49aAOorP1rWbHQdMkv7+Xy4UIAAG5nY8BVA5LE8ACr56V5dqV2f
  EXx007RpDustFt2uzH2aYqMMR7blx+NAHoWj3N/d2IuNQs1s5JG3Jb797InYOem7rkDgdMmrF5bm
  6tZYBPNAZFK+bC21091ODg1OKr31xJbWryw2c13IMYhhKBm57F2Ue/JoA4HwjZxj4meK54pp5IbG
  K3sVaadpSWK+ZIcsT3IHGOlL4YtrLxtZar4i16CO8t57maCyjl+ZLe2jJXKD+FiQzFhz054qf4eW
  Ot6edbGuaJPaT6lqM96ZvOhkQK20KmVctkAHtj3qvpug+ItK8FS+D7SxVHJlgj1QzJ5Ihdid+3O/
  eFbG3GMj72KAOf8ACusaj4r0zwV4evLmZ4Z7We+1F95DzwRSNHEhPXBO3d6ge5rqNLjh0j4uXmka
  ZCltp8ujR3U1tCNsay+ayhwo4BK8HGM8VJH4Vm8MeJNJ1LSLF72xtdJ/st4I5EWVQGDK43lVbODn
  kHJzzWl4b0G6t9Z1bxDqqouo6kyIsKNuFvAgwibu5PLMRxk8dM0Ab2oveR2Mj6fDDNdLgpFK5RX5
  5G4A4JHQ461U0LXbTX7Jp7cPHJFIYbi3lGJIJB1Rx6j8iORWrjNeX6reHwz8cNOMR22viC2EVxGO
  jSKSqv8AX7o+hoA9RqpqWpWekadPf31wkFrAu+SRugH9T7Vb7V5Z41u21v4o+GPCbHNkj/brmPtI
  VDMoPsApP40Ad9oWoXuqWrXtzYmzglIa2ikbMuzHDSDopPXbzjvzwNWkXgUtAHn/AI1+JcfhaWC1
  t9KurieebyUmlQxwbgQGwTyxHtx713svmeRJ5W3zNp2b84z2zjtXkHx3+54Y/wCv9v5LXsPagDE0
  PxAupXNzpt3D9k1azx9oti24FT0kRv4kPY9R0IBrcry/4qXL+G9W8N+K7Y7Zbe6NrPj/AJaQuMlT
  7cMa9NjdZI1dTlWAIPtQBS1fSodYtRaXTyfZi2ZYkcr5ox91iMHGecDriuZ8D6VHpWq69HpZkXQP
  ORbSNnLKJAuJfLz/AA5wPqDV3xbda+Ghs9I0Oe+tpATdSxXkUDY/uKWOee5A6dOemj4elv5dNH27
  SF0kodkVoJllKoAMElfl/AUAbB6Vgat4h+z6nDoumwC81eZPM8ottSCPp5krdh6DqT09a1766jsb
  C5u5c+XBE0rY9FBJ/lXnnwdaXVdK1bxNeEPe6pfOXf0ROFUewycUAekpnb8xGe+K5nxB4tGma5Ya
  Bp9sLzWL4FkiZ9scUYzmRyATjg8Dk4rqBXkXiWKbwv8AGnT/ABPfq39jXUH2ZrkDKQNt2/Mew6H8
  T6GgDrdb8Rav4Ttk1HWLe1u9LDqlxNZh0e3BOAxRidy564IPtWlq+uy6bYQ6tBbi80kJ5lw8JJkS
  M4IkVf4lA6jrjkZ6Vi/EPXNJl+HmtIt9bTGa0dYhFIHLMRxjGe/epPhZ5svwx0YXI3ZidQGHBj3s
  F/DbigDrLW6gvbSK5tpUmgmUOkiHIZT0INeYXE9peeAPEnibVWQ6oJLlIHY/vLR0YrFHH3UghTxg
  knnNTfDG+fTvEHifwexLQ6bctLZrn7sbHlB7DK4+prZvPBo8SeIIr7U7C1s9PglE32ZFUzXco+60
  zDjaOygnPc9qAMrxNqq2V14b1DX4ftmlixZr21VN5ilZVxM8fdR8y5PQmuo8FWU1nors8yPbXFxJ
  cWkSSeYsMDHKIG6EAenAzgdKqrYanpfibXLs6YdUtdUWIxskkYMe1CvluHI+XvkZ6nitLwhoUnh3
  wvZaXK6s8IYtsJKqWYttXPYZwPpQBu5orH1zUNV0+KF9K0R9VkZiHRLmOHYMdcv1544rG/4STxh/
  0INx/wCDW3/xoA7GiuZ0vW/El5qUUGoeEZtPtWzvuW1CGQJgEj5VOTk4HHrXSjpQAtctrXip7bxF
  Z+G9KtkutWuUMzeY+2O2iH8bkcn2UdfaupPSvI/Aszah8avGl1PzJAPIQnsofaP0QUAddd+J73w7
  rGn2XiCO2NrqL+Tb31tuVUl7I6sSRnswP1FW/FfiqHw1b2qCBrvUL6YQWdojbTK59T2UZ5Ncz8cI
  w3w7eXo8V3AyMOoJbHH51zceoS678ZPBf2vkLpEdyAf77Qs5P5gUAd5rHiHW/C1gmq6xbWVzpysq
  3X2LeHtgxxu+YkOoPX7prX1rxPpmheG5NeuZt1ksYdGj5Mm77oX1JzUPjaBLjwLr0bgFRYTNz6qh
  YfqK8Ju9TuL/AOHHgLTpmLQvqTxMD/EqSKq5/BzQB7R/a3ir+wv7aOmWOPL8/wDs3e/nbMZx5n3d
  +O23GeM1s6Brtl4k0S21bT5C1vcLkBhhlPdSOxBrR2rs27Rt6Y9uleU/BG4cJ4n00E+Ra6iWjHYb
  i44/74FAG1qfxMjtvF2j+H7XSrom/nRPtNyhiUxk43IDy31OBXfqeBXjvxAGPjV4I+q/+jDXsQHA
  oA4jxr4q8Q6Xdw2fhfQTq84XfdEglYgfujgjk8n6Y9aq+D9e8da3rgTXtBj0jT4Y2dm2HdMx4VQS
  xx3Jx6CsDUbX4saFqGoX+niwvrWe5ebyIwshC5wBghWPygDAJ6Vu+AfiTD4wmn0i/tDp+sRIS0ak
  gOBwSueQR3BoAg174i60b25tfCPhi41eK1cxzXhVvK3jqq4xnHrnrU3gL4nJ4s1CbSNQsDp+rRKW
  8vJKvj7wGeQw9DXc2Gn2+l2EFlaRiO3gXaijn/8AWT1+ted6Doi6z8YtV8VWygadaR/ZUlUYE8+3
  bIV9QOQT6j60AdVd6vrl+8qeHNPtJIo2KfbL+ZkjkYcEIqgswB43cDjjPWsLR/H+pQ+Mo/CvinSY
  rG/nG62ntpS8UvUjrzzg8+owQK63WNQvNMs91hotzqLheI4HjQDHYlmH6A15T4X8TWWr/FPz/Fmn
  XFjryj7Pp8Mg/dQDk7fUuck7jxzxigD2S9u47K0e4lWRkXHyxRl2Yk4AAHJOTXHeFfiAfFPi7VNI
  j0yazisIgWNycSl9+0gqOF+nJruG+6a8e+Hn/JZ/HH++3/o2gD0Pxb4ps/CWkC+ukeV5HEVvbx/f
  mkPRR6fXtVa4u/Flrph1A2OmTuieY+nxu4kxjJVZD8pb/gIFct8aNJ1G5sNF1ixgkuE0q686eGMZ
  YrlSGx7FMfjXa2XizQr/AEtNQh1S0MDoHP70bl9QV6g9sYoAk8OeIbDxRokGqae5MMuQVYYaNh1V
  h2IrJfxrHfazLo3h20Oq3kBxczCTZbW/s0mDk+yg/hXiem+JrzQvA3iw2EjRx32qiC2deNm4Ozke
  h2BR+Nep+DbHVfCPga0Sz8NmeRo/PuF+1Ik0jsMn5cEcDgDdn8aBnd2C36xsdQktmkOMC3RgF9ss
  Tn68Via14zs9M1aLRbK3l1PWphlbK2Iyo/vSMeEX3P5VmP8AEnTL7wHq3iDS94nsYyr2864eKU8K
  GHpkj9a5j4TWWq2/h+78QjSze6jqcrO9xc3IjMiA8beD1OTzjt2FAj07T/7ZfMmo/YYgRxDBvcqf
  dyRn/vkUVjeHvHmn61qF5pVzDLpmq2X+vtLojgZHKsOGHI/MUUAcP8QJEHxu8FksBt2bucYzI1ex
  L75/GuZl+HXhGeUyzaBZyynrJIpZj+JOa6CysrfTrSO1tYxHDHnagJOMnPf60AecfHTTpbzwElxE
  pYWd0krgDopVkJ/DdXoem3SXmlWd1GdyTQJIpHoVBp9/Z2+oWU1ndxLLbzoY5I26Mp4IrmvDFlfe
  G1Hhm7imubBAw0+9UZHlcnypcfdZegPRh6HigDI8U+P9BGqy+H5tcSwij4vp41dnPrFGVBwf7zdu
  g56WbD4i+EV+waP4emF1LI6W9vaW8Loqj1JZQAAMmtH/AIVr4MY7j4c08k8k+V1/Wrem+B/DGjXy
  X2naJZW11GCEljjwy5GDg0AQ/EK5S2+HmvyOQFaykjBPqw2j9SK53wLba9o3wu0RNL0+0ubqUtM8
  d3cNAFR2ZgchWyeRxjvWp4g0q68Z30elTQS22gWs4ku3kG1rx1ORGg67AeS3foPWuxjRURURQqKM
  AAcAelAHI/2l4/8A+hd0P/wav/8AGqtabe+MZdQhTUtE0m3syT5ssGovI6jB6KYxnnHeunxSYFAA
  eleT21u2l/tG3Ek3Canp5aEnoxCoCB9PKNes1zXi3ww2tpZ31jKlvrOmy+fZTMPlz3Rv9lhwfTrQ
  B0goIzVHSNQl1GxEtxZT2VwjbJoJhyrjrg9GX0YdRV+gBMUYpaKAKWppfSWoj0+aOGZ3AaZ13eWv
  chehPYZ45z2xXMWWp6pYfEQeHJtQfUrSbTjeF5okWW3YPtAJRVBVucZGeOtavivxZpvhHSxd6jcx
  RtK/lwJI23e59wCQB1JwcCuf8D694WvtTuhY67b6pr98PPu5UVlLBRgKgI+VFBwB+J5JoA78dK8m
  8bW7ar8a/B1nB8z2qG6kx/Cofdk/98frXp2oXv2Cye4EE9wVwFhgTc7k8AAfXueB1Nc/4a8OXFtq
  194j1nY2s6hhCiHclrCPuxKe/qT3NAHV9q8l8QW7aZ8f/DmpSgiC9t2gRu28K64+vzCvWq5/xb4Y
  h8TaUkPmm3vbaQT2d0BkwyryD7jsR3FAG3cXEVpay3M8ixwxIZJHY4CqBkk+2KraRrGn69pseoaZ
  dJc2smQsiZwSDgjB5HPrVXR7u41XTpbbV9ONvdoPKuYmXdFJkclG6Mh/MdCKu6bpllpFjHZadaxW
  ttHnZFEu1Rk5PFAHk/x3I2+GB3+3Mce2Fr2EGufv/AnhjVLt7q/0W1uZ3YszzAsck5PU8c+ladnp
  1hoVhIlnb+VAmZCibm6DsOT0HQUAec/HJTfaBoukxAtcXmpKsaDqflZT/wChivULaLyLaKLOdiBc
  /QYrk7HQ7nXfFUXifWLdreK0Qx6ZYyfejz96WTsHPZew689OxFACEZoxS0UAUdbs21HQdQsUOGub
  aSEZ9WUj+tcF8DpMeBJLNwVntb2aOVD1Ukg4P516UelcZJpFx4W8S3eu6XbSXGnakQdRtIhl45B0
  mjH8X+0o57jPSgDodQ8QaVpeo2On3t7FDd3zFLaJs5kPt6ckDnuauXUkEcDtdPGkIHzNKQFx754q
  vc6Tp2o3lpfXdlBNc2jb7eWSMFoie4PakvtC0vU7iOe/sILl4xtXzk3gDOeh4oA828QFvH9wvhvw
  tCkOieYranqsUQSNwDny4yAN5zzxxnHbr6hZWUGnafBZWqBIII1ijUdlAwKmjijijVI0VEUYVVGA
  B7CsnX77UoLdbTSLQzahcgrHI4xDAO8kjegzwoySePcAHn/w/t2vfi5411iPm2ST7MG7Fsrkfh5f
  6ivWaw/Cnhq28K6JHp1u7SvuMk87j5ppW5Zz9TW5QAYooooAyNcn1y3ihOiWFneSFiJFurpoQoxw
  QQrZrG/tHx//ANC5of8A4NX/APjVdhiigDmtMvfF82oxR6poulW9kc+ZLBqDyOvBxhTGAecDrXSD
  pS4ooAzG8QaSuvroRvohqbRecLbJ3FPX098da4LTbM+FfjXqL3AEdn4gg3W0rcKZlILJn+9ncQO+
  a9COj6a2sLqxsoDqCx+ULnYN4T0z6VJfadZ6lam2vbaK4hJB2SLuGR0PsfegDgvi5HLrOj6Z4Zsh
  vv8AU71NqDqsacs5HZRxzWf4r0lfC3jzwn4lRdumWyLp1zJ2hXaURm9FwxGfavR9P0LTNLkeSzs4
  opXGGkAyxHpuOTj2q5NBFcQvDNGkkTjDI6hlYehB60Acv8RNTSx8BaptO+W7gNtbxry0ryfKFUdy
  cmuB8S+BbzSvhX4feCAy3+hyreTxxjLMGO6QD3BC/wDfJr1Sz8M6Lp9wlxa6dBHLH/q2258v/dz9
  38MVq7RQBmLrunv4e/txbmI2H2f7R5275duM9a474P6JPp3hq61O7haK41e5a6COMMsZJ2ZH4k/j
  XWf8InoP2gzf2Vbbi28rs+Td1zt+7n3xWwFAoA8d+IBz8a/BPsUz7fvTXq+papZ6Nps2oahOtvaw
  LuklbOFHTtz1NYkvw88JzzefNoVpJN/z0cFm/MnNbcWlWMOm/wBnLbIbTBUwv86kE5IOc560AM0v
  WNO1uxW+029iurZv+WkbZA9j6H2NedaXog1f423XiXT0xpllD5clwo+Se4KFWCn+LAIyR3FdraeC
  fDVhM8tpo1pAz/f8tNob6gcH8a20gijhEMcapGF2hFGAB6ADpQB5/wCK/H+gjVJPD02uLYRJxfTx
  q7PjvFGVBwx/ib+EHjnpYsPiL4QUWGj+HpluppJEt7e0t4XQKCepLKAABkmtE/DTwYxJPhvTyTyS
  Ysk1b03wN4Y0e/jvtP0OytrqPISWOPDLkYOPwoAt6R4h0jX1mOlX8N15DmOURnlGBxyDz+PSuA8Z
  aIPFHxR8Ox6cm6TTf3+o3K9I0Dho0J/vEhsDrhq7dvBnhxtRfUBo9ql4/wB6aNNjH1yVxWvbWlva
  ReVbwpEmc7UUAE+v196AJT901478OyD8ZvG5ByC7EY7/AL2vW72xt9RtHtbqPzIXxuXJGcHPb6Vg
  wfD3wnbS+bb6HaQy/wDPSMFW/MHNAHRsRjJ6Drk9K8/8VeIIWeXRvB9nBfeIbgGMzW6KVsweC8kg
  GFI7DOf69xf6ZZapAIL63SeINu2P0J9x3p9pYWlhAILO1ht4R0jhjCL+QoA8n8UfDGSz+EsWk6aD
  dX9lN9tlKjmeQgh8D6Hgegr0jw9rdrrvh601OCRfLkiDPz/q2A+ZT6EHIOa18Csm48LaJd3Dzzab
  A0khzIQuBIfVgOG/GgDyXwz4Ul8TDx/c2n7vTdUleGxYjCSsshcMP9nIAz712/wqvd3gq20qdGh1
  DSi1pdW78OhBOCR6EHOehrtooIoIkihjWONAFVEGAoHQADoKo3+gaVqU4nu7KKSYDHm4Kvj03DBI
  9qAPHPFfhi88a/FXVRokgRLS0ijublThfN/uZ7nBHH+zRXtlnp9pp9sttZ20UEKnISNAoz6/X3oo
  C5//2Q==
  ">.The probability that exactly 3 of n cows are <i>treifot,</i> <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAAyAVUDASIA
  AhEBAxEB/8QAHQABAAMBAQEBAQEAAAAAAAAAAAUGBwgEAwECCf/EAEkQAAEDAwMCAwQGBQgHCQAA
  AAECAwQABREGBxIhMQgTQRQiUWEVFzJCcYEJFiNX0iRSgpGSlaGxOGeFlqbB40Zyc4OUorO1xP/E
  ABYBAQEBAAAAAAAAAAAAAAAAAAABAv/EABcRAQEBAQAAAAAAAAAAAAAAAAABQRH/2gAMAwEAAhED
  EQA/AOy6Vmd4352ts0wwrxqCbbpQGSzKss5pYHx4qZBqWk7p6OiRRLmu3yFGI5efK09PZax8ea2Q
  nHzzQXalRWmdR2DU9v8ApDT14g3SLnBcivJcCT8FY+yfkcGpWgUqNtV9s91uFwgW24My5FuWluWl
  o8gys5PAqHTl06pzkdMgZFVDU+9W1um7iu33bWMFEptXFxthDkgoV6hXlJUAfkaDQaVAaL1ppXWc
  JczS98h3RpsgOBlXvtk9uSDhSc/MCva/frPH1Cxp+RcGWbpJZL7Edw8VPIBIJRnoojHUAkgdSMGg
  kqUpQKUpQKUpQKUryXm6W6y2x+53adHgwo6eTr77gQhA+ZNB9Y8SLGekvR4zLLsp0PSFtthKnlhC
  UBayPtK4IQnJ64SkdgK+1fGDKZmwmJsZSlMSG0utqUgpJSoZBIIBHQ9iM19qBSlKBSo3Tl+s+ooB
  nWW4MzWEuKaWWz1bWk4UhaT1SoeqSAakqBSlKBSlfKYZAiPGIlpUkNq8oOqIQV493kQCQM4zgGg+
  tKyfWV51bo/Wei46dUqvq7/dUwplpchMtpQyUkuSGeCQ4lLfQkLUvoRk1rFApSo2732z2iXAiXG4
  Mx5NwfDENknLj6z6JSOpA7k9gOpIFBJUpUNG1Vp2Vqp3SsW8RH70zHVJehtL5rabSpKSV46JOVp6
  EgnPbFBM0pSgUpSgUpUbaL9Z7vMnwrdcGZEq3PFiYwDhxhfwUk9QD3B7EdQTQSVKUoFKV4b41dH4
  Xk2mUxEfWsBT7qOflI+8Up7KV6DPQZyc44kPdSubd1d0dxNutdP6csDh1017IzIeMmCkvQVrKx5a
  zGShPvBIUMpzg/nSi8Vz9IY02H9EvhCQ4tM5ClY6kAxyB+XI/wBddakAjB7Vyb+kN/7Df7Q//NXQ
  EqXuk60URbDo2G4R0dcvUmQE/PgIrefw5Chjn7WKRt74zrAzo9Ihx7/7L9JQ2PdaUHnFtuZSOnZI
  c+R61pm+etLpM1nYtoNIzVw7xfTzuM5o+/Ch4JWUH0cKUrOfQAeqgRJaB2ebtevJG4er72vUmqn8
  +W95AZjxQU8cNN5J6J90Ent6Z61lOzMld/8AGxrmfLJW5BamtNE/dDTzUdOP6NB0AvQ9tjaKi6Os
  a1WezoKUSExiUuvM9StHMEEKWcBS/tEFXYkETEOwWKDZhZ4tnt7FsSjh7KiOgNcfgU4wak6y/dra
  m6a8hSow3H1JbmXuXGGgM+ygH7qktoQtaf8AvrUaIyPZPSqk+Ku/3zQbJY0PCU8y8810jPKU0AWW
  z2UA97wA6AIHxGeiNxtIW/WmmXrTMUqPISfOgzWjh2HIT1Q82odQoH4HqMj1rlzZnVeutod5oe02
  qpaptolSURGG1KK0N+af2TrJPUJKiAU9uqugIrsei1lfh33Dnaws1xsGpQhvVenJBhXRI6B0pJSH
  QPmUqBx6jPQECtUrlXSUpdh8e18tcYlLN1bWh5I7HlERJz/aT/jXVVClKo+vtu/1uvDVy/XrXFg8
  uOlj2ayXb2ZheFKVzUnicrPLBPwSn4VXvqS/1u7s/wC8f/TojWaVk31Jf63d2f8AeP8A6dazQfw+
  60ww4+84ltptJWtajgJSBkkn4VhW109W9evLjrW5oUvSOn5ns1ggLH7N2QBlUpxP3lhJTxB+zz6d
  QSbh4nro/aNhdWTI6ilxcRMbI78XnUNK/wAFmoXwaxW4/h+sbyAAqU9KdX8yJC0f5IFFbFWXeKbU
  lx0pslfbvabm9An8ER4qmOjqnXVpbTxPcEcirp192tRql7raBZ3AY09Fl3JcSHaL5HuzzKWeftfk
  8sNE5HFJKup69u1EUHW8jUDW5OzWj4l+u7MtZdl3XhMWnzo8ZlJIfAOHCtfQ8s9c/E53KqDeduV3
  Pd1OvlahlRwmxfQ6IbDISpCS95q3Eu591SuieicgZwoHBF+oMV35Ezbi4x93tMsnDbzUbUkJBw3P
  jKPFDih28xCiAFd/eGegwdcsF1g32yQrzbHw/CnMIfYcH3kKGR+B69vSqtv5DbnbKayZcSClNnkP
  DPxbQVj/ABSKofgguz9y2PRGfUVC23J+I3n0SQh3H9bporclEJSVKIAAySfSqt9Y2hTqNjTreq7U
  9dHg4Ux2Xw4UhCFOL5lOQjCUqPvEdqssyNGmxHocyO1JjPtqaeZdQFocQoYUlST0IIJBB71xwnSu
  nE+PAabbs0JuzBzzBBbaCWM/Rvm44Dpx59SnGD2xihHQ43y2mN6FoGuLaZJXwCsL8nP/AI3Hy8fP
  litBdQ3JirbK1eW6gp5NuFJwR3SpJBB+BBz8KyrxQ6GsmpdoL1MdgR0T7NCcmw5KWwFtBpPNSAR9
  0pBGO3Y9wKpfhej3HcDZW1CVqrUlof07cn4bD1smBsuthLa0pcStKkuJSF8QFA4AxQTrWmpWjPEx
  YHbLc7lPg6kt0wXNmfJXKXHTHSFIUlxwlYSVrQkAn1PxAG4VE2WwRbbKdnuPybhcnm0tOzpakqdU
  hJyEjiEpQnJJ4oSkE9SM9alqIrO6GsbfoLQ1y1RchzbiN/smQcF51RwhA/EkdfQZPpVQ8P8AYZ02
  zI3K1aoS9UahZD4Woe7CiK95phkH7CeJCjjqSeucZrMv0gt0fZ05pSzJUQxLlyJLg9CppCEp/wDm
  VXTNqitwbZFgsgJbjsoaQB6BKQB/lRXk1PZGdQW36OkzrjEjqXyd9iklhbqcEFBWn3gk5yeJSeg6
  4yDyp4ULZGsvim13Z4XMRYEa4RmeaipXBE5lKck9ScAda6/rk3w2/wCmFuV/tX/7BqhHWVK+U2VG
  hQ35s2QzGix21OvPPLCENoSMqUpR6AAAkk9qrH1m7bfvC0l/fMf+OiJjUWo7BpyJ7Vf71b7WzjIX
  LkJaB/Dkep+QqtT93ttoGnIeoZeroDVum8/ZVkLK3gham1FLYTzICkqGQnHSqP4vLVpu8bD3fVbU
  G2TprbMNUG6NtoW55S5LeODo6lBS4ojBweRPrX38HdjtDWwtmnpt0YyriZPtbqmwpToTJdQEkn7o
  SkdO3c+porRtC670jriG7K0rfYtzQyQHUo5Jcbz25IUApIPXBI64NU7xA2WfbrOdzNJkRtTadbLy
  1Ae7Nhg5dYdA+0kJyoeoKemCc1ieobaztV4zbA1plAhW+/Lj+ZEa6NhuS4plaAB93mnmB2BAx2Fd
  b3mG3cLRNgOpCm5MdbKwfUKSQf8AOgp9t11YtU7LyNcNl1Ftctb78ltD6m3GShCvMb5oIUlQIICk
  kHsR6VhzcCbC8HofvLLkq5X5SUW9h9wrU49KfAQ8ck5XwIKSScJQD0KjUf4N4DeuNqtdbfXOZNjW
  5x+O4pyKtKXUh0KCgkqSoDPkAHoehP410hq3QVm1HabDbXn50JmwzY82CYi0pKXGUlLYIUlQIAPw
  /Og/pEy4aZs9rs8DTV81GiLDbYVKhvxAOTY4EK899tRV7uTgHv3zUxdLe1fLP7LMNxhB5AUoR5i4
  7zZI7c2VjqM+iiPxr026FGt8JuHDaDTDYwlIJJ6nJJJ6kkkkk9SSSepqvXfR783VH0/F1lqm2LLS
  W1Q4sptURQHr5TrawFH1KcGiM72Qtl60hrncHR0B567WeBMiSIr0tYLqVyGlOOJU53WR7nf5H71K
  16yWiBZoq48BooDrinnnFqK3HnFfaWtR6qUcDqfQAdgBSg5Q/SCXOBKuOkLdHlsOyoqJi32kLBU2
  FlgJ5Ads8Ff1V1nZ7nAvFuYuNslsS4r6Att1pYUlQIyOoqvfVltt+73SX9zR/wCCpbTumNN6c8/9
  XtPWmz+0cfP9hhNsebxzx5cAM45Kxntk/GipauXtIW9WifHFd40keWxqWI+9EWroHPN4vK/97Tif
  y+ddQ1n+8+3ytZwLfdLNJbt+qrE+JdnmqHuhYIJbc+KFYGfh369QSJXdLXtk25043f7+1OchuSkR
  f5IyHFJUoKIUQSMJASeufgBkmvfpvV+mNR6fTfrNfIMq2lPJT4dCQ38lg4KCPUKwRUfo2/xdaWST
  b75Y1wrlGCWbtaZzPJLaznGCRxcbVglKxkEfPIr1W/Qmh7fIEiBo3TsR4HIcYtjKFA/iE5oMgRpN
  3c/xIxNw246m9K6cYaZhy1JKRcn21LWFNfzm0rX9vseAxnOR0DSqXuVqa7wmxpvSFvXcNUz2SY3J
  JEeEgkp9ofXjCUJIOE9VKIwAeuAxLa6ArVnjQ1lqxgcoNk5sl0fZ80NiME/mEun+jXUNUzZ7b+37
  daQRZorypkx5wyLhNcHvyn1faWfgPQD4D1JJNzoVR9faa3AvN4alaU3M/VaEmOltyJ9BMTObgUol
  zm4QRkFI49vdz61Xv1E3n/fz/wAIxP4q1mlBk36ibz/v5/4RifxVrNKUFM3w02/q3aXUlgioLkmT
  CUqOgffdbIcQn81IA/OqJ4J7o3N2NjQAr9ra50iM4g90lS/NGR/5n+dbdWPzNPS9rNwbjrSwQX5m
  lL6Qq/QIrZW7CeBJEtpA6rR1VzSnqMkjIAAK2ClfGDKjzYTE2I6l2PIbS604nstChkEfiCK/Zsli
  FDfmSnUtR2G1OuuKPRCUjJJ+QAojLt7bWmHpG/36XrLUUS7llz6EZtlwejcXwj9iw2w2rEhal4zz
  CyeRxxGALxt2vUDmgrCvViUpvyreybiEgDD/AAHPIHQHOc46ZzjpWYbtDbDUe0M3chFut/0rc7WH
  LRco7aEXNUpTWI7Tbrf7TzQopTwCjjBB6A1pe2Ld/a26063qpxTl9TbWBcFLOVF7yxz5H1VnOT8c
  0EB4j7ozaNjdXSXlhIdty4qcnup79kB/WuoDwfackad2Pthltlt+6POXEoPcJXgIP5oQg/nXy1Xa
  ZW9V8gQFMyIu3trkiTIfcSW13p9OQlDaTghhOTlZ+0T7vYKGxtNoaaS00hKG0JCUpSMBIHYAfCiv
  6rj++3a3WP8ASAquV1lNxYiHG21vOHCUFy2JQkk+g5KGSeg7muwK5KdQ09+kNW06hDjSwULSoApU
  DaMEEetCNe8TGrYVr2wuNhhOCbfNRMG326DHPmPP+b7qlBI68Qkq69s4HrUr4edCvbe7WW2wzOP0
  i4VSp3E5AeXjKc+vFISnPrxzVpsuk9K2SWuZZtNWa2yV5CnokFplas98qSkE1M0QpSlBzp48tNSL
  ptza9RRmy59DTCH8D7LTwCSr+2lsf0q3XRl1ZvukbPemFhbc6CzISQf56Af+deu822DebTLtVzjI
  kwpbKmX2VjotChgisv2xauW1s1vb2+e0y9PPSSNOXcIK0pC1Z9kfIHuLCieKj7qgcDBHGitYedaY
  aU684hptIypS1AAD5k1yH4aLrbXPFvr2YidHMeeLmYjnmDi/ymtrTxPrlIJ6egrq6/2KyaghohX+
  zW67RUOB1DM2Mh9CVgEBQSsEA4URn5n41CfVltt+73SX9zR/4KCzzYsadDehzI7MmLIbU08y6gLQ
  4hQwpKknoQQSCD3qsfVltt+73SX9zR/4Ks8KLGhQ2IUKOzGix20tMssoCENoSMJSlI6AAAAAdq+t
  EZB4sosaF4atRQoUdmNFjtwmmWWUBCG0JlMBKUpHQAAAADtXj8HN+s72w1pt6LhHEq2KkiY0pYSp
  oKkOOBRB+6UqHXt3HoakvF+QPDtqjJ7+yY/9WzUR4XtMaav/AIetJuX3T1ouq2faw2qbCbeKP5Y8
  ehWDii4rdntf1teKT9eYKS7pTSrbcZicB+zlyG+SgGz94BxalZGRhI/nCugNX3Rmx6Uu95kLCGoM
  J6Qok+iEFX/KvfDjRoUVuLDjtR47SeLbTSAhCB8AB0ArKdz2btui+7t9YRIhafbkBGobypBSCEKy
  YsfP21kj3lD3U4wcklNBTvAZpyRbdurtqGQ2UC8TQljP32mQU8v7anB/RroyvHY7XAslmh2e1xkR
  oMNlLLDSeyEJGAPn+PrXsohSlKBSlKBSlKBSlKBSlKBSlKBSlKBSlKBSlKBSlKBX4oBSSlQBBGCD
  60pQQVo0Xo6z3NV0tGk7Db5688pUW3NNOnPfK0pB/wAanqUoFKUoKxu3JkwtqtXTIch2NJYsc11l
  5pZQttaWFlKkqHUEEAgjtX+Z4vF3F4+mRdJwufPzPbBIX5/P+dzzyz880pRqP9MNpJMmbtVpGZMk
  OyZL9jhOvPOrK1uLUwgqUpR6kkkkk96s9KUZKUpQKUpQKUpQKUpQcWePe63RO5VrsyblMTbVWVl9
  UMPq8kuee+OZRnjywAM4z0FfvgJu11XuTc7Mu5zVWxuyPPohl9RYS557A5hGeIVhShnGep+NKUax
  2lSlKMlKUoFKUoFKUoP/2Q==
  ">.Substituting p = .1, n = 100, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0a
  HBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIy
  MjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAAiAiYDASIA
  AhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQA
  AAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3
  ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWm
  p6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEA
  AwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSEx
  BhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElK
  U1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3
  uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwDr9e8c
  +KdA8Y6V4emstHlOpuiw3CNKAMtt5XPY+9dNqF74t0q0e8FhpupxxKWeG3d4ZSB127twJ9q4L4nT
  R2/xd8EzTSJHFG6M7uwVVAmGSSegr0PVPG2hafZPLFfw304B8u2s28+SRuwCpk0ATeFfFem+L9HX
  UdOZwobZLFIMPE46qRTT4i+1+IG0jSoBdNbMPt9wWxFbDrsz/FIR/COnc9q8z8PW2q/Dn4Y+INdv
  ovs2oX0geC3Y8xE5C59D8xOPYV3nwz0pdO8AaZuG+a7jN1PIeTI8nzZPvgigCe51vXtQEh8NaVaz
  QIxUXd/cGKOUg87AqliP9o4B7Z61i6F8Q71/Fw8LeJtIXTdTkGYXhl8yKXjIwevIBwfbHFdRrWtN
  oVqDBo2oX7BMrFZQggY7ZJAFeX+EfEmkeK/ic2o69FPZa5Ev2fT7OVcRxKM8ZPJk+ZuoHXigD0zW
  dfbw/dwzX9uBpEpWNr1Gz9ncnA8xeyHgbh0PX1rdVgwDAgg8gjvVbUdPg1TTbmwuYw8FxE0Tqe4I
  xXBfB3W7i+0C80a9kMlzo1wbbe3Vo+Qufpgj8KAPSKKiuZWgtpZUieZkQsI0xucgZwM8ZPSuRHjX
  Vcf8iJ4i/KD/AOOUAdnRXG/8Jrqv/QieIvyt/wD45XU6fdSXthBcy2s1o8iBmgmxvjPocEjP0NAF
  mufHiE3viBtJ0qBbk2rD7fclsRW/fYD/ABSEfwjgdz2Nf4geIJPDPgrUdSgOLlU8uAns7HAP4ZJ/
  Covhrpa6X4C0sfemuYvtU0h6yPJ8xY+/IoA6v8K4/wCIkMdv4U1XVpr2+jW1sZPLghnaJGkIwrNt
  wSckcZxXZGvO/io2paroD6DpujaleGeaEzyQxDy/KDBmG4kZPAGKAKt/dX3hH4S6Bpmnu0er34tr
  CKQ8mOabl257jLY98U/xRpVn4TufCU+iwCC+l1iGzkmBPmXMTq+8St1fOActnB6YrT8VWN34g0/Q
  tWsdOuvN0rU47xrKdRHK8a5DBQTjdg5GTzinvp154q8XaVql3ZTWWlaQHlgiuRtkuLhhgMVydqoM
  4zySfSgCh4ahg8eanrur6xBHeafBevp+n20w3RpHGAGkCngszHr1GMDFW/hZfT3nhSdJJXlhttQu
  ba2eRizGFHwgyeTgcfhVDQLfXfCnh6/8PWmkXE999puGsrsbRbssjFld2zldu7lcZ+XjOa63wl4e
  h8K+GLDRoZPN+zx4eUjHmOTlm/Ek/pQA3Xtcl8PmK8ubUvpA4ubiMkvb5PDsvdPUjkdcEVswypNE
  ssTq8bgMrKchgehB9KLiCO5t5IJkDxSKUdT0IIwRXmvwm1Sa3m13wjcSF/7GumW2JPIhLEbfoCOP
  r7UAenVk67r1n4ftFnut7ySuIreCJd0k8h6Ig7k/kO+K1c15boF4fFfxp1e6mO610GA29oh6K7Ha
  z/U4cflQB6Ppz3s1lHJf28dvctktFG+8IM8AtjkgdSOM5xVs9KWg9KAPPvGXxQtfDMsNrbadc3Vx
  NL5SSuhjgBDANhyPmI9vzrubo3H2SY2ixtcBT5aykhS3YEjkCvIfjwMHwv8A9fj/APsleyjp+NAG
  PoXiCDW1uIjE9rf2j+Xd2cpG+Fu3ThlPUMOCPyqXXTZJpzSahDPPbI25ooYXl34B4ZVByvseOma4
  H4g3h8J+PPDfiSD5Y7pjYXqjpJHkEE+4yT+Fd1daxPa6r9j/ALG1GaIoGW6hVGjJP8J+bIP1GKAO
  f+HRgu4tV1bTjHFpF9cg2lnGf+PfaNr5XojMeSo6V3B6Vg+F9DbRbS8aQKs19eS3kkaHKxlzwo9c
  ADJ7nNbNzPHa2stxKcRxIXY+wGT/ACoAx9a8RJp13b6ZZwG91e6BaG1VtoCjrJI38KD16noATWzD
  5nlL5u3zMDdt6Zxzj2rzT4Ryy6+2u+LbzDXV/dmFDnPlxIOEHsM16cKAOd8VeLbfw19itxA93qV/
  KIbS0RgDI2Rkk/wqMjJqtqmr+JNC02TVLuwsb22hXfcQWbussaD7xUtw+OvbpXG/E1Z9H+JXhXxR
  cox0m3KwyyAZWJtzZJ9Mh8/8BNemX+oacmjTXNxdW/2J4jmRpBsZSOx75oAqHXm1Dw7BrWgQLqMM
  iiURB9jOnOQvbePQ8cYyOtXNG1iy13TYtQ0+bzIJMjkYZWHBVgejA8Edq84+Aks7+C7yOQnyo70i
  LPuik/rmpbO9bwv8brvR0O2w12AXQjzws+Dlh9SrZ/8ArUAb3lw694/1az1VI5rLS7aHyLSYAxs0
  gLNKVPB6bRngYPeqHhTT7bxd4QubK/eSfSItUuUhiLnbNAkh8tSc5KDtz2FX/FWhXHia5FnZ2Edq
  7KYZ9XmUb44T95IgDlifU8DPrWnPp0nh7wpDpPhy1KuqrbW54IiycGVvpyx9T9aAMHwz4N8PW/i2
  41TSNKgtbfTw1rHIhYmWY/6xuSeF+6Pct6Cu/HSqmladBpOmW9jbAiKBAoJ5J9SfcnJPualu5Wt7
  SeZIXneONnWKPG5yBnaMkDJ6DNAE9Fcb/wAJrqv/AEIniL8rf/45R/wmuq/9CJ4i/K3/APjlAHZU
  VWsLh7uyguJLaW2eWMO0E2N8ZI+62CRkexqzQBV1C+t9NsZ727lEVvBGZJJD0VR1rmNJ17xB4j0o
  axpdhZ29lLk20V67+bOo6MSvCA9utYnx1vpbP4dtFGSBdXSRPj0AZ8fmgrutAt0tfDum28YwkdrG
  oA/3RQBS8LeKLXxRYSzRRPb3VtKYLq0lPzwSjgqfUeh71n2/iq71/XL/AE3w7BbvDpzCO5v7ksY/
  M/uIq8sR3OQK4G11OXQPiv4+jtshDp8t5tHTekaOD9cua6D4Gwqvw/M/WSe8mZ2PViCBz+VAHS6R
  4qeXxJceG9Xt0tdVijE0RjctFcxf3kJ5GOcqemOtauvap/Y+jXN4kL3EyLiG3jBLTSHhUAHOSf61
  5r8UJ20r4jeCNTh4laZoWx3Xei4P/fw1v/EC38aSXulz+Fbe2lS13yyCZlyXxtGFbrgFufegDmR8
  RPiUeB8P3BPQmKbGfeuuufGVxY6npXhhIUv/ABHcwCScbhFDCNuWZiM++AOTXH6d8X9W0bVk0zxt
  oRsixx9oiQqVH94qchh6lT+FWfiH4c1Gz1iD4h+GruHzreJWnR3AV0AwGBJAIKnBGee3NAHU+IfE
  XiDwlpzavf2Vjf6fEQLgWjPHLGpONwDZDcn2q7qXjnRtN8IReJXmeSynVTAqL88rN0QD14P0wa85
  l8cL8ULaLwqZYdCNxt+2NctuaXBz5cI45OAfmwe2DVr4s+F5NN8DaHHpMEj2GjTZljXltmB85/EH
  J/2qAO6S98Wvpo1BdO04MyeYNPaZ/NxjO3zPu7vwxnvV3wv4nsfFekC/st6FXMU0MnDwyDqrD/Oa
  n0vXdO1XRYdVt7qE2skYkL7xhMjJBPYj3rzL4W3Xm/Enxslm4bT5JzKhT7pPmHBH4E0AdX44+I1n
  4Os5HFjd3k6nyxsjKxK+MhWkPGcc4GTXV6Vdtf6TZ3jqEaeCOUqDwCyg4/WvOPj2P+Lepyf+P6L/
  ANBeu+8Of8ixpX/XnD/6AtACa/4i0vw1pzX2q3S28IOF7s7f3VUck1R07VNe1mNLqLSotNtHwU+3
  OWmdfUxrwn0JJry3Tr6bx18YLu+uLSa+sNHLC0tkxsBU4UsWIAJILfUD0r0ebx7ZaZrNvpeu2N5p
  MtycQTXGx4ZD6B0JAPTrigC74p8VW3hi3tgYXu7+9lEFnZxkBpnOO/YDIyaoav4h1zwxp66trFlZ
  z6ehX7ULJn8y3UnG75uHA74wa5HWZm1H9onQ7KXmKytS6Kem7Y75+uQPyr0XxZbx3PhDWIZVBVrK
  bIPshP8ASgBdT8SaZpPh1tdnn32PlrJG0Y3GXd90KO5ORgV5lqPxe8TacwvbjwTcW2lM3yyXBdWI
  7fNjaM1p/CKKPX/h/pD326U6VdyeSrH5dwGVJHfbvOPSu48Vz2Nr4U1SbUtptFtnEiuM7sjAGO5J
  wB70AHhfxLY+LNCg1WwLCNyVeN+GjcdVP+eRiud8a+MfEGj6jFZeG/Dsuruqbrp1RysRP3V+XuRz
  z2x60fCPw3eeHPAsUd+hS6u5DdPG3VMqoAPvhQT9a5GfVPir4Ylu7yTRbS8tJp3uHWJRKQCe5Uhu
  AAB1wAB2oA3PD/iD4heJb6S3utFTQIYoy/nzW7t5jZACgP7ZPHoKK0/AnxM03xiksEkQsNRhTfJC
  75VlzjcrcZGccHkZooGcr8StrfGHwOGwQZEyD6eeK9hjiijPyRoh/wBlQK4+/wDhd4e1S/S+v5NU
  uLtMbJn1CXcuORjnj8K6uwsl0+0W3We4nCkkPcSmR+T3Y8mgRxPxlsZbv4b35iGTBJFOwHXarc/z
  roPBFzHP4E0KRCCosIVz/uoFP6itq7t4by0ltriMSQyoUdD0ZSMEflXFeFYJPBch8LaizNp7yMdK
  u2GVdWOTC57OCTjP3gePSgDsrS+tNQgFxZXMNzCSQJIZA65HXkV5X460JdY+LvhldLjAvYgLi/kj
  H3IkcFC/ucMBnk8V2WmfD3Q9GvJrrSheWBmOXjtrt0jP/AM4resdKs9OEn2WERtK26R+rSH1Zjya
  ALhOBk/WvI/gvC0useL9TUHyJ78oh9SHkb+Tiu48Vatdrbvo2ix+drV3GVj/ALlsh4M0h7Ac4HUn
  gCrPhPw3a+FPD1rpNoSyxDMkhHMjn7zH6mgDWupJIraSSKEzSKhZYgQC5A4GTwM9Oa5AeKPFeP8A
  kn1//wCDC1/+KrtCM0YoA4z/AISjxX/0T6//APBjbf8AxVdXp89xc6fBPdWbWc7oGkt3dXMZ/ukr
  wfwqzRQB578aLGW8+G960QJ+zyxTsB/dDYP866TwXOlx4I0KSM5X7BCufcIAf5Vr3dpBe2c1rcxi
  SCZDHIh6MpGCK4zwkkvgyUeFdQZmszIx0m8I+WRGOfJY/wALg5wO46elAHdU0rn0pQc0tACY4o20
  tFAGXqcWqzOiafc21pGFJeeWLzWz2AXIAHUkk1m+BtdvvEOgG81CGJZEuZYFmhUrHcojYEqA8hW5
  /Ki/j8O+K11G0vwk0emSmG4DTFFQlFYk4YcYbHPcGqfw0uL+48J77yWaa3F1MthLOMO9qGxEW454
  7+mKAOxbpXkPw2ga6+KvjfU4ubcTNEG7El8j9FNegeJNcl0+2Fnp0ButYuVItbdex6eY56Ki9ST9
  BzTPBfhWLwl4fSxEgmuZHM11cYx5srfePsOw9qAOgryP4ZQtp3xP8b2E/EryiVc91Mjtn8pBXruK
  4rxJoNzp/ii18ZaPA09zBGYL+0T71zbnuvq69QO+MfUA6+7vbawtnubyeK3gjGXllcKq/UngVIkq
  SoskbKyMAVZTkEHoQaxb2y0Xxv4bNvOBdaddYbKsUOQc+xVgR07GtWzsrfT7GCztYxHb28axRID9
  1VGAPyFAHkXx4ILeF8H/AJfH/wDZK9kU5z9a5HWvhvoXiK7Fxqsmo3LqxaMNfSBY8nOFAOAP8K3L
  a2tvD+mytLfXD28YMjy3k5kKKB/ePOOKAPNfjnC19F4a02Ebp7m/IRR15Xb/ADYV62F4Ge1cNp+k
  z+KPGcPivUIHgsLGMxaVbyrh3z96dlP3c/wg84wTiu7HSgAqjrVq99oWoWkX+sntpIl+rKQP51ep
  D0oA8v8AgQ4Hgae2PEsN9KHU9RkL/hXpF3qFlYtCt3dwW5nfy4hLIF8xv7q56n2FcWNPbwN4rvtZ
  hjZtA1Zg96EUk2c4/wCWmB/yzbJyR069K6HWfDWi+KW0+41G3W5FnIJ7ZlkIGTg9jyDgHHsKANea
  KK4jaKaNZI3GGR1DAj3BrzLxndabaB/DHhDSLGTxDfKY2+ywIotUPDO7AfKcH8M59K9B1TRrbV1S
  O5kuREucxw3DxB8/3tpGaXS9D0zRIDDpljBaoTlhEgBY+pPU/jQBR8HeGYfCXhi00mJg7RqWlkAx
  vkPLH868/wDEcLaj+0L4ejg5NpZiSX2AMjf+zCvT9V1a20exa6ui+3IVEjUs8jHoqqOSx9K5zwh4
  cuoNW1LxRrMapq+pkAQg7vssIxtjyOCcAZx3FAHYgYFBGaWigAFQXkksNnPLBAZ5UjZkhVgpkYDh
  QTwM9Mn1qejFAHF/8JR4r/6J9f8A/gxtf/iqP+Eo8V/9E+v/APwY2v8A8VXaUYoArWE01xY281za
  taTyRhpIHcMYmI5UkcHHqOKkubqCytpLm6mjggjG55ZGCqo9STwKlxVHWNIste0m40zUIvNtLhds
  iBipPIIwRyCCAaAOO+LGknxP8OLhtPK3LQlbuHyjuEigHO0jr8rE10nhLU4dW8JaTfQMGSa1jOQe
  4ABH4EGr+maXaaRpltp1jF5VrbxiONMk4A9z1rL/AOEO0uN5zam7s4rhy80FpcvFG7HqdoOAT3xi
  gDg/DGijxL4/8c6wf+PGZZNMilHRztCOQe4AUVofBgvY+G9R0G7Xy77TL+RJoz1G7kH6HnBr0Kw0
  600uyisrG3jt7aIYSKMYVRVG/wDDOnX2oDUdsttfhPLNzaymKRl/usR94fXNAHCeNLI+I/i34W0y
  Jd6adG17dEfwLuBGfTJQYr0eDU7C6u5rSC9tpbmA4mhjlVnjP+0AciodL0Kw0gzvaxHz7ht088jF
  5ZT0BZjyf5CsW7+HHhy611tbS3ntdRZzI09pcvCSx6k7T1Pf1oAxPjNaWl74JWEwibUXuYksEVcu
  0hYBlXvyu7PbpXJQ295F8SPCHhXxA7HT7GwjaKJz+7mnEZOT2Yhvl/4DXsVrodjbXa3mxprtV2i4
  ncyOB6Anp+GKbrPh3SvEEMcWp2cdwIzujY5Dxt6qw5B+hoA5L4neDNG1rw3f6rNFHb6hZ27TR3S/
  KTtGdreoOMc8jtVn4VX+par8PrCfVS8kp3ojy8tJGDhSfXjjPtWtL4M0y6iWG/lvr+2Ugi3u7t5I
  yR0ypPP45rVvNLgvLFbPfNBCuMC2kMRwO2V6D2oA47xffeHPCsX+haLp8/iC64s7SC1QyyOejEAZ
  Cg8kmrHw28Iz+FdDml1Fg+rahJ9ou2B+6ey/hk59ya6DSvDWj6K7y2FhFFNJ/rJiC0j/AFdssfzr
  V20AeXfHs5+Hif8AX9H/AOgvXe+Hf+RZ0r/rzh/9AWszxD4C0fxRITq0l/PFu3rALx1jQ4xkKDj/
  APWa09F0G30K3NvbXN7JCFVEjubhpRGBnAXd06/oKAPO/hvZnwv4/wDFXh+7UpLcOt3as3SWIFuV
  9fv846YNS/HOBLrwnp9vEnmX8uoIlsijLsSpBAH5V6FqmhafrHkteQbpoG3QzoxSSI+qsORVe18L
  6dbahHqEgmu72JSsU93K0rRg9QueF/DmgDzjWNNn8N/E7wdrt6cxTW6afczk8LN5ZTk9Bnd1+td/
  431CLTfA+tXUxAVbSRfxYbQPzNa+oaZZatYy2V/bR3NtKMPHIMg//X96y08IaZvg+0Nd3kduweGG
  7uXljjYdCFJwSO2c4oAxfhtpQ8J/DazGpMts/ltd3BkOBHu559MKBmqkXiXw54i1GLUdV1/SorC2
  fzLKxlvIwSw6TSjPX+6p+71PPTq/EfhnT/FOnCw1M3DWu/e0cU7Rhz2Dbeo74rlf+FJ+Cv8Anzu/
  /A2T/GgDorbxt4fu7i6S31O2kt7SNZLi7Eq+RHuOFUvnGTg8e1bcM0VxEk0EqSxOMo6HcGHqCK5v
  T/hz4Z03QrzRorAyWN44kmSaRnLMMYOScjGMj3p2j+AtH0KF4NOk1CG2c5a3F7IYz68Z4/DFAHlu
  ueCZ/FHxT1seH2W3t4IkNzOnCCc43Jkd+hI9qK9ysrC1062W2s4I4IV6JGuB7n6+9FAFiiiigBG7
  UhAbqAec80UUAKv3aWiigBu0Bs4GT1OOtA60UUAOooooAKKKKACmsoY4IB78jvRRQAq0tFFABRRR
  QBzeo6HpE/iCC5m0uxknkI3yvboWb6kjJrogAowBgDgAdqKKAAqM7sDJ4zilHSiigBaQ9aKKAGhV
  UcADJycCn0UUAFIyhhhgCD2NFFADf4afRRQAUh6GiigBMAjGOKFAHAAAHAAoooAdRRRQAwgHJIBI
  JIPpSr96iigB1FFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUA
  FFFFABRRRQAUUUUAFFFFAH//2Q==
  ">.The probability that exactly 4 of n cows are <i>treifot,</i> <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAAoAUcDASIA
  AhEBAxEB/8QAHAAAAgIDAQEAAAAAAAAAAAAAAAcGCAMEBQEC/8QAVRAAAQMDAgMEBQMKDxEAAAAA
  AQIDBAUGEQASByExCBMUQSIyUWGBFRdxCRYYIzNCV5Wi0iRDRlJVVmOCg4WlsbTE0yU0NTdJU2dy
  dpGTlqGy0ePw/8QAFgEBAQEAAAAAAAAAAAAAAAAAAAEC/8QAGBEBAQEBAQAAAAAAAAAAAAAAAAFB
  ESH/2gAMAwEAAhEDEQA/AGfxa493RwxqcSFcnDWIoTEKXGfi3AVtuBJAUBmOCCMjqB1GmNKrfEuN
  F8QbAoko7cliLcqi79A7yMhOf32PfpB/VDf1Dfxh/VtWy0UveHPFu3Lxrsq2lxp9CuWHkP0mpNhD
  vLqUEEhYHXlzxzxjnqdVGbDpsB+fUJTMWJHbLjzzqwlDaQMkknkBqq3E9Pyv237Xatn05cIRDUls
  89uxS1u7iP3EpSc+4e7Uq4s1lfEXtAULg+04r5Ap6kz68hJ5SVIR3qWle1GNgx7XPakaHDjsO6hd
  8R+qwqVMi0cqSKfLkjYqcnnlxLZ9JLfTaVYKs5wBjMk142hDbaW20pQhIASlIwAB0AGovWpD9x1N
  626a84zBYITV5jSilQyM+GbUOi1AgqUOaEkYwpQKSIfH430ifx2h8L6RTFTQ6HEyKn4jahtxDK3S
  lKNp3jCAN24DJOM45y+870jWhU4RrsF9ihytrSqwlQUzGeUrCUPDqhJ5Yc5pycHb1NcokdiJ9UJa
  ixWW2GGUBtpttISlCRR8BIA6ADy1aut0yDWqRLpNTjokwpjKmX2ljktChgjRa20KStIUlQUkjIIO
  QRr3SF7L11T4NcuTg/XZa5My2H3E059w+k7ESvaAf9XKCPcsDonT60QaNcOv3haNvzEQq/dNDpMp
  bYdQzNqDTC1IJICglagSMpIz7j7Nc/5zeG34QrS/HMf8/QSzRqJ/Obw2/CFaX45j/n6lmgNRm+b0
  pVqJiR32359WqCy1TqXESFyJa/PaCQEpHVS1EJSOp117iq0Og0CoVuoL2RIEZyS8odQhCSo49+Bp
  Kdlfxt7Tbh4vXEnvKlU5SoNPQTlMSI3g7G/YCo4PtKM+ZyD0guSHYTDsuOmNIW2lTrKXN4bWRzSF
  YG7ByM4GdE6XFgw3Zk6SzFjMpK3XnnAhCEjqVKPID3nWbSZ7ZUvuuCUumR2u9qVbmxqTATuxl15w
  ZH0ltLg+OgaUm47ejLpqJFepbKqqQKcHJbaTMJAI7rJ+2ZBGNueo11NIS+4MJ3jnwrs5S2G4ls0+
  TXp61EJS220gMsHJ9VKXAevlp96CKWje0Ss1uZbdRhPUW4oSe8ep0hYUXGicJeZWOTrZ9o5g8lAH
  Ur0ne1XSpbFkx+IFBcMW4bUkIlxpCB6SmVKCHW1e1BBBIPUJI8zrqo4y0JrhRSr/AJFKrUqLNiqe
  ebp0JcgR1IO10LXyQgBWQCtSc4JHQ4Bm6NJmw+Oqr6oM6bath1mpz48pTKISHm0JDYQlSXHXl7W2
  9xUoBAK1egTjHTzh9x8gVu/FWLdlsTrQr5c7thiU8HUOLxkI3bU4Uoc08iFeR5jI4atxVhmjw0OK
  ackyn1hmJFax3kh0gkITn3Akk8kgEnAB0ruByqxXOJ/EK5a3OExcOW1RIvd57mP3Sd77befvQtSe
  Z5nGTzOvq/2OMFFuWvXhR3bAdo8WKpcZVWMsvxoyGwpxIDYCBuUkqJ5k4SCcJSBg7JlOvOJYpqFw
  iiCnVpSqrFMbvfFrcfUVLU9u9DGNu3bnl1OinRqM3zelKtRMSO+2/Pq1QWWqdS4iQuRLX57QSAlI
  6qWohKR1OuvcVWh0GgVCt1BeyJAjOSXlDqEISVHHvwNJTsr+NvabcPF64k95UqnKVBp6CcpiRG8H
  Y37AVHB9pRnzOSHcJyY1HRUKwWKbsZDkrvHx3bBxlQLhwCAcjdy0tKPx4s24OKdLsK2S9VnZing7
  Pb9CO13bK3PRJGXCdmOQA55BPTTRnRIk6MY02KxKYUpKi282FpJSoKScHlkEAj2EA6qewy0x9UP7
  thpDSMlW1CcDJpJJPxJJ0WLbaNGjRBo1FeJV802wqIatVKbWpzCULWr5OgrfDaU4yVqGENjmOa1D
  PPHQ4gdh8dVX1QZ021bDrNTnx5SmUQkPNoSGwhKkuOvL2tt7ipQCAVq9AnGOgObUUq17RKJeUagV
  +E9TY9QUlul1NawqNKdIyWVH9LcznaFclAcjnlqB8PuPkCt34qxbstidaFfLndsMSng6hxeMhG7a
  nClDmnkQryPMZY3Ee1YN62VVLaqCU93MYKW3CObLo5ocHvSoA/DQSHUZpd/2dVkVJylV2PUEU1SU
  STESp4Bat21CdoPeKO1XooyeXTmNL7srcRZd12HNplyvn5btpzw051xXpLaAOxxR9voKST5lGfPU
  S4f1CRReBt+cX6g4Uy65InTqclQx3PeK7lr4lYSB7EgY6nJeLAWncNGuu34tft+cidTZQUWX0pUn
  dtUUnkoAgggjBA6a3KjNiU6A9OnPojxmEFbrizgJSPPS+4XUWq0nhRQLetapU2BIpjCWZzk2muSm
  1vFIW6EbHmuji1AnKhkEdQdY79onFqbKob1BnWRPEJK3ZSKmxKYYckbstuJabUs+gnoFLI3eljIT
  tIjU9+sXl2ibepVUDjFHp1IkVg0pYwEhe6O0p791O5Stv3gwOu4k1yeBI4l13i9dl31RVoqiCaaJ
  U1xxIKkqitkARAo42FakFRWefPAGjRUV+qG/qG/jD+rasBKsmqy2izL4k3k40oYUlswmCf37UZCh
  8CNVz7YK6/xGqNvMWrY15y49LRILshdvymkrU6W8BIUgK5BvzA66s5Yt1NXTTvECj12kyG0IL8eq
  Ux6KpKlA8klxISvBBztJxyzjI0MYLD4e2jZCX1W9SUMSZJzJlurU9IfJOTvcWSo5PPGcZ8tV54Ar
  XJ7ZvEJ2VkuNpqKW8+xMtpKfydWv1WqrUtXDvtj064nUlui3i2uOHvvUSFJAKCfaXEtq/hPcdCGl
  x+4gs8ObJYqzyZQE2c3A76M2hxxjehay4lCyEqUEtqwCcZIJyAQVhRe1Rwqo9MZp0C37tbYaBxmM
  wpSiTlSlKL+VKUSSVHmSSTzOrJaNEUK+d22/srPnW8DVvkT/ADHdN+J/vHw/q79vr8/W6e/lq6nD
  i76bfllwLrpDEtiFO7zum5SEpdGxxTZyEqUOqDjBPLGkJ9bFyfZ4fXJ9b1W+RP2R8E54b/Bez7rj
  b6/o9evLrqzhIAJJwB1Oi1VGlLXE+qDTWYxw2+hSXseYNOSs/lgatdqtXZ6parz4/XxxX2lVKbkO
  waa6ejyvRRvT7g0hP/EHs1ZXQrh1+z7RuCYibX7WodWlIbDSHptPafWlAJISFLSSBlROPefbrn/N
  lw2/B7aX4mj/AJmtO/uE3D+/Kw1V7roHyjNZjpjNu+Mfaw2FKUE4bWkdVqOcZ56j32OHBj9pv8py
  /wC10Et+bLht+D20vxNH/M1LNKb7HDgx+03+U5f9rps6IVvaweeY7Pl1LjkhZaYQcfrVSGkq/JJ1
  qdjtCE9nm3SjGVrllf0+KdH8wGp/xDtxm7rGrNtPKCE1GG4wlZGQhZHoK+CsH4aU3YvnPxrCq1k1
  RpUar25VHWZEdfrISs7gfo3BwfD36Lh7649y2xQrkepTtbp6JqqTORUIO9agGZCAQhzAIBIycZyN
  djWKXIYiRXpcp5tiOyhTjrriglKEpGSok8gABnOiItdFBsCnV9fEC5m6XDmNxG4DlRqEru2Ush3e
  hBC1Bv7oQQSM5CefIal2qwcdhLv1ViyKg261T6/dkKNRqe4CkpgpJW9McSf0xwBO0HmhsgcitY1Z
  /QQ7ji2h3gzeaXMYFDmKGfaGVEf9QNIbs8vvPdjW/m3iShhmqoZz5J8GlX/cpWmv2q6+1QeB9dQV
  fompoTTozY6uLdOFAe30N5+GuHbtmv2L2Qq5RJrfd1BduVCVNT5pdcYcUUn3pG1P73RXI7AwHzP1
  Y45/XA9/R4+oh264ngb3sWu0xGyrOh1tK0esSy40pr4hTisa3uxLczlD4a1ZmbQ6w/Tl1hxbc2BC
  cl/be5ZCm1NtBTicAIIVt2ncRkY1MHLPrfFTjRTL0uKjS6NaluJHyXEnI2SZrwVu7xTZ5to3BJwr
  mQhIxzOBp4z4kWfCehTozMqK+gtvMPNhbbiCMFKknkQR5HXsSPHhxGYkRhqPHYQltpppAShtCRgJ
  SByAAAAA1l0aIVvaweeY7Pl1LjkhZaYQcfrVSGkq/JJ1qdjtCE9nm3SjGVrllf0+KdH8wGp/xDtx
  m7rGrNtPKCE1GG4wlZGQhZHoK+CsH4aU3YvnPxrCq1k1RpUar25VHWZEdfrISs7gfo3BwfD36Lh7
  6qb/AJRP/wC/YjVpq1UWKTTHqhJZmPNM43IiRHJLpyQBtbbSpauZ8gcDJOACdVLaeuFXa9+ctdgX
  ui3+9KO8NAklzb4Lw4XsCM43eljrjyzy0IuBpTfPb/oi4s/8uf8As0zaNUWKtTWahGamNNO52olx
  HYzowSDubdSlaeY8wMjBHIg63NELy+K39cnZ7uqsmj1akeIt+o/oOqxu4ktbWnU+mjJxnbuHPmCD
  56XnYGA+Z+rHHP64Hv6PH02uM/8AievT/Z+f/R16QnYluZyh8NaszNodYfpy6w4tubAhOS/tvcsh
  Tam2gpxOAEEK27TuIyMaLjR7dcTwN72LXaYjZVnQ62laPWJZcaU18QpxWNW10iHLPrfFTjRTL0uK
  jS6NaluJHyXEnI2SZrwVu7xTZ5to3BJwrmQhIxzOHbV6hEpNKl1Se8lmJDZW++4rohCUlSj/ALgd
  BVfszISe07xPpBaQ7TnflBLrS0hSFhM1KQCDyI2qUMe/VoVUKhqoQoKqNTlUgNhsQDGR4fYDkJ7v
  G3GfLGkL2MLcmvNXPxMqbCmnbjmL8IlXUtd4pa1fQVqx/B6sXoVjisMRYzcaMy2ww0kIbbbSEpQk
  cgAByAHs1k0aNEalKplNpTDjFLp0SC046p5aIzKW0qcUcqWQkDKj5nqdGtvRoDRo0aA1HuIVn0a+
  bYfoNbaUWVkLZebO12O6PVdbV5KH/kHIJGjRoMXD9N3Rqe7S7vEaVIhFLbFUjrAFQRg4Wpvq24MD
  cOYJOUnyEm0aNAahfEmk3Tc6E2vSZCKPRpbX906qlwGR3ZJCmGEeS1Dq4rkkK5AnoaNBIbVoFJte
  34dBocNEOnw2+7ZaT5eZJPUknJJPMkk66ejRoDRo0aA0aNGgNL69LJntXe1xAslUePcbbXcTor6i
  iPVo/L7W4QDscGBtcwcYAOR0NGgncFx56Ew9IjKivONpU4ypYUWlEZKSRyODyyOXLWjdtvUm6rdl
  2/XI7simzEhEhpuQ4yVpCgrG9tSVAZAzg8xkHkSNGjQVmvamUifxust2LQuLr1r0lqY5PmLauB1S
  H1t7Gu6Usl1GCMlTWMggEkdLTU+M3CgR4bKnltsNJaQp51TrhCQACpayVKVy5qUSSeZJOjRoF/Cs
  up3TecS8L9bZQilrKqJQ23A41EVn7u8ro48cDAHooxyJPPWXtC3JQaDwouSJV6tDhSalR5seCw66
  A5IcUyU7UJ6q5rTnHTIzo0aLCm7Btx0FixKhbL9XhtVp+svSGIS3Ql11vw7XpISeasd2vOM4CSTq
  zejRoUaNGjRBpfXpZM9q72uIFkqjx7jba7idFfUUR6tH5fa3CAdjgwNrmDjAByOho0E7guPPQmHp
  EZUV5xtKnGVLCi0ojJSSORweWRy5azaNGgNGjRoF72hbkoNB4UXJEq9WhwpNSo82PBYddAckOKZK
  dqE9Vc1pzjpkZ0puwbcdBYsSoWy/V4bVafrL0hiEt0Jddb8O16SEnmrHdrzjOAkk6NGjXPFm9L/i
  Ha9ev2om3Kg4mmWa2pC5pZezJqpGFdyMfcmgfWPrKxgYHMmjRlOafDi0+AxAgx2o0WO2lplltO1L
  aEjASAOgA1n0aNAaNGjQGjRo0H//2Q==
  ">.Substituting p = .1, n = 100, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0a
  HBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIy
  MjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAAjAiYDASIA
  AhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQA
  AAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3
  ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWm
  p6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEA
  AwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSEx
  BhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElK
  U1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3
  uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD0bUPi
  dYaVrdvo99omtw39yyrBEYYj5hY7RgiTB5461qX3jBNKhNxqeiaxaWyjLzmBJUQereU7ED8K8/8A
  iZx8XfAv/XeP/wBKEr129nt7azmnu3RLeNC0jSHChe+aAItM1Sx1ixjvtOuorm2kHyyRtkH/AAPs
  ai1HW7LT7q1s5XaS7umxDbQrvkYd2x2Ud2OAK8l+FV82g+GPFviCVHTRVmMtqp4Dbd33fqCi/Ue1
  dd8MLee+0WXxVqZ8zVNYcyFz/wAs4QSEjX0UYzj3oA70cCub8QeNdH8Patp2l3kkr3moSpFFFCoY
  ruYKGbkYXJ+vtWnqWoPbGO2tUWW+nyIo26ADq7Y6KP14HU15D8SLBLD4g+Bk3tLK90rzTOBukYzx
  ZJ/kB0A4oA9c1DWrPS7u2gvmeFLltkc7L+639kLdFY9s9frxWpVLU9OtdX0650+9iWW3uEMciN3B
  /wA5rifhr4huJY9V8O6rOZb3QpjCZnPMsIJCsffA5/CgD0OiuRHxQ8EEf8jNp3/f3/61H/C0PBH/
  AEM2nf8Af3/61AHXUVT0vVbHWtPjv9Nuo7q0lzsliOVbBIOD9QRS6lfQ6Zp1zf3JIgtomlfHXCjJ
  oAr6rrdlpLQJcO73Fw2yC3hXfLKe+1R2Hc9B3NafavO/hj5+vwXvjLUhuvdQlaOAHkQW6nARPQZy
  T6kV6JQBka2NVECvp93a2sKK7TyzQmRwAMjYMgZ65z7VzPhLxReQ/C0+KfENy852TXZOwKfL3HYo
  AHcAfnVn4m+JbLQvBGsK99BDfS2jpBCZAJGLfKCF6nB/lWBr1nBqHwJOn6HNHepbWlusi20gkJVN
  jOvy5525460AWNU1DxJpngF/GVzqksd/HEl22nBF+zrGxH7kjG4nacFs9efatWXW7vxD4uk0DTbq
  SytLK0jub+4iAMpeQZjiUkELx8xOCcYFZvi28s/GWh2HhfQrmO7GoSQ/aXt23Lb2ykM7OR90nAUA
  4JJ9qTQLyy8NeNvGZ1m7is3uLiG4gadtvmwCLA2Z+9g5GBk54oA2fCOr3k2v+I9AvLl7v+ybiIQ3
  MigO8ckYcB8AAkHIzgZGK6HVdTg0eyN5dLKbdWAkeOMv5a93YDnaO5GcVzXw/wBNuwNZ8QX0MkE+
  t3puY4ZFw8cCjbEGHZtoyR712TLkYwCMYoAbBPDcwJPBKksUihkdGBVgehBHUVJXmfh7UD4V+Jd7
  4NJI0u9T7ZpyHpCxyXjX/ZyGIHYj3r0ygBD0rO0/W7LU726t7J3nFsdks6L+6D90DdCw7gZx35rj
  /idr93bLpfhrTJmhv9bnEJlU/NFFkBiPc5wPxrttL0210jTbfT7KFYra3QRxoo6Af17/AI0AXKzN
  a1/SfD9r9p1a/gtIu3mNy3so6k/QVp1578XtMsW+H+sX7WkJuwIAs5QF1HmoMA9uCenrQB2Wiaxa
  69otrqll5n2a5TzI/MXa2M9x26UyDW7OfV59KLPFexDf5Uq7TIn99OzL2yOnfFYnwv8A+SaaB/16
  L/M034i6ZLc+GJtSsWMWqaVm8s51+8rL95foy5BFAHXEgAmuMu/EV9e+N9E0rTnEWnSLNczzYy06
  RgDA9ELMOerY445MUOqar458B6dqGgvYxTXBH2qK837CFyHT5OeWH4isjwqnibUPiPqV9eHRTFp8
  cem3BtllCgYMmIs9CCwDZ9MCgD1AGmySJFG0kjqiKCWZjgADqSadivNPF2oyeJPiBpfgaNytht+1
  6mFOPNQfMsZ9jgZ+tAHd6Vq9trds11Y+Y9tu2xzMhVZR/eTPVffoe2a0M4HNNjRY41RFCqowABgA
  VjeMJLyHwdrMlgCbtbOQxbeudvagCJPF1ld3lza6Va3mqPatsnezRfLjb+7vdlUt7AmrGj+JNO1u
  a5traSSK9tSBPaXCGOWLPQlT29xkVyfwTkt3+G1ssJBkS4mE2Ou7dkZ99pWsjx/dtofxb8I6lafL
  Ldg2txt/5aRmRVwfX7/6CgD0my1qyvNTutOV3ivbbl4JV2sUzgOv95D6j8cGq/izXD4d8NXeppEJ
  powqRRk8NIzBVz7ZIz7VhfEuzni8Pf8ACQ6a3larozfaYJR1ZM4dD6qR1HtVttR0Xxj8PY7zUZBB
  p9/Crv8APtaNwc/Kf7wZePXHSgCvdXep+G9X8PpdarNfnU7k2tzFIqhQxQkPGAAVAYdMng+tQf8A
  CZx634om0iw1K1sbO1kEM13IymS4l7xwg8cd2554Aqno2iavr/iFdf1Ge8+w6fC8WlR3iqksrspB
  ndQAFz24z0JrEtltl+EEXhq2RJPEF6DDJaqP3qXLP88jjqu05bcfQYPSgD2CGMxQpGXdyoA3Ock+
  5p9RW6NFbRRu29lQKW9SB1rH1nxp4c8P3q2eraxaWdwyCQRzPglSSAfpkH8qAN2iuS/4Wh4I/wCh
  m07/AL+//Wq/o/jTw34gvTZ6TrNpeXAQyGOF8kKCAT+ooA3qKKKAM3Wtc07w/YG91O4EMO4IvBZn
  Y9FVRyxPoKzpfGFpZG3Oq2N/pkFy4jiuLuNRGWPQMVZthP8AtYrh9euzrnx70TR5Tm10yE3AjPQy
  bC+4j6bfyrt/HenR6n4F1q2lUFTZyOM+qjcD+lAHR5BGRXPN410D/hJLXw/DfpcajcMyiOD5xHtU
  sd5HA4U8dfasb4Was3iT4b2f20mZ41e0lLfxheBn/gJArjrmytbD9pHR4LS3it4vswOyJAq58mbn
  A+lAHtdcr4x+IGieCDaLqpnaS53FI4FVmAGMkgsOOa6k9K8TuvihYaf4g1F9e8K3UqPcGOGeaMZE
  a/KoVZFAx1PB5LUAdb4e+LWj+Ktdt9J0eyvpJpNzSPMqIscajJbhjnsMe9dFP4r09NXk0m0WfUNR
  iXdLb2iB/KH+2xIVfoSDVTwh4h8L+I4XudAFukqACaIQrFKgPqMZx79K8x0PVLr4UeN9VtfEVvM+
  napN5iX6qWyQSQ3vw3I6jrzQB6knjXTY9Zh0jUYbvS764/1CXsaqs3srqzKT7Zqr4v8AiLofgm5t
  oNUF08twhdVt0Vtqg4ycsO9cn4rtX+KuqaTb6DzpFjOZJ9W6IWwMpH3YgfhnFegHwrpUmt3WsXVs
  l3eTqqBrhVcRIo4VARwM5J7kmgDh/wDhf3hD/nhqn/fmP/4uu18JeK7LxhpT6lp8N1HbrKYgblAp
  YgDJGCcjnH4Gqfi5dJ8P+Fr7UU0uxadE2W6G3T55WO1B0/vEVe8LaMnhrwrY6czKDBCDM5wAXPLs
  fxJoA1NQmtLewnlvmjW0SNjMZcbdmOc57YrK8I+KdM8W6TJe6THMltDO1uBLGEyVCnIAJ4wwx0+l
  UtWDa3o99eygrp8VtK1tERjzm2N+9Yen90f8CPbHK/AE4+H1xn/oIP8A+i46AO38V+K9M8HaT/aW
  qNJ5RkEapEAXdj2AJHpnrXHWnxu8Pape29hpdjqVxfXMixQxuiIrMTjltxwO+cdqpeMvH8GieNbi
  LU/DVzf2FpEscUrJ+7DNhnYbl2n+EZyOhHeug8HeMPB3ii5VNLt4LXUFG4QS26Rye5Ujg/gaAOh1
  LxNp+m6nDpn7661KZd6Wdqm+Tb/ePICj3Yio4fFdiNWh0u/gutNvZ8+RHeIFE2OoR1LKT7ZzXn3w
  4uZJPiz40GoZ+3s2Iw/Xy1kIwPbb5dbnxogR/h7Ldg7bizuoJYJBwUYuFyPwY/lQB6GTxXm+qfG3
  wppOrXOnyC+mkt5DE7wxoyFh1wS4z+VXb7xVep8JLbXYbWe4vrmziwkCktvcAFuASMcnOK5LR/in
  4NDx2Wp+Gm00oApeS3SUDt83Ab8cUAei+EPFsHjLTptRsbSeGxWXyopJyA0hABY4BOACcdTVHxf8
  SNB8FXlva6mbh550MipborFVBxk5YY9q6TTZrCXTIZ9NMDWTpvia3A2FfbHFeNQ/FXTLbULqTxB4
  TuvMuLhmE00SltmcIoV1GAFA4BPJJ70AddpHxZs/E941p4c0e9vJ44zJKJmSIKuQOoLZOTRXQ+E9
  a8N69Zve6B9mxwsqJEI5EPYOoGf6UUAed/FKJZ/it4JicuFklRSUcowBnTowIIPuOa9Jm8HaHdEf
  bLaa9VTkJeXc1wn/AHy7kfpXJeIvhzr/AIj8RWGtz+IrGC4sCpt0i09iqkNuycyZPIFeg6et8lqF
  1Ga3muATl7eJo1I7cFjz+NAHHfFi38r4W6nFaRrHHH5PyIuAqCRc8DtitH4cEH4ceHtvT7FGD9cc
  1v6pp8Gq6XdafdLuguYmikHsRiuN+HRn0CCfwbqhxe2DtJauelzbM2Q6+uCcEduKAMm3134jWl1c
  zf8ACBJcTSud07aggyoJ2gDPyqB29yeprhvHWs+KL3xj4YuNW8MpYXsEwNpbC5Di5PmxkDI+78wU
  f8Cr6MGMVxfi/wACP4n8T6Dq6aitsNKlEjRGEv5mJEfGdwx93HQ9aANnwzf6zqWkmfXNIXS7zzGX
  7OJhJ8oxhsj15/KvOfCBK/HjxhIDi3SF/MPbP7r/AAb9a9R1jWLPQtLn1C9k2QwjOAMl27Ko7sTw
  B3rlfhx4avNLttQ1vVo/K1bWZzcTRk5MKEkqh9xnmgBR8TfARAI1aDBGR/okn/xFL/ws3wF/0Frf
  /wABZP8A4iu4x7mjHvQBR0XVLDWdLiv9LmWazlLCN1QqDhiDwQD1B7VgfE5JZPhxrqw53/Zs8egY
  E/pmutAxUV3bQ3lpNa3CB4ZkaORT3UjBH5GgDj/hIyt8MNE244jcHHr5jV21eefD6Gbwld3ngzUG
  OY5HudNmbgXEB6gf7SnqPfPSvQ6AIntoZWDSQxuwGAWUE4pYoI4QRFGkYJyQqgZP4VJRQBGkEcW7
  y40TccttXGT74qvqE9nZWb3t+Y1htwXLuudv09+gwOSatk4rzzxTrzx6nf3F/p+ppp2jQme2AspH
  iubgKSJGcAqETgDJ+9k9hQB0fg/xTF4u0eTUYbKe1iW4kgVZiNzbDgtx05z+VdDXDfCbZB8O9JtB
  DdJJHAJZGnt3jDtIS5KlgN4+bqM12ssqQRPLKypGilmZjgKByST2FAHkPjgSN8dvB4g++I0L467f
  MfP6Zr2EdK858L6Y/iTx7e+OLiNksUj+y6UHBBkQDDS4PQHnHrnPpXo/agDx7xwHX47+DpH/ANSU
  QKT03bpM/wA1r2AHA6VxHxH8MXes2NjqulRh9X0icXNun/PVQQWj+pxxXTaLrFrr2k2+o2bZilGS
  p4aNhwyMOzA5BFAGlXEfF3/kl2s/9sf/AEclanhLw7f+Hre9jvtdutWa4uDMjXA5jU/wjk/0HoBV
  Xxt4Z1fxZpk+kwapaWdhNsL7rVpJCVbdjO8ADIHbtQAnwwP/ABbTQP8Ar0X+Zrd17b/wj2pFvu/Z
  Jc/TYayfB2gap4Z0m20m61C0u7O2i8uJo7Zo5OuRklyCOvaqnxD1Gc6G2gaWvnaxqwMEESnlEPDy
  t6Ko7nvgUAYPwGWUfDtzJna17IY8+m1Af1Br05UVSSqgZOTgdTWT4X0GDwz4bsdItzuW3jwz4++x
  5ZvxJJrYoAQ9K8f0IPH+0drnnZy9mxiz6bYsY/Jq9hrzzxpo82keLNM8dWMTSizHk6lFGMs1ucgu
  B32gk/hQB6DnFB5GD+VZ90q6zoUy2F+Yhd25EN5AQxXcOHXseuareGtIvNC8OW+nXWpyajcxKwN1
  ODliSSOpJwOnXPvQBQg8D6XpeoXF9pF1e6SZzvnjtJVELn1KMrKO/TFcDplhF44+LMepWbz3OjaJ
  w15PIXFxMMnC54Azz8oAwue9d7feFNQ11DDrevTvZP8Afs7GMW6SD0ZslyPbIrf07TLLSLCKy0+2
  jtraIYSKJcAf59aAM7xntHgfXt5GP7Pn6+vlnH61yvwUjkPw3tTMuQbiVo9w6Ddxj9avfEa6uNQ0
  pfCmlASarq2EwDxBACC8r+i4GPfPGa6fQdHt9A0Oz0q1/wBTaxCNSRyx7k/U5P40AaG30NNEKLIZ
  Aih2GC2OT+NSUUAFczr3jHwxoGoLaazfRQXLRiQK8DudpJAOQp7g101Jj3NAHEf8LN8B/wDQWg/8
  BZP/AIitPQfGXhfXtRNno99FPdCMyFFgdDtBAJyVA6kV0uKKACkz7Gg1zh8O358cjX/7duhZC28n
  +zcfu92Pvdce/TOe+OKAOFv7Y6b+0bp95NxFqFmRG3YsI9mPr8o/MV6D4wuI7bwZrUshwosphn3K
  ED9TR4k8MWniS3tvNkkt7u0lE1pdw/6yBx3GeoPcHrVHUfDOp+ILIadreqQPp5KmeKztzE1wBztZ
  izYBI5CgZoAyfgxpkmm/DiyaVSrXUj3IB64Y4H5hc/jXN6p/ycxpH/XoP/RM1eveR5Fl5FmI4jHH
  shBXKJgYXgY4HHAIrzyf4ea/ceOoPFreILAX0ACpENPbywoVlx/rM9Gbn3oA9JzxUFzbW95A0NzD
  HPEwwySoGUj3BrO1PSrzWfDU2mz6i9pdTIFa6sQYyhBzlckkDjHX1rJ8P6B4p0a2Npc+JoNShHCP
  cWZ81B/vb/m/EUAeeL4dXwr8e9LtdADJbXUXnywqSRHGdwdfplQRnpkVr+HLz/hafiLWjrA36Hpk
  gittOyVV3O4eZJjkkBTjsM16FpHhu00u9utRLPc6nd4E95NjewHRRjhVHYCsYeBBpfiO51vw7f8A
  9nzXfN1ayRebbynOc7cgqc88HuaAPO/FPh/UfhJdx+IvDOoSnS5Zlinsp23DJ6A/3gecHqPWva9N
  vV1LSrS+RCqXMKTKrdQGUHB/Oub1XwdceKGtYvEeoRzWNvIJfsVpCYklYdC7MzMR7AiutVFSMIih
  VUYAAwAKAOL1v/ifeP8AR9FB3WumL/al2O28fLCp/Hc3uBUvjq48RxpYRaHoQ1eB3druEzrEMDGw
  Ek8gkkkd8c8VqaHoLaZqWsahcXIuLrUrgSFwm0RxqNqRjk8AZOe5JrcxQB5Vqnin4iSaTeRz+AY4
  oWgdXkGoIdilTk4z2FYHwS1PX4NLisLPQ1udGkvWM+oGcKYj5a5Gzqei/wDfXtXtepWn27Tbm1D7
  DNE8e7GcblIzj8a5fwL4Hfwd4UudFfUjcNNK8guIY/KKbkVeMk8jbnNAHXMiSIVkQMp6qwyK8N+J
  /he38P8Ai3w7qnhuEWt9eXewQQDCmQEEMAOmc4I6frXe+H/Cfijw7M6R+Lf7RsmORDqFs0jr7hw4
  P9K3Lbw1B/bS6zqErXuoxqUgeRQqW6nqI1HAJ7k5PvQBDq3gzSdV1eLWP9Is9ViGFvbOXy5CPRuC
  GGOOQa87+JED65qOn+CtNvb3UdUmkWS4klmBW2iHdlQBc98kZwPevSdS0vWtRnkSLXfsFoeALW2H
  nY/32JA+oWn6B4V0nw1FKNOgImmO6e4lYvLMfV3PJoAv6dYxaZptrYwDEVvEsSfRRisrxR4S0jxV
  pkttqFtEZCp8u42gSRN2YN1/DoapeK/Ceq65qVnqGk+JrzSJ7ZdojQb4n5zlkyMntz2q4+jaxqOm
  /YdV1eMxuuyZ7G3MLyqeCMlm25/2aAOH+AzX3/CMajHMzNZx3hW3J6Zx8+PbOD+Nen3tjaajbPb3
  ttDcQOMNHMgYEfQ1j6r4SguvCg0HSrufRoUCiKSyO1kwc49SD35yfWq+h6J4p0yy+x3niS2v1HCT
  yWREqj678N+IoA8c1DRdV8KfE/VNO8GedzbLL5aHOyNiDtP0OMZ9aK930bw7ZaLJdXERkmvbtt9z
  dzHMkp7ZPQAdgOBRQBr0YFFFABVDUtLstQ8iS6t1eS2kEsMmSGjbPVWHI9D6jrRRQBfooooAoXGm
  WV1qVvd3Fuss1sCYWfJEZJ5IHTPv1q9gZoooAWiiigAooooAoappdlqluEvLdZRE3mxkkhkcdGUj
  kH3FX6KKACiiigANeFfH/wAR6tpwi0e0uzFYXtrm4iEanf8AP/eIyOnY0UUAdj8GdZ1DW/AaTajc
  edJDMYIzsVcRqq4HygZx6nmu31LTbPVII7a+gE8BkDNGxO1iASNw7jPY8UUUAW1UKoAAAA4A7U6i
  igBMcVQt9LsrTVLm8t4FinulBnZCQJCOhI6Z98ZoooA0KKKKAEwKpWmmWdte3V7FAourlv3sxyWY
  DoMnoB6DiiigC6BiloooAKQgHrRRQBR03TLLSxPFYwLBE8pkMaE7QxHOB0X1wMCr2BRRQAtBoooA
  o2OmWVnNc3EFuqT3UheaXks5zxknnA7DoO1XQMUUUALRRRQAUUUUAFFFFABSYoooAWjFFFABRRRQ
  AYFJiiigBaKKKADFFFFACYxS0UUAFGBRRQAlLRRQAmKU0UUAFGKKKAEpaKKACiiigD//2Q==
  ">.The probability that exactly 5 of n cows are <i>treifot</i>, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAAvAXUDASIA
  AhEBAxEB/8QAHAAAAgMBAQEBAAAAAAAAAAAAAAcFBggEAwIB/8QASBAAAQMDAgQCBAYOCgMBAAAA
  AQIDBAUGEQASBxMhMQhRFBUiQQkyQmFxgRcYIzM3Rld2hZWitMTSFlJWYnKCkaGlwSQl05L/xAAV
  AQEBAAAAAAAAAAAAAAAAAAAAAf/EABgRAQEBAQEAAAAAAAAAAAAAAABBAREx/9oADAMBAAIRAxEA
  PwDZejWbePnFzi1wmqdNYktWZVIlSQ4qO+inyWlZbKdyVIMg4+OnsT303JUbiw1F5kat2TMkBOSy
  5SJUdJPlzBJWfr2/VoLto0puH/GNNTvx/h5edCVbV1MnDbXPDseV7O7La8Dun2gD3Hvz00xbquCj
  2tQJVdrs5uFT4qN7rq/9gB3JJ6ADqSdBKaNVmwK1XrigO1iq0UUaBJ2qpsV5RMvldfbeHxUFXskI
  GSkdyScCr8fL+uezaVCjWPbjtxXBLcK/RkQnpKWY6einFpaIUPaKUjJA6nyxoGdo1k5vjF4lqgtM
  BnhaIbkkhpMg2/NQGiroFlS1lIxnOVAjp1097wuatWHCp1SqsU1i3WI7bVWnspPpcZwdDJU2BtW0
  ehUE4KepwR0A4vWjXhT5kWoQWJ0GQ1JiyG0usvNKCkOIUMhQI7gjXvoDRo0aA0aNGgNGjVMvq+26
  LWIdrUOCa1dVQSVxoCHNiGWx3ffXg8toeeCVHoAdBc9GvGCJQhMCcplUvlp55ZBDZXj2toJJCc5x
  k5xr5qkNFQpsqA49IZRJZWypyO6W3UBQIJQtPVKhnoR1B6jQfEyp02HKixJlQiR5EteyM068lC3l
  YztQCcqOATga9ZsqNCiOy5slmNHZSVuOvLCEISO5Kj0A1nZuzrbjeMi2KZQqYhpNAtuRVZb61qee
  dddWWEc11ZK1kBQI3E4z09+rBdUk374nadYcpIet21aX66nxVjLcmatQSwlxPZQQlQcSD0znPYaB
  t27clu3Iw4/btepVYZaVtccgTG30oPkSgkA6/LiuS3bbYbfuKv0qjtOq2trnzG2ErPkCsjJ0o72k
  tUfxeWIaeG4qqjQqgirqQNvNYbQVtFeO+1aehPbX34YFG+Yde4uVthMibX6i/HpheTu9EpzKi22y
  jPxQVBZVjG49ToHTDlRpsRqXDkMyY7yQtp1pYWhaT2II6EfONeukl4VpBTL4m0eIAij0y9JrNPZT
  97YSSCptA7BIV1AHQbj56b1wKq6aNJXQW4TlSSkFhExSktLIIylRT1GRkZ64JBwe2g79Gqrw8vaB
  d8aYyIz1NrNNd5FUpckjnRHfnx0Ug90rHRQ+sC1aA0aNGgNecl9iLGdkyXm2GGUFxxxxQSlCQMlR
  J6AAdc69NUjjfTLqrNgSaZaUOnzZb7zQkRprxbbfjhW5xrI778BBBKfZUrqOmgl6NfNlVqoN06j3
  hb1SmugluPEqTLri8Ak4SlRJwAT9A1YNKLw+zbQr6ai+xw5oto3ZQ3zDqUePAZbcaUoHBStKQdis
  H/Q9xglu6A0a56lNh02nyKhUJLUWJHbLrzzqglDaAMkknsNVbh5dtSvR16sxKT6FaykFNPkSdyZM
  45H3ZKPkM4zjd7Ssg4SO4XHRql8T+JFCsGmvyKhHqM+S1H9IMSBFU6tLeSAtagNraMgjcojscZxq
  K8PPEmXxStKp3DJpjFNQxVnIcdhtZWQ2lppYKlHurLh6gAYx095Bk6NGjQGjRo0Bo14zhKMJ8QlM
  plctXILwJbC8ezuAwSM4zjrjVTsS+E1urTbYrkEUa6qakLlQC5vQ80eiX2F4HMaPngFJ6EA9wuWj
  Ve4hXjR7Ftp24a6maYLSglZixVvFJPbdtGEjPTcogZIGckajatxHo1MqVvQpMCq/+9lIhsPJZQW2
  X1Jzy3PbzkdiUhQBBBIxoLnrhrdYpFDgGfW6pBpkNKgkvzJCWWwT2G5RAydd2lB4iId3uPUirUuz
  qNd9v0lL0mfSpqtynnCnaFpbI2qKEb8fGOVn2SQNAx6Dddr1/neorko1V5G3nehTm3uXuzt3bFHG
  cHGe+D5aNQXBh+y6tZEO57Kt6m0WJVmwt1qJDbYVvQVJKV7ANxSrcM/6d9GgQXwhv4jfpD+G1rLW
  TfhDfxG/SH8NrQErhnakxoszvX85lQwpqXcdQfbUPIpW+QR9I0WEBeIHETxnUJy1D6XFt4RTUZjH
  VpPJcW6vKh0+UG/nPTVhvWqniT4rKLYK1cy3rXzPmMd0PyUoCwVD3hKlNowf7/np+23b1Ctunin2
  /SINLig5LUVhLYJ8zgdT8566zH4f96fGXxDErPMKajy8+XpjW39nQav1k+8+NXEKx+Jl0VSTw6fc
  pLkoRWJU2K+yTHYKkoCHcbNilFbg6Hq4epGNaw1+KSlSSlQCkkYIIyCNEJXhB4jrOvypMUWbHet+
  sPkJZZkOBbLyj8lDoA9ryCgnPQDJ6ac8qOxKiuxZLSHmHkFtxtacpWkjBBHvBGsh+NjhrQLajUu+
  rbiNUp6TN9FlsxhsQtwoUtDqUjolXsKzjvkHvknU1hSZ8yxaBMqoUKg/TIzkrcMHmqaSV5/zE6Lp
  IeGSuyLZ4jXfwZmvrdi0qS9JoxcVkpY3jLfXzC0LA8yvWidZQpwWPhCZYjZ5e087Hl6tTn9rGtX6
  GqPf3Fnh/YdYapF11/1dNejpkttehvu5bKlJCstoUO6FDGc9NV77Y/gx/bL/AIyX/wDLV9r94Wjb
  8xEKv3TQ6TKW2HUMzag0wtSCSAoJWoEjKSM/MfLUf9k3ht+UK0v1zH/n0FS+2P4Mf2y/4yX/APLT
  Z1U/sm8NvyhWl+uY/wDPq2aIiL0r0W17SqtxTRuYp0RyQpIOCvakkJHzk4A+nSd8HsaXW6FX+Jld
  X6RW7jqK0qePyWG8BKE+SQoqGB7kp8tWbxZB4+Hy6vR87+UxnH9X0lrd+znXL4PCg+Hm3NncLl7/
  AKfSnf8ArGiw3dcVdq9KoNKfq1bqUOmU9jHOlS3ktNN5UEjKlEAZJAHmSBrt0aIzHwQ4iWXXfEzx
  Krr9zUpozV0+lULnSkI9MSkKQsM7iN+5aUEY77xjVnhSadYHicvivXfNYpNMuemwF0yoS1huMpTD
  fLdZ5h9kLyEq2k5IPTT10aBIW1b03iFxVufiS8y/EpAoirfttUhtTan0Kyp2XsIBCStRSgn4ycnt
  jUVwLvijcP8AgTT7TqiHE3jRUyIy7bSgmc/I5zikhtoDctK9ySHACnCsk9DrQmlHWeO0GDX6pSYH
  DTiZXk02W5Dem0ihCTGU6g4WErDgzg9OoB0Et4dLIqFi8NWYVbUhdeqUp6qVdSCCPSXjlScjvtSE
  pz2JSSNMbURZ1cVcdtw60qi1eiGUkqEKqsBmU0Aogb0BStpOMgZzgjOD01L6DOviUqLvDPirZ/FK
  mhSG5RVTK02joJLIwoZHvVtK8E+9CPLWh2HW32W32VpcbcSFIUk5CgRkEazx4+wj7E9GJ+P69bx9
  HIfz/wBacvCnn/YutP0nPP8AUkPmZ77uQjP++iuy8qtVaLRFzqPbku4JSTgRI77TSsYJKipxQGOg
  GBk5IwO+Erwe463jxNl12LRLJgpeihgxebMUlhhKuZuU+7tJJ9lG1KG8n2uwGRoPWTfg8vx5/R/8
  ToLVUuON88Pr/iUHi1bFIi0ucfuFSpK3C2lOQCv2yd4SSNwwlQBzg9AW/N4iWVBuJ63ptxRItTZb
  Q76O9uQXErSFJ5ZIw6SCOiCo+7SZ8frEZXDKhSVJT6SishDZ94Qpl0q+rKUf7ad3DJqUxw2thmcF
  CW3R4iXwrvvDKArP150EDwtt+Sxc943vMiuwnLmmMKYiup2uIjx2uW2paT1SteVLKT1AIBwcgX/R
  o0RnPxL1t+6uKVn8GYj60QahJalVnlqILjW4kN9PJKFrx5lB92tEx2WY0duPHaQ0y0gIbQgYSlIG
  AAPcANZTqG9PwhUUyc8spHJz5erTjH+bOtX6LqpcaPwPXp+b8793XpTeAb8D1W/OB793j6bXGf8A
  A9en5vz/AN3XpS+Ab8D1W/OB793j6EaG0ar1/Va56NR2pVqWl/SmaqQltyJ6xbh7GylRLm9wEHBC
  Rt7+1n3aqVFvTi1KrEKLUuCvq6E9Ibbky/6URXfR2yoBTmxKcr2jJ2jqcY0RUPEBx3ubhtPbp7Nh
  8v0ouiFPnzEKafDZAKkttEqx7SThSknBGQPdYb5uvjQzRXLgtKyKH6vaRzRDqEhxyoONgZ3cpBSh
  Bx12b1K92M9NKn4Q38Rv0h/Da1lopZ+H/i5TuKtvyH0RPV9WgKSmbE37gN2dq0H3pOD36ggg+4mp
  eL5mZb9Lt3ihQVcis29UEtlwdA5HdyFIXjuncAMeS1eeqP4Uohj+JXiQmAnbS45mMgJ+KP8AzByx
  /wDlK8aanjECD4ebjK+4XE2fT6U1/wBZ0Kh+N9YZ4h2Rw/t+jvKabvapRnlYwpSIraec707EoOzo
  emRrqvNyN9sJZtBbivORrcosipRoqOqpMh1QYQkE9ykArKicDqonvqZ8J/P+18tX0jO/lP4z/V9I
  d2/s400tBBChVT156w/pnXTG5vM9XcmF6Pt/qZ9H5u3/AD7vn1Ds8ULPnxpqaHUxV6nGWtoUthtQ
  lrdScbeUoBQGem8gJA6kgddXXRoim8FLResfhpSbdlrbXMZSt2UW/iB1xanFJT/dSVbQfeBnRq5a
  NBljxP2fxV4rVGiij8OZkKFSkPbVS6pBDjinSjPspeIAAbHvPc60LYtYuGrU7NyWlMt6a0hG9Lsq
  O+26og7uWppxRwCPlBPxhjPXFi0aA1nK/aSeHHikoXEVSdlv3IfV897siPIUjYNx7AEpbXk+S/LW
  jdRl00Gk3PQJdCrkJubT5bex5pfvHuIPcEHBBHUEAjQcvEBi55Nm1JmzJsWFX1ND0J6SgKbSrcCc
  ggjqnIGQQCQSNL7hhcHG2GhVO4h2H6y2fe6nTZ0NCl/42i6kfWnH+HV44f0a4Lfp7tHq9ZRWYUYp
  RTZbiSJRawfYf+SpSegCxjcO4B72bQLO4+H87iFctLqV8pjMUOku8+HRGHC7z3vc5IcwAcDoG0gj
  qcqIJGmRIeZjR3JD7iGmWkFa1qOEpSBkkn3ADXpqncQrUqV5uNUWXVRCtZaAqoR425Mmccn7iV9m
  2SMbtvtKyRlI7gpfDNRHrp4nXjxmlsrRCqMl2LRt6SCtreAXMHyShCM+e8e7Wi9c9Ngw6bT49Pp8
  ZqLEjNhpllpIShtAGAAB2GujQQdfs+0bgmIm1+1qHVpSGw0h6bT2n1pQCSEhS0kgZUTj5z56j/sZ
  cNvye2l+po/8mrZo0FT+xlw2/J7aX6mj/wAmrZo0aCKu+hxLmtaqW9OyI1RiuRnCB1SFpI3D5xnI
  +caTfhBemW/Sri4YV5IYrNvVBTgbJ++R3cFK0eadwJz5LT56fOqZfdj+uatCuigzk0e6qckojTuX
  vbeaPVUd9AI3tH6QUnqDnuFz0a8YJlGEwZqWUyuWnnhkkthePa2kgEjOcZGca9tAaNGjQGknVRxM
  4XcNKvcUQWpUGoMqfWKhAWy/zX23ZDj7ikyN4SlQSvoC0R7OMnTL4lxrjm8P67DtF1tmvSILrUF1
  x0tBtxSSAoKAJSoZyD5gdu+qTdke/OJFiu2fNtR+z/WbaWKvOkzo76UMkjmpjBlaytShlILgbACi
  epGCDDtGtxrltSkXFDQtuNVILM1lK/jJQ6gLAPz4VqU1yUanRKRSIVJp7QZhwo7ceO2PkNoSEpH1
  ADXncCauujyUUFyG3UlJAYclpUppBJGVKCepwMkDIyQBkZzoEH4m4D3EjidZvCumFSw0tVSrDiOo
  jMdEgk+47d+AfetHnrREdlqPHbjsIDbTSAhCR2SkDAGqxw+sinWi1MkiQ9U61UnOdU6rJA50tz3d
  uiUDslA6JH+urVoDWMfAvNuGnyrtkUiiprELbDTMjtSENSQr7vy1N8wpQoDCwQpSe6SD0I1qjiVf
  VucPbdFduaU5HireEdkNsqcU46UqUEAJBwSEK6nA6d9ZN8D19W5atwVqh1yW5Gl3A9Bj0/DK1pcd
  CnU7CUg7cl1PU4HfJGi54eFz2HcnFW/aPUb0paaHaVDWXo9KckNvSZzxx1d5ZUhKOgGApRxkfKyH
  Ro0aINGjRoM7eJqhSLZ4kWhxmgsLdi0qS1GrIbTkoZ3nDn1pWtBPns1oWK+zKjNSozqHmHkBxtxB
  ylaSMgg+8Ea+KhDi1CC/BnR2pMWQ2pp5l1IUhxBGCkg9wRqqcPLUqlmOvUSNVETbVQgqprEjcZUE
  k/eQvstoDO3OFJ6DqOoDg43vXXMs2s2za9mTa0/VaY7FEpMyKwwzzUqbIPMdSsqAO7onByOvfC/8
  K1B4gcOLbk2vclgVBLUyqelJmx6hCcQ0FobbJWnnBWE8vd7IUSCcDPfQOjQGjRo0GTfhDfxG/SH8
  Np41Ou8TUUBVPhWMhy4C1yhP9YsCnBeMc3qrnFOfa2cvPuz79Zn8cF9W5dVxUahUSU5IlW+9Oj1A
  llSEodKmklAKgNxBaVkjp2wTrWfDe+rc4g2+quWxLckxG3jHd5jK21NuhKVFBCgM4C09Rkde+ixB
  cCOGcfhrbD8Z2WKhWqi76TVJuMc1zrhKc9dqcnGepJUemcCleMGRLrlEt/hnQ08+tXHUUEND5DDe
  SpasdkhRScn3JV5aeU4yhCfMFLKpXLVyA8SGyvHs7iASE5xnAzjVTsexW6NWZl01yb66uqegIkT1
  N7EMtDqGGEZPLaHlklR6qJPYics6hRbYtSlW9CJMenRG4yFEdVbUgFR+cnJP06ltGjQGjRo0Bo0a
  NB//2Q==
  ">.Substituting p = .1, n = 100, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0a
  HBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIy
  MjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAAjAiYDASIA
  AhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQA
  AAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3
  ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWm
  p6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEA
  AwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSEx
  BhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElK
  U1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3
  uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD0a/8A
  ibYaZrlvot7omtw39yyrDE0MX7wscDDCTB5461p3/jGLSYTcaro2r2VsvLztAkqIPVjE7ECuB+JA
  /wCLx+Bv+usf/o8V6zqNxaWmmXFxfOiWkcbGZpPu7cc5oANO1Gz1WyjvbC5juLaUZSWNsq1QXmt2
  VpqdrprO0l7c8pBEu5gmeXb+6g9T+GTXkXww1Obwv8O/Eev3KMNOWcyWUbZAcgY49iSgz7H0ruPh
  pYzN4cXX9QYy6trB+03ErddpJ2IPRQvQe9AHT6zq9poGj3WqX8hS1tozI5HJx6D1JPAFedH49+FG
  BEVrqjyHhE8qP5j2H3z3q38SPF50DVtMtp9CudSsVRrmcKh8vdyqAnaQcfM2D32ntTfCvxA8D+JL
  yK1Swg0++dh5UdzbRjc3YK4GM+3BoA7RNet4r6ysb9Hs7u7iV4lk/wBW74y0av0LD04J6itisfxJ
  ocHiHQbrTp8hpEzFIOGikHKOp7EHBrC+GXimfxP4ZzfH/iZWMhtbr/aZejfiP1zQB2tFR3FxFa28
  txPIscMSF3duiqBkk/gK5X/haHgg/wDMzad/39/+tQB11Fcl/wALQ8Ef9DNp3/f3/wCtXS2N9a6l
  Yw3tlOk9tOgeORDkMp6EUAWKzbvW7O11O30wu8l7ccrBEu5lTOC7f3VHqfwyag8Ua7F4Z8NX+ryr
  vFtEWVD/ABN0UfiSKw/hpYTDw2mt6g3m6rrB+1XMzdSD9xB6KF6DtmgDs8Gua8ZXmsaZot9qNje2
  lpbWdpJO7SQGSRnUEgDJCgHgc5610+cV5t8ZdesLbwXdaOdQto7y/kitzG0qh1RnBZiucgAA8nig
  C0fFOo+H/hRpur6g327WbqCFYkcBfNuJvuKQMcDdz7Ke9GsXeseErjw9dXOsz6gdQ1GOxvIJEQRn
  erfNGAAVwV4GTkdah8dm0utP8K6hZzRT6Pp+s28txLbsHjjjXIDErkbVOM+lS6w8fjLxp4ftdNlW
  40zSJzqN5cxENH5oGIoww4LcsSB0FAFnT9Qv/GOt6zHa6jcWGkaXcfYke1CiS4nUAyMWYHCrkAAD
  nqfSrfgHXbvXvD0r3zrJdWl5PZyTKu0TeW2A+O2Rjj1zXM+C9atPCPhDVLK+Yf21b390zWP/AC2n
  kZyY9idWDArgjj34rqfh74fuPDfg2ysr0g3777i6wc/vZGLMPwyB+FAGte61Z6dqFraXheH7UdsM
  zL+6Z+ybugY9gevatIVR1nSrXW9IudNvYw9vcIUYHt6EehBwR9K5L4X+JLrVtMvtI1SUyapo85tp
  ZG6yoCQrH34IP096AO8qte3ttp9pLd3k8cFvEpaSSRsKo9Sas5rzOS9/4TP4sNozndo+gJ58sX8M
  1zwF3DuFzke4oA9A06+TUrKO7iimjjkyUE0ZRiueDg8gHqM9jVw9KTFKelAHO+IfG2geGCsepX6C
  4cqqW0fzytk4HyjoOepwK2rmc21rLP5Ms3lqW8uJdztjso7mvGPjfp1nZTeHJba1hikmvZDI6IAz
  klD8x6nknrXtuMj86AKul6nZavYJe2Fws0D5ww4wR1BB5BB4IPIrH8b6xPovhua4t1uw7usRmtYP
  NeBWPzSbehwM4zxnFcxrt7/whPxL066jOzS/EJ8m7jH3VuBgLKB6nIB9cV21z4h0i0v5LC41G2iu
  44xK0LyAPtPcDv0PSgDI8FK11Zf2rF4rvdcs7lMR/aIo0EZB54VVOe2D0rrK5bwRpT6bp1/K0Jt0
  v9Qmu4rcjBjRj8oI7Egbsds107sqIWYgKBkk9hQBS1bWLHRrQT3s2wO4SNFBZ5XPRUUcsx9BVuFt
  8SuUZNyg7X4I9j715x4CvG8a+JdW8V3QLW9tKbLS426RJ1Zx/tNxk/hXpQzQBna1rmnaBZfbNSuV
  hi3BFGCzOx6Kqjkk+grMuvGFtpsCXWqabqen2TED7VcQqUTPTfsZmT/gQFcN8Q7l4/jD4LW8ONOQ
  hk3fd8wsQT9R+7/OvUNUs4b/AEm7s7hQ0M8LxuDzwVINAEeo6zaaZpy6hOzvZkgtNCvmKiHne2P4
  fUjp16VejminhSaKRJInUMjochgehB7ivMvgfqk+peBZLS5JdbK4MEZbn5Cobb9BuIqfwvqDeGfi
  LqXglyTp0yfbNMU/8sgeWjH+znOB2x70Ab0t7ea34uvdGtryWys9Ohja4kgwJJJZOVUEg4UKMnjn
  PtWbB4q1D/hBNWvds11e2F5PYrLbwb2bZJsEpQccDk9sik8UI8GsXE3hia6fxFfQi3kigKmFQOBL
  MSCE2gnGCCeBzWj4fj0bwVpdp4euNWtxfeWZ3M0m15mZvncZ65P40AQ+BmbUoP7Wg8XajrNoymLy
  LqGKPynBGdwVAwYeh9fpXZjoK5bwbpjW11r2piIw2+p3xngiK7SUChd5HYsQTj0xXSXNzDZWs1zc
  yLFBCjSSO3RVAySfoBQBNRXJf8LQ8Ef9DNp3/f3/AOtSf8LQ8Ef9DNp3/f3/AOtQB11FQWV7b6jZ
  Q3lpKs1vOgkikTo6kZBFT0ANdgqlmICgZJPQVzcPjOzvYp7nTbDUdRsoGKvdWsKmMkddu5gXx/sg
  1i/GbWZtH+Hd15DlJbyRbYMOuCCW/NVI/Gum8KafFpfhPSbKFQqRWsYwB32gk/mTQBb0vWLDW9Nj
  1DTrlJ7WQErIvGMdQQeQR3B6VgX/AMQdNs0klt9P1jUrZM77mxsmkiGOvznAP1GRXHeFBHB8UfGP
  hKQH+zrpvtQhU4AJ2sw47EOAR6Cu71zxj4c8L25XUNStoGjGFto2DScDgKg5oAm8M+LtG8XWb3Gk
  XRkMZAliddrxk9Mg/wA+lJqviyx0u5e1S3vtQu4wDJb6fbNM6A9N2OFz6E5rjPhXpNtc6nrfi6C6
  ttupysI7K3kDC3Utuw+OjcDjtzXcTXmh+FbA/ar22soSzSM08oDSMT8zHJyxJNAGXoHxF0HxBqra
  VEbqz1Fc/wCi3sPlOccnHJBPtmtbWPEmm6HNb291KzXd0223tYUMksp/2VHb3PHvXmlnJpHxF+K1
  nrWn30EVtpCDClts94ykkMF67Bnqe1R+Po9V8I/Eqy8cLavfaYIRDKF/5YjaVYe2c5BPGaAO/wBQ
  8bWejCOTWtO1LTbaRgguZ4kaJSem4xu238cV0kcsc8KSxSK8bqGV1OQwPQg15f4o8UWvxB8Ky6F4
  UjbULy+VPN3LsW0TcCWkJ6HjgDNXde1V/hb8LrS2SZbnUI41tYHYcGQ5JbHovOB9KAOu1XxLpWjz
  xW11c5vJv9VaQoZZpPoi5OPfpV2wvJLyJnksbm1GflFwEBYeuFY4/HBrzXwBNoXhvRRrWt6pE+s6
  mBNdXk2W2g8qm/GAB35616fbXVve28dxazRzQuMrJGwZWHsRQBz3jLx5o3giK1k1Uzs1yzCOOBVZ
  jgcnkjisTQPi5o3inXbbSdHsr6SeUku0yoixoASzEhjn6e9czqvxMsdL8Vaodb8LXU8Cz/Z4LiWM
  YEacfKHUDk7jweePSu88H+IvC3iRHuNBS3juIwPNi8hYpUBPcY6e44oAv3HirT49XfSbVbjUNRiX
  dLb2iBjEP9tiQq/QnNVbbxbpEGuLpN5a3Gk6hdtvjS7hVBcNgDIdSys3AHXPArzHRtVuvhT471eD
  xDbyvpmrTeYl+qluQSQ3v94gjqOtbnie0f4p6zo8ehk/2Pp85ln1U8KzcZSPuxGOvTOKAPStZ1rT
  9A017/UrlYIFIGSCSzHoqgckn0FZcvjG0shbyanY6hpttcOEiubuJRHuPQMVYlM/7QFcT4quTrfx
  v8NaFIc2tkpu2jPRpNrOD78KB/8ArrufG+nxal4H1q1lUFWs5G59VG4fqKANTUNTs9Js2u72dYYQ
  QMnksT0AA5JPoK5K++KOj6XMg1LS9dsrd2wtzcaeyRn365/Sqvwuv49f8DaVqmoN5l1pyyW4lkbh
  duPm56HZgZ64z61T+Injnw3daDeeHbbUbK6vb5fIH7wGKAn/AJaO4yBt69zkUAehw6pYT6Uupx3k
  LWLR+aLjeNmzGd2fSsi18XQajam80rS9U1Cz523EESKr47rvdSw9wDXM3vgW5X4NHw1pd6t1ciMT
  LIrfJO2/zCoP909B+FZfgT4m6XpegR6D4iSXTdR0yIQbJIj++C8KAP7/AE479RQB6HoHijSvEqT/
  ANnzsZrdtlxbyoUlhb0ZTyP5cUVx/gvwvqknjLXPF+owPpyajlILMsPMCZX5nxwD8g49zRQBh/E6
  JZ/i54JicuFkdFJRyjAGYdGBBB9wc16RL4O0O5ZTeWst6FOQt7dS3Cj/AIDIxH6VyfiD4deIPEPi
  Ww12bxFYwXNgVNukWnsVUht3OZMnmvQbBbxLVVv5YJbgE7ngjKIRnjAJJ6e9AHBfGeBk+GVylsgS
  OOaElUGAqBvT06V1XgxlPgjQdvT+z4Pz8tc/rV3XNIt9d0S80u6H7i6iMbEdRnoR7g4P4Vy3w6nu
  NO0t/CmqYTVNJyoB6TwEnZInqvOD6EYNAHbnBHPQ9jXjfxs8J6XbaGniGyt0tb9LhI3aFdvnBjwS
  B/EDznrXW2/g/wAR6b4hudS07xdK9rcSF3sr+AzIMnoCGG3HbGK2ZvDSanqFre61P9sa0fzLe3VN
  kMb/AN/byWYdiScUAaGhtdNoGmtfZ+2G1iM+eu/aN365rzD4Mh/+Ej8aOufs5vsL6Z8yX+mK9E8U
  a9HoGhzXO1pbt/3VpboMvPMeFVR355PoMmsz4deFJPCfhhLe6KtqFy5uLxgc5kbtn2HFAHUXU8Vt
  ayzzsFhjRnckZwoGTx9K4xfib4CI41a3/wDAST/4iu4IzRigDiB8TfAf/QWg/wDAWT/4iuu02+tN
  T023vbCQSWk8YeJwpUFT0OCARVrFFAHnHxuSV/hvcmPOFuIS+P7uf/1V1ng9lbwZoZXBH2CAcf8A
  XNata7o9vr+hXulXefJuojGxH8PofwOD+Fcz8Obm4sNKbwtqmE1TSMx4/wCe0BJ2Sr6qensRg0Ad
  sRmontYJH3vDEzYxuZATUoNLQBGkEccflpGip/dC4H5URxJCm2NVRR/CowKkpkkixxs7sFRQSzE4
  AA60AcZdeLdSbWL2PTPBt7qaWcxtzdx3ECAsACwG8huM4rrrGea4sbea4tmtZ5IwzwMwYxsRypI4
  JHtxXAeLLS68LeEdS8Q6Prt2Jo5nvkjOxoJvMkyUK45B3YBBz0r0KB2eCN3Ty3ZAWTP3SR0oAkbo
  cV4/8OxIfjD43dM+R5jA46bvMGP0zXp2va3baBo8+oXR+VBiONeWlc/dRR1LE8ACuc+G3he50HR7
  q91Ndur6rObq6XOfLzkqn4ZP4mgDta8g+FQeP4i+OY58+d9oBweuPNkP8itev4rzrVNPPg/4jr4s
  UY0jU4xa6kw6QScbJW/2SQAT2yaAPRc+1GfasTxPpFz4g8PXGn2OqTabNNt23MOcjBzjgg4I44Ir
  R062kstNtbaa5kuZIYlR55PvSEDBY+560AeS/HjlvC//AF+P/wCyV7Ivf615142+H2s+Nbu1e41y
  ytYLORpLeOOxZjyRyxMnJwB0wK7jTE1GK2K6nPazzbuHtoWjXGB1DM3Oc9DQB5f8eg7ad4eWLPnt
  fkR4652cfrivWTCjsruiF16EqCR+NeeX9j/wnXxGspoxv0Pw8xZ5v4Z7rIOxfULgZPrxXpA6UAJi
  s7xCkknhvVEhz5rWkwTHXOw4rSpD0oA8w+A5H/Cv3H8Qvpd31wteoV5x4Vsf+EE8YajocvyaVq0v
  2nTJTwokx80J9GxyPUCuk8T+H7/Xp9Ley1y60wWdyJpVgH+vXj5TyPTvkcnigC34h8N6V4osBZ6t
  aiaNW3xsCVeNv7ysOQa47xxLB4S8JSQy6xq19cXCm3srSW4BaRyMDJRVdgM5OSc9O9dvqlvqlwiJ
  p1/BZ5zvkkt/Nb228gDv1BrL0vwVp9nqo1i9nuNU1YDC3d6wYxj0RQAqD6CgCn8MvC8vhXwZbWl0
  u28mYz3C/wB1m6L+AwK5HxWHf9oPwuIM7xbKZMemZc/pivW5poreGSaaRI4o1LO7nAVR1JPYCvP/
  AAlpcmv+N9R8c3UTx2zr9l0pHGC0QGDLg9A3OPYn2oA9CWNVzhQM8nA60jwxyMC8aMV6FlBIqSig
  AAxVe8uYLOznublwkEMbSSMRkBQMk4+gNWKQjNAHDj4m+AsD/ibQf+Asn/xFL/wszwH/ANBaD/wF
  k/8AiK7jFGKAKunXlrqFhb3lk4ktZ41kicAgMpGQcHGKtUgGKzfEOmXGs6BeafaahLp888e1LmL7
  0ZyDxyPTHBHWgDiPjlp8l/8ADmWWJS32S4SZwP7uGQn/AMfrt9AuY7vw9ptxGQyS2sbAj3UU3TtH
  8jw7BpOo3L6ltt/ImlnHMwxgk/WsnTPDOq+H7H+zNI1iEacmfIW7tzLJbqf4VYMAwGeNwNAHIeFL
  Vr/46+LNUQEw2qC3LDpvKopH1/dmvS77SNN1OF4b6wtrmOQYYSxBs/mKxIfBNtZ+Fr7R7K/u7e4v
  S0s+oo+J3lY5Lkj16YHanaLo/ijTrEWd74htr9VGFuHsiJQPc78MfcigDzvwXo0nhr44atpGlO50
  xbYySqSSFVgrID7hjjPWvZJraC4GJ4Y5B6Ogb+dUNF8PWWiLcvAHluruTzbq5lOZJn9SfQdABwBW
  HpHhfxLod9cGDxT9tsJXLrb6jbNI8eTnhw4P4YxQB578RfDEHh7x14Z1Hw3CLW8vbrb5EAwu9Svz
  ADpkMQQOK6bxNr13rfxJs/AsE72untEZr6SPh5l2lvLDdgRwcc8muztvDcH9tjWr+U3moohjhdlC
  pAp6hF7E9yck+tUfEXge11rWLPW7W6l07WbPHlXcShsj+66nhhyfzoA4nxn8M00KCbxN4OuZdLu7
  JDM9vG52MijLbc9DgE4OQfauZ8b61d+Mvhp4c8QXUIX7PfNBebBhSeBuHoDj9cV67qWg67remy6b
  f61bQ2sy7JmsrUpK6nqAWdguRwcCr8PhTRoPDH/COrZI2l+UYjA3O4HkknrnPOfWgC9HDaTaesCR
  RPZyR7VjCjYUI4GOmMV5T8JFn0zxj4u0GBnbSrS4PlKTkI28gY9Mj+Vd1Y6Brekaemm6frcL2cS7
  IXu7YyTRL2G4MA2O2RTbHwPZaZ4ZvdIsby7gnvdzXGoK/wC/eQ9Xz2PsOgNAHRz28N1C0NxEksTD
  DJIoYEfQ14pceHE8LfHjRYfD6GKC8TzpYE+7Gh3Bx/u8Zx2/KvQfDvhzxToUDWs3ieHUrb+A3dmx
  ljH+/v5/HNa+leG7TTr+41OR3utTuQFlu5cbto6IoHCqPQfjmgDgPD94Pij4n1pdXUvoelyCK307
  JCu5LDzJMckgKcdhkYrB8T6FqPwivYvEPhu/kbSZphHPZTtuGTkhT6ggHB6ivRR4FGmeJLnXfDt/
  /Z893/x9W0kXmwTHOc7cgqc+h7mn6p4On8Ty2q+I7+OextpBKLK0hMSSuOhclmYgZPAI60AcVesI
  /jr4Y1tkaO31SxHl7v7xiZQv1+YfmK9K8WXEdt4Q1iaQ7VWym5+qED+dN8R+FrHxFp8EExe3mtZF
  mtLmDAe3cdCvt6joaoaj4a1TX7AabrWqwSaexHnpaW5ie4AOdrMWOAcc7QKAMb4M6XJYfDWzMyY+
  1yST7WH8JwBn6hc/Q1reNfBmi+IPD14txZQRXEcLPDdIgV42AyDkduOQeKk8ReFdQ1F9Nl0PXp9F
  ksBsjjij3wuvAw0eQDgDA9qs3Oialq+mNYavqURt5BsnFjCYTMvdSSzFQe+3FAHmfgbxPf8Ahj4H
  XWryxGXyrlo7FZM7cMVAz/shy35Vt6X4A07xp4OtdU8QTT3OsalEtw18r4aIMMqij7oUA9Metd9c
  +HtLu/D7aHLZx/2a0XlCBRtCqOmMdCOuaxdF8L6z4ZsV07TNagnsIyfJS/ti7xKTnaHR1yPqKAPN
  NM8Yaz8LtevPDGtSPqtjDGslqwJ3KD93BPQEZ47EcUV6dpvgayh1i71rVpf7U1S6URtLLGFSNB0R
  EHQcdSSaKAOrowKKKACqF7pdleXNrdT26tcWsgeGUEhkJODgjnB7joe9FFAF7ApaKKAKP9mWb6v/
  AGi8CtdxxiNJWySinOQueBnvjrV0ACiigBaKKKACiiigAqhe6XZXd1a3c1urXFrIGhlBIZMnkZHO
  D3HQ0UUAX6KKKACqmqQpPpN5FICUkhdWAJGQVOeRRRQB4t8L9SvPE2qLo2tTte6fph3WsEnRSh+T
  djl9uBjdnGK9zxRRQBSl0yyuNUhvpoFkuLdMQu5J8vJOSB0B9+tXhRRQAVHLFHPE8UsayRuNrI4y
  GB6gjuKKKAK+m2FtptilnaR+XbxFgibiQoyeBk9PbtVvFFFABgVBe28V1ZTW8oJjkUowVipIPXkc
  iiigBbW0trG1jtbWCOGCJdqRxrtVR6ACp6KKACiiigCnqWm2eqWT2l9bpPA/JRx3HIIPUEHkEcir
  ECBII1GSAoAyST09T1oooAfgUtFFAFLUtNs9UiigvoFnhEofy3ztYgHGR0I9jxVxQAoAGAO1FFAC
  0UUUAFFFFABRRRQAUUUUAFJ3oooAXFJiiigBaKKKACiiigAooooAKMUUUAFFFFABikoooAWkxRRQ
  AuKMUUUAFGKKKADFFFFAH//Z
  ">.The probability that exactly 6 of n cows are <i>treifot,</i> <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAAoAXADASIA
  AhEBAxEB/8QAHAAAAgMBAQEBAAAAAAAAAAAAAAcFBggEAwEC/8QASxAAAQMEAAQDBQMFDAcJAAAA
  AQIDBAAFBhEHEiExCBNBFCJRYXEVMoEJF0JSohgjJTNGV4KFlbHE0hYkOFaDpdNDYmNykZOhtdH/
  xAAWAQEBAQAAAAAAAAAAAAAAAAAAAQL/xAAYEQEBAQEBAAAAAAAAAAAAAAAAAUERIf/aAAwDAQAC
  EQMRAD8AZ/Frj3lHDG5xIWScNYihMQpcZ+LkBW24EkBQG44II2O4HcUxpV74lxovtBwCySjy7LEX
  JVF36DzIyE7/AKWvnSD/AChv8hv6w/w1ayope8OeLeOZjfZWNLjT7FksPYftNybCHencoIJCwO/T
  rrrrXWr1cZsO2wH59wlMxYkdsuPPOrCUNpA2SSegFZW4np+1/G/i7WM+/LhCIbktnry8ilrd5iP/
  AASlJ38h8qtXFm8r4i+ICxcH2nFfYFvUmffkJPSSpCPNS0r4o1yDXxc+KRQ4ceB5UMviP3WFapkW
  zlSRb5ckcipyeu3Etn3kt9uUq0Vb3oDW+jNMtxzDLOq7ZNdo9tiA6Spw7U4r9VCRtSz8gCa6cju1
  vxnGZ96nEMwLbFW+4EAdEITvSR8emgPpWUuDNwicUuJtx4p8T7xbYlqtLoatUKdKQ3HbdPvJSkLI
  BCE6UfipSSe2qDROPZjkeStom2TB5Ma2ODmak3uYIanU/rIaQh1ej3HME7rtzLNY2I3KEb7Afj2O
  VytKvCVBTMZ9StJQ8O6EnppzqnZ0eXuZWw5Jjt/CzYb9arqG+qzCmNvcv15CdV03u2Qb1aJdpucd
  EmFMZUy+0sdFoUNEUR1oUlaQpKgpJGwQdgivtIXwvZVPg3zJOD99lrkzMYfcTbn3D7zsRK+UA/8A
  l2gj5LA7Jp9UBRUHf8wxHH5iIV/ymx2mUtsOoZm3BphakEkBQStQJG0kb+R+FR/5zeG384WJf2zH
  /wA9BbKKqf5zeG384WJf2zH/AM9WygKrOc5pasUTEjvtvz7tcFlq3WuIkLkS1+vKCQEpHdS1EJSO
  5qXyK7Q7DYLhe7gvkiQIzkl5Q7hCElR189Ckp4V/bc2m5DxeyJPmXK5ylQbegnaYkRvR5G/gCo6P
  xKN+p2D0guSHYTDsuOmNIW2lTrKXOcNrI6pCtDm0djehuvzdGJEq2yo0Wa5BkPMrbaktoSpTKiCA
  sBQKSQeuiCOnUV0V8UoJSVKIAHck9qDOszH3EeLbC7CMhyC8i02WTe567hcFOhalFTDR8tOmkaUd
  +4hPfrvpVlz+6yc048WrhOxJfZsUC2KvWRBhxTapQKghmMpSSCEFRClJ/SSddt1EcG7nByfxT8Vr
  61JadRbI8G0RClYO0gKLoHx040e3xFd2EMotni84iuXRaWHrrZ7c/a/MOvOYbb5HuTffTiRvVB4y
  Y9uwDxPYdj+Iw2LRacktM37Tt8VIbjFTCS428Gx7qV7BSVAAkHrXzg/Gt/Go3/P8xgs3izO3J2Dj
  ttlo8yLHiNe75wbPul1xWyVkEjlABA6V5QLdJ4j8dclze1qDllsFhdx+zykn97lTXOYvONq9Uo5i
  2VDoT2J0a9PDXkuP4Z4V7XNustuImyIlN3BhRAebkiQ6VMlHfzCo6Ce55hrvQSfhgukpwZ7i6pL8
  m3Y3lUq32wvOFxTMcEFLPMSSQg7A2SQCB6CmrkEudAs0mZbbWu6SmkhSIiHUtqd6jYSpXTetkA6B
  IA2N7pc+F3E7tjPDZ2dkUcxr7kdykXu4sqHvMuPkEIPqCEhOx6EkU1aCDwnK7NmFm+07M+tSUOKZ
  kMPIKH4zyeimnUHqhYPcH6jY0anKz1xnup4R8bcfz2GS1ZMl/wBRyBhP3FKRrlf1+uEq3v4NkfpG
  tCpIUApJBB6gj1oCiobMsijYvZF3WXAus5tB15VuhLku9id8qB0TpJ95WgOnXqKVnD/xC23OZV2i
  45h19mSogaMSMgt+bJCufmUskhtlKeVOypZ3zgAE9wdlLXj5xIdwOxwYdoYRKyS9v+y2thSVKSlX
  QKdUlIJKU8w6DqSR86rEDxCN27PWsR4h4VcMNkSFJSxIflJfaPMdJUpQSAEE9OZJUAe+tEiUyCC2
  rxYY7cLpypipxeQm2Kc6JVMD+nAnf6XlOb+lFS/CJjGZKXnxepOQ5TDIFyl3Rh1mWy4tPZLLqUqj
  tkb5UpSkEfE7NMalJha1ZH4jMmyq1kLstssrVjXIR/FyZQd81fKR0UWweQn0J1TMyK7Q7DYLhe7g
  vkiQIzkl5Q7hCElR189CiIjOc0tWKJiR323592uCy1brXESFyJa/XlBICUjupaiEpHc1PxHnTbmZ
  E5lER4tJW+35oWlpWtqHNobAOxvQ3rdIzwr+25tNyHi9kSfMuVzlKg29BO0xIjejyN/AFR0fiUb9
  Tt7So7EuK7FlMtvx3kFt1pxIUhaSNFKgehBHQg0Cnv8A4gMGiZtZ8Qsj5v8AcbjcmIK3Iix7PH8x
  xKCoudlEc29J3vWiRTcrI/iIixovi/4bIix2mEE2olLaAkEie4B0HyAH4CtcUUUUVDZlkUbF7Iu6
  y4F1nNoOvKt0Jcl3sTvlQOidJPvK0B069RREzRST4f8AiFtucyrtFxzDr7MlRA0YkZBb82SFc/Mp
  ZJDbKU8qdlSzvnAAJ7+UDxCN27PWsR4h4VcMNkSFJSxIflJfaPMdJUpQSAEE9OZJUAe+tEgcMa7Z
  tEsmZRrBf4T1tj3BSW7Xc1rCo0p0jZZUf+zc3vlCuigOh30q11XuI+Kwc1wq6Y1cEp8uYwUtuEdW
  XR1Q4PmlQB/Cl34UeIU7L8HlWfIXiq/Y46Ikxbitqcb6hC1H1V7qkk+pRv1oGBbM8xC6P3Nm2X2N
  PXayhMwRgp3y1LKglAKQeZZKVAJTs7Hau/EcksmWWJm+Y9OTNt7ylJQ6EKR1SopUClQCgQQehApE
  8M7m5b+HPETjTNc5lXKZOmWpKh0ShG2mT8yVDkHwG9feNXvgzaLlauDVhsON3GBDuENpCpz02A5K
  aLrqS66gBLrW1BbmirmIGiCN9irdxKy+24JhVxyi67UxDb2lpJ0p5wnSGx8yogb9Op9KWvCO4wMu
  nRp/EG5uy8mubRlwbJLiPMQorHcJjodSEPrAIKnBzkehAGz6eIC13OTj2DtZRMgzYLWXwlXZ6NEV
  GjhglaU86FuOaTtQSSVa2odq7uJ7hyPjFgGO2ZSXpVnmrvFyebO/ZI6UFKUrI7eYSQB6632oFR+U
  N/kN/WH+GrQErCbrLaLMviTmTjShpSWzCYJ/ptRkKH4EVnPxgrv/ABGuOPMYrg2Zy49rRILshePy
  mkrU6W9BIUgK6Bv1A71pzBcqaym3e0Cz320yG0IL8e6Wx6KpKlA9ElxISvRB3yk66b1sUMeGB8Pc
  RwhL6setKGJMk7ky3VqekPknZ53FkqOz11vW/Ss88AVrk+M3iE7K2XG03FLe/gmW0lP7NavrNV2t
  auHfjHt2ROpLdlzFtccPfookKSAUE/EuJbV/xPkaEMXxWNyXfD9laYvN5gYZUeX9QPtlf7IVVf8A
  BdYbLE4JW28RoUb7RuD0hUuRyAuKKXloSkq76CUp6dupPrTmu9vh3a1S7XcGEvw5bK2H2ldloUCF
  A/UE0m+FNkv/AAXeueLzrdc75iL8lUq13G3x1SXoxVoKaeZQCv0BCkJI3s9N6AUfxlY+3hc7HuJ+
  Iast4E72WQ7EAR5qihS0KUB0J0hYO/vAgHYFaQwu7Lv+HWW+uNBpdxt7EtSB2SXG0r1+G6T/ABgx
  y9cbLnYsdhWy52jE4Er2y5XG4RlRnHlBJSltlpwBzfKpfvKSAOb5aLvhRo0CAxDitpZjRmktNIHZ
  CEjQH0AFBli1LXE/KDTWYx02+hSXteoNuSs/tgVq6s1eHq1qzPj9nHFflKrU3Idg2109nle6jnT8
  g0hP/uD4VpWhUHf8PxHIJiJt/wAWsd2lIbDSHptvafWlAJISFLSSBtROvmfjUf8Amy4bfze4l/Y0
  f/JXHn3Cbh/nl4au+V2D7Rmsx0xm3fbH2tNhSlBOm1pHdajvW+tV79zhwY/3N/5nL/6tBbfzZcNv
  5vcS/saP/kq2Upv3OHBj/c3/AJnL/wCrTZohW+LB55jw+ZUuOSFlphB1+qqQ0lX7JNcng7QhPh5x
  0o1ta5ZX9fanR/cBV/4h44zl2DXnGnlBCbjDcYSsjYQsj3FfgrR/ClN4L5z8bArthN0aVGu+OXR1
  mRHX95CVnmB+nMHB+HzouHvXHe7Tar5a3rXe7ZCucB/l82LMYS805yqChzIUCDogEbHcA12UURUo
  PDHhtAmsToPD3EosqO4l1h9mzR0ONLSdpUlQRsKBAII6gipjIsbx3JGWmcisFqvDTKuZpE+G2+lB
  +ICwdH6VK0UHlDjR4cVqLEjtR47SQhtppAShCR2AA6AfKq3kNr4f2a5Jy29WfH41y8xKW7i7CaMp
  bn6KUL5edSzrolOyfSrQtSUIK1qCUpGySdACs+cL89cyU3zi5Kxy/ZI0JT8SzRray2v7Nt7Z5SoJ
  ccRzPOkFSwjmXoIGtaBB241kVtyFuQ5bkXFHs6whxM22yIatkbBCX0IKhr1AI+dS9QuDZTY81xWD
  k2OTBMtk1BU05ylJ2CUqSQeoUCCCPiKmqDOnj7bQeFNldOvMTfG0p+hYe3/cKc/Cx9+VwxxWTKJL
  71lhuOk9+YsoJ/8Amkr4vY8nNcpwbhdaCVzZ8tU2Trr5DQHIHFD4BJeP9H51oe3RGIFvjwYqORiM
  0llpPwSkAAf+gor3PUaNZM/J5Abzg66/wf8A4mtZ1jHwL31+xystWuy3KdbnUwxIkQWFSFx1jz+T
  bSNuKSoFfVKVaKRsdd0F68f9uiOcPbBd1Np9sYu3syF66+W40tSh9NtppyYdbIWS8K8Yayq1Q7qp
  20xHX2p0dLyfNLKSSUrBG9k9aWnEbG71xuzSx212zXOz4NZnjJmSbjHVGenuHQ5G2l6WE8uxzKA+
  8r4Db5bQhptLbaEoQgBKUpGgAOwAoPK3w4dvhtQoERiJFaHK2yw2EIQPgEjoBS08WDzzHh8ypcck
  LLTCDr9VUhpKv2SaaVQPEPHGcuwa8408oITcYbjCVkbCFke4r8FaP4URQPB2hCfDzjpRra1yyv6+
  1Oj+4Cm7SI8F85+NgV2wm6NKjXfHLo6zIjr+8hKzzA/TmDg/D508ZshuHCflupeW2w2pxSWWlOrI
  SNkJQgFSj06JSCSegBNCsp+JL/bC4a/1V/8AYO1rKsf8ansgyXxFYrmdnwHNn7PZFQQ86bBJQp0M
  yVPLKUqQD2Vob11Fauxy9Rb9AM2JGucZCVltSJ9veiOAgA/ceSlRHUe8ARvY3sGi1Q71xg+zLxNt
  v5ruJ032SQ4x7TEsHmMPciinnbVzjmQdbB9QRVswHKf9LrM9cTjmQ2Dy5CmPZr3C9meXpKVc6U7O
  0Hm0D8UqHpVioojJn5PIDecHXX+D/wDE1P8Aj/t0Rzh7YLuptPtjF29mQvXXy3GlqUPpttNUXwL3
  1+xystWuy3KdbnUwxIkQWFSFx1jz+TbSNuKSoFfVKVaKRsdd00+I2N3rjdmljtrtmudnwazPGTMk
  3GOqM9PcOhyNtL0sJ5djmUB95XwGy6bXDN6XJ4b4xInlRlu2eIt8q7lwsoKt/Pe6zf4YxyeKDida
  0oSuCsz+dBG0q5ZqUpBHr0UqtS3OZBstmk3CWtEaDBjqddVrSW20J2fwAFIDwYY5NeayfiZc2FNO
  5HMX7IlXcteYpa1fQrVr/h0D6RYrGixfYKLNbk2jyy37AIqBH5CdlPl65db9NV2Q4saFEaiQ47Ma
  OykIaaaQEIQkdgAOgHyFetFEeUyNGmRXYsyO1IjupKHGnUBaFpPcEHoRXJYrFZLDGVGsVnt1qYWr
  mU1DjIZSo/EhIA3UhRQFFFFAVXuIWH2bOcYfsN7aUWVkLZebPK7HdH3XW1eih/8AoOwSKKKDy4fp
  y6Nb3bXl4jSpEIpbYukdYAuCNHS1N923Bocw6gk7SfQWaiigKpfEm05Tk6E4vaZCLPZpbX8J3VLg
  MjyySFMMI9FqHdxXRIV0BPYooLDitgtOL4/DsNjhoh2+G35bLSfT1JJ7kk7JJ6kkmpOiigKKKKAo
  oooCl9mmEz2sva4gYSqPHyNtryJ0V9RRHu0fp+9uEA8jg0OVzR1oA7HYooL3BceehMPSIyorzjaV
  OMqWFFpRGykkdDo9Njp0r2oooCiiig5L1DNxs023pcLRkx3GQsfo8ySN/hukpwGn37BuEMTAJGEX
  0ZVZxIYaZTBWIMtanVrQ6mXryQg8wJ2oKHXSSdAlFBf+BWDK4c8LrRij0lEqXGSt2W8gaSt5xalr
  5f8Augq0Og6AdKtGQSLlFs8l+z25FxnpSAxHW8GkrUSB7yzvSRvZ6E6B0CdCiigq3DvAzYrtcMry
  Cai75bdQBMmpRytstjXLHYSeqWk6HfqojZ+AvFFFBE5XklhxW0quuRXaHbIYVyB2Q6EBa9FQQnf3
  lEJJCRsnRrJ3gJyOw2e6ZPbLtd4cGZc1QUQWpDoQZCwXgUo395W3EAAdTvpRRRqTxseiiijIoooo
  F9mmEz2sva4gYSqPHyNtryJ0V9RRHu0fp+9uEA8jg0OVzR1oA7Ha9wXHnoTD0iMqK842lTjKlhRa
  URspJHQ6PTY6dKKKD2ooooConK8ksOK2lV1yK7Q7ZDCuQOyHQgLXoqCE7+8ohJISNk6NFFBk7wE5
  HYbPdMntl2u8ODMuaoKILUh0IMhYLwKUb+8rbiAAOp30rY9FFFpf8Q8Xv2e3E45cHE2zDW1IXNLL
  25N1I0ryRr+KaB+8fvK1oaHU3m3w4tvgMQIMdqNFjtpaZZbTypbQkaCQB2AFFFEe9FFFAUUUUH//
  2Q==
  ">.Substituting p = .1, n = 100, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0a
  HBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIy
  MjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAAkAiYDASIA
  AhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQA
  AAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3
  ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWm
  p6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEA
  AwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSEx
  BhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElK
  U1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3
  uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwDrvEPj
  rxT4d8W6VoM9jpEp1N0SG4R5QBucJyvbBIrqb268X6fbtcpY6XqSxjc0EEkkMjD/AGd24E+xrz/4
  pyJD8VPBEsrqkaSozOxACgTpkknoK9J1HxnoGnWjzvqdvMyjKw2ziaRz2CquSSaAGeEvF+m+MdMa
  7sfMjeJvLuLeUYeF/Q/0NOvfEedcXQ9KgF5qChXuSWxFaxnvI3949lHJ9hzXnPg211PwX4V8V+L9
  TtGtZbwma3tJOCPmYruHbLPj6Cup+Etk0fgeLU53Mt5qsr3lxM33pCTgZ/AUAdhqGo2ek2Mt9f3M
  dvbRDLyyHAA/x9qwNM8S6n4lj+06LpXk6af9Xe6gxTzR6pGPmI9yRXm/jG/m8Z/Fu18MtHPPpOnO
  DLbwDJkYDcxbsOy5PAH1r0TUfHFp4ZNvHrOj6hptk5EcdztjkhU9lJjYlePbtQBqazq19odlDfS2
  QurWJSb02+d8Q4+dUP3lHJIznHIzWraXdvf2kV3aTJNbzKHjkQ5VlPQililhu7ZJYnSWGVQysp3K
  6n09RivMvh7fNovjrxJ4KZj9kt5murJSc+WjEEqPbDA/n60Aep0Vxh8c3wY/8UN4nOD1EMP/AMco
  /wCE7vv+hF8Uf9+Yf/jtAHZ0Vm6JqcurWP2qXTL3Tm3FfIvFVZOO+FJGD9as6hdx6fp9zezf6u3i
  aV/ooz/SgDK1rxGthfwaVYwG+1e4XfHbK20Ig6ySN/Cg/MngA1uR7/LXzNu/A3beme+K81+D5l1e
  w1fxVe/PfapeMpYnOyNOAg9gSa9MHSgDO1XThqVukbXt3bRo+9zbS+WXGD8pbGcc54x061wPw4uo
  9L8A654kmaR4ri9u7xfMlZz5UfyKu5iT/B3Peut8Yanc2Gh3MVjpt9fXlxBIkC2sJcBsYG49F69/
  Q1zXh7Rp9T+EDeFWsrzTrxdN+zP9qgMY81lOSD/EN3Uj1oAp22kQal8LbzxPr0a3GsXljJfi6Ynf
  bfKWiWI9Ywo28LjJznOabpWqXvjLUPDGiX8jvapokOqamA2BcyOAEV8fw5yxXoeM1cubbWPEHgm0
  8Ix6Td6dK8EVrqFxMoWOCNQA/ltn94TjC49cnFW10ubwr48udVh026udLu9NgtY/skfmNA8RICle
  u0rjB9RzQA3Q0j0j4saro2mxLb6Y2lQ3T20Q2xpNvZdyqOFJGM4xnFdtfNdJZStZRRS3IGY45XKq
  xz0JAOPriuc8KaHeRavrHiTVYhDf6pIoS33bjbwIMIhI43HljjjJrrMUAZGha/a69bStGskF1bye
  VdWk3EkEn91h+oI4I5FbFeV+M70+D/ijoGtwnZb6qv2K+UdJACMMfddwP4V6p2oAQkYzmsDSvER1
  3VZV0yAS6Xbs0ct+zYWSQcFYh/EAerdOwzXNfGLxBcaN4SjsbKQx3WqzC2VxwVQ/eI9+QPxrtNC0
  uHRdDstNt0CRW0KxqB7Dn9c0AaNY/iHxFZ+HbNri5juZmCM4htoWkcqvU8cADPUkCtisvxJ/yLWq
  /wDXnN/6A1AGT4B8X/8ACa6JcamLT7LGl08MaF9xKhVOSfX5u1XNc19vD91FPfWwGkPtSS8Rsm3c
  nAMi/wBw8DcOh68c1xXwC/5J9N/1/wAn/oEdelX9lBqNjPZXUYkt7iNo5FPdSMGgCbzE8rzN6+Xt
  3bs8Y9c15fZeJhr9xJ4g1yG+i8LifyLEBALc/Nt86fDbmBPA4KDvzTPhhf3Oq+F9c8I3dwTc6a0l
  nHMeSI2DKv8A3yQR+VW/7I1PUvA+neDE0yezZVigv7l1AijjRgWMbfxlivAH97nFAHpEaJGioiqq
  KMBVGAB6CiWRIomkkdURBuZmOAAOpJ7UqjCgYwBxXmHxc1WaeXQ/CVtIUOs3SpcEdTFuA2/Qk8/T
  3oA7PQtek8QvJd2lqV0cArBdSEhrk5+8i44TrhjyewxydzpTLaCK0t47eBAkUShEUdABwBUh56UA
  ci/i+XUvE914f8P28VxPZLm8u52IhgPZAByze3GMUl34tu/D2r2Vl4jtoI7a+fyrfULZ28sSdkdW
  5UnsckVxPwwkbw14+8T6Dq58m9u5RNA0h2+cAzcrnrkOD+Fa3xzuLX/hA/s7zRC5a6iaOMsN+ATk
  gdcYzzQB2Ot6+3h+5iuL63/4k77UkvEbJt3JwDIv9w8DcOh68c1tmQGEyJ8427l287uO1YujQ/2z
  4H06LU08z7Xp0a3Ct33Rjd/M1x3wk1y4Olap4fvWeafQ52hQgbmaLJAHvggj8vSgCjJJbXPwmu/F
  9+ytrVzG88V3u+eGTzCsaRnqoGAMDrznNdpqOvzW+gySBgl1BZfabyXGRbAJuPHdjztX8TwOcU+C
  U1/xFFe3WmRabo1vP9pW048y8m7PIo4VR/d6nvVPxh4F1NtOuI9N13Xrj+0bwLLa+ZGYUSR8uzDa
  DtVc9/SgDp/h7ZzWfgfTPtbs93cR/aZ3dsszyHeST68iuorN0fTDo+lQ2Jvbq98oECe5ZTIR2BwA
  OOg4rHv/ABheWN/Pap4Q8Q3axPtE9vFEY5PdSZAcfhQB1VFcZ/wnd9/0Ivij/vzD/wDHa2dB1yfW
  /tBm0PVNK8rbj7eiL5mc/d2s3THPTqKANqkPSloNAHJ3/iySTxR/wjWiW0d1qMcfm3UsrlYbVe27
  HLMc9Bj600+KrnR/ENlo/iG3gi+3krZ3tux8p5B/yzYNyrdMckHNcb8FpzqureLdXn+a4nu1BY+h
  3Nj6dKvfHWMJ4KtbtcrNb38ZjcdVyG5H5CgD06SWOGFpZWCRopZ2PQAdTXkt18SPHhu5jYeArmWz
  LnyJHhmDOmflJwO45rpvEz+Idd+HlnJ4fghlvr2OCSVZWVV8sqGbrwc8DHoTXFt8UfGfhS5jj8Xe
  G1NszY86JTGT9GyUP04oA7rSPE9/Z+DLjxH4whXTcMzi2EZDRIMBVweSzHnB9al03VfEutaXHqlr
  YWFrBOnmQW93I5lZOxYrwpI7YOK5f4lzjxx8Jl1HQGe4gEyzyIo+fYoYMCPVSQSPaux8E61Za14O
  027tZUZVt0SRQwJjZVwQ3p0oAXwt4stvEgu7cwSWepWMnlXlnKwLRN2II+8p7Gm+NPEN7oGkK+la
  bJqepTOFhtY1ZiQOWY7ecAfqR6159pN6sn7RmoHTpUktp7PbM0TBlJEanqOPvAfjWp4wk+Itl4tu
  NS0DS7a605IEhiB2O7D7zfLkMMt1x12j0oApWnjr4kaleQWX/CFtY+fIsZupIZdsIJwXO7A4GTzV
  nXPiV4lj8RX9h4b8MNq1pZSeS9wiSHMgHzj5Rjg5H4UnhT4wC/1hNE8S6adLv2cRrIMhN56Kytyp
  PY8g5r02ysrbT7Vba0gSGBSSqIMAEnJP4kmgDyC5+KPxAs7aS5uvAjQQRruklkjmVVHqSRxXX3fj
  5tI8O6JNqdjv1zVwgt9OtmxuZiMZZvuj5lyT3NSeOf8Aib32jeFE5XUbjzrsf9O0RDMD7M21fxrP
  +J/ge98SQ2Wq6POsWq6YS8Ss20OuQ3BPAIIBB6UAbt9d+LbLT5L1LPSrkxoZGtI5JFcgDJCueCfw
  GaoW3xI0u++H954otFbFspVraT7wm4Cpx1ySuCPWuKtPjDd6rpi6HcQ22m63ODA19cSbbZOo34GT
  uzkAfdz3xxXfeD/Amk+GvD9tYLsviJhdGeRQQ0oGFZR0GB060AcQnxK+IzoHXwBKykZBEU+D+laH
  h74heMdU8W2Gi6n4Wj05bgNI7zCVWEaj5mAbr2H1NeqbeOlcX4TH9ueKdd8TN80If+zbE/8ATKI/
  vGHs0mf++aAKvir4iXGmanLo/h7Q7rWtShANwIUYxwZGQCVBJPt2rK8J/GA6nr6aHr+lNpd7K/lo
  2TtD9lYMAVz2PSvRdL0m20e2khtlOZZXmlkblpHY5JY9z/QCvNvG2hp4m+LPh62sUHnWMYudQmUf
  6uMOGjDH1JDYHvQB1/iXxlDoWqafo1tbte6xqDYt7YOEUD+87HoOD2JODUWs6r4o0LS59UkstNvo
  LdDJNBBJJHIEHLFWbIOBz0Fc/wDE7wbql7qNn4t0CdE1PTVGY3YLuVSWBBPGRkgg9QawV+KT+NNK
  HhsC10bUb6Mw3F3dSfuQpGD5Y6lmBOAcAepoA9IsPG+i33g8+JxO0WnohaUyD5kI4KkeucdOuRVb
  Sdb8ReINLj1awsLK0tJhvt4rx3MsidmO3hM+nNcp4z8CNpPwek0TQxNP9mkW5m4y82OXOB9QcDst
  df8AD3WrPWPA2lS20qEwWyQSoGGY3RdpBHbpn8aAJ/DHi2DX576wmt2stW09/Lu7N2DFfRlb+JT2
  NFeL+OtZurP4sahd+HplkkezjjkaBgwJ43dO/C0UAdJ8Ugj/ABW8DqwVlMyBgRkEfaE616/FaW0L
  5it4o29UQA/pXHan8K9F1m/jv9Rv9YuLuPGyVr0gpg5G3AAHPPFddp1idPtBAbu5usEkSXMm9/pn
  AoA5n4pWcl58NdaiiBLLEspA64R1c/oppPhdKk3wz0PYQdtvsOOxBINdfNGksTxyKGRwVZSMgg9q
  4LwzbH4f382g3bEaJdzmXTLtvuxs3WBz/Cc8qTwee/FAGHplmfDXx9v2u1Kwa3bO1rK3RnyrMoPr
  lCMfSun+Kgtz8NNbNxtAEIKFv+em4bce+a6TVdF0/XLVbfUbZZkRw8ZyQ0bDoysOVPuKoN4QsLiW
  FtRuLzUkgcSRRXk29FYdDtAAYj/azQBR+Gdpd2Pw60aC9V1mERYK4wVQsSoI7fKRXGaHC19+0Xr1
  1FzFbW21z2BMcafzBr0nXtci0SzBET3N7NlLSziGZJ39AOw9SeAOtZPgXwrL4esru71GRJtZ1OY3
  N7KvIDE5CKfQZP45oAhPizXgxx4B1g89ftFtz/4/R/wluv8A/Qgax/4EW3/xddntAowKAM7RL+61
  Gx8+80q40yXeV8i4dGbA6HKEjn+lReKrOXUPCWsWUIzLPZTRoPcoQK1gMUEZoA82+B0qyfDpEB+a
  O7mDD0y27+Rr0qvPrOw/4V74mv59pHhvVpBK0gHFjcdDu9I24+boCAD616ArBlDKQQeQR3oANoJ5
  pAuOmKdRQA3GOcCsjW9dj0t7W1jiNzqN65jtbZTguQMsxP8ACqjkn+ZIrYPSvOmtjq3xp1KG7llS
  K10aJYI0coXV3JcgjkDIAOMdBQB2dhDrC3DSX95aPEycQw2zKUb/AHyx3D8BWmK4jwLJd/2v4mtV
  u7i60W1vVisJLiQykHYDKoduWVXOOScYxmuwu7y3sLWS6upkhgiXc8jnAUe9AHkvxsgbUNU8I6dB
  zcTXjBVHXkoK9hFcFpOjz+JPGg8YalA8FpaxGHSbaZcPtP3pnB+7nJwDzjmu9oA8g+O1tJ9m8Oah
  g+Tb3xWQ9hu2EZ/74NeuRusiK6nKsMg+xrL8S+H7TxP4fu9IvciKdcB16ow5DD3BrK8H6lcwW0fh
  zW8R6xZR7A3RbuJeFljPcYxkdQetAHSxXtrPcy20VzC88OPNiWQFkz03DqPxql4k/wCRZ1X/AK85
  v/QGqvpvhHSNJ8RajrtpA632of69jISDzk4HbJANS69oEfiC2+zT319bwMrJIlrN5YkBxwxxn/8A
  WaAOC+ARx8Ppv+v+T/0COvU+9cr4c8Bad4UIXSb3UorfzPMa3a53xucY5BH06Y6CrHijWp7K1Oma
  PGbjXLpCttCp4iB482Q/wovXJ6ngZNAHA/CSFrjxr411OP8A495Lxo1I6EmV2/kRXsAGK5/wb4Wt
  /CPhyDTIn82QEyTzYx5kh+8fp2HsK6GgArx74nQNa/FLwTqkmRb+ekRbsCsoY/owr2Guc8a+FYfF
  vh6SwaQQ3KOJrWfGfKlXofp2NAHQ5x2qCC9tbtpVt7mGZon2SCOQMUb0OOh9jWP4Z1yTUrb7FqUf
  2bWrRQt3bMec9PMU/wASN1DD1xUnh/wlpPhmbUZdLheJ7+bzpi0hbJ56Z6Dk/nQBc1LSdK1OINql
  jaXMcXzA3ESsEx3yeleUanYWvxD8R2+ieH7KKHw1p8we/voYwqTuP+WaH+Ljjj1z2GfSr/wjpurX
  Dyam11exs24W81w3kr7BAQCPrmti2tLezt0t7WGOCCMYSOJAqqPYDigB6IsaKiKFVQAAOwryH4TQ
  tc+O/G2pp/x7yXbIp7EmV2/lXf8AijXJ7K2OnaTGbnXLpCLaFf8AlnnjzXPRUXrk9SMDNHgvwtB4
  Q8OwaZE/mzZMlxNj/WyHqfp2HsKAOhxg0EZpaKAGkVy1/wCJtatL+eC38F6peRI5C3EU0AWQeoDO
  D+YrqiM0YFAHG/8ACXa//wBCBrH/AIEW3/xytjQdWv8AVftBvtAvNJ8vbs+0yRN5mc5xsY4xgdfW
  tvA9KTbzmgBarxXtrcTzW8NzDJNCQJY0cM0ZPTcByPxqxWFpfhHSNH17U9Zs4HW91Ft1wxkJB5zw
  O2TzQBwfwvtD4a8aeLfDtwuyQypdQZ/jiyQCPXgj9atfGhJNT0XR9BtVL3mo6iixIPRQcn6DPP0r
  udV8Oafq9xb3U6yR3ttnyLqBzHLHnqAw7H0ORTbHw1ZWeof2jI095fhDGtzdSb3RD1VegUfQDNAE
  9pLYaalpo63cAnjgVI4WkUSMqjGQuckcVX8T2+nXHhfU01VY/sP2dzKX6KAM5HoR2PrWd4j+Hvh/
  xPqEWoX0EyX0QCrc28zRvgdOR6Vfj8L2TRRR30t1qKQkFFvZjIAR0JHAY+7ZoA4/4J6Pf6Z4FMl8
  jxG9nM8UbjBCbQM47ZwT+NbPij/hD/C1pJq2oaXYC5c/u0SBfNuH7KoAyT/KupvLT7XaPAJ5oN+P
  3kD7XHPY9qzNP8I6Lp199vjtPOvv+fu5dppfwZycfhigDkvhp4YvrS61HxZrsK2+pan/AKu3A2/Z
  4c5Ax2PA47Bfc139pfWeowmazuoLmMHBeGRXXPpkZqaSFJomjkUMjqVZSOCDwRXIaZ8M9D0PUHvN
  Hm1HT2fho7e7YRkehU5B/HpQByfxl0JNZ1Xw5a6dEG1u4nZAUHzCEDl29lODk9Oa9cjBWNVPJAAz
  61SstGsbCeW5hizdSjElxIxeRx6Fjzj2HHtVyaLzYJIt7pvUruQ4YZGMg9jQBxvhX/ieeKtd8St8
  0Ak/s2xPby4z87D/AHnJ57gCub+L+pXa6r4c0SWaS20W/uQL2VW2hwHUFSfQKScV6Vo2j2mhaRba
  ZYqy21um1AxyTzkknuSST+NLq2i6drti9lqlpFdWz8mORc8+oPUH3FAGR4g8DaB4j0ldPu7CKNYk
  2W8sKhHgHQbSO3t0ri/gleakLfW9IuJmuLDTrgRW0pOQDlgyqfThWx23V26eDreO1+xrqur/AGPb
  s+z/AGw42+m7G7H/AAKtbS9JsNFsI7HTbWK1to/uxxrgfX3PuaAMvxtrEmi+Fbye25vZQLe0X+9N
  Idq/kTn6A1c8OaRF4f8AD1jpcRyttCqFieWb+Jj7k5NP1LQ7TVbzTrm78xjYT/aIUD4TzMEBmHfG
  Tj0zWjt96AOa8T+KbbSJItOivLKHUrlSYzdzKiQr0Mj5IyB2UcsfbJFLTdZ8I+GdLnk/4SHT55WJ
  nurlrqN5Z3xyxAOT6ADoOBVLVPg/4a1nUp9Q1CXU7i5mbc7veH8gMcAdAPaq8HwQ8HW88cqxXzGN
  w4DXRIODnkY6UAYvjzVJdY8e+EdFvRLBoF8I7iWKT5RMxJOx/XBCDH+3XaeMfAeh+K9LkS6t44Lq
  KIiG6jUK0WBwD6r7GtrWvD2leIbIWmqWUVzCpygYYKH1UjkH6VQfwdbTWps7nU9WuLMja1vLeHay
  /wB0kAMR9TQByXwQ1TU9Q8ITxXzPLBa3HlW0rnJK4BK57hTx+NdD4nTwf4btJtZ1TTbBZT93EC+b
  O/ZQOrGugTSLW30kabYg2NuqbIxa4jMY/wBnjj/69UbLwhotlfC/Fobi+HS6u5GnkH0Zydv4YoA4
  74d+FLy41HUfFuvWSW91qQ229myYEEOQRkdjwvHt70V6aBiigBaKKKAAjNNkjSVCkiK6nqrDINFF
  ADgMUUUUANMaGQSFRvAIDY5APUZ/AUuBRRQAtFFFABRRRQA1kV1ZWUMrDBBGQRSqoRQqgBQMADtR
  RQAtFFFABWXqnh/SdXmgnv7CGeaHPlysMOgPUBhzj2oooA0IIIbaFYYIkiiQYVEUAKPYCnPGki7X
  RWXIOGGRkciiigBcCloooAKYYkZ1dkUsmdpI5GfSiigB9FFFABTRGgdnCqGYAMwHJx0zRRQA6iii
  gAooooAZ5SeYJNi7wMBscgemaeBiiigAooooAaI0EhcKA5ABbHJA6fzNKABRRQAtFFFABRRRQAUU
  UUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAH//Z
  ">.The probability that exactly 7 of n cows are <i>treifot</i>, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAAoAWIDASIA
  AhEBAxEB/8QAGwAAAwEBAQEBAAAAAAAAAAAAAAYHCAUDBAH/xABREAABAwQABAMEAwgNCAsAAAAB
  AgMEAAUGEQcSITEIE0EUIlFhFTKBCRYXI0JxgqUYJDM4RlJXg4WVodLjGSZWZ3KixNNiY4SRlJax
  srS10f/EABYBAQEBAAAAAAAAAAAAAAAAAAABAv/EABgRAQEBAQEAAAAAAAAAAAAAAAABQREh/9oA
  DAMBAAIRAxEAPwCn8WuPeUcMbnEhZJw1iKExClxn4uQFbbgSQFAbjggjY7gdxVGlXviXGi+0HALJ
  KPLssRclUXfzDzIyE7/S186gf3Q3+A39If8ADVrKip7w54t45mN9lY0uNPsWSw9h+03JsId6dygg
  kLA79OuuutdaerjNh22A/PuEpmLEjtlx551YShtIGyST0ArK3E9P0v438Xaxn35cIRDcls9eXkUt
  bvMR/wBSUpO/kPlTVxZvK+IviAsXB9pxX0Bb1Jn35CT0kqQjzUtK+KNcg18XPikUOKZH4lQXcIve
  brtcuLj0COp+FKkkNquCQD7yGz7yUKPKElWirm3oDW4R+zK/1cfrv/ArSeaYlaMtsjFku6HTbG5L
  T7kVohLb4bO0tLGurewkkDX1R11sHtRI0eHHRGiR2o7DY5UNtICUpHwAHQUGUh4ytnQ4cfrv/Aq9
  scQBBu9otmW2V7H1XhhkwpK3g7HXJU2FLiqXocjiVEgcwAWBsdfdC1xf/wA7eKOF8OG/fiNPG/3h
  I7eQwdMoV8UrcOiPkKe+I+Kwc1wq6Y1cEp8uYwUtuEdWXR1Q4PmlQB+ygYaKjXhQ4gz8xwmVZr+6
  pd/x14Q5a1nanUdQ2tR9Ve6pJPqU79astEFFcO/5hiOPzEQr/lNjtMpbYdQzNuDTC1IJICglagSN
  pI38j8K5/wCE3ht/KFiX9cx/79A2UUp/hN4bfyhYl/XMf+/TZQFLOc5pasUTEjvtvz7tcFlq3WuI
  kLkS1+vKCQEpHdS1EJSO5rr5FdodhsFwvdwXyRIEZyS8odwhCSo6+ehUU8K/tubTch4vZEnzLlc5
  SoNvQTtMSI3o8jfwBUdH4lG/U7C6QXJDsJh2XHTGkLbSp1lLnOG1kdUhWhzaOxvQ3ROlxYMN2ZOk
  sxYzKSt155wIQhI7lSj0A+Zr2qM+MqX5XBKXbI7Xm3K9zY1pgJ5tbdecGx+ctpcH20FSk5Hj0Zdt
  RIv1rZVdSBbg5LbSZhIBHlbP4zYI1y77iupUEzuDCd458K8OUthuJjNvk36etRCUtttIDLB2fqpS
  4D39KvdAqYjm0S83uZjdxhPWXIoSfMet0hYUXGidJeZWOjrZ+I6g9FAGmuo74q7VLYwmPxAsLhi5
  DikhEuNIQPeUypQQ62r4oIIJB7hJHqaf+GWWRM4wO0ZTDSEInsBa2978twEpcR9igofZQMdFeM6Q
  iHCfluIecQw2pxSWWlOOKCRshKEgqUrp0ABJPQVFk+JDHHOIsfDkYzkbTylrbeXJiFt5Kw2pSEIY
  AK1KWQlIBCfrb7UFvrnZNbXbzYZlqanvQPa2/KXIZH4xCCdL5T6KKeYBX5JIOjrVSDOeO97widGe
  yrhTfLbY5LnIieqcy4v46KEcyUq1s8pcB6Gq3ZMjst5xdjJ7fcGXbQ/HMlMknlSGwCSVb+rrR2D2
  IO+1BDc+4EcKWH7RjFgxYtXu7vaTI+kJKzFit6L8gpLhSSAQhOxorcT01ur/AG2FFttujW6CwhiJ
  FZQyw0j6qEJASlI+QAApL4WMv3uRP4h3Bpbb17CW7Y04NKj25BPkjR7KcJLqv9tI/Jp8oClOZnEJ
  Wat4hZYj14ubZQu5FhQDNuZJ+s8s9AsjfK2NqPyHWuH4juIC+HPDCZd4aki6SliHb+YbAdWCefX/
  AEUhSvhsAete/h9xVOK8L7Wh/mdutybFxuchw8zj0h4BaipXckbCf0fnQOl7u1rsdsdud5uEW3wm
  Rtx+Q6G0J+0+vyqecOuNmL8QOIszE8YYlSWIkByYu4uDy21lLjaOVCCOYg+ZvmPL27He6pMyFDme
  T7ZEYkeQ4HWvNbC/LXojmTvsdKI2OuifjWUfDO22z4veI7LKEttoTdEoQkaCQLg0AAPhRWtaKKKI
  KKQ+LnFGz8NoHtd2s9+moKEq8yFBUplJUSlIW8rTaSSO3Nvt06ja3jPGi55bhLF/xDhzeL1JUXPP
  joktssx+VxSQkvOa51lISrlQlWuYAn4hYKVIWbRBmzuIXqE9aLm5zuW0vLCmbiynuppY6c4H1mzp
  Q79R1pQ4OccbNn99lYxOtEvHMkjc/Nb5a+bn5PrhKtJPMn1SUg66jejrr+IbF/vm4X3NUVS2bvaU
  G52uS2eVxmQyCscpHYqAKft36CgodFTzw85+eI3DKDfJBQLkyoxLglI0PPQBtQHoFJKVa9ObXpVD
  oF6Pm+JSb/PsUa/Q3rjbm1OzGW1FXkAKCSFkDQVzKA5Sd/Loa+jEspsWVxJUqwzTKbhy1w5IUy40
  tl9GuZCkOJSoEbHcVIuGrrc/P+KHFS6O81pgTFQoqPQphNELcHy6kj02Se4Gun4dGLkrhGHLdNiR
  siu7671NkSYa5DCVyXCsApS43tRaCDrnBHMkkaI2VWL1FkTbTLhxJy4D77Sm0SW0hSmSRrnSD05h
  3G/XXftUE4i8CuFdutsGzWbFy5kd6fESC87cZSy2dczslafM0Q2gKX20Vcqde9V0sTd7jQnPvhuV
  tmvhZUl2HBXEbSjQ6FK3nSTvZ3zAaI6dNlS4cc2U3+4cRJAJivpMCwpV+TCSr3nh83nBzf7CG6I7
  9nwzF7VaIdsjWWEWIbCGGi4ylSilCQkbJHU6HeimCigyb90N/gN/SH/DVoCVhN1ltFmXxJzJxpQ0
  pLZhME/ptRkKH2EVnPxgrv8AxGuOPMYrg2Zy49rRILshePymkrU6W9BIUgK6Bv1A71pzBcqaym3e
  0Cz320yG0IL8e6Wx6KpKlA9ElxISvRB3yk66b1sUXHhgfD3EcIS+rHrShiTJO5Mt1anpD5J2edxZ
  Kjs9db1v0rPPAFa5PjN4hOytlxtNxS3v4JltJT/u1q+s1Xa1q4d+Me3ZE6kt2XMW1xw9+SiQpIBQ
  T8S4ltX858jQjStCiEgqUQAOpJ9KKRePEq/McMbpDxi3zJ13uYTb4yYzKllrzjyKcUUg8iUpKjzH
  oDqiF3gEDk19y7ii8CpF7nmFalH0gxiUJUn4c6gokfFNVyuRhdhiYviVqx2CP2vborcdJ1rm5Roq
  PzJ2T8zX23e4RLTapd0nvJZiQ2VvvuK7IQlJUo/9wNBlvwvLXH8UvEuAwdRCueogdtonJCP7FKrV
  1Z08GGOTXmsn4mXNhTTuRzF+yJV3LXmKWtX5itWv5utF0WuHf8PxHIJiJt/xax3aUhsNIem29p9a
  UAkhIUtJIG1E6+Z+Nc/8GXDb+T3Ev6mj/wByvjz7hNw/zy8NXfK7B9IzWY6Yzbvtj7WmwpSgnTa0
  jutR3rfWl79jhwY/0N/Wcv8A5tA2/gy4bfye4l/U0f8AuU2VJv2OHBj/AEN/Wcv/AJtVmiJb4sHn
  mPD5lS45IWWmEHX8VUhpKv8AdJr5PB2hCfDzjpRra1yyv8/tTo/9AKf+IeOM5dg15xp5QQm4w3GE
  rI2ELI9xX2K0fsqTeC+c/GwK7YTdGlRrvjl0dZkR1/WQlZ5gfzcwcH2fOi4u9cfJcYsWSO2p2929
  ExVpnIuEHnWoBmQgEIc0CASNnW9iuxX4pKVJKVAKSRogjoRREtmZHwEl5j99svM8GfvXsaYJkOX9
  hQ8lLnmJQUFzk6L97et7AO+g1TocmNNhszIchqTGfbS6y80sLQ4hQ2lSVDoQQQQR3qV5bwy4ZT7n
  ExK28PcRYkS0+0Tn2bNHQuNDSrSiFBG0rcV+LSQQQPMUDtuqsw00wy2ww2hpptIQhCE6SlIGgAB2
  AoFHji2h3gzmaXNaFjmKG/iGVEf2gVM/Ai++9wWlNuklDF6fba36J8tpX/uUqmrxV39qw8D76gq/
  bNzQm3Rmx3cW6dKA+Puc5+yuj4dsNfwXhJZrJNb8u4LQqVNT6pdcPMUn5pHKn9Gi4oVZM0P8olrQ
  +P6prWdY8yOdNtvj7dmwLY9dHmlIJisqSHFoNrSFlPMQCoIKlAbGyNetCNEeIO3RLpwTy9iY2laG
  rS/JRzDs40guII+fMkVOfBKyLrwGmW67xmpcA3WQwhiQ2FtrZKGypJSehSVKX07d67vGO75FnmKP
  YPg+N3tMi76Ym3G5296FGhsbBXtTqUlaiBrSAroT8qoHDPELfgmD2zFraouNQmtLdI0XnCSpaz+d
  RJ16DQ9KBjbQhtCUISlKEgBKUjQA+Ar9ooojKn3Qp55NvwyOknyFuzFrHpzJDIT/AGKVWp4yENxm
  m2tciUAJ18AOlRjxj4TJy7hOubbmVPT7G97alCRtS2uUh1I+zSv0KoPCLImMr4Z49fmHAv2mC35u
  j9V1I5XE/YtKh9lFw1Vk3w2/vwuJX9K//YNVp/Jr5GsFuE6VEukpJWG0t2+3vS3SSCfqNJUQOh94
  6G9DfUVlPga7kGPeIbKs0vuB5rDtN7E3yV/QEpxTXnSUPJ5kpQT2SR031oRrS9Tvoyzzbl7HLm+y
  R3H/AGaI35j73Ikq5G09OZZ1oD1JFTH8Nv8Aqi4s/wDlz/EqqQpDcyGxLaS8lt9tLiUvMqaWAobA
  UhYCknr1SoAjsQDXrREb8Us36T8L17uXscuF7VHgP+zS2vLfZ55LCuRxPXlWN6I9CCK+vwgADw7Y
  vod/a9/+Leo8X/73bKP+yf8Ay2aVvCvl6rNwJskS547f1MtmSYciBbnZqJSTIdJ/cUqLagvmTpYS
  NAEHr0LhM40RPonxsYXLtKPLkXBUB6SUdCrmeWysn+aR1+VawltoeiPNO68tbakq38CNGorw+wW+
  5Pxnm8XsztblpQ037PYbW+QXmmwkpDroBISrRUeXqQVntobo/FjImMU4bZBf33A37LBcLWzrmdUO
  VtP5yspH20Gffuez76rZmUZRPkNvQ3ED05lJeCv7EprVVRLwaYXJxThMm4XBlTU2+ve2lChpSWeU
  BoH842v9OrbQr4INls0G2O2yFaYEWA9zl2MzHQhpfP8AX2gDR5tnex131r2tdut9qgNQLXBiwYjQ
  02xGaS22gd+iUgAV9NFEeclhiVGdjSWW32HkFtxtxIUlaSNFJB6EEdNURWGIsZqLFZbYYZQG2mm0
  hKEJA0EgDoAB0AFelFAUUUUBRRRQFL3ELD7NnOMP2G9tKLKyFsvNnldjuj6rravRQ/8A0HYJFFFB
  5cP05dGt7try8RpUiEUtsXSOsAXBGjpam+7bg0OYdQSdpPoGaiigKn/EPF79ntxOOXBxNsw1tSFz
  Sy9uTdSNK8ka/cmgfrH6ytaGh1JRQPNvhxbfAYgQY7UaLHbS0yy2nlS2hI0EgDsAK96KKAooooCi
  iigKn2aYTPay9riBhKo8fI22vInRX1FEe7R+n4twgHkcGhyuaOtAHY7FFA9wXHnoTD0iMqK842lT
  jKlhRaURspJHQ6PTY6dK87vNTbbVLuC48qQmMyt4sxmVOvOBIJ5UIT1Uo60AOpNFFAq8I4N1Nhdy
  fJIq4t/yFwTZcdwEKhta0xF0eo8tvQI/jqcPdRp0oooJxCwu55TmcTMM9bZQi1rKrJY23A41EVv9
  3eV2ceOhoD3Ua6EnrVHoooPGbKjQYT82bIajRY7anXnnVhCG0JG1KUo9AAASSe1YzTmmKnxzffb9
  OwhYfM8v6QLg8nf0d5O+btrzOm+3r2ooosbMhSo06ExNhSGpMWQ2l1l5pYWhxChtKkqHQggggjvX
  tRRRBRRRQBAI0eoqcY7h12wLLHlYgyzJxO7yvNmWtbgbVbXldFPsE9C2fymumtbT/FoooKPRRRQF
  FFFBEPGXklhh8Gb1jcm7Q27zPTFXGgl0ec4gSUKK+Tvy6bX73baSO9fvg1yOwzODFkxyNd4bl5gp
  lLkwQ6PPbQZTigoo78unEde3vAUUUa54t1TnJMMumf5O19+CGo2J2qV5kS0tuc6ri6k+69II6Bsd
  0tDe97UfyaKKMqMkBIAAAA6AD0ooooCiiigKKKKAooooP//Z
  ">.Substituting p = .1, n = 100, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0a
  HBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIy
  MjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAAkAiYDASIA
  AhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQA
  AAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3
  ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWm
  p6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEA
  AwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSEx
  BhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElK
  U1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3
  uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD0XUPi
  hpml6vBpV9pOtQX1wyrDE1qv7wsdo2kPg8nHWtO78ZQ6ZCZ9T0fWLK2UZed7YSIg9WMbNgV5/wDF
  Pj4qeBv+u8ef/AiOvY5WjETtKVEQUly3QDvn2oAraZqljrFjHfaddxXNtIPlkjbIP+B9jUeq61Za
  QIBcyMZrh/LggjXfJM3oqjk47noO9eS/CS7XTj4y1RSyeHoZWlhP8Pylz8v/AADaK6f4Z+f4iS88
  Z6mu68vpGitVPIt7dTwiegJ5PrigC74o+KPh7whq/wDZmom7a58tZCIIg4UHpk5HPFYv/C+fB/8A
  c1P/AMBh/wDFV11t4J0WO+vb+8soL+9vJjJJNdRLIQOgVQRgAAAcVi+O9L0bSvC8y2Oh6aNRvXSy
  s9tnHnzZDtBHy9QMn8KANXT/AB1pF/otprO26trC6lMST3EO1VIOAXIJCqTwCePpXTgggHNZWnaF
  Z2PhuDQ/JSSzjtxbsjLw64wcj35P41x/gDVptO8Rav4HvZnlbTG32MjnLNbnGFJ77cgfT6UAejUV
  gy+N/CkMrxS+JdHjkRiro97GCpBwQRng5pv/AAnfhD/oadF/8D4v/iqAOgrN1TWrLS3t4riQm4uX
  2QW8S75JD3wo7Dueg7mq8virRToV9rFpqVreWlkjNK9tMsgBAztypPPT865j4ZRz61Z3HjHVBv1H
  U3YRZ5EFupwsaegzkn170AegDpWdqy6o6RLpk9pBknzpbhGfav8AsqCAT9TitEVgeNNbt9A8Jare
  yzRJJHayGNXcKXbbgAepyRQBieAvEN9eeC77X9dvxPGLm4eOTy1jCQRnbwB/uk8561ROpeINR8B3
  PjH+1pbGX7O95Z2KRoYViHKLJkbnLAckEYzx0qvFYFv2eX0/TitxP/YxysJ3FmZdzDjucmp/EOo2
  998MrHQdDlhub3WLWKytIom3YQqA7nHRVXcST06UAWx4tvPEWo6Do2lSGzmvtOGp306qGa3gOAFT
  PG5mOMnoBnBzVvQtWvrTx9qXha7vZL+GOxivreeYL5iAtsZGKgBucEHArG8N2lv4e+KWtwXk0cKf
  2RZJYtIQoeKNdr7SfRgM/WtDwXbvq/jLxF4vKsLO68uy09mGPMhiHzSD/ZZun0oA7LU9Qi0uwkvJ
  0laGLBkMUZcqvdsDnA6nHapLS6t721juraaOaCVQ0ckbblYHuDUzDIrzW1vv+EL+KP8Awj2duja2
  n2i0j7W85J3KvoGIJx2JFAHpdGaQdK434l+K5vCvhgyWWDqN3ILa0Ho5H3vwH64oA6FdaspdafSo
  JGmu4l3zCNdywjsHboCew61pVieFdBi8O6Bb2SEvMR5lzM3LTStyzse5Jrb7UARzzxW8DzTypFEg
  yzuwVVHqSelY2heKtI8S3F9FpN19qWydUlkUfISwJ+U9+h/+vV/UNJsdVjSPULSK6iRtwjmXcucY
  5U8Hr36V5L+z+MW/iMDoLmIAf8BegD1XUNZs9LubaG+kaFbltkczqfL39lLdFJ7Z61o96ralp1tq
  2nT2F7EsttcIUkRu4P8AWuI+G/iC6kl1Xwrqk7TX+iTGJJmPzTQ5wrH1I459xQBa1/TtdhjvtXuP
  GF5plpGWZLWztoZAqDhQC6ks7cfi2BW/4ZtdUtfD9omsXkt3qDJvmkkCggnnb8oA46fgao3J/t/x
  Slip3WGlMs1zjpJcEZjQ/wC6PnPuVrpx0oAKzZNask1qLSVkaS9dfMaKNd3lJ/ec9FB6DPXtWT4+
  8UDwj4Tu9TVQ1zxFbqw4MjdM+w5P4UvgXRTpHhq3luGaXUb5RdX078vLKwzyfbOB6YoA6Y9Kx9V8
  S6fpN7BYSNJPqFwCYrO3QySuB1OOw9yQK2T0rx74e3T3fxi8Yyagf9NGY4g/URrIRge23Z+dAHoU
  PiywOqQ6ZfRXWm3txnyIr2MKJsdlYEqT7Zz7VdvdastOvrW1vJGhN0dsMrqRGz9k3dAx7A9e1cj8
  Y7OO4+HN7cniezkinhccFW3hcg/RjWpoLReOPhxZHVIhKt/aATZ6lum4ehyNw9DQB1Q6VyFve6h4
  m1XWBb6jPp9hps32WLyFQtLKFBZ2LA/KMgBRjoeao/DPxHdX9vqPh/VZvN1PRZjbvIessYJCuffj
  n8Ko+IrW7ttT1DSvB99OdR1h/MuokVWhs8gBpmfGUJA4XqT0FAHQ+H9Wu/Evg2w1GS9WzeQMLiaI
  AbtpKkoWyFyRnPNL4M1DUL2TVlnuJbzToboJYXsqBWnTaN3IADANkBgOaLHS/C9loUHhS5m0+5Ww
  gVZLedkLKMffKk5XPJzTPh3ps+leHJLdjKLQ3cz2SSE5S3L/ACDnkDHIHoRQB11FRXNzBZWstzdT
  Rw28Sl5JZGCqijkkk9BWJ/wnfhD/AKGnRf8AwOi/+KoA6Ciuf/4Tvwh/0NGi/wDgdF/8VW1a3dtf
  Wsd1aTxXFvKu6OWJwyuPUEcEUATUUVzHxA1mXQfA2q38DbZ1i8uJv7rOQoP60ASt4vsZb65s9Nt7
  zVJbU7bk2UQZYm/ulmKgt7Ak1e0bXdP161knsJt/lOY5o2Uq8TjqrKeQa534UWEdj8N9JKD57lDc
  yN3ZnJOT+GPyrm7q7Phv9oCCKI7bfXLRBOg6M43BW+uU/U+tAHe3nirTrbVW0qAzXupKu97W0j8x
  o19XOQq/iRVdPG2mR6tBpepR3Wl3lx/qEvogiyn0VwSpPtmvK/D2tTfDL4g6za+JYpRaarL5iX+0
  sDhmKtnuvzEHHII6Vu+PY3+Jdxpej+Gh9pt7a48651Rf9RD8uNqv/E2DnA7gfgDPSNZ17TNAtUuN
  SulgWRtka4LPI391VGSx9hWZe+M7fS7b7XqOlataWQ5a5ktgyIPVgrFlH1FcD8VbTVdF8TeHfFsE
  L3thpaqsqf3GBOWb03AgZ9VGe1dDqPxB0XxB4Rni0QSajqN9A8MWmxoTMGIwS4/hUZyWPHvQI7eD
  VLC60xNSgvIXsnj8wXAcbNvrmsiDxha6hA1zpWn6lqVqCQLi2gGxsddpdlLfgDXCR/D/AF2w+CV1
  oCS+Zqckn2loEbIxkExA98hfpkmpvh18SdFg8NwaLrUy6Xf6XEIWSdSokVeBj/a6ZXqT0zQB32he
  KdJ8RiYafckz27bZ7aVDHLEemGU8j+VR3HiqwTVJdMs47nUb6AZnhs0DeUD03sSFB9s5rgdD0bWr
  rxn4i8dJZzafby20q2VtKu2SchAFZk7AlQeectU3wHljm8J6jI7br19QZ52P3jlVIJ/Hd+tAHeaX
  4n07VL6XT1aW21GFd0lndR+XKF/vAdGHupIrYkdUQu7BVUZJJ4A9TXk3xlujoOoeGPEdqdl5bXbR
  ll6tHjcVPqDgj8a6L4meKYvD3h+1EttPcR30wikji4JixucZxgZHH0JoAyX+O3g+OR0H9osFYjct
  uMH3HzdK7DRddsvE3hpdZmtvI06QmSP7Xt5RGyJCOi8rkemAa4LQvHvw01iZbebR7LTpmO0farGI
  Jn03gYH44ru9f8Nx67ptnpSlIdJ81WuYYvk8yJRkRjHRScZ9hQBzk/xq8GQ3xtRd3Mqg4M8duTH+
  ec/pXc6dqVnq9hDfWFzHcWsy7kljOQR/ntXPax8P/C99ok1k+kWVsgjOyaKEI8RA4YMOePc1yfwG
  gvI/CN7LMW+yy3hNuD04UBiPqf5UAesHpXPWvjPR7zxfN4ZtpZJb+CJpZSi5jXGMruz975hxirN/
  cTXt2dLspDGwAN1cIeYVPRV/22HT0HPpXlvhW3itf2h9dggRUijtXVVHQDbB/n86APaqwpfFuli/
  k0+zeXUL2M4khsozL5f+8w+VT9SK4L4m+LL651+y8EaNdi0muyv2y63bfLRv4c9uASe/Qd67HwxN
  4T0Oxg0LR9T07cgwI0uEMkrd2Izkk0AdLbymaBZHhkhLDJjkxuX64JH61FqF9baZZT315OsFtAhe
  SRzwoFTSSLFG0jsFRQWZmOAAOprifFaPrHgrXtRnUizj0+4azhYY3HyziVh6/wB0dhz1PABv+GPE
  tj4s0ZdU05ZvszSPGBMm1sqcHjJorkvgj/yTiH/r7uP/AEOigDnvixH5vxO8Ex73TfKi742wy5nj
  5B7H3r0W48GWF9GYdRvNTvrc/eguLxjG31UYyPY1xvi3wP4x8T+KNM1pG0O1/s0q0ERuZXyQ4f5j
  5Y7gdBXpWnvfyWwOpQW0Nzk5W3maRMeuWVT+GPxoA5L4jWMOm/CnWLXTbeO3ijgULHCgUBd654Ht
  mpfhPj/hWOh4/wCeLZ+u9q6nUrCDVNLurC5XdBcxNFIP9lhg1w/wvNxoVrfeD9S+W+0yVnh7Ca3c
  5V19RnIPoaAPQ81xNx/xP/ifbwfes9At/Pk9DcyjCD/gKAn2JFdoT6dawPCeg3OjWt9NqEsMuo39
  3JdXDxEleThVBIBwqgDp2oA6EdK8ckLr+0zD5fQ2Y8zHp5L9fxC16/LLHBE8krhI41LO7HAAHUk1
  5x4D02XWvF2teOriNlgvCYNNDjBaAYHmfQ7Rj2zQBrXGs/DiO6lS5v8Awys6uwkEjQbg2ec575zm
  mf258Mf+gj4V/wC+7euobQtKdyzabZMzEkk26Eknv0pP7A0j/oF2P/gMn+FAHBeO7vRLv4T64/hm
  fTpoA8QlNgUK58xN2dnGdv6V0Pwwx/wrXQNvT7Iv55Oa2L/w/p95ol9pQtooba8iaORYownUYzx3
  HFcr8L5bjS9LuvCep/JqOkysAp/5awMcpIvqvUe1AHf1RvtG03U5EkvtPtLpkGEM8CyFR7FgcVdB
  zS0AUrHSbDTA62Fja2iyHLi3hWPcffaBmltdLsLKeWe1sraCaY5kkiiVGf6kDJ/GrlFAHIeLdbsr
  G5gs73wjq+urs81ZLTTVuY4zkjGWIw3H5EVY8MeL4vEV5dWaaHrWmtaIrONRtVhHzdAMMecDPTp+
  FaWt6oulWIdYxLdzuIbWDPMsh6D6dST2AJqXSNP/ALOshG8nm3DsZJ5iP9ZIfvN9OgA7AAdqANA1
  458YN6+N/BLQf6/7QcY648yL/wCvXsROBXmX2L/hNfi1HqaDdo/h5PKWX+GW6ySQp77cjP096APT
  RXjvxq3/APCReDd3+o+1/NnpnzIv6Zr2IVxfxO8LT+J/C5FiM6lYyC5tR/eYdV/EfrigDtRSZFY/
  hjXIfEHh2z1GLhnTbLGfvRyjh0YdQQc1T0XTPEdr4m1i71TWIrrS7hgbK2VcGEZ78cccdTnrQB0h
  6V4x8AP9T4j/AOvqL+T16xqsmqrbAaTb2ctwcj/S5mjReOD8qsTz24+tcB8O/BPijwNLerONIvIL
  2VHkMdzIjR4yDgGIg8E8cfWgD049K8d0Lev7RuvrFnyjbkvjpny4v65r1u8vLews5ru7lWK3hQyS
  SMcBVAySa4H4Z6Lcz32seMr+FoZ9ZmLW8TjDJb5yuR6nj8B70AegxQRQ7/KiRN7F32qBuY9zjqfe
  paKKAPHv2gPN/sHR9ufK+2Nv9M7OP6167Ft8pNn3Nox9MVzXxB8Lf8Jd4RutNjIF0MTWxPQSL0H4
  8j8ak8Da1/bPhW0aVWivrVRa3kLjDRTIMMCPyP0NAHSEg8Vy2reB7DUNej1+1uLnTdXjG03VqRl1
  6YdWBDccdKls9M8RReNr7ULnV45dCkhCW9iF+aN+Oen+9znnPtVjU4/Et1O8WnXGm2NueFuJEeeX
  3IT5VB+pNAHnPxUa6ntLDwrDqd1qWsajKoWABY1VAc7nVQAeR34ABPavUNB0qPQvD9jpcRBW1gWL
  I7kDk/icms3w94L0/QbufUTLPfatcf6+/um3SN7DHCj2FbWoX9rpOnXF9eTLDbW8ZkkdjgACgDyf
  wdvHx+8VrH/qTE+/0z+6x/Nq9dhtobdCsMUcSk7iqKFGT1OB3rgPhhoV1H/avinUoXhvdbnMqROM
  NHDklQfc5/lXotAFObSrC4uluprG1kuV+7K8Ks4+hIzVoLg06igCtqElpFp9xJqDQpZrGxmafGwJ
  jndnjGOua4/+3Phj/wBBHwr/AN929dtNFHPC8UqK8bjaysAQR6EHrVH+wNI/6Bdj/wCAyf4UAcx/
  bnwx/wCgj4V/77t663TJbGbTYJNNe3eyZAYWt8eWV7bccY+lQ/2BpGf+QXY/+Ayf4Vdhhjt4VihR
  I40GFRFCgD0AFADycVxfxXs5L/4a6ukQLNGqTYHojqx/QV1GqwXlxpd5Dp1wtteyQssEzLuEbkcE
  j2NUPDum6nbeGYLDxDeR6je7GWeUDKuCTweBng4zgZoAz/hrMkvw40AqQdtmiHHqvB/lXGeJrdtU
  /aE8OxRcm0tFlkx2CmRufzH511uh+G9W8IRz2GivaXmkvI0tvBdytE9sWOSoZVbcuckZAIz1q54f
  8LHTtY1DXtRnS51i/wAK7xrtjhjXG2NAeccDJPJxQBxen3a/FjxNqtjeqB4c0hwqW68PcSksA7N1
  AG1uBjt1rD17Qtc+EFyut+Hr97nRJJQk1ncHIUnoD2OegYYOcZzXdW3gW98OeKLzWvDF1brDfHN1
  p12GEbHOco6gleSeMHqan8R+GNY8aWsOnatLaafpYlWWeO0kaaWbachd7KoUZ9iaAMPxP48ub+48
  N6FoTC3n8QQxzNcyoGMML5HAPBbhuvTHvVTX/hAljGmqeDL65sNTtkyqGU4mI/2uzH8j6V0/if4e
  WusNpV3pdz/ZmpaSFWzmVN6hVPCsvcD+p9a0X/4S+WyMAi0aG5K7ftSzyOqn+8Iyg/LfQBw2gfFu
  aT4d6nqupWyyanpjrAyj5Vmd+EJ9OQc49KfZfDy28eeF4dd1+7l/tvUY1uFuYAEWBTyihBwRgjOe
  ffpXR6Z8MNFsfBt34emaS4F63mXNycK7SdmXsMHoP8al8O6L4m8L6ZHpMcunarZQDbbyzyvbyqnZ
  WARw2OnagDjfB2v+IPCXjmPwL4in+2wyj/Q7nJLAYJXk8lSFIweQR3rtIvAVpp+tXOq6Ff3WkzXZ
  zcQwBHhkPqUYEA8npjrTdP8ABcsvjBvFeuXMU+orGIraC3UiK3QZ6E8s3J5OOp4q1fad4q1bfAdU
  s9ItWJBayRppyvs74CnHcKaAPOvFdg/jH4kaV4bgvbi/Wwbz9RmYgRwjjKhVGAcDHrk+xr2rCkbe
  2MY9qx/DvhXSvC1gbXTINhdt8szndJM395mPJNYms6N44Hik6noevWRsGVQdOvkYIoA5wVBPJ5zw
  eaAM/wCJPw/0XVfDmoalDaQ2uo2sDzrPEoXdtG4qwHByAffNTfBu9vbz4c2TXhdhHJJFAzdTGD8v
  5cj8K29T0bVfEenHT9SmgsbOUbbmOzkaSSVe6h2C7VPfC5x3FJr2lavB4YTSPB/2GwcIIkkmZlEE
  eP4AqnLe5+vNAEGtyz+KJ5/D+nTGKxH7vUr1D90d4Yz3cjhj/CD6mrOtTN4Y8Iyx6Bp/nT20IS0s
  oFyzAED5R3wDn+dec2vw9+KllbJb23jG1ihQYVFncAf+Qq63wN4S8Q6RqN5qXinV01O9eNYLd1kZ
  xHHnLD5lXBJx0HagDFsfHviDT7YQQ/DPXiMlmd3JZ2PVmPl8kmuG0rxTqtt8XdW1uPwrqE99PCyv
  paE+bECsQyfl/wBkdv4hX0eQNprz/SfA2pWPxa1TxXLPaGwu4mjjiVm80ErGORtxj5D37igDkdFt
  Vvfj/PPq1mEe4shdwQXC5MbmKPA57r84+oruviboVnrPgTU5JoUM9nA1zBKV+ZGQbuD7gEfjV/xJ
  4Rj1m+sNXs7j7FrOntut7kJuUr3R1/iU8+/NV9b0fxF4l0h9Hu5NP0+1uAEuZ7aV5ZHTuqKyqFz6
  ktigDz3Q/GHiK48HaCJfDmp6/bYlExgJO8xvtQSHacjHOO+Bngc3vEnxA8Q3/hfVrOf4fazaQz2c
  sclxIx2xKUILH5OgHP4V6rpOlWei6VbadYxCO2t0CRr7e/v3qv4k06XV/DOq6ZbtGs13aSwI0hwo
  ZlIBOMnHNAHjHw38Z63ofhCKysPBep6tCJ5XF1bthCS3I+6eR060V6b8PPC954S8Jx6VqEsEs6zy
  yFrdiVwzZHLAH9KKAOvpMAUUUALWVq+h2epvBcyeZDeWpLwXUDbZI/UA91PQqcg0UUAamKUDFFFA
  GZq+jWutrFb3zSvaq2+S3D4SbGMBx1ZQecZwe+a0FjWNAiAKqgAKBgAUUUAPooooAKydV0Kz1G4t
  71vMhvbQ5huYG2yKO657qe6nI/GiigDVAxS0UUAFFFFAHDadM+q/F3W47s749Gs7dLNOyGYFpH/3
  jtAz6cd67jGKKKAKmqWQ1Cxe0aeeFJSFZoH2PjPIDdRnpxzT7GwtNNsorOygjt7aFdscUYwFFFFA
  FmjFFFAGTDodpaa5LqdqZIJbkEXEcbYjmbs7L/eHqME981q4oooAMDOaXFFFAGVqmiWmsXFt9u8y
  a3gPmC2LfuncEYZ1/ix2B49q1AAowOlFFAC0UUUAGKym0OzTXU1eHzILtx5c/lNhbhQDjzF6MR2P
  UeuKKKANTAoxRRQAorK1LQ7PV722kvvMmitiJEtmb90Xzw7L/ER2zwPTNFFAGpiloooAKKKKACii
  igAooooAQjNGKKKAFxRiiigBMUdqKKAFpMUUUALRiiigBMUY5oooAWkxRRQAuKQiiigBcUmOaKKA
  FpMUUUALSYoooAWkIzRRQAAUUUUAf//Z
  ">.The probability that exactly 8 of n cows are <i>treifot</i>, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAAoAVcDASIA
  AhEBAxEB/8QAHAAAAgMBAQEBAAAAAAAAAAAAAAcFBggEAwEC/8QATBAAAQMEAAMFAwQLDQkBAAAA
  AQIDBAAFBhEHEiEIEzFBURQiYRUXMnEJFjdCV3aBgpWhtBglM0ZSVoOFkaKlxNIjJENjcnOTsdPR
  /8QAFQEBAQAAAAAAAAAAAAAAAAAAAAH/xAAYEQEBAQEBAAAAAAAAAAAAAAAAARFBMf/aAAwDAQAC
  EQMRAD8AZ/Frj3lHDG5xIWScNYihMQpcZ+LkBW24EkBQG44II2PEDxFMaVe+JcaL7QcAsko8uyxF
  yVRd+od5GQnf52vjSD+yG/xG/rD/AC1ayope8OeLeOZjfZWNLjT7FksPYftNybCHeniUEEhYHj06
  66611q9XGbDtsB+fcJTMWJHbLjzzqwlDaQNkknoBWVuJ6flftv4u1jPvy4QiG5LZ68vIpa3eYj/k
  lKTv4D4VauLN5XxF7QFi4PtOK+QLepM+/ISeklSEd6lpXqjXINernqkUMNvFc4h32yT8lVDftmOM
  I7yNcZxDQlNAEreCD1Q30GlK0VA70BrcJinEm5ZzHkXDBMVE60NOqabuN1nGC1JUnx7pKW3VkD1U
  lPXY8QakeN+Kzsu4RX7GLMUMy5MZIjJBCEqLa0rDfoArk5fTrSH7LXF+2YdbPmuz5C7BKt8hxMWR
  KSW0JK1lamnd/QIUpRCj0IPUjQ2DdRxgYs+dRMMz2wPYxcJ+vYZQkpkwpOzoAOhKSk76e8ka89bG
  7PmeaRsQucI32C+xY5XK0q8JUFMxnlK0lDw8UJPTTnVOzo8viUXxZstw7QHEbH4+Lxn28TsZc9pv
  7iC208VqQVhgnq5oNgAjpsk7A0TpK92yDerRLtNzjokwpjKmX2ljotChoig60KStIUlQUkjYIOwR
  X2kL2Xsqnwb5knB++y1yZmMPuJtz7h952IlfKAf+naCPgsDwTTO+c3ht+ELEv0zH/wBdEWyiqn85
  vDb8IWJfpmP/AK6kLBmGI5BMXCsGU2O7SkNl1bMK4NPrSgEAqKUKJA2oDfxHrQTlFFJbtaZ/ccUw
  6Fj2POqbv2SPGJHWhWltN9AtST5KJUlIPlzEjqKC+2jOIV8zCRYMfiPXOPAUpFzuSFBMaK6BsMpV
  /wARzetpT0SD1O+htlQHDzFrfheGWzGrahIZhMJQpYGi654rcPxUrZP11P0HLMuduhzIkOZcIkeT
  NWURWXXkoW+oAqKUJJ2ogAnQ30FeMG+WWfdJtqg3e3yrhA5fbIrMlC3o/N9HvEA7RvR1sDdJvK2k
  ZB2w8VtkVsEY3YpV2lu7J5VyOWOEj0PKhPTprmKh1r37PCot24i8Uc25mUG6X/5KhJ2AXGYDQb50
  DzBKuuvSgamZXmZYLKq6RbJLvCGVgyWIik98lnR5loSf4Qjp7gIJG9bPQ++MX20ZLY4t7sc5qdAl
  I5mnmz0PqCPEEHoQeoI0akqzyq7/ADT9p1FhbV3eLZwEyEseCI05SigqSPLmWBv/ALg/kig0NRRV
  V4lZzbcCshu10tt6nMJQtavk6Ct8NpTrZWoaQ2Oo6rUN9deB0FqopM4Hx1VnVhnTcVwO83OfHlKZ
  RCQ82hIbCEqS468vlbb5ipQCAVq9wnWvD5w+4+QL3nisFyzGJ2IX8ud2wxKeDqHF62Ec3KnSlDqn
  oQryPUbGJTtMTbjbsKtk5tMxdiZvMZWRJiEh1Vv2e8Hu9eUnl5tEbHTw3Ufg8G3SOK9uvfDJlqLh
  irU4m6rht91BlPEjug2jQSp1Pvcykjp9FR3sV3ZNOFw7QlrxbI1JTYfkJcy3xXjpibN74JVzA9HF
  IR1CDvW+bW9Go7g3ZWsb4yZvZsVd3h6WmH1R0K5mYk9e+dps+A90bUkeG0A60KKc1fFqShClrUEp
  SNqUToAetfaQPafyeddMnxng5ZJbkZ7I32zdH2jpaIil8vKD8QlZPwQB4KNENTDM1i5fcpnyDBfk
  WOLzNi8KISxJeCgChkeLiR1250TsaHN1I9s9zvE8Ftnt+UXqNAQQS20o8zr3wQge8r8g0PPVTNmt
  sGz2mLarbGbjQojSWWGkDSUISNACqpxst1vd4V5pNdgxVyvtdmI79TSS5ypZcUkc2t6CiSB5Ek0H
  54McSLfxPx64X212+TChxbkuE0JCgVuhLba+cgdE77zWtnw8euheKzz2BvuPXb8YHv2ePWhqFFFF
  JHK+0fjVhzGBjTuOZE29IkstvPT4a4iWmlrCS6ELBcWANnXIN60DQO6uDIJc6BZpMy22td0lNJCk
  REOpbU71GwlSum9bIB0CQBsb3Sn4g8acjxCKm8yeEt+OOlQCrg/LabWkE6ClMpC1IB6a5ynxAOj0
  pjcO8ysWeYrGyPHpCnYj+0qSscrjLg+k2seSh/Z1BGwQaD1wnK7NmFm+U7M+tSUOKZkMPIKH4zye
  imnUHqhYPiD9Y2NGpys9cZ7qeEfG3H89hktWTJf9xyBhP0FKRrlf1/LCVb36NkffGtCpIUApJBB6
  gjzoPGfLiwIT86bIajRY7anXnnVhKG0JG1KUT0AABO6q54lYR9qz2UN31t+zsrWhclhh10e5rnUA
  lJUpKdjagCketdnE2/jF+Ht/yDmCVwYDzrW/NwJPIPyq5R+WkNcrBKxXss2DFFvct7y6TDtxWrxa
  9pdLqkdfIJK+Yn75Sj4aADSlvmRrhAjz4TyX4slpLzLifBaFAFKh8CCDS97Scm9w+FsqXZkzFMsy
  mF3RMNRS+YIWO/CCOoPL4kfe81T6Gb5JtVv+0u72a3WpqOGmkT7O7JUtKDypUgpkNaQUga6HY0Qd
  Gqlnk993jFhmL5HIbTYJUB99QI7uPcJ6NBLSwSdpSNrDZJ2SN8xAoIfG4lnufEXFr7wfZZiWARJP
  y6/DZLUN8KbHcIWnoFvhZ2de8kfSPUCivfh/Yo+NdovIbfh6w3jUm0Il3WGydx4lwU7pCUgdEKUg
  KVy+h9OUAoFr9kN/iN/WH+WrQErCbrLaLMviTmTjShpSWzCYJ/PajIUPyEVnPtgrv/Ea448xiuDZ
  nLj2tEguyF4/KaStTpb0EhSAroG/MDxrTmC5U1lNu9oFnvtpkNoQX490tj0VSVKB6JLiQleiDvlJ
  103rYovHhgfD3EcIS+rHrShiTJO5Mt1anpD5J2edxZKjs9db1vyrPPAFa5PbN4hOytlxtNxS3v0T
  LaSn+7Wr6zVdrWrh32x7dkTqS3ZcxbXHD33qJCkgFBPqXEtq/pPgaEPvOclt2H4jc8mupX7Hb2C6
  sI+ks+CUj4qUQB8TWf8AFsGT2jLLKz/OD8lofLkWyMW5CEqYbQrRcccKeZ08wKdEgDSiAObo9OKe
  JMZ1w/vGKSJBjpuDISh4DfdrSoLQojzAUlJI8xSv4Dv5Vwuxk4Jm+LXdyNBfcVAulphuT2HW1qKy
  FJZCnEnmUojafA9QNdQVbd24l9mXJYlsu8g5BhUtwhkbPIUg+93e+rLgB3yb5T18fEbBtM+LdbVE
  ukFzvYsxhD7C9fSQtIUk/lBFIvjtY8h41NWbFMfsVxtlnjThMnXi6xVRQjSFICW2nNOrOlqP0QNh
  PXzDvsFsi2OwW+zQ+YRLfFbjM8x6hDaAlOz9QFCsv2pa4n2QaazGOm30KS9rzBtyVn++BWgvmy4b
  fg9xL9DR/wDRSW7PVrVmfH7OOK/KVWpuQ7Btrp8Hle6jnT8A0hP/AJB6VfP3OHBj+Zv+Jy//AK0F
  t+bLht+D3Ev0NH/0VIWDD8Rx+YubYMWsdplLbLS3oVvaYWpBIJSVISCRtIOvgPSqF+5w4MfzN/xO
  X/8AWrDgPCbh/gd4du+KWD5OmvR1RnHfbH3dtlSVFOnFqHihJ3rfSgvFZQ7T61u9qnhpEe37KFW9
  QB8NqnKCv1JTWr6zr2z8amojY3xKtTCnn8blpMpKR17orSpC/qStOvz9+VCNFUVyWa4xLvaId1gO
  h2JMYQ+wsffIWkKSf7DXXREPBxixQsquOUxrehF5uTLbEuVzqKnG2xpCdE6SB8AN+dQGDwuGOLXh
  7F8XlWKNd+d11yAiel2WkrVzue6pZcCSepHh8KmOJCLw5w9yJGPSm4l3VbJAhPrcCA293auRXMei
  dHR2eg8TWdeFsuy8SJvDDGoEUWC74Mlu53dEoBEl5SWwnTPm828s94tzw0UkklVBqmso9vda4l+w
  CfFOpaFy1II8dpVHKf1k1q6s1cZrYriV2osTxCKO+hY7HTOuqx1S0FLCyg/FSUtD88elFjStVLjR
  14PZpv8Am/O/Z11baqfGf7j2afi/P/Z10QpewMB8z92Ouv2wPfs8eqh264nsOb4LfbYjkuzodbSt
  H0iWXGlNflCnFaru7EuTOWPhrdmZtjvD9uXeHFtzYEJyX/te5ZCm1NtBTidAIIVy8p5iNjVXBzD7
  3xU40WzNMis0uzYpjiR8lxJyOSTNeCubvFNnq2jmCTpXUhCRrqdF6dF8slmvsQRL3aIF0jBXMGpk
  ZDyAfXlUCN17Wu3W+1Qm4NrgxYMRv6DEZpLbafqSkACumiiCsoXJa5H2QiI1J2W2UpSzv0FtUoa/
  OJrV9Zr7RNqXhvHfCOLQQRazJag3RwDoyfeTzq+ttah/R/GixpSqnxn+49mn4vz/ANnXVsBCgCCC
  D1BFLnj3f1RcByDHYVhyK73O6Wh+PHbt1okSGwXULbBU4hBQnR2Skq5ta6dRshf9gb7j12/GB79n
  j05c+yn7UbO1cvtcyG/95ISx7NZIXtL6NpUrnUnY0gcuifVSfWkd2N5Vxw/D5mLZNimWWybLvBfY
  cesMruVJcbabHMsNkI0UHZVoAHe/HWkqLfSxsvGD5TvEK2/NdxOhe1yG2PaZdg7thnnUE87iuc8q
  BvZPkAaTXaSAPbC4bAgaPyVv4/vg7Ws6yH2qXpcftW4BIt8T22Y0xblx4/OEd84JzxSjmPQbOhs9
  BuhGrsht0S8WGfap7aXIkyM4w8lQ2ChSSD+o1mb7Hu9LVZswjrKvZESIq2h5d4pLgXr46Sj9VNji
  Bm17uOKS7PhuIZM9kM9lUZsS7a5GYhlY0XHH3AGzygkjkUrZA8utSHAXhzH4ZYAxYg6iRPeWZNwk
  IHuuPKABCd9eVIASPXW9DdAs+322g8KbK6dd4m+NpT9RYe3/AOhTn4WPvyuGOKyZRJfessNx0nx5
  iygn9dJXtex5Oa5Tg3C60Ermz5apsnXXuGgOQOKHoEl4/m/GtD26IxAt8eDFRyMRmkstJ9EpAAH9
  goPxd7bbrvbnbddoEW4QntB2PKZS60vRBHMlQIOiAevmBXndbPaLtHZj3W1QZ7LDiXWm5MdDqW1p
  +ipIUCARs6I6iu6iiCuO82m1XqEYN4tkK5RVEEsS2Eutkjz5VAiuyig47ParXZoSYNntsO3REklL
  EVhLTYJ8+VIAorsooCiiigKr3ELD7NnOMP2G9tKLKyFsvNnldjuj6LravJQ//QdgkUUUHlw/Tl0a
  3u2vLxGlSIRS2xdI6wBcEaOlqb8W3Bocw6gk7SfIWaiigKpfEm05Tk6E4vaZCLPZpbX753VLgMju
  ySFMMI8lqHi4rokK6AnwKKCw4rYLTi+Pw7DY4aIdvht92y0ny8ySfEknZJPUkk1J0UUBRRRQFeM2
  LGnQ3ocxhuRGfbU2604kKStChopIPiCDqiigonDzF79gVzON28ouWFuqcdgqce1ItRO1Fk7/AIVo
  n6JB5k70djrTBoooFV2loeQS8ax02m1Trxa4+RQ5F/t8JvvHpMFBKlpDY6uJ5gglA8dDy3XnHsT+
  Zcc7DxDj2ybbLVYbVIjIenRFxZE517oEd04A4lttPMdrA2pXugjZoooGBmD+RM2coxaDEk3N5Yab
  XLd5GY4IO3V695QTr6KepJA6DZEVwzwSBhcKY57U7c71c3vabrdH0gOy3TvrodEoGzyoHQD47NFF
  BbqXvaFySw2HhRkkS73aHCk3KzzY8Fh10ByQ4pkp5UJ8VdVp3rw2N0UUWFN2DcjsLGCXDGX7vDav
  T95ekMQluhLrrfs7XvISeqtd2vet6CSTWm6KKFFFFFEFR2TWO1ZLYZlivcNuZb5jZbeaX4EeoPiC
  DogjqCARRRQVvhnacqxhCsVuzqLtZYTQFruxcAf7oEBLD6PNaR4OJ6KA6gHxutFFAUUUUBWLu0Bm
  eLXPtS4RfbbfYM21Ws24TJkd0ONN8kxbi/eTsHSVAnW6KKLGxbLdbZe7YzdLPcItwgv77qRGdS42
  vRKTpSTo6IIPoQRX4yCRcotnkv2e3IuM9KQGI63g0laiQPeWd6SN7PQnQOgToUUURVuHeBmxXa4Z
  XkE1F3y26gCZNSjlbZbGuWOwk9UtJ0PHqojZ9BeKKKAooooCiiigKKKKD//Z
  ">.Substituting p = .1, n = 100, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0a
  HBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIy
  MjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAAhAiYDASIA
  AhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQA
  AAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3
  ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWm
  p6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEA
  AwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSEx
  BhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElK
  U1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3
  uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwDrte8c
  eKtA8Y6V4ems9HlOpuiw3CmUAZfbyuex966fULzxdpVo92tjpmppECzw27yQysB127twJ9q4L4nT
  RW/xc8FTTyJHFG6M7uwVVAmGSSegr0LU/G+hWNm8kF9Ff3AH7u2sm8+SRuwATNAE3hTxZpvjDRxq
  Gns6hW2SwyDDxP3U/wCNI3iL7Z4hbR9KgF01sw+33BbEdsD/AAZ/ikI/hHTue1eaeH7XVfhx8M/E
  Ou38X2bUb6QPBbseYichc+hyxOPYV3Xwy0xdN8A6Yx+ae7T7XNIeTI8h3bj74x+VABF4xk1vxLfa
  H4et4pW08AXd5cMRFGxJAVVHLnIPcDg1V1rxpqHg6/sV8R2ls+nXknlLfWTMPKfGcOjZOMc5B7Gu
  L8QadrHwt8Xz+J9G8m70vUpSJ7N3w5LHcVA6nkkgjOM8jFW4NQ0/4zataQ3N3FZadYSeb/ZhbNzc
  NjBJPQJjI4yeucZ4Bnous+IG0G4hub2Af2NLtR7xGybdyeDIv9w5A3DoevHNbqsrqGUhlIyCDkGq
  97YW+oadPYXEYe3niMToehUjFef/AAd1q4udGv8AQr2QyXGi3BtldjyY8kL+RBFAj0qioriRoreS
  RI2lZFLBExliB0Ge56VyQ8Z6zj/kQ9f/AO+7f/45QB2VFcd/wmes/wDQh6//AN92/wD8crp9Oupb
  3T4bmazms5JF3NBMV3xn0O0kZ+hoAtVz7eIjd+IG0fSYFupLYr9vuC2IrYHnbn+KQjoo6dSR0Mfj
  vX38NeDNS1OIgTxx7Yc/32OF/LOfwql8MNMXTvAOmOSXuLxDdzyn70jyHduJ9cY/KgDr/wAK5Xx3
  BFF4b1HVLi9vY4rOzlcQQTmJHbacFtuGJzjjOK6yvP8A4ptqup+GrnQdJ0bUbuS6MYklhRfLEe4F
  hksOcDGPegDPlvL/AMH/AAY0W1sWK6veRwWsLOdxWec5Lc9SCzH64p3i7SLTwjB4XutHi8vUjrFt
  aNc5JlulfcHWVur7sZ5zg9MVq+JtOu/EWgaReWOnXKz6XqMF4LO5AjklWM/Mo5IzjkZOOKkn0698
  W+KNIv7qxnsdJ0h2uY4rkBZLi5IwvygnCoMnJ5JPpQBS0CKDxxr+v6hq0SXem2N2dOsbWZd0Q2Ae
  ZIVPBYscZPTbxVn4XXktxoGpwGR5Lay1a5tbQuxYiFSNoyeSBkgewqrolvrnhTT9X0e20ee7uZr2
  e4sblSogcSncpkYnKbSTkYPTjOa6bwh4eTwt4Zs9JWXzpIgzzTYx5krEs7fTJOPbFADte1mfQEjv
  pbQzaWn/AB9yxEmSAf39v8SjvjkdcGta2niureO4gkSSGRQ6OhyGB6EH0p8iLJG0bqGRgVZT0INe
  Y/C3UH0/XPEfg2Ry0Wl3LPZ5P3Yi33PoMrj60AeoVl65rtl4fsftd6zYZhHFFGu6SaQ8KiL3Y/5x
  WnmvLbC8bxV8crxZTutPD1uy28Z6CVsKz/XJYfhQB6Lpkt/cWSTahapazuSfJSTfsXsC3Qt644+t
  XT0NLQaAPP8Axz8TrXwjCUg066u7hnaNHZDHAHHUbyOSPRc/Wu5m842shtwnnbDs8wnaWxxnHOK8
  k/aBGPD2jf8AX63/AKLavYF+4PpQBjaF4hi1aS5spoWs9UsyFurN2yUz0ZT/ABIezfyNZvxEa7Tw
  q72jE7biFpoFk2NcRBwXiU+rDIwOvSub+KF0/hfxD4b8WWuVdJzZ3QB/1sLc7T9MMa6TX7W+PiXQ
  tbt7aS9sLSOZZIIcF0aQLtlVSQGwAR6jdx3oAoeD/sep67Nrnh5UtNDa2ED2ygJvuA2STGPuFRwc
  4Jz6YJ7w9K5Pwzo0sHiHXNbNq1jBqLRbLVsBiVGGlYDgM2enoOa6qR1jjZ2OFUEk+1AGLrniKHSJ
  bayhha81S7JFrZxthnx1ZifuoO7H8Mnite2877PH9o2CbYPMEedobHOM84rzL4WXL+J9e8S+LLr5
  pZLgWdtk/wCqiX5to9jlT+deoigDn/Fviy08KWcDyxPc3d1IIbW0iOHmc9s9gM8mq19qPirTNNfU
  ZdP0+7WJPMls7eR1lCjkhWb5WIHsM1xnxcgu9P8AE3hfxOYnl03T7hftG0ZEZ3hsn0yARmvTTq2m
  Ppgvje2xsnXd5xkXYVPvQBQtPEP9ueGYta8Oxx3okG5YZH8stg/Mmedr9ueM/nVvQ9cstf08Xlmz
  YDGOWKQbZIZBwyOvZga80+BE0r2HiCIE/ZFvg0Ppk7s/oFq7e3reFfjhaxxnbZeIrcCeMdPOXKh/
  rwo/GgDduwmt/EOXStRVZNPsdPW5S1k5SaR2ILsDwwUDGDkAtn0qr4QiXW9O1/T0uJl0mDV5YLfy
  ZCuYQFJjVhyE3E9D04GKv+K9Ju9ddLSw06GK5UFP7VuVH+jI33vLHVmPYcAHk1XvYLrwZ4e0/QvD
  WiaheRAbZJ7Xy98a9WbLsAXYnj069sUAReFtDg0Txzq0Ohq8WiLbRrPDvLRrdZJITJPOzG76iu8z
  WB4VubiayeJ9AutHghIWNLqRGkkJ5ZvlJ79yck5rckYpGzBSxAJCjqfYUASUVxo8Zazgf8UHr/8A
  33b/APxyl/4TLWf+hD1//vu3/wDjlAHY0VS0u8mv9PhuZ7KexlkBLW85UunJHO0kds8HvV2gBDXn
  /jf4n23hQLDb6ddXdxJIYkkdDHBuBww3n7xHov516Aa8b+Pw/wCJd4f/AOv1+/8AsCgD2QdK4Lx1
  8TbXwfav5On3V5cbmjVjGyQBwMkFyOT7Ln8K74V5D+0EP+KN07/r/P8A6KkoA9ZgcyQRyMMFlDH8
  RT3IC5NRWh/0KH/rmv8AIVlzE65cvaRk/wBmxMVuHH/Ldh1jB/uj+I9/u+tAHNQfERr34mweF7Sz
  U2bwNK105IZzsLAoOm046nrWnq/jJbfxVb+F9Lthd6vMhlfzJNkVumM7nPJJxzgeorhdoX9plAAF
  AssAAYH+oNW/iN4b1XRfES/EDQLiFbiBFF1BMwAYAbeMkBgVwCuQeBjmgDpvEXiPX/CFh/auoWVl
  f6bG6rcGzLxyxAnAYBiQwyR3HWtTUPGOk2HhSPxGZXlspkVoFjXLys33UUf3ify59K81l8aJ8U44
  vCsk0OgrKVN4J23STEHPlxDgdQD82D04r1G28MaZawaTBFE3k6SCLWNjkK23buPqwGcH/aNAHmmo
  fF/xPpUq3V/4KntNMdvlacurEf72Nufwr0XTfF+l6p4Wi1+3eQ20gwIwuZPMzjy9o6tnjHf6Uzx1
  PZ23gjWGvlVomtnjCEZLuwwgA7ncRjvWb8MfDNx4b8DWdpqCbbt3a6kRhnymboPqABn8aAF1LVfH
  aQPd6f4c00wgbvs016xuCPoo2Z9txqx4E8cW3jbTbiVLaS0u7RxHc27nOwnOCD3BwfxBqh408daj
  4dsLhrDw1qVy6ZH2l4sQJ/tHBJI/AD3FZHwwvfD8Pg7Vb3RZZpdRAa41A3AAkMm1mHA42ZzjHvQB
  0v8Awldzq/iS80Pw/BDIbAAXl7cE+VG56Rqq8s3XPIAwaksPFMsfij/hGtat47fUJIvOtZYWJiuk
  HXbnlWHdTn2Ncp8B1E3hHUb9+Z7nUXMjHqcKp/8AZjVb4zXB0rWvCGrQ/LcQ3rAMOpA2nH86AOq8
  X+Om0C7j0vStKudY1iRPM+zW6kiNOzOQO/YVymkfGS7h16LSvFegvpTSsFEgLDy88Asrc4z3HSvU
  LPSreyu727hUme8kDzSMck4G1R9ABwPrXnXxX0VfFWs+HNBtEDX7zNLNIo5htsYZmPYZxj1IoA9R
  LKqliQFHJJPAFcnB41Gt6jNY+GLI6kIG2z3ryeXbRH0DYJc+yj8a5X41+Ibuw0ew8P6ezrNqbFZC
  n3jGMDaPqSAa3dOuX8C+E7eCHwzqD2NpEGmkhaJpCcZZ2Tdn1J6mgDsbIXog/wBOeBpc/wDLBWCg
  f8CJrD1jxpZadq0ei2UEup6zIMrZWxHyD+9Ix4QfX8qztc+INgPhze+JdEuFnATy4SRgrKxAAYHo
  QTn8q5j4U2t7pvhmXXBot3qGo6lI0stw0kaM6A8BS7DIPJz3J9qAPS7D+2nJe/SxhBHEUJdyp92O
  AfyFFZvhzxvpXiS5urKIT2mo2hxcWV0mySPnBOOQRnuKKAOA+JRU/GHwODgjzY8g/wDXcV7CiIp+
  VVU+wxXIX3wv8Nanere3y6hcXS42zSahMWXHTB3cfhXU2Fimn2i28ctxKqkkNcTNI3P+03NAHF/G
  OxlvPhvqBiGTA8c7AdSqtz/OtzwNcJc+A9BkjYFRYwpkeqqFP6g1uXUEN1bS288YkhlUo6HoykYI
  /KuL8LQS+CJ/+EZvTI+lyTFtKuyCRhjnyHP8LA9OzZ9eKAOc8D3Mur/F3xPNrZJ1CyBisoZP+WUW
  4glAeny7Dkf3z61W+MXhbTdK06LxVpQ+warHcxrut/l80seuB/EMZyOvevTdS8LaRq93FfXVoBex
  cR3ULtHKg9A6kHHtTI/CWki8gu7mOa9uIDmF72d5/KPqoYkA+9AFzRJru40HTp79Nl5JbRvOuMbX
  KgsPzzXmXwchabxD4x1RQfs898UQ+p8yR/5OK7jxRqt4kD6PokZm1m6jIjPIS2Q8GWRuwHOB1J4A
  qz4T8N2nhPw9baTaEuIhmSUjmRz95j9TQBr3DvHbyPHEZZFUlYwQNxxwMngZ6VyA8SeMMD/i31yD
  3/4mtt/jXaEZpMUAcb/wkvi//on9x/4NLb/Gup024urrToJ72yayuXTMls0iyGM+m5eD+FWqKAOC
  +MNjLe/DfUfKBJgaOcgf3VcE1t+ArhLnwDoDpyBYRJn3VQp/UVu3FvFdW0tvOgkilQo6HoykYIri
  fC0Evge4Phq7LvpckxbSrsgkDccmBz/CwJ4PQ5PfigDu6QrmgHNLQAmOKNtLRQByd5r8994vuPDO
  nTxWk9tZrdTXEse8/McKEQkA46knpkCpPAmuajr+gyXOpJCZIruWCO4gUrHdRo2BKoOcA8/lXITW
  Wk+P7bXNa1qb7DJpk89lbvBJ5UtqicbpGHLFjztPy4wAMk57PwHNqk/gbSJNZQpftAN4Mew7cnZl
  R0O3aSO1AHRnpXkfgOFrr41eNNSj5gjzCWHQsWT/AONmvQfEOty6bbrb2Fs15q1yCLW2Xuem9z0V
  B1JP0HNV/BXhWPwpohtmmFxe3EhuLy4Ax5srdSPbsKAOjryLwLA2m/Gzxlaz/K9wDPHn+JTJv4/B
  xXruK43xT4euo9esfF2jQ+bqVipjntgcG7tz95Af7w6j1xigDrLu8t7C1kurueOC3iXc8srBVUep
  J6U6G4iuYUmhkSSJ1DI6MCrA9CD6Vkyx6P408OSW80ZuLC6G2SNt0bKQc4I4KsCB7gitHT9OtdK0
  +3sLKIQ21vGI4kB+6o6UAeT/ALQJB8P6N/1+t/6LavYEPyj6CuW1v4eaD4juPO1Zb66IYsiPfShE
  z1CqGwBWxZafa+H7CXF1cm3jXczXVy8vlqo7FiSBgUAed/HgG48NaRp8YJmudRUIo6n5GX+bivUL
  OEwWUELdUjVT9QAK4q30qbxf4wtPEd/BJDpOmqRpkEq7Xmc9Z2U8qOBtB5OAeO/eDpQAmKiu4jPZ
  zRL1eNlH4jFTUHpQB5P8BwbXw7rGnSZE9rqBEinqPkVf5oa9OvtSsdMjjkvryC2SRxGjTSBAzHoB
  nqa4670qXwj4uufE1hDJLpeoqF1SCJSzRMPuzqo+8Ou4Dnknntv6v4f0PxjY2n9o2yXlsjrcQMHI
  BOODkHkEHp3oA2JFSVCjoGVhgqRkEe4rzjxjf6VpAbRPC2j2M/ia8BSOO0tkBtweDI5A+XAz1rvd
  T0m21aJYrl7gRKSSkU7xhvY7SMimaXoWl6LE0em2EFqrHLeWgBY+pPU/jQBl+BPCqeD/AAtb6ZvE
  lwSZbmQfxyN1/AYA/CuH8dwtqHxp8F2sHLw/vnx2USb/AOSGvUdS1K20mxku7tmWNMABVLMxPAVV
  HJJPAArmPDHh66l8RX3i/WYfK1C8UQ21qSGNpbjopI43nqcdM4oA7LFG2looAQDFNlLLGzIm9gCQ
  ucZPpT6KAOLXxL4w2jPw+uQcdP7VtuP1pf8AhJfF/wD0T+5/8Gtt/jXZ0UAUtKuLu706Ce+sGsLp
  wS9s0qyGM5PG5eDxg8etWpporeF5ppEjijUs7ucBQOpJ7CnYqvf2Ftqmn3FjeRCW2uIzHLGejKRg
  igBbS9tr+0jurO4iuLeUbo5YnDKw9QR1ryL4/c6f4f8A+vx//QBXq2j6NYaDpMGmabB5NpACETcW
  6kkkk8kkkmsLWvh14f8AEVyLjVlvrplZmQPfS7Y8nJCqGwB9KAOq3DmvI/2giD4O07n/AJfz/wCi
  ZK9P0zSotKt2ghnu5UJBH2m4eYrxjALEkDisTXPh7oPiSfzdXF9dYYsqNeyhEz12qGwKANieCa70
  CW1tpzb3EtqY45gM+WxTAb8DzXBweCfiFa28cEPxBjjijUKqrpqYAFd7pOjQaNbmC3nvJI+Aoubl
  5toAwACxJArQxQB85nRvE/8AwuldNPiUf239n3f2n9lH3fKzjZ06cZrppWv5/jLoWh+J7v7Vb2Vm
  rwM67I7m4EeTJt6ZLbvptFelf8Ifo3/CWDxP9nf+1Qnl+b5zbcbdv3c46e1WtY8PaTr8Mcep2UVw
  IzujY8PGfVWHIP0NAHG/FTwdo2q+GNR1mWNLfUbKBpkuk+UttGdreueg7g4q38PdcuU+GNnqviK4
  8sRRuWuJzgmJSQrH1JH58VtSeDdKuVRL77XfQowZYby7kmjyOhKscH8al8Q+E9J8T2MNjqcUz2sT
  bliineJc4wMhSM47elAHKWvijwzrOpRavrHiLSY4oDusLB7tP3J/56yDPMhHQfwg+uTXRWnjfw9e
  zXQg1O3e2tfLWS8MqiHe+dqBs8tx2rE/4Ux4I/6Btx/4Gy//ABVa1r8O/DFp4en0JNND6fPJ5rxy
  yM5L9juJyCO3NAHSEpIhHDIwz6gj/CvI/hr4cV/Fvi/UrRNmhXDyWdqV4WQFssV9QMEZ9+K9A0/w
  XpGm2P2GD7YbLGPs0t5K8ePTBbp7VuwwRW8KQwxpFEg2qiKFVR6ADpQB5l8HY5NEg17wxeL5d5YX
  xkKHjcjKAGHqvyjn3pvxHsT4m8eeEdAhUuYpHvLnH8EIKgk+mcYHua73UPDmm6leR3ssTx3salEu
  oJGilCn+HcpBI9jUmm6Dp+lSTTW0TfaJ8edcSuZJZMdNzsSTj0oAoeKPFdj4atoknubWO8umKWyX
  EwjQnuzMeijufwHJrF0bxF4N0eK4urjxXpd1qNz+8u7o3KFpCOgUA8KOiqOnuaua38NPDPiLVJNR
  1S2uZ7lwAWN5KoUDoAA2APYVnf8ACl/BH/QOuP8AwNm/+KoA534jk36eEvHdtbzNYWdwksoZfmWE
  urK5Hodo/MZr1kTW81oJg6PbOm/fn5ShGc/TFPFpbi1Fr5SGAJ5flkZXbjGMemKxP+EK0YW5tVS5
  Sxbg2SXUggI9NmcY9ulAHjfhTwpf638NPF62cbNBc3ay6egHEpiZj8vrkbR+GK9X+G2pQaj8PtHM
  Rw9vAtrKh4KPGNpBHY8Z/Guot7WC0t47e3iSKGNQqRooCqB0AHasubwtpkl7NeQrPaXE5zNJZzvC
  ZT6ttIBPv1oA8c8XaBqfiH4u6tH4cYpLDaRG5kQ4AY4yCfU/Lx7UV7dpejafotu8On2yQq7F5COW
  kY9WZjyx9zRQBfooooAbJ92my/d/4Ev8xRRQA9e/1paKKAIh/rX+i/1p696KKAHUUUUAFFFFAB2q
  Gb7h/wB5f/QqKKAJE6/gKdRRQAUdqKKAPBNS/wCS/wAn++n8hXvC/eP+e5oooAZ/y3P+7/WpR0oo
  oAWmnrRRQBEnWX/eNT0UUAFRz/6l/wDdNFFACt1NPoooAKQ9KKKAEX7opkP+rT/dFFFAEtFFFAEU
  n3k/3hTxRRQA6iiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKA
  CiiigAooooAKKKKACiiigD//2Q==
  ">.The probability that exactly 9 of n cows are <i>treifot</i>, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAAoAWEDASIA
  AhEBAxEB/8QAHAAAAgMBAQEBAAAAAAAAAAAAAAcFBggDBAEC/8QAVRAAAQMEAAQDBQIHCAoTAAAA
  AQIDBAAFBhEHEiExCBNBFCJRYYEVMgkWF0JxgqEzUleFlaKl0hgjOEZWcoPE0dMZJCVTVWJjZ4SR
  kpOUlrGytLXj/8QAFgEBAQEAAAAAAAAAAAAAAAAAAAEC/8QAGBEBAQEBAQAAAAAAAAAAAAAAAAFB
  ESH/2gAMAwEAAhEDEQA/AGfxa495RwxucSFknDWIoTEKXGfi5AVtuBJAUBuOCCNjuB3FMaVe+Jca
  L7QcAsko8uyxFyVRd/QPMjITv9bXzpB/hDf7xv4w/wA2rWVFL3hzxbxzMb7KxpcafYslh7D9puTY
  Q707lBBIWB36dddda61erjNh22A/PuEpmLEjtlx551YShtIGyST0ArK3E9P2v438Xaxn35cIRDcl
  s9eXkUtbvMR/yJSk7+Q+VWrizeV8RfEBYuD7TivsC3qTPvyEnpJUhHmpaV8Ua5Br4ufFIocOHCMu
  byuBJvMa2yoVkGjBmy9NmYjrzOpbPVDfQcqlaKgSdAaJr2PcTJWZzJ44fY6i822C8WHbrOnexxXX
  QNlLRDbi16BHXlA6jr1G/wAeJt2bB4A5UbSlTbiYaGtNjXKypxCXNa7Dyyr6VC+DJ2I5wAs6IxT5
  rUiUiTrv5nnLUN/PkKP2UFlsvEyP+O7eEZZaHsbv0hHmQkuPpejTU9f3F4AbV0PuqSk1K5nmkbEL
  nCN9gvsWOVytKvCVBTMZ5StJQ8O6EnppzqnZ0eXuU548WfZcHxnIoqyzcoF7SiM+g6W3zNLWSD/j
  NIP0p02JcTOOGduevERt+LfbQy5JYUPdUl5oKUPl96gsaFJWkKSoKSRsEHYIr7SF8L2VT4N8yTg/
  fZa5MzGH3E259w+87ESvlAP+LtBHyWB2TT6ogoqDv+YYjj8xEK/5TY7TKW2HUMzbg0wtSCSAoJWo
  EjaSN/I/Co/8pvDb+ELEv5Zj/wBegtlFVP8AKbw2/hCxL+WY/wDXq2UBVZznNLViiYkd9t+fdrgs
  tW61xEhciWv15QSAlI7qWohKR3NS+RXaHYbBcL3cF8kSBGckvKHcIQkqOvnoUlPCv7bm03IeL2RJ
  8y5XOUqDb0E7TEiN6PI38AVHR+JRv1OwekFyQ7CYdlx0xpC20qdZS5zhtZHVIVoc2jsb0N0TpcWD
  DdmTpLMWMykrdeecCEISO5Uo9APma7UmfGVL8rglLtkdrzble5sa0wE82tuvODY/SW0uD60DSk5H
  j0ZdtRIv1rZVdSBbg5LbSZhIBHlbP9s2CNcu+4qUpCZ3BhO8c+FeHKWw3Exm3yb9PWohKW22kBlg
  7P3UpcB7+lPugqmI5tEvN7mY3cYT1lyKEnzHrdIWFFxonSXmVjo62fiOoPRQBq10nfFXapbGEx+I
  FhcMXIcUkIlxpCB7ymVKCHW1fFBBBIPcJI9TV/4ZZZEzjA7RlMNIQiewFrb3vy3ASlxH0UFD6UFj
  orjOkIhwn5biHnEMNqcUllpTjigkbIShIKlK6dAAST0FJZPiQxxziLHw5GM5G08pa23lyYhbeSsN
  qUhCGACtSlkJSAQn72+1A76pfHLKpeE8KL/k0BCVTIkdKY/MNhLji0tpUR6gFYOvlVDznjve8InR
  nsq4U3y22OS5yInqnMuL+OihHMlKtbPKXAehpkzHMS4hcN3XpEhibjV2hlS3VL5E+X3JJOuQpI9d
  FJT8qBV8Xset3DfhExlduBXmNvkw1i7qUVSpz6nkB1Ljn3loWkue4fd12HSn3WcrHi9w4lZvZW27
  xfLlw4xZ5LrEy5rRu7SUfdDfIhAcaRrXmKBJ94cx5umjaKKqczOISs1bxCyxHrxc2yhdyLCgGbcy
  T955Z6BZG+VsbUfkOtQfiO4gL4c8MJl3hqSLpKWIdv5hsB1YJ59f8VIUr4bAHrXfw+4qnFeF9rQ/
  zO3W5Ni43OQ4eZx6Q8AtRUruSNhP6vzoi6Xu7Wux2x253m4RbfCZG3H5DobQn6n1+VLzh1xsxfiB
  xFmYnjDEqSxEgOTF3FweW2spcbRyoQRzEHzN8x5e3Y73TJmQoczyfbIjEjyHA615rYX5a9Ecyd9j
  pRGx10T8ayj4Z222fF7xHZZQlttCbolCEjQSBcGgAB8KK1rRRRRBRVD4ucUbPw2ge13az36agoSr
  zIUFSmUlRKUhbytNpJI7c2+3TqN1vGeNFzy3CWL/AIhw5vF6kqLnnx0SW2WY/K4pISXnNc6ykJVy
  oSrXMAT8QcFVSFm0QZs7iF6hPWi5uc7ltLywpm4sp7qaWOnOB95s6UO/UdaqHBzjjZs/vsrGJ1ol
  45kkbn5rfLXzc/J98JVpJ5k+qSkHXUb0dS/iGxf8ZuF9zVFUtm72lBudrktnlcZkMgrHKR2KgCn6
  79BQMOil54ec/PEbhlBvkgoFyZUYlwSkaHnoA2oD0CklKtenNr0ph0Fej5viUm/z7FGv0N6425tT
  sxltRV5ACgkhZA0FcygOUnfy6GvRiWU2LK4kqVYZplNw5a4ckKZcaWy+jXMhSHEpUCNjuKUXDV1u
  fn/FDipdHea0wJioUVHoUwmiFuD5dSR6bJPcDUn4dGLkrhGHLdNiRsiu7671NkSYa5DCVyXCsApS
  43tRaCDrnBHMkkaI2UwOJ+ROYnw8v2SMNJeet8Fx5pCvulYHu7+W9b+VKHOcegY34enc7feLmZtQ
  49xN+WrcoylrQeVK+4b2rkDY93l6apxi0zJ+M3G05rLtV1jy23GnvZYS4jXkKTpSVBTzh3948wUP
  ToNbKHRiMjiLf7Rh1kvd8uXDPHXkqlzpzqFNTVtkckVhSEJ81tOtFaubXXR6J5gYH5Sch/4AR/2V
  f6aKanlN/wC9o/6qKIyf+EN/vG/jD/Nq0BKwm6y2izL4k5k40oaUlswmCf12oyFD6EVnPxgrv/Ea
  448xiuDZnLj2tEguyF4/KaStTpb0EhSAroG/UDvWnMFyprKbd7QLPfbTIbQgvx7pbHoqkqUD0SXE
  hK9EHfKTrpvWxRccMD4e4jhCX1Y9aUMSZJ3JlurU9IfJOzzuLJUdnrret+lZ54ArXJ8ZvEJ2VsuN
  puKW9/BMtpKf5tavrNV2tauHfjHt2ROpLdlzFtccPfmokKSAUE/EuJbV/lPkaEaPnRY06E/CmMNy
  I0htTTzTieZK0KGikg9wQSKVeG8H7jw/us5zh9mSrbaZrnmu2m5W/wBtZSvttCg42tPTp3JIA2To
  U1Zb6I0V6S4l1aGkKWpLTanFkAb0lKQSo/AAEn0pdX3Mc9vLK4OBYJNjvODlF0yDliR2N/nBnZdX
  +jlH17UQkvFc1esnzPFuGjGQOXq9ypIedisR0sxYgUOVKigcy+blK1EqWeVI2AObZ1XY7cxZ7JBt
  MXYjwozcZrf71CQkfsFL3g7wjh4TPmZLebk7kOXXIlUy6Pp1y83UobH5o7de50Ow6UziQASToDua
  KyjalrifhBprMY6bfQpL2vUG3JWf54FaurNXh6tasz4/ZxxX5Sq1NyHYNtdPZ5Xuo50/INIT/wB4
  PhWlaFQd/wAPxHIJiJt/xax3aUhsNIem29p9aUAkhIUtJIG1E6+Z+NR/5MuG38HuJfyNH/qV48+4
  TcP88vDV3yuwfaM1mOmM277Y+1psKUoJ02tI7rUd631qvf2OHBj/AAN/pOX/AK2gtv5MuG38HuJf
  yNH/AKlWylN/Y4cGP8Df6Tl/62mzRCt8WDzzHh8ypcckLLTCDr96qQ0lX80mvJ4O0IT4ecdKNbWu
  WV/p9qdH/oBV/wCIeOM5dg15xp5QQm4w3GErI2ELI9xX0Vo/SlN4L5z8bArthN0aVGu+OXR1mRHX
  95CVnmB/RzBwfT50XD3qHyXGLFkj1qdvdvRNVaZyLhB51qAZkIBCHNAgEjZ1vYqYqFz2dcbXg1/u
  dnZL9yiWyS/DaCebneQ0pSE69dqAGqIgL/F4XR+I7F3v02wMZZIiIgsomXBKXXWQ55iEJZUvSvf9
  4Hl3sDr0GrzWa338Ol+B+XcJz8S5SLlj7sqS+4Q4+/dS0VKUo/eLiXvXulKPQCnXwfdnv8JsPfui
  3HJ7ljhLkrcO1KcLCCoq+e97+dB5+OLaHeDOZpc1oWOYob+IZUR+0Cln4EX33uC0pt0koYvT7bW/
  RPltK/8AcpVWrxV39qw8D76gq/2zc0Jt0Zsd3FunSgPj7nOfpUj4dsNfwXhJZrJNb8u4LQqVNT6p
  dcPMUn5pHKn9Wi4YVZM0P9kS1ofH+ia1nWPMjnTbb4+3ZsC2PXR5pSCYrKkhxaDa0hZTzEAqCCpQ
  GxsjXrQjRHiDt0S6cE8vYmNpWhq0vyUcw7ONILiCPnzJFLbwTxGbxwNXFvUGPOixr28YjclpLiUg
  IbVtIUCAQpTnX5mrBxju+RZ5ij2D4Pjd7TIu+mJtxudvehRobGwV7U6lJWoga0gK6E/KmBwzxC34
  Jg9sxa2qLjUJrS3SNF5wkqWs/pUSdeg0PSgsaQEpCUgAAaAHpX2iiiMqfhCnnk2/DI6SfIW7MWse
  nMkMhP7FKrU8ZCG4zTbWuRKAE6+AHSkx4x8Jk5dwnXNtzKnp9je9tShI2pbXKQ6kfTSv1KYPCLIm
  Mr4Z49fmHAv2mC35uj911I5XE/RaVD6UXFqrJvht/uwuJX8a/wD2DVafya+RrBbhOlRLpKSVhtLd
  vt70t0kgn7jSVEDofeOhvQ31FZT4Gu5Bj3iGyrNL7geaw7TexN8lf2BKcU150lDyeZKUE9kkdN9a
  Ea0vU77Ms825exy5vskdx/2aI35j73Ikq5G09OZZ1oD1JFLH8tv/ADRcWf8Ay5/+lNSFIbmQ2JbS
  XktvtpcSl5lTSwFDYCkLAUk9eqVAEdiAa60Qm/FLN+0/C9e7l7HLhe1R4D/s0try32eeSwrkcT15
  VjeiPQgivX4QAB4dsX0O/te//FvUeL/+52yj/on/AMtmqt4V8vVZuBNkiXPHb+plsyTDkQLc7NRK
  SZDpP7ilRbUF8ydLCRoAg9ehcUzjRE+yfGxhcu0o8uRcFQHpJR0KuZ5bKyf8kjr8q1hLbQ9Eead1
  5a21JVv4EaNJXh9gt9yfjPN4vZna3LShpv2ew2t8gvNNhJSHXQCQlWio8vUgrPbQ2x+LGRMYpw2y
  C/vuBv2WC4WtnXM6ocraf0lZSPrQZ9/B7PvqtmZRlE+Q29DcQPTmUl4K/YlNaqpJeDTC5OKcJk3C
  4Mqam31720oUNKSzygNA/pG1/r07aFeCDZbNBtjtshWmBFgPc5djMx0IaXz/AH9oA0ebZ3sdd9a7
  Wu3W+1QGoFrgxYMRoabYjNJbbQO/RKQAK9NFEea4wINyimLcYUaZHJCi0+0lxBI7HRBFd20IabS2
  2hKEJACUpGgAPQCv1RQFFFFAUUUUBVe4hYfZs5xh+w3tpRZWQtl5s8rsd0fddbV6KH+kHYJFFFBy
  4fpy6Nb3bXl4jSpEIpbYukdYAuCNHS1N923Bocw6gk7SfQWaiigKpfEm05Tk6E4vaZCLPZpbX+6d
  1S4DI8skhTDCPRah3cV0SFdAT2KKCw4rYLTi+Pw7DY4aIdvht+Wy0n09SSe5JOySepJJqToooCii
  igKKKKApfZphM9rL2uIGEqjx8jba8idFfUUR7tH6f2twgHkcGhyuaOtAHY7FFBe4Ljz0Jh6RGVFe
  cbSpxlSwotKI2UkjodHpsdOlePKLo9ZbFJujFrmXRUflUqLERzPLRzAL5E/nKCSVBPrrXrRRQK2/
  Yfj+fsqteO4YmwWu5vtuZDdH7IbbJlMIWHDGSlaEPKU4pKQpZHKE82iSQKcTaENtpbbQlCEgBKUj
  QAHYAUUUC6hYXc8pzOJmGetsoRa1lVksbbgcaiK3+7vK7OPHQ0B7qNdCT1pj0UUHGbKjQYT82bIa
  jRY7anXnnVhCG0JG1KUo9AAASSe1YzTmmKnxzfjb9uwhYfM8v7QLg8nf2d5O+btrzOm+3r2ooosb
  MhSo06ExNhSGpMWQ2l1l5pYWhxChtKkqHQggggjvXaiiiCiiigCARo9RS4x3DrtgWWPKxBlmTid3
  lebMta3A2q2vK6KfYJ6Fs/nNdNa2n97RRQMeiiigKKKKBIeMvJLDD4M3rG5N2ht3memKuNBLo85x
  AkoUV8nfl02v3u20kd6++DXI7DM4MWTHI13huXmCmUuTBDo89tBlOKCijvy6cR17e8BRRRrnh3Uu
  ckwy6Z/k7X44IajYnapXmRLS25zquLqT7r0gjoGx3S0N73tR/NoooyYyQEgAAADoAPSiiigKKKKA
  ooooCiiig//Z
  ">.Substituting p = .1, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0a
  HBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIy
  MjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAAhAiYDASIA
  AhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQA
  AAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3
  ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWm
  p6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEA
  AwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSEx
  BhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElK
  U1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3
  uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwDrvEHj
  nxV4f8YaV4fms9HlOpuiQ3CtKANzhOVz2JHeupv7rxdpts90llpeprGNzQQPJDIwHXbu3An2Nef/
  ABRmjt/iz4JmmkSOKORHd3YKqgTqSST0Fei6j420GxtHli1CG9mA+S3sm8+SRuwCpk80AO8J+LtO
  8Y6Sb6wLoY28ueCUYeJ/Q/40s3iL7Rr50XSYBdXEBVr2YtiK1U9iR1cjoo+pIrzfwxaar8P/AAB4
  m8S6jB9mvb5/MgtXPMZJbbu9Duc8egFdl8KtPFl4BsLhyXudQ3XlxK3LSM5zk++MCgC1feLXk8U/
  8Ixolul1qMcfm3UsrlYbVeMbscsxyPlHr1qHVPFd94VubQ+Ire2Om3UohF/aFgsLnoJEbJAPqD26
  Vxvh2Q+FfjZr8OsN5CawGktJ5TtST594AY8ZwSPqv0rofjJd2K/DnULaa4hFxKYvJjLDcxEikkD6
  Z5oA6XXddk0ERX89t5uj4xc3ERJe3yeHK/xJ6kcjrgitqGWOeFJonWSN1DI6nIYHoQa5rwP5t/8A
  DvRl1Bd7S2KJIr91xjn8MVzHwm1OW1vNe8H3Dlxo9ywtSTkiIsRt+gOMfX2oA9QopD0rjv8AhMtZ
  BI/4QTXz/wACt/8A45QB2VFcd/wmes/9CHr/AP33b/8Axyui0e/n1LT1ubnTbnT5GYg29yVLjB6n
  aSOfrQBfrn9U8RGDVo9E0yBb3VpE8x4922O3jz/rJW7D0A5b9a0Nb1NNG0S+1KQZW2geUg98Dgfn
  XE/B2GS68L3PiC8bzdQ1e7kmmlbqQp2qv0GDigD0QZ4z1rP1bTl1C3US3l5bQxne4tpfLMgA6Mw5
  x9CK0q5nxtf6hb+Hb210rTL29vrm2kjgNuilUcjALEkY65/CgDkfBGor4e+E2reJ2WWTzpbq/QSy
  FiyhisYyxz0VepqPWdFSx+Et14gvwX8SCzF62ok/vo5zhgEb+FRkKFHGB0NX7Xw/d6z8G5PCosbq
  wvIrBLcC8QKrSAZ4IJyMjGferOqWWp+MtIsNAl0m606y3xNqUlyVGUTB8qPaTv3EYzwAKAIIrubx
  f43h0a+G7TtM02G7vbfkJNcygFVcd1Vedp4JPI4qz4UKWHxH8U6LYARaZBFbTJbpxHDK4O4IOigg
  A4HFSRWV94b8d67qiaZc31pq8VuyNaBWaJ4k8sowJGAQAQ3TrWl4Q0G6006nquqbBqmr3P2ieNG3
  LCoG2OMHvtUDJ7mgDY1ae/tLE3Gn2a3kqNuaDfsZ05yEPTd6A8HpkUmi6zZa9pseoWEu+F8qQw2s
  jDqrA8hgeCDWieleWQ3p8K/HGbTIzt0/X4FnMfZZwCNw9yVIP1HpQB6nUF5dQWNrLdXUqRW8Sl5J
  HOAqjqTU2a8t+Jd82reL/DPgwMRa3lws94o/jRTkKfY7W/IUAdxoOsXOuxPfJZNb6bIAbR5iRLMP
  75X+FT2B5PXitsdKRVCqFUAAcADsKWgDnPFPi6z8K2EtzcW13cvHH5hjtoS21c4yzdFGe5P507wX
  4ifxX4WtdZe2Fubh5QIg27aFkZRz64FVfiSMfDjxD/14yfyrP+D3/JMNK/35/wD0c9AG4viE2fiF
  dI1WBbZrlmNhcK2Y7jH8Gf4ZAP4e/UHtVnxNqE2k+FtV1G2QPPa2ks0YIyNyqSM+3FY3xM0sal4C
  1JlytxaJ9rgkU4MbxncCD64Bp3g3xM3iLwHZatPA887xmOeOJQxZ1O1uOnPX8aAOa1SG10Xw14a1
  G0cSa7d3lrtu92ZblpMGQM3VlILcdBx6V6ivSuE0PwOp8RJruoWVtZpa7l0/TbflIMnmRyOC59uB
  711mtaimj6Je6jIAUtYHlIPfAzigChqviLyNVj0TTIBe6tInmNFu2pbx9PMlbsPQDk9vWt4Zx715
  z8HoZbvw3d+IrxjJf6vdySyyHrtU7VX6DBr0egDmPE3i6LQr2w0u3tze6vqL7La1V9ox3d2/hUf0
  qDWNa8QeHNOfVb+ysr2xh+a5SyZ1liTuw3ZD47jiuL8a+d4d+MmgeJ75WGkMgt2nIOyFiHU7j2++
  D+Br0jXtQ06Pw1eT3F1bi2ktnAdpBtfKnAHrn2oAZNrclz4ft9Y0O3XUrdwJfLV9rvHjnZnguPQ4
  zgjir2j6vZa5pcOoafMJbeUZB6EEdVI7EHgg9K89+BElxJ8PXSYnZHeOsWf7u1Sf/Hi1N0u9Phj4
  2ajoUZ26frMQvEjzwk23kj03FW/T0oA24Ut/EHjjxBDrEcc1ppIhit7afBjAdN7SlTwSSdoJ6beM
  c0ngW8Go+BLmS6kd7BZ7qOF2kYE2yu2z5s5+7xnPal8VeHp/Fd2bW2sI7JGUw3WqyqPOMWeY4gDk
  5/vNwATS+LLSfS/Ak3h3w5pF3O0tqbWEW6rtjQ/KxLEjnBJ9zQA/4X2nk+C4LvaynUJZLzDEkqrs
  di5PPChRzXaVQ0WNIdGsoY4ZYY44UjWOZdrqAMYI7HipdSupbLT57mG0mu5I13LBDjfIfQZIGfrQ
  BaorjR4y1kf8yHr/AP31b/8Axyg+M9Z/6EPX/wDvu3/+OUAdlRUNpK09pDM8LwvIgZonxuQkZKnH
  GR04qagArz/xz8T7bwlEY4NOuru5ZmjSRkMcAcdRvI+Yj0XP1r0CvHP2gf8AkAaL/wBfr/8Aos0A
  etXN3DZWUl3cyLHDGm52PYf1+lcxe6p40uITcaP4fsEhxlI9Qu2SZx2+VRtX6FvriusKK8Sh1DDg
  4IzyOQa5bxV4uvPD9rO1n4b1TUXjH34owIhx1JyWx9BQBV8EePx4pvL7S73T5NO1ix/19szbgQDg
  kH2JAI9x1q/4g8WDTNZsNB0+2F7rN9lo4S+1Iox1kkPOBweAMnFcb8I9S0LVr7VtTWWVvEt4xlvF
  lUKAu7pEBn5Ace/TNQ+EZjqnx68U3E/LWsLQxf7IVlTj8KBnX6n4o1DwrcWb+IobVtNupRCb20LA
  W7noHVs/Kf7wPHpXYZBGQa4f4uQJP8MtYLjJjRJF9iHUf1NSeDdfkl+Hegzupnvp7cRQxZ+aVlyu
  SewAXJPb8qBF3xt4tj8JaDcXqwG5uVUGOEdBlgoZj2XJAz+Aq94U1iXX/CumatPEkUt3brKyISVU
  nsM81yHxO082fwu1iWaQTXc8lu002Mbj5yYAHZR0A/qTW98Nufhv4eH/AE4x/wAqAMrx18TbXwjb
  sIdOury4LNGjlDHAHA5Bc9SPRc/hXeRNviRjgEqDXkX7QYI8LaV/1+t/6KevW4f+PeL/AHB/KgDM
  8Q+JtK8L2P2zVLnylY7Y41G6SVv7qqOSaradqHiDVVS5/syDTLZjlUvHLzsvuq4CHHbJrynwre3P
  jn4oah4huLGe/ttNytlCpURx8kJkscA4Bb6n2r0lvH9lY67Bo+uWN3pFxc/8e8lxsaGX6OpIBzxz
  igCTxf4zi8Li3tbewuNT1W6ybexthlio6s3XC/zrhIvjPq2k6rHbeKvDEunQSfxLvDqv97DD5gPa
  vV4dLtYdVudSVS13cRpEzsc7UXOFX0GSSfUmuD+MVkNY0PTdEtYVm1W8vU+yoOqgZ3sT2UDqaANH
  xt43vdDj0yDw/pw1fUL8GVIkDN+5AHz/AC885Fct/wALE+Jf/QgSf9+pq9P0fRbXSbK0ijjRp4LS
  K0M+35nRBgAn06n8ai8Uawnh7w1f6oyhmgiPlJ/fkPCL+LECgDidG+I+sy+FNe1vWtIgtDp8gt4I
  EL7pZ+mwg+7KOPespfiL8SWUMvgFyCMgiKbmu/8AC/hmHS/DGmWd5Ck11AftMjyKCftDZZn+uWPN
  dHwBzQB5d4Y8e+MdW8XWmi6r4Zh05JI2mkaTzFYRgYyAf9rArtvEnie08NWEtzcQXdwUiaQxW0Jc
  7R1JPRR7k1j+DB/bOsa34qYZS6m+x2R/6d4SRkezPuP4Vs+MQf8AhCNfP/UNuP8A0W1AFPwJ4qbx
  j4eOrNai2VriSJIw+4hVOASfWiuc+Bv/ACTmP/r7n/8AQhRQBifEwqfi94GzjHnR5z6eetexIiKf
  lVVPsMVyF/8AC/w1ql6t5fpqFzdLjbNJqExZcdMHdx+FdTp9gmnWq28c1xKqkkNcTNK3P+0xzQBy
  PxasZb34baqsKktEI5yB1IRwx/QVf+HU6T/DnQGQ5C2SRnHqo2n9RXSzxRzwPFKgeN1KspGQwIwR
  XD+G7WTwHdvoFzvbRLicvpl3gkRFzzBJ/dOeVPQ5PfigDsL+zsL22KahbW1xAvzFbiNXUe/zDFeT
  65a2/wAQNXi8NeF7OC38P20ofU9Rt4QkchHSNCBhu/TjP0r0q/8AC2laxctPqUMl2Dj9zNM7RDH/
  AEzzt/StS3tYLSBILaGOGFBhY41CqPoBQAQW8VraxW8KBIokEaKP4VAwBXk3w4ga6+LXjbVI8m3W
  Vod3YlpM/wDshr0HxHrU9hB9i0yBrrWLlSLaAcKvbzJG6Ki9z36DJqLwX4Wh8JaAlisnn3Mjma6u
  MYMsrfeP07AelAHRN0rjf+Ek8XhiB8P7kjPX+1Lb/GuzIzSYoA43/hJfF/8A0T+5/wDBrbf410ej
  Xd9e6cs2o6Y+m3JZgbd5klIAPB3JxyOa0KKAOd8dWMupeBtbs4RmSWzkCj1IGf6Vh/Bq4Sf4Zaci
  9YpJkYeh8xm/9mFd6QD1rgdNsf8AhXuu3qqrf8I1qU3nK6qT9hnPBDAdI24w3YgA+tAHf0hGaAc0
  tACAY6UYpaKAOZ1TXbh/FMHhnTSiXbWpvJ7iRdwhj3bVwvG5i2fYYNVvDeualL4n1rw5qrwXE+nr
  FNFdQx+X5scgOAyZOGBB6dan1bRtKvddOuR6tJp2padF5FxcQOnERG/ZIrqVxzkHH41neBdGddU1
  zxJOkynVp1FuJyfMMEY2q7Z6FzlsdgRQB3NeQeKoTqfx/wDDMMHLWtsssuP4QGdv5EfnXqGq6rb6
  RYtdXO8jIRI4kLvI56Kqjkk1znhHw3cw6vqPijWY1XV9SIUQhgwtYRjbHnueBk/hQB2A4FeReMoW
  s/jt4S1GXiCdBCrdtw8wY/8AIgr17Fc3408KJ4q0dIY5vs+oWsguLK5xnypV5GfY9DQB0mcCq1jq
  VlqcTS2N3BdRq5jZ4ZA4DDqDjvWboGttq1s0F9bNaapbqFvLVx91vVT0ZDzhh/OpPD/hjSPDFpLa
  6RaC2hmlMrqGLZY8d+2AAB2FAGZ8SWH/AArfxD/14yfyrP8Ag8cfDDSv9+f/ANHPXQa94V07xLEY
  dSe8a3K7Wgiu5IkbnPKqQCai0HwdpfhlVTS2vYoF3EQPeSPHk9TtYkZoAd42nS38Da7JIcD7BMv4
  lCB+prm/gxZS2nw3smlGPtEss6g/3S2B/KrHiyK48aufDFgXTTllH9q3uMKFU58lCfvOSBnHC9/S
  uztLaGytIbW2jEcEKCONF6KoGAPyoAnFc/44sZdT8Ea1ZwjMktnIqgdScZ/pXQUhGaAOA+DFwk/w
  z09V+9FLMjD0JkZv5MK7V9TsY9Qj0+S8gW9lUvHbmQCRlHUhepFcZp9gfh7r18URj4a1OXzt6gn7
  DOeCGA6Rtx83YgZ9a6afwxo914jtvEMtoranbxmOKfceF57ZwfvHn3NAGlNDDcRPFPEksbDDI6hl
  P1Bry7xjcWEzSeEfBWl2L6xdjy7ma1gRUtIj94u6jgkduoB9cV6JqmhWWsbFvTcPEowYkuHRG/3g
  pGfxqbTtI0/SLUW2nWcFpCOdkKBQfrjrQBS8LeH7fwv4bs9ItzuWBPmfGN7nlm/E5rzrWoG1H9oz
  SBDz9islklx2AEh5/wC+xXp2r6rb6NZfaJllkZjsighQtJM56KoHUn8h1NYHg/wxc2N9qPiLWQn9
  t6q+6REO5beIfdiB74AGT7fmAdeBgUbaWigBAMVV1Ke5ttOnms7Jr24RCY7dZFQyH03NwPxq3Qea
  AOLHiTxeDx8P7n/waW3+NB8SeMMf8k+uf/Brbf412eMUuKAIbSSWW0hknhMErIGeIsG2MRyuRwcH
  jI9KbeX1rp9rJdXlxFb28Yy8srhVUe5PSrFZ+taLYeINKm0zUoPOtJsb03FTkHIII5BBANAFyKaO
  eFJoXWSJ1DK6HIYHoQe9eQftBHOg6L/1+P8A+izXrVjYW2m2FvY2cQitreMRxIOiqBgCud1v4d6B
  4jufO1Zb66IZmRXvpQiZ67VDYA+lAHRXN5bWNk1zeXEVvbxKDJLKwVVHuT0p8Usc8KSwyLJG4yro
  dwYeoI61lx+F9NGkXGlXAuLyynUK8V5cPMMDoAWJI6A/XmquleBtG0WA21h9uhtSSTb/AG2Ux89f
  lLYH4UAcJ4e0ET/HTWdW0xAml2iFJpE4R52QBkHuCSTVjTbM+GvjxfmcbbfXbVntpD0ZwQzL9cg8
  e4r0+1sbaxtkt7WFIIU+6kahQKg1XRdP1q1W31C2SZFYOhPDIw6MrDlSPUUAcj8XrrZ8PruyjUvc
  X8sVrBGvLO5cHAHf7tZsfw/8VWUGnLovipNLWDT4rZ4vsol+cZLkE9ix7egrtrbwxptvfRXrrPdX
  UIIhlu53maLPXbuJ2/Uc1sYoA8P+IHhzxrp/gq+udX8ZrqNijReZa/Ylj35kUD5h0wSD+FdP8JdL
  1620DT7+810XWkzWC/ZtP8gL5GSCPn6tgAj8a7nXNEsPEWkzaXqUTS2kxUuiuUJ2sGHI56gVJpel
  2mjaXbabZI0drbRiOJSxYhR05PJoA8s/aCIPhbSv+v1v/RT16zCR9njB/uD+Vc3rnw80HxJcebq4
  vrrDFkRr6UImeu1Q2BW1pWjwaPbmC3nvJI+NouLl5doAwACxOBQB5v8ACe0bwx4g8TeF7xfLulnW
  5g3f8tYcbQy+o6fTNL8crYX2i6JZ2yF9Tn1AJbIoyxyhzj8dlejanoOnas8Mt1Bm4gJMM8bFJYs9
  drrgiorPw1ptlfjUPLknvlQotzdStNIi+ilicD6UAVdX8QWfhPQLebVLiMTFUhjV5QvmyYAxk8Ad
  yx4A5rC0fxD4UtLqbVdS8V6Nc6vcrtklW7TbEnURRjPCj82PJrV8Q/D3w94pv1vNXguZ5VQIgF1I
  qoPZQQB7+tZP/CmPBB/5h1x/4Gy//FUAdZoet2uv6d/aFjva0aR0ilYYEoU43L/skg4z6Vz3iM/2
  7420Tw+vzW1oTql6O2E4iU/VznHcL7V1Om6ba6Tptvp9lH5VtbxiONMk4A9zyT71FbaNZWmqXupR
  Rt9rvQgmkZy2QgwoAPQDJ4HqaANAVzPjzVJ9N8Lzx2Z/0++ZbK0HcySHaD+Ay30FdNVC90ay1C/s
  Ly6jZ5bCRpLf5yFVyMbsdCcE4z0yaADRtLh0XRbLTLcYhtYViX1OB1PuetUvGLD/AIQjXx/1Dbj/
  ANFtW52xWPrnhux8RQfZ7+S8+zlSrxQ3TxK4P94KRn8aAOM+BxH/AArmP/r7n/8AQhRXQ6Z8P9G0
  W2Nvpc2p2kBbcY4dQmC59cbqKAOqooooAbJ9002b/Vn6j+dFFAD17/WloooAiH+ub/dH8zT16/hR
  RQA6iiigAooooAKim/1Mn+4f5UUUAOTqPpT6KKACiiigDwzxv/yXax/69of5tXuI7/WiigBjf6yP
  /e/oaevSiigB1Ie1FFAEX/LwP90f1qVelFFAC0HpRRQBEn3P+BN/OpF6UUUALRRRQBHL/qpP90/y
  qQdBRRQAUUUUARSf66H6n+RqQdDRRQAvaiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAK
  KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigD//2Q==
  ">.Thus, if the general incidence of <i>treifot</i> among cows is 10%—which reflects the most liberal definition of <i>mi’ut ha-maẓui</i> —the statistical probability that in any sample of one hundred cows at least ten cows will be <i>treifot</i> is1 - .00002656 - .00029513 - .00015231966 - .0058916025 - .015874595 -. 0338658038 - .0595787289 - .0888952464 - .1148230266 - .1304162771 which equals .5487.Hence, statistically, in more than a majority of such samples at least ten cows will be <i>treifot</i>.</i>
  The probability that at least two of the cows are <i>treifot</i> is far greater than the probability that the percent of <i>treifot</i> in the sample is as great as the percent of <i>treifot</i> among all cows.11<i class="footnote">The probability that at least two of the animals are <i>treifot</i> is .9997. That this is the case may be demonstrated as follows:The probability that exactly 0 of n cows is a <i>treifah</i>, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAAoATcDASIA
  AhEBAxEB/8QAHAAAAgMBAQEBAAAAAAAAAAAAAAcFBggEAwEC/8QATBAAAQMDAgMEAwcQCAcAAAAA
  AQIDBAUGEQAhBxIxCBNBURQiYQkVMkJxgaEWFxgkN0ZSV3aCg4WVosTSIyVWkaWxtNMzY3J1k7LR
  /8QAFQEBAQAAAAAAAAAAAAAAAAAAAAH/xAAYEQEBAQEBAAAAAAAAAAAAAAAAAUERMf/aAAwDAQAC
  EQMRAD8AZ/Frj3dHDGpxIVycNYihMQpcZ+LcBW24EkBQGY4IIyOoHUaY0qt8S40X0g2BRJR5cliL
  cqi78g7yMhOfzse3SD90N+8b9Yfw2tZaKXvDni3bl412VbS40+hXLDyH6TUmwh3bqUEEhYHXbfG+
  Mb6vVRmw6bAfn1CUzFiR2y4886sJQ2kDJJJ2A1lbien337b9rtWz68uEIhqS2d+XkUtbvMR/ySlJ
  z7B7NWrizWV8Re0BQuD7TiveCnqTPryEnaSpCO9S0rzRjkGPNzzSNDhx2HdQu+I/VYVKmRaOVJFP
  lyRyKnJ3y4ls+slvpylWCrOcAYzX2+Jiq5d9RtixKGmvv0ohNRmvzPRYUdZJAb7wIWpa8g7JQQMH
  fXfxtuByz+EFx1qCQw/EgluMUjHduLIbbI+RSgfm1RexNS2oPA6NUEp/papOkSHFnqrlX3Q3/R/S
  dBc6BxHadvj6hropK7fuFxrv4jZkB+PNb3yWXcJ5iMHKVJSdjt11JXneka0KnCNdgvsUOVytKrCV
  BTMZ5SsJQ8OqEnbDm6cnB5epTPbnQ/SqLZ15U5fcVKlVfkYeT1SVI7wfLuyPp09YvvbelisLmxUP
  06t05C3GV7hTbrYOP7lddBNIUlaQpKgpJGQQcgjX3SF7L11T4NcuTg/XZa5My2H3E059w+s7ESvl
  AP8A05QR7FgdE6fWiDRqDr94Wjb8xEKv3TQ6TKW2HUMzag0wtSCSAoJWoEjKSM+w+Wo/65vDb8YV
  pftmP/PoLZo1U/rm8NvxhWl+2Y/8+rZoDVZvm9KVaiYkd9t+fVqgstU6lxEhciWvx5QSAlI6qWoh
  KR1Ope4qtDoNAqFbqC+SJAjOSXlDqEISVHHtwNJTsr+m3tNuHi9cSe8qVTlKg09BOUxIjeDyN+QK
  jg+ZRnxOQekFyQ7CYdlx0xpC20qdZS5zhtZG6QrA5sHIzgZ1x3LXaRbVFk1quz2oMCK2XHnnM4Sk
  DJ2G52B2AJ1JaS/bIkuq4RN25CQhVRuerQ6LEURkpU64FKx44KWyk/LoGBUeIFo09y3ETKsW1XKt
  tuk/azpEhTiQpAJCfUJBB9fl8fI6tGkJfy6b9kTw8tqTIQ3TrTo0muSeYZ51LKYzAAG5UF9Ejckg
  AdNPvQVS0b2iVmtzLbqMJ6i3FCT3j1OkLCi40ThLzKxs62fMbg7KAOrXpO9qulS2LJj8QKC4Ytw2
  pIRLjSED1lMqUEOtq80EEEg9QkjxOrNQ+J1LqHCil38KdVZbExkFcamw3JTrbgCg4nCBslKkKHMr
  A2G+40F80aSfD/tC02+ZVWi25Z1dmSogaMSMgt97JCufmUskhtlKeVOSpZzzgAE9fKB2hG6dfrVo
  8Q7KqFmyJCkpYkPykvtHmOEqUoJACCduZJUAeuMEgcWPi/XZ8u6rZ4Z0SW9Dl3A4t6oymFFLkeA0
  OZzlUN0qcwUBQ6b+ODqNRGplldoK2qBbMJmnQa3RpXp8Vgcra1skKbeUnxc+GkrO5B3J17PRkx+1
  kio1FSW2nrOLUBbhwlS0SsupST8YJUCR5KzrysKM5enHGtcSE+vQqXC95KK78WUoK5n3keaQrmQF
  DZW/lopv6+LUlCFLWoJSkZUonAA89fdIHtP3POqlz2zwcoktyM9cb7Zqj7RwtERS+XlB9oSsn2IA
  6KOiGpZl6xbvqUz3hgvyKHF5mxWFEJYkvBQBQyOriRvlzZORgc25Htft92nYtM9PuitRoCCCW2lH
  mde9iED1lfMMDxxqZo1Ng0ekxaVTYzcaFEaSyw0gYShCRgAaqnGynU93hXek12DFXK+p2Yjv1NJL
  nKllxSRzYzgKJIHgSToPzwY4kU/ifb1QrtLp8mFDi1JcJoSFArdCW2185A2TnvMYyenXfAvGs89g
  b7j1W/KB7/Tx9aG0KNGjSZ4v9oOhcPJyqY/bFwy5vMtLRdimLHd5ThRQ44MrTn4yUqB653Ggc2uC
  4Jc6BRpMym0tdUlNJCkREOpbU7uMhKlbZxkgHAJAGRnOlnfHF2v0WkOV+hcL65W7faT3i6iuQiMF
  N9e8S0Qp0oxvzKSkY36b6s3CDiVb3E23FVehqdacYWG5cR7AdjrIyAcbFJ3wobHB6EEAJmybro14
  Ub3zoz61JQ4pmQw8gofjPJ2U06g7oWD1B+UZGDovG7ras+Aidc1Zi0xhxRS2Xles4obkJSMqUfYA
  dJbjPVTwj422/fsMlqiXL9o3Awn4ClIxyv4/DCVZz5NkfGOrD2khJr8+xLCpz4berlbTJdVgKAjR
  k944SOhG6Tg7Ep0Uw6le1r024YVAnVVMeoTgv0dC2XORZQnnWnvOXkCgnBKSQcEbbjRpY3FIpJ7Q
  tIiSir3rs6hOSOUIU6tcuWru+XlAKlqLYKydz1J8To0QvPdDfvG/WH8NrQEqyarLaLMviTeTjShh
  SWzCYJ/PajIUPmI1nPtgrr/Eao28xatjXnLj0tEguyF2/KaStTpbwEhSArYN+IHXWnLFupq6ad6Q
  KPXaTIbQgvx6pTHoqkqUDskuJCV4IOeUnG2cZGi48LD4e2jZCX1W9SUMSZJzJlurU9IfJOTzuLJU
  cnfGcZ8NZ54ArXJ7ZvEJ2VkuNpqKW8+SZbSU/u61frNVWpauHfbHp1xOpLdFvFtccPfFRIUkAoJ8
  y4ltX6T2HQhkdqeC9UOAF1x2ElSkRm3yB+C0824r6EHUd2OnkO9nq3kJIJZcloX7D6S6r/JQ016h
  EjVCBIgTGUvRpLSmXm1dFoUCFJPsIJGktwmpFc4MSqtalRpdVq9pyJaplJqdPirlOMcwAUy802C4
  D6qSFJSQTnpnACE7fbyE8KaLHJ9ddcbWB7EsPA/+w07OG9PepPDu2qVJSUvQ6TFjuA9QpDKUn6Rp
  Z3ha9S4vcRLdkzqTNplmW84qUr3wYLD1SkEpIQGVeulsBIyVhJPMoAeIdZIAJJwB1OgyjSlrie6D
  TWYxw2+hSXseINOSs/vga1drNXZ6parz4/XxxX5SqlNyHYNNdPR5Xqo50+wNIT/5B5a0roVB1+z7
  RuCYibX7WodWlIbDSHptPafWlAJISFLSSBlROPafPUf9bLht+L20v2NH/k1x39wm4f35WGqvddA9
  8ZrMdMZt30x9rDYUpQThtaR1Wo5xnfVe+xw4Mf2N/wATl/7ugtv1suG34vbS/Y0f+TVs0pvscODH
  9jf8Tl/7umzohW9rB55js+XUuOSFlphBx+CqQ0lX7pOuTsdoQns826UYytcsr+X0p0f5Aav/ABDt
  xm7rGrNtPKCE1GG4wlZGQhZHqK+ZWD82lN2L5z8awqtZNUaVGq9uVR1mRHX8JCVnmB+TmDg+b26L
  h765Z9Np1Qciuz6fElriPB+Mp9lKyy4AQFoJHqqAJ3G++urXNVZrFNpkqoylFLEVlbzpHglKSo/Q
  NEQlaqFiUe6Ysuszrbp1wSmhHjOy3WGpbreThtBUQtSck7DbJOrJrOVhRGa32V7zv66orMmp3VT6
  nVJy3khRS0lLiWG0k7hCENoKR4E5G++mvwElzp3BOy5lScW7KeokVTjizlS8tJwonxJGDnxzoPXj
  i2h3gzeaXMYFDmKGfMMqI+kDSz7CT7z3BWU08SUM1l9toH8EttKP7ylatXarr7VB4H11BV9s1NCa
  dGbHVxbpwoDz9TnPzakeztZr9i8JKNRJrfd1BaFSpqfFLrh5ik+1I5U/m6LhHe55AZvg43/q/wDi
  dT/b/p0Rzh7QKuptPpjFW9GQvG/duNLUofJltOqL2F66/Q5V2rXRalOpzqYYkSILCpC46x3/ACZa
  RlxSVAr3SlWCkZG+dNPiNbda43XpQ6a7RqnR7GozxkzJNRjqjPT3DgcjbS8LCeXI5lAfCV5DI0yr
  PpNPubhdaouukQaq4qlRHnW50ZDwDxZSVKwsHfJO+rdHZajsIYYaQ002kJQhCQlKQOgAHQa+toQ0
  2lttCUIQAlKUjAAHQAa/WiDWUKktcj3QiI1JyW2UpSznyFNUoY/OJ1q/Wa+0TSl2bx3sji0EEUsy
  WoNUcA2ZPrJ51fK2tQ/R+3RY0pqp8Z/uPXp+T8//AE69WwEKAIIIO4I0uePdfVFsG4LdhUG4qvU6
  pSH48dunUiRIbBdQtsFTiEFCcHJKSrmxjbcZIX/YG+49Vvyge/08fTlv66fqRo7VS+py4a/3khLH
  o1EhekvoylSudScjCBy4J81J89I7sbyqjZ9nzLWua1Lspk2XWC+w49QZXcqS4202OZYbIRgoOSrA
  AOc9caS0W+ljReMHvnWIVN+tdxOhelyG2PSZdA7thnnUE87iuc8qBnJPgAdJv3Q3rYx/7h/Da1lr
  Jvuhv3jfrD+G0J61ipCVIKCkFJGCkjYjy1kzskxjSO0ZxDodPBRSowltBKfgjupgQ1+6V4+fTzqf
  EOpxaAoM2Lc71y91yppqac4pnvsY3lAdz3efjBeceGdtRPZx4YzLCotRqlwutyLnrr/pNRWg5S1u
  SGwfHdSiSNiT4gA6Cl9vttB4U0V047xNcbSn5Cw9n/IabnDKFDqNi2bXqhT4r1XRQovJLcZSp5rn
  YQVhKyMpBPUA76UHa9jyb1umxuF1IJXNny1TZON+4aA5A4oeQSXj+b7daHp0RiBT48GKjkYjNJZa
  T5JSAAP7hoPFNIpKawusppkIVNbYaVMDCe/UgdEleOYp9mcaNdujRBo0aNAar3EKz6NfNsP0GttK
  LKyFsvNnldjuj4LravBQ/wDoOQSNGjQeXD9N3Rqe7S7vEaVIhFLbFUjrAFQRg4Wpvq24MDmG4JOU
  nwFm0aNAapfEmk3Tc6E2vSZCKPRpbX9Z1VLgMjuySFMMI8FqHVxWyQrYE9DRoLDatApNr2/DoNDh
  oh0+G33bLSfDxJJ6kk5JJ3JJOpPRo0Bo0aNAaNGjQGl9elkz2rva4gWSqPHuNtruJ0V9RRHq0fb+
  jcIB5HBgcrmDjAByOho0F7guPPQmHpEZUV5xtKnGVLCi0ojJSSNjg7ZG22uK7KWa3a1WoqXQ0Z8F
  6KHD8XvEFOfmzo0aBC0+BdNW7PtK4MRrbrVLrq4zdIq0uTCWiHDjJVh59L5Hdvc6AQlLalElYzgA
  60HR6fFpNIh0qC33cSFHRHYR+ChCQlI/uA0aNBQoVl1O6bziXhfrbKEUtZVRKG24HGois/8AHeV0
  ceOBgD1UY2JO+rjddyUG1aSqq3FVodMhhXIHZDoQFrwVBCc/CUQkkJGScHRo0GTuwTcdBo9UuemV
  arw4MypqgogtSHQgyFgvApRn4SsuIAA3OdtbH0aNFo0aNGiDUdc1DpVy0GZQq3DbmU+Y2W3ml9CP
  MHqCDggjcEAjRo0Fb4Z0m6rYQq1as6irUWE0BS6sXAH+6BASw+jxWkdHE7KA3APW66NGgNGjRoDW
  N+3rclBrNWtilUmrQ50ylqnInNMOhZjrUWUhC8dFZbWCnqMb6NGixrO1rjoN00sVO3avDqkPm5C7
  GdCwleASlWPgqAUDg77jXtcEipRaPJfo9ORUZ6UgMR1vBpK1Egess5wkZydicA4BOBo0aIq3Duwz
  QqtULruCair3bVQBMmpRytstjHLHYSd0tJwOu6iMnyF40aNAaNGjQf/Z
  ">.Substituting p = .1, n = 100, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0a
  HBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIy
  MjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAApAiYDASIA
  AhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQA
  AAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3
  ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWm
  p6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEA
  AwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSEx
  BhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElK
  U1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3
  uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD3ManY
  EZF7bEevnL/jSx6lYysFjvbd29FlUn+deE/GvQNJ03X/AA9LZada27XRcSiKJVD7XjA4Ax/Ea9ku
  fCHh28gaGXRLDaePkgVCPoVAIoA3c0V474W1bUPC3xZuPBLXk95pUwL24uHLvBlPMA3HnGAQfwrq
  PEniK7vvE9v4N0KcwXsqebfXijJtIf8AZ/2zkYJ6ZHrQB3KurZ2sDg4OD0NQzXltbsqzXEUbN0Du
  FJ/OuQ8V6pa/DrwFPNp8YV0xHAHYsXlb+JieWPUk1H4K8KWknha2vdbt49R1TUIxcXU92gkbL87R
  n7oAOMDFAHcggjINCsrDKkEeoNeX+G9Wn8L/ABMvfBM08kmmXCfaNN81izQ5XcYwTzt4YD/dHrWl
  4oebwJc/8JNpyu2lSSgarZA5UBj/AK+MfwsCeR0P1oA7+iq9ndw31pDdW8iyQTIHjdTwynkGrFAB
  RRUVxNFbwPNK6pHGpdmY4CgDJJ9sUASMyopZmAAGSSelLXnmg3EvxFvZtXufMTw5azmOxs84F069
  ZZfUA9F6ZznOK9CFAHKfEcwQ+B9Vu7m4uYore2kcLBM0W9yu1ASuDjcRxke+elcjYac+mfBmO41a
  a5t7Cz0wyrZxTNE0zlS26R1w3zMeEBHB+bJOB2vjfwzL4u0NNJS8FtA9xHJcZTd5kanJQc8ZOPyq
  Px14XuPFvhKXQ7W9SyEskZeRo9wKIwOMfUD8qAIvhhpMmjfDjRbWYkzNB5z56gyEvj8mArrqy9F0
  +602yEV5qMl9MTkyGNY1UYwFVF4Uce9aVACq6tnawODg4PelrhvFsdz4SaTxZpCs0KuG1WxB+SeM
  nBlUfwyL1yOo69K67S9RttX0u21CzlElvcRiRGHcH+R9qALdFFcv438Wx+E9JSSOH7RqN3IILK2z
  /rJDx+QyM/UDvQB0+5d23cN2M4zzilrn/DmgS6XE15qNw15rNyoN1dMeM9diDoqDsB6ZPNdBQAU0
  OpYqGBYYyM8isvW7LVb+1EGmamunFg2+cQiRxxxtzwO+Tj0ry34DXl1qEniW6vLiW4neS3LSSsWY
  8SdzQB7OWUMFLAE9BnrS1la5odtrll5EzSwyxnfBcQttkhfsyn19uh6GsPwb4ouNRur/AMP6zsXX
  dLbbMVGFuI/4ZVHYHjI9frQBr65r0ejLbRJC1zfXkvk2tqjAGRupJJ6KByT2qLR9emutZvNG1C2i
  t9QtY0nxDKZI5InyAykhTkEEEEVhWqvqnxl1CWXmLSNOjihU9FeU7mYe5Xj8BSaC7al8W/El/Fzb
  WdpBp+7sZMl2x9MigDvaKQ9K4jxL4ju73xNbeDdBnMN9MnnXt4oybSHH8I/vnjHpkHvQB26srZ2k
  HBwcHoaWqOlaVa6NYx2dnGViXJJZizOx6szHkk9yal1G8j0/Trm9l/1dvE0rY9FBJ/QUASzTw26b
  5pUjT+87BR+tEU0U6b4ZEkT+8jAj9K8r+GgPjs6n4n19VvJPtJt7a3lG6K3QKGwqHj+MDJ54qX4h
  QN4E+xeLPD6C1CTrDe2kfyxXEbdMoOA2eMigD1IOpYqGBI6jPSlritat5b/SYfF3hqQx6mlusypk
  lLuLGTFIo68ZwRyDW34V8SWfivw/b6tZZVJPlkjY5aJx95T7g/nwaAIfGHiGbwvoU2qppzXscKlp
  QsyptA+vXJ4wMmrM2twWHhsazqm21jSATTKG3bcgHaOOTyB9a534kE30ehaApOdT1KJXx2jj+dj9
  PlA/GovHitf694Q0P/l1utRM0yjjKwrvAPtntQBqweJr6LUNMh1XS0srfVGZLYiffIj7dwWVcAKS
  AcbS3IxU934huJdZn0jRbSK8u7ZA9080xjih3fdQkBiXI5xjgdetYnjCRr7x34Q0qDJaGeTUZsfw
  Ii4XPsSSKg+G9wsHgO88TXjjzL+e41CZ2PQBiAM+g24FAHX+Hdbh8Q6NFqEKNFuLJJExyY3VirKf
  oQa1a4r4VW00PgS1nnRke8mmvArDBAkcsP0NdrQAUUUUAFFFFABSMyopZmAUdSTgClrykX7+Pvin
  d6PK7HQNFUtJbqSFuJgdo346gEnA6fL70AeoQXVvcgmCeKUDr5bhsflU1ecfETR00LQT4l8Owx6f
  qOmOshNugRZYicMrqOGGDnn0qp4v+I8sfw60zUdHIj1DWsRQY58pujn6g8D60Aelve2scwhe4hWU
  9ELgMfwqeuT0rwLotv4fisryxivJnjH2i5nXdNI5HzNvPzA56YIxWB8O9fu4Nf1rwXqly9xNpcjG
  0nlbLvCDgBj3IBU/ifSgD0l5EiQvI6qo6sxwBTILmC5XdBNHKo7owYfpXlmiXjfEX4iao92Wk0HR
  v3dvaE/u5ZckF3H8XQkA8dKtfEWzPg+ytfFfh2JLS4s5lS5hhGyK4ibjDqOOuMHGRmgD06oJLy2h
  lEUtxEkh6KzgE/hXC+OvHy6T8PbbWNKYfadUVFsyeSm4ZLEeoGfxxWh4d8FaXB4agi1Szjvr2eIS
  XdxdL5kkjsMt8x5GM4GMYxQB2NNeRI0Luyqo6knAFeaeBNbutL8Z6x4Gv7iSdLUmbT5Zmy/lcfIS
  euARj8ar6j4R1b4l3Vxfahq9xp2hLI0djaQLzIinHmODx8xBIz2xQB6jFcQzqTFKkgHdGB/lUlfN
  XiLw7rPwf1ux1PS9Seaylc7SBt3kctG6jg5Gef619Bz6va2OhNq17IILZIBNIzfwggHH64oA0s1E
  tzA0xhWaMyjqgcZ/KvLtC1HWvilfXF209xpfhWBzGkMDFJrtv9pxyB0zj1x616Jp+haXpQUWNhBC
  VGA6oC2PdjyfzoA0iQBkniqh1Kxz/wAflv8A9/V/xrE8YXiGyi0Zb5LKbVC0AnZ1UxR4y7DdwTjg
  D1b2rz1P2fdDkRXTXLxlI4ZYYSD75xzQB69ZahZ6jHJJZ3Ec6RyGNmjbIDDqM/jVmuZt49I+HXgm
  OOSZksNPixvYDc5Jz0HViT0Fcn4bk1r4mSPq2qTT6f4cWQrbWFtIUa4x/FI45I+negD05LmCSRo0
  mjZ1+8quCR9RRVSw0fTdLQLZWVvb4G3ckYBI9z1P4migDyD49kjV/CxAJIM2AO/zw8V6bNrWttGw
  tPCl55p6G5u7dEH1KyMf0ry/48zxnXPDKB1LxeY7gHlQXiwT6dD+Ve3RSJJGHRlZG5Vgcgj2NAHC
  eFPAd3ZeKbvxX4hu4rnWLjISO3B8qAHjAJ5Jxx09etcx8Gbg634r8W67MS08sqAE9ldnbH0GxR+F
  ey1438IbY+H/ABp4t8PTDbKjo6A/xIrMAfxDqfxoAk/aDeQeF9NVT8hun3f9+zivVNKCro9kq/dF
  vGB9NormPid4Yl8V+C7mztV3XkRE8C92I6r+IzVzwHrsOt+D7GXcFubeIQXUTcNFIgwQw7dKAPPP
  HJZPj/4WMOQzR2+7Hp5sgP6V6j4rtEv/AAhq1tIMq9nJx9FJ/pXCaZYf8JX8Z7zxFFiTStIjFtDP
  1WWYKQQp77SzcjuB613fiy8j0/wjq91IcJHaSdfdSB/OgDz74R3mpa38Lp9PstR+w3lrOYYLpoRN
  5ana4G1uDwSK6MeGvHX/AEUGP/wSQ/8AxVc78JdG1ay+FssumvbwahfTedA91GXjCjauSAQSCFOP
  qK6H7F8TM/8AIY8N/wDgDN/8XQAv/CNeO/8AooMf/gkh/wDiqx/iPPqugfCm6t9R1X+0L25nW3a6
  S3WDKOxONqkjoMe+a1/sXxL/AOgx4b/8AJv/AIusz4haJrepfCq7TVZLW51K1kF0TZxMiFVJ6Akn
  hSSfpQB0fw4tUtPh1oMaDAezSU/VxvP6tXU1ynw1vEvfhzoUiMDstVhPsU+Q/wDoNdXnNAEVxKIY
  JJWDsqKWIRC7EAZ4A5J9h1rB8PeMtP8AEup6lYWVtfxS6dsE/wBrtzDgvnAw3zZwpPIHUV0TYxz0
  ryzwHrtpY6LrHiW7WWSbXdXnktLeCMyTTIp2KqqOuNp56DPNAHqY6UtZmhancatpq3Vzpd3prliB
  b3e3zAB0J2kjmtOgClq1ol/pN7ZyKGS4geJge4KkV5j8BNVkuvCt7p8jFhZ3AKEnorjOPpkH869P
  1a8TT9HvbyQgJbwPKSfRVJrzL4C6VJaeFLzUJFKi9uB5ef4lQYz+ZP5UAes14x4huTq/7Q+h6bLk
  w2Cqyr23eW0ufrkL+Qr2evGvE9qdE+P3h/V5Bi3vwsQc9N+wxYz+K/nQB7KKKTIrC0Xxdouv6lqF
  hp10ZbiwfZMpQr3IyCeoyCOPSgDerxH9nn/VeIv9+3/lJXtckscMTSSOqIoyzMcAD1JrxT9nsgJ4
  hUn5i1uQO+MSf40Ae3GvGfFlydB+Pnh+8gO030McMwB++GYxYP5Kfwr2WvG/FVsde+PmgWcI3f2f
  DHNMR/CFYyAn/wAdH4igD0O98O3X9uzavpOoJZXNzAsFyJbfzVcKTtZRuXDDJ5ORjtV7QdDtfD+n
  Cztd7ZdpZZpDl5pGOWdj3JNadLQAGvGfg1cHW/F/i3XZiTPJIignsrsxx9BsFezGvG/hFanw/wCO
  PFvh+YbZEZXQHjciswB/EOp/GgD2SoL21ivrGe0mBMU8bROB3VgQf0NRapqVpo+mXGoXsoitrdDJ
  I5GcAVBoOu2HiPSYdT02YyW0uQCylSCDggg9CDQB5v4H03WvhrqOoaTf6ZeX2j3Moltr2xhM20jj
  50XLDIx2PI79aZ8Z9eW48Drbx2F5HHPdR4muI/KyVO7aEbDk8f3ce9eja34m0bQYDJqOoQwsPuxb
  syOewVByT+Fef2mhar8RfFdtr+uWkljoFkc2NjMMSTd97jsDgHn0x60Adx4LsZdM8E6NZz8Sw2ka
  uD2OM1518JrhtO8f+LvDqk/Z0nklROylJNhx9Qy/lXsf8PPFeO/CW2bUvHni7xIgP2aWeSKNuzF5
  N5x9Aq/mKAO0u/Cus3ni+x199bs9tiskcFr/AGexUI/XJ83JfAAz09q1td0H+1pbC7guPs1/p8xm
  tpim9QSpVlZcjKkHsQemCK2RS0AYekeHVsNQu9Uu5/tmqXYCSTmPYqRjpGi5O1e/Uknkk1kr4FYa
  YdCOpkeHjK0n2NYMSFCxYxGTdzHknjaDjjNdlRQAyKNIoljjUIigKqqMAAdAKzNfsdW1CxSHRtYX
  SrkSBmnNqtxuTByu1iAOSDn2961qyfEEWuS2KDw/c2Nvd+aC73kLSIUwcgBSDnOOfrQBzv8AwjXj
  v/ooMf8A4JIv/iqP+Ea8d/8ARQY//BJD/wDFUn2L4mf9Bjw3/wCAM3/xdH2L4l/9Bjw3/wCAM3/x
  dAHYafDdW+nW0N7di7ukjVZbgRiPzWA5baOFz6VZqtp63iadbrqEkMl4I1EzwqVRnxyVB5AzU0ki
  xozuQqKCST2A5JoAU/1rxX4HFn8SeLZJf9dvjyT15klz/SvT/DfivSPFlrPcaPcmZIJPKfKFSDjI
  4PYjkGuC0Wz/AOEF+LupR3WItL15S1rO3CeaDuEZPQHlgM9ePWgDs/iKAfhz4iB6fYJT+O0188K0
  r6V4Gjlz5Qv5guemPtEY/kTXunxUv/L8GXGlWw83UdUK2tvbL9+TcRuIHXAGeelcr4u+H13ZfDfQ
  xp8fn6joRE7ogyZcnc+PXBH44oA9jrw9S8f7SGpiDhWt3L4/69VP/oVeuaPrtjrGgW+rW9zGbaSI
  OzlgAhx8wb0I75rgPh/pj61448Q+NZEP2O5ka3sWZceagIBce2FAHrk+lAGZ+z6WbTded/8AWG4i
  3H32En9c12HxcAPwv1rPZI8fXzUrnfBNofA3xE1rQb0iK11Rhc6fK3Cy4J+QHpuAOMe1anxbnlvf
  D0HhuwAm1PVbhEjgH3tinczn0UELyaAPJvErzP8ADb4fLITtLXOc+0yhf0r6cGAMV5f4+8AyT/DP
  TrDSkMt3oio8aKOZQFw4HufvfUV23hrX7XXvDNrqcMyENEPOGceW4HzK3oQc9aAPL7pnX9pu28vO
  GiQPj0+zt/XFe0gBRgDA6Yry7wbpzeIfidrnjIrnT4ybSxk7TYAUuvsAMZ9zVrxx8R9I0vUR4fOo
  zW7Ef6Zc2sRkeFf7i46Oc9f4R74oAPEumD4h+LLHToudF0WYy3twOkk+MCFD3IH3j0Gcdayvj1qE
  ln4X03T4vkhubgl9vGQi5C/Tn9BV2x+LvgPSdOjsrH7bFbwrhI1snH6nueuTTfGei3/xG+GdvqEN
  mIb8Mbq2ts/MYjkbT/tFcH68UDN7w54U1fSvCOn6dDrz2UsUA4t7WJkDHk5DqxbknPIz7dsjw547
  1a18cT+DfFCQyXYOLa8gTYJeMjcucDI9O4I966zwd4gg8Q+G7S6Vgt0kYjuoSMPFKowysvUcjv2r
  z5tOPiz47rqlh8+n6PHGk1yvKNKob5AehOWwfTBoEdX41+G2neN7q3ury/vbeW3iMcYhKlME5yQR
  1/GvLNc8JeMPhYf7Y0fV5LjTlYCRkyAM9BJGcjB6Z5/CvYNK+IHhrV9Yn0m11FReQu0flyqY95U4
  O0ng1D8Q9QgTwnqGlxp9p1HUIGtra0jG6R3cYB29gOpJ44oA8l+I3jKXxl4T8LiFfI+2ySefGrHA
  mQqmPplwwz7eleuWPhjV7Dw9aWFn4iezlt4VjRYbOJogQP7rAsf++hXnmtfC/UbD4Z6ULRBcavpk
  7XksKc792Cyr6kbV+uDXrXh7XrPxBoVvqVtKrK6AyLnmN8fMrDsQc9aBnCeGfiZNb61qHh7xiILW
  9sc4uogQkoBA5HYnIII4Iz0xyVzNx4Q/4WT8S9f1C2k8rSYQkS3YGVllVVUhT/EOGOfp60UAeoSf
  DrwhMxaXw5p8jHqzxbj+ZrW0rQtM0OJ4tMsorWNyCyxDAJAwOPpWlRQIK4vxR4buotdtPF2hRB9V
  tF8u4tgcfbIO6A9nAzjP+FdpQelAFDSdUttYsEu7UvsYkMsiFHRh1VlPII71WvfC+iajdG5utLtp
  Z2+9IUwW+uOv41qnr/wKn0AQ29rDaW6QW8KRRIMKiKAAPoK4jxJb3Pj2f/hH7MSQ6JFKP7SvSCvm
  lT/qYs/e5HzN0HTk13h6Gmp3oAZa20VnbRW1vGscMSBEReigdBU1FFABTZEWRCjqGVhggjIIp1FA
  Hn+k2T/DrU7iyZXbwxfTebbzKpb7DKeqPjkIeMN0B69c1346UjdD9DQPvH6CgDnvHmrDQ/A+s6iW
  IaK0cIR/fYbV/VhXnehwyfDDW/DX9r3udL1PTRZTTTELHaXCnzAAeiq24g+pySeK7L4p/wDIlSf9
  fMH/AKGKq/Fb/kWdI/7DFp/6EaAO3sruC/s4rq1fzIJRuRwCNw9easEgUifcFI/Q/SgDifFa3njF
  5PC+l+ZFZFwuqX5UhUQEExR5+856HHAGcnPFdbp2n22ladb2FnEI7a3QRxoOwFWR0H1p1ABXO+Mf
  Clv4t0VrN5DBdROJrS5Ucwyjofp2P+NdFRQBz3hrW7m+jNhq9ubXWrVQLiIg7JO3mRt0ZG9uRnBx
  WlZaNp2nXN1c2Vjb2890++eSJArSN6k96unr+BpV+6PpQBm6t4f0rXUjTVLGG7SPO1ZRkDOM8fgK
  zIfh94Tt3EkHh+xhcc7o49p/SumooAytb1mLRrNXME1zcytst7WBdzzPjoPQepPAHNYvg3wrPpVx
  f65q7JJrmpvvnKHKwJ/DEp7gdz3/AAFdZ/y0p9ABRRRQAHpXF+KfDV2muWni3Q4g+q2a+XPbZC/b
  IO6ZPRhnjPt7Y7Sg0AZdjeWHiTR2dY/NtpQ0csNxEVYHoyOrdCOhFWbDTbTS7KOzsLaO2toxhIol
  2qv0FWR9406gDLs/Duj2ExmtdMtYpSSTIsY3fmea0+lLSHqKAOW8ValfXaS+H9BjL6ncptknYERW
  cbDl3b1xnao5J54HNaXhjw7Z+FtBt9JssmOIZeRvvSOfvMfcn+grV/vfUU4feNAC0UUUAFFFFABR
  RRQAUUUUAFNZdwIIyDwRTqKAKGm6Pp+jwPBptjBaRO5kZIUCgsep471NeWFrqNs1teW0VxC3VJVD
  A++DVmigDL0/w7pGlTGax063hlP/AC0Vfm+mTyBWnilooAxJvCWgz3LXEukWbSsdzHywNx9wODWw
  saxoqIoVV4AUYAFPooAqX+m2ep25gvrWG5iznbKgYA+o9DVfTdA0vSWZ7Cwgt3YYZkX5iPTPXFad
  FACYrGufCeg3ly9zPpNq8rnLtsxvPqcdfxraooAjjhjhiWOKNURBhVUYAHsK5t/hz4Okdnk8N6az
  sSzM0IJJPUk11FFAHKn4beDP+hZ0z/vwK6WKFIIUiiRUjRQqqOgAGABUtFAGRfeGNF1K4Nxd6ZbS
  zHrIUwzfUjr+NX7WytrK2W3tbeKGFekcahQPyqxRQBhX/g3w5qd39qvdEsJ7gncZXhG7Prn1q/Za
  TY6cuLS0hhOMblXnHpnrir1FADcVkXfhXQ725a4uNKtXlY5Z9mC31x1/GtmigCGC2htYEgghjiiQ
  YVI1Cqv0AoqaigD/2Q==
  ">.The probability that exactly 1 of n cows is a <i>treifah</i>, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQEBLAEsAAD/2wBDAAUDBAQEAwUEBAQFBQUGBwwIBwcHBw8LCwkMEQ8SEhEP
  ERETFhwXExQaFRERGCEYGh0dHx8fExciJCIeJBweHx7/2wBDAQUFBQcGBw4ICA4eFBEUHh4eHh4e
  Hh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh4eHh7/wAARCAAsAV4DASIA
  AhEBAxEB/8QAHQAAAgMAAwEBAAAAAAAAAAAAAAcFBggBAwQCCf/EAEwQAAEDBAECAwUDBggKCwAA
  AAECAwQABQYREgchCBMxFCJBUWEWMnEJFTiBhaEXI0JGhJG0xChXZ5KipaaxtdIkJSZEUnKCk5TB
  0f/EABYBAQEBAAAAAAAAAAAAAAAAAAABAv/EABYRAQEBAAAAAAAAAAAAAAAAAAABQf/aAAwDAQAC
  EQMRAD8A2XRWKvHDhGL4nc8YnY3ZolqNxRKTIaithttRaLXEhA7A/wAYfQfKtPyukvTR+L7OMIsk
  cBPEORYqY7o+ocb4q39d7oq7UVmO3ZPkfSLxE27pxJvc++YrfCz7C3PdLr0TzlKbQErPfQcSRr04
  99b702es/UNeHR7bZbHFbuOV318RbTDWfcCiQC65ruG07H4/gCQQw6KrOAYs5jcB1243WVeb3O4u
  XG4SFHbyxvSUI9G207IShIAAPxJJpdeKi49RbJ08vN+x/IINmtcIspKY7SlTH0uLQ2T5pIDWlL7B
  IJIG+Q3oA66KVXhJfek+H3G5El5x55xUxS3HFFSlEzHtkk9yan+puM3ea0Mlw+c5Bym3Mn2YFZLE
  1sEqMZ5G+KkqJOldiknYI70F2oqn9Ic+tvUXDmb5BbVGkIUWJ0NZ9+K+n7yD9PiD8QR6HYFwoCii
  igKKKKAoopOu5JP6q59c8Qxu4PwMSsSg3e7lFWUPTniSPZmVjuhHY8lp7nWgQCCQcVRuTw2Z9ilR
  pNym22OU8npER/yXUISQpQCx3SCAQSNEAnRB0a9kGLHgwmIURpLUeO2lpptPohCRoAfgAKgeqNnv
  GQ9O79YLBJjRblcoLkRl+QpQQ35g4qUSkE7CSrWh66/GgRXhEtFzv3TK4Soc+52iy3i9S5r0oPqM
  yQOQbSy26vakoSlscndlZJIBSUlVTPgztMX2PPMtt6pJtt6yN5u2+fIW+pUVgqS2srWSpRJWobJJ
  931pkWjDZ9g6Gt4LYJcZi5xbAq3xpSgUtCT5JT5p0CQC4So9ie9RHh5wDK+nmFW7Hr/kFslxoEdb
  TMO3QyhsKW4XFOLdWStxWyodghOidpJ0QDPoopadWbffMaEnqPhrjq50JsO3a1LcJYucZA973fRD
  yEjaVpGyBxIV2FAy6KhMEyi05nikDJLI8XYU1vkkK7KbUOykKHwUkgg/hU3QFFFFAVQuqvUKZg8i
  0stYpMuqLpMZgsSEy2WmvaHVKCW9ElwnSSSQjWiO+9gX2lJ1E/7QeITAcZHvMWePJv8ALR+A8pg/
  qc3/AF0DbooooCik67kk/qrn1zxDG7g/AxKxKDd7uUVZQ9OeJI9mZWO6EdjyWnudaBAIJawjKgWX
  2Ozx2EqjRvKhsuLKWwUp0hJIBIT2AJAJ186D2UVkXq9d+oUHxKYJjeSZS3MgSLlbZrcOAyqPGQFS
  +PEoKiVkFs+8on17a9K11QFFFFAUUUUBRVJ6m4zd5rQyXD5zkHKbcyfZgVksTWwSoxnkb4qSok6V
  2KSdgjvXd0hz629RcOZvkFtUaQhRYnQ1n34r6fvIP0+IPxBHodgBcKKT+BZllOQdRc9s8i8xFWuy
  SURmZDMINJjABRdUrkpRU4DpCdnj7q1EduJ9/RbOJErpDBy3P8hgxkTJTwjzJymogW15iktBX3Uc
  iEk9gNjVA0aWN76n3qH1EewqDgUyRNbhrnIW9cGkB2OHfKDiQ35hAKvQK4nXqBTAsd5s99g+32O6
  wLpE5FHnw5CHm+Q9RySSNj5UsulH/X/WrqRlx99iJIYsENX/AIfITyfH/uFJoFR+UN/mN+0P7tWg
  JXUW1NtFUSwZlOd17rTeMzWyr6cnWkJH61Cs/flDSN4ON9x7edf/ABq1mCCAQdg9waLjPuO9N8tz
  nrkz1Uzq3IsUG3cBarSXkuvkN7LZcKdpTpSisje99ta71X8InqzTxxXuZKV5jWOxJDERB7hvyuLB
  1+K3XFfiqtRVlXo9FXjvjaza3ywULuDMx1rf8oOutSBr/wBO/wCqg1VSm8X/AOjtlH9E/tbNNmlN
  4v8A9HbKP6J/a2aJB4QP0dsX/pf9reps0pvCB+jti/8AS/7W9TZoVl7pTcF4l4yszxBlRRBvanH/
  ACR2SHuAkpUB9ErdH6/pWoayti0Vd78fN7uUYFTNraWt5QHYcYaI5/0l1qmi1R8+/hb/ADw19gvs
  P+bPZ0+b+e/avP8AO5K3x8r3eHHhrfffL6VXv8JL/JN/rCrDn2S9QLNeGouKdM/tTCVHS45L/PrE
  Pg4VKBb4OAk6ASeXp72vhVe+3fWf/EN/tdE/5aA/wkv8k3+sKbNKb7d9Z/8AEN/tdE/5abNEUrrt
  kT+KdIclvsRwtyWIRQw4D3Q44Q2hQ+oUsH9VUfwUWtqD0KhzUJHmXKbJkuK+JKV+UP3NCrN4mrS/
  euhGWQoyStxMRMnQ9SGXEOn9yDUH4M5jcnw/2VlCgVRH5TK/oS+tf+5YouHHRRRRFQtPU7ALvlqM
  TtOV2243pQc/6LEcLxT5YJXyUkFKdaPqR8qt9JLE1s37xbZvf3OCY2KWCJZ0uq0EhbxMhat/NOik
  n4Uw8I6iYXm1yulvxO/x7u9aigTFRkLLbZXy46c1wXvgr7pPpQWquFpStCkLSFJUNKBGwR8q5ooM
  v+DW5u2bP8+6blxRiwZjsiKgn7hbeLLn9Y8r/NrUFZW8KMVdz8Q/UzKGAVQw9KaC9diX5ZcT+5o1
  qmi1j/xys32wT7K4zmOQyYN6MtTtvdkhMdnyy2QlCG0pBTp3Xvcj27k7NOTOehNizGyuG93u9y8i
  Ujk3dXJi+LbutjiwD5SG9/yUpB18d96VP5Q3+Y37Q/u1avW60hkvrcQlpKeZWVAJCdb3v5UGbPCB
  nmUypWSdOMifVcbnY0LXCVJeOxwX5a2luaJ4hZRo6JAJ7EACmHhWK59F6xXnNskiYyY10iNQ20Q7
  k+47CZbBPFIWwkOcl6J2Ua7kb9KWfhEtTt66q9QepbTak2ubMkx4ayNB3zZHnK1/5Qlv/OrT9Ciq
  V12yJ/FOkOS32I4W5LEIoYcB7occIbQofUKWD+qrrS48TVpfvXQjLIUZJW4mImToepDLiHT+5Boi
  s+Ci1tQehUOahI8y5TZMlxXxJSvyh+5oU7aTngzmNyfD/ZWUKBVEflMr+hL61/7linHQrJviS/TC
  6a/sr/iDtayrJviS/TC6a/sr/iDtayotFQ2ZWSVkFkXbomQXWxOqPISrctCXfQjiStKtJ2Qfd0ew
  7jvulXro/wDnO8Tbl/Cj1Ohe1yHH/Zol/wDLYZ5qKuDaeB4oG9AfAAVbcBxb7I2d22/aPIb/AOZI
  U/7Te5vtL6NpSnglWhpA47A+alfOiMt+Di33LP5eZnIsmvr8XUL2ttE5xDssnz+IW8D5gQPf2lKh
  y5DfYaPZ1zxWd0BymzZ106uU6NbJsgsyoD0hTjZWBy4K2drQtIV97ZBTsH017vyeShyzhOxyIgED
  ff8A7z/+1Z/HM4u6Y3ieG25sybxc7yl2NHR3WoJbW36fAFTqe/0Pyo1p/Y9c2L3YLdeYoIYnxWpT
  QPqEuICh+41mvpVcXMS8ZOaYgweMK9lx8M70kPeWJKVfTSVuD9f0rR2H2kWDErPYgsOC3QGInIfy
  vLbSjf7qzVi0Vd78fN7uUYFTNraWt5QHYcYaI5/0l0SGjinTbIbN0WyPG1T7anKsgVOflzG1LLBf
  kFQB5FIVoI4jfHsd9j8bl00x+bjOI26zzZDKlRIbMZDMcktNBtAT7pIClFR2oqIG9+g1VmooiNyN
  y+tWxSsch22XPJ0lE+UthoDR94qQ2snR120N/MVTOgeJZPheKyrPk6bO7KdmuzVzIEtx0yXHVlSy
  tK2kcNDiBoq39Nd2LRQJvOvD3Yc5uDU/Ks1zS5vsoKGiuTFQlsE7ISlMcJG/oPgKYeD4y9i1vMBW
  T32+MJQhDIui2VqZCQRpKm20KO9jfIq+6Na77sNFAUm+u2GXNjKrF1bxGGuXe8eVqbCa+/Oh9wtK
  fmsJUsAfEK+JABclFBGYtfrVk1hiXyyS0S4MtsLacT+8EfBQPYg9wQRVR6odLYvUNl6HfMuyhi1u
  qQv82w3YzbAKQNerJWruOXvKPc9taAFus9gs1nn3Cda7ezDfuTgdllkFKXXACOZT93kd9yBs9t70
  Kk6Cg9MemMfp9Gj2+y5fk79pYUtSbdMcjOMEr3vuGAtPvHlpKh39dgkGc6h5dbsLxt27Tgt55R8q
  FDaHJ6Y+r7jLaR3Uon5eg2fQVYqjX7FaH8hYyB+Ay7dI7BYYkODkppBJJCN9kk77kaJGgToCgXXh
  06eXDE7XdMkygIVlmSSVTLjxOwwFKKg0D8wVEnXxOu/EGmvRRQFFFFAUUUUHy62h1pbTqErbWkpU
  lQ2FA+oIpGdMraei/UO44bOUpGI5JK9psMxZ/i2JJGlRXD8FFITxJ+9wHqSQHrXivlptl8tT9qvE
  GPOgyE8XWH0BSFD8D8fiD6g0HtrhakoQpa1BKUjZJOgBXVBjMwoTEOOlSWWG0tNhSyohKRobJJJ7
  D1J3XRf7a3ebFcLQ8/IjtzozkZbrCglxsLSUlSCQQFDewSD3+FBj3CrzckNwc4yeFFc6e53nEw3B
  LhVycSvbcQvfD2dC21bSrYOjy2OIrZcaOxGZQzGYaZbQkJShtASlIHoAB6CqXP6WYlN6Qjpc/Hf+
  z6YiYyNODzkcSFJcCiNcwoct61v4a7VN4RitqxCyptdrVNeSSFOyJspch99YSE8luLJJOkga7AAA
  AAdqCdqj9Y8xdxjHDBszS52U3ZKo1lgMjk448RrzCPghG+SlHQGvUbq8VGxrFaI9+l35qAyLpLQl
  t6URycKEjQQCfup7b4jQ339e9BUOgXTpnprgDFmWtD9zkLMm5Pp7hbygOwJ78UgBI+eie2zTBooo
  Mm/lDf5jftD+7U9Lj0ugzbMcfcyrKm8fKfLNqbmNhryvTyvNLZfKNduPma129Kxl4ouoN+zLqNcL
  Ndkw0Q8cuU2FBQw2UngHeBUsknaiG0b9B27AVrPwtdQr91I6dyr1kSIaZkW5LhBUZsoC0JaaWFKB
  J94lw+mh6dqLhk49ZrXj9mjWaywWYMCKjgyw0nSUj/7JOySe5JJNe+iiiCvl1tDrS2nUJW2tJSpK
  hsKB9QRX1RQIrplbT0X6h3HDZylIxHJJXtNhmLP8WxJI0qK4fgopCeJP3uA9SSA75rLkiE/HalPR
  HHW1IS+yElbRI0Fp5hSdj1HIEbHcEdq6L5abZfLU/arxBjzoMhPF1h9AUhQ/A/H4g+oNd8GMzChM
  Q46VJZYbS02FLKiEpGhskknsPUndAn8h8PGP5BlLWUXjNc2lXhlSFMyjLjJU0UHkjiEsAJ0e40B3
  701Mctkq0wDFl3653tZWVJkTwyHEjQHD+JbbSR2J2QTsnvrQElRQFFFKbxT9Qb9046cxrzjqYftk
  u5IhFchsrDaFNOrKkgEe8C2Nb2PXtQIvwLY4zepGXTEXO52ufDTDSxLgvBK0pX5/NJStKm1glCDp
  STopGtVpvHOnFitWVOZZMk3G+5CpvykXG6OpccZb7+62lCUttjufupHqfmaxP4XeoV+w3qJAstpR
  Dch5JcoUKcmQ2VEILvAKQQRpQDq9b2O/cGv0KotV3qHl1uwvG3btOC3nlHyoUNocnpj6vuMtpHdS
  ifl6DZ9BVK8OnTy4Yna7pkmUBCssySSqZceJ2GApRUGgfmCok6+J134g0xX7FaH8hYyB+Ay7dI7B
  YYkODkppBJJCN9kk77kaJGgToCpKiCiiigKKKKD/2Q==
  ">.Substituting p = .1, n = 100, <img src="data:image/jpg;base64,/9j/4AAQSkZJRgABAQAAAQABAAD/2wBDAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0a
  HBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDL/2wBDAQkJCQwLDBgNDRgyIRwhMjIyMjIy
  MjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjL/wAARCAAmAiYDASIA
  AhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQA
  AAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3
  ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWm
  p6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEA
  AwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSEx
  BhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElK
  U1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3
  uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD1Kb4l
  eEba4+zz6zHDPx+6khkV+enBXNWm8b+HoiPP1D7MCQA1zBLEv5uoFebfGRFXx34NcABmlAJxyQJ4
  sfzNezTwRXUEkE8SSxOCGRwGDD0INABb3MN1Ak9vNHNC4yrxsGVh7EcGlubiG1t3nuJkihjG55HY
  Kqj1JPSvHvhnJLpfxQ8T+HrB2Oiwl3SPOViYMoGPT7zD/gPtW5pN8PiN4xvpJMP4d0WURwQfwXVx
  /wA9H/vKMcDp0NAHoltcRXVtFcQuHilQOjDowIyDUFxqthb6jbafLeQpeXJIhgL/ADvhSxIHXoDz
  U1zAbi1kgEskO9SoeI4ZfcHBwa8Pt9Mt9J/aSsra2MxjKGQmaVpGLNbyE5ZiSeaAPbry7gsLOS6u
  pVigiXc8jdFHqfapoZY5oUlidXjcBldSCGB6EEdRSsAwIIBB4INecm//AOEA8d2ukE7fD2tlmtkP
  S0uM/Mq+iNkHHYnigD0iikHSlzigAopM/X8qM/X8qAFoPSisvxFrtr4c0G71a7J8q3jLbR1Y9lHu
  TQBZur+0s5IIri4jjkuH8uFGODI3oB3/AKVZ9q4rwBYXGo2SeLdZxNq2pJvjz921gJyscY/hGME9
  ya7YcUAZmrjV2ES6XLZwISfPmuVZyi/7KggE/UiuX8AeI7++8D3niHX75ZY/tFxIsgjEaxwRnb0H
  b5WPJPXrW5441VdF8E6zfk4MVpJt5xliMD8cmuPu9Mms/wBnRrG0TMw0YMwQZzldz/zagC1FqHiL
  WPBN14uGrS6c/wBnkvLKxSJGiWJQSol3AsxYDJIK4zx0qVvG17q9j4Ws9KWO31PX4BcSSMu9bWJV
  BkYA/eOeFz9ai8S6hHb/AAkttP0srPd6pYxWFhFGQTKZECnHsFLEnoKqaJo0ehfFbS7CZsra+GI7
  e0Y8CR0fEm31OME/WgDYj1PUdA+IWmeH7jUp9Ss9UtJpUa5VPNhkj5PzIqgqR2I4I612c8qW9vJN
  K22ONS7tjOABkmuDso/+Ek+L8+sQnfpuh2JskmByr3LnLhT0O1Tg+hr0Dr2oAjtLu3vrWO5tZ45o
  JF3JJGwZWHqCKmrzbWr3/hXni+yu4sJ4f1qYxXUH8NvcHpKg7Bv4h04z1r0gEEAg0ALVa/vrXTrN
  7m8uI4IU6vI2B7D3J9O9SzzxW1tLPM4SKNC7u3RQBkk1554Lnfx7qtx4rv0zYW0zQ6Tav92PH3pS
  O7nPXt0FAHoqkEAjuKdSChhkYoAp6jq1hpYhN9eQ2/nSLHEJHALsxAAA6nkjpUt1cw2VpLc3DiOG
  JS7ueigdzXhnxI0eDSviT4SaKa5mluJ4mmkuZ2kLMs6AHngdegAHtXvJGR0zQA23niuYEmglSWKR
  QyOjBlYHoQR1FK+7B24zjjNec3V+Ph944s7QHZ4d112Cxfw2lzkZK+iNkZHTJJHSvQbu6isrOe6n
  bbFDG0jn0CjJ/QUAc14V1zWNX1rXra/+x+Rp06W0bWqMAz7dz8sf4cgdK60Vx3wztpI/BkN/ccXO
  qTS38p9TIxIP/fO2uxoAKqW1/aXdxcQW9xHLJbMEmCNnY3off2ri/HHiK7fWtM8HaNO0Goao2Z7l
  PvW8HO4r6MQDg9q7HS9Ks9F0+Gw0+BYbeIYVR1PqSe5PUk8mgC8elZN/4j0nTbn7Lc3qC5xu8iNW
  kkA9SqgkD3Ip/iPUZNI8M6nqMSb5LW1kmRcdSqkiuF+CiLceE7rVZnM2oXl7Kbm4Y5d9pwMn0749
  6AO803WtN1hHbT7yKcxnEiqfmQ+jKeR+IqaLULSW/msUuIzdQqHkhz8yqehx6e9eYfFO5PhLxFoH
  i2wxFO0xtrsKMCeLG7DeuAGxXY+LtGfV9GGp6Y/2fWLJDcWNyvUHGdjeqMOCP8KAOobpXFp4g1fW
  bLUtW0mS1g0+zeVIFnhLm7MYO5iQRsUkEAgE960fBPiqHxf4Zt9UjQRzH93PF/zzkHUfTuPY1xGr
  R6x4a0l/BOiy2t3calJILNQrebBBISXaUdAFyQGzz6UAdUfGL6jZ+H4tJiQX2twmdPOyVt4goLsw
  HUjIAHGTVzRtYvP+ElvvD+pSQzzwW8d1FcRR+XvRiQQVycEEevINZF74dXwzb6Hq1pf20EehWRtZ
  /tmRHJAQATlRkMCMjjnNM8AWupapqeqeMNVhMD6kEhs4CpUpbpnaSDyC2c/lQB346UUmeKM/X8qA
  FopM/X8qMigBaKKRiApJ6CgCjqes6do8aPf3kUHmHbGrH5nPoqjlj9BUema/perTSQWd4jzxjLws
  GSRR6lGAbHvivOPh5dHxj8QPEPiW6/epZsLSxVukKEnlR2JAB/GrvxgWXSNP0vxXYfu9R0y7VRIv
  BeN+Ch9QSBQB6W7qil2YBQMkk8D3rB/4TXw6XA/tWHyy2wT4byc+nmY2frXn/wATPEsmsad4a0HT
  pmjXX3jeZlPJiJUBfoSTn/d969QOj2DaN/Y5tYzp5h8jyMfLsxjH5UAXlYMoYEEHkEVm6l4h0rSZ
  0t7y8RLiQbkhUF5CPUIoLY98V5p8N/Fcuk+H/EekajK0p8OtIYi55MSlgF+gYYHsfatL4OQSaho2
  o+KL4+bqWqXb75m6iNeAoPYA7v09KAO903WNP1iF5bC7juFRtrhDyjejA8qfY07U9WsNGtDdajeQ
  WsA43yuFBPoPU+wrzXx1eHwd8SPD/iC3/dwaiTZ36L0lUEfMw7kBs59q27zwFB4u8QXep+KFmmto
  nMNhYiUqiRjq52nO5jk9RxigDf0bxn4d8QTmDS9YtbmYc+Wr4Y/QHBP4VukhVJJAA5r57+KfgOy8
  Ff2dr3h15rbdciMReYWKSYLKysef4SOprpvjH4g1Kz8C6VaBmhl1Ihbpl+U4CBmT2yT+lAHfv408
  PIWI1KNo0OGmjR3iUj1dQV/Wtq3uIbuBLi3mjmhkG5JI2DKw9QRwRVfTbC0sdItrC0jQWkUKxooA
  wVxjp715l4Pvz4Z+Let+DoiRpk+bq1h6iFygchfQYJ/ECgD1DUtRs9J06a/v7hLe1hG6SV+ijOP5
  kVyj/FfwQkZb/hIbY4UthVck49PlrL8e+JfCr65D4c8S3DCwWIXE8Sh8O5OI1ZkIIAGW9/l9KND8
  E/C7XALjSLS0u/LIbCXkrEYP8Sl+n1FAHf6e0E9qt5DbmH7WqzOGjCuSVABf/awFHPoB2plxq1hb
  6jbafLeQpeXOfJgLDe+AWOB16Kefapri3+0WskCyyQ712h4iAy+4ODg14jbabb6T+0lZWtsZTHsM
  mZpWkYs1vIT8zEnrQB7qOBXOap498K6Pdm1v9dsop1OGj8zcVPocZwa1tUsX1Gya0W5kgSQgSPGc
  OU7gHsT0z6E1XfS9E0/R3t5LKwg05E+dHjURAd92ePzoAs6Zq2n6zaC6068huoDxvhcMAfQ+h+tF
  eW/CzQLq08V6/q9hHLbeG7gslkrgjzhuBV1B52gbgD33UUAUvjSHPjTwcI2CuZDtZhkA+dFgkZGR
  7ZH1FelXOleI7yJoX8R29ujcM9np+yTHszyOB9cV5f8AEr+1/EfirQr7SvDety22mne7tZOm8+Yj
  YAPPRO+K9k0rUl1W089bW8tiG2mO7gaJwcA9D1HPUcdaAOaXwzp3gfwZrkmjxyfafsc00lzK+6WV
  wjEFm+vpgc1gfAaNF8BzOANzX0gJ+irivTLq2ju7Sa2mG6KaNo3HqCMH+deZfCOObw9da/4RvTtu
  bK68+PPHmRNwHHscD86APU68Uu/+TnbH/rgP/SeSvZbm5W1tpZ2SV1jUsVijLucdgo5J9hXitwNU
  k+Ntv4pXw3rn9lxARFzZNuOImTdt64y31xQB7hXkfx+/deHNHu0/1sV/8hHX/Vs381FeqWd2l7aR
  3CRzRq4yFniaNx9VYAj8a8x+KVs/inxT4a8J2vzOZWvLk/8APOIfLk/Ub6APTNOme4021mk+/JCj
  N9SoJrP8QeGNJ8TRQxarBLKkLF4/LnkiIJGD9wjP41rxRrFEsaDaqgBR6AdKydf8Mad4lihi1EXB
  WFiyeRcyQnJGOShGfxoAwv8AhVHg7/nwvP8AwY3P/wAXSr8KvB6sGFhd5ByP+Jjc/wDxdH/Cq/DH
  9zU//Brcf/F0q/CzwwrBgmp5ByP+Jpcf/F0AdpXk/wAfrmWHwTawoTsmvAHH+6jMP1Ar1iuL+KPh
  qXxN4Hu7a3TfdQEXEKjqxXqo+oyKAOl0WNYtEsI0ACrbxgY/3RV4jIrnPAerprfgnSrxTlxAsUo7
  q6fKwP0IrpKAMnVfDOia5Ikmq6TZ3rou1TcRB8D05qXTNB0rRYZItL062s45CC6QRhAccDgVo0UA
  Zdj4c0bTbp7qx0qztp3zukihVW568j1qXU9H03WIUi1Kwt7tEO5VmjDbT6jPSr9VNR0601Wzks72
  ETW743xknDY55xQBU02/0NJxo+m3NkJIIt/2W3df3aA4ztXpyR+daw6V5t4AsbaTx/431K1to4La
  K4h02FYkCqvlJ8/T1JU16QDgUAeafHONH+Hu5hyt5DtP1JBrsvCN1LfeDtFup8mWaxhd89clBmuJ
  +MXm6xbaL4UsvnvNSvA+0fwRp1c+wz+lej2FnHp+n21nCMRQRLEn0AxQBzHxSuZLX4aa5JFkMYVj
  OPRnVT+hNQfCNFT4Y6KVABaN2b3PmN/hXR+I9HTX/DmoaU7BRdQNGGP8LY+U/gcGuO+D13KnhKXQ
  7tTFfaTcvbzRHqmTuH4HJoA9EyKMiuX8V6x4g0u70mPQ9FGoxXE+y7fcR5Kcc8dOpOTkDGO9dODh
  SeuPSgDxb4uf8lJ8Ff8AXZP/AEojr2sV4f8AEIapr/jbw/qWm+HNbltNNdGldrF0LYlVztBwei98
  c17Jp2ox6lbGdILqABtpS6gaF89ejAZHPXpQB5p8fUUeDLC4H+sj1BAp+sbn/wBlFdqul2/irwtp
  66p9paOe1jeVIrmSLeWQZDbCMjnoa4z4swyeJdW8OeEbP5p7m4NzNj/lnGo27j7ct+XvXqNvDHb2
  8cES7Y41CKPQAYFAFTStJttF0+OxshKLaIBY1lneTYAMAAsSQABwOgq92paKAPGdGla8/aR1Uzcm
  3tnSPPYBI8Y/76avZq8n16yPhj406R4iYbdP1VTZzSdklIwM/XC16nPJIltK8MfmSqpKpnG444Ge
  2aAFnhiuYJIJkDxSKUdW6MDwRXA+HvBWs+CL27j0C7s7rSLl/M+yXxdHibp8rqGzx6j/ABO94N1b
  XdX0SS68Q6R/Zl4szosIz8yDo2Dk+o98Z71Bfa7r99G0GgeH50kbgXepkQRJ77OXb6YFAHnXxSTU
  fEnibw/4XeW2e5ll817a2UnyVPBZnbk/LuP3RgD3Fe1xxLHAkQ+6qhOfTpXKeEvA0Xh+6udW1C7b
  UtdvObi8kGAP9lB2H88enFdHqmoQaVpN1qFy4SG2haV2PQADNAHk3wVkca14v04Fhbpc7htOMHfI
  vB7cKK9XsNGsNMkmktbZUmnOZpmJaSQ/7THJP4muC+DGhz2Xh+71q8jMdxq85nVWGCsfJH5kk/jX
  plAGdeaJp+oXUVxe24uGiIaNJWLIrDowQ/Ln3xmtADApaKAILq3ju7aa3mBMUqNG4DEEqRg8jkcG
  uPHwo8HAD/QLz/wY3P8A8XXY3VtHd2s1tLu8uVGjbaxU4IwcEcjr1Fch/wAKq8Mf3NT/APBrcf8A
  xdACf8Kp8H/8+F5/4Mbn/wCLrotD0LT/AA7YfYdMikit95fbJM8hyevLkntXPf8ACq/DH9zU/wDw
  a3H/AMXXRaJoNl4esPsVgJhDvL/vp3lbJxn5nJPagDTqOb5oXA6lTj8qkNcx4W1jxDql9rEWt6IN
  Ohtp9lpICf3y8889egORxzjtQBwfwCQwWHiC3k/1sV4iuD1yEwf1Bre+Nrqvw2ud3e5gx/32D/Sn
  waDeeDfG9/rGnWct3o2qjddQW43S28oOd4TqynJzjJHpS+J9Lu/iDc6bpq2d1aaJbzi4vJ7qIxNN
  gcRojfNznkkACgDznUbKew8T/C1rkMqrbWsZ3eolyf0YV9DZ6Vx3j3wWfE+i262Dpb6lp8gmsnbh
  QRj5D6A4HPsKsp4o1E6Ztbw7qQ1jZj7N5P7rzP8ArrnZtz3znHagDxzT7Oe+1L4pzWwJQxXCjb6m
  YsP0Br0v4LOr/DOx2/wzTg/9/WP8iK0PAvgseGvDs9vftHcX9+7TXzj7rM38A9QAT+ZrM8LaZf8A
  w8uL/SpLO6vdCmmNxZ3FrGZXhyOY3Rfm7cEAigDnvjzE90PDNtFzLLdyKgHXJCgfqa9haRIoy8jK
  iKMszHAAHUk1wkug3Xi3x3p+uX9pLaaTpKZtYbgbZJ5ic7yvVVHHXk4HFYfxA8ReLLnVRpujeE76
  80yBv37TW77Lph2+UglB/wCPfTqAb/8AZo8ea9Z6rdxkeH9OcvZROMG8l/57Ef8APMY+Ud+T0PO1
  4v8ACVh4x0N9NvWeM7t8M6fejf1Hr7jvXm3/AAn/AMUlTC+BQoAwNtnNx9Bvr0XRJtQ0jwlaXGsp
  e3mpSjzJ0hiLuHc52hR90Dp6DFAFfQrPxZo2lw6ddTaRexWyCOO7Z5I32AYG5dpBIHfI/rXBeA7e
  bxN8YNb8StMtxbWYMK3McexJHK7BtGTgbQT1PbPWuv1m28V+MYW06C3/AOEe0qXie4mkV7qVD1VU
  UkJnvk5rpND8P2PhfQ49N0mBUiiUkBjzI/qx9TQBX1fwX4c1+eSfU9HtLi4dQpnKYkIHT5hzxXjX
  j7wJJ8Orm18S+Gb2eGETBCrPlomPQZ/iU9MHP416LoXjPxONQks/E3g+9txk7LqwjaaLr0I5P4jP
  0FWPE+g3nj0WumzQvY6HHMs9xJLgTXG3oqL1QepbB9B3oA6bQNROr+H9O1Jo/La7to5in90soOP1
  rya6/wCTnrL/AK4D/wBJpK9hcw6Zpx2Qv5FvHhYoIy7bQOAqjk/QV4zMNUf43W/ilfDeuf2XGojL
  mybcR5TJu29cZYe+O1AHtx+uK5LxR8PNI8Vxv9uuNRWRiSpW8cohPojEr+GKu67q2sL4ZOo+HdLa
  7vMqy2t2jQttzhvlODn0FRaP4rn1KxRrrw7rFle4w1vJbHGfaQ4XH1IoA8a0nxNrnwr8T6l4euJH
  1CzjUGOJiSBnBV1/u5BOR9KK9U0fwQLnxHqXibxBDBLe3oEcVr99LeIYwCf4n4GSOPT1ooA7faPQ
  UAYoooAWsDXvDY1G8ttVsJxZ6zaZEFzt3K6nrHIP4kP5jqKKKANi1857SFrgIsxQGQRklQ2OcE84
  qXYM5xzRRQBBfG5SylNksJuMYjExITPvjnFZWgeGotHkub6edrzVrxt13eSLgvjoqj+FB2UfqaKK
  AN4DAooooAKKKKACkPSiigDm4PD0uja/Jf6NLHFZXsoa+spAdhf/AJ6x4+63qOh9jzXRr1oooAdR
  RRQAVznjPxhZeCtFGp30FxNGZBGqQBSdxBIzkjjiiigDzf4H+N7G+F3ojwXP9qXl1PfySkL5Zzjv
  nOcYHSvZbh5EtpWhCtKqFkDnCk44yR2zRRQBiaL4Z+xahPrGpTi91m5UJJPtwkSdo4l/hQfmTya6
  GiigArm9W8OSLq41/RpY7bU1Ty51kB8q7jH8Lgcgjsw5HuKKKAOhUAjkc9TTsUUUAJsHt+VQ3hnj
  s5WtViM+0+WJSQm7tnHOKKKAMjQfDSaVc3OpXc5vNYvMfabtlxwOiIP4UHYfic1vgYFFFABRRRQB
  Q1jRrHXtLm07UYRNbTDDL0IPYg9iOxqvoVrqljbtZ6jeR3qw4WC5wVkdO3mDpuHqOvXAoooA1tvp
  QFxRRQAh9653WPD83iW9SDUZ0/sOFlf7HHnNy4wR5p/uA/wjr3PaiigDokQRqFUAKBgADAAp1FFA
  BRRRQAUUUUAFFFFABSbaKKADaO9G2iigAxSbfpRRQA6kIzRRQAbaTaPaiigBdo9BRtBoooAMUtFF
  ADdopQMUUUABGaTYPb8qKKAF2ijaKKKAFHFFFFAH/9k=
  ">.Accordingly, 1 - .000002656 - .00029513 = .9997.</i>
  Hence, since the milk of all cows in the herd is combined in the processing procedure with the result that non-kosher milk is diffused proportionately in kosher milk, ten percent of the milk in any milk container is non-kosher. Ten percent is much higher than the sixty to one ratio of kosher to non-kosher ingredients required for nullification of a non-kosher foodstuff. Indeed, since forbidden foodstuffs present in concentration greater than one in sixty, or 1.67 percent, are not subject to nullification, were the percentage of <i>treifot</i> determined to be as low as two percent the identical problem would exist.
  The problem was first presented in print by R. Gavriel Yehudah Illowitz in a contribution to a Torah journal published by the Karlin-Stolin community, <i>Kovez Bet Aharon ve-Yisra'el</i>, (no. 104), vol. 14, no. 2 (Tevet 5763) and republished anonymously in an Israeli journal published in Kiryat Sanz, <i>Ha-Be'er</i> (no. 21-22), vol. 4, no. 3-4 (Adar-Nisan 5763). A series of responses, each seeking to establish the permissibility of the milk in question, was published in <i>Kovez Bet Aharon ve-Yisra'el</i> (no. 106), vol. 18, no. 4 (Nisan-Iyar 5763) and (no. 108), vol. 18, no. 6 (Av-Elul 5763) as well as in <i>Ha-Be'er</i> (no. 21-22), vol. 4, no. 3-4 (Nisan 5763), (no. 23) vol. 5, no. 1 (Tishri 5764), and (no. 24) vol. 5, no. 2 (Adar 5764). A rejoinder by Rabbi Illowitz affirming his earlier negative view was published in <i>Kovez Bet Aharon ve-Yisra'el</i> (no. 107), vol. 18, no. 5 (Sivan-Tammuz 5763). An additional response in support of the negative view authored by R. Judah Moskowitz appeared in <i>Ha-Be'er</i> (no. 23), vol. 5, no. 1 (Tishri 5764). The authors of the articles expressing a permissive view are R. Zalman Nehemiah Goldberg, R. Asher Weiss, R. Levi Yitzchak Halperin, R. Menasheh Klein, R. Asher Ze'ev Schreiber, R. Gavriel Yudewitz and R. Yosef Meir Feldman. A similar permissive view is also expressed by R. Eliyahu Schlesinger, <i>Bet Hillel</i> (no. 15), vol. 4, no. 3 (Av 5763) and R. Dov Landau, <i>Be-Netiv ha-Halav</i>, I, 5963.
  II. Does Halakhah Take Cognizance of Minor Sets?
  The crucial issue in determining the <i>kashrut</i> status of commercially produced milk is whether the principle of <i>rov</i> serves to exclude any halakhic consideration of the existence of the minor set or whether the existence of the minor set is acknowledged but nevertheless disregarded as a matter of procedure. If the former is the case, the statistical probability of the presence of a minor set large enough to render nullification impossible need not be considered. If the latter is true, it follows that, when it is known that, statistically, significant representation of the minor set must occur, nullification is rendered impossible.
  A. The Sources
  1. Teshuvot Mishkenot Ya'akov
  In support of his argument that statistical probability can serve to render the principle of nullification nugatory Rabbi Illowitz cites a comment of <i>Teshuvot Mishkenot Ya'akov, Yoreh De'ah</i>, no. 16. <i>Mishkenot Ya'akov</i> formulates the rather novel thesis that, although by talmudic times a significant minority of animals were <i>treifot</i>, in earlier times, i.e., during the Temple period, internal <i>treifot</i> were so extremely rare as to be virtually nonexistent. In establishing the validity of that assertion <i>Mishkenot Ya'akov</i> cites a statement of the Mishnah, <i>Pesaḥim</i> 64a. The Mishnah records that, in addition to sprinkling the blood of each paschal offering on the altar, R. Judah maintains that, after the slaughter of all individual sacrifices was completed, the priest filled a container with blood collected from the floor of the Temple and poured it upon the altar. The Gemara, <i>Pesaḥim</i> 65a, explains that R. Judah was concerned that, due to the pressure of the large number of paschal sacrifices, a priest might have inadvertently failed to sprinkle the blood of one or more of those animals. Since the pooled blood on the ground came from each of the sacrificed animals, pouring from a utensil in which blood from all paschal sacrifices had been mingled served to assure that some portion of the blood of each sacrificial animal reached the altar. <i>Mishkenot Ya'akov</i> questions the efficacy of that procedure: It is forbidden to sprinkle the blood of a <i>treifah</i> on the altar. Contrary to the majority view, R. Judah himself maintains that nullification is possible only through admixture with an entirely different substance; thus, for R. Judah, blood cannot be nullified even by admixture with copious quantities of other blood. Accordingly, objects <i>Mishkenot Ya'akov</i>, if, in actuality, even one of the sacrificed animals was a <i>treifah</i>, the expedient of pouring a cup of commingled blood should not have been permitted because of the prohibition against sprinkling the forbidden blood of a <i>treifah</i>. Therefore, deduces <i>Mishkenot Ya'akov</i>, in antiquity, there must have been no cause to fear that even a minority of the animals were <i>treifot</i>.12<i class="footnote">See also <i>Teshuvot Mishkenot Ya’akov</i>, no. 17. <i>Mishkenot Ya’akov</i>’s contentions are rebutted by R. Ephraim Zalman Margulies, <i>Teshuvot Bet Efrayim</i>, no. 6.</i>
  Rabbi Illowitz notes that <i>Mishkenot Ya'akov</i> himself recognized that the blood of the paschal sacrifices was sprinkled on the altar before examination of the lungs. Accordingly, he argues, if the import of the principle "the majority of animals are kosher"is that the statistical incidence of the minority need not be a matter of concern, <i>Mishkenot Ya'akov</i>'s argument is entirely lacking in cogency: The statistical occurrence of <i>treifot</i> may have been no less than in later times, but, nevertheless, the principle serves to determine that each and every animal in question is a member of the major set. Accordingly, deduces Rabbi Illowitz, <i>Mishkenot Ya'akov</i> must have assumed as a matter of course that the statistical probability of the presence of a member of the minor set of <i>treifot</i> cannot be ignored. Only if that unstated premise is accepted, argues Rabbi Illowitz, does <i>Mishkenot Ya'akov</i>'s argument acquire cogency.
  In his contribution to the Tishri 5764 issue of <i>Ha-Be'er</i>, R. Asher Ze'ev Schreiber deftly refutes that argument. It is indeed the case that the sacrificial blood was sprinkled before the animal was incised. But the cup of blood to which the Gemara refers was collected only after the sacrifice of all the paschal offerings was complete. By that time examination of the lungs had certainly been carried out. According to R. Judah, had even a single animal been found to be a <i>treifah</i>, the expedient of sprinkling the commingled blood of all the sacrificed animals would not have been available. Accordingly, <i>Mishkenot Ya'akov</i>'s deduction that <i>treifot</i> were so rare that it was possible to slaughter so many animals without discovering a single <i>treifah</i> is not based upon a halakhic assumption that the statistical probability of the occurrence of a <i>treifah</i> cannot be ignored but upon actual empirical observation that there were no <i>treifot</i>.
  2. Shulḥan Arukh
  There is, however, another, more compelling source that does serve to support the thesis that application of the principle of <i>rov</i> entails ignoring completely the statistical likelihood of the presence of a member of the minor set. R. Levi Yitzchak Halperin cites a ruling of <i>Shulḥan Arukh</i> from which it is readily deduced that, in sanctioning the eating of cheese, the statistical probability that a portion of the milk too great to be nullified came from a <i>treifah</i> is to be disregarded. <i>Shulḥan Arukh, Yoreh De'ah</i> 81:2, rules that cheese made from milk of an animal later found to be a <i>treifah</i> is non-kosher. <i>Shulḥan Arukh</i> then adds the comment that, if the cheese was made from milk collected from the entire herd, the milk is permissible provided that the herd consisted of at least sixty kosher cows. The grounds for that ruling are obvious: the non-kosher milk used in production of the cheese in question is nullified in sixty parts of kosher milk. However, nullification in an admixture containing a ratio of sixty to one is cogent only if there is a halakhic presumption that each of the sixty live cows is a non-<i>treifah</i>. Since examination for internal <i>treifot</i> is not possible while the cow is yet alive, the rule formulated by <i>Shulḥan Arukh</i> flows from the halakhic presumption that, unless shown to be otherwise, each animal for which a determination is sought is a member of the <i>rov</i>, or major set of animals that are kosher. Yet the requirement for examination of lungs of slaughtered animals for the absence of adhesions because of the presence of a <i>mi'ut ha-mazui</i>, i.e., a significant minority of animals that are <i>treifot</i>, reflects recognition that, statistically, there must be some non-kosher animals in a herd of sixty dairy cows. Since milk collected from one of those cows is known to be non-kosher, the cheese made from the combined milk of all sixty-one cows is kosher only if all of the remaining cows are non-<i>treifot</i>. However, if even a single additional cow in the herd is a <i>treifah</i>, not to speak of a <i>mi'ut ha-mazui</i> of <i>treifot</i>, the milk of the <i>treifah</i> could not have been nullified by virtue of admixture with the requisite quantity of kosher milk and hence the cheese made from such milk should be forbidden. <i>Shulḥan Arukh</i>'s ruling, then, clearly reflects the notion that the principle declaring that "the majority of animals are kosher" is applied to each animal individually and statistical exceptions are to be disregarded. If the principle is applied to each animal individually, <i>Shulḥan Arukh</i>'s ruling extends equally to a herd of sixty, six hundred or six thousand cows and even the extremely high statistical probability of the presence of a significant quantity of milk derived from non-kosher cows is to be ignored.
  3. Tevu'at Shor
  The same principle is also reflected in a later source, <i>Tevu'at Shor</i> 1:26. <i>Tevu'at Shor's</i> discussion focuses upon the hypothetical case of a slaughterer who, after a period of service, was found to be ignorant of the halakhic provisions governing examination of lungs. <i>Tevu'at Shor</i> compares the case to a situation in which the lung is lost before examination and rules that any already slaughtered meat and the utensils in which it was cooked are permissible.13<i class="footnote">R. Ephraim Zalman Margulies, <i>Rosh Efrayim</i> 39:171, is in basic agreement with the ruling of <i>Tevu’at Shor</i> but finds reason to distinguish that situation from the case of a slaughterer suspected of purposely ignoring <i>treifot</i>. Cf., the note appended to <i>Teshuvot Ohev Mishpat</i>, <i>Yoreh De’ah</i>, no. 1. See, however, <i>Teshuvot Ge’onei Batra’i,</i> no. 14, <i>Teshuvot Ẓemaḥ Ẓedek</i>, no. 73, and <i>Teshuvot Avodat ha-Gershuni</i>, no. 54, who adopt a contrary view. Rema, <i>Yoreh De’ah</i> 1:1, also appears to rule that meat slaughtered by such an individual must be retrospectively ruled to be non-kosher. R. Raphael Joseph Chazan, <i>Teshuvot Ḥikrei Lev, Yoreh De’ah</i>, no. 12, expresses astonishment with regard to that position since the situation is comparable to that in which a lung has been lost. R. Akiva Eger, <i>Ḥiddushei R. Akiva Eger, Yoreh De’ah</i> 1:1, similarly objects that the case is similar to that of a lost lung. Rabbi Illowitz’ thesis does not serve to explain the position of the authorities who forbid the meat since they do so even if only the meat of a single animal is subject to doubt.</i> <i>Tevu'at Shor</i> adds the further comment:
  There is no reason to say [that] since, in the present situation, [the slaughterer] examined a superfluity of animals, it is almost certain that there were [some] <i>treifot</i> among them in which case [this situation] is not comparable to [that] of a lost lung and therefore the utensils should be deemed non-kosher and similarly meat of many animals should not be given to a single person to eat, [nevertheless,] it seems to me that one need not be concerned, just as we eat even from each of many animals in a herd and similarly we cook milk even from each of the animals in pots and we are not concerned.
  The thrust of that comment is that, in relying upon the principle of <i>rov</i>, we are not at all concerned with the fact that, statistically, at least some of the animals in a large herd must be <i>treifot</i>. The fact that, empirically, it is "close to certain" that some of those animals are not kosher is halakhically irrelevant.14<i class="footnote">In his contribution to <i>Ha-Be’er</i>, Tishri 5764, Rabbi Judah Moskowitz attempts to interpret <i>Tevu’at Shor</i>’s comments in a manner that, in this writer’s opinion, is not compatible with the plain meaning of the comment cited.</i>
  4. The Status of a Minor Set with Regard to Milk
  Rabbi Halperin endeavors to show that, with regard to resolution of doubt, the status of milk is somewhat different from that of meat. <i>Hagahot Asheri, Beizah</i> 4:15, rules that, in areas is which the majority of animals are found to be non-kosher, it is forbidden to slaughter on <i>Yom Tov</i>. Slaughtering, like cooking, is permitted only for the purpose of preparing meat for consumption on <i>Yom Tov</i>. When the majority of animals are kosher, the presumption with regard to each act of slaughter is that it is undertaken for the purpose of preparing food to be eaten on <i>Yom Tov</i>. But in cases in which the majority of animals will not be fit for consumption by Jews, the act of slaughter cannot be regarded as designed to make meat available for eating. Nevertheless, no authority rules that it is forbidden to drink milk in a locale in which a majority of animals are found to be non-kosher. Rabbi Halperin suggests that milk has a <i>ḥezkat hetter</i>, i.e., milk is presumptively permissible unless there is evidence to the contrary, whereas the meat of a living animal is certainly forbidden while the animal is yet alive.15<i class="footnote"><i>Teshuvot Rashi</i> (New York, 5703), nos. 59-60, quotes the opinions of his “teachers” who ruled that the meat of an animal whose lungs have been lost is not permissible. Rashi comments in astonishment that, since “yesterday we drank the milk,” the meat cannot today be forbidden. Rashi’s comments are briefly cited by <i>Or Zaru’a</i>, I sec. 411. See also Rashba, <i>Ḥullin</i> 9a. The position of the teachers of Rashi does not however convincingly serve to establish that milk has a different status with regard to a known <i>treifah</i>; it can only establish that there is a different standard with regard to an unknown presence of a possible member of a <i>mi’ut ha-maẓui</i>. See also R. Abraham Samuel Benjamin Sofer, <i>Teshuvot Ketav Sofer, Yoreh De’ah</i>, no. 44.</i> Rabbi Halperin argues that the <i>ḥezkat hetter</i> is sufficient to overcome the statistical probability of the presence of a <i>treifah</i>.
  In his rejoinder published in <i>Kovez Bet Aharon ve-Yisra'el</i>, vol. 18, no. 4, Rabbi Illowitz quite cogently points out that a <i>ḥezkat hetter</i> is of no avail when it is known with certainty that the majority of animals are <i>treifot</i> and that, moreover, since the normative position is that a <i>ḥazakah</i> with regard to a particular prohibition, e.g., the limbs or organs of a living animal, does not create a <i>ḥazakah</i> with regard to any other prohibition, the meat should similarly be prohibited.
  The apparent paradox reflected in permitting the milk of animals that may not be slaughtered on <i>Yom Tov</i> is readily resolvable. The ruling of <i>Hagahot Asheri</i> regarding slaughter of animals on <i>Yom Tov</i> is limited by <i>Sha'agat Aryeh</i>, no. 64, to cases in which the majority of animals are not actually <i>treifot</i> but are afflicted with anomalies which render the animal <i>treifah</i> because of our inability to resolve matters of halakhic doubt or because of stringencies imposed to prevent actual transgression. Thus, in the situation described by <i>Hagahot Asheri</i>, the milk is permitted because there is no actual majority of non-kosher animals and also because of a "double doubt," <i>viz.</i>, perhaps the animal from which the milk is derived has no anomaly and, if the animal does suffer from an anomaly, perhaps the anomaly does not render it an actual <i>treifah</i>. However, with regard to slaughtering the animal on <i>Yom Tov</i>, since the meat will not actually be eaten, its slaughter is forbidden.16<i class="footnote">See also <i>Teshuvot R. Akiva Eger</i>, no. 5.</i>
  5. Hatam Sofer
  The halakhic stance with regard to ignoring the presence of a minor set is stated in even more explicit terms by R. Moses Sofer, <i>Teshuvot Hatam Sofer, Even ha-Ezer</i>, I, no. 5. The rules of evidence provide that a single witness has no credibility to testify counter to a <i>ḥazakah</i>. Thus, for example, the testimony of a single witness to the effect that a husband has died cannot, in and of itself, establish the wife's capacity to contract a second marriage. A <i>ḥazakah</i> of this nature is the presumption that a halakhic status has not changed unless there is evidence to that effect; the testimony of a single witness does not constitute sufficiently strong countervailing evidence to overcome the presumption established by a <i>ḥazakah</i>.
  There is another procedural rule to the effect that a <i>ḥazakah</i> can be overcome on the basis of a countervailing <i>rov</i>. If so, argue R. Ezekiel Landau, <i>Teshuvot Noda bi-Yehudah, Mahadura Kamma, Even ha-Ezer</i>, no. 69, and <i>Pnei Yehoshu'a, Kiddushin</i> 63b, since <i>rov</i> is assigned greater weight than <i>ḥazakah</i> and <i>ḥazakah</i> is accorded greater weight than a single witness, it should follow that <i>rov</i> should be given greater weight than a single witness. Or, to express the matter somewhat differently, just as a single witness has no credibility in the presence of a <i>ḥazakah</i>, he should, <i>a fortiori</i>, not have credibility to testify contrary to a <i>rov</i>. Yet a single witness does have standing to testify that he recognizes one piece of meat among nine other pieces of meat as being kosher even when it is known that nine other pieces of meat are definitely non-kosher. Similarly, a slaughterer has credibility to identify one animal as kosher despite the fact that the majority of animals in the slaughterhouse are non-kosher. In effect, a single witness can be heard to testify that an unknown entity is not to be assigned to the major set.
  <i>Hatam Sofer</i> resolves the problem by distinguishing between application of the principle of <i>rov</i> in the sense of a majority in a situation in which there is a known antithetical minority and application of the principle of <i>rov</i> in a situation in which there is no identifiable minority. The testimony of a single witness in contradiction to application of the principle of <i>rov</i>, he asserts, is accepted only when the presence of a minority is established and known with certainty. In such a situation the witness is simply identifying one entity as a member of the minor set whose existence has already been established. However, the type of <i>rov</i> that prevails over a <i>ḥazakah</i> is a <i>rov</i> with no known exceptions, e.g., the majority of animals are kosher. That <i>rov</i>, properly understood, establishes that the majority of unexamined animals are kosher. Animals known to be non-kosher are not part of the quandary and hence are excluded from the <i>rov</i>. That type of <i>rov</i> carries greater weight than a <i>ḥazakah</i> and a single witness would have no credibility to contradict a <i>rov</i> of that nature because of the complete absence of a known minor set. Hence, concludes <i>Hatam Sofer</i>, "it is possible that, at present, <i>there is no single treifah in the world other than those known to us</i>." <i>Hatam Sofer</i> quite obviously accepts the proposition that, with regard to unknown entities, Halakhah completely disregards the statistical probability of the existence of a minor set.
  B. The Theoretical Basis
  The heretofore cited sources are culled from statements of latter-day authorities and demonstrate that, in applying the principle of <i>rov</i>, those authorities treat the minor set as devoid of members. However, an unequivocal logical reason supporting the position that the probability of existence of a minor set is to be entirely ignored for purposes of halakhic determinations remains to be demonstrated.
  On first analysis it would seem that such a conclusion follows only on the basis of recognition that application of two rather different, but complementary, principles of <i>rov</i> yield that result. As earlier indicated, the principle that "the majority of animals are kosher" renders it permissible to partake of meat even though it is impossible to examine the animal for perforation of the trachea and the esophagus. The Torah, in effect, bids us to rely upon the presumption that the animal before us is one of the majority of animals that is not a <i>treifah</i>. That <i>rov</i> applies to all the animals of the world or, at the minimum, to all the animals of the locale from which the slaughtered animal has been culled. Of course, on the basis of past observation, one can take notice of the fact that <i>treifot</i> have existed in the past and hence unidentified <i>treifot</i> are quite likely to exist at the present. That <i>rov</i> is termed a "<i>rubba de-leta kamman</i> – a <i>rov</i> that is not present before us," i.e., a <i>rov</i> unaccompanied by a parallel antithetical minor set containing identifiable members.
  There is, however, another entirely distinguishable principle of <i>rov</i> that is applied when the question is whether a given entity became detached from a major set composed of identifiable members or from a corresponding minor set also composed of identifiable members. Since all members of those two sets were identifiable at least at one time, that <i>rov</i> is known as a "<i>rubba de-ita kamman</i>—a <i>rov</i> that is present before us." The latter principle of <i>rov</i> is derived directly from the biblical phrase "according to the majority to determine" (Exodus 23:2). The paradigmatic example is meat found in the street in a city having nine kosher butcher stores and one non-kosher establishment. The meat that has been found has clearly become separated from its original source. In that situation there is a major set comprised of nine members and a minor set having a single member. Application of the principle of <i>rov</i> yields the determination that the meat in question has become separated from a member of the major class and hence the meat is ruled to be kosher. The talmudic formulation of that principle is "<i>Kol de-parish me-rubba parish</i>—All that are separated are separated from the majority."
  Once it is known that there exists a <i>rubba de-leta kamman</i> establishing that the majority of all animals are not <i>treifot</i>, each and every animal presented for a determination of its status must be regarded as non-<i>treifah</i> despite the statistical probability that some animals are <i>treifot</i>. The universe does indeed include both <i>treifot</i> and non-<i>treifot</i>. But any animal whose status is subject to question must be judged to have become detached from the major set rather than from the minor set. Thus, the empirical existence of the minor set is of no halakhic import, statistical reality notwithstanding.17<i class="footnote">See the discussion of R. Shimon Shkop, <i>Sha’arei Yosher, sha’ar</i> 3, chap. 1.</i>
  There remains, however, a further problem. The principle of <i>kol deparish</i> does not apply to the case of a person living in a city having nine kosher meat establishments and one non-kosher store who has purchased meat in a store but cannot remember whether the establishment he patronized sold kosher or non-kosher meat. The principle of <i>rov</i> applies only when doubt arises after the meat in question has become "separated" or detached from its point of origin; however, if the doubt arises while the subject of doubt is <i>kavu'a</i>, i.e., at its point of origin, rather than subsequent to separation, the principle of <i>rov</i> does not apply. How, then, can <i>kol deparish</i> be applied in situations in which a purchaser acquires an animal by visiting a herd and selecting the animal or animals he desires? <i>Teshuvot R. Akiva Eger</i>, <i>Mahadura Tinyana</i>, no. 103, sec. 12, resolves that problem by formulating a hypothesis to the effect that a situation of <i>kavu'a</i> constitutes an exception to the principle of <i>rov</i> only if the existence of a non-kosher animal at the point of origin is known with certainty; otherwise, the principle of <i>rov</i> continues to govern.
  However, earlier, in section 11 of the same responsum, R. Akiva Eger raises an even more perplexing issue. The principle of <i>kol de-parish</i>, he asserts, can be applied to a calf born after the mother has been acquired by the purchaser. The status of the calf as a <i>treifah</i> or as a non-<i>treifah</i> is questionable from the very moment that the calf comes into existence; that question arises at the calf's point of origin, i.e., at the moment of birth or even earlier, not upon separation from other members of a set.
  That observation should necessarily lead to one of two conclusions:
  1) The principle of <i>kol de-parish</i> is indeed not applicable to large herds of dairy cows calved and raised by a single dairyman and hence the statistical incidence of <i>treifot</i> must be taken into consideration; or 2) even without the principle of <i>kol de-parish</i>, the principle of <i>rubba de-leta kamman</i> serves, in and of itself, to establish that each and every animal of doubtful status is adjudicated to be a member of the majority and the statistical probability of the presence of at least some members of the minority must be ignored.
  III. Conclusion
  The application of the principle of <i>rov</i> is a halakhic principle rather than an empirical determination. Rabbi Menasheh Klein, in his contribution to the Nisan 5768 issue of <i>Ha-Be'er</i>, probably overstates the case in declaring, "But, in truth, the law of <i>rov</i> is a positive commandment"18<i class="footnote">Rabbi Klein’s assertion appeared earlier in his <i>Mishneh Halakhot, Mahadura Tinyana</i>, VI, no. 25. In that responsum Rabbi Klein adds that, according to some authorities, a person who refuses to eat a foodstuff in which a forbidden food has become nullified is in the category of “one who does not believe in the Torah of God; rather, it is a <i>miẓvah</i> to eat the permitted together with the prohibited that has been mingled in it for such is the will of the Creator, blessed be He, in order to extract the holy sparks. See <i>Bnei Yissaskhar</i>, II, 95a.” Cf. the conflicting authorities cited by <i>Pitḥei Teshuvah, Yoreh De’ah</i> 116:10. Nevertheless, Rabbi Klein asserts that there are grounds to refrain from eating nullified food for fear of <i>timtum ha-lev</i>, i.e., the spiritual and/or physical harm occasioned by consumption of such foods. Rabbi Klein repeats those statements in <i>Mishneh Halakhot, Mahadura Tinyana</i>, VI, no. 126 as well as in VIII, no. 104. For a further discussion of <i>timtum ha-lev</i> with regard to foods rendered permissible by reason of nullification see <i>Mishneh Halakhot</i>, VII, no. 104 and R. Yosef Yitzchak Lerner, <i>Shemirat ha-Guf ve-ha-Nefesh</i> (New York, 5748), introduction, chap. 13, Cf., <i>Mishneh Halakhot</i>, V, no. 101.Rabbi Klein implies that <i>timtum ha-lev</i> is associated not only with prohibited foodstuffs that have been nullified but also with food products sanctioned by virtue of the principle of <i>kol de-parish</i>. Cf., the anecdote concerning the <i>Ḥafeẓ Ḥayyim</i> cited in chapter seven of this volume, pp. 243-244. R. Moshe Sternbuch, <i>Teshuvot ve-Hanhagot</i>, IV, no. 184, suggests that <i>timtum ha-lev</i> may result from consuming the meat of animals or fowl produced by genetic engineering employing genes of non-kosher species.See, however, <i>Me-Shulḥan Gevoha, Va-Yikra</i>, p. 91, who reports that R. Yitzchak Ze’ev Soloveitchik asserted that forbidden foods do not cause <i>timtum ha-lev</i> unless consumption of the foodstuff is accompanied by an actual halakhic infraction.For a further discussion of <i>timtum ha-lev</i> in the absence of halakhic prohibition see this writer’s <i>Bioethical Dilemmas</i>, II (Southfield, MI, 2006), 160, note 32. For a review of sources regarding <i>timtum ha-lev</i> in circumstances of <i>force majeure</i> see R. Gedaliah Oberlander’s discussion in <i>Or Yisra’el</i> (no. 45), vol. 12, no. 1 (Tishri 5767).</i> but he is entirely correct in stating that with regard to any particular animal "we are commanded" to rule in accordance with <i>rov</i>. Those comments echo the declaration of R. Moses Sofer, <i>Teshuvot Hatam Sofer, Oraḥ Hayyim</i>, no. 83: "For He who commanded and prohibited the <i>treifah</i> commanded us to rely upon <i>rov</i>."
  Writing with regard to an entirely different subject, R. Moshe Feinstein, <i>Dibberot Mosheh</i>, <i>Bava Kamma</i>, addenda, responsum no. 1, sec. 12,19<i class="footnote">This responsum is also published in R. Ya’akov Breisch’s <i>Teshuvot Ḥelkat Ya’akov</i>, III, no. 52, sec. 13.</i> declares unequivocally that, in applying the principle of <i>kol de-parish</i>, each entity is to be regarded as a member of the major set "and there is not even one" that is a member of the minor set with the result that "<i>sevarot bnei adam</i>," i.e., presumptions of the human intellect, are to be totally ignored. Rabbi Feinstein concludes with the sharp statement: "He who does not wish to rely upon … the law of <i>parish me-rubba</i> because of reasons of human intellect is as if he disputes the Torah, Heaven forfend."
  
  Chapter 10 Formula Fed Veal
  
  <i>Had Boaz known that the Holy One, blessed be He, would write of him "and he extended parched corn to her" (Ruth 2:14), he would have given her fattened calves.</i>
  RUTH RABBAH 5:6
  I. Humane Treatment
  Animal rights groups have long protested the condition under which animals and poultry are raised for commercial markets. Methods employed in raising calves for veal have also been an ongoing focus of attention. Animal rights groups charge that the calves are raised in inhumane conditions. Not surprisingly, those contentions have been denied by the meat industry. Nevertheless, the most successful animal rights boycott in the United States began more than twenty years ago with publicization of photographs of formula-fed veal calves tethered in tiny crates in which they were virtually immobilized. As a result, consumption of veal plummeted from an average of four pounds a year per person to approximately half a pound per year. A recent feature article in the Dining section of the <i>New York Times</i>1<i class="footnote"><i>New York Times</i>, April 18, 2007, p. F1.</i> reports the growth of a yet nascent movement within the industry to raise pastured calves that are suckled and later fed grass and grain. Meat derived from such calves is not only "certified humane" but is reportedly more tender and more flavorful than formula-fed veal.
  The meat of calves raised on grass or grain is rosy or pink in color. Formula-fed veal, in contrast, is prized for its white color which is the result of a lack of iron. R. Moshe Feinstein, <i>Iggerot Mosheh</i>, <i>Even ha-Ezer</i>, IV, no. 92, addresses the propriety of inhumane treatment of animals in order to achieve that purpose.
  Although wantonly causing pain to animals is forbidden, the prohibition against causing pain to animals is generally regarded as inapplicable in situations in which there is human benefit, at least when the benefit is substantial in nature.2<i class="footnote">For a fuller discussion see this writer’s “Animal Experimentation,” <i>Contemporary Halakhic Problems</i>, III (New York, 1989), 217-229.</i> Nevertheless, asserts <i>Iggerot Mosheh</i>, infliction of pain upon animals can be sanctioned only for a recognized and legitimate human benefit. Even when normatively permitted, avoidance of causing pain to animals is regarded as an act of piety,3<i class="footnote">See <i>ibid</i>., pp. 229-231.</i> at least in most circumstances.4<i class="footnote">See <i>ibid</i>., pp. 231-235.</i>
  Thus, although giving vent to rage or anger has a calming effect, Jewish law does not recognize that need as legitimizing either destruction of property, even one's own, or infliction of pain upon animals.5<i class="footnote">See <i>Shabbat</i> 105b.</i> <i>Iggerot Mosheh</i> asserts that such an act, when not performed to advance a legitimate human goal, is forbidden even if the act would redound to a person's financial advantage. Trapping an animal in order to sell its pelt is permitted; however, an act of wanton cruelty that is not designed to yield a human benefit is forbidden even if one receives remuneration for inflicting such pain upon the animal. The distinction is quite simple. Fur is used to satisfy a legitimate human need for clothing and hence money may be accepted in order to satisfy another person's need for the pelt. Wanton infliction of pain does not satisfy a legitimate human need and does not acquire legitimacy simply because one is paid for performing such an act.6<i class="footnote">An analogous situation not discussed by <i>Iggerot Mosheh</i> or other authorities is that of a student required to perform a laboratory experiment involving infliction of pain upon an animal solely in order to satisfy course requirements. It may well be the case that a student intent upon a career in the biological sciences may justify such conduct on the grounds that the knowledge and/or skills derived from performance of the experiment is causally related to human benefit that will result from his or her acquisition of such knowledge and/or skill. However, the situation of a liberal arts student engaged simply in fulfilling a science requirement necessary to obtain a baccalaureate degree is comparable to that of a person who engages in an inhumane act in order to earn a fee.</i> Indeed, that conclusion is entirely cogent: otherwise forbidden acts of cruelty could be rendered permissible simply on the basis of a pact between two persons who agree to pay one another for engaging in such conduct.
  <i>Iggerot Mosheh</i> concedes that formula-feeding might be sanctioned if the purpose were to make the animals gain weight or to make the meat more tasty; both purposes reflect normal and legitimate human needs. <i>Iggerot Mosheh</i> asserts that white veal is neither tastier nor more healthful than other veal; any claim of that nature he declares to be prohibited as fraud. <i>Iggerot Mosheh</i> does not recognize any enhanced aesthetic appeal that may be associated with white veal as satisfying a legitimate human need and hence he rules that raising calves in an inhumane manner is forbidden.
  <i>Iggerot Mosheh</i>'s halakhic points are certainly well-considered. However, he seems to have been provided with less than complete information. Some consumers do indeed prefer formula-fed veal because of its whiteness. However, feeding calves formula rather than grass or grain also significantly reduces the cost of producing veal. Moreover, there is no gainsaying the fact that the taste is entirely different. The taste of grass- or grain-fed veal is similar to mild beef; formula-fed veal has a much blander taste. Some consumers have a marked preference for one over the other. <i>De gustibus non disputandum est</i>. Whether the economic and/or gastronomical benefit is sufficient to warrant the incremental discomfort caused the calves is another matter.7<i class="footnote">See <i>Contemporary Halakhic Problems</i>, III, 225-229.</i>
  II. Kashrut Problems
  <i>Iggerot Mosheh</i> does, however, report that formula-fed calves have an inordinately high incidence of lumbar adhesions that render animals non-kosher. Ordinarily, there is no obligation to excise the internal organs, other than the lungs, of a slaughtered animal because disqualifying abnormalities are relatively rare. However, asserts <i>Iggerot Mosheh</i>, since it has been demonstrated that the conditions under which formula-fed calves are raised result in an extremely high rate of lung anomalies, it may well be the case that the diet and/or the living conditions to which those animals are subjected causes abnormalities in other organs as well. Accordingly, <i>Iggerot Mosheh</i> rules that the meat of formula-fed calves may not be eaten unless the internal organs are properly examined and that pious individuals should not rely even upon such examinations. Oral reports received subsequent to publication of that responsum indicate that such examinations have been carried out upon large numbers of formula-fed calves. Those examinations have not revealed a disproportionate rate of anomalies in organs other than the lung and serve to demonstrate that the concern expressed by <i>Iggerot Mosheh</i> is without empirical basis.
  More recently, questions have been raised with regard to the permissibility of eating formula-fed veal because of the ingredients of the formula. In addition to other ingredients, the formula contains significant quantities of animal fat and milk combined in powder form. The powdered formula is mixed with hot water poured into a large vat and stirred by means of an electrical implement. Rema, <i>Yoreh De'ah</i> 60:1, rules that the meat of an animal that "all its life8<i class="footnote">The phrase “all its life” is understood quite literally by <i>Teshuvot Pnei Yehoshu’a</i>, I, <i>Yoreh De’ah</i>, no. 4. However, <i>Shakh</i>, <i>Yoreh De’ah</i> 60:5, understands the phrase as connoting only that the animal’s primary development took place during the period that it was fed forbidden foodstuffs. That is also the position of <i>Teshuvot Ma’aset Binyamin</i>, addenda, <i>Ḥiddushei Dinim</i>, sec. 3, and of R. Shalom Mordecai Schwadron, <i>Da’at Torah</i>, <i>Yoreh De’ah</i> 60:5. See also R. Shalom Toibes, <i>Teshuvot She’ilat Shalom</i>, <i>Mahadura Tinyana</i>, no. 153. <i>Shakh</i> further comments that the meat of the animal is forbidden if the animal “ate forbidden foods each morning all its life” even though otherwise it ate permitted foods.</i> has been fed only forbidden foods" may not be eaten.9<i class="footnote"><i>Knesset ha-Gedolah</i>, followed by <i>Teshuvot Ne’ot Desha</i>, no. 37, and <i>Ikkarei ha-Dat</i>, no. 8, sec. 6, rule that the meat of the animal is permissible if it is fed permissible food for a period of twenty-four hours prior to slaughter. That view is disputed by <i>Da’at Torah</i>, <i>Yoreh De’ah</i> 60:5, and is impliedly rejected by <i>Pri Megadim</i>, <i>Yoreh De’ah</i>, <i>Siftei Da’at</i> 60:5. Cf., <i>Levushei Serad</i>, IV, sec. 81.</i> Calves fed formula containing a mixture of prohibited meat and milk, it has been contended, are animals raised solely upon non-kosher food and hence, according to Rema, their meat should be prohibited. In addition, the problem is further complicated by the fact that the cooking of milk and meat together does not simply yield a foodstuff that is forbidden for consumption but results in a product which is <i>assur be-hana'ah</i>, i.e., from which no benefit may be derived.
  Rabbi Aaron Teitelbaum, the <i>Nierbarter Rav</i>, who is responsible for <i>kashrut</i> supervision of major purveyors of meat, has authored a permissive responsum dealing with those issues. Although that responsum is as yet unpublished it has been circulated among rabbinic scholars as has a rebuttal composed by R. Ya'akov Yechezkel Sofer of Toronto, Canada.
  Rema, relying upon earlier authorities, limits the prohibition against eating the meat of animals that have consumed forbidden foods solely to those animals that have been fed forbidden foods exclusively because only in such cases is the animal regarded as a "<i>yozei</i>" or "product" of the forbidden food. However, if the animal consumes both permitted and non-permitted foods, another principle, <i>viz</i>., <i>zeh va-zeh gorem</i>, becomes operative. The principle of <i>zeh va-zeh gorem</i> establishes that the product of two distinct causes, one permissible and the other non-permissible, is not prohibited since it is not the <i>yozei</i> solely of the product of a forbidden entity. In the case of formula-fed calves, the animal's development is attributable to two separate causes, one permissible and one non-permissible, and hence its meat is permitted.10<i class="footnote">The majority of authorities, including <i>Pri Megadim</i>, <i>Yoreh De’ah</i>, <i>Siftei Da’at</i> 60:5 and <i>Da’at Torah</i>, <i>Yoreh De’ah</i> 60:5, rule that, although the meat is permitted, nevertheless, it is forbidden to cause a situation of <i>zeh va-zeh gorem</i> by feeding the animal forbidden foods. Cf., however, <i>Shakh</i>, <i>Yoreh De’ah</i> 60:5, who asserts that, according to <i>Mordekhai</i>, such a practice is entirely permissible. <i>Teshuvot Bet Ya’akov</i>, no. 1, rules that an animal raised on permitted foodstuffs that has been fed a quantity of only forbidden food that can sustain the animal for a twenty-four hour period should not be slaughtered during that twenty-four hour period.</i>
  Rambam, in his <i>Commentary on the Mishnah</i>, <i>Temurah</i> 6:5 and in <i>Hilkhot Ma'akhalot Assurot</i> 3:14, clearly rejects the notion that an animal is the "<i>yozei</i>" or "product" of the food it consumes in ruling that animals raised even on foods from which no benefit may be derived are permitted. Rambam rules that the meat of the animal is permitted because the forbidden foodstuff becomes "changed," i.e., it metamorphoses in the process of digestion,11<i class="footnote">Cf., <i>Da’at Torah</i>, <i>Yoreh De’ah</i> 60:5, who suggests that Rambam’s notion of “change” may be limited to other foodstuffs that are “changed” into flesh or fat but forbidden meat that becomes part of the body of the animal that consumes such meat does not undergo a similar metamorphosis and remains meat.</i> and hence, for purposes of Halakhah, the forbidden substance ceases to exist before it is turned into flesh.12<i class="footnote">See also R. Moshe Feinstein, <i>Iggerot Mosheh</i>, <i>Oraḥ Ḥayyim</i>, I, no. 174, who adopts this view.</i> Rema, however, apparently maintains that nutrients present in the forbidden food are absorbed and metabolized before the food becomes "changed."
  <i>Shakh</i>, <i>Yoreh De'ah</i> 60:5, disagrees with Rema with regard to animals that have been raised entirely on food forbidden for a Jew to eat but agrees that there is a prohibition if the animal is raised on <i>issurei hana'ah</i>, i.e., food from which it is forbidden to derive any benefit whatsoever, even if the animal consumes both permitted and non-permitted foodstuffs. <i>Shakh</i> also notes that, even in such circumstances, <i>Shulḥan Arukh</i>, <i>Yoreh De'ah</i> 142:11, rules that, <i>post factum</i>, the animal's meat is permissible unless the animal's "primary development" had resulted from consumption of <i>issurei hana'ah</i>.13<i class="footnote">See <i>Taz</i>, <i>Yoreh De’ah</i> 142:17 <i>Arukh ha-Shulḥan</i>, <i>Yoreh De’ah</i> 60:6, argues that the benefit derived from <i>issurei hana’ah</i> consumed by an animal is an “unusual” benefit insofar as the foodstuff is concerned and hence the meat of the animal is permissible in all circumstances unless the foodstuff was used for idolatrous purposes in which case even “unusual” benefit is forbidden. An earlier authority, <i>Knesset ha-Gedolah,</i> cites Maharit as having been inclined to limit the prohibition to animals that have been raised on foodstuffs of <i>avodah zarah</i>. That opinion is also cited by <i>Kereti u-Peleti</i>, <i>Peleti</i> 60:1. Cf., however, the authorities cited <i>infra</i>, note 16, who maintain that the benefit derived from feeding animals such foodstuffs is not at all “unusual.”</i>
  Basing himself upon <i>Issur ve-Hetter he-Arukh</i> 47:9, <i>Pri Megadim</i>, <i>Siftei Da'at</i> 60:5, rules that "if it is [widely] known" that the animal has been raised solely on only non-kosher food its meat is forbidden because of <i>mar'it ayin</i>, i.e., possible confusion on the part of the onlooker.14<i class="footnote">See also <i>Darkei Mosheh</i>, <i>Yoreh De’ah</i> 81:6. However, that view is disputed by <i>Iggerot Mosheh</i>, <i>Oraḥ Ḥayyim</i>, I, no. 147, <i>anaf</i> 6, and was earlier questioned by <i>Da’at Torah</i>, <i>Yoreh De’ah</i> 60:5.</i>
  Rabbi Teitelbaum asserts that formula-fed veal is permissible even though the animals are raised on forbidden foods for three separate reasons:
  1) Despite the fact that <i>Bi'ur ha-Gra</i>, <i>Yoreh De'ah</i> 60:1, concurs in the ruling of Rema, the majority of later authorities, including <i>Pri Hadash</i>, <i>Yoreh De'ah</i> 60:5; <i>Kereti u-Peleti</i> 60:1; <i>Teshuvot Pnei Yehoshu'a</i>, I, <i>Yoreh De'ah</i>, no. 4; <i>Da'at Torah</i>, <i>Yoreh De'ah</i> 60:5; <i>Nishmat Adam</i>, <i>Hilkhot Pesaḥ</i>, sec. 9; <i>Arukh ha-Shulḥan</i>, <i>Yoreh De'ah</i> 60:7; as well as R. Yitzchak Yosef, <i>Yalkut Yizḥak</i>, X (Jerusalem, 5749), pp. 143-147, accept the permissive view of <i>Shakh</i>.15<i class="footnote">It should be noted, however, that <i>Bi’ur ha-Gra</i>, <i>Yoreh De’ah</i> 60:1, cites <i>Sifri</i>, <i>Parashat Re’eh</i>, in support of Rema’s ruling. Cf., <i>Iggerot Mosheh</i>, <i>Oraḥ Ḥayyim</i>, I, no. 174, <i>anaf</i> 7, who rejects Gra’s interpretation of <i>Sifri</i>.</i> In addition, a number of latter-day authorities extend that ruling to include even foodstuffs from which no ordinary or usual benefit may be derived but from which "unusual" benefit may be derived.16<i class="footnote">Cf., however, R. Judah Aszod, <i>Teshuvot Yehudah Ya’aleh</i>, <i>Oraḥ Ḥayyim</i>, no. 127, who points out that even “unusual” benefit is forbidden by rabbinic decree. Cf. also, <i>Iggerot Mosheh</i>, <i>Oraḥ Ḥayyim</i>, I, no. 147, <i>anaf</i> 5, who challenges the contention that such benefit is “unusual.” That objection was earlier raised by R. Chaim Sofer, <i>Teshuvot Maḥaneh Ḥayyim</i>, <i>Yoreh De’ah</i>, II, 20, s.v. <i>yoẓei</i>, in the name of R. Shlomoh Ganzfried and prompted a rather novel reinterpretation of <i>Kereti</i>’s comment by <i>Maḥaneh Ḥayyim</i>.</i> Those authorities assert that eating meat of an animal fed such food products is not considered to be a benefit "usually" derived from the forbidden foodstuff.
  2) Rabbi Teitelbaum reports that even formula-fed calves are permitted to suckle for the first seven to ten days of their lives.17<i class="footnote">Presumably, the calves are also provided with water in addition to the formula on which they are raised. Water, however, is not a nutrient and, although it is certainly a major component of every cell in the animal’s body, for halakhic purposes, it is not regarded as a fattening agent that produces body tissue. Were that not the case, in light of the fact that all animals require hydration in addition to nutrition, the halakhic discussions concerning animals raised on prohibited foodstuffs would be contrafactual. Cf., R. Moshe Chaim Schmerler of Zurich, <i>Mesorah</i>, no. XV (Tishri, 5759).</i> Thus, they have not been fed forbidden foods exclusively "all their lives."
  3) In addition to the forbidden meat and milk derivatives, the formula also contains a variety of innocuous ingredients. R. Dov Berish Weidenfeld, <i>Teshuvot Dovev Meisharim</i>, III, no. 50, rules that, even according to Rema, if the mixture fed to the animal contains only the "taste" of a forbidden food, i.e., the forbidden food has not been nullified by sixty equivalent measures of permitted food, the meat of the animal is permitted because the kosher portion of the mixture also contributes to the fattening of the animal18<i class="footnote">This is so despite the principle of <i>ḥatikhah na’asit neveilah</i>, i.e., a permitted food that absorbs the taste of a forbidden food becomes itself intrinsically forbidden. See <i>Shulḥan Arukh</i>, <i>Yoreh De’ah</i> 92:3. According to many authorities, including Rema, <i>Yoreh De’ah</i> 93:4, that rule applies not only to milk and meat but to other forbidden foods as well. Although milk and meat that are cooked together become intrinsically forbidden as a matter of biblical law, according to most authorities, foodstuffs that absorb other types of forbidden foods become intrinsically forbidden only by virtue of rabbinic decree. See <i>Taz</i>, <i>Yoreh De’ah</i> 92:11 and <i>Shakh</i>, <i>Yoreh De’ah</i> 92:12. Cf., however, <i>Maharik</i>, no. 192, and sources cited by <i>Pri Megadim</i>, <i>Yoreh De’ah</i>, <i>Siftei Da’at</i> 92:12. Similarly, according to most authorities, foodstuffs that absorb the forbidden taste of milk and meat become intrinsically forbidden by virtue of rabbinic law. Although the “taste” of the absorbed milk and meat is biblically prohibited, the intrinsic prohibition of the underlying substance of a foodstuff that has absorbed milk and meat is that of rabbinically proscribed milk and meat. See <i>Shakh</i>, <i>Yoreh De’ah</i> 94:22. [<i>Shakh</i> also cites the conflicting view of <i>Baḥ</i>, <i>Yoreh De’ah</i> 96:3, who maintains that foodstuffs that absorb milk and meat become intrinsically forbidden by virtue of biblical law in the same manner as the commingled meat and milk itself.] Since rabbinically forbidden milk and meat is not <i>assur be-hana’ah</i>, the portion of the formula that is neither meat nor dairy serves as a <i>gorem</i>, or cause, from which it is not forbidden to derive benefit and hence the principle of <i>zeh va-zeh gorem</i> remains applicable.However, <i>Dovev Meisharim</i> apparently fails to take cognizance of Rema’s ruling, <i>Yoreh De’ah</i> 93:4, to the effect that the principle of <i>ḥatikhah na’asit neveilah</i>, i.e., the entire substance of a foodstuff that absorbs a forbidden taste becomes intrinsically forbidden, applies not only to milk and meat cooked together but also, at least by virtue of rabbinic decree, to a mixture containing any forbidden substance. If so, to the extent that the mixture fed to the animals contains even the “taste” of non-kosher food, the entire mixture is intrinsically forbidden and hence, according to Rema, the meat of the animal should be forbidden since it has been raised exclusively on an intrinsically forbidden foodstuff.R. Shalom Toibes, <i>Teshuvot She’ilat Shalom</i>, <i>Mahadura Tinyana</i>, no. 152, adds the interesting observation that, other than with regard to milk and meat that have been cooked together, in the opinion of some authorities, the principle of <i>ḥatikhah na’asit neveilah</i> is not applicable in instances in which the admixture occurred “in the hands of a non-Jew,” i.e., while owned by a non-Jew. See <i>Teshuvot Rema</i>, no. 54 and <i>Pri Megadim</i>, <i>Yoreh De’ah</i>, <i>Siftei Da’at</i> 103:14, s.v. <i>ve-hineh</i>.</i> with the result that the animal has not been raised on forbidden food exclusively.19<i class="footnote">Although <i>Dovev Meisharim</i> addresses a situation in which the major portion of the mixture is kosher, his reasoning would seem to apply equally to instances in which only a minor portion of the ingredients is kosher.</i>
  Rabbi Teitelbaum's first considerations may not be germane because <i>Shakh</i>'s permissive view may not be applicable to formula-fed veal. <i>Shakh</i> permits the consumption of meat of animals raised on forbidden foods but he does not sanction eating meat of animals raised on food from which no benefit may be derived. Since the formula fed to the calves contains both meat and dairy ingredients that are combined and cooked together, it may be the case that it is forbidden to derive benefit from the mixture.20<i class="footnote">It should be noted that milk and meat of a member of a non-kosher species cooked together do not become <i>assur be-hana’ah</i>. See <i>Shulḥan Arukh</i>, <i>Yoreh De’ah</i> 87:3. Rambam, <i>Commentary on the Mishnah</i>, <i>Keritut</i> 3:4, maintains that such is also the case with regard to the meat of an animal of a kosher species that has not been properly slaughtered. That view is accepted by R. Ezekiel Landau, <i>Dagul me-Revavah</i>, <i>Yoreh De’ah</i> 87:3. If so, since the tallow contained in the formula in question is derived from non-kosher sources, the formula would not become <i>assur be-hana’ah</i>. However, <i>Pri Megadim</i>, introduction to <i>Hilkhot Basar be-Ḥalav</i>, s.v. <i>od</i>, and R. Moses Sofer, <i>Teshuvot Ḥatam Sofer</i>, <i>Yoreh De’ah</i>, no. 92, reject that view. Hence, according to those authorities, tallow obtained from a non-kosher species, when cooked with milk, becomes <i>assur be-hana’ah</i>. Moreover, R. Akiva Eger, <i>Beiẓah</i>, <i>ma’arakhah zayin</i>, asserts that, even for Rambam, it is only the non-kosher meat that does not become <i>assur be-hana’ah</i>, but that no benefit may be derived from the portion of that mixture consisting of milk cooked with such meat.</i> Even "unusual" forms of benefit may not be derived from meat and milk that have been commingled in cooking.21<i class="footnote">The meat products used in the formula are certainly derived from non-kosher animals. Nevertheless, as already indicated, <i>supra</i>, note 20, although R. Ezekiel Landau, <i>Dagul me-Revavah</i>, <i>Yoreh De’ah</i> 87:3, maintains that the milk cooked with non-kosher meat is not <i>assur be-hana’ah,</i> that view is disputed by <i>Pri Megadim</i>, introduction to <i>Hilkhot Basar be-Ḥalav</i>, s.v. <i>od</i> and <i>Teshuvot Ḥatam Sofer</i>, <i>Yoreh De’ah</i>, no. 92.</i> Hence the crucial issue is whether the formula contains milk and meat derivatives that have been "cooked" together.
  In describing the process used in preparing the formula Rabbi Teitelbaum is somewhat unclear. In his introductory description Rabbi Teitelbaum stresses that hot water is released into a large vat and only afterwards is the powder mix stirred into the water. If so, "cooking" takes place in a <i>keli sheni</i>, i.e., a secondary utensil not in contact with fire. The general rule is that "cooking," as generally understood in halakhic contexts, cannot take place in a secondary utensil. Even though, with regard to the prohibition against cooking that is prohibited on <i>Shabbat</i>, foodstuffs that are "easily cooked" may not be heated in a <i>keli sheni</i>, <i>Teshuvot Maharil</i>, no. 82, indicates that the concern is limited to violation of Sabbath restrictions but does not apply in other areas of Halakhah. Hence, since it is forbidden to derive benefit only from milk and meat that have been cooked together, milk and meat heated in a secondary utensil cannot become <i>assur be-hana'ah</i>.
  However, later in his discussion Rabbi Teitelbaum indicates that, at times, additional hot water is poured into the mixed formula already present in the vat in order to raise its temperature. As recorded by <i>Tosafot</i>, <i>Zevaḥim</i> 95b, s.v. <i>irah</i>, there is a controversy with regard to whether a hot liquid poured from a primary utensil, i.e., a utensil heated on a stove, can effect the "cooking" of at least the upper surface to the depth of a "peel" of the substance on which it is poured. However, <i>Pri Megadim</i>, <i>Yoreh De'ah</i>, <i>Mishbezot Zahav</i> 88:9, maintains that the controversy is limited to a hot liquid falling upon a solid food but that all concur that no "cooking" occurs if the hot liquid is poured into a cold liquid. Similarly, no "cooking" occurs if the hot liquid falls upon a powder. In addition, R. Jacob of Lissa, <i>Havvat Da'at</i>, <i>Bi'urim</i> 91:7, maintains that a "peel" cannot become <i>assur be-hana'ah</i> because cooking only the surface of a foodstuff is not a natural or usual form of "cooking."
  Moreover, even if the thickness of a "peel" is cooked and becomes <i>assur be-hana'ah</i>, many authorities maintain that if the peel falls into other food and is not recognizable in the resultant mixture it is permitted to derive benefit from the mixture. Other authorities maintain that the forbidden "peel" is nullified only in a mixture sixty times as great. In addition, although some authorities differ, <i>Shakh</i>, <i>Yoreh De'ah</i> 94:22, rules that if a mixture of cooked milk and meat becomes commingled with other food the resultant mixture is not treated as milk and meat but as a conventionally prohibited foodstuff and hence that mixture is not rendered <i>assur be-hana'ah</i>. Furthermore, even those who differ with <i>Shakh</i> regarding this issue recognize that the resultant mixture is prohibited only by reason of rabbinic decree, rather than by biblical law. Hence, since there is no <i>issur hana'ah</i> associated with milk and meat forbidden only by virtue of the rabbinic prohibition, the resultant mixture may be fed to animals.
  In light of Rabbi Teitelbaum's premises, his discussion of whether or not the formula is <i>assur be-hana'ah</i> seems to be somewhat superfluous. Rabbi Teitelbaum accepts the position of <i>Shakh</i> who maintains that there is no restriction even <i>ante factum</i> upon eating meat of an animal that has been raised on forbidden foods and that the problem exists only with regard to animals raised on feed that is <i>assur be-hana'ah</i>. He further asserts that formula-fed calves are not raised all their lives on food that is <i>assur behana'ah</i> because they suckle cow's milk the first seven to ten days of their lives and also because the major portion of the formula is comprised of kosher ingredients. Accordingly, it follows that, even were the formula to be deemed <i>assur be-hana'ah</i>, the resultant problem would certainly affect the farmer who wishes to derive benefit from the use of such formula in feeding his cows. However, since under such circumstances the meat is permitted <i>post factum</i> on the basis of the principle of <i>zeh va-zeh gorem</i>, the consumer encounters no problem in purchasing the meat.
  However, a complication with regard to the <i>post factum</i> status of such animals arises from the fact that the proprietors of many, and probably the majority, of slaughterhouses and meat processing plants are non-Jews. That is certainly true with regard to the cattlemen who raise calves. As is evident from <i>Shakh</i>'s analysis of the ruling of <i>Shulḥan Arukh</i>, <i>Yorah De'ah</i> 142:11, the meat of an animal that has been raised both on <i>issurei hana'ah</i> and on permissible substances is, <i>post factum</i>, permitted on the basis of the principle of <i>zeh va-zeh gorem</i>. However, purchasing such an animal from a non-Jew might well be considered to be <i>ante factum</i> since refraining from doing so will result in no actual financial loss to a Jew. R. Ovadiah Yosef, <i>Yabi'a Omer</i>, VII, <i>Yoreh De'ah</i>, no. 7, sec. 1, demonstrates that the overwhelming majority of halakhic decisors regard a situation of that nature as <i>post factum</i>.22<i class="footnote">See also R. Yitzchak Yosef, <i>Yalkut Yosef</i>, X (Jerusalem, 5769), 147.</i> It is certainly clear that if no other veal is available, or is not available at a reasonable price, all authorities would agree that the situation is to be regarded as <i>post factum</i>.
  More significantly, Rabbi Teitelbaum's contention that the problem is obviated since the fact that the calves suckle after birth for a period of seven to ten days invokes a consideration that is certainly subject to dispute. <i>Shakh</i> himself asserts that Rema's reference to an animal that has consumed forbidden foods "all its life" is not to be understood literally but as connoting that the animal's primary development was the product of forbidden foodstuffs.23<i class="footnote">See <i>supra</i>, note 9.</i>
  Rabbi Sofer's objections are based on a number of considerations that would render the formula <i>assur be-hana'ah</i>:
  1) <i>Pri Megadim</i>, <i>Oraḥ Hayyim</i> 318:30, maintains that "easily cooked" foodstuffs cooked in a secondary utensil are considered to be "cooked" not only for purposes of Sabbath prohibitions but also with regard to the prohibited cooking of milk and meat.
  2) If the stream of hot water remains attached to its source and the powder is poured into the water, some authorities maintain that not only a "peel" of the powder is "cooked" but that the entire quantity of powder added in that manner is deemed to be "cooked." Those authorities maintain that, in effect, each particle of powder is discrete and, since its thickness is less than that of a "peel," each particle becomes cooked sequentially.
  3) Even if the entire quantity of powder is placed in the vat before the hot water is introduced so that only a "peel" is "cooked" by the hot water, each successive "peel" is "cooked" as the powder is stirred while water is constantly added.
  Rabbi Sofer further objects that it is forbidden to feed animals foodstuffs that contain even a "taste" of <i>issurei hana'ah</i>. That objection is well-taken insofar as feeding animals is concerned. Direct benefit may not be derived from foodstuffs containing a "taste" of <i>issurei hana'ah</i> because the "taste" of the forbidden substance cannot be separated from the food it permeates. However, as stated by <i>Pri Megadim</i> in his introduction to <i>Hilkhot Basar be-Halav</i>, permitted food containing a small quantity of <i>issurei hana'ah</i> may be sold to a non-Jew provided that the value of the <i>issurei hana'ah</i> is deducted from the purchase price. Similarly, insofar as an animal raised on such feed is concerned, the animal should be deemed to have been raised on a mixture of <i>issurei hana'ah</i> and kosher food with the result that, <i>post factum</i>, the meat of the animal is permitted on the basis of the principle of <i>zeh va-zeh gorem</i>. Rabbi Sofer fails to address the additional consideration that the calves are fed cow's milk exclusively during the period immediately following birth.
  III. Conclusions
  Resolution of the question of the permissibility of veal derived from formula-fed calves is dependent upon adjudication of a number of halakhic issues:
  1. The primary issue is whether the formula is <i>assur be-hana'ah</i> as a form of milk and meat that has been cooked together. That is unlikely to be the case for a number of reasons:
  a) It may well be the case that the "cooking" of all but a small portion of the produced formula takes place in a secondary utensil.
  b) A substantial portion of the formula consists of entirely innocuous ingredients. The consensus of opinion is that <i>ḥatikhah na'asit neveilah</i> does not apply to otherwise permissible foodstuffs that absorb commingled milk and meat. Accordingly, the principle of <i>zeh va-zeh gorem</i> would serve to negate the prohibition against deriving benefit from commingled milk and meat.
  The contention that the calves are suckled for seven to ten days during the initial post-partum period is probably not a sufficient factor to render the principle of <i>zeh va-zeh gorem</i> applicable but, in light of the foregoing that consideration is superfluous.
  2. Assuming that the formula is not <i>assur be-hana'ah</i>, the second issue is whether the meat of an animal raised exclusively on forbidden foods is prohibited. That issue would arise by virtue of the presence of non-kosher tallow in the formula even if the formula did not include milk derivatives. A permissive ruling with regard to that question is contingent upon:
  a) acceptance of <i>Shakh</i>'s view, in opposition to that of Rema and <i>Bi'ur ha-Gra</i>, that the meat of an animal raised exclusively on forbidden foods is not forbidden; or
  b) recognition that the formula contains innocuous ingredients in addition to the forbidden milk and meat derivatives coupled with rejection of the opinion that <i>ḥatikhah na'asit neveilah</i> applies at least by virtue of rabbinic decree to forbidden substances other than commingled milk and meat;24<i class="footnote">See <i>supra</i>, note 18.</i> or
  c) the unlikely determination that suckling during the post-partum period is a factor sufficient enough to render the principle of <i>zeh va-zeh gorem</i> applicable.
  Assuming that there is no problem of <i>issur hana'ah</i> resulting from biblically prohibited cooked milk and meat, a kashrut supervisory authority would be justified in relying upon the position of <i>Shakh</i> since it appears to be accepted by the majority of latter-day authorities. However, certification of veal raised under such conditions as <i>glatt</i> kosher is more problematic.
  The term <i>glatt</i>, or the Hebrew <i>ḥalak</i>, in its literal meaning refers to the fact that the lungs have been found to be "smooth," i.e., no adhesions are present that might compromise the <i>kashrut</i> of the animal. The reason for that standard is that the presence of an adhesion requires a determination that the adhesion is not of a nature that would render the animal non-kosher. Such a determination usually requires adjudication between conflicting halakhic opinions and also presents issues with regard to the method employed in removal of the adhesions prior to examination for a possible perforation.
  The Gemara, <i>Hullin</i> 37b, cites Ezekiel 4:14; "… I have not eaten of <i>neveilah</i> or <i>treifah</i> from my youth until now," and offers a remarkable interpretation. <i>Neveilah</i> and <i>treifah</i> are forbidden to all Jews. It would have been unthinkable for Ezekiel to have violated those prohibitions. Hence, his almost boastful comment would have been entirely superfluous. Accordingly, the Gemara understands Ezekiel to have exclaimed, "I have not eaten of an animal with regard to which a scholar ruled," i.e., Ezekiel, as an act of piety, refused to eat meat whose <i>kashrut</i> was the subject of any doubt even if it was ruled to be kosher by a competent scholar.
  The term <i>glatt</i> in common parlance has acquired the connotation that food described in that manner is not in the category of <i>behemah she-horah bah ḥakham</i>, i.e., it is not a foodstuff whose <i>kashrut</i> was subject to question and whose acceptability is contingent upon a permissive determination by a scholar. That expanded connotation of the term <i>glatt</i> is entirely understandable since the piety adopted by Ezekiel and emulated by others was assuredly not limited to lumbar adhesions. Adhesions of the lungs are simply the most common problem requiring an opinion of a scholar with regard to the <i>kashrut</i> of the animal. Adjudication of a controversy between Rema and <i>Shakh</i> certainly entails a "ruling of a scholar" which those who adopt a <i>glatt</i> standard of <i>kashrut</i> would be unwilling to entertain.
  There is one expedient that would serve to remove virtually all questions with regard to the permissibility of veal derived from formula-fed calves. A formula that contains no forbidden ingredients has been devised and patented.25<i class="footnote">U.S. Patent no. 6,348,222 (issued February 19, 2002).</i> However, for reasons that are not clear but which presumably are economic in nature, that product has not been adopted by the veal industry. <i>Levushei Serad</i>, IV, sec. 81, cited by <i>Pitḥei Teshuvah</i>, <i>Yoreh De'ah</i> 60:2, suggests that even if an animal is raised entirely upon forbidden foodstuffs its meat is permissible if the animal is fed permissible food exclusively "two or three days" prior to slaughter.26<i class="footnote">See <i>supra</i>, note 9.</i> Although that authority does not make a final determination with regard to this matter, he strongly inclines to that view.27<i class="footnote">See also R. Benjamin Adler, <i>Kashrut u-Treifot be-Of</i> 9:6.</i> If so, feeding the calves kosher formula at least during the two- or three-day period preceding slaughter would obviate the problem.28<i class="footnote"><i>Levushei Serad</i> does, however, suggest that it may be forbidden intentionally to generate a <i>zeh va-zeh gorem</i> in the same manner that it is forbidden intentionally to nullify a forbidden substance. There is, however, no restriction preventing a non-Jew from acting in that manner before selling the animal to a Jew.</i>
  
  Chapter 11 Issues Concerning Kohanim
  
  <i>Beloved are the Kohanim for, when God bestows a sobriquet upon them, He always refers to them as ministering angels.</i>
  SIFRI, PARASHAT KORAH 46
  Priestly Identity
  Marriage between a person of Aharonic descent and a divorcée is expressly forbidden by Leviticus 21:7. As recorded by <i>Shulḥan Arukh</i>, <i>Even ha-Ezer</i> 6:8, marriage between a <i>kohen</i> and a female convert is also prohibited. The late Rabbi Moshe Feinstein, of blessed memory, is widely quoted as having ruled that <i>ba'alei teshuvah</i>, i.e., newly-observant young men, whose fathers were not observant may be permitted to marry divorcées or converts to Judaism despite the fact that they have held themselves out as <i>kohanim</i>. The reasoning, it is reported, is that the assumption of priestly descent on the part of such a person is derived entirely from his father's genealogical claim. Since the father was, and remains, non-observant, his testimony, it is claimed, as a matter of Halakhah, can be given no credence.
  This issue is discussed in two remarkably brief responsa published in <i>Iggerot Mosheh</i>, <i>Even ha-Ezer</i>, IV, no. 11 and no. 39. The particular situation discussed by <i>Iggerot Mosheh</i> in his first responsum, dated 5721, involved a man already married to a convert who presumed that he was a <i>kohen</i> on the basis of the fact that on a single occasion on which his father was "called to [the Torah at] a <i>bar mizvah</i> celebration" the father stated that he was a <i>kohen</i>. <i>Iggerot Mosheh</i> describes the father as totally non-observant and in the same sentence adds that the father "also ran away from his father's house in his youth because of rebellion against his father" so that "presumably (<i>lefi ha-da'at</i>) he could not properly know a matter such as this." <i>Iggerot Mosheh</i> further takes note of the fact that a female cousin declared that her father, a brother of the gentleman's father, challenged her cousin's claim to priestly descent declaring, "How is that possible that you are a <i>kohen</i> since my father, who is your father's brother, is not a <i>kohen</i>?" Without further analysis of the grounds for his ruling, <i>Iggerot Mosheh</i> proceeds to declare that since the gentleman's priestly identity was established solely on the basis of the declaration of his father "who has no credibility" with regard to that matter, the son is not obliged to divorce his wife. The discussion does not reveal whether that conclusion is based solely upon the father's lack of religious observance, upon the father's unreliable recollection due to the particular circumstances of his relationship with his family, upon the cousin's contradictory evidence or upon a combination of those considerations.
  <i>Iggerot Mosheh</i>'s ruling was understood as based upon the confluence of those considerations in a decision of a Jerusalem <i>bet din</i> established for purposes of ruling on matters of personal status. The decision of that <i>bet din</i>, comprised of the respected Sephardic authority, the late R. Eliyahu Abba Sha'ul, R. Abraham Dov Levine and Rabbi Eliyahu Shlesinger is published in <i>Piskei Din me-Bet ha-Din le-Dinei Mammonot u-Birur Yahadut shel ha-Rabbanut ha-Rashit le-Yerushalayim</i>, vol. I (5755), part 2, pp. 43-54.
  1. Credibility of a Parent
  It is entirely likely that, were the matter to hinge entirely upon the father's credibility, Rabbi Feinstein would have maintained that the father's lack of observance would be sufficient, in and of itself, to dismiss his testimony. In doing so Rabbi Feinstein would simply have been relying upon his own opinion as announced in <i>Iggerot Mosheh</i>, <i>Even ha-Ezer</i>, I, no. 82, <i>anaf</i> 11 and reiterated in <i>Even ha-Ezer</i>, IV, no. 32, <i>anaf</i> 7, to the effect that even a <i>tinok ha-nishbah bein ha-akum</i>, i.e., a child held in captivity among the pagans, has no credibility. That understanding of Rabbi Feinstein's position is bolstered, but not unequivocally confirmed, by his discussion in the second of those responsa, <i>Even ha-Ezer</i>, IV, no. 39. Surprisingly, the Jerusalem <i>bet din</i> fails to make any reference to the latter responsum.
  The latter responsum, written in 5738, addresses the status of a newly observant man married to a non-Jewish woman with whom he had a child. The woman in question was apparently eligible for conversion as a righteous proselyte but would have been forbidden to marry a <i>kohen</i>. The gentleman's grandfather immigrated to this country from behind the Iron Curtain in 1923. The gentleman in question presumed himself to be a <i>kohen</i> solely on the basis of information imparted to him by his father.
  The family reportedly had "no knowledge whatsoever of Torah or Judaism." Since the father "knew nothing" and it is clear that the grandfather made no attempt whatsoever to educate his child "even in some matters" of a Jewish nature, <i>Iggerot Mosheh</i> deduces that the grandfather was both a Communist and an atheist and was already a "thoroughly evil" person (<i>rasha gamur</i>) when he arrived in this country. Accordingly, concludes <i>Iggerot Mosheh</i>, the grandfather had no credibility to claim priestly descent. That statement, standing alone, would certainly indicate that the grandfather's lack of observance, or his atheism, was dispositive with regard to his non-credibility.
  <i>Iggerot Mosheh</i>, however, adds a number of additional comments. The progenitor's testimony is not to be accepted, asserts <i>Iggerot Mosheh</i>, "particularly since [the grandson] did not hear this from his grandfather who perhaps may have at least studied in a <i>ḥeder</i> before the Communist conquest even though there were already many irreligious people (<i>ḥofshim</i>) in the cities of the Ukraine in the fourth and fifth year of this century … nevertheless, they certainly knew a little even though they had no credibility." The reader can only register puzzlement with regard to the thrust of those comments: If such individuals lack credibility what difference can be effected by their knowledge of Jewish practices or by the grandfather's attendance at a <i>ḥeder</i>? The statement seems to reflect equivocation between a statutory lack of credibility and lack of credibility due to sheer ignorance.
  Moreover, <i>Iggerot Mosheh</i> places weight upon the fact that the couple had already established a non-halakhic marital relationship which, he maintains, should not be disturbed on the basis of the available information. Also incorporated in that statement is <i>Iggerot Mosheh</i>'s sensitivity to the fact that if the gentleman's status as a <i>kohen</i> is confirmed and he is informed that marriage to this woman subsequent to her conversion is forbidden "it is certain that one must be concerned" that, in all likelihood he will return to his prior non-observant lifestyle. At the same time, <i>Iggerot Mosheh</i> expresses a strong inclination, but not a certainty, that such a person may be permitted to marry a convert or a divorcée even in the absence of those factors.
  In his first responsum <i>Iggerot Mosheh</i> dismisses the statement of the father, not because he was a heretic, but because he was a Sabbathdesecrator. The principle formulated by the Gemara, <i>Eruvin</i> 69b, is that public desecration of the Sabbath is tantamount to heresy. That principle is predicated upon the premise that open and notorious desecration of the Sabbath is evidence that the transgressor denies that God created the universe over a period of six days and rested on the seventh. That talmudic presumption went unchallenged until the mid-nineteenth century. R. Jacob Ettlinger, <i>Teshuvot Binyan Ẓion ha-Hadashot</i>, no. 23, reports a socioreligious phenomenon, novel in nineteenth-century Germany but all too familiar in twentieth-century America, viz., the existence of countless numbers of Jews who offered Sabbath prayers and recited <i>kiddush</i> each Sabbath eve but then proceeded to desecrate the Sabbath by engaging in all manner of forbidden activity. Is it logical, queries <i>Binyan Ẓion</i>, for a person who denies creation to devoutly recite "And the heaven and earth and all their hosts were complete. And on the seventh day God completed the world which He made and He rested on the seventh day…." (Genesis 2:1-2). The notion of a believing Sabbath-desecrator might have been an oxymoron in a bygone age but in the modern world, argues <i>Binyan Ẓion</i>, it is a new reality. In the modern age, whether because of financial duress or other factors, that phenomenon is all too real. Others, who are raised in irreligious homes, know no better and have the status of "a child who was held in captivity among pagans." Accordingly, in light of the changed realia, rules <i>Binyan Ẓion</i>, the talmudic presumption of heresy does not attach itself to such persons.
  <i>Iggerot Mosheh</i>, <i>Oraḥ Hayyim</i>, I, no. 33 reaches the same conclusion but on somewhat different grounds. <i>Iggerot Mosheh</i> queries why a person who violates the Sabbath in private should be viewed less severely than one who desecrates the Sabbath in public. In private, he submits, it may be assumed that the trespasser acts out of weakness and desire rather than because of disbelief. When the transgression is performed in public onlookers observe an act of heresy whereas the individual's motives and intent are concealed from them. In our day, contends <i>Iggerot Mosheh</i>, since "it is known that the majority of Sabbath-violators [act] because of desire for lucre" onlookers do not perceive such acts to be heretical in nature. Hence there is no longer a distinction between Sabbath desecration, in private and in public. R. David Zevi Hoffmann, <i>Melamed le-Ho'il</i>, <i>Oraḥ Hayyim</i>, no. 28, reasons simply that, in an age of rampant Sabbath desecration, people have become desensitized to the severity of the transgression. Since they perceive no need to conceal their actions, argues <i>Melamed le-Ho'il</i>, public desecration of the Sabbath, in our day, is no more onerous than private transgression in times gone by.
  Although disputed by others,1<i class="footnote">See, for example, R. Chaim Eleazar Shapiro, <i>Teshuvot Minḥat Elazar</i>, I, no. 74 and II, no. 23.</i> the basic position, albeit sometimes formulated on the basis of different halakhic nuances, has been espoused by a long list of rabbinic authorities.2<i class="footnote">See, for example, R. Shalom Mordecai Schwadron, <i>Teshuvot Maharsham</i>, I, no. 121; R. David Zevi Hoffmann, <i>Melamed le-Ho’il</i>, <i>Oraḥ Ḥayyim</i>, no. 29; and R. Joseph Eliyahu Henkin, <i>Peirushei Ivra</i>, chap. 5, sec. 4.</i> Even those who disagree concede that a person who grows up among non-observant Jews has the status of a "<i>tinok ha-nishbah bein ha-akum</i>," i.e., "a child held in captivity among the pagans" who is not to be treated as a heretic since he never knew better. In effect, invincible ignorance is a shield against the halakhic sanctions and disqualifications suffered by a heretic with the result that a person who was raised in a non-observant environment and denied a meaningful Jewish education is to be regarded as having the status of "a child held in captivity."
  <i>Iggerot Mosheh</i>, <i>Even ha-Ezer</i>, I, no. 82, <i>anaf</i> 11 and IV, no. 39, maintains, however, that, although such individuals suffer no other sanction, they are disqualified from serving as witnesses, not because of inherent "wickedness," but for the simple reason that such persons have no compelling reason to be truthful. The testimony of a believing Jew is accepted because he recognizes perjury to be a sin. A person who has no knowledge of the binding nature of the Sinaitic revelation has no such constraint; hence his testimony can be given no credence.3<i class="footnote">See also R. Moshe Feinstein, <i>Iggerot Mosheh</i>, <i>Yoreh De’ah</i>, II, no. 43 and <i>Even ha-Ezer</i>, IV, no. 37, <i>anaf</i> 7. Cf., however, R. Samuel Walkin, <i>Teshuvot Zekan Aharon</i>, I, no. 74, who asserts that people are truthful by nature in situations in which they have nothing to gain from a falsehood.</i> It would follow that a person having knowledge of the prohibition against false testimony, but who professes no commitment to the observance of any of the commandments, would, according to <i>Iggerot Mosheh</i>, be similarly disqualified as a witness since he is under no imperative to be truthful. R. Yehudah Leib Diskin, <i>Teshuvot Maharil Diskin</i> (Jerusalem, 5691), p. 108, sec. 5, seems to adopt a similar view. That is also clearly the opinion of <i>Teshuvot Maharshakh</i>, III, no. 15, who comments with regard to those in the category of a <i>tinok ha-nishbah</i>: "It is not proper to accept them as witnesses for after all they transgress many commandments…. [Is] it conceivable that [such a person] be accepted for testimony since he does not act in the manner of proper Jews?"
  A contradictory position was espoused by the Israeli Rabbinical Court of Appeals in 5708 in an opinion issued by R. Ben-Zion Uziel, R. Isaac ha-Levi Herzog and R. Meshullem Roth published in <i>Osef Piskei Din Rabbaniyim</i> (Jerusalem, 5710), pp. 137-138, a slim volume that served as a precursor to the <i>Piskei Din Rabbaniyim</i>. The case involved a woman divorced from a <i>kohen</i> who claimed that she and her former, but now deceased, husband had remarried and hence she was entitled to the rights and prerogatives of a widow. Since marriage between a <i>kohen</i> and a divorcée is forbidden the marriage was solemnized without benefit of clergy but in the presence of witnesses. At issue was a challenge to the validity of the marriage by virtue of the fact that the witnesses were non-observant and, in the case of one witness, there was testimony that he had been seen smoking in public on <i>Shabbat</i>. The <i>bet din</i> refused, albeit somewhat tentatively, to disqualify the witnesses arguing that "in a time and a place in which non-observance has multiplied and spread … " transgressions of this nature do not undermine the credibility of witnesses. In the situation of such great 'hiding of the face' [the witnesses] are virtually inadvertent transgressors." Disqualification of transgressors or witnesses is because of fear of false testimony "and accordingly the credibility of the witness must be assessed in accordance with the circumstances of the time and the place" (<i>ibid.</i>, p. 137). The <i>bet din</i> was prepared to accept such testimony, not only in instances of invincible ignorance, but even in situations in which the improper conduct might be attributed to sociological factors. Quite clearly, the <i>bet din</i>, unlike <i>Iggerot Mosheh</i>, assumed that witnesses abjure perjury not solely because of a divine mandate to do so but because of the innate, natural human condition. According to that view a non-observant father would have credibility to testify with regard to his son's status as a <i>kohen</i>.
  Indeed, it would be entirely consistent even with <i>Iggerot Mosheh</i>'s position to assert that, although individuals who do not accept the veracity of revelation at Sinai have no credibility, nevertheless, those who accept the basic phenomenon but, due to ignorance of its provisions, economic pressures, unrestrained desire or convenience are largely nonobservant, retain credibility concerning matters with regard to which they are scrupulous. The individual described in <i>Iggerot Mosheh</i>'s first responsum was totally non-observant; the individual described in the second responsum is depicted as an atheist. Neither responsum addresses the phenomenon of the somewhat knowledgeable, selectively-observant, professing Jew.
  Moreover, in addressing an entirely different issue, <i>Iggerot Mosheh</i> himself asserts that, at times, even an avowed atheist has credibility with regards to matters of religious law. Rabbi Feinstein, <i>Iggerot Moshe</i>, <i>Yoreh De'ah</i>, I, no. 54, reports that on a visit to Moscow in 1936 he was consulted with regard to an agonizing question. Circumstances forced many elderly and infirm individuals to live with their children. Those children, living under the influence of the Communist state, did not observe <i>kashrut</i> and "the majority denied God and His Torah." Nevertheless, they accommodated their aged parents by purchasing kosher food and providing kosher utensils on their behalf. The issue presented to Rabbi Feinstein was the credibility of those children with regard to <i>kashrut</i>.
  <i>Iggerot Mosheh</i> responded that a parent might indeed rely upon his child's guarantees of <i>kashrut</i> provided that the parent was firmly convinced that the "daughter or daughter-in-law" would not cause the parent to transgress. Such conviction would arise, asserts <i>Iggerot Mosheh</i>, if the parent "tested" the homemaker on a number of occasions and ascertained that no attempt was made to fool the parent because the child "did not want to cause him pain or because it was the nature [of the child] not to cause others to act contrary to their will."
  In explaining his ruling, <i>Iggerot Mosheh</i> argues that rules governing disqualification of witnesses pertain only to acceptance of testimony with regard to individuals whose nature is unknown. Statements of such individuals are accepted on the basis of "belief" in the truth of their testimony rather than on the basis of knowledge. Citing an anecdote recorded in <i>Ketubot</i> 85a, <i>Iggerot Mosheh</i> asserts that, in a situation in which a person has personal knowledge of another individual's unchallengeable probity, his acceptance of the latter's statement is a matter of "knowledge" rather than "belief" and hence he may act on such statements even if that person is not qualified to serve as a witness.4<i class="footnote">See also <i>Iggerot Mosheh</i>, <i>Yoreh De’ah</i>, II, no. 43. The incident recorded in <i>Ketubot</i> 85a may, however, reflect a much more circumscribed rule, <i>viz.</i>, that a <i>dayyan</i> dare not decide a financial matter on the basis of objectively admissible evidence when he is convinced of, or even suspects, the unreliability of the evidence presented. In instances in which exoneration is contingent upon an oath and the <i>dayyan</i> suspects that the defendant may swear falsely he must assign the oath to the plaintiff. See Rambam, <i>Hilkhot Sanhedrin</i> 24:1 and <i>Kesef Mishneh</i>, <i>ad locum</i>. It is in that context that the Gemara reports that Rava acted upon the declaration of the daughter of Rav Ḥisda whose testimony could not be formally accepted. Cf., also, the opinion of Rif cited by <i>Kesef Mishneh</i>, <i>loc. cit.</i>, who states that this principle is not fully applicable in our day.</i>
  It would then follow that, even according to <i>Iggerot Mosheh</i>, if the son is absolutely certain that his father, despite the fact that the latter is a Sabbath-violator or even an atheist, would not lie with regard to matters of religious observance, the son whose priestly genealogy has been disclosed by his father, is deemed to "know" that fact to be true and must comport himself accordingly.
  The Jerusalem <i>bet din</i>, in its decision, failed to address another issue affecting disqualification of witnesses. <i>Bet Yosef</i>, <i>Even ha-Ezer</i> 42, s.v. <i>maz'ati</i>, describes an incident involving a person who entered into a marriage in the presence of witnesses who were known to have committed serious transgressions. One authority, a certain R. Chaim he-Arukh, declared the marriage to be null and void. The Sages of Tilutula challenged the ruling and placed R. Chaim he-Arukh under a ban until such time as he would rescind that ruling. Their position was that a person cannot be disqualified unless testimony establishing his wrongdoing is heard in his presence by a properly constituted <i>bet din</i>. Since that had not occurred they maintained that the marriage was valid.5<i class="footnote">See <i>Teshuvot ha-Ralbaḥ</i>, no. 136 and <i>Teshuvot Noda bi-Yehudah</i>, <i>Even ha-Ezer</i>, <i>Mahadurah Kamma</i>, no. 72, who maintain that testimony to disqualify an individual from serving as a witness may be heard only in his presence. Cf., the contradictory view of <i>Teshuvot ha-Rivash</i>, no. 266. See also sources cited in <i>Koveẓ ha-Poskim</i>, V (Brooklyn, 5738), 105-111, 115 and 147-148.</i>
  A similar position is espoused by <i>Teshuvot ha-Baḥ</i>, no. 102; <i>Urim ve-Tumim</i>, <i>Hoshen Misphat</i>, <i>Urim</i> 28:3 and <i>Tumim</i> 87:27; <i>Teshuvot Hatam Sofer</i>, <i>Yoreh De'ah</i>, no. 11; <i>Teshuvot Hikrei Lev</i>, <i>Yoreh De'ah</i>, no. 1; <i>Netivot ha-Mishpat</i>, <i>Bi'urim</i> 46:22, cited by <i>Pitḥei Teshuvah</i>, <i>Hoshen Mishpat</i> 28:1; <i>Teshuvot Pnei Aryeh</i>, no. 28, cited by <i>Pitḥei Teshuvah</i>, <i>Even ha-Ezer</i> 42:18; and R. Samuel Walkin, <i>Teshuvot Zekan Aharon</i>, I, no. 74.6<i class="footnote">Cf., however, <i>Teshuvot Ḥavvot Ya’ir</i>, addenda and <i>Keẓot ha-Ḥoshen</i> 28:8 who adopt an opposing view. See also R. Chaim Halberstam, <i>Teshuvot Divrei Ḥayyim</i>, <i>Likkutim ve-Hashmatot</i>, no. 33 and R. Meir Arak, <i>Teshuvot Imrei Yosher</i>, II, no. 36.R. Moshe Schick, <i>Maharam Shik al Taryag Miẓvot</i>, no. 37, sec. 7, asserts that common knowledge regarding an evildoer’s transgressions establishes a <i>ḥazakah</i> and hence no <i>bet din</i> proceedings are required to disqualify him as a witness. See also <i>Iggerot Mosheh</i>, <i>Yoreh De’ah</i>, I, no. 160; <i>Even ha-Ezer</i>, I, no. 135; and <i>Even ha-Ezer</i>, IV, no. 11, sec. 3, who makes the same point with regard to the disqualification of Conservative clergy. See also <i>Iggerot Mosheh</i>, <i>Even ha-Ezer</i>, no. 85. Cf., however, <i>Teshuvot Pnei Aryeh</i>, no. 28, who asserts that common knowledge regarding transgression is not sufficient to disqualify a person from serving as a witness.</i>
  2. Personal Credibility—Shavya Anafsheih
  The Jerusalem <i>Bet Din le-Birur Yahadut</i> addressed a case similar to that brought to the attention of <i>Iggerot Mosheh</i> and ruled in a different manner in at least one salient aspect. The case involved a newly-observant Russian couple. The wife had previously been married to a non-Jew and hence was disqualified from marrying a <i>kohen</i>. The husband had been informed of his family's priestly status by his non-observant father.
  The general rule is that a person has standing, even when contradicted by trustworthy witnesses, to declare that certain matters are prohibited to him. Thus a person may not eat a foodstuff he has declared to be non-kosher even though others are fully entitled to rely upon the contradicting testimony of two witnesses. The resultant principle is known as "<i>Shavya anafsheih ḥatikha de-issura</i>—He has established upon himself a prohibited piece." Similarly, a person has credibility to declare himself a <i>mamzer</i> and consequently forbidden to marry a Jewess of legitimate birth. Although such credibility is generally limited to oneself, nevertheless, as recorded by Rambam, <i>Hilkhot Issurei Bi'ah</i> 15:15-16, a father is granted specific authority to declare his son to be a <i>mamzer</i>, most particularly when he testifies directly with regard to his son's status. That authority is derived from the biblical verse "<i>yakir</i>— he shall recognize" (Deuteronomy 21:17). If the father testifies that he himself is a <i>mamzer</i> and, by implication, that status is shared by his son as well, some authorities maintain that the father's testimony creates only a doubtful status with regard to the son.
  The Jerusalem <i>bet din</i> raised an intriguing issue in questioning whether the rule of "<i>yakir</i>" is limited to disqualification as a <i>mamzer</i> or whether it applies also to acknowledging a son as a <i>kohen</i> who is then bound by attendant marital restrictions. A person certainly has credibility to declare himself a <i>kohen</i> for purposes of being forbidden to marry a divorcée. The question is whether or not he can generate the selfsame prohibition with regard to his son. If yes, argued the <i>bet din</i>, the father's lack of observance is of no consequence. A father is disqualified from serving as a witness in any matter affecting his son. Nevertheless, despite the paternal-filial relationship that exists between them, the rule of "<i>yakir</i>" endows the father with credibility despite such ostensive disqualification; similarly, that principle establishes credibility even in face of other disqualifications as a witness.
  Moreover, argued the <i>bet din</i>, the son who had declared himself a <i>kohen</i>, albeit on the strength of his father's declaration, has created his own <i>shavya anafsheih</i> quite apart from the implications flowing from the father's declaration.7<i class="footnote">That position was reiterated by Rabbi Levine in a later opinion published in <i>Piskei Din me-Bet ha-Din le-Dinei Mammonot u-Birur Yahadut shel ha-Rabbanut ha-Rashit le-Yerushalayim</i>, vol. III (5755), part 2, pp. 274-276 and 280.</i> <i>Teshuvot ha-Radvaz</i>, I, no. 362, advances the thesis that <i>shavya anafsheih</i> is established not only on the basis of alleged personal knowledge but also on the basis of the acceptance of statements of others. That position is also reflected in the comments of <i>Tosafot</i>, <i>Kiddushin</i> 66a, s.v. <i>Rava</i>, as well as the ruling of <i>Teshuvot Hatam Sofer</i>, <i>Even ha-Ezer</i>, I, no. 78, cited by <i>Pitḥei Teshuvah</i>, <i>Even ha-Ezer</i> 115:25.8<i class="footnote">These sources seem to have been unknown to R. Eliyahu Schlesinger, <i>Piskei Din</i>, <i>ibid.</i>, p. 279.</i>
  <i>Iggerot Mosheh</i> dismissed the issue of <i>shavya anafsheih</i> in the first case brought to his attention because the son's presumption of his status as a <i>kohen</i> was predicated entirely upon the statements of his non-observant father. The Jerusalem <i>bet din</i> chose to interpret <i>Iggerot Mosheh</i>'s comment regarding the father's lack of observance, not as a reference to that factor <i>per se</i>, but in the context of the total fact pattern, i.e., because of the father's non-observance he had no knowledge of what priesthood connotes and may therefore have been engaged in idle speculation. Accordingly, the <i>bet din</i> declared that a different result would obtain in a situation in which a non-observant father explicitly presents the information in the context of transmitting a family tradition and the report is accepted by the son as credible. That interpretation would appear to be compelled unless <i>Iggerot Mosheh</i> is to be understood as rejecting the notion that <i>shavya anafsheih</i> may arise from reliance upon statements of others.
  However, the Jerusalem <i>bet din</i> acknowledged that <i>Iggerot Mosheh</i> may have dismissed the issue of <i>shavya anafsheih</i> for a different reason. The <i>bet din</i> presumed that the principle of <i>shavya anefsheih ḥatikha deissura</i> is limited to actual and intended declaration of a prohibition or <i>issura</i> rather than to the establishment of facts that give rise to a prohibition. Accordingly, in the case at hand, the <i>bet din</i> ruled that <i>shavya anafsheih</i> would be applicable only if the person making such representations was cognizant of the fact that, as a result of declaring himself to be a <i>kohen</i>, he would be forbidden to marry a divorcée. That view is consistent with the position of <i>Teshuvot Shev Ya'akov</i>, cited by <i>Kezot ha</i>-<i>Hoshen</i> 80:2, who maintains that <i>shavya anafsheih</i> is a form of vow and hence is applicable only if the person understands the consequences of his statement; otherwise, it is a vow undertaken in error and therefore of no effect.9<i class="footnote">See also <i>Sha’ar ha-Melekh</i>, <i>Hilkhot Ishut</i> 9:15; R. Ezekiel Landau, <i>Teshuvot Noda Bi-Yehudah</i>, <i>Even ha-Ezer</i>, <i>Tinyana</i>, no. 23, s.v. <i>u-mah she-raẓah le-ḥadesh</i>; R. Joseph Saul Nathanson, <i>Teshuvot Sho’el u-Meshiv</i>, vol. 1, no. 1; <i>Amudei Esh</i>, no. 4, <i>klal</i> 2; R. Abraham I. Ableman, <i>Teshuvot Zikhron Yehonatan</i>, p. 165a; R. Chaim Chizkiyahu Medini<i>, Sedei Ḥemed</i>, <i>ma’arekhet ha-shin</i>, <i>klal</i> 2; <i>idem</i>., <i>Teshuvot Or Li</i>, no. 129, sec. 9 and R. Shalom Mordecai Schwadron, <i>Da’at Torah, Yoreh De’ah</i> 1:69.</i> <i>Kezot ha-Hoshen</i> himself rejects that view and advances the thesis that <i>shavya anefsheih</i> simply reflects the credibility that the Torah extends to a person's statements insofar as they affect himself. Accordingly, <i>Kezot ha-Hoshen</i> regards ignorance of the implications of such a statement to be inconsequential. <i>Hatam Sofer</i>, in his previously-cited responsum, adopts a position compatible with that of <i>Shev Ya'akov</i> in ruling that a husband who declares that his wife has committed adultery may not continue the marital relationship even though, when making the statement, the husband was unaware of the halakhic rule forbidding an adulterous wife to her husband.10<i class="footnote">The Jerusalem <i>bet din</i> seems to cite <i>Teshuvot Ḥatam Sofer</i>, <i>Even ha-Ezer</i>, I, no. 78, s.v. <i>hinei</i>, as adopting the position that <i>shavya anafsheih</i> is operative only if the individual making the pronouncement is aware of its halakhic implications. In point of fact, <i>Ḥatam Sofer</i> asserts something rather different. <i>Ḥatam Sofer</i> declares that a <i>kohen</i> who announced that he believes his wife’s statement that she was raped may recant upon learning that, if such were the case, she would be prohibited to him. <i>Ḥatam Sofer</i>’s rationale is not that failure to appreciate the consequence defeats <i>shavya anafsheih</i> but that, once the <i>kohen</i> becomes aware of the prohibition against continued marital relations, he realizes that there may have been a motive for his wife to fabricate the account. The fact that <i>Ḥatam Sofer</i> found it necessary to advance that explanation tends to indicate that he rejects the view of <i>Shev Ya’akov</i>. Cf., <i>Teshuvot Ḥatam Sofer</i>, <i>Yoreh De’ah</i>, no. 185, in which <i>Ḥatam Sofer</i> cites both theories regarding the basis of <i>shavya anafsheih</i>. See also <i>Teshuvot Ḥatam Sofer</i>, <i>Even ha-Ezer</i>, I, no. 28, in which <i>Ḥatam Sofer</i> describes <i>shavya anafsheih</i> either as a vow or as a reflection of the principle that a witness does not have capacity to recant.</i>
  3. Hazakah
  Another factor that must be considered with regard to the question of priestly identity is the possibility that investigation would disclose the existence of a <i>ḥazakah</i> with regard to the family's priestly status. In this context, the notion of <i>ḥazakah</i> is nothing other than common knowledge within a community of the status of an individual or of his forebears. Thus, although the father may never have attended synagogue and hence never have been accorded Torah honors, it may be the case that there are records in the archives of the European community from which the family hailed indicating that the grandfather was a <i>kohen</i>. Discovery of such information subsequent to marriage to a woman forbidden to a <i>kohen</i> would be disastrous. Although the matter has not received attention in the responsa literature, a gravestone indicating that the grandfather was a <i>kohen</i> might be deemed evidence of "common knowledge" of such status within the community.11<i class="footnote">R. Joseph B. Soloveitchik maintained that a gravestone symbol is sufficient in and of itself to establish priestly identity. See “Surrendering to the Almighty,” <i>Light</i>, 17 <i>Kislev</i> 5736, p. 15. A transcript of the lecture upon which that article is based is available at  That consideration becomes even more compelling if, as often was the case in European communities, gravestone inscriptions were supervised by the local <i>ḥevra kaddisha</i> (burial society).12<i class="footnote">For an example of control over gravestone inscriptions see the statute of the <i>ḥevra kaddisha</i> of Eisenstadt cited by R. Joseph Schwartz in a note to <i>Hadrat Kodesh</i>: <i>Ẓion le-Nefesh Ḥayah</i>, sec. 9, appended to <i>Zikaron le-Mosheh</i> (Arda’a, 5698).Even at present, when cemeteries are owned by the <i>ḥevra kaddisha</i>, it exercises control over gravestone inscriptions as evidenced by several instances of litigation in recent years. In a recent case, an Israeli <i>ḥevra kaddisha</i> sought to ban the use of a civil date. See <i>Fredrikah Shavit</i> v. <i>Ḥevra Kaddisha Gaḥasha Rishon le-Ẓion</i>, <i>Piskei Din shel Bet ha-Mishpat ha-Elyon</i>, vol. 53, part 3 (1977), pp. 600-663. Similar issues were earlier adjudicated by the Israeli Supreme Court in <i>Gidon</i> v. <i>Ḥevra Kaddisha Gaḥasha</i>, <i>Piskei Din shel Bet ha-Mishpat ha-Elyon</i>, vol. 27, part 1 (1973), pp. 10-25 and <i>Ḥevra Kaddisha Gaḥasha “Kehillot Yerushalayim”</i> v. <i>Lionel Aryeh Kestenbaum</i>, <i>Piskei Din shel Bet ha-Mishpat ha-Elyon</i>, vol. 46, part 2 (1992), pp. 464-538.More recently a London <i>ḥevra kaddisha</i> forbade a description of the deceased as a socialist and was sued in a British court. See <i>The Jewish Chronicle</i>, December 23, 2005, p. 3. That matter has apparently become moot since, upon consultation with the London <i>bet din</i>, the <i>ḥevra kaddisha</i> relented and permitted the inscription. See <i>The Jewish Chronicle</i>, February 17, 2006, p. 34.</i>
  Whether the existence of a <i>ketubah</i> indicating that the father or grandfather was a <i>kohen</i> is evidence of priestly status is a matter of controversy among early authorities. Rambam, <i>Hilkhot Issurei Bi'ah</i> 20:9, regards such evidence as sufficient to create at least doubtful status and to entitle such an individual to partake of priestly offerings obliged by rabbinic, and distinct from biblical, decree. Re'ah, cited by Ran, <i>Ketubot</i> 24b, and Ritva, <i>ad locum</i>, express doubt with regard to whether any evidentiary weight is to be assigned to such instruments. <i>Shakh</i>, <i>Hoshen Mishpat</i> 49:15, explains that the underlying issue is whether the attesting witnesses testify only to the substantive provisions recorded in the document or whether they testify to ancillary matters contained therein as well.
  The <i>bet din</i> also opined that the family name "Cohen" is sufficient to establish a presumption of priestly status.13<i class="footnote">That position is reiterated by the <i>bet din</i> in <i>Piskei Din</i>, <i>ibid.</i>, vol. 3, part 2, p. 290.</i> The <i>bet din</i> reasoned that, since the "vast majority" of individuals bearing the name "Cohen" are <i>kohanim</i>, ascription of that name to the family generates a <i>ḥazakah</i> or presumptive evidence of priestly status. In a footnote the <i>bet din</i> takes cognizance of the opinion of R. Israel Meir Lau, <i>Yaḥel Yisra'el</i>, I, no. 30, who maintains that, absent a family tradition to that effect or a <i>ḥazakah</i> established by individual members of the family, the surname alone is not dispositive. His primary consideration is that, in some European communities, marriages were not registered in a timely fashion with the result that, not infrequently, children were registered as bearing the mother's family name.
  In a response to an inquiry by the <i>bet din</i> in a different case published by the Jerusalem <i>bet din</i> in <i>Piskei Din</i>, <i>ibid.</i>, III, part 2, p. 291, R. Samuel ha-Levi Woszner declares that the family name is of no consequence in determining priestly status.
  Marriage of a Kohen and the Daughter of a Non-Jew
  In a decision issued on 19 Iyar 5766, the Rabbinical District Court of Rehovot confirmed its earlier ruling declaring an applicant for a marriage license to be eligible to contract a marriage as a Jewess but denied her permission to marry a <i>kohen</i> on the grounds that she was the issue of a union between a Jewish woman and a non-Jewish man.14<i class="footnote">Index no. 321328494-15-1.</i>
  The decision, on the surface, was routine and unexceptionable. Denial of the application for a marriage license was based upon an explicit provision of Halakhah recorded twice in <i>Shulḥan Arukh</i>, <i>Even ha-Ezer</i> 4:5 and 4:19 and implicit in the ruling of <i>Shulḥan Arukh</i>, <i>Even ha-Ezer</i> 7:17. The Rehovot <i>bet din</i> composed a detailed analysis of the basis of that ruling only because the case was remanded to them on appeal for further consideration by the Supreme Rabbinical Court of Appeals.
  The Gemara, <i>Yevamot</i> 45a, posits a controversy among the <i>Amora'im</i> regarding the status of the progeny of a union of a non-Jewish man and a Jewish woman. One opinion is that the child born of such a liaison is a <i>mamzer</i> while a second opinion maintains that the child is legitimate but, if a female, is forbidden to marry a priest. The Gemara, <i>Yevamot</i> 45b, concludes, "The law is that if a non-Jew or a slave consorts with a daughter of Israel the child is legitimate." Ostensibly, that definitive ruling represents an adjudication of the earlier controversy and the term "<i>kosher</i>" or "legitimate" is used as the antonym of "<i>mamzer</i>" or "bastard" with the implication that the child is legitimate but nevertheless forbidden to a <i>kohen</i> as had earlier been stated. Rosh understands the Gemara in that manner and rules that the daughter is certainly forbidden to a <i>kohen</i>.
  Rif, however, states that some authorities rule that the daughter is not only legitimate but permitted to a <i>kohen</i> as well. Presumably, the rationale underlying that position is that, since there is no halakhically recognized paternal-filial relationship between the biological father and the child, there is no basis for disqualifying her from marrying a <i>kohen</i>. The term <i>kosher</i> is understood by the authorities cited by Rif as "legitimate" in the fullest sense of the term and the dictum is understood by him, not as adjudication between two conflicting views, but as the expression of a third, previously unexpressed position, which is pronounced to be normative.
  Ramban, <i>Yevamot</i> 45a, similarly asserts that the Gemara records three distinct views with regard to the status of such progeny and, although the Gemara clearly rules that the child is not a <i>mamzer</i>, there is no definitive ruling with regard to whether a daughter of such a liaison may marry a <i>kohen</i>. Since the issue remains a matter of doubt, Ramban rules that such a girl may not enter into marriage with a <i>kohen</i> but, <i>post factum</i>, if such a marriage has been celebrated, they cannot be compelled to divorce.15<i class="footnote">Rambam’s position is a matter of some dispute. Rambam, <i>Hilkhot Issurei Bi’ah</i> 15:3, rules that the child is legitimate but makes no reference to the status of a daughter <i>vis-à-vis</i> a <i>kohen</i>. Accordingly, <i>Maggid Mishneh</i>, <i>ad locum</i>, presumes that Rambam regards the daughter to be permitted to a <i>kohen</i>. However, <i>Mishneh le-Melekh</i>, <i>ad locum</i>, advances evidence indicating that Rambam maintains that she is forbidden to a <i>kohen</i>.</i> Ramban ascribes that view to Rif as well. The view that Rif regards the matter as remaining in doubt is supported by the published text of Rif.16<i class="footnote">Cf., however, <i>Nemukei Yosef</i>, <i>ad locum</i>, and R. Yechiel Ya’akov Weinberg, <i>Teshuvot Seridei Esh</i> (Jerusalem, 5726), III, no. 54, reprinted in <i>Teshuvot Seridei Esh</i> (Jerusalem, 5763), I, no. 72. For a comprehensive survey of the positions of early-day authorities regarding this issue see <i>idem</i>, <i>Teshuvot Seridei Esh</i> (Jerusalem, 5763), I, no. 71.</i>
  As previously noted, the prohibition against marriage of such a woman to a <i>kohen</i> is explicitly codified by <i>Shulḥan Arukh</i> in two places, <i>Even ha-Ezer</i> 4:5 and <i>Even ha-Ezer</i> 4:19. Both <i>Helkat Meḥokek</i>, <i>Even ha-Ezer</i> 4:3 and 7:26, and <i>Bet Shmu'el</i>, <i>Even ha-Ezer</i> 4:2 and 7:39, confirm Ramban's view in ruling that the daughter of such a liaison is forbidden to marry a <i>kohen</i> but, if already married, the couple need not be compelled to divorce.17<i class="footnote"><i>Shulḥan Arukh</i> rules simply that such a marriage is forbidden but does not expressly address the <i>post factum</i> status of such a couple. R. Chaim Joseph David Azulai, <i>Birkei Yosef</i>, <i>Even ha-Ezer</i> 4:19, regards that omission as indicating that <i>Shulḥan Arukh</i> maintains that even after the fact they are forbidden to live together as man and wife. See also <i>Dibberot Eliyahu</i>, IV, no. 31. Cf., however, R. Shalom Mashash, <i>Shemesh u-Magen</i>, III, <i>Even ha-Ezer</i>, no. 58.</i> That ruling is explicitly accepted by R. Moshe Feinstein, <i>Iggerot Mosheh</i>, <i>Even ha-Ezer</i>, I, no. 6. R. Ovadiah Yosef, <i>Teshuvot Yabi'a Omer</i>, VII, <i>Even ha-Ezer</i>, no. 60, is in fundamental agreement with that position but rules that, since Ramban regards the status of such a woman to be doubtful, in the presence of a second "doubt," e.g., the status of the prospective groom as a <i>kohen</i> is itself doubtful, the marriage may be permitted on the basis of <i>sefek sefeika</i> or "double doubt."
  The case heard by the <i>bet din</i> of Rehovot involved one additional consideration. Before applying for a marriage license the couple had been living together for a period of approximately one-and-a-half years and had announced that, if a license was denied, they would continue to do so without benefit of marriage. The issue of a <i>kohen</i> already living with the daughter of a Jewish mother and a non-Jewish father pursuant to civil marriage has been analyzed by R. Moshe Feinstein in his earlier-cited responsum18<i class="footnote"><i>Iggerot Mosheh</i> forbids marriage even in such circumstances. A similar decision is published in <i>Piskei Din shel ha-Rabbanut ha-Rashit le-Yerushalayim</i>, III (5755), part 2, pp. 293-297.</i> and by the late Sephardic Chief Rabbi of Jerusalem, R. Shalom Mashash, <i>Shemesh u-Magen</i>, III, <i>Even ha-Ezer</i>, nos. 55 and 58.
  Ramban's position that the status of such a woman is unresolved led him to rule that she may not marry a <i>kohen</i> but, if married, the couple cannot be compelled to divorce. That ruling seems somewhat contradictory. The couple is forbidden to marry because of the possibility of transgression. The selfsame possibility of transgression persists within an already contracted marital union. If so, would not logical consistency lead to the conclusion that, if married, they must divorce? A number of latter-day authorities have endeavored to resolve that perplexity.
  One possible solution is that the prohibition is rabbinic in nature and the edict was not designed to apply <i>post factum</i>. However, the most cogent resolution is that advanced by <i>Sha'ar ha-Melekh</i>, <i>Hilkhot Issurei Bi'ah</i> 15:3. A divorce executed under duress by a <i>bet din</i> is valid only if there are halakhic grounds for coercion. However, asserts <i>Sha'ar ha-Melekh</i>, if the transgression is doubtful, it follows that the grounds for coercion are also doubtful. If so, argues <i>Sha'ar ha-Melekh</i>, a <i>get</i> cannot be compelled in cases of doubt born of halakhic controversy because its validity would perforce be a matter of doubt with the result that the woman could not be permitted to remarry on the strength of such a <i>get</i>. Nevertheless, as a matter of religious law, the parties themselves are forbidden to continue to live together as man and wife. <i>Iggerot Mosheh</i> accepts <i>Sha'ar ha-Melekh</i>'s explanation in ruling that a religious ceremony may not be performed for such a couple despite the fact that they have been living together in an ongoing relationship pursuant to civil marriage.19<i class="footnote">Cf., however, R. Ovadiah Yosef, <i>Teshuvot Yabi’a Omer</i>, VII, <i>Even ha-Ezer</i>, no. 9, sec. 6, who cites an oral report regarding Sephardic authorities who permitted performance of a marriage pursuant to a civil marriage in France. They considered the matter to be <i>post factum</i> because France, at the time, did not allow civil divorce and remarriage. Sanction of a marriage under such circumstances is certainly not consistent with the position of <i>Sha’ar ha-Melekh</i>.</i> <i>Sha'ar ha-Melekh</i>'s resolution of the problem is particularly compelling if it is accepted that the prohibition is biblical in nature.
  Rabbi Mashash, on the other hand, in his previously cited responsum, was quite willing to permit marriage of such couples not only in situations in which a civil marriage had already taken place20<i class="footnote">See also Rabbi Mashash’s position in another case published in <i>Piskei Din shel ha-Rabbanut ha-Rashit le-Yerushalayim</i>, vol. VI (5759), part 2, p. 183.</i> but also in situations in which the parties had already been sharing an abode.21<i class="footnote">R. Joshua Ehrenberg, <i>Teshuvot Dvar Yehoshu’a</i>, III, <i>Even ha-Ezer</i>, no. 8, permits such a marriage even in a situation in which the couple is engaged on the basis of the consideration that no other man may be willing to marry the woman.</i> He bases his conclusion upon two considerations:
  1) A Jew suspected of consorting with a non-Jewish woman may not marry her subsequent to her conversion. However, if they have already married he is not obligated to execute a <i>get</i>. There is significant controversy with regard to whether, in such a situation, civil marriage constitutes a <i>post factum</i> condition. Those opinions have been analyzed in detail by this writer in <i>Contemporary Halakhic Problems</i>, I (New York, 1977), 286-290. Rabbi Mashash chooses to cite only the permissive opinions and entirely ignores the opposing position. He then summarily equates the case of a union between a <i>kohen</i> and the daughter of a non-Jew with that of a Jew suspected of having had a sexual relationship with a non-Jewish woman.
  Apart from the controversy with regard to the effect of civil marriage in the latter situation, according to <i>Sha'ar ha-Melekh</i>'s explanation of the rationale underlying that provision in the case of a <i>kohen</i>, the two situations are obviously entirely dissimilar.
  Rashi, <i>Yevamot</i> 24b, explains that the prohibition against a Jew marrying a convert with whom he is suspected of having had a liaison was promulgated in order to safeguard the honor and reputation of the husband since marriage under such circumstances is likely to lend credence to rumors of previous immorality. Accordingly, some authorities conclude that the prohibition does not extend to situations in which the couple have been living together publicly since in such cases previous immoral conduct is an established verity.22<i class="footnote">See this writer’s <i>Contemporary Halakhic Problems</i>, I (New York, 1977), 286, note 42.</i> <i>Teshuvot ha-Rashba</i>, I, no. 1,205, explains that, if the couple are suspected of having lived together previously, their marriage subsequent to conversion will lead to suspicion that the conversion itself was insincerely contrived for purposes of marriage. Accordingly, a number of authorities assert that, according to Rashba, if the couple have already established a permanent conjugal relationship, there can be no grounds for the suspicion that the conversion was insincerely entered into merely for the sake of marriage.23<i class="footnote">See <i>ibid</i>., p. 289.</i> Quite obviously, according to these lines of reasoning, in the case of a converted paramour, civil marriage or a longstanding conjugal relationship is not equated with a valid marriage as creating a <i>post factum</i> situation but is regarded as rendering the prohibition entirely nugatory.
  2) Rabbi Mashash asserts, again quite unequivocally, that in both instances the prohibition is rabbinic in nature and hence lenience is in order in matters of doubt. It is certainly the case that the prohibition against marrying a proselyte in the situation described is rabbinic in nature but it is far from clear that such is the case with regard to a <i>kohen</i> and the daughter of a non-Jewish father. Indeed, if the prohibition is merely rabbinic and if, as Ramban maintains, the permissibility of such a marriage is a matter of doubt, it is difficult to understand why the marriage is not permitted even <i>ab initio</i>. R. Moshe Feinstein, <i>Iggerot Mosheh</i>, <i>Even ha-Ezer</i>, I, no. 5, depicts the prohibition as based upon doubt with regard to a biblical transgression. Nevertheless, <i>Teshuvot Rema mi-Panu</i>, no. 124, <i>Helkat Meḥokek</i>, <i>Even ha-Ezer</i> 7:26, <i>Teshuvot R. Akiva Eger</i>, no. 91 and <i>Bet Me'ir</i>, <i>Even ha-Ezer</i> 4:5, regard the prohibition as rabbinic in nature.24<i class="footnote">The question is contingent upon whether the <i>kal va-ḥomer</i> formulated by the Gemara, <i>Yevamot</i> 45a, in establishing the prohibition is a compelling <i>a fortiori</i> argument or a mnemonic device associated with a rabbinic edict.</i>
  R. Shlomoh Amar, <i>Shema Shlomoh</i>, V, <i>Even ha-Ezer</i>, no. 8, endorses Rabbi Mashash's ruling and adds that, in a sexually promiscuous age such as ours, it is quite possible that the groom's mother may have engaged in sexual conduct that would serve to deprive any future progeny of the priesthood.
  The <i>bet din</i> of Rehovot took note of the opinions of Rabbis Mashash and Amar but declined to accept them as authoritative. The matter was appealed to the Supreme Rabbinical Court of Appeals and, in a brief decision, the Court of Appeals simply cited the responsa of Rabbis Mashash and Amar, reversed the ruling of the rabbinical district court and "requested" the rabbinate of Rehovot to perform the marriage.25<i class="footnote">Index no. 321328494-12-1.</i> One of the members of the panel hearing this case was R. Shlomoh Amar.
  The halakhic authority of the Israeli Supreme Rabbinical Court of Appeals to reverse decisions of district courts is not at all clear. That issue has been discussed by this writer in <i>Contemporary Halakhic Problems</i>, IV (New York, 1995), 17-45. The theories offered in support of such authority are much more persuasive in matters of jurisprudence than in matters of religious law. The general rule "If a scholar has prohibited another scholar dare not permit" (<i>Berakhot</i> 63b, <i>Avodah Zarah</i> 77b, <i>Hullin</i> 44b and <i>Niddah</i> 20b) admits of exceptions in instances of clear error. It does, however, bar exercise of purely subjective discretion in choosing one set of precedents over another.
  It is difficult to believe that any objective scholar would find Rabbi Mashash's reasoning absolutely compelling. It is impossible to fault the <i>bet din</i> of Rehovot for assaying greater weight to the responsa of Rabbis Feinstein and Yosef than to those of Rabbis Mashash and Amar. It is quite easy to conclude that the lower court's decision was overruled simply because Rabbi Amar chose to follow his own previously announced opinion.
  Such a course of action might well be legitimate were Rabbi Amar able to allege a lack of cogency in the opposing view or to show that the weight of precedent contradicted the decision of the lower court. This he made no attempt to do. Mere preference for one set of precedents over another without a clear elucidation of doctrinal support is not the manner in which appellate review is carried out in the common law judicial system from which the Israeli system of rabbinic appellate review was adopted. Such a procedure strikes this writer as a misuse of appellate power.
  Furthermore, there is some question with regard to whether the appellate jurisdiction of the Supreme Rabbinical Court extends to "non-adversarial" matters. In an unpublished decision issued in 5702, Index no. 1/46/701, the Supreme Rabbinical Court held that it had appellate authority in all matters. However, in a subsequent unpublished decision issued in 5716 the Supreme Rabbinical Court ruled that it had no authority to hear appeals in "non-adversarial" matters, i.e., in determining issues of Jewish religious law since such matters are not within the ambit of authority granted to the <i>bet din</i> by virtue of the applicable Israeli civil law. The issue before the court involved the conversion of a minor child by its Jewish father in face of the announced opposition of its non-Jewish mother. The district court declared that it was not acting by virtue of the powers vested in a <i>bet din</i> to adjudicate disputes but was simply announcing a matter of religious law. The Supreme Rabbinical Court ruled that such matters are not subject to appeal.26<i class="footnote">See Eliav Shochetman, <i>Seder ha-Din (Civil Practice in Jewish Law)</i> (Jerusalem, 5748), p. 450.</i> The question of whether a daughter of a non-Jewish father is permitted to marry a <i>kohen</i> is similarly a non-adversarial issue of religious law.27<i class="footnote">For a comprehensive discussion of the issues involved in determining the ambit of the appellate authority of the Supreme Rabbinical Court see this writer’s <i>Contemporary Halakhic Problems</i>, IV (New York, 1995), 17-45.</i>
  Moreover, as cogently demonstrated by <i>Iggerot Mosheh</i>, even in circumstances in which the couple will continue to cohabit even without benefit of marriage, celebration of a wedding ceremony is tantamount to rabbinic license to commit an even graver transgression. According to many early-day authorities, cohabitation outside of marriage constitutes a rabbinic infraction. Rambam, <i>Hilkhot Issurei Bi'ah</i> 17:12, maintains that priestly conjugal prohibitions are biblically binding only within a marital relationship. Thus, according to Rambam, performance of a wedding ceremony transforms what would otherwise be a rabbinic infraction into a more serious biblical transgression.
  It is, however, not inconceivable that the same result might have been achieved on the basis of other considerations. As pointed out by Rabbi Feinstein in his earlier-cited responsum, in Jewish law, cohabitation for purposes of marriage creates a marital relationship when such is the intention of the parties. Whether or not such intention is presumed in a relationship tantamount to common law marriage depends upon whether the presumption that couples do not engage in fornication when the option of a marital union is available is applicable in situations of the nature described. An analysis of that very complex issue is beyond the scope of the present discussion. Suffice it to say that a <i>bet din</i>, in appropriate circumstances, might reasonably have concluded that the existence of a valid halakhic marriage necessitating a <i>get</i> for its dissolution was at least a matter of doubt. Since, in the case of a <i>kohen</i> married to the daughter of a non-Jew, a <i>get</i> cannot be compelled, it should follow that the recalcitrant parties may not be prevented from regularizing their relationship by means of a formal marriage.
  Sonograms and the Unborn Kohen
  The verse "Speak unto the priests, the sons of Aaron and say unto them…." (Leviticus 21: 1) serves as a prefatory comment to the prohibition against priestly defilement through contact with a corpse. The <i>Gemara</i>, <i>Yevamot</i> 114a, takes note of the double expression "speak" and "say unto them" in declaring that Moses was instructed to transmit two prohibitions: 1) a commandment addressed to adult priests forbidding them to defile themselves; and 2) a directive commanding adults not to defile young priests who have not yet attained the age of halakhic majority. The latter prohibition is recorded in <i>Shulḥan Arukh</i>, <i>Yoreh De'ah</i> 373:1. Rema adds that the prohibition is limited to the performance of an overt act on the part of an adult causing defilement of the minor. As explained by <i>Shakh</i>, <i>Yoreh De'ah</i> 373:1, intervention to restrain a child from defiling himself is mandated only by virtue of a father's obligation with regard to <i>ḥinnukh</i>, i.e., "training," and hence is limited to the child's father and, moreover, becomes applicable only at an appropriate age. The opinion of <i>Baḥ</i>, <i>Yoreh De'ah</i> 373, to the effect that even an infant must be prevented from defiling himself is not accepted by other authorities. However, <i>Magen Avraham</i>, <i>Oraḥ Hayyim</i> 343:2, and <i>Mishnah Berurah</i> 343:3 emphasize that actively causing defilement is forbidden even in the case of a child completely lacking in reason.
  R. Eliezer ben Judah of Worms, <i>Sefer Rokeaḥ</i>, no. 366, declares that the pregnant wife of a <i>kohen</i> may enter a building in which a corpse is present by virtue of application of the principle of <i>sefek sefeika</i>, i.e., "double doubt." There are two separate contingencies that would render defilement permissible, <i>viz.</i>, perhaps the pregnant woman will miscarry because the fetus she is gestating is not viable and, even if the fetus is viable, perhaps it is a female. There is no prohibition against defiling either a non-viable fetus or a daughter of a <i>kohen</i>. However, <i>Rokeaḥ</i> certainly maintains that, in principle, it is forbidden to defile a fetus. <i>Rokeaḥ</i>'s ruling is confirmed by <i>Shakh</i>, <i>Yoreh De'ah</i> 371:1.
  Although <i>Rokeaḥ</i>'s position is accepted as normative, there are reasons for stringency on the part of the wife of a <i>kohen</i> in refraining from coming into contact with a corpse. R. Chaim Chizkiyahu Medini, <i>Sedei Hemed</i>, <i>ma'arekhet ha-samakh</i>, sec. 71, cites a number of authorities who assert that, although it is permissible to rely upon a "double doubt," it is nevertheless a meritorious act of piety not to do so.
  <i>Rokeaḥ</i>'s construction of this <i>sefek sefeika</i> is challenged by <i>Pri Megadim</i>, <i>Oraḥ Hayyim</i>, <i>Eshel Avraham</i> 343:2. <i>Tosafot</i>, <i>Ketubot</i> 9a, s.v. <i>ve-i ba'it eima</i>, declare that a doubt resolved by virtue of the principle of <i>rov</i>, or majority, does not constitute a cognizable doubt at all. <i>Tosafot</i> opine that the occurrence of rape generally becomes known to the public. Hence, in a significant majority of all sexual acts that do not give rise to reports of rape, intercourse is willful. Accordingly, declare <i>Tosafot</i>, the contingency of rape cannot be deemed a matter of doubt for the purpose of constructing a <i>sefek sefeika</i>. Similarly, argues <i>Pri Megadim</i>, the majority of pregnancies result in viable neonates; hence the possibility of miscarriage should not constitute a doubt for purposes of generating a <i>sefek sefeika</i>.
  R. Moshe Sofer, <i>Teshuvot Hatam Sofer</i>, <i>Yoreh De'ah</i>, no. 354, resolves that difficulty by pointing out that the principle of <i>rov</i> serves, in and of itself, to determine that in the majority of cases the woman commits no transgression in causing her fetus to be defiled. Fifty percent of all fetuses are female28<i class="footnote">This is empirically imprecise. Historically, the ratio of male births to female births has been 106 to 100. In recent years there has been a decline in the proportion of male births in North America and Western Europe. Those declines, although statistically significant, are small, with the result that there remains a plurality of male births over female births. See, for example, P. Morcelli, P.M. Thouth, P. E. Serrari et al., “Changes in Human Sex Ratio,” <i>Lancet</i>, vol. 355, no. 9218 (May 24, 2000), pp. 1858-1863. However, since the number of miscarriages and stillbirths is far higher than 6%, <i>Ḥatam Sofer</i>’s crucial point, <i>viz.</i>, that in the majority of pregnancies the woman commits no transgression, is indeed valid.</i> while some fetuses are not viable. The result is that in any given pregnancy there is less than a fifty percent chance that defiling the fetus entails a forbidden act.29<i class="footnote">See also R. Joseph Babad, <i>Minḥat Ḥinnukh</i>, no. 263.</i> Although <i>Rokeaḥ</i> invokes the principle of <i>sefek sefeika</i> rather than <i>rov</i>, <i>Hatam Sofer</i> asserts that the principle involved is indeed that of <i>rov</i> and that <i>Rokeaḥ</i>'s nomenclature is merely unfelicitous. There are indeed two separate factors to be considered—and hence <i>Rokeaḥ</i>'s use of the term <i>sefek sefeika</i>—the possibility of a female fetus and the possibility of a nonviable fetus; however the two factors combine not to generate a <i>sefek sefeika</i>, but to generate a <i>rov</i>.
  Acceptance of the thesis that <i>Rokeaḥ</i> is expressing the concept of <i>rov</i> rather than <i>sefek sefeika</i> yields yet another ramification. The putative practice of not relying upon a "double doubt" does not apply to not relying upon <i>rov</i>. <i>Hatam Sofer</i> himself, <i>Oraḥ Hayyim</i>, no. 83, forcefully comments, "He who commanded and admonished us with regard to the <i>treifah</i> is He who commanded us to rely upon <i>rov</i>."30<i class="footnote">See also <i>Pitḥei Teshuvah</i>, <i>Yoreh De’ah</i> 116:10 as well as R. Moses Luria, <i>Avnei Shoham</i>, <i>Ḥullin</i> (Jerusalem, 5747), I, no.1.</i>
  <i>Magen Avraham</i>, <i>Oraḥ Hayyim</i> 346:2, takes issue with the basic principle upon which <i>Rokeaḥ</i>'s comment is based. As noted, <i>Rokeaḥ</i> assumes that it is forbidden to cause defilement not only to a minor but also to a fetus. <i>Magen Avraham</i> challenges that assumption on technical grounds. One of the principles governing defilement as recorded in the Mishnah, <i>Oholot</i> 5:4, is that a "swallowed" (<i>balu'ah</i>) object can neither defile nor become defiled. Thus, for example, if a person swallows a metal ring and then enters a room in which a corpse is present, the ring is not subject to the regulations governing ritual impurity. The fetus, argues <i>Magen Avraham</i>, is "swallowed" within the mother's body and hence is not subject to defilement. If so, there is no reason why the pregnant wife of a <i>kohen</i> may not come into contact with a corpse.31<i class="footnote"><i>Pri Megadim</i> suggests that “swallowed” objects are subject to defilement by virtue of rabbinic decree. <i>Teshuvot Ḥatam Sofer</i>, <i>Yoreh De’ah</i>, no. 354, dismisses that contention with the observation that, assuming there is rabbinic defilement of that nature, such defilement is not necessarily prohibited even to the <i>kohen</i> just as, for example, there is no prohibition against a <i>kohen</i> defiling himself by contact with a metal object defiled by a corpse. There is certainly no evidence of a specific rabbinic decree forbidding <i>tum’ah balu’ah</i> to a <i>kohen</i>.</i>
  R. David ibn Zimra, <i>Teshuvot Radvaz</i>, I, no. 200, resolves the problem by postulating that <i>Rokeaḥ</i>'s concern was limited to situations involving a pregnant woman approaching term. In such situations there is reason to fear that labor may commence at any time and hence that the child's head may enter the birth canal in the early stages of parturition while the woman is yet under the same roof as the corpse. Since, for purposes of Jewish law, the child is considered to be "born" when the major portion of the forehead enters the birth canal, the child would no longer be deemed "swallowed" within the mother's abdomen and hence would become subject to defilement. A similar observation is made by <i>Netiv Hayyim</i> in a marginal comment on <i>Magen Avraham</i>, <i>ad locum</i>. According to those authorities, <i>Rokeaḥ</i> had no need to justify contact with a corpse on the part of a pregnant woman during earlier periods of gestation since the fetus will assuredly remain entirely within her body; the need for justification was limited to the final stage of pregnancy when there is reason to fear that the infant's head may emerge.
  R. Jacob Emden, <i>She'ilat Ya'avez</i>, II, no. 177, advances a completely opposite interpretation of <i>Rokeaḥ</i>'s position. <i>She'ilat Ya'avez</i> contends that <i>Rokeaḥ</i>'s comment is designed to justify contact with a corpse only during early stages of pregnancy but that, according to <i>Rokeaḥ</i>, the pregnant wife of a <i>kohen</i> is indeed forbidden contact with a corpse during the ninth month of pregnancy. As does <i>Hatam Sofer</i>, <i>She'ilat Ya'avez</i> dismisses the notion of <i>sefek sefeika</i> on the grounds that, since the majority of pregnancies do not result in either a miscarriage or stillbirth, the possibility of a non-viable fetus does not constitute a halakhically cognizable doubt. Hence contact with a corpse during the ninth month, when emergence of the forehead is a distinct possibility, is forbidden. <i>Rokeaḥ</i>, he contends, was concerned with establishing the permissibility of such contact only during earlier stages of pregnancy.
  Such justification is necessary, argues <i>She'ilat Ya'avez</i>, because, in order to obviate error and confusion, the Gemara, <i>Hullin</i> 72a, in a different context, extends the rules cogently applicable during the ninth month of pregnancy to earlier stages of pregnancy as well. Ostensibly, that consideration should also apply in the case of a pregnant wife of a <i>kohen</i>. Accordingly, explains <i>She'ilat Ya'avez</i>, <i>Rokeaḥ</i> intended to explain that contact with a corpse is not prohibited during earlier stages of pregnancy lest the pregnant woman assume that such contact is permitted during the final stages of pregnancy as well. <i>She'ilat Ya'avez</i> asserts that <i>Rokeaḥ</i> reasons that no such ban is applicable because of the remote nature of the concern, i.e., there is no actual concern in the earlier stages of pregnancy because the fetus' head is unlikely to emerge from the uterus and, even were the woman to place herself in such a situation during the final stages of pregnancy, the fetus may be either female or nonviable. In effect, as does <i>Hatam Sofer</i>, albeit for a different reason, <i>She'ilat Ya'avez</i> understands <i>Rokeaḥ</i> as declaring simply that the contingency that is the subject of concern is remote and hence not a matter of halakhic concern, but not as contending that the principle of <i>sefek sefeika</i> is technically applicable.
  R. Chaim Joseph David Azulai, <i>Birkei Yosef</i>, <i>Oraḥ Hayyim</i> 343:4, rejects the applicability of the principle of <i>tum'ah balu'ah</i> on the basis of the talmudic opinion, <i>Gittin</i> 23b, that maintains that the fetus is deemed to be an integral part of the mother's body (<i>ubbar yerekh imo</i>). As recorded by R. Jacob Ettlinger, <i>Teshuvot Binyan Ẓion</i>, no. 96, R. Pinchas Shiffer, writing to Rabbi Ettlinger, explains that <i>Birkei Yosef</i>'s contention is that the fetus cannot be considered to be "swallowed" since it is an integral part of the mother's body. Nevertheless, since the gender of the fetus is male rather than female, the fetus is subject to the rules governing defilement.
  <i>Binyan Ẓion</i> dismissively rejects that understanding of <i>Birkei Yosef</i> on the grounds that to declare the fetus to be an integral part of the mother's body but to have the status of a male is a contradictory assertion. <i>Hatam Sofer</i>, in his previously cited responsum, without citing <i>Birkei Yosef</i>, reports that he had earlier considered the issue of <i>tum'ah balu'ah</i> as inapplicable, at least prior to commencement of labor, because of the principle <i>ubbar yerekh imo</i>. However, upon reflection, <i>Hatam Sofer</i> declares that he has reversed his earlier view because of the pronouncement of the Gemara, <i>Yevamot</i> 67a, that "a fetus in the womb of a non-<i>kohen</i> is a non-<i>kohen</i>," i.e., priestly status, and hence the applicability of the prohibition against defilement, devolves upon the fetus only upon parturition.
  R. Moshe Sternbuch, <i>Teshuvot ve-Hanhagot</i>, I, <i>Yoreh De'ah</i>, no. 679, quotes the <i>Brisker Rav</i>, R. Yitzchak Ze'ev Soloveitchik, as advancing an entirely opposite view in stating that a male fetus does indeed have the status of a <i>kohen</i>. The statement of the Gemara to the effect that the fetus is not a <i>kohen</i> is limited, he contends, to laws of <i>terumah</i>, <i>viz.</i>, although the mother of a <i>kohen</i> may partake of <i>terumah</i>, a women carrying a fetus who is a <i>kohen</i> may not partake of <i>terumah</i>. The fetal <i>kohen</i>, according to this analysis, enjoys the status of a <i>kohen</i>, particularly with regard to matters of defilement, but is not a <i>kohen</i> who can cause his mother to become eligible to partake of <i>terumah</i>.
  <i>Minḥat Hinnukh</i>, no. 263, espouses a diametrically opposed position in maintaining that even were the fetus to be regarded as rendering its mother eligible to partake of <i>terumah</i>, the fetus itself, since it is an integral part of the mother's body, is not subject to strictures concerning defilement.
  <i>Binyan Ẓion</i> himself offers a novel interpretation of <i>Birkei Yosef</i>'s position. The fetus, as an integral part of the mother's body, becomes defiled together with its mother. But since there is no prohibition against defilement of the mother, such defilement as part of the mother's body is entirely innocuous. However, defilement to which a fetus is subjected during pregnancy persists subsequent to parturition. Accordingly, contends <i>Binyan Ẓion</i>, <i>Rokeaḥ</i> was concerned that upon birth it becomes retroactively clear that, in coming into contact with a corpse while pregnant, the mother caused prohibited future defilement of a <i>kohen</i>. Although no transgression is incurred until the fetus is actually separated from the mother, the earlier act causing future defilement of a <i>kohen</i> would be prohibited save for the principle of <i>sefek sefeika</i>.
  <i>Avnei Milu'im</i> 82:1 dismisses the issue of <i>tum'ah balu'ah</i> by citing the rule that immersion of a pregnant woman for purposes of conversion <i>ipso facto</i> constitutes immersion of the child as well despite the fact that the mother's body serves as a <i>ḥazizah</i>, or interposition, preventing the water of the <i>mikveh</i> from coming into direct contact with the child. The Gemara, <i>Yevamot</i> 88b, explains that the mother's body is not an interposition because "that is the way it grows" (<i>hainu reviteih</i>), i.e., only a foreign object constitutes an interposition whereas the mother's body in which the fetus gestates is hardly alien to the fetus.
  In effect, <i>Avnei Milu'im</i> postulates that the reason a "swallowed" object is not subject to defilement is because the object in which it is encased serves as an interposition between the object and the source of defilement. The mother's body, however, does not constitute such an interposition.
  R. Samuel ha-Levi Woszner, <i>Teshuvot Shevet ha-Levi</i>, II, <i>Yoreh De'ah</i>, no. 25, challenges <i>Avnei Milu'im</i>'s equation of the principle of <i>tum'ah balu'ah</i> with the notion of interposition. The Mishnah, <i>Niddah</i> 43b, declares that a newborn child can become defiled through contact with a corpse. The Mishnah clearly excludes defilement of the fetus <i>in utero</i>. As explained by <i>Mishnah Aḥaronah</i>, <i>ad locum</i>, the reason is that prior to parturition the fetus is "swallowed" within the womb and, accordingly, is not subject to defilement. Hence, <i>Magen Avraham</i>'s puzzlement with regard to <i>Rokeaḥ</i>'s comment is well-founded.
  R. Elchanan Wasserman, <i>Kovez Shi'urim</i>, II, no. 41, resolves the problem in a highly original manner. Reb Elchanan maintains that a <i>kohen</i> is forbidden contact with a corpse even in circumstances in which he does not becomes defiled. In support of that thesis he cites the halakhic provision prohibiting a <i>kohen</i> from being in the same building as a <i>goses</i>, i.e., a moribund person, despite the fact that the <i>kohen</i> does not become defiled by virtue of his presence in that building. A fetal <i>kohen</i> "swallowed" in his mother's body, contends Reb Elchanan, is indeed in the same building as the corpse and is forbidden to be in that building even though he does not thereby become defiled.32<i class="footnote">Reb Elchanan clearly maintains that <i>tum’ah balu’ah</i> is not rooted in the concept of interposition, but is simply a divine edict.</i>
  At present, many obstetricians perform sonograms upon pregnant women for cogent medical reasons. Although performed for other purposes, such sonograms generally reveal the gender of the fetus. If the fetus is a male, and its gender is disclosed to the mother, <i>Rokeaḥ</i>'s <i>sefek sefeika</i> no longer exists since the fetus is known with virtual certainty to be male. Obviously, according to Radvaz, <i>Hatam Sofer</i> and <i>Netiv Yam</i> who assert that <i>Rokeaḥ</i> was concerned only with regard to a woman in labor or approaching term, there is no problem with regard to defilement during earlier stages of pregnancy. However, according to the authorities who regard a <i>sefek sefeika</i> to be necessary even during earlier stages of gestation, the wife of a <i>kohen</i> whose sonogram has revealed that she is carrying a male fetus would be forbidden to come into contact with a corpse other than in situations of <i>pikuaḥ nefesh</i>.33<i class="footnote">Delivery itself, in this writer’s opinion, may take place in a hospital because parturition is categorized by Halakhah as a threat to the life of the mother and the danger, again in this writer’s opinion, is significantly mitigated by delivery in a hospital. See, for example, Jenny W.Y. Pang, James B. Heffelfinger, Greg J. Huang <i>et al.</i>, “Outcomes of Planned Home Births in Washington State: 1989-1996,” <i>Obstetrics and Gynecology</i>, vol. 100, no. 2 (August, 2002), pp. 253-259. See also S.L. Barron, P. Blain, C.H.W. Bullough <i>et al.</i>, “Collaborative Survey of Perinatal Loss in Planned and Unplanned Home Births,” <i>British Medical Journal</i>, vol. 313, no. 7068 (November 23, 1996), pp. 1306-1309, and “Digests: Health of Mothers, Babies May Be Compromised in Planned Home Births,” <i>Perspectives on Sexual and Reproductive Health</i>, vol. 34, no. 6 (December, 2002), pp. 320-321. This view is indeed disputed by some. See, for example, O. Olsen, “Meta-Analysis of the Safety of Home Birth,” <i>Birth</i>, vol. 24, no. 1 (March, 1997), pp. 4-14; Ursula Ackerman-Liebich, Thomas Voegeli, Katherin Gunter-Witt, “Home Versus Hospital Deliveries: Follow Up Study of Matching Pairs for Procedures and Outcome,” <i>British Medical Journal</i>, vol. 313, no. 7068 (November 23, 1996), pp. 1315-1318; and Kenneth C. Johnson and Betty-Annie Daviss “Outcomes of Planned Homebirths with Certified Professional Midwives: Large Prospective Study in North America,” <i>British Medical Journal</i>, vol. 330, no. 7505 (June 18, 2005), pp. 1416-1423. A patient suffering from a life-threatening condition—and labor and childbirth are categorized by Halakhah as life-threatening—may seek the ministration of a medical practitioner in whom he or she has greater confidence, despite attendant violation of religious law that would be incurred, even if an equally qualified practitioner might render the same care without such violation. See R. Joshua Neuwirth, <i>Shemirat Shabbat ke-Hilkhatah</i>, 2nd ed. (Jerusalem, 5739) 32:38.</i>
  The situation of a woman who has undergone a sonogram but who has not been informed of the fetus' gender is somewhat different. To her, there is a <i>sefek sefeika</i> but one of those doubts might be resolved by means of a simple inquiry. <i>Pri Megadim</i>, <i>Siftei Kohen</i>, <i>Yoreh De'ah</i> 110, <i>Kelalei Sefek Sefeika</i> 35:66, rules that when "mere inquiry" would resolve the doubt, it is obligatory to solicit such information. The crucial question, now that sonograms are available, and if they are to be regarded as posing no hazard to the health of either the mother or the fetus,34<i class="footnote">Basic research on harnessing the Doppler effect in the use of ultrasound waves for medical diagnostics was performed more than sixty years ago by the Viennese neurologist K.T. Dussik. See K. Dussik, “Über die Möglichkeit hochfrequente mechanische Schwingungen als diagnostiches Hilfsmittel zu verwerten,” <i>Zeitschrift für de gesamte Neurologie und Psychiatrie</i>, vol. 174 (1942), pp. 153-168. As a result ultrasound became a primary exploration technique in obstetrics and gynecology and gained wide acceptance because of the absence of any known side-effects. See J.P. Schaaps, “L’Ethographie et la Gynecologie-Obstetrique: Du Bistable au 3D,” <i>Revue Médicale de Liège</i>, vol. 54, no. 5 (2002), pp. 401-408; G.C. Mason, R.J. Lilford, J. Porter <i>et al.</i>, “Randomised Comparison of Routine Versus Highly Selective Use of Ultrasound in Low Risk Pregnancies,” <i>British Journal of Obstetrics and Gynecology</i>, vol. 100, no. 2 (February, 1993), pp.130-133; and J.S. Abramowicz, G. Kossoff, K. Marsal <i>et al.</i>, “Safety Statement, 2000 (reconfirmed 2003),” International Society of Ultrasound in Obstetrics and Gynecology (ISUOG), <i>Ultrasound in Obstetrics and Gynecology</i>, vol. 21, no. 1 (January, 2003), p. 100. Although four separate studies revealed an increase in left-handedness related to prenatal ultrasound exposure, that phenomenon has been dismissed as innocuous by some researchers. See K.A. Salveson, “Ultrasound and Left-handedness: A Sinister Association?” <i>Ultrasound in Obstetrics and Gynecology</i>, vol. 19, no. 3 (March, 2002), pp. 217-221; and J.S. Abramowicz, G. Kossoff, G. Marsal <i>et al.</i>, “Literature Review by the ISUOG Bioeffects and Safety Committee,” <i>Ultrasound in Obstetrics and Gynecology</i>, vol. 19, no. 3 (March, 2002), pp. 318-319. Others, however, have expressed concern that the phenomenon may be a sign of effect of ultrasound upon the brain. See H. Kieler, S. Cnattingius, B. Haglund <i>et al.</i>, “Sinistrality—a Side-Effect of Prenatal Sonography: A Comparative Study of Young Men,” <i>Epidemiology</i>, vol. 12, no. 6 (November, 2001), pp. 618-623; and <i>idem</i>, “Ultrasound and Adverse Effects,” <i>Ultrasound in Obstetrics and Gynecology</i>, vol. 20, no.1 (July, 2002), pp. 101-102.In recent years the power outputs of clinical devices have increased with the result that the potential for both thermal and non-thermal insults has become greater. The risk of inducing thermal effects is greater in the second and third trimesters while sheer stresses from radiation pressure may be of concern in early gestation. The likelihood of producing undesired biological effects can be enhanced by new procedures such as use of gas encapsulated echocontrast agents. See S.B. Barnett and D. Maulik, “Guidelines and Recommendations for Safe Use of Doppler Ultrasound in Perinatal Applications,” <i>Journal of Maternal-Fetal Medicine</i>, vol. 10, no. 2 (April, 2001), pp. 75-84; M.C. Ziskin and S.B. Barnett, “Ultrasound and the Developing Central Nervous System,” <i>Ultrasound in Medicine and Biology</i>, vol. 27, no. 7 (July, 2001), pp. 875-876; and S.B. Barnett, “Routine Ultrasound Scanning in First Trimester: What Are the Risks?” <i>Seminars in Ultrasound, CT and MR</i>, vol. 23, no. 5 (October, 2002), pp. 387-391. One recent study concludes that the data available to date suggest that diagnostic ultrasound has no adverse effect on embryogenesis or fetal growth and that thermal effect probably does not influence fetal development but, while endorsing B and M mode during the first trimester, concludes that color, pulsed or power Doppler be performed with caution. See R. Hershkovitz, E. Sheiner and M. Mazor, “Ultrasound in Obstetrics: A Review of Safety,” <i>European Journal of Obstetrics and Gynecology and Reproductive Biology</i>, vol. 101, no. 1 (February 10, 2002), pp. 15-18. A number of researchers have advised that obstetrical ultrasound be undertaken only for medical reasons and that exposure should be kept as low as reasonably achievable. See, for example, S. Westin and L. S. Bakketeig, “Unnecessary Use of Ultrasound in Pregnancy Should be Avoided. Probably Safe, But New Evidence Suggests Caution,” <i>Scandinavian Journal of Primary Health Care</i>, vol. 21, no. 2 (June, 2003), pp. 65-67; S. Bly and M.C. Van den Hof, “Obstetric Ultrasound Biological Effects and Safety,” <i>Journal of Obstetrics and Gynecology, Canada</i>, vol. 27, no. 6 (June, 2005), pp. 572-580; and L. Morin and M.C. Van den Hof, “Ultrasound Evaluation of First Trimester Pregnancy Complications,” <i>ibid.</i>, pp. 581-591.</i> is whether the pregnant wife of a <i>kohen</i> is obligated to undergo a sonogram before coming into contact with a corpse. <i>Shakh</i>, <i>Yoreh De'ah</i> 110, <i>Kelalei Sefek Sefeika</i>, sec. 35:66, records two opinions with regard to whether a <i>sefek sefeika</i> may be relied upon in circumstances in which the "double doubt" may be dispelled on the basis of examination or investigation. <i>Shakh</i> rules that it is necessary to be stringent "when [it is] possible and no loss is incurred." R. Shlomoh ha-Kohen of Vilna, <i>Teshuvot Binyan Shlomoh</i>, no. 13, similarly maintains that a <i>sefek sefeika</i> that may be resolved cannot be relied upon.35<i class="footnote">See also <i>Sedei Ḥemed</i>, <i>ma’arekhet ha-samakh</i>, no. 29.</i> Nevertheless <i>Shakh</i> rules that no investigation is necessary if such investigation would entail expense. A sonogram, presumably, does entail such an expense.
  R. Ezekiel Landau, <i>Teshuvot Noda bi-Yehudah</i>, <i>Mahadura Kamma</i>, <i>Yoreh De'ah</i>, no. 43 and no. 97, maintains that even the authorities who require resolution of the doubt posed by a <i>sefek sefeika</i> when such doubt can be resolved by inquiry or investigation do so only when both doubts can be resolved, but not if one of the doubts will persist even subsequent to the investigation. In the case of a pregnant woman, the doubt with regard to the viability of the fetus would indeed remain since it is not dispelled by means of a sonogram. However, according to <i>Hatam Sofer</i>'s understanding of <i>Rokeaḥ</i>, since the vast majority of fetuses are viable, the operative consideration is not <i>sefek sefeika</i> but the consideration that the majority of fetuses are not viable males. According to <i>Hatam Sofer</i>, the principle actually relied upon is that of <i>rov</i>. The general rule is that a <i>rov</i> may be relied upon even when examination is possible except in instances in which there is a <i>mi'ut ha-mazui</i>, i.e., a significant number of exceptions to the <i>rov</i>.36<i class="footnote">For an explication of <i>mi’ut ha-maẓui</i> see chapter 7 of this volume, pp. 245-246.</i> The incidence of viable male fetuses is certainly a <i>mi'ut ha-mazui</i> and, accordingly, examination for gender should be required according to <i>Hatam Sofer</i>. If the fetus is shown to be male, the principle of <i>rov</i> serves to establish that the overwhelming majority of fetuses are viable.
  Examination for a <i>mi'ut ha-mazui</i> is required only when such examination is possible. Thus, for example, <i>Shulḥan Arukh</i>, <i>Yoreh De'ah</i> 39:2 rules that a slaughtered animal whose lungs have been lost or misplaced prior to examination for adhesions37<i class="footnote">Examination of the lungs of a slaughtered animal is the paradigm for the requirement of examination in instances of a <i>mi’ut ha-maẓui</i>. See, for example, Rashi, <i>Ḥullin</i> 12a, s.v. <i>Pesaḥ</i>.</i> may be regarded as kosher on the basis of <i>rov</i>.38<i class="footnote">Cf., Rema, <i>Yoreh De’ah</i> 39:2. Rema’s stringency, however, is not applicable to the matter under discussion. See <i>Shakh</i>, <i>Yoreh De’ah</i> 39:8.</i> It is certainly not permissible to perform an ultrasound test solely to determine gender if the procedure is accompanied by any cogent medical risk,39<i class="footnote">See <i>supra</i>, note 34.</i> with the result that such examination is, in the halakhic sense, "impossible." A determination regarding the safety of the procedure can be made only by a physician familiar with the available data and is subject to revision as more information becomes available.
  
  Chapter 12 The Case of the Poisoned Sandwich
  
  
  
  Chapter 13 Cadavers on Display
  
  
  
  Chapter 14 A $25,000,000 Funeral
  
  