Carl Friedrich Gauss
Johann Carl Friedrich Gauss (German: Gauß [kaʁl ˈfʁiːdʁɪç ˈɡaʊs] (listen); Latin: Carolus Fridericus Gauss; 30 April 1777 – 23 February 1855) was a German mathematician, geodesist, and physicist who made significant contributions to many fields in mathematics and science. Gauss ranks among history's most influential mathematicians.Gauss was a child prodigy in mathematics, attended Collegium Carolinum, and, while studying at the University of Göttingen, made several important mathematical discoveries. At the age of 21, Gauss completed his magnum opus, Disquisitiones Arithmeticae. He was director of the astronomical observatory in Göttingen for nearly half a century, from 1807 until his death in 1855.
Gauss published the second and third complete proofs of the fundamental theorem of algebra, made important contributions to number theory and developed the theories of binary and ternary quadratic forms. He is also credited with inventing the fast Fourier transform algorithm and was instrumental in the discovery of the dwarf planet Ceres. His work on the motion of planetoids disturbed by large planets led to the introduction of the Gaussian gravitational constant and the method of least squares, which is still used in all sciences to minimize measurement error.
Furthermore, Gauss invented the heliotrope in 1821, a magnetometer in 1833, and alongside Wilhelm Eduard Weber, invented the first electromagnetic telegraph in 1833.
Gauss was a careful author and refused to publish incomplete work, and though having published extensively, he left a lot of posthumous works of important content. Gauss was known to dislike teaching, but some of his students became influential mathematicians. He believed that the act of learning, not possession of knowledge, granted the greatest enjoyment.
Gauss married twice and had six children, and died of a heart attack in 1855 in Göttingen.

Biography
Family, youth and education
Johann Carl Friedrich Gauss was born on 30 April 1777 in Brunswick (Braunschweig), in the Duchy of Brunswick-Wolfenbüttel (now part of Lower Saxony, Germany), to a family of lower social status. His father Gebhard Dietrich Gauss (1744–1808) worked in several jobs as butcher, bricklayer, gardener, and in addition as treasurer of a death benefit fund; Gauss characterized his father as an honourable and respected man, but rough and dominating at home. He was experienced in writing and calculating, but his wife Dorothea (1743–1839), Carl Friedrich's mother, was nearly illiterate. He was christened and confirmed in a church near the school he attended as a child. He had one elder brother from his father's first marriage.
Gauss was a child prodigy in the field of mathematics. When the elementary teachers noticed his intellectual abilities, they brought him to the attention of the Duke of Brunswick, who sent him to the Collegium Carolinum in Brunswick, which he attended from 1792 to 1795 with Eberhard August Wilhelm von Zimmermann as one of his teachers. Thereafter the Duke granted him the resources for studies at the Hanoverian University of Göttingen until 1798, where he studied mathematics, sciences and classical languages as well. One of his professors in mathematics was Abraham Gotthelf Kästner, whom Gauss called "the leading mathematician among poets, and the leading poet among mathematicians", because of his writing epigrams; Gauss depicted him by a drawing showing a lecture scene, when he produced errors in most simple calculations. Astronomy was taught by Karl Felix von Seyffer (1762–1822), with whom Gauss stayed in correspondence after graduation; Olbers and Gauss mocked him in their correspondence. In contrast to them, Gauss thought highly of Georg Christoph Lichtenberg, his teacher of physics, and of Christian Gottlob Heyne, whose lecture in classics Gauss attended with pleasure.Though being a registered student at university, it is obvious that he was a self-taught student in mathematics, when he independently rediscovered several important theorems. He succeeded with a breakthrough in a geometrical problem that had occupied mathematicians since the days of the Ancient Greeks, when he showed in 1796 that a regular polygon can be constructed by compass and straightedge. This discovery was subject of his first publication and ultimately led Gauss to choose mathematics instead of philology as a career. Gauss' mathematical diary shows that, in the same year, he was very productive in number theory, too. He discovered a construction of the heptadecagon, advanced modular arithmetic, found the first proof of the quadratic reciprocity law, and dealt with the prime number theorem.
Thus from that time he got many ideas for his mathematical opus magnum Disquisitiones arithmeticae, published in 1801.

Private scholar
Gauss graduated as Doctor of Philosophy in 1799, not in Göttingen as sometimes mentioned, but on the Duke's special request at the University of Helmstedt, the only state university of the Duchy. There Johann Friedrich Pfaff assessed the doctoral thesis, and Gauss got the degree in absentia without further oral examination as usually requested. The next years the Duke granted his costs of living as a private scholar in Brunswick. He showed his gratitude and loyalty to the Duke when he refused several calls from the Russian Academy of science in St. Peterburg and the Landshut university. The Duke of Brunswick had promised him the foundation of an observatory in Brunswick in 1804, and architect Peter Joseph Krahe made preliminary designs, but one of Napoleon's wars cancelled those plans: the Duke was mortally wounded in the Battle of Jena–Auerstedt in 1806, the Duchy was abolished in the following year, and Gauss' financial support stopped. Thus he followed a call to the University of Göttingen, then an institution of the newly founded Kingdom of Westphalia under Jérôme Bonaparte, as full professor and director of the astronomical observatory.
When Gauss studied the determination of asteroid orbits he established contact with the astronomical community of Bremen and Lilienthal, especially Wilhelm Olbers, Karl Ludwig Harding and Friedrich Wilhelm Bessel with regards to an informal society then named Celestial police. One of their aims was the discovery of further planets, and they assembled data of asteroids and comets as basic fund for Gauss' calculations. Thereby Gauss developed new powerful methods for the determination of orbits, later published in his astronomical opus magnum Theoria motus corporum coelestium (1809).

Professor in Göttingen
Gauss arrived at Göttingen in November 1807, and in the following years he was confronted with the demand for two thousand Francs from the Westphalian government as war contribution. Without having yet received his salary, he could not raise this enormous amount. Both Olbers and Laplace wanted to help him with payment, but Gauss refused it. Finally an anonymous person of Frankfurt, later discovered to be the Prince-primate Dalberg, paid the sum.
Gauss took on the directorate of the 60-years-old observatory, founded in 1748 by Prince-elector George II and built on a converted fortification tower, with usable, but partly out-of-date instruments. Harding, who had been extraordinary professor for astronomy since 1805 as Seyffer's successor, cared for the instruments, and gave most of the lessons in astronomy, a task that Gauss always detested. The construction of a new observatory had been approved by Prince-elector George III in principle since 1802, and the Westphalian government continued the planning, but the building was not finished until October 1816 with new competitive instruments, for instance two meridian circles from Repsold and Reichenbach, and a heliometer from Fraunhofer.In 1810, Gauss got the opportunity to move to Berlin as full member of the Prussian Academy of Science, with great liberty of research in mathematics, but no obligation to give lectures. Gauss refused, possibly because of his difficult family conditions. In the 1820s another call came from Berlin, but Gauss refused once more, and thereby could improve his material situation.
Gauss remained mentally active into his old age, even while suffering from gout and general unhappiness. At the age of 62, he began to teach himself Russian, presumably to understand the writings of Lobachevsky on non-Euclidean geometry. His last observation was the Solar eclipse of July 28, 1851. On 23 February 1855, Gauss died of a heart attack in Göttingen; he is interred in the Albani Cemetery there. Heinrich Ewald, Gauss' son-in-law, and Wolfgang Sartorius von Waltershausen, Gauss' close friend and biographer, gave eulogies at his funeral.

Gauss' brain
The day after Gauss' death his brain was removed, preserved and studied by Rudolf Wagner, who found its mass to be slightly above average, at 1,492 grams (52.6 oz). The cerebral area was determined by Wagner's son Hermann in his doctoral thesis to be 219,588 square millimetres (340.362 sq in). Highly developed convolutions were also found, which in the early 20th century were suggested as the explanation for his genius. After various previous investigations, a magnetic resonance study of 1998, done at the Max Planck Institute for Biophysical Chemistry in Göttingen, gave no results which could be used to explain his mathematical abilities.In 2013, a neurobiologist of the same institute discovered that Gauss' brain had been mixed up, by reason of wrong labelling, with that of the physician Conrad Heinrich Fuchs, who died in Göttingen in the same year as Gauss. A further investigation showed no remarkable anomalies in the brains of either person. Thus all investigations on Gauss' brain until 1998, except the first ones of Rudolf and Hermann Wagner, actually refer to the brain of Fuchs.

Religious views
Gauss was nominally a member of the St. Albans parish of the Evangelical Lutheran church in Göttingen. Dunnington describes Gauss' religious views as follows:

For him science was the means of exposing the immortal nucleus of the human soul. In the days of his full strength, it furnished him recreation and, by the prospects which it opened up to him, gave consolation. Toward the end of his life, it brought him confidence. Gauss' God was not a cold and distant figment of metaphysics, nor a distorted caricature of embittered theology. To man is not vouchsafed that fullness of knowledge which would warrant his arrogantly holding that his blurred vision is the full light and that there can be none other which might report the truth as does his. For Gauss, not he who mumbles his creed, but he who lives it, is accepted. He believed that a life worthily spent here on earth is the best, the only, preparation for heaven. Religion is not a question of literature, but of life. God's revelation is continuous, not contained in tablets of stone or sacred parchment. A book is inspired when it inspires. The unshakeable idea of personal continuance after death, the firm belief in a last regulator of things, in an eternal, just, omniscient, omnipotent God, formed the basis of his religious life, which harmonized completely with his scientific research.
Apart from his correspondence, not many details are known about Gauss' personal creed. Many biographers of Gauss disagree about his religious stance, with Bühler and others considering him a deist with very unorthodox views, while Dunnington (admitting that Gauss did not believe literally in all Christian dogmas and that it is unknown what he believed on most doctrinal and confessional questions) points out that he was, at least, a nominal Lutheran.In connection to this, there is a record of a conversation between Rudolf Wagner and Gauss, in which they discussed William Whewell's book Of the Plurality of Worlds. In this work, Whewell had discarded the possibility of existing life in other planets, on the basis of theological arguments, but this was a position with which both Wagner and Gauss disagreed. Later Wagner explained that he did not fully believe in the Bible, though he confessed that he "envied" those who were able to easily believe. This later led them to discuss the topic of faith, and in some other religious remarks, Gauss said that he had been more influenced by theologians like Lutheran minister Paul Gerhardt than by Moses. Other religious influences included Wilhelm Braubach, Johann Peter Süssmilch, and the New Testament. Two religious works which Gauss read frequently were Braubach's Seelenlehre (Gießen, 1843) and Süssmilch's Göttliche Ordnung (1756); he also devoted considerable time to the New Testament in the original Greek.
Dunnington further elaborates on Gauss' religious views by writing: Gauss's religious consciousness was based on an insatiable thirst for truth and a deep feeling of justice extending to intellectual as well as material goods. He conceived spiritual life in the whole universe as a great system of law penetrated by eternal truth, and from this source he gained the firm confidence that death does not end all.
Gauss believed in an omniscient source of creation, but he claimed that belief or a lack of it did not affect his mathematics.Though he was not a church-goer, Gauss strongly upheld religious tolerance, believing "that one is not justified in disturbing another's religious belief, in which they find consolation for earthly sorrows in time of trouble." When his son Eugene announced that he wanted to become a Christian missionary, Gauss approved of this, saying that regardless of the problems within religious organizations, missionary work was "a highly honorable" task.

Family
On 9 October 1805, Gauss married Johanna Osthoff (1780–1809), and had two sons and a daughter with her: Joseph (1806–1873), Wilhelmina (1808–1840) and Louis (1809–1810). Johanna died on 11 October 1809 one month after Louis' birth, who himself died a few months later. Gauss plunged into a depression from which he never fully recovered. Soon after her death he wrote a last letter to his dead wife in the style of an ancient threnody, the most personal document of Gauss.He then married Wilhelmine (Minna) Waldeck (1788–1831), a friend of his first wife, on 4 August 1810 and had three more children : Eugen (later Eugene) (1811–1896), Wilhelm (later William) (1813–1879) and Therese (1816–1864). Minna Gauss died on 12 September 1831 after serious illness for more than a decennium, possibly caused by tuberculosis. Then Therese took over the household and cared for Gauss for the rest of his life. His mother Dorothea Gauss lived in his house from 1817 until her death in 1839. The daughter Wilhelmina married the orientalist Heinrich Ewald and died at the age of 42, possibly by tuberculosis. Therese married the actor Constantin Staufenau after her father's death and died at 47, possibly by tuberculosis.Gauss was never quite the same without his first wife, and just like his father, grew to dominate his children. Gauss eventually had conflicts with his sons, because he did not want any of them to enter mathematics or science for "fear of lowering the family name", as he believed none of them would surpass his own achievements.
Still being a school boy, the eldest son Joseph helped his father as assistant during his survey campaign in summer 1821. After a short time at university, he joined the Hanoverian Army in 1824 and assisted in surveying again in 1829. Later in the 1830s he was responsible for the enlargement of the survey network to the western parts of the Kingdom. But in all these years staying in a low military rank (since 1834 as Premier-Leutnant) with a very small salary, he needed financial support from his father, especially since his marriage in 1840. Thus he left the service and, with the background of his geodetical qualifications, engaged as director of the Royal Hanoverian State Railways with the construction of the railway network. In 1836 Joseph Gauss had studied the railroad system in the US for some months.Eugen shared a good measure of Gauss' talent in computation and languages, but had a vivacious and sometimes rebellious character. He wanted to study philology, whereas Gauss wanted him to become a lawyer. Having run up debts and caused a scandal in public, he suddenly left Göttingen under dramatic circumstances in September 1830 and emigrated via Bremen to the United States. After having wasted the little money he had taken for starting, his father refused further financial support. Thus he joined the army for five years, and thereafter worked for the American Fur Company in the Midwest, where he learned the Sioux language. Later, he moved to Missouri and became a successful businessman. It took many years for Eugene's success to counteract his reputation among Gauss' friends and colleagues.Wilhelm married a niece of the astronomer Bessel and also moved to Missouri in 1837, starting as a farmer and later becoming wealthy in the shoe business in St. Louis. Eugene and William are progenitors of numerous descendents in America, but the German Gauss issue descends from Joseph as the Gauss daughters had no children.

Personality
Though he did take a few students, Gauss was known to dislike teaching. Several of his students became influential mathematicians, among them Richard Dedekind and Bernhard Riemann.
On Gauss' recommendation, Friedrich Wilhelm Bessel was awarded an honorary doctoral degree from Göttingen University in March 1811; they had been friends since 1804. Before she died, Sophie Germain was recommended by Gauss to receive an honorary degree, but she never received it.Gauss was an ardent perfectionist, a hard worker, and refused to publish work which he did not consider complete and above criticism. This was in keeping with his personal motto pauca sed matura ("few, but ripe"). His personal diary indicates that he had made several important mathematical discoveries years or decades before his contemporaries published them. Eric Temple Bell said that if Gauss had published all of his discoveries in a timely manner, he would have advanced mathematics by fifty years.
Gauss usually declined to present the intuition behind his often very elegant proofs and erased all traces of how he discovered them. This is justified, if unsatisfactorily, by Gauss in his Disquisitiones Arithmeticae, where he states that all analysis (in other words, the paths one traveled to reach the solution of a problem) must be suppressed for sake of brevity.
Gauss summarized his views on the pursuit of knowledge in a letter to Farkas Bolyai dated 2 September 1808 as follows:

It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment. When I have clarified and exhausted a subject, then I turn away from it, in order to go into darkness again. The never-satisfied man is so strange; if he has completed a structure, then it is not in order to dwell in it peacefully, but in order to begin another. I imagine the world conqueror must feel thus, who, after one kingdom is scarcely conquered, stretches out his arms for others.
Gauss supported the Welf monarchy and opposed Napoleon, whom he saw as an outgrowth of revolution.

Scientific work
Algebra, number theory, complex analysis
In his doctoral thesis from 1799 Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. Mathematicians including Jean le Rond d'Alembert had produced false proofs before him, and Gauss' dissertation contains a critique of d'Alembert's work. He subsequently produced three other proofs, the last one in 1849 being generally rigorous. His attempts clarified the concept of complex numbers considerably along the way. Gauss was the first one who used the symbol i for representing the imaginary number.

The entries in Gauss' Mathematical diary indicate that he was busy with subject of number theory at least since 1796. A detailed study of previous researches showed him that some of his findings had been already done by other scholars. In the years 1798 and 1799 Gauss wrote a voluminous compilation of all these results in the famous Disquisitiones Arithmeticae, published in 1801, that was fundamental in consolidating number theory as a discipline and covered both elementary and algebraic number theory. Therein he introduces, among other things, the triple bar symbol ≡ for congruence and uses it in a clean presentation of modular arithmetic. It deals with the unique factorization theorem and the primitive root modulo n. In the main chapters, Gauss presents two proofs of the law of quadratic reciprocity, which allows mathematicians to determine the solvability of any quadratic equation in modular arithmetic, and develops the theories of binary and ternary quadratic forms. He also proved this triangular case of the Fermat polygonal number theorem for n = 3.
In the last chapter of the Disquisitiones Gauss gives his proof for the construction of a regular heptadecagon (17-sided polygon) with straightedge and compass by reducing this geometrical to an algebraic problem. The construction is possible, if the number of its sides is the product of distinct Fermat primes and a power of 2.It appears that Gauss already knew the class number formula in 1801.
In the same year, he published a result on the number of solutions of polynomials with coefficients in finite fields, which 150 years later led to the Weil conjectures.
Furthermore, he dealt with continued fractions and the prime number theorem. In addition, he proved Fermat's Last Theorem for n = 5 and Kepler conjecture for regular arrangements.
He also explained the pentagramma mirificum, and developed algorithms for determining the date of Easter and the date of Pesach.
He invented the Cooley–Tukey FFT algorithm for calculating the discrete Fourier transforms when calculating the orbits of Pallas and Juno in 1805, 160 years before Cooley and Tukey. Gauss developed it as a trigonometric interpolation method. His paper, Theoria Interpolationis Methodo Nova Tractata, was published only posthumously in 1876 in Volume 3 of his collected works, preceded by the first presentation by Joseph Fourier on the subject in 1807.His work "Disquisitiones generales circa series infinitam..." on a special infinite series was the first one with an exact inquiry of convergence in the history of mathematics.

Astronomy
On 1 January 1801, Italian astronomer Giuseppe Piazzi discovered the dwarf planet Ceres. Piazzi could track Ceres for only somewhat more than a month, following it for three degrees across the night sky. Then it disappeared temporarily behind the glare of the Sun. Several months later, when Ceres should have reappeared, Piazzi could not locate it: the mathematical tools of the time were not able to extrapolate a position from such a scant amount of data – three degrees represent less than 1% of the total orbit. Gauss heard about the problem and tackled it. After three months of intense work, he predicted a position for Ceres in December 1801 and this turned out to be accurate within a half-degree when it was rediscovered by Franz Xaver von Zach on 7/31 December at Gotha, and indepently by Heinrich Olbers on 1/2 January in Bremen. This confirmation eventually led to the classification of Ceres as minor-planet designation 1 Ceres: the first minor-planet ever discovered.Gauss's method involved determining a conic section in space, given one focus (the Sun) and the conic's intersection with three given lines (lines of sight from the Earth, which is itself moving on an ellipse, to the planet) and given the time it takes the planet to traverse the arcs determined by these lines (from which the lengths of the arcs can be calculated by Kepler's Second Law). This problem leads to an equation of the eighth degree, of which one solution, the Earth's orbit, is known. The solution sought is then separated from the remaining six based on physical conditions. In this work, Gauss used comprehensive approximation methods which he created for that purpose.Zach noted that "without the intelligent work and calculations of Doctor Gauss we might not have found Ceres again". The discovery of Ceres led Gauss to his work on a theory of the motion of planetoids disturbed by large planets, eventually published in 1809 as Theoria motus corporum coelestium in sectionibus conicis solem ambientum. In the process, he so streamlined the cumbersome mathematics of 18th-century orbital prediction that his work remains a cornerstone of astronomical computation. It introduced the Gaussian gravitational constant.

Theory of errors
It is likely that Gauss used the method of least squares for calculating the orbit of Ceres to minimize the impact of measurement error. The method was published first by Adrien-Marie Legendre in 1805, but Gauss claimed in Theoria motus that he had been using it since 1794 or 1795. In the history of statistics, this disagreement is called the "priority dispute over the discovery of the method of least squares." Gauss' concept of priority as the first to discover, not the first to publish differed from that of his scientific contemporaries. Gauss proved the method under the assumption of normally distributed errors (Gauss–Markov theorem) in his paper Theoria combinationis observationum erroribus minimis obnoxiae from 1821.

Arc measurement and geodetic survey
Gauss was busy with geodetic problems since 1799, when he helped Karl Ludwig von Lecoq with calculations during his survey in Westphalia. Later since 1804, he taught himself some geodetic practise with a sextant in Brunswick, and Göttingen.Since 1816, his former student Heinrich Christian Schumacher, then professor in Copenhagen, but living in Altona (Holstein) near Hamburg, made a triangulation of the Jutland peninsula from Skagen in the north to Lauenburg in the south. The aim was not only the foundation of map production, but also the determination of the geodetic arc of that distance. Schumacher asked Gauss to continue this work further to the south and said he could find support for this project directly from the government of Hanover. Finally in May 1820, King George IV gave the order to Gauss.Gauss and Schumacher had yet determined some angles between Lüneburg, Hamburg, and Lauenburg for the geodetic connection in October 1818. During the summers of 1821 until 1825 Gauss directed the triangulation personally, that reached from Thuringia in the south to the river Elbe in the north. The triangel between Hoher Hagen, Großer Inselsberg in the Thuringian Forest, and Brocken in the Harz mountains was the largest one Gauss had ever measured with a maximum side of 107 km (66.5 miles). In the thin populated Lüneburg Heath, without significant natural summits or artificial buildings, he had great difficulties to find suitable triangulation points, sometimes cutting lanes through the vegetation was necessary or even the erection of signal towers.For pointing signals, Gauss invented a new instrument with movable mirrors and a small telescope that reflects the sunbeams to the triangulation points, and named it heliotrope. Another suitable construction for the same purpose was a sextant with an additional mirror which he named vice heliotrope. Gauss got assistance by soldiers of the Hanoveran army, among them his eldest son Joseph. Gauss took part in the baseline measurement (Braak Base Line) of Schumacher in the village Braak near Hamburg in 1820, and used the result for the evaluation of his triangulation.The arc measurement needed a precise astronomical determination of two points in the network. Gauss and Schumacher used the favourite occasion that both observatories in Göttingen and in Altona, in the garden of Schumacher's house, laid nearly in the same longitude. The latitude was measured with both their own instruments and a zenith sector of Ramsden that was transported to both observatories.An additional result was a better value of flattening of the approximative earth ellipsoid. Gauss developed the universal transverse Mercator projection of the ellipsoidal shaped earth (what he named conform projection) for representing geodetical data in plane charts.
When the arc measurement was finished, Gauss intended the enlargement of the triangulation to the west to get a survey of the whole Kingdom of Hanover. The practical work was directed by three army officers, among them Lieutenant Joseph Gauss. The complete data evaluation laid in the hands of Carl Friedrich Gauss, who applied his mathematical inventions as the method of least squares and his elimination method to it. The project was finished in 1844, but Gauss did not publish a final report of the project and his method of projection; this work was not done until 1866.In 1828, when studying differences in latitude, Gauss first defined a physical approximation for the figure of the Earth as the surface everywhere perpendicular to the direction of gravity; later his doctoral student Johann Benedict Listing called this the geoid.

Theorema Egregium
The geodetic survey of Hanover fueled Gauss' interest in differential geometry and topology, fields of mathematics dealing with curves and surfaces. This led in 1828 to an important theorem, the Theorema Egregium (remarkable theorem), establishing an important property of the notion of curvature. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring angles and distances on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space or 2-dimensional space.
A consequence is the impossibility of an isometric transformation between surfaces of different Gaussian curvature. This means practically that a sphere or an ellipsoid cannot be transformed to a plane without distortion, which causes a fundamental problem in designing projections for geographical maps.

Non-Euclidean geometries
Gauss claimed to have discovered the possibility of non-Euclidean geometries but never published it. He is the one who coined the term "non-Euclidean geometry". This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory. Research on these geometries led to, among other things, Einstein's theory of general relativity, which describes the universe as non-Euclidean.
Gauss' friend Farkas Bolyai with whom he had sworn "brotherhood and the banner of truth" as a student, had tried in vain for many years to prove the parallel postulate from Euclid's other axioms of geometry. Bolyai's son Janos discovered non-Euclidean geometry in 1829 and published his work in 1832. After seeing it, Gauss wrote to Farkas Bolyai: "To praise it would amount to praising myself. For the entire content of the work ... coincides almost exactly with my own meditations which have occupied my mind for the past thirty or thirty-five years." This statement put a strain on his relationship with Janos Bolyai who thought that Gauss was stealing his idea.Letters from Gauss years before 1829 reveal him obscurely discussing the problem of parallel lines. Dunnington argues that Gauss was in fact in full possession of non-Euclidean geometry long before it was published by Bolyai, but that he refused to publish any of it because of his fear of controversy.In 1854, Gauss selected the topic for Bernhard Riemann's inaugural lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen from three proposals. On the way home from Riemann's lecture, Weber reported that Gauss was full of praise and excitement.

Magnetism
Gauss' interest in magnetism is obvious since the first decennium of the 19th century. Since 1826, when Alexander von Humboldt visited him in Göttingen, both scientists began intensive research on geomagnetism, partly independent, partly in productive cooperation. In 1828, Gauss was Humboldt's personal guest during the conference of the Society of German Natural Scientists and Physicians in Berlin, where he got acquaintance with the physicist Wilhelm Weber.When Weber got the chair for physics in Göttingen by Gauss' recommendation in 1831, both of them started a fruitful collaboration, leading to a new knowledge of magnetism with a representation for the unit of magnetism in terms of mass, charge, and time. They founded the Magnetic Association (German: "Magnetischer Verein"), an international working group of several observatories, which supported measurements of Earth's magnetic field in many regions of the world with equal methods at arranged dates in the years 1836 to 1841. In 1836, Humboldt was helpful to organize the worldwide spread of observatories including the British dominions with a letter to the Duke of Sussex, then president of the Royal Society, wherein he asked for support for a program of global resaerch based on Gauss' methods. The dates, times, and intervalls of observations were determined in advance, the Göttingen mean time was used as standard. Finally 61 station participated in this global program. Gauss and Weber founded a series for the publication of the results, six volumes were edited between 1837 and 1843. Weber's departure to Leipzig in 1843 as late effect of the Göttingen Seven affair marked the end of Magnetic Assiciation activity.Following Humboldt's example, Gauss ordered a magnetic observatory to be built in the garden of his observatory, but both scientists differed over instrumental equipment; Gauss preferred stationary instruments, which he thaught to give more precise results, whereas Humboldt was accustomed to movable instruments. Gauss was interested in the temporal and spatial variation of magnetic declination, inclination, and intensity, but discriminated Humboldt's concept of magnetic intensity to the terms of "horizontal" and "vertical" intensity. Together with Weber, he developed methods of measuring the components of intensity of the magnetic field, and constructed a suitable magnetometer to measure absolute values of the strength of the Earth's magnetic field, not more relative ones that depended on the apparatus. With this work, Gauss was the first one who derived a non-mechanical quantity by basic nmechanical quantities.Gauss carried out a "General Theory of Terrestrial Magnetism" (1839), in what he believed to describe the nature of magnetic Force; following Felix Klein, this work is actually a presentation of observations by use of spherical harmonics rather than a physical theory. The theory predicted the existence of exactly two magnetic poles on the earth, thus Hansteen's idea of four magnetic poles became obsolete, and the data allowed to determine their location with rather good precision. In his "General theorems concerning the attractive and repulsive forces acting in reciprocal proportions of quadratic distances" (1840) Gauss gave the baseline of a theory of the magnetic potential, based on Lagrange, Laplace, and Poisson; it seems rather unlikely that he had knowledge of the previous works of George Green on this subject.
Gauss got a remarkable influence on the begin of geophysics in Russia, when Adolph Theodor Kupffer, one of his former students, founded a magnetic observatory in St. Petersburg, following the eample of the observatory in Götttingen, and similar Ivan Simonov in Kazan.Furthermore they found the rules for branches electric circuits, later benamed as Kirchhoff's circuit laws, and made inquiries on electromagnetism. They constructed the first electromechanical telegraph in 1833, which connected the observatory with the institute for physics in the town centre of Göttingen, but they did not care for any further development of this invention with regard to commercial purposes.It was during this time that Gauss formulated his namesake law describing the electric flux through a closed surface.

Optics
In his influential Dioptrical Investigations (1840), Gauss gave the first systematic analysis on the formation of images under a paraxial approximation (Gaussian optics). Gauss demonstrated, that under a paraxial approximation an optical system can be characterized by its cardinal points, and he derived the Gaussian lens formula.

Anecdotes
Several stories of his early genius have been reported. Carl Friedrich Gauss' mother had never recorded the date of his birth, remembering only that he had been born on a Wednesday, eight days before the Feast of the Ascension, which occurs 39 days after Easter. Gauss later solved this puzzle about his birthdate in the context of finding the date of Easter, deriving methods to compute the date in both past and future years.In his memorial on Gauss, Wolfgang Sartorius von Waltershausen tells a story about the three-years-aged Gauss, who corrected a math error his father made. The most popular story, also told by Sartorius, tells of a school exercise: the teacher J.G. Büttner and his assistant Martin Bartels ordered students to add an arithmetic series. Out of about a hundred pupils, Gauss was the first to solve the problem correctly by a significant margin. Although (or because) Sartorius gave no details, in the course of time many versions of this story have been created, with more and more details regarding the nature of the series – the most frequent being the classical problem of adding together all the integers from 1 to 100 – and the circumstances in the classroom.Gauss referred to mathematics as "the queen of sciences" and arithmetics as "the queen of mathematics", and supposedly once espoused a belief in the necessity of immediately understanding Euler's identity as a benchmark pursuant to becoming a first-class mathematician.

Honours and awards
The first membership of a scientific society was given to Gauss in 1802 by the Russian Academy of Sciences. Further memberships (corresponding, foreign or full) were from the Academy of Sciences in Göttingen (1802), the French Academy of Sciences (1804/ 1820), the Royal Society of London (1804), the Royal Prussian Academy in Berlin (1810), the National Academy of Science in Verona (1810), the Royal Society of Edinburgh (1820), the Bavarian Academy of Sciences of Munich (1820), the Royal Danish Academy in Copenhagen (1821), the Royal Astronomical Society in London (1821), the Royal Swedish Academy of Sciences (1821), the American Academy of Arts and Sciences in Boston (1822),  the Royal Bohemian Society of Sciences in Prague (1833), the Royal Society of Sciences in Uppsala (1843), the Royal Irish Academy in Dublin (1843), the Royal Institute of the Netherlands (1845), the Spanish Royal Academy of Sciences in Madrid (1850), the Russian Geographical Society (1851), the Imperial Academy of Sciences in Vienna (1848), the American Philosophical Society (1853), the Cambridge Philosophical Society, and the Royal Hollandish Society of Sciences in Haarlem.Gauss was honorary member of the University of Kasan and the Philosophical Faculty of the University of Prague since 1849.
Gauss received the Lalande Prize in 1809 and the Copley Medal in 1838.Gauss was appointed Knight of the Legion of Honour in 1837 and was one of the first members of the Order Pour le Merite (Civil class) when it was established in 1842. Furthermore, he received the Order of the Crown of Westphalia (1810), the Order of the Dannebrog (1817), the Royal Guelphic Order (1815), the Order of the Polar Star (1844), the Order of Henry the Lion (1849), and the Bavarian Maximilian Order for Science and Art (1853).The Kings of Hanover appointed him the honorary titles "Hofrath" (1816) and "Geheimer Hofrath" (1845). On occasion of his golden doctor degree jubilee he got the Honorary citizenship of both towns of Brunswick and Göttingen in 1849. Soon after his death a medal was issued by order of King George V of Hanover with the back side inscription : GEORGIVS V REX HANNOVERAE MATHEMATICORVM PRINCIPI and the circumscription : ACADEMIAE SVAE GEORGIAE AVGVSTAE DECORI AETERNO.The ″Gauss-Gesellschaft Göttingen″ (Gauss Society) was founded in 1964 for researches on life and work of Carl Friedrich Gauss and related persons and edits the ″Mitteilungen der Gauss-Gesellschaft″ (Communications of the Gauss Society).

Writings
Mathematics
1799: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse [New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors of the first or second degree]. Helmstedt: C. G. Fleckeisen. 1866. (Doctoral thesis on the fundamental theorem of algebra, University of Helmstedt) Original book
1816: "Demonstratio nova altera theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 3: 107–134.
1816: "Theorematis de resolubilitate functionum algebraicarum integrarum in factores reales demonstratio tertia". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 3: 135–142.
1850: "Beiträge zur Theorie der algebraischen Gleichungen". Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen. 4: 34–35.
Die vier Gauss'schen Beweise für die Zerlegung ganzer algebraischer Funktionen in reelle Faktoren ersten und zweiten Grades. (1799–1849) [The four Gaussian proofs of the fundamental theorem of algebra]. Translated by Netto. Leipzig: Wilhelm Engelmann. 1890. (German)
1801: Disquisitiones Arithmeticae. Leipzig: Gerh. Fleischer jun.
Gauss, Carl Friedrich (1986). Disquisitiones Arithmeticae & other papers on number theory. Translated by Clarke, Arthur A. (2nd, corrected ed.). New York: Springer. doi:10.1007/978-1-4939-7560-0. ISBN 978-0-387-96254-2.
1808: "Theorematis arithmetici demonstratio nova". Commentationes Societatis Regiae Scientiarum Gottingensis. Comm. Math. 16: 69–74. (Introduces Gauss's lemma, uses it in the third proof of quadratic reciprocity)
1809: Theoria motus corporum coelestium in sectionibus conicis solem ambientium (in Latin). Hamburg: Friedrich Perthes & Johann Heinrich Besser.
Theory of the Motion of Heavenly Bodies Moving about the Sun in Conic Sections. Translated by Davis, Charles Henry. Little, Brown & Co. 1857.
1811: "Disquisitio de elementis ellipticis Palladis ex oppositionibus annorum 1803, 1804, 1805, 1806, 1807, 1808, 1809". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Math. 1: 1–26. (Orbit of Pallas)
1811: "Summatio quarundam serierum singularium". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 1: 1–40. (Determination of the sign of the quadratic Gauss sum, uses this to give the fourth proof of quadratic reciprocity)
1812: "Disquisitiones generales circa seriem infinitam 
  
    
      
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1815: "Methodus nova integralium valores per approximationem inveniendi". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 3: 39–76.
1818: "Theorematis fundamentalis in doctrina de residuis quadraticis demonstrationes et ampliationes novae". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 4: 3–20. (Fifth and sixth proofs of quadratic reciprocity)
1821: "Theoria combinationis observationum erroribus minimis obnoxiae. Pars Prior". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 5: 33–62.
1823: "Theoria combinationis observationum erroribus minimis obnoxiae. Pars Posterior". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 5: 63–90.
1828: Gauss, Carl Friedrich (1828). "Supplementum theoriae combinationis observationum erroribus minimis obnoxiae". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. Dieterich. 6: 57–98. Bibcode:1828stco.book.....G. (Three essays concerning the calculation of probabilities as the basis of the Gaussian law of error propagation)
Gauss, Carl Friedrich; Stewart, G. W. (1995). Theory of the Combination of Observations Least Subject to Errors. Part One, Part Two, Supplement (Classics in Applied Mathematics). Translated by G. W. Stewart. Philadelphia: Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611971248. ISBN 978-0-89871-347-3.
1827: "Disquisitiones generales circa superficies curvas". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 6: 99–146.
General Investigations of Curved Surfaces (PDF). Translated by J. C. Morehead and A. M. Hiltebeitel. The Princeton University Library. 1902.
1828: "Theoria residuorum biquadraticorum, Commentatio prima". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 6: 27–56.
1832: "Theoria residuorum biquadraticorum, Commentatio secunda". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. Comm. Class. Math. 7: 89–148. (Introduces the Gaussian integers, states (without proof) the law of biquadratic reciprocity, proves the supplementary law for 1 + i)
1845: "Untersuchungen über Gegenstände der Höheren Geodäsie. Erste Abhandlung". Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Zweiter Band: 3–46.
1847: "Untersuchungen über Gegenstände der Höheren Geodäsie. Zweite Abhandlung". Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen. Dritter Band: 3–44.
Klein, Felix, ed. (1903), "Gauß' wissenschaftliches Tagebuch 1796–1814", Mathematische Annalen (in Latin and German), 57: 1–34, doi:10.1007/BF01449013, S2CID 119641638
Jeremy Gray (1984). "A commentary on Gauss's mathematical diary, 1796–1814". Expositiones Mathematicae. 2: 97–130.

Physics
1832: "Principia generalia theoriae figurae fluidorum fluidorum in statu aequilibrii". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. 7: 39–88.
1832: "Intensitas vis magneticae terrestris ad mensuram absolutam revocata". Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores. 8: 3–44.
The Intensity of the Earth's Magnetic Force Reduced to Absolute Measurement. Translated by Susan P. Johnson.
1836: H.C. Schumacher (ed.). "Erdmagnetismus und Magnetometer". Jahrbuch für 1836 (in German). Tübingen: J.G.Cotta'sche Buchhandlung. 1836: 1–47.
1840: Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhältnis des Quadrats der Entfernung wirkenden Anziehungs- und Abstoßungskräfte [General Theorems concerning the attractive and repulsive Forces acting in reciprocal Proportions of quadratic Distances] (in German). Leipzig: Weidmannsche Buchhandlung. 1840.
1843: "Dioptrische Untersuchungen" [Dioptrical Investigations]. Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen (in German). Erster Band: 1–34.

together with Wilhelm Weber
Weber, Wilhelm Eduard; Gauss, Carl Friedrich (1837–1839). Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1836–1838 (in German). Göttingen: Dieterichsche Buchhandlung.
Weber, Wilhelm Eduard; Gauss, Carl Friedrich (1840–1843). Resultate aus den Beobachtungen des magnetischen Vereins im Jahre 1839–1841 (in German). Leipzig: Weidmannsche Verlagsbuchhandlung.
Weber, Wilhelm Eduard; Gauss, Carl Friedrich (1840). Atlas des Erdmagnetismus nach den Elementen der Theorie entworfen. Supplement zu den Resultaten aus den Beobachtungen des magnetischen Vereins (in German). Leipzig: Weidmannsche Verlagsbuchhandlung.

Collected Works
Königlich Preußische Akademie der Wissenschaften, ed. (1863–1933). Carl Friedrich Gauss. Werke (in Latin and German). Vol. 1–12. Göttingen: (diverse publishers).

Correspondence
Königlich Preußische Akademie der Wissenschaften, ed. (1880). Briefwechsel zwischen Gauss und Bessel (in German). Leipzig: Wilhelm Engelmann. (letters from December 1804 to August 1844)
Schwemin, Friedhelm, ed. (2014). Der Briefwechsel zwischen Carl Friedrich Gauß und Johann Elert Bode. Acta Historica Astronomica (in German). Vol. 53. Leipzig: Akademische Verlaganstalt. ISBN 978-3-944913-43-8. (letters from February 1802 to October 1826)
Schoenberg, Erich; Perlick, Alfons (1955). Unbekannte Briefe von C. F. Gauß und Fr. W. Bessel. Abhandlungen der Bayerischen Akademie der Wissenschaften, Math.-nat. Klasse, Neue Folge, No. 71 (in German). Munich: Verlag der Bayerischen Akademie der Wissenschaften. pp. 5–21. (letters to Boguslawski from February 1835 to January 1848)
Franz Schmidt, Paul Stäckel, ed. (1899). Briefwechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai (in German). Leipzig: B. G. Teubner. (letters from September 1797 to February 1853; added letters of other correspondents)
Axel Wittmann, ed. (2018). Obgleich und indeßen. Der Briefwechsel zwischen Carl Friedrich Gauss und Johann Franz Encke (in German). Remagen: Verlag Kessel. ISBN 978-3945941379. (letters from June 1810 to June 1854)
Clemens Schaefer, ed. (1927). Briefwechsel zwischen Carl Friedrich Gauss und Christian Ludwig Gerling (in German). Berlin: Otto Elsner. (letters from June 1810 to June 1854)
Karl Christian Bruhns, ed. (1877). Briefe zwischen A. v. Humboldt und Gauss (in German). Leipzig: Wilhelm Engelmann. (letters from July 1807 to December 1854; added letters of other correspondents)
Reich, Karin; Roussanova, Elena (2018). Karl Kreil und der Erdmagnetismus. Seine Korrespondenz mit Carl Friedrich Gauß im historischen Kontext. Veröffentlichungen der Kommission für Geschichte der Naturwissenschaften, Mathematik und Medizin, No. 68 (in German). Vienna: Verlag der Österreichischen Akademie der Wissenschaften. (letters from 1835 to 1843)
Gerardy, Theo, ed. (1959). Briefwechsel zwischen Carl Friedrich Gauß und Carl Ludwig von Lecoq. Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, No. 4 (in German). Göttingen: Vandenhoeck & Ruprecht. pp. 37–63. (letters from February 1799 to September 1800)
Carl Schilling, ed. (1900). Briefwechsel zwischen Olbers und Gauss: Erste Abtheilung. Wilhelm Olbers. Sein Leben und seine Werke. Zweiter Band (in German). Berlin: Julius Springer. (letters from January 1802 to October 1819)
Carl Schilling, ed. (1909). Briefwechsel zwischen Olbers und Gauss: Zweite Abtheilung. Wilhelm Olbers. Sein Leben und seine Werke. Zweiter Band (in German). Berlin: Julius Springer. (letters from January 1820 to May 1839; added letters of other correspondents)
Christian August Friedrich Peters, ed. (1860–1865). Briefwechsel zwischen C. F. Gauss und H. C. Schumacher (in German). Altona: Gustav Esch.
Volumes 1+2 (letters from April 1808 to March 1836)
Volumes 3+4 (letters from March 1836 to April 1845)
Volumes 5+6 (letters from April 1845 to November 1850)
Poser, Hans, ed. (1987). Briefwechsel zwischen Carl Friedrich Gauß und Eberhard August Zimmermann. Abhandlungen der Akademie der Wissenschaften in Göttingen, Mathematisch-Physikalische Klasse, Folge 3, No. 39 (in German). Göttingen: Vandenhoeck & Ruprecht. ISBN 978-3525821169. (letters from 1795 to 1815)
			
			
		
		
			
			
		
		
			
			
		
		
			
			
		
		
			
			
		
The Göttingen Academy of Sciences and Humanities provides a complete collection of the yet known letters from and to Carl Friedrich Gauss that is accessible online. Written estate from Carl Friedrich Gauss and family members can also be found in the municipal archive of Brunswick.

Names and Commemorations
List of things named after Carl Friedrich Gauss

References
Notes
Citations
Sources
Further reading


== External links ==