- Good evening.
Just a brief housekeeping note.
I just wanted to ask everyone to turn off
their cell phones if they
have them, thank you.
I'm Nat Silver, guest curator
of the exhibition here
Piero della Francesca in America.
And I'd like to welcome
you to The Frick Collection
this evening.
Tonight we are delighted to be joined
by Professor James R.
Banker who will deliver
the second in three
Wednesday night lectures
dedicated to the celebrated
renaissance painter
Piero della Francesca.
This series is made
possible by the generous
and continuing support of the
Robert H. Smith Foundation,
support for which we
are profoundly grateful
here at The Frick.
Before we begin I would
just like to mention
that tonight's event is being broadcast
live on the internet.
The Frick's FORA TV channel
has made it possible
to reach a global audience
and I extend my welcome
to our online viewers.
It gives me the greatest
pleasure to welcome
James Banker, Professor
Emeritus in Italian renaissance
history at North Carolina
State University.
Beyond his celebrated career in teaching
that spanned over 40 years,
Professor Banker has served
on scientific committees
for several exhibitions
dedicated to Piero della
Francesca in Italy,
held fellowships at Villa Itati
and the American Academy in Rome.
And lectured widely on early modern Italy.
His contributions to the
field of renaissance studies
are distinguished by a
long lost of publications
including the books Death in
the Community: Memorialization
and Confraternities in an Italian Commune
in the Late Middle Ages, and
The Culture of San Sepolcro
during the Youth of Piero
della Francesca in 2004.
It was while researching his
first book on confraternities
in San Sepolcro that
Professor Banker uncovered
a wealth of documents
pertaining to the lives
of it's 15th century citizens.
This trail of clues
revealed precious details
about the most celebrated one of them all:
Piero della Francesca.
Professor Banker's
discoveries in the archives
of San Sepolcro and
Florence are encompassing
and profound.
Thanks to just one of
his remarkable findings
made in 2001, and I
emphasize from the point
of view of an art
historian, this was a really
incredible finding, we can
today reimagine the original
shape of the Sant'Agostino altarpiece,
six fragments of which are
today in our exhibition
at The Frick Collection.
Professor Banker has
transformed our understanding
of Piero della Francesca's
career and we eagerly
await his forthcoming
monograph on the artist
which is soon to be published
by Oxford University Press.
Throughout the course of this project,
Jim has been a guiding
light, leading with his
learned erudition and his
infectious enthusiasm for Piero.
I'm thrilled to welcome him here tonight
to deliver a lecture
entitled Three Geniuses
and a Franciscan Friar.
Please join me in welcoming
Professor James R. Banker.
(audience applauding)
- Thank you Nat.
We are here tonight because
of Nat Silver's inspiration
to have an exhibition of the paintings
of Piero della Francesca in America
here at The Frick Museum.
I wish to thank him
and The Frick Director,
Doctor Ian Wardropper, for
the invitation to write
an essay in the catalogue
of the exhibition
and to be here with you tonight.
I also wish to express
my gratitude to the staff
of The Frick for their
ceaseless efforts on my behalf.
Tonight I shall mention
several of Piero's paintings,
and give an outline of his life.
But I shall concentrate
more on his passion
for geometry and Archimedes,
that supplemented
and eventually surpassed
his commitment to painting.
I wish also to share
with you how Piero became
one of the preeminent
geometricians of the 15th century
and how his knowledge
of Archimedes made its
way into the mind of Leonardo.
The most advanced and
intriguing mathematical
knowledge that are tracing from Piero
to Leonardo derives from Archimedes.
Let me also note here at
the beginning that much
of what I shall be
recounting today on Peiro
builds on the earlier seminal research
of Marshal Clagett and
the more recent research
of the last decade, largely undertaken
in Europe.
A few introductory
words on the two persons
other than Piero and Leonardo
in our discussion tonight.
Archimedes, died 212 BCE, is generally
regarded as the most
important mathematician
of ancient Greece and Rome,
and one of the geniuses in
the history of the west.
He was also an inventor,
being credited with
the Archimedean screw, among his many
other inventions.
These are two in the mirth dimension,
as are his conceptual leaps in geometry.
Several of the latter
will appeal as we discuss
Piero's contacts with the
writings of Archimedes.
The less well known Luca Paciloli was born
in Piero's town of Borgo Santo Sepolcro
around 1447.
A full generation after Piero.
Paciloli became a
Franciscan friar and taught
commercial and Greek
mathematics in universities
and courts of Italy, eventually
in Milan with Leonardo
in the 1490's.
Paciloli published books, real
books from printing presses
on renaissance mathematics.
To understand the mathematical
achievements of Piero,
Luca and Leonardo, we
must note the existence
of two cultures in the renaissance.
One culture was founded
on knowledge of Latin
and a university education.
Men in this tradition,
usually from noble families,
sought church and government
positions where they
used their linguistic skills
to produce Latin documents
and to reproduce Latin treatises.
They also attended to demean all those
who worked with their
hands, viewing that activity
as socially denigrating
and ultimately destructive
of the minds of artisans.
The other culture was composed of persons
who were called illiterate because
they could not read Latin.
This group could only read and write
in the vernacular and were from
merchant or artisan families.
An artisan worked with
his hands with which
he eventually proved himself
by producing a masterpiece.
Painters and sculptors
fell into this category
of artisans.
Socially inferior because
they worked with their hands.
The bifurcation of culture
underwent a fundamental change
during Piero's lifetime, in
part because of the creations
of the artist.
Many of the greatest
renaissance achievements
were made by men who
crossed the barriers between
the two cultures.
For us tonight, Piero and Leonardo.
Though not educated in the university
or capable of writing Latin treatises,
Piero and Leonardo...
could read Latin and write Latin phrases.
And they eventually absorbed
university-based learning.
Their uniting of the two
cultures enabled them
to make important innovations
in art and thought.
The division into two
cultures finds its extreme
expression in mathematics.
In the universities, Greek
mathematics and geometry
made up two of the seven liberal arts
and was said to be the underlying basis
for two others, music and astronomy.
Geometry and optics were largely based
on the writings of the Greek mathematician
Euclid who had been translated into Arabic
in the early middle ages
and from Greek to Latin
in the 12th and 13th
centuries in Western Europe.
But mathematical instruction
in the universities
was largely limited to medical students
who would employ it for
determining the astrological
basis for treatment.
On the other hand, instruction
in the Italian dialects
in the vernacular in
commercial mathematics
flourished in the renaissance.
Just after 1200, Leonardo da Pisa,
usually called Fibonacci,
brought Arabic commercial
mathematics and Arabic numerals to Europe
and became the father of
a mathematics that dealt
with fractions, interest, square roots,
currency exchange and algebra.
Together called abaco,
this practical instruction
lasting few years for
young boys introduced them
to problems that merchants
of the renaissance
encountered daily.
Abaco instructors produced textbooks
and this form of training
was found in the large
commercial centers like Florence, Venice,
and Luca.
But not in Piero's Borgo San Sepolcro.
And this presents us with a problem.
No one knows where Piero picked up
his early knowledge of mathematics.
Abaco would have come
easily enough to him,
in part because his father,
as a leather merchant and tax collector,
would have had many financial
account books.
It is probably that
Piero kept the books with
his father as a youth in the 1420's.
This would explain why the 16th century
painter Giorgio Vasari
who wrote biographies
of all the important
painters and sculptors
of the renaissance, asserted
that Piero concentrated
on mathematics until at 15 years of age
he turned to painting.
On the other hand, it was
indelibly a long process
for Piero or anyone
else in the 15th century
to understand and to
imitate the procedures
of Greek geometry.
Employing axoms and postulates,
Greek geometricians
logically deduced proof
based on proportion,
similarity and congruence.
They seldom used numbers.
Rather they begin and ended
with verbal statements,
usually about lines, planes and angles.
So words, not numbers.
The geometrician Euclid
flourished around 300 BCE,
summarized the Greek
logical deductive method
in his elements of geometry which served
as a geometrical textbook
for over 2,000 years
in the west.
Piero possessed his own
copy of this treatise
by the 1450's or '60's,
and from the elements
learned Greek geometry.
This prepared him for Archimedes.
We begin our closer analysis
of Piero in the 1450's.
By this time, Piero was in his forties.
He was completing a two-decade journey
through much of central
Italy after having left his
home town in 1439.
He painted The Baptism
of Christ there prior
to his departure.
The St. Jerome in the Desert in Ancona
in 1450.
The St. Jerome and the
Donor Jacopo Anastagi.
The fresco Sigismondo Malatesta Before
St. Sigismundus.
And a portrait of the ruler of Rimini,
Sigismondo Malatesta.
All three in Rimini in 1451.
And for much of the 1450's, Piero painted
the frescos in the church of San Francesco
in Arezzo.
Piero left Tuscany for a second sojourn
in Rome where he painted frescos already
destroyed by the 16th
century in the Vatican
for Pope Pious II.
More important, while in Rome in 1458-59,
he renewed his acquaintance
with the papal official
and architect of the
urban renewal of Rome,
Francesco Chario del Borgo,
who's father had married Piero's aunt.
Cousins once removed.
We want to tarry a bit
on Francesco del Borgo,
because he played a
significant role in the life
of Piero.
Francesco helped facilitate
a papal-led initiative
to revive Greek science through
translations of Greek
mathematical treatises
into Latin.
In the 1450's, Pope Nicolas
V, in large Francesco Patracas
earlier humanistic program
that emphasizes literature
in history.
The Pope commissioned new translations
of Greek treatises, especially
significant for us tonight,
a translation of the works of Archimedes.
We now know that this newly...
excuse me.
We now know that this
newly translated Latin
version which we shall call
the Paris Manuscript because it is today
in the Biblioteca Nacional in Paris,
was placed in the Vatican library.
It was immediately lent to members of the
intellectual elite.
In 1458, Francesco del Borgo removed
the Archimedes translation
from the Vatican
library without leaving the customary note
indicating the borrower.
Perhaps he chose not to
sign out the manuscript
because he did not intend to return it.
In any event, he never did.
Francesco commissioned a scribe to copy
the text of Archimedes
in another manuscript
which eventually was found in Urbino
and hence we shall call
it the Urbino Manuscript.
Because he was an architect and maker
of architectural plans,
Francesco probably drew
the 200 or so geometrical
designs that illustrate
the text of the Urbino manuscript.
He apparently intended
that he would be the owner
of this manuscript
because in this miniature
his name appears, probably
invisible to us tonight,
on the horizontal bar of the capital A.
In several other miniatures,
the unknown scribe
or painter has depicted what appears to be
the same man, in 15th century dress,
studying with books and tools.
The Italian historian of mathematics,
Argente Totchi, has suggested
that the miniaturist
has given us here the likeness
of Francesco del Borgo.
And one more image from that manuscript.
Francesco and Piero were
from Borgo San Sepolcro
and were related by marriage.
And both had an interest
in ancient geometry
and most important as we
shall see in a moment,
Piero came into possession
of the two Archimedean
manuscripts discussed
above that were first
in the possession of Francesco del Borgo.
For these reasons, we
can conclude that Piero
shared his appreciation, excuse me,
that Francesco shared his appreciation
and his knowledge of Archimedes with Piero
in Rome.
Piero returned to Borgo
San Sepolcro in 1459
from Rome.
At this point he entered an
intense period of painting
in his native town.
He completed several of his
most important paintings
in the years 1459-62.
The St. Louis of Toulouse,
the St. Julian if he
had not painted it a little earlier...
and he completed painting
of the Altarpiece
for the Confraternity of Santa Maria
della Misericordia.
In the same years he executed the frescos
of The Resurrection and the
Madonna del Parto,
as well as preparing or
beginning the painting
of the Poliptih for the
Church of Sant'Agostino,
commissioned back in 1454.
In the middle of the 1460's,
Piero was resident in Borgo San Sepolcro
and in the neighboring town of Arezzo
where he painted the Santa Maria Magdalene
in the Cathedral of Arezzo
and several other lost
paintings in and around
the cities.
In these years he also
completed the first of his
mathematical studies, the Trattato d'Abaco
or the Treatise on Abaco.
I say completed because it was obviously
the fruit of an extended period of study
and reflection.
Piero informs us early
in the work that someone
who had great authority and who could not
be denied asked him to write a textbook
on abaco.
The first portion of the
book is what traditionally
had been presented in textbooks going back
two and a half centuries:
fractions, roots,
squares, algebra, and the rule of three.
Piero apparently thought
that boys preparing
to be merchants needed a great deal
of instruction in the rule of three,
as he gave many examples.
The rule of three is a basic calculation
which we would set up as A is to B
as C is to D.
Piero discusses it in verbal phrases,
writing "though the rule of three
"says one should multiple
the thing one wants
"to know by what is dissimilar to it,
"and one divides the result
by the remaining thing.
"And the number that comes from this
"is of a nature of that
which is dissimilar
"to the first term, and the divisor
"is always similar to the thing which one
"wants to know about.
"For example, seven lengths of cloth
"are worth nine lira.
"What will five lengths be worth?
"Do it like this," he says.
"Multiply the quantity you want to know
"by that quantity which seven lengths
"of cloth are worth, that is, nine lira.
"Five times nine makes 45,
"divided by seven and what comes out,
"the answer, is 6 lira and a remainder
"of three sevenths."
I go into some detail on the rule on three
in part because Piero did so, but also
because it is illustrative of a quality
of Piero's mind.
The rule of three is
basically solving a problem
by proportion.
Proportion is at the center
of all the activities
of Piero.
Here in mathematics but also in geometry,
in the organization of pictorial space
and especially in perspective, proportion
is the underlying principle.
The second and more important part
of the Treatise on Abaco
is Pierro's discussion
on geometry.
Which had been a minor
part of earlier textbooks
and then only for practical problems like
measuring distances.
Piero elevated the Abaco
discourse by shifting
almost exclusive to theoretical problems.
He discussed the polygons
of plain geometry,
concentrated on solving problems of
the relationship of
angles to sides and areas.
He was especially
interested in the regular...
bodies sometimes called regular solids.
And irregular bodies.
I shall delay our discussion of these more
important propositions until we examine
Piero's second and later
mathematical treatise.
In his treatise on Abaco,
he already demonstrates
a working knowledge of Greek geometrical
methodology and of Euclid's elements.
Piero was encouraged by
the Greek to contemplate
the importance of proportion.
It is a central idea of Euclid as well.
Piero mentions Archimedes and says he has
taken material from him,
but it is clear that he is
citing the Greek concepts
through one or more
medieval commentator.
Given my earlier discussion
of Piero's interaction
with Francesco de Borgo,
and the latter's interest
in Archimedes, one might expect that Piero
would have celebrated Archimedes in his
Tratatto.
But he does not.
Piero's relationship at
this point to Archimedes
is complicated.
He does not cite him directly.
He does not cite the text of Archimedes.
This is strange unless
we take the most likely
explanation - that in
Tuscany in the 1460's he had
as yet no copy of Archimedes in hand.
Just after having completed
his Treatise on Abaco
in the mid-60's, Pierro
painted three masterpieces
in the late 1460's.
The large and beautiful
Sant'Antonio Alterpiece
in Perugia.
The Flagellation, probably
realized in Urbino.
The Sant'Agostino Alterpiece.
For this photo montage of Sant'Agostino,
I thank Elaina Squeelatoni
and Juacamo Guatsini,
young Italian scholars
who present their findings
in the exhibition catalogue.
In the 1460's Piero was
at heights of his powers
and realized his most
Piero-esque paintings.
That is monumental figures placed into
a fully realized perspectival space.
He positioned sculpted human beings
in an imagined but real historical world.
In this period, Piero also began
to accumulate materials
for his second book,
his Prospectiva Pingendi,
or On Perspective in Painting.
Which I believe was being
completed in the mid-1470's.
We shall not comment on
this highly original work
here except to note that
he does mention Archimedes
at all in the treatise.
In the period of the 1470's Piero divided
his time between Bargo San Sepolcro
and the marque town of Orbino.
In the former, San Sepolcro,
he painted his only
large classical figure, the Hercules,
for his family home.
In Orbino he received three commissions
from the ruler of town,
Federico de Montefeltro,
the so-called Uffizi
Diptych with portraits
of Federico and his wife Battista Sforza.
The Brera Altarpiece of Madonna and Child
with Saints and Angels,
and the Senigallia Madonna
and Child with Angels.
Late in the 1470's, Piero
made a radical change
in his activities, almost
abandoning painting.
He executed only two extent
paintings thereafter.
Instead, he finished On
Perspective in Painting.
In the vernacular, specific
to Borgo San Sepolcro,
and then set up a scriptorium
with all the equipment
and assistants, scribes
and at least one translator
necessary to produce at
least eight manuscripts.
Most notably, five copies
of his On Perspective.
This constituted a substantial enterprise
that would have consumed a great deal
of his time and energy.
His participation in the
making of the manuscripts
differs from one to the other.
In this manuscript, in
Parma, he wrote the Italian
text completely in his
hand, as well as drawing
all the diagrams, over 100.
These he had created to
instruct an apprentice
on how to draw paintings in perspective.
In the Reggio Amelia example he copied
two pages of the text in Italian and again
drew all the diagrams.
And in the main script,
in the Ambrosiana library
in Milan, Piero has added
corrections in Latin
to the Latin text as well as again
drawing all the instructional diagrams.
Finally in a Latin version
of the On Perspective,
today in Bordeaux, Piero
made Latin corrections
in the margins of the Latin text.
Why he decided to supervise and intervene
in the production of all these copies
of his book On Perspective is one of
the many mysteries that
perplexed observers
of Piero's life.
One of the more important discoveries
of the last decade concerning Piero
is that his fascination
with Archimedes went far
beyond reading about him
in the comments of others.
Piero in the late 1470's gained access
to manuscripts of the work of Archimedes,
the Paris and Orbino
manuscripts, that Francesco
de Borgo had possessed in the 1450's.
Borrowing portions of
both of these manuscripts,
Piero sat down and copied in his own hand
all the treatises of Archimedes and drew
approximately 225 geometrical
diagrams all extent
today in a manuscript in
the Riccadiana Library
in Florence.
Again, this copying of the manuscript was
a time and energy consuming project
in part because of the
need of first gaining
the intellectual expertise necessary
to understand the Latin text, and then
to link the Latin phrases
with the geometrical
images.
I shall raise here the question of whether
Piero thought in Latin or in Italian while
dealing with the geometry of Archimedes.
Here we have no proof,
but because Piero never
wrote a sequence of Latin sentences
or a succession of Latin paragraphs,
I am inclined to hypothesize that he
reconceptualized Latin
in his native Italian.
If so, he would have had in many cases
to invent Italian words for
the difficult Latin geometrical
terms and to rethink the procedures
of Archimedes in Italian.
It is clear that he
understood the diagrams
in the two manuscripts.
In certain instances,
Piero discovered that
the diagrams in the two manuscripts
did not follow the logic of the text.
And he had to correct the errors he found
in the geometry of the diagrams.
This demonstrates that
Piero was not simply
a scribe copying the words and diagrams,
but a geometrician thinking
through the implications
of the Latin text.
Note that this,
the first page of the manuscript,
has no title, or author.
Or the scribe who wrote the manuscript.
How do we know that it is Piero's?
In notes to his last will and testament
we see the G,
I hope we can all see it here.
The A, and let's see if we can find...
and let's see if we can find an E,
which is difficult from
my angle, I'm sorry.
Which is written like a Z.
So the E is like a Z.
Now we move back to the
Archimedes manuscript.
And we see in this blow
up of the first lines
the same letters: the
G, the A and especially
the E as a Z.
I trust we can locate those.
The fact that the hand in the testamentary
notes and in the manuscript is the same
and the fact that there's such a beauty
in the diagrams that Piero drew
at least convinces me that we have here
a complete manuscript of Archimedes
written by Piero.
And I especially note the spirals
at the bottom of the page.
Access to and possession of manuscripts
of Archimedes was limited
in the renaissance.
Only a few enjoyed that
privilege, and they were
Latin university elite or humanists with
a university education.
We should note that there
are only five or six
extent Latin manuscripts of Archimedes
from the 15th century.
Piero was probably the only
person in the 15th century
to hold in his hands
three of these manuscripts
of Archimedes.
That Piero was one of the
most important geometricians
of the 15th century is even more clear
when we examine his third book.
The Libellus de Quinque
Corporibus Regularibus,
or The Little Book on
the Five Regular Bodies.
Piero tells us that he wrote this book
late in life, thus in
the 1480's, and in part
to keep the Torp War
at bay when it invaded
his aged mind.
He also tells us at the
beginning of the book,
and one of the few passages where he says
anything personal, that
he wished to obtain
the fame similar to that
of the ancient painters
and sculptors.
In writing the Libellus,
Piero began a whole
new genre of mathematical literature.
No one previously had isolated the subject
of the five
regular bodies in a book.
Not that the regular
bodies had been neglected,
they were discussed from Plato and Euclid
to the medieval writers.
First, some definitions.
A regular polygon has equal
sides and equal angles,
thus one of the chief
subjects of plane geometry.
A regular body is a
solid geometrical shape.
A polyhedron made of a specific
type of irregular polygon.
A cube, for example.
In each regular body, all sides and angles
are the same and have identical faces.
When combined to create a solid,
the polygons of their
regular body form themselves
identically at their vertices.
The simplest regular
body is a tetrahedron,
a solid form with four
equilateral triangles.
Another simple regular body is the cube.
We have here six squares.
All line segments are equal, the faces are
the same, all the angles are 90 degrees
and are arranged around their
vertices in the same manner.
The other three regular
bodies are more complex.
The octahedron has eight equilateral
triangles.
The dodecahedron has 12
faces of regular pentagons.
And the last is the
icosahedron composed of 20
equilateral triangles.
Several Greek writers
were intrigued with these
five regular bodies, as was Piero.
Plato, in his late work The Timaeus,
viewed the five regular
bodies as the building blocks
of the universe.
With four of the bodies
equated with earth,
water, air, fire, and the
fifth, the most complex
icosahedron with the ether of the heavens.
But Piero would have none of this.
He was interested in the regular bodies
as theoretical geometrical constructions
which could be used to
understand other elements
of geometry: areas, measurement of size,
and especially the relationship
of the regular bodies
to the spheres that would be placed inside
or around the regular bodies.
Here, Piero used the formula of Archimedes
of three (mumbles) for the relationship
of the diameter to the circumference.
And the circle's area, along with many
other borrowings.
In the Libellus, Piero cited Archimedes
with such accuracy that he must have had
his copy of the works of
the Greek geometrician
in hand, which he would consult to cite
specific propositions by title and number.
Despite the title of The
Little Book on the Five
Regular Bodies, the most
sophisticated geometry
is in the last section
where Piero discusses
the 13 Archimedean or irregular bodies.
A solid or polyhedron
was irregular when it was
composed of either two or
three regular polygons,
or stated differently,
each was composed of two
or three types of
equilateral and equal angular
polygons.
The fact that each of
the 13 had more than one
regular polygon made each irregular.
Archimedes is credited
with first identifying
these forms.
Until the modern world, knowledge of how
to construct several of these solids
was unknown.
Already in his earlier Treatise on Abaco,
Piero had discussed two
of the 13 regular bodies.
In one of these discussions
he took a regular
tetrahedron and cut off
the vertices in the middle
of the sides and then
connected the two vertices
with lines.
He had constructed an irregular body
called a truncated tetrahedron
with four equilateral
triangles and four equilateral hexagons.
In the second, he
constructed a truncated cube
by cutting the vertices of, or cutting
the sides of the cube one third down
the sides and connecting the two points.
Thereby producing an irregular body
of six pentagons and eight
equilateral triangles.
In the Libellus he discussed
how to construct three others.
In the 15th century
Piero is one of the few
who wrote on these irregular solids.
The truncated tetrahedron
and truncated cube were
relatively easy to make
but the construction of other
more difficult irregular
bodies require the use of the golden mean,
and thus not a whole number.
For two irregular bodies,
he was the only person
in the 15th century who
knew how to make them.
Where he learned how to construct the two
is unknown.
The one ancient Greek source on this
was not available to
him in the 15th century.
I would suggest that Piero
was able to construct
these irregular bodies
because of his distinctive
visual acuity.
He brought his graphic
skills from his preparatory
drawings for paintings
to his study of geometry.
This enabled him to see
and construct solutions
rather than to read them
from an ancient text.
In one final example of his
geometrical sophistication,
Piero imagined and solved a famous problem
in the history of
mathematics which originated
with Archimedes in his treatise on method
that was only recovered
in the 20th century.
Hence Piero could not
have known of the problem
from Archimedes and in fact there is no
surviving explanation from Archimedes
himself on how to solve the problem.
In a simplified form,
Archimedes challenged a fellow
geometrician to determine the common value
of two crossing cylinders.
This problem led Piero
into a long discussion
of measuring cones,
cylinders and ellipses.
Because he had no available
ancient or contemporary source
that raised or solved this
problem, in all likelihood
Piero undertook to imagine
and to solve the problem
from his knowledge of the architecture
of crossing vaults...
which were very common
in the 15th century.
They were seen in buildings,
drawings and paintings.
In fact, in his On
Perspective in Painting,
Piero drew such a vault.
This Archimedean problem
and its use of a drawing
from Piero's treatise and his
graphic skills for painting
are an excellent example
of the way Piero combined
Latin verbal culture of the university
with the vernacular
visual culture of the workshop.
The emphasis on geometry
and writing by Piero
is the other side of
the reality that Piero
had largely abandoned
painting in his later years.
It may have been that he
received no commission
that interested him or challenged him.
The one painting he surely
executed in the 1480's,
The Nativity, never left his
home until the 19th century
and was probably intended for his home.
The other possible
painting from this period
around 1480 is in the nearby oval room.
The Williamstown Madonna and Child
with Angels from around 1480 was also
probably intended for a home.
In the last 15 years of his
life, Piero concentrated
his attention on his hometown and family
and apparently did not receive
or accept in that period
a commission for a publicly
displayed painting.
This turn inward to Borgo San Sepolcro
and his family tracked an
emphasis for home settings
or at least non-public
settings in his late paintings.
The Book of the Dead of Borgo San Sepolcro
record the death of their famous citizen.
Piero, October 12th 1492.
In this Book of the
Dead we can barely read
the words in the middle
below the Italian word
for October, Piero de Benedeto Francesca
and here they say that he
was the famous painter.
Giorgio Vasari, a half century later,
opened his biography of Piero by noting
that his mathematical accomplishments went
unnoticed because they were,
"they were falsely
claimed by the presumption
"of one who sought to
conceal his asses skin
"under the honorable spoils of the lion.
"Hence the true craftsmen
is robbed of the honor
"that is due to his labors as has happened
"to Piero della Francesca
of Borgo San Selpolcro.
"He, having been held a rare master
"of the difficulties of
drawing regular bodies,
"as well as of arithmetic
and geometry, was yet not
"able to bring to light his
noble labors and the many
"books written by him
which are still preserved
"in the Borgo, his native place.
"The very man who should have striven
"with all his might to increase the glory
"and fame of Piero from
whom he had learned
"all that he knew was
impious and malignant
"enough to seek to blot out
the name of his teacher.
"And to usurp for himself the honor that
"was due to the other,
"publishing under his own
name Friar Luca Paciloli
"del Borgo, all the labors
of that good man Piero."
There has been a scholarly debate
on Luca Pacioli's culpability.
But on Luca Pacioli's theft
of Piero's intellectual
property.
I tend to agree with
Asari, that the ass stole
the lion's spoils.
But we shall forgo a
discussion of this debate.
But I do want to note
very quickly some items
in this image.
First on the bottom right, note the book.
The closed book, which
while we can't see it,
does have abbreviations
indicating the book
is by Luca Pacioli.
Also take a look at the
open book that Pacioli
is pointing to.
A historian of mathematics has figured out
that this is actually
the Elements of Euclid
and he's able to identify
the precise proposition.
Lastly let us notice on the left...
the irregular body.
It's transparent and if we
could see clearly enough
we could see that it
reflects the two personages
in the painting.
This is an irregular body
of eight regular triangles
and 18 squares.
I have no interest tonight
in inditing for plagiarism
Pacioli, but I wish to
describe what he took
from Piero, especially
the painter's insights
from Archimedes.
As I initially indicated,
Pacioli became a Franciscan friar
and a university professor in mathematics.
Already in 1478 he was
a lecturer in Peruga
on commercial mathematics and eventually
he taught mathematics in the courts
or universities Rome, Naples, Florence,
Milan, Bologna and elsewhere.
In 1494, Pacioli
published in Venice a book
entitled the Summa of
Arithmetic, Geometry,
Proportions and Proportionality.
In this work, Piero
disseminated a broad swath
of medieval and renaissance mathematics,
which became the textbook of mathematics
for 16th century Europe.
In the section entitled
Particularas Tractatas
in the Summa, Pacioli borrowed directly
from the geometrical portions of Piero's
treatise on Abaco.
Pacioli laid out 56 exercises on regular
and irregular bodies, borrowed from
Piero's treatise.
About 50% of them are taken nearly
word for word from Piero.
Pacioli continued to lecture
at various universities
and courts until his death in 1517.
In 1498 and 1512 he published
his Divine Proportion.
Here he finds a mystical signs in this
Divine Proportion which is the same as
the golden section.
In this book, Pacioli has taken Piero's
Latin Libellus, his little
book on the five regular bodies
and translated the whole book into Italian
and published it as his own without even
hinting Piero was the author.
He also has another section on the regular
and irregular bodies in
the book where he repeats
much of this same material.
This time taken from Piero's
earlier Treatise on Abaco.
Pacioli does praise Piero
as "the monarch of painting"
and seems proud to claim
him as a co-citizen
of Borgo San Sepolcro.
But Pacioli carefully
avoids even suggesting
that Piero had any mathematical interests.
He does note that in
Piero's book on painting,
that nine of out ten books are devoted
to proportion but he does not acknowledge
that his own interests in
proportion and geometry
has any relationship to
Piero's concentration
on proportion in either
painting or geometry.
Pacioli had another important role to play
in the dissemination of the knowledge
and teachings of Archimedes.
Pacioli, like Piero, was
fascinated with irregular
and irregular bodies.
In the late 1480's
Pacioli made or supervised
the construction of models
of the regular bodies
which he presented to the son
of Federico de Montefeltro
and ruler of Urbino Guido Baldo.
In the years 1496 to 1499,
Luca Paciloli and Leonardo da Vinci
were brought into the circle
of scholars and artists
recruited by Ludovico
Sforza, ruler of Milan.
There, the artists and
the mathematician became
friends and collaborators.
Leonardo at that point
was already established
as an important painter
in Florence and made
claims to Sforza of his abilities
in military engineering.
At the time of his entrance into the court
in Milan, an examination of Leonardo's
early notebooks show that
he was a poor mathematician
but with the aid of Paciloli he became
an expert geometrician and a great admirer
of Euclid and Archimedes.
Even before their convergence in Milan
in 1496, Leonardo had
purchased Paciloli's Summa
on Mathematics.
Leonardo says at one point in his journals
"I learned multiplication of the roots
"from Master Luca."
We do not have time to
discuss Leonardo's astute
borrowings from Archimedes,
it is clear though
that he possessed one of the manuscripts
of Archimedes in the family of manuscripts
discussed above, either the Paris, Urbino
or Piero's manuscript.
During the three years in Milan,
Leonardo borrowed ideas on proportion
directly from Paciloli.
They both elevate the
importance of proportion.
For example, in 1497, Leonardo says
"proportion is not only found in numbers
"and in measurements, but truly also
"in sound, weights and clocks."
This is taken nearly word for
word from Paciloli's Summa.
There are more borrowings
that show that Leonardo
depended on Paciloli for his schooling
and mathematics.
Paciloli published 59
drawings of the regular
and irregular bodies in
the divine proportion.
And in the early 1500's,
he presented wooden models of these bodies
to the government of Florence.
Paciloli also included the 59 designs
in his Divine Proportion.
In that book he stated
that his close relationship
to Leonardo led the painter to assist him
by drawing the regular
and irregular bodies
and we see several here,
drawn by Leonardo.
By the early 1500's Leonardo began to show
his own interests in Archimedes.
For one among reasons for his interest
is that Leonardo became
interested in the problem
of squaring the circle.
He was also searching for a copy
of the works of Archimedes and made
mental notes to himself
on where his manuscripts
were located.
On one occasion he said that the brother
of a cleric had a copy
of a complete Archimedes
in Rome and that it had been
stolen in 1502 from the Urbino Library
by Cesare Borgia, the warrior son
of Pope Alexander VI.
This manuscript was likely the one that
Francesco del Borgo
commissioned in 1457/58
that we named the Urbino Manuscript.
On another occasion Leonardo notes that
one could retrieve a
manuscript of Archimedes
from the Bishop of
Pagua, we'll ignore this,
and more importantly, "from Vitellozzo,
"the one from Borgo San Sepolcro."
Now which manuscript is Leonardo speaking
of as being in Borgo San Sepolcro?
It is probably not the one from Urbino,
the one stolen by Cesare Borgia,
that was in Rome.
The Vitellozzo that Leonardo mentions
was the rular of Citta di Castello
ten miles south of Borgo San Sepolcro.
Vitellozzo Vitelli was also
a captain in Cesare Borgia's
mercenary army.
He was made famous by Machiavelli
who recounted in the prints that in 1502
Vitellizzo was treacherously married by,
murdered by Borgia.
(audience laughing)
Earlier in that year
Leonardo as cartographer
for Borgia had accompanied Borgia
and Vitellozzo as the
passed through the area
around Borgo San Sepolcro
and Citta di Castello.
Hence, Leonardo probably
gained his knowledge
of the Archimedes manuscript through
Vitellozzo or while in the region of the
Upper Tiber around Borgo San Sepolcro.
It is possible that the
manuscript that Leonardo
was referring to was the
Paris one that Francesco
del Borgo had taken
from the Vatican library
and never returned.
It might have been sent
to Borgo San Sepolcro
after Francesco's death
as part of his patrimony.
We know that Piero at one
point had access to it.
It is also possible
that Leonardo knew from
Vitellozzo of the manuscript of Archimedes
that Piero had copied in his own hand
and that remained in the hands
of the painter's nephews.
Vitellozzo and Piero
are documented as having
at least a passing familiarity.
Late in life, Piero served
as a witness for Vitellozzo
in a notarial document.
One hopes the mystery of how Leonardo
knew of the Borgo San Sepolcro manuscript
of Archimedes is on its
way of being solved.
Piero was one of the
first in modern Europe
to possess an intense
interest in Archimedes.
By citing the Greek
geometrician in his two
mathematical treatises
and eventually copying
approximately the Greek's 150 pages
and over 200 geometrical
designs in his own hands.
Piero took five important
ideas from his much
revered books of Euclid and Archimedes.
The methods of proof,
the five regular bodies,
the golden section, and the
Archimedean or irregular bodies.
Especially going beyond
all his contemporaries
in his ability to discuss
and draw the regular bodies.
Lastly, Piero took from
Euclid and Archimedes
the importance of proportion in geometry
and applied it to his
painting, especially in
emphasizing the central
importance of perspective
in painting.
Piero also began a way
of thinking in geometry
that broke with great precedence.
His practices laid the
basis for an element
of modern science.
The Greeks wrote their
geometrical treatises
in grammatical phrases
and made their proofs
arguing from assumed
axioms and postulates.
These proofs were not
established with numbers
but through similarity and congruence.
Nurtured initially in
commercial mathematics
in Abaco, Piero was
accustomed to constructing
problems through numbers.
He added, he subtracted, he divided
and so on in Arabic numbers.
When Piero came to write on geometry
he combined the two
mathematics which I shall
remind you derive from
two cultural traditions.
By placing the numbers in procedures
of Abaco within the form of argumentation
of the Greeks, Piero
succeeded in numerizing
geometry.
It was just another step
to say that this numerical
geometry expressed the
underlying structure
of nature.
Certainly, Luca Paciloli
believed and wrote this
but whether Piero did as
well is an open problem
that researchers will want
to address in the future.
Thank you.
(audience applauding)
