Perspective, Painters, and Sailors
Plague came to Europe through Genoa in 1347 from Kaffa on the Black Sea where
Genoa had maintained a trading post with China and the East. The disease was carried
by fleas and spread by rats and was particularly ugly and particularly lethal. In a few
years, plague wiped out almost half the population of Europe and, when it was done, it
left the survivors a good deal richer since there was no decrease in the amount of property
so the amount of property per person nearly doubled.
Italy recovered the quickest and wealth concentrated in the cities. And the survivors,
perhaps because they had a healthy understanding that life was transient, were inclined to
spend money on luxuries.
They were, however, still living under the Aristotelian construct of science which said
that every thing in nature had an individual place in the order of things (the Great Chain
of Being). The earth was at the center of the universe and the heavenly bodies rotated
around it in fixed crystal spheres. In painting, reality was unimportant. Saints were
larger than normal humans because they were more important. Demons and devils were
smaller because God gave them less importance. There was no background, but simply a
matte of gold paint. That was because there was no such thing as empty space since God
filled everything and was symbolized by gold. Gold symbolized God because it was the
king of metals and late medieval chemists devoted their lives and virtually all their
energies to finding the philosopher’s stone that would change base metals to gold.
Specifically, these paintings seem unrealistic because they entirely lack perspective.
The fresco in the slide is by Andrea da Firenze (1366-68) and is in the chapter house of
Santa Maria Novella in Florence. It was paid for by the merchant Buonamico di Lapo
Guidalotti to celebrate his escape from the plague.’
Gold wasn’t the only thing at the top of a hierarchy. In fact the Great Chain of Being
placed everything in a hierarchy with God above the saints above humans above animals
above plants above stones. Within the groups there were hierarchies as well. The lion
was the king of beasts and the eagle the king of birds. Within humans, kings were above
nobles who were above merchants who were above peasants who were above slaves.
Men were always above women. The hierarchy never changed and one never moved
from one hierarchy to another. A comfortable anti-evolutionary point of view, especially
if you were satisfied with your position in the order.
As one might expect, the late medieval mind was entirely taken with symmetry. There
were four elements (earth, air, fire, and water), four cardinal directions, 4 seasons, and 4
ages of man. The heavens rotated in a crystal sphere because the sphere was the perfect
solid and crystal the perfect substance.
Italy also had a cultural bent toward mathematics—possible because they were close to
the sea and were traders and they had more universities than any other area in Europe.
The Italians were captivated by numbers and by Pythagorean symmetry. The peninsula
also lay directly between Europe with its gold, wood, furs, leather, grain, and Slavs and
the routes to the east with spices, silk, cotton, and perfumes. They were busy getting rich
trading and running ships.
The Florentines were a bit of an exception because they, unlike Genoa, Venice, and
Rome, did not have direct access to the sea. They did however become Italy’s and
Europe’s bankers, in no small part because of the work of the mathematician Leonardo
Fibonacci whom you might know for his series which is a thing of mathematical beauty
in itself. Much more important to the Florentines was the fact that he brought the city the
Arab method of recording income and expense in duplicate and reconciling the two sets
of entries exactly—double entry accounting.
Double-entry bookkeeping system ensures the integrity of the financial values recorded
in a financial accounting system. It does this by ensuring that each individual transaction
is recorded in at least two different nominal ledgers (sections) of the financial accounting
system and so implementing a double checking system for every transaction. It does this
by first identifying values as either a Debit or a Credit value. A Debit value will always
be recorded on the debit side (left hand side) of a nominal ledger account and the credit
value will be recorded on the credit side (right hand side) of a nominal ledger account. A
nominal ledger has both a Debit (left) side and a Credit (right) side. If the values on the
debit side are greater than the value of the credit side of the nominal ledger then that
nominal ledger is said to have a debit balance.
Each transaction must be recorded on the Debit side of one nominal ledger and that same
transaction and value is also recorded on the Credit side of another nominal ledger hence
the expression Double-Entry (entered in two locations) one debit and one credit. This
ensures that when the nominal ledgers (sometimes known as accounts) are placed in a list
which has two columns, the left column for listing nominal ledgers with Debit balances
and the right column for ledgers with Credit balances, then the total of all the Debit
values will equal the total of all the Credit balances. If this does not happen that may
mean that one of the transactions was not recorded twice, i.e. once as a debit and once as
a credit as required in the double-entry bookkeeping system.
And the accountants were mathematicians. Witness the painting of Paccioli who became
famous for his text on double entry accounting teaching geometry from Euclid.
Before Fibonacci, merchants had recorded each transaction as a separate entity with notes
below the transaction to include costs of the transactions, interest on notes, and
repayment or payment terms. No one knew exactly how much they had at any given
moment. Fibonacci’s accounting let the Florentines keep exact track of their finances and
manipulate them accordingly. The Medici family in particular became phenomenally
wealthy and influential as bankers. By 1397, they were Europe’s financiers.
Giovani di Bicci, increased the wealth of the family through his creation of the Medici
Bank, and became one of the richest men in the city of Florence. Although he never held
any political charge, he gained strong popular support for the family through his support
for the introduction of a proportional taxing system. Giovani's son Cosimo the Elder took
over in 1434 as gran maestro, and the Medici became unofficial heads of state of the
The "senior" branch of the family — those descended from Cosimo the Elder — ruled
until the assassination of Alessandro de' Medici, first Duke of Florence, in 1537. This
century-long rule was only interrupted on two occasions (between 1494–1512 and 1527–
1530), when popular revolts sent the Medici into exile. Power then passed to the "junior"
branch — those descended from Lorenzo the Elder, younger son of Giovanni di Bicci,
starting with his great-great-grandson Cosimo I the Great. Cosimo and his father started
the Medici foundations in banking, manufacturing - including a form of franchises -
wealth, art, cultural patronage, and in the Papacy that ensured their success for
generations. At least half, probably more, of Florence's people were employed by them
and their foundational branches in business.
Finance and trade worked. Venice grew to 100,000 as did Naples. Florence had 40,000
and Rome the same (both equal to Paris). As they grew, the bureaucracy also exploded.
Between 1350 and 1400, the number of Florentine bureaucrats tripled and the number of
lawyers, accountants, and notaries grew a similar amount. The public debt also exploded
and, by 1400, was 7.5 times the wealth of the entire city.
Between 1410 and 1420, Florence underwent a dramatic change coincident with the rise
to prominence of Paolo del Pozzo Toscanelli.
Paolo dal Pozzo Toscanelli
Mathematician, astronomer, and cosmographer, b. at Florence in 1397; d. there, 10 May,
1482. Toscanelli, who was one of the most distinguished scientists of the fifteenth
century, was the son of the Florentine physician Dominic Toscanelli and came from a
family of wealth spice traders. The family had 19 servants, 2 horses, and a mule.
He began his mathematical studies at Florence under Giovanni dell'Abacco. At the age of
eighteen he entered the University of Padua where he studied mathematics, philosophy,
Padua had been under Venetian rule since 1404 and, since Venice was on poor terms with
Rome, the university was not nearly as bound to Orthodox Church doctrine as other parts
of Europe. Also, a number of the supporters of the Spanish Moslem Averroes had fled to
Padua in the preceding two centuries and brought Arabic mathematics with them.
After getting a degree in medicine, Toscanelli returned home. At Florence Toscanelli
took up scientific studies which brought him together with the most learned men in
Florence, other cities of Italy and other countries. He may indeed be said to have been the
centre of the learned world of that era. His contemporaries pronounced him one of the
most distinguished mathematicians of his time.
At a friend’s dinner partying 1424, he met Filippo Brunelleschi, a local architect and
builder who had been hired to build the dome on the city’s new cathedral. Brunelleschi
was a tradesman and an architect by practice only. No one in Italy had actually used
geometry in architecture when Toscanelli introduced the idea to him. Using Euclidean
geometry, Brunelleschi was able to build a dome without internal columns larger than
had ever been done—even by the Romans. It was finished in 1436 and the moderns had
finally done something demonstrably better than the ancients.
Toscanelli had also studied optics from works of al Hazen translated by Biagio in Padua.
Al Hazen had studied Aristotle, Euclid, and Ptolemy. The common wisdom had been that
the eye sent out rays toward and object that were reflected back and recaptured. That
made no sense to al Hazen because he saw no way the eye could produce enough rays to
illuminate the whole of a visual field with every blink of the eye. He decided that light
must be emitted from a source such as the sun or a candle and that light would reflect off
an object into the lens of the eye.
Alhazen comes up with one idea. He brings an element of sensitivity without any element
of direction, the eye can see light and colors mixed and confused and an organized vision
links one point of the object to one impressed point of the eye from the optical diagram of
the eye of Galen and considering that the several different transparent layers are
concentric, the crystalline at the centre of the eyeball where the essential of the visual
faculties stands; Alhazen takes into consideration Euclid’s pyramidal perspective but
draws it with rays going from various part of the object and flowing at the surface of the
eye and therefore places the summit of the pyramid at the centre of the eye. Colors and
different points of the object are spread in a regular way along the rays to the cornea and
pupil. They get printed on the first layer of the crystalline. This picture in the same order,
similar to the object is therefore in the eye with all the elements to realize a vision
knowing the existence and the nervous structure of the retina why Alhazen admits that
printing is made at the first layer of the crystalline? The answer to that question is
evidence? It is due to the difficulty to get the rays through the crystalline.
Said another way, the eye was the apex of a pyramid of light. It was a short jump from
there to the idea of artistic perspective. In fact, Biagio’s lectures were titled ―On
Perspective‖ although by that he meant the perception of objects.
Toscanelli explained al Hazen’s theory to Brunelleschi with the idea that one might use
the converging ray idea to make three dimensional drawings of architectural plans for
prospective clients and sell more business for the architect.
But there was one more step.
In 1291, the government of Venice banned the furnaces from central Venice, moving
them instead to the island of Murano. This was due to the fear of fire in the wooden
structures of a crowded Venice and also to isolate the master glassblowers to prevent
their sharing valuable glassmaking secrets. The glassblowers became virtual prisoners on
Murano, where the penalties for divulging glassblowing secrets could include death. One
of the products that were being made in Murano was a flat lead backed piece of glass that
served as a near perfect mirror. Toscanelli (among others) had noticed that the reflection
in a mirror seemed to have depth and that one could exaggerate the depth by rotating the
Brunelleschi put a mirror six feet in front of the main door of the cathedral at Florence
and aimed it at the Baptistery across the plaza. The then painted over the reflection
caught in the mirror. He next drilled a hole in the middle of the mirror and turned it
around and mounted it in the same place. He had people stare through the hole at the
actual building and then step back and look at the painting on the glass. The observers
were stunned at the depth and the similarity to the real thing. It was the first perspective
painting and the first time people had created an image that actually looked like real life.
The first commissioned perspective painting was The Trinity by Brunelleschi’s friend
Masaccio in Santa Maria Novella which is still in the chapel in Florence. Look especially
at the lines in the vaulted ceiling which are in mathematically exact perspective. It is still
possible to see the lines scratched in the plaster that the artist used to make the lines.
Parenthetically, the vanishing point of the fresco is at exactly 5’3‖ high which was the
average height of the fifteenth century Florentine.
Alberti put the method in writing (in Latin) and showed how one could use gauze to
create a grid to look through and divide up a scene into small squares that could each be
copied and the whole would come out in perspective. He also showed how one could tie
strings to the parts of an object and bring them to a gauze grid to make perspective lines.
It is hard to understand the philosophical change this brought. Since Aristotle, the
position of objects (and people) in space was only important insofar as it affected their
relationship to God. Now things in space could be accurately portrayed in relationship to
one another. In a way, it removed the unique and special relationship to the deity but it
brought a whole new world of interactions with physical surroundings.
Things could be measured and reduced to formulae rather than just memorizing an
Aristotelian ―essence.‖ And the measurements could be standardized and were the same
for everyone and everything. Beyond that, if relationships could be reduced to formulae,
there must be laws that governed those relationships (laws of nature) and it was at least
theoretically possible to learn what those laws were. One no longer had to explain things
that happened as merely God’s will. Events were definably the result of laws that one
could decipher and understand.
Not only science changed from memorizing categories and hierarchies to learning
generalizable laws. Art moved away from depicting saints to depicting normal life and
the subjects of many Florentine paintings were either religious scenes with actual
Florentines in the scenarios to pure portraits of wealthy citizens. Religious tracts were
supplemented by (or even replaced by) biographies of famous citizens and even novellas.
Perspective also spread to more mundane and useful representations like maps.
Remember that Toscanelli’s family were spice traders and had a lively interest in
accurate maps for their trading ships. So did the king of the maritime Portuguese who
were beginning to push down the west coast of Africa in hopes of finding a sea route to
Meanwhile, the Turks were gradually surrounding the 1000 year old capital of the
Eastern Empire at Constantinople. The city also lay across the entrance to the Black Sea
and the entrepôts at the western end of the Silk Road. By 1439, they were a serious threat
and the Eastern Roman Emperor Manuel II Paleologus was near panic. Pope Eugenius
IV was not too anxious to help out the rival branch of the Church but he agreed to a
meeting at Ferrara to talk about reuniting the Church in return for help saving
Constantinople. The meeting started at Ferrara with 700 delegates from the Eastern
Empire alone (note the black man in the forefront of the painting) in 1438 and moved to
Florence in 1439.
Toscanelli did not want the Turks in Constantinople because it would wreck the family
business but he also recognized that there needed to be a fall back plan. The Portuguese
had been interested in better navigation for some time. The Florentine interest in map
making started around 1400 when a group of Florentine businessmen had visited
Constantinople and come home with a copy of Ptolemy’s Geographica. The book caused
a sensation because, unlike the existing portolans that were divided by lines of prevailing
winds and currents, the Geographica was divided into grids—remember Ptolemy’s
latitudes determined by the length of the longest day of the year. Points on the Earth—
even in places one had never been—could be measured in distance from one another.
Portuguese interest in accurate maps dated to their voyages down the African coast.
There had been a scientific expedition to the Canaries in 1416, Madeira had been
colonized in 1420, the Azores discovered and the settled between 1427 and 1432, and
Cape Bojador rounded in 1434 into the ―Sea of Darkness.‖ Prince Henry the Navigator’s
brother Dom Pedro had visited Florence in 1425 partially to collect a debt but also to look
into the city’s map making technology. Portuguese interest heightened further with their
discovery of gold in what came to be called the Gold Coast in 1441—not to mention the
new slave trade they were just building.
Grids exactly like Alberti’s for painting could be used to represent the whole Earth with
places related to one another in exact proportion. Grid maps allowed navigation out of
sight of known landmarks. Portolans only kept you oriented to places you had already
Portuguese navigators were also getting in trouble as they went farther south because
they were losing the North Star which vanished below the horizon when one crosses the
Equator. The usual way to navigate was to sail due north or south until Polaris was a
given altitude above the horizon and then turn straight east or west to where you wanted
to go. It all worked as long as you had the pole star to determine latitude. ―Sailing the
line.‖ Grid line made it possible to sail the lines using the height of the sun at noon.
In a June 25, 1474 letter from Toscanelli to his friend Fernan Martins de Roriz, the Canon
of Lisbon and the Confessor to the King of Portugal, ―I am pleased to hear the King is
interested in a shorter route than the African one now being attempted. . . I enclose a
chart showing all the islands from Ireland to India and South to Guinea . . . the straight
lines across the map show the distance East-West . . . the others show the distance North-
South . . . if you go West from Lisbon . . .you get to the fine and noble city of Quinsay
(Cathay, China) . . and . . . . to Chipango . . . most fertile in gold, pearls, and precious
stones, and they cover the temples and the royal residences with solid gold . . .‖
He actually sent another copy of the chart in 1483 with an Italian captain sailing to
Lisbon who could get no interest from the King or his advisors. The captain then tried the
Spanish whose new monarchs finally agreed to help him. He pasted the map to the
flyleaf of his atlas when he sailed in 1492.