This article presents some quotes carefully extracted
from the book ‘The science of Leonardo: inside the mind of the great genius of
the Renaissance’, Fritjof Capra, Anchor Books, 1st Edition, December
2008.
1) …in the collection of his notes on painting, known
as Trattato della pittura (Treatise on
Painting), he writes: The science of painting extends to all the colors of
the surface of bodies, and to shapes of the bodies enclosed by those surfaces…
[Painting] with philosophic and subtle speculation considers all the qualities
of forms… Truly this is science, the legitimate daughter of nature, because
painting is born of nature.
2) Nature as a whole was alive for Leonardo. He saw the
patterns and processes in the microcosm as being similar to those in the
macrocosm. He frequently drew analogies between human anatomy and the structure
of the Earth, as in the following beautiful passage from Codex Leiscester: We
may say that the Earth has vital force of growth, and that its flesh is the
soil; its bones are the successive strata of the rocks which form the
mountains; its cartilage is the porous rock, its blood the veins of the waters.
The lake of blood that lies around the heart is the ocean. Its breathing is the
increase and decrease of the blood in the pulses, just as the in the Earth it
is the ebb and flow of the sea.
3) As a painter, Leonardo felt that he should use
gestures to portray the frames of mind and emotions that provoked them. He
asserted that, in the painting of a human figure, the most important task was
to “express in gesture the passion of its soul.” Indeed, to portray the body’s
expression of the human spirit was the artist’s highest aspiration, in Leonardo’s
view. And it was one in which he himself excelled, as the paintings of his
mature period attest. As art historian Irma Richter explains in the
introductory comments to her classical selections from the Notebooks, for
Leonardo, “the human body was an outward and visible expression of the soul; it
was shaped by its spirit.” We shall see that this vision of soul and spirit,
unmarred by the mind-body split that René Descartes would introduce in the
seventeenth century, is perfectly consistent with the conception of the
“embodied mind” in today’s cognitive science.
4) Leonardo did not pursue science and engineering to
dominate nature, as Francis Bacon would advocate a century later. He had a deep
respect for life, a special compassion for animals, and great awe and reverence
for nature’s complexity and abundance. While a brilliant inventor and designer
himself, he always thought that nature’s ingenuity was vastly superior to human
design. He felt that he would be wise to respect nature and learn from her. It
is an attitude that has reemerged today in the practice of ecological design. Leonardo’s
synthesis of art and science is infused with a deep awareness of ecology and
systems thinking. It is not surprising that he spoke with great disdain of the so-called
“abbreviators”, the reductionists of his time: The abbreviators of works do
injury to knowledge and to love… Of what value is he who, in order to
abbreviate the parts of those things of which he professes to give complete
knowledge, leaves out the greater part of the things of which the whole is
composed?... Oh human stupidity!... You don’t see that you are falling into the
same error as one who strips a tree of its adornment of branches full of
leaves, intermingled with fragrant flowers and fruit, in order to demonstrate
that the tree is good for making planks.
5) Leonardo’s physical beauty in his youth and middle
aged years must have been exceptional, as it is mentioned by all his
contemporary commentators, even though this was not customary at the time. An
anonymous writer called the Anonimo Gaddiano exclaimed, “He was so unusual and
many-sided that nature seemed to have produced a miracle in him, not only in
the beauty of his person, but in many gifts with which she endowed him and
which he fully mastered.” Others marveled at the unique combination of physical
strength and grace seemed to embody. Many authors, including Vasari, referred
to him with the ultimate epithet – il
divino.
6) Throughout his life, Leonardo displayed an air of
serene self-confidence, which helped him to overcome professional setbacks and
disappointments with equanimity and allowed him to calmly pursue his research
even during times of great political turbulence. He was aware of his unique
genius and skill, yet he never boasted about them. Nowhere in his Notebooks
does he vaunt the originality of his inventions and discoveries, nor does he
flaunt the superiority of his ideas, even as he explains how they differ from
traditional beliefs. This lack of arrogance and ego was remarkable indeed. Another
quality that distinguished him was his passion for life and for all living
things.
7) The artist’s fascination with the grotesque forms
also led him to devise the most extravagant, and often quite macabre, practical
jokes, which delighted the courtiers in Milan and Rome. At the papal court in
Rome, Vasari tells us that Leonardo obtained a large lizard to which he
attached “with a mixture of quicksilver some wings, made from the scales stripped
from other lizards, which quivered as it walked along. Then, after he had given
it eyes, horns, and a beard he tamed the creature, and keeping it in a box he
used to show it to his friends and frightened the life out of them.”
8) During Leonardo’s time, the term “genius” did not
have our modern meaning of a person endowed with extraordinary intellectual and
creative powers. The latin word genius
originated in Roman religion, where it donate the spirit of the gens, the family. It was understood as a
guardian spirit, first associated with individuals and then also with people
and places. The extraordinary achievements of artists and scientists were
attributed to their genius, or attendant spirit. This meaning of genius was
prevalent throughout the Middle Ages and the Renaissance. In the eighteen
century, the meaning of the word changed to its familiar modern meaning to
denote these individuals themselves, as in the phrase “Newton is a genius”.
9) The first is an intense curiosity and great
enthusiasm for discovery and understanding. This was indeed an outstanding
quality of Leonardo, whom Kenneth Clark called “the most relentlessly curious
man in history.” Another striking sign of genius is an extraordinary capacity
of intense concentration over long periods of time. Isaac Newton apparently was
able to hold a mathematical problem in his mind for weeks until it surrendered
to his mental powers. When asked how he made his remarkable discoveries, Newton
is reported to have replied, “I keep the subject constantly before me and wait
until the first dawnings open little by little into the full light.” Leonardo
seems to have worked in a very similar way, and most of the time not only on
one but on several problems simultaneously.
10) Indeed the Italian humanists were so bold as to
compare artistic creations to the creations of God. This comparison was first
applied to the creativity of poets, and was then extended, especially by
Leonardo, to the painter’s creative power: If the painter wants to see beauties
that make him fall in love, he is the lord who can generate them, and if he
wants to see monstrous things that frighten, or funny things that make him
laugh, or things that truly arouse compassion, he is their lord and God… In
fact, whatever there is in the universe, by essence, presence, or imagination,
he has it first in his mind and than in his hands.
11) He wished to achieve relief through the scientific
use of the light and shade. According to Leonardo, such an achievement is “the
soul of painting”. Leonardo’s technique of using light and shade to give his
figures “great vigor and relief,” as Vasari put it, culminated in his
celebrated creation of sfumato, the
subtle melting of shades that eventually became the unifying principle of the
paintings. “Leonardo’s sfumato was
the power behind the poetry of his paintings,” Arasse claimed, “and the mystery
that seems to emanate from them.”
12) Leonardo could have not developed his mastery of
chiaroscuro, nor his characteristic sfumato style, without a major advance in
Renaissance paint – the use of oil-based paints. Oil painting makes it possible
to put layers of paint on top of each other without blurring the colors
(provided the layers are allowed to dry individually), to go back over work
again, and to mix paints at ease, all of which were essential for Leonardo to
achieve his special effects of relief and sfumato.
13) Over the years, Leonardo achieved a sublime
mastery in applying the finest layers of paint to create the luminous color
tones that give his paintings their special magic. As Serge Bramly describes
it, “The light passes through his paintings as if through stained glass,
straight on to the printed surface beneath, which reflects it back, thus
creating the impression that it emanates from the figures themselves.
14) On the other hand, Leonardo’s completed
masterpieces always involved radical innovations at several levels – artistic,
philosophical, and scientific. For example, the Virgin of the Rocks was not only revolutionary in its rendering of
light and dark. It also represented a complex and controversial meditation on
the destiny of Christ, expressed through the gestures and relative positions of
the four protagonists, as well as in the intricate symbolism of the surrounding
rocks and vegetation.
15) In a similar vein, Vasari refers to Leonardo as
“Florentine painter and sculptor” in the title of his biography. And yet, we
have no known sculpture from Leonardo’s hand. His reputation rests on a single
piece of work: a monumental bronze horse that was never cast, but which
occupied Leonardo intensely for over ten years.
16) Good designers have the ability to think
systematically and to synthesize. They excel at visualizing things, at
organizing known elements into new configurations, at creating new
relationships; and they are skillful in conveying these mental processes in the
form of drawings almost as rapidly as they occur. Leonardo, off course,
possessed all these abilities to a very high degree. In addition, he had an
uncanny knack of perceiving and solving technical problems – another key
characteristic of a good designer – so much so, in fact, that it was almost
second nature to him.
17) What made Leonardo unique as a designer and
engineer, however, was that many of the novel designs he presented in his
Notebooks involved technological advances that would not be realized until
several centuries latter. And second, he was the only man among the famous
Renaissance engineers who made the transition from engineering to science. Like
painting, engineering became a “mental discourse” for him. To know how something worked was not enough for
Leonardo, he also needed to know why.
Thus an inevitable process was set in motion, which led him from technology and
engineering to pure science. As art historian Kenneth Clark notes, we can see
the process at work in Leonardo’s manuscripts: First, there are questions about
the construction of certain machines, then… questions about the first
principles of dynamics; finally questions which had never been asked before
about winds, clouds, the age of the earth, generation, the human heart. Mere
curiosity has become profound scientific research, independent of the technical
interests which had preceded it.
18) In other words, the problems Leonardo addresses
are theoretical problems of architectural design. The questions he asks are the
same questions he explores throughout his science of organic forms – questions
about patterns, spatial organization, rhythm, and flow. The notes accompanying
his drawings (written in his customary mirror writing, and hence intended for
himself) can be seen as fragments of a treatise on architecture that Leonardo,
according to Heydenreich, may have intended to compose.
19) In view of Leonardo’s central focus on
understanding nature’s forms, both in the macro- and the microcosm, it is not
surprising that he emphasized similarities between architectural structures and
structures in nature, especially in human anatomy. In fact, this linking of
architecture and anatomy goes back to antiquity and was common among
Renaissance architects, who recognized the analogy between a good architect and
a good doctor. As Leonardo explained, “Doctors, teachers, and those who nurse
the sick should understand what man is, what is life, what is health, and in
what manner a parity and concordance of the elements maintains it… The same is
also needed for the ailing cathedral, that is, a doctor-architect who
understands well what buildings is and from what rules the correct way of
building derives.”
20) Leonardo’s science, by contrast, cannot be reduced
to a single foundation, as we have seen. Its strength does not derive from a
single trunk, but from the complex interconnectedness of the branches of many
trees. For Leonardo, recognizing the numerous patterns of relationships in
nature was the hallmark of a universal science. Today, we, too, sense a greater
need for such universal, or systemic, knowledge, which is one of the reasons
why Leonardo’s unified vision of the world is so relevant to our time.
21) Leonardo showed greatly artistic talent early in
his youth; his synthesis of art and science was also foreshadowed early on.
This is vividly illustrated in a story related by Vasari. When Piero da Vinci
was asked by a peasant to have a “buckler” (a small wooden shield) decorated
with a painting in Florence, he did not give the shield to a Florentine artist
but instead asked his son to paint something on it. Leonardo decided to paint a
terrifying monster. “To do what he wanted,” writes Vasari, “Leonardo carried
into a room of his own, which no one else entered except himself, a number of
lizards, crickets, serpents, butterflies, locusts, bats, and various strange
creatures of this nature. From all these he took and assembled different parts
to create a fearsome and horrible monster… He depicted the creature emerging
from a dark cleft of a rock, belching forth venom from its open throat, fire
from its yes, and smoke from its nostrils in so macabre a fashion that the
effect was altogether monstrous and horrible. Leonardo took so long over the
work that the stench of dead animals in his room became unbearable, although he
himself failed to notice because of his great love of painting.” When Ser Piero
came to see the finished painting, Leonardo went back into the room, put the
buckler on an easel in the light, and shaded the window. Then he asked Piero to
come and see it. When his eyes fell on it, Piero was completely taken by
surprise and gave a sudden start, not realizing that he was looking at the
buckler and that the form he saw was, in fact, painted on it. As he backed
away, Leonardo stopped him and said: ‘This work certainly serves its purpose.
It has produced the right reaction, so now you can take it away.’”
22) Other inventions he created from that time
involved fire and a hot air. In addition to the self-regulating spit mentioned
earlier, Leonardo invented a method of creating a vacuum to raise water by
means of a fire burning in a closed bucket, based on the observation that a
burning flame consumes air. During these early years he also developed his
first versions of a diving apparatus. During a visit to Vinci he designed an
olive press with more efficient leverage than the presses used at the time.
While he was engaged in these multiple projects of invention, design, and
engineering, Leonardo also painted his Annunciation,
two Madonnas, and the portrait of Ginevra de’ Benci.
23) He drew [a] long series of diagrams showing the
effect of light falling on spheres and cylinders, crossing, reflecting,
intersecting with endlessly variety… The calculations are so complex and abstruse
that we feel in them, almost for the first time, Leonardo’s tendency to pursue
research for its own sake, rather than as an aid to his art.
24) He was asked by Ludovico to paint a portrait of
the Moor’s mistress, the young and lovely Cecilia Gallerani. Leonardo painted
her holding an ermine, a symbol of purity and moderation which, because of its
Greek name, gale, was also a veiled
allusion to her name, Gallerani. Lady
with an Ermine, as it is called today, was a highly original portrait in
which Leonardo invented a new pose, with the model looking over her shoulder
with an air of surprise and subdued delight, caused, perhaps, by the unexpected
arrival of her lover. Her gesture is graceful and elegant, and is echoed in the
animal’s twisting movement.
25) For Leonardo himself, the 1940s were a period of
intense creative activity. With two major projects – the equestrian statue and The Last Supper – his artistic career
was at its peak, he was consulted repeatedly as an expert on architectural
design, and he embarked on extensive and systematic research in mathematics,
optics, mechanics and the theory of human fly.
26) Leonardo’s research in statics and dynamics was
concerned not only with the workings of machines but also, and even more
important, with understanding the human body and its movements. For example, he
investigated the body’s ability to generate various amount of forces in several
positions. One of the key aims was to find out how a human pilot might generate
enough force to lift a flying machine off the ground by flapping its mechanical
wings. In his studies of machines during that period, Leonardo began to
separate individual mechanisms – levers, gears, bearings, couplings, etc. –
from the machines in which they were embedded. This conceptual separation did
not arise again in engineering until the eighteenth century. In fact, Leonardo
planned (and may even have written) a treatise on Elements of Machines, perhaps influenced by his discussions with
Fazio Cardano of Euclid’s celebrated Elements
of Geometry in Pavia.
27) Leonardo’s Last
Supper, generally considered the first painting of the High Renaissance
(the period of art between, approximately 1495 and 1520), is dramatically
different from earlier representations of the subject. Indeed, it became famous
throughout Europe immediately after his completion and was copied innumerable
times. The firstly highly imaginative feature one notices is the way Leonardo
integrated the fresco into the architecture of the refectory. Demonstrating his
mastery of geometry, Leonardo contrived a series of visual paradoxes to create
an elaborate illusion – a complex perspective that made the room of the Last Supper look like a refectory
itself, in which the monks ate their meals. One consequence of this complex
perspective is that from every viewing position in the room, the spectator is
drawn into the drama of the picture’s narrative with equal force. And dramatic
it is. Whereas traditionally the Last
Supper was pictured at the moment of communion, a moment of calm,
individual meditation for each apostle, Leonardo chose the ominous moment when
Jesus says, “One of you will betray me.” The words of Christ have stirred up
the solemn company, creating powerful waves of emotion. However, the effect is
far from chaotic. The apostles are clearly organized into four groups of three
figures, with Judas forming one of the groups together with Peter and John.
This is another striking compositional innovation. Traditionally, Judas was
pictured sitting on the other side of the table, facing the apostles, with his
back to the spectator. Leonardo had no need to identify the traitor by
isolating him in this way. By given the apostles carefully expressive gestures,
which together cover a wide range of emotions, the artist made sure that we
immediately recognize Judas, as he shrinks back into the dark of John’s shadow,
nervously clutching his bag of silver. The depiction of the apostles as
embodiments of individual emotional states and the integration of Judas into
the dramatic narrative were so revolutionary that after Leonardo, no
self-respecting artist could go back to the previous static configuration.
28) Soon after they began their study sessions,
Leonardo and Fra Luca decided to collaborate on a book, titled De divina proportione, to be written by
Pacioli and illustrated by Leonardo. The book, presented to Ludovico as a
lavish manuscript and eventually published in Venice, contains an extensive
review of the role of proportion in architecture and anatomy – and in
particular of the golden section, or “divine proportion” – as well as detailed
discussions of the five regular polyhedra known as the Platonic solids. It
features over sixty illustrations by Leonardo, including superb drawings of the
Platonic solids in both solid and skeletal forms, testimony to his exceptional
ability to visualize abstract geometric forms. What further distinguishes this
work is that it is the only collection of drawings by Leonardo published during
his lifetime.
29) In the Madonna
and Child with Saint Anne, as the paint is called today, Leonardo had again
broken new ground with both his composition and the theological interpretation
of a traditional religious theme. Rather than presenting Mary and her mother,
Saint Anne, in static configuration – seated next to each other with Jesus in
Mary’s arms between them, or with Saint Anne seated higher in a majestic,
hierarchical composition – Leonardo upset tradition by adding a lamb as a fourth
figure. Jesus, having slipped to the ground, reaches for the lamb as Mary tries
to restrain him, and Saint Anne seems to hold her back. The theological message
embodied in Leonardo’s highly original composition can be seen as a
continuation of his long meditation on the destiny of Christ, which he had
begun with the Virgin of the Rocks. Mary,
in an anxious gesture, attempts to pull her soon away from the lamb, the symbol
of Passion, while Saint Anne, representing Mother Church, knows that Mary’s
gesture is futile – the Passion is Christ’s destiny and cannot be avoid.
30) When he had built flight machines in Milan and
tested them in his workshop in Corte Vecchia, Leonardo’s main concern had been
to find out how human pilot could flap mechanical wings with enough force and
velocity to compress the air underneath and be lifted up. For these tests he
had designed various types of wings modeled after those of birds, bats, and
flying fish. Now, ten years later, he embarked on careful and methodical
observations of the flight of birds. He spent hours in the hills surrounding
Florence, near Fiesole, observing the behavior of birds in flight, and filled
several Notebooks with drawings and comments that analyzed the birds’ turning
maneuvers, their ability to maintain their equilibrium in the wind, and the
detailed mechanisms of active flight. His aim was to design a flying machine
that would be able, like a bird, to maneuver with agility, keep its balance in
the wind, and move its wings with enough force to allow it to fly.
31) In his Anatomical Studies, Leonardo gives a vivid
description of the dreadful conditions under which he had to work. As there
were no chemicals to preserve the cadavers, they would begin to decompose
before he had time to examine and draw them properly. To avoid accusations of
heresy, he worked at night, lighting his dissection room by candles, which must
have made the experience even more macabre. “You will perhaps be impeded by the
fear of living through the night hours in the company of these corpses,
quartered and flayed and frightening to behold.”
32) One will see darkly gloomy air beaten by the rush
of different and convoluting winds, which are mingled with the weight of
continuous rain, and which are carrying helter-skelter an infinite number of
branches torn from the trees, entangled with countless autumn leaves. The
ancient trees will be seen uprooted and thorn to pieces by the fury of the
winds… Oh how many will you see closing their ears with their hands to shut out
the tremendous noises made in the darkened air by the raging of the winds…
Others, with gestures of hopelessness, took their own lives, despairing of
being able to endure such suffering; and of these, some flung themselves from
high rocks, others strangled themselves from high rocks, others strangled
themselves with their own hands.
33) The drawings that illustrate his apocalyptic
narrative are dark, violent, menacing, and disturbing. Nonetheless, they are
astonishingly accurate in their renderings of water and air turbulence.
Throughout his life, Leonardo had carefully studied the forms of waves, eddies,
waterfalls, vortices, and air currents. Here, in old age, he summed up his
knowledge of turbulence. Beyond their expressive emotional power, the deluged
drawings can be seen as sophisticated mathematical diagrams, presenting a
visual catalog of turbulent flows that would not look out of place in a modern
textbook on fluid dynamics.
34) In Leonardo’s mind, his science of living forms
was certainly an integrated whole. At the end of his life, his problems were no
longer conceptual; they were simply the limitations of time and energy. As he
wrote several years before his death, “I have been impeded neither by avarice nor
by negligence, but only by time.” And yet, Leonardo never gave up. In June 1518
he wrote what may have been the last entry in his Notebooks: “I shall go on.”
35) Nor was he perturbed by contemplating his
approaching death. “Just well-spent day brings a happy sleep,” he had written
thirty years earlier, “so a well-employed life brings a happy death.”
36) A few days after completing his will, on May 2,
1519, Leonardo da Vinci died in the manor of Cloux – according to legend, in
the arms of the king of France.
37) To appreciate Leonardo’s science, it is important
to understand the cultural and intellectual context in which he created it.
Scientific ideas do not occur in a vacuum. They are always shaped by the
technologies available at the time. The entire constellation of concepts,
values, perceptions, and practices – the “scientific paradigm” in the terminology
of science historian Thomas Kuhn – provides the context that is necessary for
scientists to pose the great questions, organize their subjects, and define
legitimate problems and solutions. All science is built upon such an
intellectual and cultural foundation. Hence, when we recognize ancient or
medieval ideas reflected in Leonardo’s scientific writings, this do not mean
that he was less of a scientist, Leonardo consulted the traditional texts and
used their conceptual framework as his starting point. He then tested the
traditional ideas against his own scientific observations. And, in accordance
with scientific method, he did not hesitate to modify the old theories when his
experiments contradicted them.
38) The leading figure in the movement to weave the
philosophy of Aristotle into Christian teachings was Saint Thomas Aquinas, one
of the towering intellects of the Middle Ages. Aquinas taught that there could
be no conflict between faith and reason, but the two books on which they were
based – the Bible and the “book of nature” – were both authored by God. Aquinas
produced a vast body of precise, detailed, and systematic philosophical
writings in which he integrated Aristotle’s encyclopedic works and medieval
Christian theology into a magnificent whole. The dark side of this seamless
fusion of science and theology was that any contradiction by future scientists
would necessarily have to be seen as heresy. In this way, Thomas Aquinas
enshrined in his writings the potential for conflicts between science and
religion – which indeed arose three centuries later in Leonardo’s anatomical
research, reached a dramatic climax with the trial of Galileo, and have
continued to the present day.
39) A few years later, at the height of his anatomical
work in Milan, Leonardo added a technical note about the reproduction of his
drawings to his famous assertion of the superiority of drawing over writing. He
insisted that his anatomical drawings should be printed from copper plates,
which would be more expensive than woodcuts but much more effective in
rendering the fine details of his work. “I beg you who come after me”, he wrote
on the sheet that contains his magnificent drawings of the vertebral column,
“not let avarice constrain you to make the prints in [wood].”
40) The conception of the Renaissance worldview was
the conceptions of the universe that had been developed in classical Greek
science: that the world was a kosmos,
an ordered and harmonious structure. From its beginnings in the sixth century
B.C., Greek philosophy and the science understood the order of the cosmos to be
that of a living organism rather than a mechanical system. This meant that all
its parts had an innate purpose to contribute to the harmonious functioning of
the whole, and that objects moved naturally toward their proper places in the
universe. Such an explanation of natural phenomena in terms of their goals, or
purposes, is known as teleology, from the Greek telos (purpose). It permeated virtually all of Greek philosophy and
science. The view of the cosmos as an organism also implied for the Greeks that
its general properties are reflected in each of its parts. This analogy between
macrocosm and microcosm, in particular between the Earth and the human body,
was articulated most eloquently by Plato in his Timaeus in the fourth century B.C., but it can also be found in the
teachings of the Pythagoreans in other earlier schools. Over time, this idea
acquired the authority of common knowledge, which continued throughout the
Middle Ages and into Renaissance. In early Greek philosophy, the ultimate
moving force and source of all life was indentified with the soul, and its
metaphor was that of the breath of life. Indeed, the root meaning of both the
Greek psyche and the Latin anima is “breath”. Closely associated
with that moving force – the breath of life that leaves the body at death – was
the idea of knowing. For the early Greek philosophers, the soul was both the
source of movement and life, and that which perceives and knows. Because of the
fundamental analogy between micro- and macrocosm, the individual soul was
thought to be part of the force that moves the entire universe, and accordingly
the knowing of an individual was seen as part of a universal process of
knowing. Plato called it the anima mundi,
the “world soul”.
41) The culmination of the early phase of Greek mathematics
was reached around 300 B.C. with Euclid, who presented all of the geometry and
other mathematics known in his days in a systematic, orderly sequence in his
celebrated Elements. The thirteen
volumes of this classical textbook were not only widely read during the
Renaissance, but remained the foundation for the teaching of geometry until the
end of the nineteenth century.
42) Health, according to the Hippocratic writings,
requires a state of balance among environmental influences, the way in which we
live, and the various components of human nature. One of the most important
volumes in the Hippocratic Corpus, the book on Airs, Waters and Places, represents what we might now call a
treatise on human ecology. It shows in greater detail how the well-being of
individuals is influenced by environmental factors – the quality of air, water,
and food, the topography of the land, and general living habits. During the
last two decades of the fifteenth century, this and several other volumes from the
Hippocratic Corpus were available to scholars in Latin, most of them derived
from Arabic translations.
43) Leonardo da Vinci shared with his fellow humanists
their great confidence in the capabilities of the human individual, their
passion for voyages and exploration, and their excitement about the rediscovery
of the classical texts of antiquity. But he differed dramatically from most of
them by refusing to blindly accept the teachings of the classical authorities.
He studied them carefully, but then he tested them by subjecting them to
rigorous comparisons with his own experiments and his direct observations of
nature. In doing so, I would argue, Leonardo single-handedly developed a new
approach to knowledge, known today as scientific method.
44) All scientific models and theories are limited and
approximate. This realization has become crucial to the contemporary
understanding of science. Twentieth-century science has shown repeatedly that
all natural phenomena are ultimately interconnected, and that their essential
properties, in fact, derive from their relationships to other things. Hence, in
order to explain any one of them completely, we have to understand all the
others, which is obviously impossible. This insight has forced us to abandon
the Cartesian belief in the certainty of scientific knowledge and to realize
that science, to put into bluntly, we never deal with truth, in the sense of a
precise correspondence between our descriptions and the described phenomena. We
always deal with limited and approximate knowledge. This may sound frustrating,
but for many scientists the fact that we can formulate approximate models and
theories to describe an endless web of interconnected phenomena, and that we
are able to systematically improve our models and approximations over time, is
a source of confidence and strength. As the great biochemist Louis Pasteur put
it, “Science advances through tentative answers to a series of more and more
subtle questions which reach deeper and deeper into the essence of all natural
phenomena.”
45) “All our knowledge has its origins in the senses,”
he noted in his first Notebook, the Codex Trivulzianos. “Wisdom is the daughter
of experience,” we read in the Codex Forster, and in his Treatise on Painting, Leonardo asserted: “To me it seems that those
sciences are vain and full of errors that are not born of experience, mother of
all certainty… that is to say, which do not at their beginning middle, or end
pass through any of the five senses.” Such an approach to the study of nature
was unheard-of in Leonardo’s day, and would fully emerge again only in the
seventeenth century, the era of the Scientific Revolution.
46) He recognized that learning from skilled masters
was important in the arts, but he also observed that such masters were rare. “The
surer way,” he suggested, “is to go to the objects of nature, rather than those
that are imitated with great deterioration, and so acquire sad habits; for he
who can go to the well does not go to the water jar.”
47) He was deeply aware of the fundamental
interconnectedness of all phenomena and of the interdependence and mutual
generation of all parts of an organic whole, which Immanuel Kant in the
eighteenth century would define as “self-organization.” In the Codex
Atlanticus, Leonardo eloquently summarized his profound understanding of life’s
basic processes by paraphrasing a statement by the Ionian philosopher
Anaxagoras: “Everything comes from everything, and everything is made of
everything, and everything turns into everything, because that which exists in
the elements is made up of the elements.”
48) Only in the twentieth century did the limits of
Newtonian science become fully apparent, and the mechanistic Cartesian
worldview begin to give way to a holistic and ecological view not unlike that
developed by Leonardo da Vinci. With the rise of systemic thinking and its
emphasis on networks, complexity, and patterns of organization, we can know
more fully appreciate the power of Leonardo’s science and its relevance for our
modern era. Leonardo’s science is a science of qualities, of shapes and
proportions, rather than absolute quantities. He preferred to depict the forms of nature in his
drawings rather than describe their
shapes, and he analyzed them in terms of their proportions rather than measured
quantities. Proportion was seen by Renaissance artists as the essence of
harmony and beauty. Leonardo filled many pages of his Notebooks with elaborate
diagrams of proportions between the various parts of the human figure, and he
drew corresponding diagrams to analyze the body of the horse.
49) Leonardo was always impressed by the great
diversity and variety of living forms. “Nature is so delightful and abundant in
its variations,” he wrote in a passage about how to paint trees, “that among
trees of the same kind there would not be found one plant that resembles
another nearby, and this is not only of the plant as a whole, but among the
branches, the leaves, and the fruit, not one will be found that looks precisely
like another.”
50) Leonardo was fascinated by water in all its
manifestations. He recognized its fundamental role as life’s medium and vital
fluid, as the matrix of all organic forms. “It is the expansion and the humor
of all living bodies,” he wrote. Without it nothing retains its original form.”
Throughout his life, he strove to understand the mysterious processes
underlying the creation of nature’s forms by studying the movements of water
through earth and air.
51) At the center of Leonardo’s investigations of
turbulence lies the water vortex, or whirlpool. Throughout the Notebooks, there
are countless drawings of eddies and whirlpools of all sizes and types – in the
currents of rivers and lakes, behind piers and jetties, in the basin of
waterfalls, and behind objects of various shapes immersed in flowing water.
These often very beautiful drawings are testimony to Leonardo’s endless
fascination with the ever-changing and yet stable nature of this fundamental
type of turbulence. I believe that this fascination came from a deep intuition
that the dynamics of vortices, combining scalability and change, embody an
essential characteristic of the living forms.
52) To investigate the mechanics of muscles, tendons,
and bones, Leonardo immersed himself in a long study of the “science of
weights,” known today as statics, which is concerned with the analysis of loads
and forces on physical systems in static equilibrium, such as balances, levers,
and pulleys. In the Renaissance this knowledge was very important for
architects and engineers, as it is today, and the medieval science of weights
comprised a large collection of works compiled in the late thirteenth and
fourteenth centuries.
53) Leonardo applied his knowledge of mechanics not
only to his investigations of the movements of the human body, but also to his
studies of machines. Indeed, the uniqueness of his genius lay in his synthesis
of art, science, and design. In his lifetime, he was famous as an artist, and
also as a brilliant mechanical engineer who invented and designed countless
machines and mechanical devices, often involving innovations that were
centuries ahead of his time.
54) Based on these designs, British engineers recently
built a glider and tested it successfully in a flight from the chalk cliffs in
southeast England know as the Sussex Downs. This maiden flight of “Leonardo’s
glider,” reportedly, exceeded the first attempts by the Wright brothers in
1900. Although the machines with movable mechanical wings were not destined to
fly, the models built from Leonardo’s designs are extraordinary testimonies to
his genius as a scientist and engineer. In the words of art historian Martin Kemp:
“Using mechanical systems, the wings flap with much of the sinuous and menacing
grace of a gigantic bird prey… [Leonardo’s] designs retain their conceptual
power as archetypal expressions of man’s desire to emulate the birds, and
remain capable of inspiring a sense of wonder even in a modern audience, for
whom the sight of tons of metal flying through the air has become a matter of
routine.”
55) Leonardo’s careful and patient studies of the
movements of the heart and the flow of blood, undertaken in old age, are the
culmination of his anatomical work. He not only understood and pictured the
heart like no one before him, but also observed subtleties in its actions and
in the flow of blood that would elude medical researchers for centuries.
56) Leonardo’s success in cardiac anatomy [is] so
great that there are aspects of the work which are not yet equaled by modern
anatomical illustration… His consistent practice of illustration of the heart
and its valves, both in systole and in diastole, with a comparison of the
position of the parts, has rarely if ever been performed in any anatomical
textbook.
57) Leonardo’s embryological drawings are graceful and
touching revelations of the mysteries surrounding the origins of human life. In
the words of physician Sherwin Nuland, “[His] depiction of a five-month fetus
in the womb is a thing of beauty… It stands as a masterwork of art, and,
considering the very little that was at the time understood of embryology, a
masterwork of science perception as well.” Leonardo knew very well that,
ultimately, the nature and origin of life would remain a mystery, no matter how
brilliant his scientific mind was. “Nature is full of infinite causes that have
never occurred in experience,” he declared in his late forties, and as he got
older his sense of mystery deepened. Nearly all the figures in his last
paintings have that smile that expresses the ineffable, often combined with a
pointing finger. “Mystery to Leonardo”, wrote Kenneth Clark, “was a shadow, a
smile and a finger pointing into darkness.”
58) Leonardo’s approach to mathematics was that of a
scientist, not a mathematician. He wanted to use mathematical language to
provide consistency and rigor to the descriptions of his scientific
observations. However, in his time there was no mathematical language
appropriate to express the kind of science he was pursuing – explorations of
the forms of nature in their movements and transformations. And so Leonardo
used his powers of visualization and his great intuition to experiment with new
techniques that foreshadowed branches of mathematics that would not be
developed until centuries later. These include the theory of functions and
fields of integral calculus and topology.
59) The really important mathematics for him was
geometry, which is evident from his praise of the eye as “the prince of
mathematics.”
60) Like most mathematicians of his time, Leonardo
frequently used geometrical figures to represent algebraic relationships. A
simple but very ingenious example is his pervasive use of triangles and
pyramids to illustrate arithmetic progressions and, more generally, what we now
call linear functions. He was familiar with the use of pyramids to represent
linear proportions from his studies of perspective, where he observed that “All
the things transmit to the eye their image by means of a pyramid of lines. By
‘pyramid of lines’ I mean those lines which, starting from the edges of the
surface of each object, converge from a distance and meet in a single point…
placed in the eye.”
61) Leonardo realized very early on that the
mathematics of his time was inappropriate for recording the most important
results of his scientific research – the description of nature’s living forms
in their ceaseless movements and transmutations. Instead of mathematics, he
frequently used his exceptional drawing facility to graphically document his
observations in pictures that are often strikingly beautiful while, at the same
time, they take the place of mathematical diagrams. His celebrated drawing of
“Water falling upon water”, for example, is not a realistic snapshot of a jet
of water falling into a pond, but an elaborate diagram of Leonardo’s analysis
of several types of turbulence caused by the impact of the jet.
62) Arasse makes an interesting point: Whenever
Leonardo rendered objects in their sharp outlines, these pictures represented
conceptual models rather than realistic images. And whenever he produced
realistic images of objects, he blurred the outlines with his famous sfumato
technique, in order to represent them as they actually appear to the human eye.
63) What Leonardo found especially attractive in
geometry was its ability to deal with continuous variables. “The mathematical
sciences… are only two,” he wrote in the Codex Madrid, “of which the first is
arithmetic, the second is geometry. One encompasses the discontinuous
quantities [i.e., variables] the other the continuous.” It was evident to Leonardo
that a mathematic of continuous quantities would be needed to describe the
incessant movements and transformations in nature. In the seventeenth century,
mathematicians developed the theory of functions and the differential calculus
for that very purpose. Instead of these sophisticated mathematical tools,
Leonardo had only geometry at his disposal, but he expanded it and experimented
with new interpretations and new forms of geometry that foreshadowed subsequent
developments.
64) In the course of his explorations of circles and
squares, Leonardo tried his hand at the problem of squaring the circle, which
had fascinated mathematicians since antiquity. In its classical form, the
challenge is to construct a square with an area equal of that of a given
circle, and to do so by using only ruler and compass. We know today that this
is not possible, but countless professional and amateur mathematicians have
tried. Leonardo worked on the problem repeatedly over a period of more than a
dozen years. In one particular attempt, he worked by candlelight through the
night, and by dawn he believed that he had finally found the solution. “On the
night of St. Andrew,” he excitedly recorded in his Notebook, “I found the end
of squaring the circle; and at the end of the light of the candle, of the
night, and of the paper on which I was writing, it was completed; at the end of
the hour.” However, as the day progressed, he came to the realization that this
attempt, too, was futile.
65) When we look at Leonardo’s geometry from the point
of view of present-day mathematics, and in particular from the perspective of
complexity theory, we can see that he developed the beginnings of the branch of
mathematics now known as topology. Like Leonardo’s geometry, topology is a
geometry of continuous transformations, or mappings, in which certain
properties of geometric figures are preserved. For example, a sphere can be
transformed into a cube or a cylinder, all of which have similar continuous
surfaces. A doughnut (torus), by contrast, is topologically different because
of the hole in its center. The torus can be transformed, for example, into a
coffee cup where the hole now appears in the handle. In the words of historian
of mathematics Morris Kline: Topology is concerned with those properties of
geometric figures that remain invariant when the figures are bent, stretched,
shrunk, or deformed in any way that does not create new points or fuse existing
points. The transformations presupposes, in other words, that there is a
one-to-one correspondence between the points of the original figure and the
points of the transformed figure, and that transformation carries nearby points
into nearby points. This latter property is called continuity.
66) During the last twelve years of his life, Leonardo
spent a great deal of time mapping and exploring the transformations of his
“geometry done with motion.” Several times he wrote of his intention to present
the results of these studies in one or more treatises. During the years he
spent in Rome, and while he was summing up his knowledge of complex turbulent
flows in his famous deluge drawings, Leonardo produced a magnificent compendium
of topological transformations, titled De
ludo geometrico (On the Game of Geometry), on large double folio in the
Codex Atlanticus. He drew 176 diagrams displaying a bewildering variety of
geometric forms, built from intersecting circles, triangles and squares – row
after row of crescents, rosettes and other floral patterns, paired leaves,
pinwheels, and curvilinear stars. Previous this endless interplay of geometric
motifs was often interpreted as the playful doodling of an aging artist – “a
mere intellectual pastime,” in the words of Kenneth Clark. Such assessments
were made because art historians were generally not aware of the mathematical
significance of Leonardo’s geometry of transformations. Close examination of
the double folio shows that its geometric forms, regardless of how complex and
fanciful, are all based upon strict topological principles.
67) Since Leonardo’s science was a science of
qualities, of organic forms and their movements and transformations, the
mathematical “necessity” he saw in nature was not one expressed in quantities
and numerical relationships, but one of geometric shapes continually
transforming themselves according to rigorous laws and principles.
“Mathematical” for Leonardo referred above all to the logic, rigor, and
coherence according to which nature has shaped, and is continually reshaping,
her organic forms. This meaning of “mathematical” is quite different from the
one understood by scientists during the Scientific Revolution and the
subsequent three hundred years. However, it is not unlike the understanding of
some of the leading mathematicians today. The recent development of complexity
theory has generated a new mathematical language in which the dynamics of
complex systems – including the turbulent flows and growth patterns of plants
studied by Leonardo – are no longer represented by algebraic relationships, but
instead by geometric shapes, like the computer-generated strange attractors or
fractals, which are analyzed in terms of topological concepts. This new
mathematics, naturally, is far more abstract and sophisticated than anything Leonardo
could have imagined in the fifteenth and sixteenth centuries. But is used in
the same spirit in which he developed his “geometry done with motion” – to show
with mathematical rigor how complex natural phenomena are shaped and
transformed by the “necessity” of physical forces. The mathematics of
complexity has led to a new appreciation of geometry and to the broad
realization that mathematics is much more than formulas and equations. Like
Leonardo da Vinci five hundred years ago, modern mathematicians today are
showing us that understanding of patterns, relationships, and transformations
is crucial to understand the living world around us, and that all questions of
pattern, order, and coherence are ultimately mathematical.
68) From perspective, he proceeded in two opposite
directions – outward and inward, as it were. He explored the geometry of light
rays, the interplay of light and shadow, and the very nature of light, and he
also studied the anatomy of the eye, the physiology of vision, and the pathways
of sensory impressions along the nerves to the “seat of the soul”. To a modern
intellectual, used to exasperating fragmentation of academic disciplines, it is
amazing to see how Leonardo moved swiftly from perspective and the effects of light
and shade to the nature of light, the pathways of the optic nerves, and the
actions of the soul. Unencumbered by the mind-body split that Descartes would
introduce 150 years later, Leonardo did not separate epistemology (the theory
of knowledge) from ontology (the theory of what exists in the world), nor
indeed philosophy from science and art. His wide-ranging examinations of the
entire process of perception led him to formulate highly original ideas about
the relationship between physical reality and cognitive processes – the
“actions of the soul”, in his language – which have reemerged only very
recently with the development of a post-Cartesian science of cognition.
69) As architectural historian James Ackerman points
out, the geometry of perspective developed by the Florentine artists was the
first scientific conception of three-dimensional space: As a method of
constructing an abstract space in which any body can be related mathematically
to any other body, the perspective of the artists was a preamble to modern
physics and astronomy. Perhaps, the influence was indirect and unconsciously
transmitted, but the fact remains that artists were the first to conceive a
generalized mathematical model of space and that it constituted an essential step
in the evolution from medieval symbolism to the modern image of the universe.
70) Leonardo demonstrated his throughout understanding
of linear perspective not only in his art, but also in his scientific drawings.
While he was conducting his experiments on the geometry of perspective, he also
investigated the anatomical connections between the eye and the brain. He
documented his findings in a series of magnificent pictures of the human skull,
in which the foreshortening of visual perspective is employed with great
effect. Leonardo combined this technique with delicate renderings of light and
shade to create a vivid sense of space within the skull, in which he exhibited
anatomical structures that had never been seen before and located them with
complete accuracy in three dimensions.
71) From the earliest years in Verrocchio’s workshop,
Leonardo was familiar with the grinding of lenses and the use of concave
mirrors to focus sunlight for welding. Throughout his life he tried to improve
the design of these burning mirrors, and when he became seriously interested in
the theory of optics, he undertook careful studies of their geometries. He was
fascinated by the intricate intersections of the reflected rays, which he
explored in a series of precise and beautiful diagrams, tracing their pathways
from parallel beams of light through their reflections to the focal point (or
points). He showed that in spherical mirrors, the rays are focused in an area
along the central axis, whereas parabolic mirrors are true “mirrors of fire”,
focusing all the rays in a single point. He also made several attempts to solve
Alhazen’s problem, and late in his life, while experimenting with parabolic
mirrors in Rome, found an ingenious solution by employing an instrument with
hinged rods.
72) According to Leonardo, shadow is the central
element in the science of painting. It allows the painter to effectively
represent solid bodies in relief, emerging from the backgrounds of the painted
surface. His poetic definition of shadow in Codex Atlanticus is clearly written
from the artist’s point of view: Every opaque body is surrounded, and its whole
surface is enveloped, in shadow and light…. Besides this, shadows have in
themselves various degrees of darkness, because they are caused by the absence
of a variable amount of the luminous rays…. They clothe the bodies to which
they are applied.
73) As Kenneth Clark has remarked, “The calculations
are so complex and abstruse that we feel in them, almost for the first time,
Leonardo’s tendency to pursue research for its own sake, rather than as an aid
to his art.”
74) He marveled at the swift velocity of light: “Look
at the light of the candle and consider its beauty,” he wrote. “Blink your eye
and look at it again. What you see of it was not there before, and what was
there before is not anymore.” But he also realized that, however fast light
moves, its velocity is not infinite. He asserted that the speed of sound is
greater than that of elastic waves in earth, and that light moves faster than
sound, but that the mind moves even faster than light. “The mind jumps in an
instant from East to the West,” he noted, “and all the other immaterial things
have velocities that are by a long way inferior.”
75) The structure of the eye and the process of vision
were natural wonders for Leonardo that never ceased to amaze him. “What
language can express this marvel?” he writes about the eyeball, before continue
with rare expression of religious awe: “Certainly none. This is where human
discourse turns directly to the contemplation of the divine.” In the Treatise on Painting, Leonardo waxes
enthusiastic about the human eye: Don’t you see that the eye embraces the
beauty of the whole world? It is the master of astronomy, it practices
cosmography, it counsels and corrects all human arts; its transports man to
different parts of the world. [The eye] is the prince of mathematics; its
sciences are most certain. It has measured the heights and sizes of the stars;
it has discovered the elements and their locations…. It has created
architecture, perspective, and divine painting…. [The eye] is the window of the
human body, through which [the soul] contemplates and enjoys the beauty of the
world.
76) “The pupil of the eye,” he concluded, “changes to
as many different sizes as there are differences in the degrees of brightness
and darkness of the objects which present themselves before it…. Nature has
equipped the visual faculty, when irritated by excessive light, with the
contraction of the pupil… and here nature works like someone who, having too
much light in his house, closes half of a window, and more or less according to
necessity.” And then he added: “You can observe that in nocturnal animals such
as cats, screech owls, tawny owls and others, which have the pupil small at
midday and very large at night.”
77) During the last two decades of the twentieth
century, however, a novel conception of the nature of mind and consciousness
emerged in the life sciences, which finally overcame the Cartesian division
between mind and body. The decisive advance has been to reject the view of mind
as a thing; to realize that mind and consciousness are not entities but
processes. In the past twenty-five years the study of mind from this new
perspective has blossomed into a rich interdisciplinary field known as cognitive
science, which transcends the traditional frameworks of biology, psychology,
and epistemology. One of the central insights of cognitive science is the
identification of cognition, the process of knowing, with the process of life. Accordingly,
the interactions of a living organism – plant, animal, or human – with its
environment are understood as cognitive interactions. Thus life and cognition
become inseparably connected. Mind, – or, more accurately, mental activity – is
immanent in matter at all levels of life. This new conception represents a
radical expansion of the concept of cognition and, implicitly, the concept of
mind. In the new view, cognition involves the entire process of life –
including perception, emotion, and behavior – and does not even necessarily
require a brain and a nervous system.
78) At the end of life, the reverse process takes
place: “While I thought I was learning how to live, I had been learning how to
die,” Leonardo wrote movingly late in his life.
79) Nature, as a whole was alive and animated for
Leonardo, a world in continual flux and development, in the macrocosm of the Earth
as in the microcosm of the human body.
80) Leonardo himself never boasted about his unique
talents and skills, and in his thousands of pages of manuscripts he never
vaunted the originality of so many of his ideas and discoveries. But he was
well aware of his exceptional stature. In the Codex Madrid, in the midst of extensive
discussions of the laws of mechanics, we find two lines that can stand as his
own definitive epitaph: Read me, O
reader, if in my words you find delight, for rarely in the world will one such
as I be born again.
Synthesis Ratio :: 274 total of pages read / 23
typewritten pages :: 11.9
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