Fritjof Capra (21 page)

Several centuries earlier the Babylonians had developed a different approach to solving mathematical problems, now known as algebra, which began with simple arithmetic operations and then evolved into more abstract formulations with numbers represented by letters. The Greeks learned these numerical and algebraic methods together with Babylonian astronomy, but they transformed them into their geometrical language and continued to see mathematical problems in terms of geometry. Plato’s Academy, the principal Greek school of natural philosophy for nine centuries, is said to have had a sign above its entrance, “Let no one enter here who does not know geometry.”

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 day 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. About one hundred years after Euclid, Greek mathematics reached its final climax with Archimedes, a brilliant mathematician who wrote many important treatises in what we would now call mathematical physics. But he was never as popular as Euclid. His mathematical work was so advanced that it was not understood until many centuries later, and his great fame as an inventor eclipsed his reputation as a mathematician.

With the rise of Islam during the seventh and subsequent centuries, the Arab world became the center of mathematical studies. Arab mathematicians translated and synthesized the Greek texts and also commented on important influences from Mesopotamia and India. Of particular importance was the work of Muhammad al-Khwarzimi in the ninth century, whose
Kitab al jabr
was the most influential work on algebra from this period. The Arabic
al jabr
(binding together) in its title is the root of our modern word “algebra.”
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Two centuries later, Persia produced an outstanding algebraist in the poet Omar Khayyam, the world-renowned author of the
Rubaiyat
, who was famous in his time for classifying cubic equations and solving many of them. Another Islamic scholar of that period who was very influential in the Renaissance was the Arab mathematician Alhazen (Ibn al-Haitham), who wrote a brilliant treatise on the “science of perspective,” which included detailed discussions of geometrical optics and of the geometrical principles of vision and the eye’s anatomy.

In the Renaissance, thus, mathematicians had access to two different approaches for solving mathematical problems, geometry and algebra. However, until the seventeenth century, geometry was considered to be more fundamental. All algebraic reasoning was justified in terms of geometrical figures in the tradition of Greek mathematics. In the seventeenth century, this dependence of algebra on geometry was reversed by René Descartes, the founder of modern philosophy and a brilliant mathematician, who invented a method for associating algebraic equations with curves and surfaces.
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This method, now known as analytic geometry, involves using Cartesian coordinates, the system invented by Descartes and named after him. Long before Descartes, however, the fields of geometry and algebra were related because both of them were necessary for the development of an accurate science of astronomy.

For astronomy was surely the principal physical science throughout antiquity. The Babylonians successfully applied their numerical methods to compile astronomical tables. The Greeks used their geometrical approach to construct elaborate cosmological models, involving the use of trigonometry—which the Greek astronomers had learned from Hindu mathematicians—to determine the distances between celestial bodies from their observed angular positions.

When the conquests of Alexander the Great made the observations and mathematical methods of the Babylonian astronomers available to the Greeks, they found it impossible to reconcile this improved data with their Platonic idea of circular planetary orbits. Several Greek astronomers therefore abandoned the Platonic-Aristotelian view and began to devise complex geocentric systems of cycles and epicycles to account for the movements of the sun, moon, and planets. The culmination of this development was reached in the second century
A.D.
with the Ptolemaic system, which predicted the motion of the planets with considerable accuracy.

Ptolemy’s thirteen-volume treatise,
He mathematike syntaxis (The Mathematical Collection)
summarized much of this ancient astronomical knowledge. It remained the authoritative text on astronomy for fourteen centuries. (It is indicative of the prestige of Islamic science that the text was known throughout the Middle Ages and the Renaissance under its Arabic title,
Almagest
.) Ptolemy also published the
Geography
, which contained detailed discussions of cartographic techniques and an elaborate map of the known world. The book was printed in the fifteenth century under the title
Cosmography
and became the most popular geographical book printed from movable type during the Renaissance.

NATURAL HISTORY

Throughout antiquity and in the centuries that followed, the study of the living world was known as natural history, and those who pursued it were known as naturalists. This was often an amateur activity rather than a professional occupation. It was only in the nineteenth century that the term “biology” began to be widely used, and even then, biologists often continued to be called “naturalists.”

In the fifteenth century, books about natural history still tended to display some fascination with the fabulous, often imaginary beasts that had populated medieval bestiaries. At the time of Leonardo, the rediscovery of classical natural history texts, together with the explorations of new floras and faunas in the Americas, began to stimulate more serious interest in the study of living things. The ideas of the ancient natural philosophers about plants and animals were represented in great detail in the encyclopedic works of Aristotle, Theophrastus, Pliny the Elder, and Dioscorides.
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Aristotle was the classical author most widely available to Renaissance scholars. His numerous works included several treatises on animals, including the
Historia animalium (History of Animals)
and
De anima (Of the Soul).
While Aristotle’s observations of plants were less accurate than his observations of animals, his disciple and successor Theophrastus was a keen botanical observer. His treatise
De historia plantarum (Of the History of Plants)
was a pioneering work that made Theophrastus famous as the “father of botany.”

In the first century
A.D.
, the Roman naturalist Pliny the Elder (Gaius Plinius) wrote a monumental encyclopedia titled
Natural History
, comprising 37 books in which almost 500 Greek and Roman authors are cited. It became the favorite scientific encyclopedia in the Middle Ages, not only because of its rich content but also because it was written in an informal style. While it lacked scientific rigor, it was much easier and more pleasant to read than the learned volumes of Aristotle and the other Greek philosophers. For most Renaissance humanists, Pliny’s name meant natural history itself. And his encyclopedia was the most convenient entry point to further research.

Botany, from ancient times up to the end of the sixteenth century, was often considered a subdiscipline of medicine, since plants were mainly studied for their use in the healing arts. For centuries the authoritative text in this field was the
Materia Medica
by the Greek physician Dioscorides, who was a contemporary of Pliny.

MEDICINE AND ANATOMY

In prehistoric cultures around the world, the origin of illness and the process of healing were associated with forces belonging to the spirit world, and a great variety of healing rituals and practices were developed to deal with illness accordingly.
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In Western medicine, a revolutionary change occurred in Greece in the fifth century
B.C.
, with the emergence of the scientific medical tradition associated with Hippocrates. There is no doubt that a famous physician by that name practiced and taught medicine around 400
B.C.
on the island of Cos, but the voluminous writings attributed to him, known as the Hippocratic Corpus, were probably written by several authors at different times.

At the core of Hippocratic medicine was the conviction that illnesses are not caused by supernatural forces, but are natural phenomena that can be studied scientifically and influenced by therapeutic procedures and wise management of one’s life.
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Thus medicine should be practiced as a scientific discipline and should include the prevention of illness, as well as its diagnosis and treatment. This attitude has formed the basis of scientific medicine to the present day.

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 great 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.
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The culmination of anatomical knowledge in antiquity was reached in the second century
A.D.
with Galen (Claudius Galenus), a Greek physician who resided chiefly in Rome, where he had a large practice. His work in anatomy and physiology, based partly on dissections of animals, greatly increased the ancient knowledge of the arteries, brain, nerves, and spinal cord. Galen wrote over one hundred treatises in which he summarized and systematized the medical knowledge of his time in accordance with his own theories. By the end of the ninth century, all his works had been translated into Arabic, and Latin translations followed in due course. The authority of the Galenic teachings was unchallenged until Leonardo’s time, although they were not founded on detailed knowledge of human organs. His dogmatic doctrines actually impeded medical progress. Nor was Galen successful in correlating his medical theories with corresponding therapies.

The medical bible throughout the Middle Ages and the Renaissance was the
Canon of Medicine
, written by the physician and philosopher Avicenna (Ibn Sina) in the eleventh century. A vast encyclopedia that codified the complete Greek and Arabic medical knowledge, Avicenna’s
Canon
was more elaborate than Galen’s works and had the advantage of being a single monumental opus rather than a collection dispersed in many separate treatises.

Medical teaching at the great universities was based on the classical texts of Hippocrates, Galen, and Avicenna, and concentrated on interpreting the classics, without questioning them or comparing them with clinical experience. Practicing physicians, on the other hand, many of them without medical degrees, used their own eclectic combinations of therapies.
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The best of them simply relied on the Hippocratic notions of clean living and the ability of the body to heal itself.

As medical theory and practice increasingly diverged, human anatomy gradually became an independent field of study. Leonardo da Vinci, who became the greatest Renaissance anatomist, never practiced medicine. In fact, Leonardo had a very low opinion of doctors. “Strive to preserve your health,” he wrote on a sheet of anatomical drawings, “in which you will be the more successful the more you are wary of physicians.”
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One of the earliest texts on anatomy was the
Anatomia
by Mondino de’ Luzzi, a professor at Bologna in the fourteenth century. He was one of the few medieval teachers who actually performed anatomical dissections himself.
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His text, much influenced by the Arab interpreters of Galen, gave rudimentary instructions for dissections without, however, specifying the exact position and nature of individual organs. Yet, because of its succinctness and utility, Mondino’s
Anatomia
was a standard textbook in medical schools in the fourteenth and fifteenth centuries.

LEONARDO AND THE CLASSICS

During the years of his extensive self-education in Milan,
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Leonardo familiarized himself with the principal classical texts. He not only accumulated a considerable personal library, but also consulted classical manuscripts in the private libraries of wealthy aristocrats and monasteries whenever he had an opportunity, or borrowed them from other scholars. His Notebooks are full of reminders to himself to borrow or consult certain books. Since he had only the most rudimentary knowledge of Latin, he studied Italian translations whenever he could obtain them, or sought out scholars who could help him with the Latin texts.

We know from Leonardo’s own accounts that he knew Plato’s
Timaeus
well. He also owned several of Aristotle’s works, in particular the
Physics
. His knowledge of the mathematical writings of Plato, Pythagoras, Archimedes, and Euclid was derived mostly from Luca Pacioli’s famous Renaissance textbook, which was written in Italian. When Leonardo and Pacioli became friends, Pacioli helped Leonardo deepen his understanding of mathematics, particularly geometry, by guiding him through the complete Latin edition of Euclid’s
Elements
.
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Leonardo’s interest in astronomy was largely confined to studying optical effects in the visual perception of the heavenly bodies. But he was well aware of the Ptolemaic model of planetary motions. He owned several books on astronomy and cartography, including Ptolemy’s celebrated
Cosmography
and a work by the Arabian astronomer Albumazar (Abu-Mashar).
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With regard to natural history, Leonardo, like most Renaissance humanists, was well acquainted with the works of Aristotle, Pliny the Elder, and Dioscorides. He studied an Italian edition of Pliny’s encyclopedic
Natural History
, printed in Venice in 1476, and read Dioscorides’ popular
Materia Medica
. His own work in botany, however, went far beyond those classical texts.
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