## 09 Jan Of all books from Western civilization, only the Bible has received more intense scrutiny than Euclid’s Elements. (WILLIAM DUNHAM) | Part A’

The Academy produced many capable mathematicians and one indisputably great one, Eudoxus of Cnidos.

A century and a half passed between Hippocrates and Euclid. During this span, Greek civilization grew and matured, enriched by the writings of Plato and Aristotle, of Aristophanes and Thucydides, even as it underwent the turmoil of the Peloponnesian Wars and the glory of the Greek empire under Alexander the Great.

By 300 B.C., Greek culture had spread across the Mediterranean world and beyond.

In the West, Greece reigned supreme.

The period from 440 B.C. to 300 B.C. saw a number of individuals contribute significantly to the development of mathematics.

Among these were Plato (427-347 B.C.) and Eudoxus (ca. 408-355 B.C.), although only the latter was truly a mathematician.

Plato, the great philosopher of Athens, deserves mention here not so much for the mathematics he created as for the enthusiasm and status he imparted to the subject. As a youth, Plato had studied in Athens under Socrates and is of course our primary source of information about his esteemed teacher. For a number of years Plato roamed the world, meeting the great thinkers and formulating his own philosophical positions. In 387 B.C., he returned to his native Athens and founded the Academy. Devoted to learning and contemplation, the Academy attracted talented scholars from near and far, and under Plato’s guidance it became the intellectual center of the classical world.

Of the many subjects studied at the Academy, none was more highly regarded than mathematics. The subject certainly appealed to Plato’s sense of beauty and order and represented an abstract, ideal world unsullied by the humdrum demands of day-to-day existence. Moreover, Plato considered mathematics to be the perfect training for the mind, its logical rigor demanding the ultimate in concentration, cleverness, and care. Legend has it that across the arched entryway to his prestigious Academy were the words ‘ ‘Let no man ignorant of geometry enter here.” Explicit sexism notwithstanding, this motto reflected the view that only those who had first demonstrated a mathematical maturity were capable of facing the intellectual challenge of the Academy. We might say that Plato regarded geometry as the ideal entrance requirement, the Scholastic Aptitude Test of his day.

Although very little Original mathematics is now attributed to Plato, the Academy produced many capable mathematicians and one indisputably great one, Eudoxus of Cnidos. Eudoxus came to Athens about the time the Academy was being created and attended the lectures of Plato himself. Eudoxus’ poverty forced him to live in Piraeus, on the outskirts of Athens, and make the daily round-trip journey to and from the Academy, thus distinguishing him as one of the first commuters (although we are unsure whether he had to pay out -of-city-state tuition) . Later in his career, he traveled to Egypt and returned to his native Cnidos, all the while assimilating the discoveries of science and constantly extending its frontiers.

Particularly interested in astronomy, Eudoxus devised complex explanations of lunar and planetary motion whose influence was felt until the Copernican revolution in the sixteenth century.

Never willing to accept divine or mystical explanations for natural phenomena, he instead tried to subject them to observation and rational analysis.

In mathematics, Eudoxus is remembered for two major contributions.

One was his theory of proportion, and the other his method of exhaustion.

The former provided a logical victory over the impasse created by the Pythagoreans’ discovery of incommensurable magnitudes. This impasse was especially apparent in geometric theorems about similar triangles, theorems that had initially been proved under the assumption that any two magnitudes were commensurable. When this assumption was destroyed, so too were the existing proofs of some of geometry’s foremost theorems. What resulted is sometimes called the “logical scandal” of Greek geometry. That is, while people still believed that the theorems were correct as stated, they no longer were in possession of sound proofs with which to support this belief. It was Eudoxus who developed a valid theory of proportions and thereby supplied the long-sought proofs. His theory, which must have brought a collective sigh of relief from the Greek mathematical world, is now most readily found in Book V of Euclid’s Elements.

Eudoxus’ other great contribution, the method of exhaustion, found immediate application in the determination of areas and volumes of the more sophisticated geometric figures. The general strategy was to approach an irregular figure by means of a succession of known elementary ones, each providing a better approximation than its predecessor.

**Journey Through Genius: THE GREAT THEOREMS OF MATHEMATICS **

**William Dunham**

Image Α: https://en.wikipedia.org/wiki/Euclid%27s_Elements

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