Biography - Eudoxus of Cnidus

"Willingly would I burn to death like Phaeton, were this the price for reaching the sun and learning its shape, its size and its substance." Attributed to Eudoxus by C. B. Boyer in A History of Mathematics, New York 1968. Though this quote may not have been uttered by the famous mathematician, it certainly does describe Eudoxus's lifelong drive for knowledge.

Biography

Board a plane headed to the modern day Republic of Turkey and chances are your touchdown destination will be in its most populous city of Istanbul, where nine million people make their home. A day's drive south will take you from this flurry of human civilization to a place seemingly forgotten by the world. The architectural ruins which inhabit this ancient city are the only signs that more than two millennia past, men and women who called themselves Carians walked the streets of Cnidus, a city remembered in antiquity for giving the world the mathematician Eudoxus.

Born around the year 408 B.C. (a happy medium between the extremes of 410 B.C. and 405 B.C. given by varying sources) to the physician Aeschines of Cnidus, Eudoxus is remembered not only for his contributions to the world of mathematics but for his work in areas including astronomy, geography, medicine, and philosophy. Throughout his life, he constantly strove for knowledge in any area in which he could find a teacher. Eudoxus began his journey to enlightenment in the sciences when he traveled to Tarentum, in modern day Italy, to study under the philosopher and statesman Archytas. From Archytas, Eudoxus first learned mathematics. Archytas was interested in the problem of duplicating the cube, and it is probable that Eudoxus garnered his interest in mathematics from Archytas. He may also have learned number theory and music theory from Archytas. Eudoxus traveled to Sicily, an island off the southern coast of Italy, during his stay in Tarentum; there, he learned medicine from a man variously identified in sources as Philistium, Philiston, and Philistion the Sicilian, about whom apparently little else is known.

In the year circa 387 B.C., at the age of 23, Eudoxus traveled with the physician Theomedon to Athens, where he planned to study with the followers of Socrates. There, he became the pupil of the great philosopher and mathematician Plato. Eudoxus studied with Plato for several months until a disagreement caused a rift to open between student and master. Eudoxus was a hedonist; he believed that the supreme good in life was in pleasure because, according to an account in Aristotle's Nicomachaean Ethics, "all beings sought it and endeavored to escape its contrary, pain" and "it is an end in itself, not a relative good." Plato, however, held that wisdom came before pleasure. Aristotle, who probably knew Eudoxus personally, paints a picture of Eudoxus that is not nearly as bold as the exaggerated hedonist Plato would have had his followers believe:

"His arguments about pleasure carried conviction more on account of the perfection of his character than through their contents. Eudoxus passed indeed for a man of remarkable moderation. Again he did not seem to embrace these arguments as being a friend of pleasure, but because he regarded them as conforming to the truth."

The tale of this falling out from the Plato's point of view is given in that philosopher's Philebus.

During his time at the Academy, the school founded by Plato in Athens, Eudoxus lived in an apartment in the harbor district, known as the Piraeus; he was too poor to afford living quarters anywhere else. As with all ancient knowledge, specific facts and figures are open to debate, but it is known that Eudoxus walked from his apartment in the Piraeus a number of miles, anywhere from two to seven, every day to study at the Academy. It is frequently given that Eudoxus walked this distance because, in his own peculiar way, he found pleasure in the walk; however, the most reasonable inference is that he found pleasure in the knowledge he would gain at the Academy and was willing to walk this distance to gain that knowledge. He lived in the Piraeus and trudged back and forth between his living space and Athens for two months while studying at the Academy.

Due to Eudoxus's poverty, his friends in Cnidus - after the mathematician's falling out with Plato - raised funds to send Eudoxus with Chrysippus the physician to Heliopolis (Greek for town of the sun) in Egypt; he was given letters of introduction recommending him to the priests there. There he studied and gained a knowledge of astronomy as well as a wider viewpoint on mathematics as a whole. Eudoxus lived in Heliopolis for 16 months before he packed his bags and traveled north to Cyzicus, then part of Mysia, in modern day Turkey. While based in Cyzicus, he traveled south to the court of Mausolus, who was satrap or governor - and practical ruler - of a large portion of the Persian empire. During his extensive travels, Eudoxus worked as a Sophist and gathered a body of students of his own; he founded a school in Cyzicus (the name of which was either the School of Eudoxus or has been forgotten, depending upon the source).

Around the year 365 B.C. Eudoxus returned with his students to Athens, and here his story is again debated. Some sources say that he became a colleague of his estranged former teacher, Plato; an equal number insist that Eudoxus founded another school in Athens which Plato became rather envious of for its success. It could even be true that he both created a new school and lectured at the Academy. It seems unlikely that Eudoxus would return a school where he was apparently unwelcome and teach under a man who disagreed so fundamentally with him, so perhaps the founding of a new school is the more reasonable argument.

After some time in Athens Eudoxus, always light of foot, returned to his native Cnidus where a democracy had been established. He served on the city assembly in a prestigious position as a lawmaker in order to compose legislation for Cnidus. He built an observatory and observed the star Canopus, as told by Hipparchus in his commentary on Eudoxus's writings. He continued writing and lecturing on astronomy, meteorology, and theology. He even wrote the tale of his journeys in a seven volume (unfortunately lost) text entitled Circuit of the Sun. He had one son, Aristagoras, and three daughters: Actis, Philitis, and Delphis. Eudoxus perished approximately in the year 355 B.C. (also given as 347 B.C.), and none of his direct written works have survived the years between then and the present.

Works

Some of Eudoxus's work, thankfully, has survived vicariously through a few other sources. The most predominant of these sources is the Elements by Euclid. Books V and XII of the Elements contain much material known or believed to have descended from Eudoxus. Book V of the Elements, a "book of proportions"

- pa > qb iff pc > qd

- pa = qb iff pc = qd

- pa < qb iff pc < qd

This may seem quite simple and logical today. Students are taught multiplication and division in grade school; they understand positive and negative numbers; they are given a wealth of knowledge concerning irrational and continuous numbers derived from two thousand years of work in the field. Indeed, it may seem as though this work on proportions is rather elementary.

Eudoxus, however, did not have this knowledge with which to work - the great mathematician Eudoxus did not have his math spoon-fed to him in grade school. He derived his theory of proportions entirely from scratch. "He could not use multiplication or division to define proportion because it was part of his program to define multiplication and division in terms of proportion."

1. The volume of a pyramid is one third the volume of a prism of the equal base and height

2. The volume of a cone is one third the volume of a cylinder of equal base and height

As mentioned in the beginning of this writing, Eudoxus studied with Archytas in Tarentum, who was interested in the problem of duplicating the cube. Eratosthenes wrote in his history of the cube duplicating problem that Eudoxus solved the problem using curved lines. His proof, lost to antiquity, was reconstructed by the mathematician Paul Tannery centuries later; however, without the original proof, Tannery's guess remains just that. [University of St. Andrews]

At the edge of mathematics in the field of mathematical astronomy, Eudoxus was the developer of an important early model of the visible universe in terms of spheres. He believed that the universe was comprised of 27 different spheres which held all of the stars, the planets, and the sun. The stars in the outer sphere rotated independently of the inner spheres, as did each of the planets and the Sun. Each body required four separate spheres to rotate it independently from the other spheres. While Eudoxus's geometrical description of the universe was, as is known today, inaccurate, it was reasonable enough to lead a number of his contemporaries to declare that it would stand the test of time with only minor changes. This model did not account for changes in the observed diameter of the planets and the moon especially; in later years, astronomers would update Eudoxus's model to allow for eccentricity of the spherical orbits in order to accommodate observed differences between the model and the actual universe.

Conclusion

It is stated in the biography of Eudoxus by Diogenes Laertius that, while the mathematician was in Heliopolis, he was licked by a bull identified as the Egyptian god Apis. This was interpreted to be a sign that death would be quick coming, but that Eudoxus's life would be illustrious beforehand. Diogenes gives the following epigram for Eudoxus:

'Tis said, that while at Memphis wise Eudoxus

Learnt his own fate from th' holy fair-horned bull;

He said indeed no word, bulls do not speak;

Nor had kind nature e'er calf Apis gifted

With an articulately speaking mouth.

But standing on one side he lick'd his cloak,

Showing by this most plainly-in brief time

You shall put off your life. So death came soon,

When he had just seen three and fifty times

The Pleiads rise to warn the mariners.

The great mathematician's life was arguably of reasonable length, but it was definitely an illustrious one. Without the work of Eudoxus, mathematics may have halted for any number of years on the problem of irrational numbers. It is for this reason that Eudoxus is often remembered as the greatest of the ancient mathematicians.

Reference Books

Anglin, W. S. Mathematics: A Concise History and Philosophy. New York, 1994

Huxley, G. L. Dictionary of Scientific Bibliography. New York, 1970-1990 p465-7

Yonge, C. D., translator. The Lives and Opinions of Eminent Philosophers, by Diogenes Laertius. London, 1853

Christianidis, Jean. Classics in the History of Greek Mathematics. New York, 2004 p97-8