## Greek astronomer and mathematician Eudoxus of Cnidus

Initially, he followed his colleagues, the doctors, on their tours. Thus, he traveled to Magna Graecia, where he attended courses by the famous Pythagorean mathematician Archytas of Tarentum, who was also a military leader holding some administrative post there, and by Philistion in Sicily (428-347 BC). At some point, however, a wealthy doctor, Theomedon (385 BC), impressed by his abilities, paid for his expenses to go to Athens for studies at Plato’s Academy, which was founded in 387 BC.

Thus, at the age of twenty-three, he found himself in Piraeus, where fish and oil were very cheap, and so due to lack of money, he resided there. From there, he went to Athens daily to listen to Plato and other Socratic philosophers. Returning to Cnidus, he left with the doctor Chrysippus for Egypt (380 BC), equipped with a letter of introduction from Agesilaus, to Pharaoh Nectanebo. He was introduced to the priesthood of Heliopolis, where he was initiated into the wisdom and science of the Egyptian priests.

According to Diogenes Laertius, he studied mathematics, astronomy, mechanics, music, and medicine there. He stayed there for sixteen months and wrote his first significant work “Octaeteris,” which referred to a calendar based on an eight-year cycle, possibly stemming from the study of the planet Venus. From Egypt, he brought knowledge of astronomy and proposed a reform of the Greek calendar, which met great success and support.

After returning from Egypt, he stayed for a while near Mausolus, the ruler of Halicarnassus. Then, in 378 BC, he returned to Greece and founded the famous School of Cyzicus, which brought him great fame, where he taught philosophy, geometry, arithmetic, grammar, music, rhetoric, and geography. After a few years, he returned to Athens, followed by his students (Menaechemus who solved the problem of angle trisection, Deinostratus who invented the quadrature of the circle, and the Athenian from Cyzicus), transferring the seat of his School there. He gained great fame throughout Greece as a legislator and for some time even competed with Plato.

It seems that at some point, he was persuaded by Plato to dissolve his School and teach astronomy, mathematics, and medicine at the Academy. Finally, he returned to his homeland where he was received with great honors and made laws. His few biographical details are known mainly from the texts of Diogenes Laertius in the 3rd century AD, and from his works, only a few fragments have remained. For the rest of his work, we have information from Archimedes, Eratosthenes, Eudemus, Proclus, Eutocius, and the Byzantine John Philoponus.

Eudoxus of Cnidus was a student of Euclid and became known for the development of an early method of integration, for using proportions in solving problems, and for using formulas to measure three-dimensional shapes. He discovered, that is, the theory of proportions (the philosopher Proclus said that he added three proportions to the already known ones) and the method of exhaustion (Archimedes mentioned his discovery, but with a different name because it was used later) which were his two main contributions to mathematics.

It is generally accepted that Euclid used Eudoxus’s work in the text of his Elements, something that is mentioned by the book’s commentators. The apparent deadlock of incommensurables was largely circumvented, given that products and their ratios could be used through the theory of proportions. The use of ratios instead of fractions had some advantages, as one could formulate rules and use them in some theorems without needing to use pi, which is irrational. Thus, the ratio was the most fundamental relation between magnitudes, and the theory of proportions allowed different ratios to be compared with each other.

It is very likely that Euclid’s axiomatic method was initially developed by Eudoxus. With the method of exhaustion he applied for calculating areas and volumes, Eudoxus showed that it is not necessary to assume the “existence” of infinitely small quantities. But for the purposes of mathematics, one can reach a magnitude as small as desired, through continued division of a given size.

According to Archimedes, Eudoxus used this method to prove that the volumes of pyramids and cones are equal to 1/3 of the volumes of prisms and cylinders, respectively, with the same bases and heights. From this perspective, Eudoxus, along with Archimedes, is considered a founder of integral calculus. He also proved that the areas of two circles are proportional to the squares of their diameters.

He had formulated the definition of equal ratios, which was epoch-making and allowed mathematicians to handle irrational numbers with the same rigor as rationals. This was essentially the starting point of a modern theory of irrationals. Eudoxus seems to have written a book titled “On Cutting” and was probably in scientific contact with Theaetetus, implying the study of incommensurable magnitudes he had engaged in.

He also dealt with the Delian Problem (the problem of doubling the cube, which from antiquity until the 19th century occupied all mathematicians and not only) which was characterized as unsolvable, since the solution using only the ruler and compass was and still is impossible. However, he used certain curves – about which, unfortunately, we do not know much – and managed to solve it.