Knidos, approx. 406 - 355 B.C.

Outstanding among the mathematicians and astronomers of Plato's time, Eudoxus the Knidius (Knidos, Asia Minor, about 400 BC - Knidos, about 350 BC), significantly advanced the theory of numbers beyond the Pythagorean tradition, proving the existence of asymmetric quantities and devising various techniques for measuring curved surfaces. Moreover, with the system of concentric spheres he invented, he gave the first systematic explanation of the movements of the Sun, Moon and planets, emphasizing once again the insistence of the ancient Greeks on spherical perfection. He also introduced geometry into the science of astronomy and was the first to emphasize the necessary interaction between observation and theory that has since characterized the development of astronomy.

Eudoxus, son of Aeschines, studied mathematics and medicine at a school whose reputation rivaled that of Hippocrates of Kos. A wealthy doctor, impressed by his abilities, paid for his travel to Athens to study at Plato's Academy. He lived sixteen months in Egypt during the reign of Nectanebo I (380 - 363 BC). In Heliopolis, a present-day suburb of Cairo, he was initiated into the wisdom of the priesthood, which included astronomy. There he also wrote the Octaeterida, his first major work, which referred to a calendar based on an eight-year cycle, perhaps after studying the planet Venus. He then traveled to the region of the Propontis sea professing to be a teacher and then returned to Athens, where he gained great fame throughout Greece as a legislator. His few biographical details are known to us mainly from the writings of Diogenes of Laertius in the 3rd AD. century.

Eudoxus' two main contributions to mathematics are the theory of ratios, found in book V, and the method of exhaustion in book XII. The philosopher Proclus attributes the theory of proportions to Eudoxus and Archimedes attributes the method of exhaustion to him. It is also possible that Euclid's axiomatic method was originally developed by Eudoxus.

Although the geometric method of calculating the distance of the Earth from the Sun and the distance of the Earth from the Moon is usually attributed to Aristarchus of Samos (he flourished in the 3rd century BC), there is a possibility that it was discovered by Eudoxus. Eudoxus provided a solution to another astronomical problem, namely the mathematical explanation of the apparent movements of the Sun, the Moon and the five known planets. The seven heavenly bodies were in the following order of distance from the stationary Earth: Moon, Mercury, Venus, Sun, Mars, Jupiter, and Saturn.

Eudoxus made a model of 27 spheres. On the outer sphere he placed all the fixed stars to take into account their daily motion from East to West. He devised a set of interlocking spheres for each of the remaining heavenly bodies, namely three spheres for the Sun, three for the Moon, and four for each of the remaining five planets. Each planet moved in the middle of its three spheres. Each sphere had the appropriate axial tilt and rotational speed. The outer sphere described the diurnal movement from East to West.