Eudoxus of Cnidus (408-355 BCE) was a mathematician, astronomer, and philosopher. (If you want to be thorough, he was also a theologian, meteorologist, Doctor, geographer, and did some government work).

He was born in Cnidus, on the Resadiye peninsula in what is now Turkey. His father was Aischines, but his mother's name does not seem to have been recorded. He traveled to Athens to study, where he had a couple of famous teachers, Archytas (mathematics) and Plato (Philosophy and astronomy). Then then traveled the world for a bit, stopping in Cyzicus for a while to set up a School, which was quite well know at the time. He also wrote a number of works on various subjects, but unfortunately all have since been lost.

After he finished his studies in Athens, Eudoxus traveled to Egypt where he studied astronomy with the priests in the city of Heliopolis. After returning to Cnidus (by circuitous rout) he built an observatory. From here he observed the movements of the constellations, on which he wrote two books Mirror and Phaenomena. He also developed a theory of planetary movement, based on the Pythagorean theory of the spheres. Eudoxus' homocentric sphere system is quite complex, and a complete description should be left to that node. To put it simply, Eudoxus proposed that there were a number of perfect spheres rotating about the earth (upon which the sun and stars resided), and that the axis of each passed through the earth. But to explain some of the confusing movements of celestial bodies, he added the idea that the axes were not fixed but were rotating ('wobbling', so to speak. Confused? Visualize the axis as a strait line passing through the earth. The ends of this axis are describing small figure eights, even as the celestial sphere continues to rotate around the axis. Hope that helped).

While the homocentric sphere system is Eudoxus' best known idea, overall he is probably better know as a mathematician than an astronomer. Unfortunately I am no good at math, and can't explain his ideas in any great detail. I'll give you a brief overview here, and leave it to someone else to fill in the actual mathematics.

The Axiom of Eudoxus became the basis of Book V of Euclid's Elements. It states "Magnitudes are said to have a ratio to one another which is capable, when a multiple of either may exceed the other." This solves the problem of how one might compare lines of rational and irrational lengths. Eudoxus also gave conditions under which two ratios could be considered equal:

"Magnitudes are said to be of the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and the third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or are alike less than the latter equimultiples taken in corresponding order."
Together these allowed mathematicians to start working with irrational numbers.

The method of exhaustion put forth by Eudoxus also appears in Euclid's Elements (Book X, Proposition 1). It's over my head, but it is well noded here.

Eudoxus came up with proofs for the formulas for the volume of a pyramid and cone. He also worked on the classic puzzle of the duplication of the cube; it's possible that he used the kampyle curve while searching for a solution to this problem, and because of this it is know as the Kampyle of Eudoxus.