Eudoxus of Cnidus

Eudoxus of Cnidus was a fourth-century BCE Greek mathematician and astronomer. His theory of proportions, method of exhaustion, and concentric-sphere model shaped Greek mathematics and astronomy.

Theory of proportions: A general account of ratios that compares magnitudes without requiring them to share a common unit. Eudoxus’s treatment of ratios underlies Book V of Euclid’s Elements.
Concentric spheres: A geometric model of nested spheres sharing a center, used in ancient astronomy to represent celestial motion. Eudoxus modeled each planet’s motions through combinations of nested rotating spheres.
Plato: An Athenian philosopher whose dialogues and Academy influenced Greek philosophy and mathematics. Ancient accounts connect Eudoxus’s astronomical work with Plato’s interest in uniform celestial motion.
Aristotle: A Greek philosopher whose writings addressed logic, natural science, ethics, and politics. Aristotle described a system of celestial spheres associated with Eudoxus and modified by Callippus.
Cavalieri's principle: A geometric principle equating areas or volumes by comparing corresponding cross-sections. It offers a later route to results that ancient geometers established through exhaustion.
Method of exhaustion: A technique for finding areas and volumes by successively approximating them with inscribed or circumscribed figures. Eudoxus used this method to prove geometric results involving curved figures.
Hippopede: A figure-eight curve generated by the intersection of a sphere with another sphere rotating about a tilted axis. The combined rotations of Eudoxus’s spheres could trace this path for planetary motion.
Academy: The philosophical school founded by Plato in Athens in the fourth century BCE. Eudoxus was associated with the Academy during the period of his mathematical and astronomical work.
Callippus: A fourth-century BCE Greek astronomer who revised the number of spheres in Eudoxus’s planetary system. His added spheres refined Eudoxus’s model of planetary motions.
Integral calculus: A branch of mathematics concerned with accumulation, areas, volumes, and antiderivatives. The exhaustion method anticipates its area-and-volume problems without using limits or symbolic integration.

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