MoM Ep3: Eudoxus of Connitus

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This podcast episode of Men of Mathematics explores the life and mathematical contributions of Eudoxus of Connitus, highlighting his solutions to the crisis of incommensurables and his development of the method of exhaustion.

Key Topics:

• The crisis of incommensurables in Greek mathematics
• Eudoxus's theory of proportions
• The method of exhaustion
• Eudoxus's cosmological model
• Eudoxus's influence on future mathematicians, including Archimedes, Newton, and Leibniz

Summary:

Eudoxus of Connitus was a mathematician born around 408 BC who made significant contributions to geometry, astronomy, geography, medicine, and philosophy. He lived in poverty, walking miles to attend Plato's Academy. He eventually became a distinguished colleague of Plato. His work is known through references by Euclid, Aristotle, Archimedes, and later commentators, as none of his writings survive.

Eudoxus addressed the crisis of incommensurables, which arose from the discovery that the diagonal of a unit square (the square root of 2) cannot be expressed as a ratio of whole numbers. This discovery challenged the Pythagorean worldview, which was based on the belief that all ratios could be expressed as ratios of whole numbers. To resolve this, Eudoxus developed a theory of proportions that defined equality of ratios in terms of comparisons, rather than relying on whole numbers. Two ratios, A to B and C to D, are equal if, for all positive integers M and N, when M times A is greater than N times B, then M times C is greater than N times D, and similarly for equality and less than. This definition works for both commensurable and incommensurable quantities. This approach anticipated modern mathematics by over 2,000 years.

Eudoxus also developed the method of exhaustion, a technique for calculating areas and volumes of curved figures by approximating them with polygons. As the number of sides of the polygons increases, they exhaust more and more of the curved area. This method relies on the Archimedean axiom, which Eudoxus is credited with, stating that for any two magnitudes A and B, there exists a positive integer n such that n times A is greater than B. Using this method, Eudoxus proved several geometric theorems, including the relationship between the area of a circle and the square of its diameter, the volume of a cone and a cylinder, the volume of a pyramid and a prism, and the relationship between spheres and the cubes of their diameters.

Beyond pure mathematics, Eudoxus proposed the first mathematical model of the cosmos, which consisted of 27 concentric rotating spheres centered on Earth. This model aimed to explain the observed motions of the sun, moon, and planets, including complex motions like planetary retrograde.

Eudoxus's work established a standard of rigor in Greek mathematics. His method of exhaustion influenced subsequent mathematicians, including Archimedes, who used it to calculate areas and volumes. The ideas behind the method of exhaustion were later formalized and extended by Newton and Leibniz in the development of calculus. The debate about the legitimacy of infinitesimals, which violate Eudoxus's Archimedean axiom, continues to this day. Eudoxus's theory of proportions is preserved in Book V of Euclid's Elements, considered one of the greatest achievements of Greek mathematics.