Eudoxus of Cnidus Boyer, in Essay

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He invented a planetary system, which consisted of spheres, the earth being still at the center, and twenty-seven concentric spheres rotating around the earth.

Actually, most of his accomplishments are difficult to explain at all to the nonprofessional, since they involve the complicated fields of math and astronomy. But, for those who work in those areas, Eudoxus accomplishments are extraordinary. However, what his work does is make the work today so much easier. Those who labor in those fields know the practicalities, complexities, and almost impossibility of what Eudoxus did.

Eudoxus made important contributions to the theory of proportion, where he made a definition allowing possibly irrational lengths to be compared in a similar way to the method of cross multiplying used today. A major difficulty had arisen in mathematics by the time of Eudoxus, namely the fact that certain lengths were not comparable. The theory developed by Eudoxus is set out in Euclid's Elements Book V. Definition 4 in that Book is called the Axiom of Eudoxus. (Encyclopaedia Brittanica, 2008)

G.L. Huxley, writes, in his "Dictionary of Scientific Biography,"

It is difficult to exaggerate the significance of Eudoxus' theory, for it amounts to a rigorous definition of real number. Number theory was allowed to advance again, after the paralysis imposed on it by the Pythagorean discovery of irrationals, to the inestimable benefit of all subsequent mathematics."

Similarly, Eudoxus's theory of incommensurable magnitudes (magnitudes lacking a common measure) and the method of exhaustion (its modern name) influenced Books X and XII of the Elements.

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Eudoxus also contributed a solution to the problem of doubling the cube -- that is, the construction of a cube with twice the volume of a given cube. (Encyclopaedia Brittanica, 2008)

Perhaps Eudoxus's greatest fame stems from his being the first to attempt a geometric model of the motions of the Sun, the Moon, and the five planets known in antiquity.

Eudoxus also wrote an ethnographical work ("Circuit of the Earth") of which fragments survive. It is plausible that Eudoxus also divided the spherical Earth into the familiar six sections (northern and southern tropical, temperate, and arctic zones) according to a division of the celestial sphere. (Encyclopaedia Brittanica, 2008)

Conclusion

He is the most innovative Greek mathematician before Archimedes. His work forms the foundation for the most advanced discussions in Euclid's Elements and set the stage for Archimedes' study of volumes and surfaces. The theory of proportions is the first completely articulated theory of magnitudes. Although most astronomers seem to have abandoned his astronomical views by the middle of the 2nd century B.C., his principle that every celestial motion is uniform and circular about the centre endured until the time of the 17th-century astronomer Johannes Kepler. (Encyclopaedia Brittanica, 2008)

Eudoxus was a genius mathematician and astronomer who substantially advanced proportion theory, contributed to the identification of constellations and thus to the development of observational astronomy in the Greek world, and established the first sophisticated, geometrical model of celestial motion. He also wrote on geography and probably contributed to philosophical discussions in Plato's Academy......

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