Eudoxus of Cnidus: Difference between revisions

(Add figures explaining his planetary model and a reference)
==Mathematics<!--Linked from 'Galileo Galilei'-->==
Eudoxus is considered by some to be the greatest of [[Classical Greece|classical Greek]] mathematicians, and in all [[Ancient Greece|antiquity]] second only to [[Archimedes]].<ref name=calinger>{{Cite book |last=Calinger |first=Ronald |title=Classics of Mathematics |publisher=Moore Publishing Company, Inc. |year=1982 |location=Oak Park, Illinois |page=75 |isbn=0-935610-13-8}}</ref> He rigorously developed [[AntiphonSophistic (person)works of Antiphon|Antiphon]]'s [[method of exhaustion]], a precursor to the [[integral calculus]] which was also used in a masterly way by Archimedes in the following century. In applying the method, Eudoxus proved such mathematical statements as: areas of circles are to one another as the squares of their radii, volumes of spheres are to one another as the cubes of their radii, the volume of a pyramid is one-third the volume of a [[Prism (geometry)|prism]] with the same base and altitude, and the volume of a cone is one-third that of the corresponding cylinder.<ref name="Kline">Morris Kline, ''Mathematical Thought from Ancient to Modern Times'' Oxford University Press, 1972 pp. 48–50</ref>
Eudoxus introduced the idea of non-quantified mathematical [[Magnitude (mathematics)|magnitude]] to describe and work with continuous geometrical entities such as lines, angles, areas and volumes, thereby avoiding the use of [[irrational number]]s. In doing so, he reversed a [[Pythagoreanism|Pythagorean]] emphasis on number and arithmetic, focusing instead on geometrical concepts as the basis of rigorous mathematics. Some Pythagoreans, such as Eudoxus' teacher [[Archytas]], had believed that only arithmetic could provide a basis for proofs. Induced by the need to understand and operate with [[Commensurability (mathematics)|incommensurable]] quantities, Eudoxus established what may have been the first deductive organization of mathematics on the basis of explicit [[axiom]]s. The change in focus by Eudoxus stimulated a divide in mathematics which lasted two thousand years. In combination with a Greek intellectual attitude unconcerned with practical problems, there followed a significant retreat from the development of techniques in arithmetic and algebra.<ref name="Kline" />