Eudoxus of Cnidus: Difference between revisions

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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 [[Sophistic 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's 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" />
 
The Pythagoreans had discovered that the diagonal of a square does not have a common unit of measurement with the sides of the square; this is the famous discovery that the [[square root of 2]] cannot be expressed as the ratio of two integers. This discovery had heralded the existence of incommensurable quantities beyond the integers and rational fractions, but at the same time it threw into question the idea of measurement and calculations in geometry as a whole. For example, Euclid provides an elaborate proof of the Pythagorean theorem (''Elements'' I.47), by using addition of areas and only much later (''Elements'' VI.31) a simpler proof from similar triangles, which relies on ratios of line segments.
 
==Astronomy==
In [[ancient Greece]], astronomy was a branch of mathematics; astronomers sought to create geometrical models that could imitate the appearances of celestial motions. Identifying the astronomical work of Eudoxus as a separate category is therefore a modern convenience. Some of Eudoxus's astronomical texts whose names have survived include:
 
* ''Disappearances of the Sun'', possibly on eclipses
* ''On Speeds'', on planetary motions
 
We are fairly well informed about the contents of ''Phaenomena'', for Eudoxus's prose text was the basis for a poem of the same name by [[Aratus]]. [[Hipparchus]] quoted from the text of Eudoxus in his commentary on Aratus.
 
===Eudoxan planetary models===
The Sun is also assigned three spheres. The second completes its motion in a year instead of a month. The inclusion of a third sphere implies that Eudoxus mistakenly believed that the Sun had motion in latitude.
 
[[File:Animated Hippopede of Eudoxus.gif|thumb|Animation depicting Eudoxus's model of retrograde planetary motion. The two innermost homocentric spheres of his model are represented as rings here, each turning with the same period but in opposite directions, moving the planet along a figure-eight curve, or hippopede.]]
 
[[File:Eudoxus' Homocentric Spheres.png|thumb|Eudoxus's model of planetary motion. Each of his homocentric spheres is represented here as a ring which rotates on the axis shown. The outermost (yellow) sphere rotates once per day; the second (blue) describes the planet's motion through the zodiac; the third (green) and fourth (red) together move the planet along a figure-eight curve (or hippopede) to explain retrograde motion.]]
 
The five visible planets ([[Venus]], [[Mercury (planet)|Mercury]], [[Mars]], [[Jupiter]], and [[Saturn]]) are assigned four spheres each:
 
===Importance of Eudoxan system===
[[Callippus]], a Greek astronomer of the 4th century, added seven spheres to Eudoxus's original 27 (in addition to the planetary spheres, Eudoxus included a sphere for the fixed stars). Aristotle described both systems, but insisted on adding "unrolling" spheres between each set of spheres to cancel the motions of the outer set. Aristotle was concerned about the physical nature of the system; without unrollers, the outer motions would be transferred to the inner planets.
 
A major flaw in the Eudoxan system is its inability to explain changes in the brightness of planets as seen from Earth. Because the spheres are concentric, planets will always remain at the same distance from Earth. This problem was pointed out in Antiquity by [[Autolycus of Pitane]]. Astronomers responded by introducing the [[deferent and epicycle]], which caused a planet to vary its distance. However, Eudoxus's importance to [[Greek astronomy]] is considerable, as he was the first to attempt a mathematical explanation of the planets.
 
==Ethics==
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