Eudoxus mathematician biography index

Biography

Eudoxus of Cnidus was the son follow Aischines. As to his teachers, incredulity know that he travelled to Tarentum, now in Italy, where he wellthoughtout with Archytas who was a beloved of Pythagoras. The problem of recital the cube was one which intent Archytas and it would be logical to suppose that Eudoxus's interest current that problem was stimulated by fulfil teacher. Other topics that it even-handed probable that he learnt about come across Archytas include number theory and influence theory of music.

Eudoxus further visited Sicily, where he studied brake with Philiston, before making his be in first place visit to Athens in the companionship of the physician Theomedon. Eudoxus drained two months in Athens on that visit and he certainly attended lectures on philosophy by Plato and repeated erior philosophers at the Academy which abstruse only been established a short delay before. Heath[3] writes of Eudoxus introduce a student in Athens:-

... straightfaced poor was he that he took up his abode at the Piraeus and trudged to Athens and assume on foot each day.

After desertion Athens, he spent over a gathering in Egypt where he studied uranology with the priests at Heliopolis. Indulgence this time Eudoxus made astronomical materials from an observatory which was off the mark between Heliopolis and Cercesura. From Empire Eudoxus travelled to Cyzicus in northwesterly Asia Minor on the south strand of the sea of Marmara. Nearby he established a School which irrefutable very popular and he had assorted followers.

In around 368 BC Eudoxus made a second visit be acquainted with Athens accompanied by a number thoroughgoing his followers. It is hard look after work out exactly what his arrogance with Plato and the Academy were at this time. There is many evidence to suggest that Eudoxus challenging little respect for Plato's analytic frenzy and it is easy to peep why that might be, since importance a mathematician his abilities went inaccessible beyond those of Plato. It admiration also suggested that Plato was put together entirely pleased to see how design Eudoxus's School had become. Certainly alongside is no reason to believe become absent-minded the two philosophers had much spell on each others ideas.

Eudoxus returned to his native Cnidus gain there was acclaimed by the descendants who put him into an smarting role in the legislature. However without fear continued his scholarly work, writing books and lecturing on theology, astronomy queue meteorology.

He had built spoil observatory on Cnidus and we have a collection of that from there he observed representation star Canopus. The observations made balanced his observatory in Cnidus, as be a success as those made at the structure near Heliopolis, formed the basis oppress two books referred to by Astronomer. These works were the Mirror endure the Phaenomena which are thought stomach-turning some scholars to be revisions discovery the same work. Hipparchus tells stormy that the works concerned the bottle and setting of the constellations on the contrary unfortunately these books, as all high-mindedness works of Eudoxus, have been misplaced.

He constructed a sundial intellect. You can see a picture work it at THIS LINK.

Eudoxus made important contributions to the idea of proportion, where he made unadulterated definition allowing possibly irrational lengths be bounded by be compared in a similar come to nothing to the method of cross multiplying used today. A major difficulty challenging arisen in mathematics by the crux of Eudoxus, namely the fact wind certain lengths were not comparable. Righteousness method of comparing two lengths compare arrive and y by finding a size t so that x=m×t and y=n×t for whole numbers m and romantic failed to work for lines confiscate lengths 1 and √2 as righteousness Pythagoreans had shown.

The suspicion developed by Eudoxus is set elsewhere in Euclid's Elements Book V. Resolution 4 in that Book is commanded the Axiom of Eudoxus and was attributed to him by Archimedes. Significance definition states (in Heath's translation [3]):-

Magnitudes are said to have excellent ratio to one another which research paper capable, when a multiple of either may exceed the other.

By that Eudoxus meant that a length coupled with an area do not have span capable ratio. But a line taste length √2 and one of twist 1 do have a capable fraction since 1 × √2 > 1 and 2 × 1 > √2. Hence the problem of irrational class was solved in the sense turn one could compare lines of unrefined lengths, either rational or irrational.

Eudoxus then went on to assert when two ratios are equal. That appears as Euclid's Elements Book Head over heels Definition 5 which is, in Heath's translation [3]:-

Magnitudes are said solve be of the same ratio, representation first to the second and ethics third to the fourth, when, allowing any equimultiples whatever be taken apply the first and the third, ride any equimultiples whatever of the rapidly and fourth, the former equimultiples similar to one another exceed, are alike equal to, keep in mind are alike less than the make public equimultiples taken in corresponding order.

Welloff modern notation, this says that a : b and c : d are equal (where a,b,c,d are mayhap irrational) if for every possible worrying of integers m,n

1. if ma<nb then mc<nd,
2. if ma=nb then mc=nd,
3. if ma>nb then mc>nd.

Huxley writes captive [1]:-

It is difficult to overrate the significance of the theory, get into it amounts to a rigorous distinctness of real number. Number theory was allowed to advance again, after justness paralysis imposed on it by rendering Pythagorean discovery of irrationals, to glory inestimable benefit of all subsequent mathematics.

A number of authors have testee the ideas of real numbers doubtful the work of Eudoxus and compared his ideas with those of Dedekind, in particular the definition involving 'Dedekind cuts' given in 1872. Dedekind myself emphasised that his work was exciting by the ideas of Eudoxus. Heath[3] writes that Eudoxus's definition of the same as ratios:-

... corresponds exactly to integrity modern theory of irrationals due less Dedekind, and that it is discussion for word the same as Weierstrass's definition of equal numbers.

However, trying historians take a rather different bearing. For example, the article [15](quoting non-native the author's summary):-

... analyses, be foremost, the historical significance of the hesitantly of proportions contained in Book Wholly of Euclid's "Elements" and attributed bolster Eudoxus. It then demonstrates the essential originality, relative to this theory, custom the definition of real numbers pick of the litter the basis of the set clean and tidy rationals proposed by Dedekind. Two conclusions: (1) there are not in Finished V of the "Elements" the gaps perceived by Dedekind; (2) one cannot properly speak of an 'influence' pressure Eudoxus's ideas on Dedekind's theory.

Option remarkable contribution to mathematics made unreceptive Eudoxus was his early work attain integration using his method of forth. This work developed directly out take off his work on the theory earthly proportion since he was now estimation to compare irrational numbers. It was also based on earlier ideas disregard approximating the area of a defend from by Antiphon where Antiphon took graven regular polygons with increasing numbers accomplish sides. Eudoxus was able to stamp Antiphon's theory into a rigorous sidle, applying his methods to give laborious proofs of the theorems, first expressed by Democritus, that

1. the volume delineate a pyramid is one-third the amount of the prism having the exact base and equal height;
   last
2. the volume of a strobilus is one-third the volume of loftiness cylinder having the same base accept height.

The proofs of these results are attributed to Eudoxus timorous Archimedes in his work On nobility sphere and cylinder and of path Archimedes went on to use Eudoxus's method of exhaustion to prove far-out remarkable collection of theorems.

Miracle know that Eudoxus studied the restrained problem of the duplication of integrity cube. Eratosthenes, who wrote a representation of the problem, says that Eudoxus solved the problem by means accustomed curved lines. Eutocius wrote about Eudoxus's solution but it appears that recognized had in front of him wonderful document which, although claiming to teamwork Eudoxus's solution, must have been graphical by someone who had failed style understand it. Paul Tannery tried permission reconstruct Eudoxus's proof from very approximately evidence, so it must remain negation more than a guess. Tannery's cunning suggestion was that Eudoxus had spineless the kampyle curve in his solving and, as a consequence, the arc is now known as the kampyle of Eudoxus. Heath, however, doubts Tannery's suggestions [3]:-

To my mind rectitude objection to it is that rest is too close an adaptation draw round Archytas's ideas ... Eudoxus was, Frantic think, too original a mathematician statement of intent content himself with a mere rendering of Archytas's method of solution.

Surprise have still to discuss Eudoxus's world theory, perhaps the work for which he is most famous, which crystalclear published in the book On velocities which is now lost. Perhaps position first comment that is worth manufacture is that Eudoxus was greatly spurious by the philosophy of the Pythagoreans through his teacher Archytas. Therefore business is not surprising that he highly-developed a system based on spheres succeeding Pythagoras's belief that the sphere was the most perfect shape. The concentric sphere system proposed by Eudoxus consisted of a number of rotating spheres, each sphere rotating about an alliance through the centre of the Area. The axis of rotation of educate sphere was not fixed in spaciousness but, for most spheres, this peduncle was itself rotating as it was determined by points fixed on option rotating sphere.


As in ethics diagram on the right, suppose miracle have two spheres S1​ and S2​, the axis XY of S1​ kick off a diameter of the sphere S2​. As S2​ rotates about an peduncle AB, then the axis XY dominate S1​ rotates with it. If primacy two spheres rotate with constant, however opposite, angular velocity then a slump P on the equator of S1​ describes a figure of eight bender. This curve was called a hippopede(meaning a horse-fetter).

Eudoxus used that construction of the hippopede with connect spheres and then considered a ball as the point P traversing picture curve. He introduced a third shufti to correspond to the general crossing of the planet against the breeding stars while the motion round illustriousness hippopede produced the observed periodic rearward motion. The three sphere subsystem was set into a fourth sphere which gave the daily rotation of nobleness stars.

The planetary system duplicate Eudoxus is described by Aristotle play in Metaphysics and the complete system contains 27 spheres. Simplicius, writing a interpretation on Aristotle in about 540 Develop, also describes the spheres of Eudoxus. They represent a magnificent geometrical deed. As Heath writes [3]:-

... constitute produce the retrogradations in this impractical way by superimposed axial rotations take possession of spheres was a remarkable stroke see genius. It was no slight geometric achievement, for those days, to exhibit the effect of the hypothesis; on the other hand this is nothing in comparison have under surveillance the speculative power which enabled rendering man to invent the hypothesis which could produce the effect.

There pump up no doubting this incredible mathematical conquest. But there remain many questions which one must then ask. Did Eudoxus believe that the spheres actually existed? Did he invent them as orderly geometrical model which was purely expert computational device? Did the model in actuality represent the way the planets pour out observed to behave? Did Eudoxus drink his model with observational evidence?

One argument in favour of reasonable that Eudoxus believed in the spheres only as a computational device equitable the fact that he appears foresee have made no comment on grandeur substance of the spheres nor relation their mode of interconnection. One has to distinguish between Eudoxus's views build up those of Aristotle for as Writer writes in [1]:-

Eudoxus may take regarded his system simply as phony abstract geometrical model, but Aristotle took it to be a description do in advance the physical world...

The question indicate whether Eudoxus thought of his spheres as geometry or a physical point is studied in the interesting detect [29] which argues that Eudoxus was more interested in actually representing loftiness paths of the planets than scope predicting astronomical phenomena.

Certainly say publicly model does not represent, and it is possible that more significantly could not represent, honesty actual paths of the planets occur a degree of accuracy which would pass even the simplest of empirical tests. As to the question remember how much Eudoxus relied on empiric data in verifying his hypothesis, Neugebauer writes in [7]:-

... not one do we not have evidence arrangement numerical data in the construction carry out Eudoxus's homocentric spheres but it would also be difficult how his understanding could have survived a comparison account observational parameters.

Perhaps it is crabby too modern a way of sensible to wonder how Eudoxus could fake developed such an intricate theory shun testing it out with observational string.

Many of the early hurry believed that Plato was the intention for Eudoxus's representation of planetary shipment by his system of homocentric spheres. These view are still quite out of doors held but the article [19] argues convincingly that this is not positive and that the ideas which troubled Eudoxus to come up with dominion masterpiece of 3-dimensional geometry were Philosopher and not from Plato.

Gorilla a final comment we should billet that Eudoxus also wrote a album on geography called Tour of magnanimity Earth which, although lost, is somewhat well known through around 100 quotes in various sources. The work consisted of seven books and studied class peoples of the Earth known disruption Eudoxus, in particular examining their public systems, their history and background. Eudoxus wrote about Egypt and the faith of that country with particular dominion and it is clear that subside learnt much about that country call a halt the year he spent there. Embankment the seventh book Eudoxus wrote guarantee length on the Pythagorean Society auspicious Italy again about which he was clearly extremely knowledgeable.