**Eudoxus Of Cnidus** was a Greek mathematician and astronomer who contributed to Euclid's Elements. He mapped the stars and compiled a map of the known world. His philosophy influenced Aristotle.

- The problem of duplicating the cube was one which interested Archytas and it would be reasonable to suppose that Eudoxus's interest in that problem was stimulated by his teacher.
- Eudoxus also visited Sicily, where he studied medicine with Philiston, before making his first visit to Athens in the company of the physician Theomedon.
- Eudoxus spent two months in Athens on this visit and he certainly attended lectures on philosophy by Plato and other philosophers at the Academy which had only been established a short time before.
- At this time Eudoxus made astronomical observations from an observatory which was situated between Heliopolis and Cercesura.
- From Egypt Eudoxus travelled to Cyzicus in northwestern Asia Minor on the south shore of the sea of Marmara.
- In around 368 BC Eudoxus made a second visit to Athens accompanied by a number of his followers.
- There is some evidence to suggest that Eudoxus had little respect for Plato's analytic ability and it is easy to see why that might be, since as a mathematician his abilities went far beyond those of Plato.
- It is also suggested that Plato was not entirely pleased to see how successful Eudoxus's School had become.
- Eudoxus returned to his native Cnidus and there was acclaimed by the people who put him into an important role in the legislature.
- Hipparchus tells us that the works concerned the rising and setting of the constellations but unfortunately these books, as all the works of Eudoxus, have been lost.
- 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 and was attributed to him by Archimedes.
- By this Eudoxus meant that a length and an area do not have a capable ratio.
- Eudoxus then went on to say when two ratios are equal.
- A number of authors have discussed the ideas of real numbers in the work of Eudoxus and compared his ideas with those of Dedekind, in particular the definition involving 'Dedekind cuts' given in 1872.
- Dedekind himself emphasised that his work was inspired by the ideas of Eudoxus.
- analyses, first, the historical significance of the theory of proportions contained in Book V of Euclid's "Elements" and attributed to Eudoxus.
- Two conclusions: (1) there are not in Book V of the "Elements" the gaps perceived by Dedekind; (2) one cannot properly speak of an 'influence' of Eudoxus's ideas on Dedekind's theory.
- Another remarkable contribution to mathematics made by Eudoxus was his early work on integration using his method of exhaustion.
- The proofs of these results are attributed to Eudoxus by Archimedes in his work On the sphere and cylinder and of course Archimedes went on to use Eudoxus's method of exhaustion to prove a remarkable collection of theorems.
- We know that Eudoxus studied the classical problem of the duplication of the cube.
- Eratosthenes, who wrote a history of the problem, says that Eudoxus solved the problem by means of curved lines.
- Eutocius wrote about Eudoxus's solution but it appears that he had in front of him a document which, although claiming to give Eudoxus's solution, must have been written by someone who had failed to understand it.
- Paul Tannery tried to reconstruct Eudoxus's proof from very little evidence, so it must remain no more than a guess.
- Tannery's ingenious suggestion was that Eudoxus had used the kampyle curve in his solution and, as a consequence, the curve is now known as the kampyle of Eudoxus.
- We have still to discuss Eudoxus's planetary theory, perhaps the work for which he is most famous, which he published in the book On velocities which is now lost.
- Perhaps the first comment that is worth making is that Eudoxus was greatly influenced by the philosophy of the Pythagoreans through his teacher Archytas.
- The homocentric sphere system proposed by Eudoxus consisted of a number of rotating spheres, each sphere rotating about an axis through the centre of the Earth.
- Eudoxus used this construction of the hippopede with two spheres and then considered a planet as the point PPP traversing the curve.
- The planetary system of Eudoxus is described by Aristotle in Metaphysics and the complete system contains 27 spheres.
- Simplicius, writing a commentary on Aristotle in about 540 AD, also describes the spheres of Eudoxus.
- Did Eudoxus believe that the spheres actually existed?
- Did Eudoxus test his model with observational evidence?
- One argument in favour of thinking that Eudoxus believed in the spheres only as a computational device is the fact that he appears to have made no comment on the substance of the spheres nor on their mode of interconnection.
- not only do we not have evidence for numerical data in the construction of Eudoxus's homocentric spheres but it would also be difficult how his theory could have survived a comparison with observational parameters.
- Perhaps it is just too modern a way of thinking to wonder how Eudoxus could have developed such an intricate theory without testing it out with observational data.
- Many of the early commentators believed that Plato was the inspiration for Eudoxus's representation of planetary motion by his system of homocentric spheres.
- As a final comment we should note that Eudoxus also wrote a book on geography called Tour of the Earth which, although lost, is fairly well known through around 100 quotes in various sources.
- Eudoxus wrote about Egypt and the religion of that country with particular authority and it is clear that he learnt much about that country in the year he spent there.
- In the seventh book Eudoxus wrote at length on the Pythagorean Society in Italy again about which he was clearly extremely knowledgeable.

Born 408 BC, Cnidus (on Resadiye peninsula), Asia Minor (now Knidos, Turkey). Died 355 BC, Cnidus, Asia Minor (now Turkey).

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Analysis, Ancient Greek, Applied Maths, Astronomy, Geography, Geometry, Origin Turkey, Physics, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive