Eudoxus of Cnidus
Born: c. 408 BC
Birthplace: Cnidus, (on Resadiye peninsula), Asia Minor (now Knidos, Turkey)
Died: c. 355 BC
Location of death: Cnidus, Asia Minor (now Turkey)
Cause of death: unspecified
Gender: Male
Religion: Pagan
Race or Ethnicity: White
Occupation: Mathematician, Astronomer
Nationality: Ancient Greece
Executive summary: Influential Greek mathematician
Eudoxus of Cnidus was a Greek astronomer, mathematician, scholar and student of Plato. He was the son of Aischines. He travelled to Tarentum, now in Italy, where he studied with Archytas , a follower of Pythagoras, from whom he learned mathematics.
Eudoxus also visited Sicily, where he studied medicine with Philiston, before making his first visit to Athens in the company of the physician Theomedon around 387 BC at the age of 23. 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. To attend Plato's lectures, he walked the seven miles (11 km) each direction, each day.
After leaving Athens, he spent over a year in Egypt where he studied astronomy and mathematics with the priests at Heliopolis. He lived there for 16 months. 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. There he established a School where he taught physics and which proved very popular and had many followers.
In around 368 BC Eudoxus made a second visit to Athens accompanied by a number of his followers. Eudoxus returned to his native Cnidus and there was acclaimed by the people who put him into an important role in the legislature. However he continued his scholarly work, writing books and lecturing on theology, astronomy and meteorology.
He had built an observatory on Cnidus and we know that from there he observed the star Canopus. He had one son, Aristagoras, and three daughters, Actis, Philtis and Delphis.
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 method of comparing two lengths x and y by finding a length t so that x = m × t and y = n × t for whole numbers m and n failed to work for lines of lengths 1 and √2 as the Pythagoreans had shown.
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 byArchimedes. The definition states (in Heath's translation):
Magnitudes are said to have a ratio to one another which is capable, when a multiple of either may exceed the other.
By this, Eudoxus meant that a length and an area do not have a capable ratio. But a line of length √2 and one of length 1 do have a capable ratio since 1 × √2 > 1 and 2 × 1 > √2. Hence the problem of irrational lengths was solved in the sense that one could compare lines of any lengths, either rational or irrational.
Eudoxus then went on to say when two ratios are equal. This appears as Euclid's Elements Book V Definition 5 which is, in Heath's translation:
Magnitudes are said to be of the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and the third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or are alike less than the latter equimultiples taken in corresponding order.
In modern notation, this says that a : b and c : d are equal (where a, b, c, d are possibly irrational) if for every possible pair of integers m, n
i. if ma < nb then mc < nd,
ii. if ma = nb then mc = nd,
iii. if ma > nb then mc > nd.
Another remarkable contribution to mathematics made by Eudoxus was his early work on integration using his method of exhaustion. This work developed directly out of his work on the theory of proportion since he was now able to compare irrational numbers. It was also based on earlier ideas of approximating the area of a circle by Antiphon where Antiphon took inscribed regular polygons with increasing numbers of sides. Eudoxus was able to make Antiphon's theory into a rigorous one, applying his methods to give rigorous proofs of the theorems, first stated by Democritus, that
i. the volume of a pyramid is one-third the volume of the prism having the same base and equal height; and
ii. the volume of a cone is one-third the volume of the cylinder having the same base and height.
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.
Eudoxus's planetary theory, perhaps the work for which he is most famous, which he published in the book On velocities is now lost. 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. The axis of rotation of each sphere was not fixed in space but, for most spheres, this axis was itself rotating as it was determined by points fixed on another rotating sphere. In mathematical astronomy, his fame is due to the introduction of the astronomical globe, and his early contributions to understanding the movement of the planets. Strabo states that he discovered that the solar year is longer than 365 days by 6 hours; Vitruvius that he invented a sundial.
Some of Eudoxus' astronomical texts whose names have survived include:
· Disappearances of the Sun, possibly on eclipses
· Oktaeteris (Ὀκταετηρίς), on an eight-year lunisolar cycle of the calendar
· Phaenomena (Φαινόμενα) and Entropon (Ἔντροπον), on spherical astronomy, probably based on observations made by Eudoxus in Egypt and Cnidus. The Phaenomena of Aratus is a poetical account of the astronomical observations of Eudoxus.
· On Speeds, on planetary motions
As in the diagram illustrated above, suppose we have two spheres S1 and S2, the axis XY of S1 being a diameter of the sphere S2. As S2 rotates about an axis AB, then the axis XY of S1 rotates with it. If the two spheres rotate with constant, but opposite, angular velocity then a point P on the equator of S1 describes a figure of eight curve. This curve was called a hippopede(meaning a horse-fetter).
Eudoxus used this construction of the hippopede with two spheres and then considered a planet as the point P traversing the curve. He introduced a third sphere to correspond to the general motion of the planet against the background stars while the motion round the hippopede produced the observed periodic retrograde motion. The three sphere subsystem was set into a fourth sphere which gave the daily rotation of the stars.
The planetary system of Eudoxus is described by Aristotle in Metaphysics and the complete system contains 27 spheres. Simplicius, writing a commentary onAristotle in about 540 AD, also describes the spheres of Eudoxus. They represent a magnificent geometrical achievement.
According to Aristotle's Ethics, Eudoxus held that pleasure was the chief good, because (1) all beings sought it and endeavored to escape its contrary, pain; (2) it is an end in itself, not a relative good.
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. The work consisted of seven books and studied the peoples of the Earth known to Eudoxus, in particular examining their political systems, their history and background. 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.
REFERENCES:
http://www.gap-system.org/~history/Biographies/Eudoxus.html
http://www.nndb.com/people/658/000096370/
