Eudoxus 

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Eudoxus of Cnidus Born 408 BC in Cnidus, now Knidos, Turkey, died 355 BC in Cnidus Eudoxus studied mathematics from Archytas, who was Pythagoras' follower. One of the questions that Eudoxus was interested in was the problem of duplicating the cube. He was also interested in number theory and the theory of music. Another of his interests was geography, and he wrote a book called Tour of the Earth, but there is no surviving copy of it. One of the most interesting and important contributions to mathematics by Eudoxus is his work on the theory of proportion. He looked at various types of lengths and where others have come to a standstill in front of a problem of how to compare lengths which could be measured by rational and irrational numbers, Eudoxus came to a solution: he made a definition allowing possibility of using irrational lengths and comparing them with the rational by using the method of cross multiplication. As we know, irrational numbers are those which cannot be written or expressed as a fraction of two rational numbers. They occur in nature and in mathematics quite often  Pi is an irrational number, as is . Eudoxus' theory appears in Euclid's Elements, Book V, definition 4: it states that
What this really means is that you cannot compare lengths and areas, or areas and volumes, but you can compare lengths between themselves (and for that matter, areas and volumes) even if one of the lengths is an irrational number. For example: two lines, one of length 1 and one of length have a capable ratio because 1x>1 and 2x1>. Can you think of some other irrational numbers? Try comparing them with other lengths. For example calculate the diagonal of a square. Try by working it out using Pythgoras' Theorem. What is it? Compare it with the side of the square. Do the same with the diagonal of a cube.

See more on Greek mathematics. See on Pythagoras and Pythagoreans. Irrational numbers? Some of them can be found here. Click on the picture to see a worksheet on irrational numbers. Euclid and his Elements are keen to be rediscovered by you! See some other famous mathematicians here, or even a page where some of them appear when they were kids. 

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