Eudoxus lived from 408-355 BC, and loved to travel, searching for knowledge. He was famous for his works with irrational numbers, even when Pythagorus shied away from them. He also founded and developed the earlier terachings of infitesimal calculus, furthuring the work he did with irrational numbers. Many of Euclid's Elements were founded first by none other than Eudoxus.

Eudoxus was the son of Aischines, and lived in Cnidus (current day Turkey). He travelled to Tarentum (now Italy) and there he studied with Archytas — a follower of Pythagoras — on the problems of duplicating the cube, and also number theory and the theory of music.

He also travelled to Sicily to study medicine before making his first journey to Athens. He spent two months there where he attended philosophy lectures by Plato and other philosophers of the Academy. After leaving Athens, he left to Egypt where he studied astronomy with Heliopolis' priests. He then moved on to Cyzicus (northwest Asia Minor, on the south shore of the sea of Marmara) where he gained followers after establishing a very popular School.

It wasnt until about 368 BC that he returned to Athens accompanied with many of his followers. The relationship between Plato and Eudoxus went sour at this point; Eudoxus lost respect for Plato's innacurate indings, and Plato became jealous of the popularity of Eudoxus' School. After this short visit in Athens, he returned to Cnidus, where the people there set him up in the legeslature. Even in this role he continued his schoolwork, writing books and lecturing on theology, astronomy, and even meteorology.

After a while, he built an observatory in Cnidus and it is from there, with his findings from Heliopolis, that the Mirror and the Phaenomena by Hipparchus were based off of. The works by Hipparchus tell us that the findings consisted of the rising and setting of constellations by Eudoxus — however all of Eudoxus' findings have been lost.

In mathematics, he made important discoveries: he made a meathod closely similar to what we use as cross multiplication; comparable lengths that were variables, he found that area and length dont have capable ratios, yet a line length √2
and 1 have a comparable ratios, such that 1 * √2 > 1 and 2 * 1 > √2, therefore giving a solution to lengths — whether the values rational or irrational.

He made many more findings with using and comparing rational and irrational numbers, for the use in algebra, as well as ion calculus.

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