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By W. W. Rouse Ball

This article is still one of many clearest, so much authoritative and such a lot exact works within the box. the traditional background treats enormous quantities of figures and faculties instrumental within the improvement of arithmetic, from the Phoenicians to such 19th-century giants as Grassman, Galois, and Riemann.

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CH. III] THE SCHOOLS OF ATHENS AND CYZICUS 35 Plato. The next philosopher of the Athenian school who requires mention here was Plato. , and was, as is well known, a pupil for eight years of Socrates; much of the teaching of the latter is inferred from Plato’s dialogues. c. Plato left Athens, and being possessed of considerable wealth he spent some years in travelling; it was during this time that he studied mathematics. He visited Egypt with Eudoxus, and Strabo says that in his time the apartments they occupied at Heliopolis were still shewn.

III] THE SCHOOLS OF ATHENS AND CYZICUS 37 (that is, so that AB : AH = AH : HB) is solved in Euc. ii, 11, and probably was known to the Pythagoreans at an early date. If we denote AB by l, AH by a, and HB by b, the theorems that Eudoxus proved are equivalent to the following algebraical identities. (i) (a + 21 l)2 = 5( 12 l)2 . (ii) Conversely, if (i) be true, and AH be taken equal to a, then AB will be divided at H in a golden section. (iii) (b + 21 a)2 = 5( 12 a2 ). (iv) l2 + b2 = 3a2 . (v) l + a : l = l : a, which gives another golden section.

There is some reason to think that he believed the earth to be a disc-like body floating on water. c. But though this prophecy and its fulfilment gave extraordinary prestige to his teaching, and secured him the name of one of the seven sages of Greece, it is most likely that he only made CH. II] IONIAN AND PYTHAGOREAN SCHOOLS 14 use of one of the Egyptian or Chaldaean registers which stated that solar eclipses recur at intervals of about 18 years 11 days. Among the pupils of Thales were Anaximander, Anaximenes, Mamercus, and Mandryatus.

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