By W. W. Rouse Ball

This article continues to be one of many clearest, such a lot authoritative and such a lot exact works within the box. the normal background treats countless numbers of figures and colleges instrumental within the improvement of arithmetic, from the Phoenicians to such 19th-century giants as Grassman, Galois, and Riemann.

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II] IONIAN AND PYTHAGOREAN SCHOOLS 24 take OA as the axis of x, and the generator through O as axis of z, then, with the usual notation in polar co-ordinates, and if a be the radius of the cylinder, we have for the equation of the surface described by the semicircle, r = 2a sin θ; for that of the cylinder, r sin θ = 2a cos φ; and for that of the cone, sin θ cos φ = 12 . These three surfaces cut in a point such that sin3 θ = 12 , and, therefore, if ρ be the projection of OP on the base of the cylinder, then ρ3 = (r sin θ)3 = 2a3 .

It is said that his predecessors occasionally intercalated a month to keep the seasons in their customary places, and if so they must have realized that the year contains, on the average, more than 360 days. 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.

Iv. v. vi. 1896–1899. CH. II] IONIAN AND PYTHAGOREAN SCHOOLS 19 similar, Similarly Hence ∴ BC : AB = AB : BD. BC : AC = AC : DC. 2 AB + AC 2 = BC(BD + DC) = BC 2 . This proof requires a knowledge of the results of Euc. ii, 2, vi, 4, and vi, 17, with all of which Pythagoras was acquainted. (iv) Pythagoras is credited by some writers with the discovery of the theorems Euc. i, 44, and i, 45, and with giving a solution of the problem Euc. ii, 14. It is said that on the discovery of the necessary construction for the problem last mentioned he sacrificed an ox, but as his school had all things in common the liberality was less striking than it seems at first.