By W. W. Rouse Ball

This article continues to be one of many clearest, so much authoritative and such a lot exact works within the box. the normal heritage treats hundreds of thousands 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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On AB, therefore, by Euc. xii, 2, area 1 2 on BC = area 1 2 on AC + area on AB. 1 2 Take away the common parts ∴ area ABC = sum of areas of lunes AECD and AF BG. Hence the area of the lune AECD is equal to half that of the triangle ABC. B C E F D A O (β) He next inscribed half a regular hexagon ABCD in a semicircle whose centre was O, and on OA, AB, BC, and CD as diameters described semicircles of which those on OA and AB are drawn in the figure. Then AD is double any of the lines OA, AB, BC, and CD, ∴ sq.

It seems most likely that they were tortuous curves; whatever they were, he applied them in explaining the apparent motions of the planets as seen from the earth. CH. III] THE SCHOOLS OF ATHENS AND CYZICUS 38 Eudoxus constructed an orrery, and wrote a treatise on practical astronomy, in which he supposed a number of moving spheres to which the sun, moon, and stars were attached, and which by their rotation produced the effects observed. In all he required twenty-seven spheres. As observations became more accurate, subsequent astronomers who accepted the theory had continually to introduce fresh spheres to make the theory agree with the facts.

Amongst the most important of the philosophers who resided at Athens and prepared the way for the Athenian school I may mention Anaxagoras of Clazomenae, who was almost the last philosopher of the Ionian school. c. , and there taught the results of the Ionian philosophy. Like all members of that school he was much interested in astronomy. He asserted that 1 The history of these schools is discussed at length in G. Loria’s Le Scienze Esatte nell’ Antica Grecia, Modena, 1893–1900; in G. J. Allman’s Greek Geometry from Thales to Euclid, Dublin, 1889; and in J.