A short account of the history of mathematics by W. W. Rouse Ball

By W. W. Rouse Ball

This article continues to be one of many clearest, so much authoritative and so much actual works within the box. the normal 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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It will be convenient to begin by describing their treatment of geometry and arithmetic. First, as to their geometry. Pythagoras probably knew and taught the substance of what is contained in the first two books of Euclid about parallels, triangles, and parallelograms, and was acquainted with a few other isolated theorems including some elementary propositions on irrational magnitudes; but it is suspected that many of his proofs were not rigorous, and in particular that the converse of a theorem was sometimes assumed without a proof.

II] IONIAN AND PYTHAGOREAN SCHOOLS 18 ras of the first of these theorems. 1 A F E B K G D H C (α) Any square ABCD can be split up, as in Euc. ii, 4, into two squares BK and DK and two equal rectangles AK and CK: that is, it is equal to the square on F K, the square on EK, and four times the triangle AEF . But, if points be taken, G on BC, H on CD, and E on DA, so that BG, CH, and DE are each equal to AF , it can be easily shown that EF GH is a square, and that the triangles AEF , BF G, CGH, and DHE are equal: thus the square ABCD is also equal to the square on EF and four times the triangle AEF .

On AC + sq. 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.

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