By W. W. Rouse Ball

ISBN-10: 1418185272

ISBN-13: 9781418185275

This article is still one of many clearest, such a lot authoritative and so much exact works within the box. the traditional background 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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The numbers (2n2 + 2n + 1), (2n2 + 2n), and (2n + 1) possessed special importance as representing the hypotenuse and two sides of a right-angled triangle: Cantor thinks that Pythagoras knew this fact before discovering the geometrical proposition Euc. i, 47. A more general expression for such numbers is (m2 +n2 ), 2mn, and (m2 −n2 ), or multiples of them: it will be noticed that the result obtained by Pythagoras can be deduced from these expressions by assuming m = n + 1; at a later time Archytas and Plato gave rules which are equivalent to taking n = 1; Diophantus knew the general expressions.

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 . Hence the square on EF is equal to the sum of the squares on F K and EK. A B D C (β) Let ABC be a right-angled triangle, A being the right angle. Draw AD perpendicular to BC. The triangles ABC and DBA are 1 A collection of a hundred proofs of Euc.

Several of the leaders of the Athenian school were among his pupils and friends, and it is believed that much of their work was due to his inspiration. The Pythagoreans at first made no attempt to apply their knowledge to mechanics, but Archytas is said to have treated it with the aid of geometry. He is alleged to have invented and worked out the theory of the pulley, and is credited with the construction of a flying bird and some other ingenious mechanical toys. He introduced various mechanical devices for constructing curves and solving problems.

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