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 the right angle a perpendicular to the hypotenuse, and comparing the two triangles thus obtained with the given triangle to which they are similar. This proof was unknown in Europe till Wallis rediscovered it. The Brahmins never inquired into the properties of figures. They considered only metrical relations applicable in practical life. In the Greek sense, the Brahmins never had a science of geometry. Of interest is the formula given by Brahmagupta for the area of a triangle in terms of its sides. In the great work attributed to Heron the Elder this formula is first found. Whether the Indians themselves invented it, or whether they borrowed it from Heron, is a disputed question. Several theorems are given by Brahmagupta on quadrilaterals which are true only of those which can be inscribed on a circle—a limitation which he omits to state. Among these is the proposition of Ptolemæus, that the product of the diagonals is equal to the sum of the products of the opposite sides. The Hindoos were familiar with the calculation of the areas of circles and their segments, of the length of chords and perimeters of regular inscribed polygons. An old Indian tradition makes $$\scriptstyle{\pi=3}$$, also $$\scriptstyle{=\sqrt{10}}$$; but Aryabhatta gives the value $$\scriptstyle{\frac{31416}{10000}}$$. Bhaskara gives two values,—the 'accurate,' $$\scriptstyle{\frac{3927}{1250}}$$, and the 'inaccurate,' Archimedean value, $$\scriptstyle{\frac{22}{7}}$$. A commentator on Lilavati says that these values were calculated by beginning with a regular inscribed hexagon, and applying repeatedly the formula $$\scriptstyle{AD=\sqrt{2-\sqrt{4-\overline{AB^2}}}}$$, wherein AB is the side of the given polygon, and AD that of one with double the number of sides. In this way were obtained the perimeters of the inscribed polygons of 12, 24, 48, 96, 192, 384 sides. Taking the radius = 100, the perimeter of the last one gives the value which Aryabhatta used for $$\scriptstyle{\pi}$$.

Greater taste than for geometry was shown by the Hindoos for trigonometry. Like the Babylonians and Greeks, they