Page:Indian mathematics, Kaye (1915).djvu/17

 the rule $\scriptstyle{c^2=a^2+b^2}$; the third gives in a geometrical form the sides of the rectangle as $$\scriptstyle{a\sqrt{2}}$$ and $\scriptstyle{\frac{a\sqrt{2}}{2}}$;|undefined the fourth rule gives a geometrical construction for $$\scriptstyle{ab=(b+\frac{a-b}{2})^2-(\frac{a-b}{2})^2}$$ and corresponds to, II, 5; the fifth is not perfectly clear but evidently corresponds to , II, 4. 7. .—According to the altar building ritual of the period it was, under certain circumstances, necessary to square the circle, and consequently we have recorded in the S'ulvasūtras attempts at the solution of this problem, and its connection with altar ritual reminds us of the celebrated problem. The solutions offered are very crude although in one case there is pretence of accuracy. Denoting by a the side of the square and by d the diameter of the circle whose area is supposed to be $$\scriptstyle{a^2}$$ the rules given may be expressed by Neither of the first two rules, which are given by both  Āpastamba and Baudhāyana, is of particular value or interest. The third is given by Baudhāyana only and is evidently obtained from $$\scriptstyle{(\alpha)}$$ by utilising the value for $$\scriptstyle{\sqrt{s}}$$ given in paragraph 5 above. We thus have which, neglecting the last term, is the value given in rule $\scriptstyle{(\gamma)}$. This implies a knowledge of the process of converting a fraction into partial fractions with unit numerators, a knowledge most certainly not possessed by the composers of the S'ulvasūtras; for as says there is nothing in