Page:Indian mathematics, Kaye (1915).djvu/15

 Kātyāyana gives no such rational examples but gives (with Āpastamba and Baudhāyana) the hypotenuse corresponding to sides equal to the side and diagonal of a square, i.e., the triangle $\scriptstyle{a,\ a\sqrt{2},\ a\sqrt{3}}$,|undefined and he alone gives $\scriptstyle{1^2+3^2=10}$, and $\scriptstyle{2^2+6^2=40}$. There is no indication that the S'ulvasūtra rational examples were obtained from any general rule. Incidentally is given an arithmetical value of the diagonal of a square which may be represented by This has been much commented upon but, given a scale of measures based upon the change ratios 3, 4, and 34 (and  Baudhāyana actually gives such a scale) the result is only an expression of a direct measurement; and for a side of six feet it is accurate to about $\scriptstyle{\frac{1}{7}}$th|undefined of an inch; or it is possible that the result was obtained by the approximation $$\scriptstyle{\sqrt{a^2+b}=a+\frac{b}{2a}}$$ by 's R-process, but it is quite certain that no such process was known to the authors of the S'ulvasūtras. The only noteworthy character of the fraction is the form with its unit numerators. Neither the value itself nor this form of fraction occurs in any later Indian work. There is one other point connected with the theorem to be noted, viz., the occurrence of an indication of the formation of a square by the successive addition of gnomons. The text relating to this is as follows: "Two hundred and twenty-five of these bricks constitute the sevenfold agni with aratni and pradesa." "To these sixty-four more are to be added. With these bricks a square is formed. The side of the square consists of sixteen bricks. Thirty-three