Indian Mathematics/The Sulvasūtras

by George Rusby Kaye

II.

3. For the purpose of discussion three periods in the history of Hindu mathematics may be considered:

(I) The S'ulvasūtra period with upper limit c. A.D. 200;

(II) The astronomical period c. A.D. 400—600.

(III) The Hindu mathematical period proper, A.D. 600—1200.

Such a division into periods does not, of course, perfectly represent the facts, but it is a useful division and serves the purposes of exposition with sufficient accuracy. We might have prefixed an earlier, or Vedic, period but the literature of the Vedic age does not exhibit anything of a mathematical nature beyond a few measures and numbers used quite informally. It is a remarkable fact that the second and third of our periods have no connection whatever with the first or S'ulvasūtra period. The later Indian mathematicians completely ignored the mathematical contents of the S'ulvasūtras. They not only never refer to them but do not even utilise the results given therein. We can go even further and state that no Indian writer earlier than the nineteenth century is known to have referred to the S'ulvasūtras as containing anything of mathematical value. This disconnection will be illustrated as we proceed and it will be seen that the works of the third period may be considered as a direct development from those of the second.

4. The S'ulvasutra period.—The term S'ulvasūtra means 'the rules of the cord' and is the name given to the supplements of the Kalpasūtras which treat of the construction of sacrificial altars. The period in which the S'ulvasūtras were composed has been variously fixed by various authors. Max Müller gave the period as lying between 500 and 200 B.C.; R. C. Dutt gave 800 B.C.; Bühler places the origin of the Apastamba school as probably somewhere within the last four centuries before the Christian era, and Baudhāyana somewhat earlier; Macdonnell gives the limits as 500 B.C. and A.D. 200, and so on. As a matter of fact the dates are not known and those suggested by different authorities must be used with the greatest circumspection. It must also be borne in mind that the contents of the S'ulvasūtras, as known to us, are taken from quite modern manuscripts; and that in matters of detail they have probably been extensively edited. The editions of Āpastamba, Baudhāyana and Kātyāyana which have been used for the following notes, indeed, differ from each other to a very considerable extent.

The S'ulvasūtras are not primarily mathematical but are rules ancillary to religious ritual—they have not a mathematical but a religious aim. No proofs or demonstrations are given and indeed in the presentation there is nothing mathematical beyond the bare facts. Those of the rules that contain mathematical notions relate to (1) the construction of squares and rectangles, (2) the relation of the diagonal to the sides, (3) equivalent rectangles and squares, (4) equivalent circles and squares.

5. In connection with (1) and (2) the Pythagorean theorem is stated quite generally. It is illustrated by a number of examples which may be summarised thus:

Āpastamba.	⁠	Baudhāyana.
${\begin{aligned}\\\scriptstyle {3^{2}+\ 4^{2}}&\scriptstyle {=\ 5^{2}}\\\scriptstyle {12^{2}+16^{2}}&\scriptstyle {=20^{2}}\\\scriptstyle {15^{2}+20^{2}}&\scriptstyle {=25^{2}}\\\scriptstyle {5^{2}+12^{2}}&\scriptstyle {=13^{2}}\\\scriptstyle {15^{2}+36^{2}}&\scriptstyle {=39^{2}}\\\scriptstyle {8^{2}+15^{2}}&\scriptstyle {=17^{2}}\\\scriptstyle {12^{2}+35^{2}}&\scriptstyle {=37^{2}}\\\end{aligned}}$		${\begin{aligned}\\\scriptstyle {3^{2}+\ 4^{2}}&\scriptstyle {=\ 5^{2}}\\\scriptstyle {5^{2}+12^{2}}&\scriptstyle {=13^{2}}\\\scriptstyle {8^{2}+15^{2}}&\scriptstyle {=17^{2}}\\\scriptstyle {7^{2}+24^{2}}&\scriptstyle {=25^{2}}\\\scriptstyle {12^{2}+35^{2}}&\scriptstyle {=37^{2}}\\\scriptstyle {15^{2}+36^{2}}&\scriptstyle {=39^{2}}\\\end{aligned}}$

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 $\scriptstyle {a,\ a{\sqrt {2}},\ a{\sqrt {3}}}$, 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

$\scriptstyle {{\sqrt {2}}=1+{\frac {1}{3}}+{\frac {1}{3.4}}-{\frac {1}{3.4.34.}}}$

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 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 Tannery'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 Pythagorean 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 bricks still remain and these are placed on all sides round the borders."

This subject is never again referred to in Indian mathematical works.

The questions (a) whether the Indians of this period had completely realised the generality of the Pythagorean theorem, and (b) whether they had a sound notion of the irrational have been much discussed; but the ritualists who composed the S'ulvasūtras were not interested in the Pythagorean theorem beyond their own actual wants, and it is quite certain that even as late as the 12th century no Indian mathematician gives evidence of a complete understanding of the irrational. Further, at no period did the Indians develop any real theory of geometry, and a comparatively modern Indian work denies the possibility of any proof of the Pythagorean theorem other than experience.

The fanciful suggestion of Bürk that possibly Pythagoras obtained his geometrical knowledge from India is not supported by any actual evidence. The Chinese had acquaintance with the theorem over a thousand years B.C., and the Egyptians as early as 2000 B.C.

6. Problems relating to equivalent squares and rectangles are involved in the prescribed altar constructions and consequently the S'ulvasūtras give constructions, by help of the Pythagorean theorem, of

(1) a square equal to the sum of two squares;

(2) a square equal to the difference of two squares;

(3) a rectangle equal to a given square;

(4) a square equal to a given rectangle;

(5) the decrease of a square into a smaller square.

Again we have to remark the significant fact that none of these geometrical constructions occur in any later Indian work. The first two are direct geometrical applications of 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 $\scriptstyle {\frac {a{\sqrt {2}}}{2}}$; the fourth rule gives a geometrical construction for $\scriptstyle {ab=(b+{\frac {a-b}{2}})^{2}-({\frac {a-b}{2}})^{2}}$ and corresponds to Euclid, II, 5; the fifth is not perfectly clear but evidently corresponds to Euclid, II, 4.

7. The Circle.—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 Delian 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 ${\begin{aligned}\\&\scriptstyle {(\alpha )\quad d=a+{\frac {1}{3}}(a{\sqrt {2}}-a)}\\&\scriptstyle {(\beta )\quad a=d-{\frac {2}{15}}d}\\&\scriptstyle {(\gamma )\quad a=d(1-{\frac {1}{8}}+{\frac {1}{8.29}}-{\frac {1}{8.29.6}}+{\frac {1}{8.29.6.8}})}\\\end{aligned}}$ 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 ${\begin{matrix}\scriptstyle {a}\\\scriptstyle {d}\end{matrix}}\scriptstyle {={\frac {3}{2+{\sqrt {2}}}}={\frac {3}{2+{\frac {577}{408}}}}={\frac {1224}{1393}}}$
$\scriptstyle {=1-{\frac {1}{8}}+{\frac {1}{8.29}}-{\frac {1}{8.29.6}}+{\frac {1}{8.29.6.8}}-{\frac {41}{8.29.6.8.1393}}}$ 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 Thibaut says there is nothing in these rules which would justify the assumption that they were expert in long calculations; and there is no indication in any other work that the Indians were ever acquainted with the process and in no later works are fractions expressed in this manner.

It is worthy of note that later Indian mathematicians record no attempts at the solution of the problem of squaring the circle and never refer to those recorded in the S'ulvasūtras.