232 G-. Thibaufc— On the S'ulvasutras. [Xo. 3, 



parts had to remain unchanged. A look at the outlines of the different 

 chitis is sufficient to show that all this could not be accomplished without 

 a certain amount of geometrical knowledge. Squares had to be found 

 which would be equal to two or more given squares, or equal to the differ- 

 ence of two given squares ; oblongs had to be turned into squares and 

 squares into oblongs ; triangles had to be constructed equal to given 

 squares or oblongs, and so on. The last task and not the least was that of 

 finding a circle, the area of which might equal as closely as possible that 

 of a given square. 



Nor were all these problems suggested only by the substitution of the 

 more complicated forms of the agni for the primitive chaturasras'yena, al- 

 though this operation doubtless called for the greatest exertion of ingenuity ; 

 the solution of some of them was required for the simplest sacrificial con- 

 structions. Whenever a figure with right angles, square or oblong, had to 

 be drawn on the ground, care had to be taken that the sides really stood 

 at right angles on each other ; for would the ahavaniya fire have carried 

 up the offerings of the sacrificer to the gods if its hearth had not the shape 

 of a perfect square ? There was an ancient precept that the vedi at the 

 sautramani sacrifice was to be the third part of the vedi at the soma sacri- 

 fices, and the vedi at the pitriyajna its ninth part; consequently a method 

 had to be found out by which it was possible to get the exact third and 

 ninth part of a given figure. And when, according to the opinion of some 

 theologians, the garhapatya had to be constructed in a square shape, ac- 

 cording to the opinion of others as a circle, the difference of the opinions 

 referred only to the shape, not to the size, and consequently there arose 

 the want of a rule for turning a square into a circle. 



The results of the endeavours of the priests to accomplish tasks of this 

 nature are contained in the paribhasha sutras of the S'ulvasutras. The 

 most important among these is, to use our terms, that referring to the 

 hypotenuse of the rectangular triangle. The geometrical proposition, the 

 discovery of which the Greeks ascribed to Pythagoras, was known to the 

 old acharyas, in its essence at least. They express it, it is true, in words 

 very different from those familiar to us ; but we must remember that they 

 were interested in geometrical truths only as far as they were of practical 

 use, and that they accordingly gave to them the most practical expression. 

 What they wanted was, in the first place, a rule enabling them to draw 

 a square of double the size of another square, and in the second place 

 a rule teaching how to draw a square equal to any two given squares, and 

 according to that want they worded their knowledge. The result is, that 

 we have two propositions instead of one, and that these propositions speak 

 of squares and oblongs instead of the rectangular triangle. 



