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
