(London :  Kegan Paul, Trench, Trübner & Co.,  1910.)

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 ``` 168 ALBERUNPS INDIA. circumfer- that it is thricc its diameter. So the Matsya-Purdjut diameter, says, after it has mentioned the diameters of the sun aud moon in yojanas : " The circumference is thrice the diameter." The Aditya-Purdna says, after it has mentioned the breadth of the Dvipas, i.e. the islands and of their surrounding seas : " The circumference is thrice the diameter." The same occurs also in the Vdyu-Purdna. In later times, however, Hindus have become aware of the fraction following after the three wholes. According to Brahmagupta, the circumference is 3^ times the diameter; but he finds this number by a method peculiar to himself. He says : " As the root of 10 is nearly 3^-, the relation between the diameter and its circumference is like the relation between i and the root of 10." Then he multiplies the diameter by itself, the product by 10, and of this product he takes the root. Then the circumference is solid, i.e. consists of integers, in the same way as the root of ten. This calculation, however, makes the fraction larger than it really is. Archimedes defined it to be something between l-^ and -fi. Brahmagupta relates with regard to Aryabhata, criticising him, that he fi:jted the circumference as 3393 ; that he fixed the dia¬ meter in one place as 1080, in another place as 1050, According to the first statement, the relation between diameter and circumference would be like i : 3xW- This fraction (xW) ^^ byxV smaller than \. However, as regards the second statement, it contains no doubt a blunder in the text, not of the author ; for according to the text, the relation would be like i : 3^ and some¬ thing over. Pulisa employs this relation in his calculations in the proportion of I : 3 xVVo- This fraction is here by so much smaller than one- seventh as it is according to Aryabhata, i.e. by yV- ```