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-