122 Journal of the Asiatic Society of Bengal. {March, 1908. 
of the diagonals. It appears that the latter case only is here 
intended; but Brahmagupta gives the more general case also. 
Here is Colebrooke’s translation of Brahmagupta’s rule: “ At 
the intersection of the diagonals, or the junction of a diagonal and 
a Porpendion an the upper and lower portions of the ae or 
ml 
the complement of the segment ( , iv., 25.) Chrishna’s 
example is an isosceles re and it may be remarked that the 
isosceles trapezium was a favourite figure with Ahmes and that 
eron devoted nine eek of his geometry to it: in the 
(2) Universally the area of a figure is cate by multiply- 
ing the si 
(b) The — of the sixth part of the circumference is ope to 
the semi-diamet 
(a) Rodet tape that this means that the area of a recti- 
lineal recdanie may be obtained by decomposing it into a succession of 
tra 
(b) “Euclid IV., 15. Heron gives a rule that the sides of a 
polygon inscribed in a circle is equal to three diameters divided 
y th ides. 
of the regular hexa; 
10. Add Ss, to one hundred, multiply by eight and add again 
siaty-two thousand: the result is the approximate value of the cir- 
cumference when the diameter is twenty thousan 
This gives m = 62832/20000 = 3927/1250= 32 ps5 =3'1416. 
A great deal has been made of this statement on account of 
its — accuracy, and it has often been said that this aaa 
result was the discovery of the Hindus, if not of Aryabhata him- 
self. But this cannot be true. According to Albiruni (I., 168) 
Pulisa employed the ratio of 1: 37745. Archimedes prov ved that 
the ratio is less than 3} and greater than 374 ; Ptolemy used the 
value 377/120 (=3- 1418). Brahmagupta gives the values 3 and 
10; M. ibn Musa not only gives the value 62832/20000, but 
a gives a summary of Archimedes’ proof, and it is absolutely cer- 
tain that M.ibn Musa did not copy this from the Hindus, Accord- 
ing to Albiruni (I., 169) Ya’kub ibn Tarik used 344%,.  Bhaskara 
gives 3927/1250. (See Journ. Asiatic Soc. Bengal, 1907, p . 500). 
Brahmagupta finds fault with Aryabhata the elder for using 
in one place the value 3393/1080 ( =3-1416), 7 .. Ptolemy’s value, 
and in another 3393/1050 ( =3°23...). No early Hindu mathema- 
tician quotes Aryabhata as using the value given in the text, Yn 
practical applications ' where the value of z is required, the Hindu 


1 Calculated from these practical applications the value of the ratio 
would be: Aryabhata r=1°7; Brahmagupta ; Bhaskara 7=3. See also 
the Surya Siddhanta (Ed. Bargess, E. J., Am _ Or. Soe. i 58) and the Paneha 
Siddhantika (Ed. Thibaut, iv., 1),and Warren’s Kala Sankalita (p. 92). 
