Vol. Ill, No. 7.] Notes on Indian Mathematics, 501 



[N.S.'] 



lies between 22/7 and 228/71. Oi*dinarily lie used, as \ve do 

 now, 22/7. Ptolemy's value is 377/120 ^'hich equals 3-141666. 



177 



Pulisa gave 3 ,--— = 3*1416. The method of Alimes is not 



unlike that given in the Sulvasutras. M. ibn Musa finds the 

 area of the circle by treating it as the limit of an equilatei-al 

 polygon of many sides : this method certainly does not point to 

 Brabmagupta as the soiirce of his knowledge. 



VII. 



I 



Not many years ago the opinions of the eai'ly orientalists 

 appeared to be corroborated to some extent by the discovery of 

 the Bakhshali manuscript, or ratfier by Dr. Hoernle's statements 

 regarding the manuscript ; but, when it is understood, that Dr. 

 Hoernle's conclusions were based upon false premises, it Avill be 

 seen that there are other possible explanations than those he 

 gives. 



. Now, in his discussion on the age and origin of the Bakh- 

 shali arithmetic, Dr. Hoeride makes the following assump- 

 tions : — 



■ * ■ 



" That Indian arithmetic and algebra, at least, are of entirely 

 native origin. " (Ind. Antiq. xvii., 37). 



" That the Hindus did not ^^t their elements of the arithme- 

 tical science from the Greeks. '" (ib.) 



" That this principle (of value of position) was known in 

 India as early as A.D. 500." (Ind. Antiq. xii., 37). 



From such assumptions he finally concludes that the woik is^ 

 a very ancient one of pm^ely Hindu origin, and even suggests that 

 it helps to prove the early use of the modem system of notation 

 among the early Hindus. He protects himself from giving a 

 rather crude illustration of the vicious circle by a qualifying 

 clause, but, in actuality, that is the form of his reasoning. 



In addition to the statements given above, Dr. Hoernle writes, 

 "The MS. is written in the so-called Sarada chai'actei's, which 

 are still used in Kashmir and Avhieh, as they occur on the coins of 

 the Maharajas of Kashmir, are of a not inconsiderable age. 

 . . , . I have not obsei'ved these (some of the forms of 

 lettei^) in other MSS. written in the Sarada characteis. Hence 

 I am inclined to look on them as evidence of great age in the 

 Bakhshali MS. ; and as the "West Indus Districts were early lost 

 to Hindu civilization through Muhammadan conquests, during 

 which it was a common practice to bury MSS. to save them fix)m 

 destruction .... The Bakhshali MS. may be referred to the 8tl 

 or 9th century A.D 



1 



" The methods (of setting out the problems, etc.) differs 

 considerably from that used in other Hindu mathematical treatises, 

 f.y., in those of Bhaskara and Brahraagupta . • . . '* This nega- 

 tive sign is the most remarkable difFerence between the Bakhshali 

 MS. and the works of Bhaskara and othei^s. The MS. uses a cross +, 

 while the sign that is commonly used is a dot, placed above the 



