110 Dr. Hoeriile — On a lirch-larh Manttscript, [Aua., 



5 



between them \ e. g., ^ (= |) : multiplication, by placing one quantity 



5 



beside the other ; e. g., ^ 32 (= f x 32 = 20): addition, by writing yw 



(abbreviated iov yuta " added") before or after the additive quantity and 

 placing the latter either hy the side of or below the other quantity ; e. g., 11 

 5 yu or 11 ytt 5 (= 11 + 5 = IG) : subtraction, by writing the negative 

 sign -f after the subtractive quantity and placing the latter beside or below 

 the other quantity ; e. g., 1 (= 1 — i = f), or 11 7 + (= 11 — 



3 + 

 7 = 4). This negative sign is the most remarkable difference between the 

 Bakhshali MS. and the works of Bhaskara and others. The MS. uses a cross 

 + (exactly resembling our modern plus -sign), while the sign which is com- 

 monly used is a dot, placed above the quantity ; e. g.,11 7 (= 11 — 7 = 4). 

 I may add that the cijjher is used (as in the Lilavati) to indicate an unknown 



quantity, the value of which is sought ; e. g.,0 5 yu mu 



11 1 



f for Y + - = -^ and J — J = Y ; here a? = 11, y 



abbreviated for mulada " square.") It is, however, also employed, in the 

 usual way as the tenth figure of the decimal notation. A proportion is 



16 1 ^^^^ ^^ ( ^^^ ^ ' TT = 30 : 65 ; plia abbrevia- 

 ted for plialam\. All these peculiarities of method, terminology and nota- 

 tion, differing as they do from those in common use since the time of 

 Brahmagupta (about 628 A. D.) and Aryabhata (about 500 A. D.), 

 whose mathematical treatises are the earliest known, tend to show that the 

 work contained in the Bakhshali MS. is more ancient than any of those 

 I have just mentioned. 



" There is another remarkable feature in the MS., which points in 

 the same direction, namely, the language in which it is written. 

 This is what is now commonly called the Gatha dialect, because it was 

 first noticed in ancient Buddhist works (such as the Lalita Vistara) written 

 in verses or gathas. The term, Gatha dialect, however, is no more appro- 

 priate now, because that dialect is now known to be also used in ancient 

 Buddliist works, which are partly written in prose, such as the Mahavastu, 

 of which M. Senart has just published an excellent edition. However 

 that may be, it is generally admitted that this species of language is a 

 very ancient one. It is a kind of ungrammatical Sanskrit, (judged, that 

 is, by the standard of what is commonly called Sanskrit), interspersed to a 

 large extent with ancient Prakrit or Pali forms. There is some dispute 

 as to the exact origin, time and locality of this species of ancient irregu- 



