502 Journal oj the Asiatic Society of Bengal. [July, 1907. 



quantity .... Here, therefore, there appears to be a mark of 

 great antiquity. As to its (the + ) origin I am unable to suggest 

 any satisfactory explanation. I have been informed by Dr. 

 Tliibaut^ .... that Diaphantus used the letter i|r reversed (thus ^) 

 to indicate the negative quantity .... Now the Bakbsh^li 

 arithmetic is written in the Sloka measure ; and this circumstance 

 carries its composition back to a time anterior to that change of 

 literary fashion in the 5th century A.D.^ .... Again the foreign 

 terms dinara (Latin denarius) and dramma (Greek drachme^ occur 

 in both (i.e. also in Brahmagupta} .... the decimal system 

 of notation 8 is employed . . . etc., etc.*' Notwithstanding Dr. 

 Hoernle's conclusion every one of these poirj^s seems to me to 

 emphasize the fact that this work is not of pure Indian origin : 

 clearer evidence for a non-Indian origin could not be given. 



Further, Dr. Hoernle has failed to note on one most important 

 matter. He explains that a mixed number is shown in the Bakh- 

 shali MS. by placing ' the three numbers under one another; thus 



14 



1 means 1 — |and 1 means 4^, but he does not explain that 

 3+ 2 



this mode of writing fractions is peculiarly Arabic* That the 

 Bakhshali MS. exhibits such characteristic difference from the old 

 Hindu treatises ; that the only points of resemblance are admitted- 

 ly of foreign origin^; that the MS. was found in that part of 

 Lidia that was dominated at the time when it was written by 

 a foreign race ; and that this foreign race was, at the period in 

 question, superior to the Hindus as regards their knowledge of 



1 Ib it possible that Dr. Hoernle had not read the introduction to Cole- 

 brooke's Lilavati ? (p. xii., Ed. 1817). Colebrooke there gives a fall explana- 

 tion of the use of the inverted ^ by Diaphantus). 



^ Is this sound argument ? 



8 That is, the modern system with place-values. 



* Woepcke, p. 497. 



6 The only resemblance between the matter of the Bakhshali manuscript 

 and Brahmagupta*8 work, that Dr. Hoernle points out, lies in the fiftieth sutra 

 of the MS. and ch, iviii. §84 of Brahmagupta's algebra. Peculiar signifi* 

 cance attaches to this problem, for it was fully dealt with by Diophantus and 

 fully expounded in the algebra of Alkarkhi which was based on that of Dio- 

 phantus (Woepcke's Extrait duFahhri). The problem given in the Bakhshdli 

 MS. may be expressed in modern notation thus : — Solve x + 5 — rn^, a?— 7«n2 

 (not as Dr. Hoernle puts it^ — aj + S^as^^ a; ^7=*'^). Alkarkhi gives at least four 

 problems of the same kind and a great number of allied types. The solution 

 given in the MS. is as follows : — *' The sum of the additive and subtractive 

 numbers is 12; the half of it is 6; lessened by two is 4; its half is 2; its 

 square is 4; this is added to the eubtractive number and becomes 11. This 



is the number." This solution is based upon the fact that J ] p 



+ (a + b) is a perfect square. This formula is given by Alkarkhi (p. 63). In 

 the present case a — 6, b = 7 and p »^ 2. 



This rather remarkable coincidence unmistakably points to Diophantus 

 as one of the ultimate sources of both Brahmagupta'a work and the Bakhshali 

 arithmetio. 



