Vol. Ill, No. 7.] Notes mi Indian Mathematics* 487 



IN.S.] 



with my notes above. The very doubtful examples are from old- 

 fashioned lithographs not reproduced by merely mechanical process- 

 es. Although one may not attach great weight to the forms of 

 symbols as evidence, yet an examination of this table almost 

 drives one to conclude that example (Z) is the earliest that is not 

 2>rima facie suspicious. 



The earliest known inscription that contains a complete set of 

 figures is of A.D. 1050 (Ind. Antiq. xii., 202) and the next that I have 

 jcome across is of A.D. 1114 (Epigr. Ind. i,, 34). In the former of 

 these (Table II) the remarkable variations of the 6,8 and 9 that 

 occur are noticeable, and also the peculiar symbol for seven ^ ; in the 

 other the form of the second ' eight ' is probably accidental. I am 

 inclined to accept the second set of figures as typical of the 

 period and I am doubtful about the first set, the lithograph is 

 so beautifully clear ; but neither helps us to attain any immediate 

 solution. 



Biihler in his Indian PalseograpJiy gives some of the sym- 

 bols quoted above as authentic examples (see Table III). There 



cannot be any doubt about the examples in rows i, ii, iii, and v 

 being extremely unreliable. They are not in the remotest degree 

 authentic. A careful examination of his table leads to the same 

 conclusion as that given above. Bumell states that examples in 

 South India do not occur before the year 1,000 A.D. He quotes some 

 Nagari tenth century figures, but does not give a reference. On 

 palaeographic grounds we are forced to fix the 9th century A.D. 

 as the earliest period in which the modern place- value system of 

 notation may have been in use in India.* This earliest period 

 depends upon one inscription only. If this inscription, on further 

 light being thrown upon it, proves unreliable (as it possibly will), 

 then we shall have to fix the tenth century as the earliest period. 

 Even for the tenth centiuy there is not an excessive amount of 

 good evidence, and it is within the bounds of possibility that we 

 may have finally to turn to the 11th centmy for evidence of the 

 use of our modem system in India. 



IV. 



As stated above, the object of this paper is not to establish any 

 particular theory, but to re-open the question by showing that the 

 premises of the earlier orientalists were, in many cases, unsound 

 and that their conclusions as to the Indian origin of our notation, 

 and on Indian mathematics generally, were possibly wrong. It is 

 therefore necessary to consider several minor points whose import- 

 ance has possibly been exaggerated. For example, Bayley notices 

 the use of the abacus ^ in early times, and the principle it involves 



^ Buhler quoteB this example, bnt omits these peculiarities. 



* Of course * no evidence ' is not * proof ' and it i8 po$gtble that the new 

 notation was in use long before it appeared in inscriptions. 



S Bayley makes the following remarkable statement : ** The use of the 

 ' abacus ' is still common in every village bazar in India, and has been uni- 



