494 Journal of the Asiofic Society of Bengal. [Julj, 1907. 



The rale is algebraical in character, that is, it is perfectly 

 general and applies to all possible notations. There is absolutely 

 nothing in it that can lead one to suppose that it was meant to 

 apply in particular to a notation with ' place-values ' and a zero. 

 The method given here was known to the Grreeks and was admir- 

 ably expressed by Theoii of Alexandria while Euclid gave excellent 

 geometrical solutions for the operation. Brahmagupta does not give 

 any rale for the extraction of square roots, although^ he gives 

 identically the same rule for cube roots as is given by Arjabhata. 

 On turning to Rodet*s notes we find the cause of his erroneous con- 

 clusions in the following statement. He says, " Pour rendre comple- 



tement intelliglbles les expressions . . . dont Ary abhata fait usage dans 

 ces regies, je vais reproduire un extrait des comnaentateurs de la 

 Lilavati.,. et in di quant le precede pratiqne snivi par les Indiens 

 pour operer Textraction des racines," Of course, if one relies upon 

 a commentator of the Lildvati for enlightenment as to the mathe- 

 matical practice in the time of Aryabhata, wrong impressions are 

 likely to result. As I have stated before, there is not in any part 

 of Aryabhata*s work the remotest indication of a knowledge of a 

 notation with * place-values* ; on the other hand there is plenty 

 of evidence in the opposite direction. 



The use of the new notation is not indicated in the rules for the 

 fundamental operations given by Brahmagupta ; but there is one 

 point about multiplication that is possibly worth noting. He 

 says, '* If the multiplicator be too great or too small the mul- 

 tiplicand is to be multiplied by the excess or defect as put ; and 

 the product of the multiplicand so put is added or subtracted." 

 The commentator Kfisbna (sixteenth century a.d.) misunder- 

 stood ^ this rule which will be found in most modern textbooks 

 and which must have been particularly useful with a notatioti 

 without place-values. 



In the absence of any detailed workings * of examples by the 

 early Hindu mathematicians it would be difficult to come to defi- 

 nite conclusions regarding the notations they used if we had no 

 other evidence. But fortunately there is plenty of other evidence 

 that points to no uncertain conclusion. Sir R. Temple has, for 

 example, shown us that the old ideas of notation still prevail, to a 

 very great extent, among those in India who have not come in 

 contact with foreign systems. This is, practically, the proof ab- 

 solute that the new notation is not of Indian origin. The chief 

 virtue of the new notation is, it is claimed, that it simplifies 

 enormously arithmetical operations. Consequently we could not 

 possibly give the credit of the invention to those who did not use 



i Such miaunderstaBdings are extremely common among the Hindu com- 

 mentators. 



2 ''This omigsion is still the characteristic of the unskilful worker in 

 ftrithmetio, who will, if possible, show np the question and answer; but 

 from a noble scorn of details, or a desire to keep secret the mysterious pro- 

 cess omits the steps of the work and gives no inkling of the method." The 

 Story 0/ Arithmetic's. Carrington. 



