Vol. Ill, No, 7.] Notes on Indian Mathematics, 493 



'[N.S.I 



•etc. " Albiruni gives a list of these numerical woi'ds and then 



says: "As far as I have seen and heard the Hindus, they do 



not go beyond twenty-five with this kind of numerical notation, '' 



In Albiruni's list the following are the equivalents of 1 and 



10, 2 and 20 : 



1 Adi, 9a9in, indus, ^ita, urvai*a, dhara, pitaniah, 



candra, sitamea, ra9mi, 

 10. . , . . Di^, apa, kendu, ravanaf*ara. 



2 Yama, A^vin, ravicandrau, locana, axi, Dasra, 



Yamala, pasca, netra. 



20 Nakha, kriti. 



V. 



For operations involving large quantities the old notation 

 (i\e, notations without place-values) were clumsy in that a large 

 number of symbols were required ' and there are certain problems 

 that have no meaning, apart from the idea of place value. For in- 



stance the sum of the digits of -y ^ would be "J 



and nothing else. Problems that involve such ideas as * the sum of 

 the digits of twenty-five is seven ' connote the idea of place- 

 value. The following well-known example given by Jamblichus 

 (circ. 360 A.D.) is a distinct proof that he had perfectly clear 

 ideas on the ^ value of position. ' " If the digits of any three be 

 added together, and the digits of their sum be added together, and 

 -so on, the final sum Avill be six,"* Jamblichus also tells us that 

 iihe Pythagoreans called ten the ' unit of the second coui'se, a hun- 

 dred the * unit of third course,' and so on. 



In early Hindu mathematics we find no such pi'oblems as that 

 given by Jamblichus. We can go further and state with perfect 

 truth that, in the whole range of Hindu mathematics, there is not 

 the slightest indication of the use of any idea of place-valuQ 

 before the tenth century A.D. Rodet, however, attempted to 

 «how that Aryabhata's rule for the extraction of the square root 

 implied a knowledge of the value of position. 



The rule in question is as follows : 



* 



^JlH mgj «SEf ^TiT5fr% ^ 



root 



of the square, after having subtracted from this squared part the 



«quai 



1 The Hindus employed some 400 different flymbols in their 'numerical 

 word ' notation. See Rice, etc. < 



2 Gow'a History of Greek Mathematics. Example : Take 25, 26, 27 of 

 which the highest 27 is divisible by 3; then 2 + 5 + 2 + 6 + 2 + 7 = 24 and 2 + 4 

 = 6. The proposition can be made more general. 



3 Rodet translates the term ^|«|^^ by '^ distance d'une place* or *k 



intervalle d'une place ou d'an rang.* 



