-#-^ 



Vol. Ill, No. 7.] Notes on Indian Mathematics. 479 



IN.S.] 



HW^T^ (IT = 6, ▼ = 40, ^ =s 300, ^ = 1000, le. 1346, ) Epigr. Ind. 

 iii., 40, 



^T^^T^ (^=2, W=40, ^=400, IT = 1000, i.e. 1442,) Epigr. Ind. 

 vi., 112. 



iniirr#% (»( = 5, ^ = 10, ^ = 300, ^ = 1000, le. 1315,) Epigr. Ind. 

 iii., 229. 



These examples follow the system described in Biihler's 



Indian Palgeogi-aphy. The series k «... n = l ,..♦ o, <... 



..•» = 1 0, jp ,,..'m. = l 5 and y ^ = 1 9. 



This is practically the same as the so-called ^yabhata's notation 

 modified by the principle of 'place value/ L. D. Bamett gives 

 some examples from Pali MSS. (Joum. Roy. Asiatic Soc, 1901, 

 p. 21) from which he deduces the following values ; k, t, p, y = l ; 

 kh, th, ph, r=2 ; g (and presumably dor b) = 3; bh (and gh, 

 dh ?) = 4 ; m, n (and n ?) = 5 ; 1 = 7: zero initially a, internally n. 

 The use of a = o, he says, is not clear, and he premises that the 

 reduplication of consonants does not affect their . value. For 

 example, gu^aggara m=:2352 ; ratt^akkhaya m~ 

 1222; al ap p a y a n = 1170 ; b h an u v a k k h a ip = 2404 ; 

 gammakhakk e = 1253# 



Here again we have the smaller elements first (t.e., on the 

 left). Bumell expresses this idea by saying: "The order of the 



letters is from right to left." (Burnell, South Indian Palaeography 

 79). Of course, this is not quite right. The order of the letters is 

 theorder of the script, ^ e., from left to right, but the numbera 

 were always expressed with the smaller elements first and not, as 

 is the custom now, with the higher elements first. Bumell describes 

 another system ( Burnell, loc. cit.) which " is only applied to number- 

 ing pages of MSS. ; it was used a good deal in Malabar .... 

 By this system the consonants (with short a, and in their usual 

 order) stand for 1, 2, etc., up to 34, and then they are repeated 

 with long a, eg.j K a =35, kha=36 and so on. " 



Albiiuni tells us that Brahmagupta invented another system 



of notation generally designated by the term ' numerical words. ' 

 Albiruni said : " If you want to write one, express it by every- 

 thing which is unique, as the earth, the moon ; two by everything 

 Avhich is double, as e.g.y black or white; and soon/' It is, 

 however, very doubtful whether this system was invented by 

 Brahmagupta. It was probably introduced into India from the 

 East. Fleet says : " The earliest epigraphical instance at present 

 available in India itself is the record of Saka Samvat 867 (A.D. 

 945-6) for the accession of the eastern Chalukya King Amma II. " 

 (Corp. Inscript. Ind., p. 73) ; while, according to Dr. Liiders, the 

 earliest instance is the stone inscription of Chandamahas§naat 

 Dholpur, dated Vikrama Samvat 898 (Epigr. Ind. iv,, 335) ; and 

 Biihler quotes the Gicacole inscription (-AD. 641) which, however, 

 is now known to be spurious (Ind. Antiq. xxx., 214), and the Kadab 

 inscription (AJD. 813) which is deemed doubtful by Liiders (Epigr. 

 Ind. i^f., 335) and Fleet (K.D.). 



