782 



SCIENCE 



[N. S. Vol. XLIII. No. 1118 



balanced treatment of the general field of 

 native Hindu mathematics. In this antici- 

 pation he will, however, be disappointed. Mr. 

 Kaye's mind runs rather to monographs than 

 to treatises, and these monographs are gen- 

 erally of real value to those who are less for- 

 timately situated with respect to the sources 

 of information. But in the present instance he 

 has given us a monograph with a rather pre- 

 tentious title which does not seem quite worthy 

 of his undoubted powers. It is, of course, 

 characterized by Mr. Kaye's prejudice against 

 any claims of originality on the part of the 

 Hindu scholars, but this feature is rather less 

 obtrusive than in his other monographs, and 

 in any case a reader can overlook a bias of this 

 kind if he is presented with the evidence in 

 such a fashion as to allow of its being weighed 

 by himself. But the work is by no means an 

 exhaustive presentation of Indian mathematics 

 and it contains but little that is not already 

 familiar to the students of history. 



Mr. Kaye divides the subject into three his- 

 toric periods: (1) the S'ulvasutra period, ex- 

 tending to about A.D. 200; (2) the astronomical 

 period, extending from about a.d. 400 to 600; 

 (3) the Hindu mathematical period proper, 

 extending from a.d. 600 to 1200, after which 

 there was no native mathematics worth men- 

 tioning. 



The word S'ulvasutra means " the rules of 

 the cord," and applies to certain verses treat- 

 ing of the construction of altars. The con- 

 nection with the Egyptian " rope-stretchers " 

 (harpedonaptse) will occur to every one who 

 has considered the history of ancient geom- 

 etry, and, like so many parallels of this kind, 

 is suggestive of the early intercourse between 

 the East and the West. The dates of the 

 S'ulvasutras are uncertain, and the manu- 

 scripts show many variations due to different 

 scribes, but we know to a certainty enough 

 about them to render their study of interest 

 in the history of mathematics. The Pytha- 

 gorean theorem is stated with some general- 

 ity, although there is nothing to show whether 

 it was an independent product of India, or 

 came from China, where it seems to have been 

 already known, or worked its way eastward 



from the Mediterranean civilization, perhaps 

 at the time of the visit of the forces of Alex- 

 ander. The unit fraction is in evidence, as it 

 was in Egypt and Babylon two thousand years 

 earlier. The mensuration of the circle is also 

 a feature of the S'ulvasiitras. The most inter- 

 esting suggestion made by Mr. Kaye in this 

 connection relates to a circle of diameter d 

 and area a^, namely, that the relation 



d = a + i(o-sl2 -a), 



which is given in the editions of Apastamba 

 and Baudhayana, led to the relation 



a=d{l-l 



+A-. 



■ + : 



8,29 8.29.6 ' 8.29.6.8; 

 through the substitution for V2 of 



1 + ^ + ^ 

 ^ 3 ^ 3.4 



3.4.34 ' 



which value is given earlier in the S'ulvasutras. 

 Mr. Kaye asserts, however, that this substitu- 

 tion was beyond the powers of the Hindu 

 mathematicians of that period, and it is a fact 

 that we have no other evidence of any ability 

 to make such a substitution. 



As far as our present knowledge goes, there 

 is a gap between the S'ulvasutra mathematics 

 and the first distinct treatises on the subject, 

 such as the Siirya Siddhanta, the anonymous 

 astronomical classic of about a.d. 400. This 

 work was included in the great collection made 

 by Varaha Mihira in the sixth century, and the 

 evidence seems to show that, by this time, more 

 or less of Greek mathematics was known in 

 India. Ptolemy's influence seems to be evi- 

 dent in the table of chords given by Varaha 

 Mihira, but the earliest known use of the sine 

 occurs in the Hindu works of this period. 



Mr. Kaye summarizes the work of Aryabhata 

 in a satisfactory manner, making no mention 

 of the younger mathematician of the same 

 name to whom he devoted some attention a few 

 years ago. Indeed, his statement that " the 

 Indian works record distinct advances on 

 what is left of the Greek analysis " is perhaps 

 the most outspoken statement in favor of the 

 Hindu algebraists to be found in any of his 

 writings. 



