[49] L. von Schroeder, _Pythagoras und die Inder_, Leipzig, 1884; H. Vogt, "Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale gekannt?" _Bibliotheca Mathematica_, Vol. VII (3), pp. 6-20; A. Buerk, loc. cit.; Max Simon, _Geschichte der Mathematik im Altertum_, Berlin, 1909, pp. 137-165; three S[=u]tras are translated in part by Thibaut, _Journal of the Asiatic Society of Bengal_, 1875, and one appeared in _The Pandit_, 1875; Beppo Levi, "Osservazioni e congetture sopra la geometria degli indiani," _Bibliotheca Mathematica_, Vol. IX (3), 1908, pp. 97-105. [50] Loc. cit.; also _Indiens Literatur und Cultur_, Leipzig, 1887. [51] It is generally agreed that the name of the river Sindhu, corrupted by western peoples to Hindhu, Indos, Indus, is the root of Hindustan and of India. Reclus, _Asia_, English ed., Vol. III, p. 14. [52] See the comments of Oppert, _On the Original Inhabitants of Bharatavar[s.]a or India_, London, 1893, p. 1. [53] A. Hillebrandt, _Alt-Indien_, Breslau, 1899, p. 111. Fragmentary records relate that Kh[=a]ravela, king of Kali[.n]ga, learned as a boy _lekh[=a]_ (writing), _ga[n.]an[=a]_ (reckoning), and _r[=u]pa_ (arithmetic applied to monetary affairs and mensuration), probably in the 5th century B.C. [Buehler, _Indische Palaeographie_, Strassburg, 1896, p. 5.] [54] R. C. Dutt, _A History of Civilization in Ancient India_, London, 1893, Vol. I, p. 174. [55] The Buddha. The date of his birth is uncertain. Sir Edwin Arnold put it c. 620 B.C. [56] I.e. 100.10^7. [57] There is some uncertainty about this limit. [58] This problem deserves more study than has yet been given it. A beginning may be made with Comte Goblet d'Alviella, _Ce que l'Inde doit a la Grece_, Paris, 1897, and H. G. Keene's review, "The Greeks in India," in the _Calcutta Review_, Vol. CXIV, 1902, p. 1. See also F. Woepeke, _Propagation_, p. 253; G. R. Kaye, loc. cit., p. 475 seq., and "The Source of Hindu Mathematics," _Journal of the Royal Asiatic Society_, July, 1910, pp. 749-760; G. Thibaut, _Astronomie, Astrologie und Mathematik_, pp. 43-50 and 76-79. It will be discussed more fully in Chapter VI. [59] I.e. to 100,000. The lakh is still the common large unit in India, like the myriad in ancient Greece and the million in the West. [60] This again suggests the _Psammites_, or _De harenae numero_ as it is called in the 1544 edition of the _Opera_ of Archimedes, a work in which the great Syracusan proposes to show to the king "by geometric proofs which you can follow, that the numbers which have been named by us ... are sufficient to exceed not only the number of a sand-heap as large as the whole earth, but one as large as the universe." For a list of early editions of this work see D. E. Smith, _Rara Arithmetica_, Boston, 1909, p. 227. [61] I.e. the Wise. [62] Sir Monier Monier-Williams, _Indian Wisdom_, 4th ed., London, 1893, pp. 144, 177. See also J. C. Marshman, _Abridgment of the History of India_, London, 1893, p. 2.