[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.

[63] For a list and for some description of these works see R. C. Dutt, _A History of Civilization in Ancient India_, Vol. II, p. 121.

[64] Professor Ramkrishna Gopal Bhandarkar fixes the date as the fifth century B.C. ["Consideration of the Date of the Mah[=a]bh[=a]rata," in the _Journal of the Bombay Branch of the R. A. Soc._, Bombay, 1873, Vol. X, p.

2.].

[65] Marshman, loc. cit., p. 2.

[66] A. C. Burnell, _South Indian Palaeography_, 2d ed., London, 1878, p. 1, seq.

[67] This extensive subject of palpable arithmetic, essentially the history of the abacus, deserves to be treated in a work by itself.

[68] The following are the leading sources of information upon this subject: G. Buhler, _Indische Palaeographie_, particularly chap. vi; A. C.

Burnell, _South Indian Palaeography_, 2d ed., London, 1878, where tables of the various Indian numerals are given in Plate XXIII; E. C. Bayley, "On the Genealogy of Modern Numerals," _Journal of the Royal Asiatic Society_, Vol.

XIV, part 3, and Vol. XV, part 1, and reprint, London, 1882; I. Taylor, in _The Academy_, January 28, 1882, with a repet.i.tion of his argument in his work _The Alphabet_, London, 1883, Vol. II, p. 265, based on Bayley; G. R.

Kaye, loc. cit., in some respects one of the most critical articles thus far published; J. C. Fleet, _Corpus inscriptionum Indicarum_, London, 1888, Vol. III, with facsimiles of many Indian inscriptions, and _Indian Epigraphy_, Oxford, 1907, reprinted from the _Imperial Gazetteer of India_, Vol. II, pp. 1-88, 1907; G. Thibaut, loc. cit., _Astronomie_ etc.; R.

Caldwell, _Comparative Grammar of the Dravidian Languages_, London, 1856, p. 262 seq.; and _Epigraphia Indica_ (official publication of the government of India), Vols. I-IX. Another work of Buhler's, _On the Origin of the Indian Br[=a]hma Alphabet_, is also of value.

[69] The earliest work on the subject was by James Prinsep, "On the Inscriptions of Piyadasi or A['s]oka," etc., _Journal of the Asiatic Society of Bengal_, 1838, following a preliminary suggestion in the same journal in 1837. See also "A['s]oka Notes," by V. A. Smith, _The Indian Antiquary_, Vol. x.x.xVII, 1908, p. 24 seq., Vol. x.x.xVIII, pp. 151-159, June, 1909; _The Early History of India_, 2d ed., Oxford, 1908, p. 154; J. F.

Fleet, "The Last Words of A['s]oka," _Journal of the Royal Asiatic Society_, October, 1909, pp. 981-1016; E. Senart, _Les inscriptions de Piyadasi_, 2 vols., Paris, 1887.

[70] For a discussion of the minor details of this system, see Buhler, loc.

cit., p. 73.

[71] Julius Euting, _Nabataische Inschriften aus Arabien_, Berlin, 1885, pp. 96-97, with a table of numerals.

[72] For the five princ.i.p.al theories see Buhler, loc. cit., p. 10.

[73] Bayley, loc. cit., reprint p. 3.

[74] Buhler, loc. cit.; _Epigraphia Indica_, Vol. III, p. 134; _Indian Antiquary_, Vol. VI, p. 155 seq., and Vol. X, p. 107.

[75] Pandit Bhagav[=a]nl[=a]l Indr[=a]j[=i], "On Ancient N[=a]g[=a]ri Numeration; from an Inscription at N[=a]negh[=a]t," _Journal of the Bombay Branch of the Royal Asiatic Society_, 1876, Vol. XII, p. 404.

[76] Ib., p. 405. He gives also a plate and an interpretation of each numeral.

[77] These may be compared with Buhler's drawings, loc. cit.; with Bayley, loc. cit., p. 337 and plates; and with Bayley's article in the _Encyclopaedia Britannica_, 9th ed., art. "Numerals."

[78] E. Senart, "The Inscriptions in the Caves at Nasik," _Epigraphia Indica_, Vol. VIII, pp. 59-96; "The Inscriptions in the Cave at Karle,"

_Epigraphia Indica_, Vol. VII, pp. 47-74; Buhler, _Palaeographie_, Tafel IX.

[79] See Fleet, loc. cit. See also T. Benfey, _Sanskrit Grammar_, London, 1863, p. 217; M. R. Kale, _Higher Sanskrit Grammar_, 2d ed., Bombay, 1898, p. 110, and other authorities as cited.

[80] Kharo[s.][t.]h[=i] numerals, A['s]oka inscriptions, c. 250 B.C.

Senart, _Notes d'epigraphie indienne_. Given by Buhler, loc. cit., Tafel I.

[81] Same, ['S]aka inscriptions, probably of the first century B.C. Senart, loc. cit.; Buhler, loc. cit.

[82] Br[=a]hm[=i] numerals, A['s]oka inscriptions, c. 250 B.C. _Indian Antiquary_, Vol. VI, p. 155 seq.

[83] Same, N[=a]n[=a] Gh[=a]t inscriptions, c. 150 B.C. Bhagav[=a]nl[=a]l Indr[=a]j[=i], _On Ancient N[=a]gar[=i] Numeration_, loc. cit. Copied from a squeeze of the original.

[84] Same, Nasik inscription, c. 100 B.C. Burgess, _Archeological Survey Report, Western India_; Senart, _Epigraphia Indica_, Vol. VII, pp. 47-79, and Vol. VIII, pp. 59-96.

[85] K[s.]atrapa coins, c. 200 A.D. _Journal of the Royal Asiatic Society_, 1890, p. 639.

[86] Ku[s.]ana inscriptions, c. 150 A.D. _Epigraphia Indica_, Vol. I, p.

381, and Vol. II, p. 201.

[87] Gupta Inscriptions, c. 300 A.D. to 450 A.D. Fleet, loc. cit., Vol.

III.

[88] Valhab[=i], c. 600 A.D. _Corpus_, Vol. III.

[89] Bendall's Table of Numerals, in _Cat. Sansk. Budd. MSS._, British Museum.

[90] _Indian Antiquary_, Vol. XIII, 120; _Epigraphia Indica_, Vol. III, 127 ff.

[91] Fleet, loc. cit.

[92] Bayley, loc. cit., p. 335.

[93] From a copper plate of 493 A.D., found at K[=a]r[=i]tal[=a][=i], Central India. [Fleet, loc. cit., Plate XVI.] It should be stated, however, that many of these copper plates, being deeds of property, have forged dates so as to give the appearance of antiquity of t.i.tle. On the other hand, as Colebrooke long ago pointed out, a successful forgery has to imitate the writing of the period in question, so that it becomes evidence well worth considering, as shown in Chapter III.

[94] From a copper plate of 510 A.D., found at Majhgaw[=a]in, Central India. [Fleet, loc. cit., Plate XIV.]

[95] From an inscription of 588 A.D., found at B[=o]dh-Gay[=a], Bengal Presidency. [Fleet, loc. cit., Plate XXIV.]

[96] From a copper plate of 571 A.D., found at M[=a]liy[=a], Bombay Presidency. [Fleet, loc. cit., Plate XXIV.]

[97] From a Bijayaga[d.]h pillar inscription of 372 A.D. [Fleet, loc. cit., Plate x.x.xVI, C.]

[98] From a copper plate of 434 A.D. [_Indian Antiquary_, Vol. I, p. 60.]

[99] Gadhwa inscription, c. 417 A.D. [Fleet, loc. cit., Plate IV, D.]

[100] K[=a]r[=i]tal[=a][=i] plate of 493 A.D., referred to above.

[101] It seems evident that the Chinese four, curiously enough called "eight in the mouth," is only a cursive [4 vertical strokes].