[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. Buehler, _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 repetition 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 Buehler'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. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, 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 Buehler, loc. cit., p. 73. [71] Julius Euting, _Nabataeische Inschriften aus Arabien_, Berlin, 1885, pp. 96-97, with a table of numerals. [72] For the five principal theories see Buehler, loc. cit., p. 10. [73] Bayley, loc. cit., reprint p. 3. [74] Buehler, loc. cit.; _Epigraphia Indica_, Vol. III, p. 134; _Indian Antiquary_, Vol. VI, p. 155 seq., and Vol. X, p. 107.