[250] Woepcke, loc. cit., pp. 462, 262. [251] Woepcke, loc. cit., p. 240. _[H.]is[=a]b-al-[.G]ob[=a]r_, by an anonymous author, probably Ab[=u] Sahl Dunash ibn Tamim, is given by Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca Mathematica_, 1896, p. 26. [252] Steinschneider in the _Abhandlungen_, Vol. III, p. 110. [253] See his _Grammaire arabe_, Vol. I, Paris, 1810, plate VIII; Gerhardt, _Etudes_, pp. 9-11, and _Entstehung_ etc., p. 8; I. F. Weidler, _Spicilegium observationum ad historiam notarum numeralium pertinentium_, Wittenberg, 1755, speaks of the "figura cifrarum Saracenicarum" as being different from that of the "characterum Boethianorum," which are similar to the "vulgar" or common numerals; see also Humboldt, loc. cit. [254] Gerhardt mentions it in his _Entstehung_ etc., p. 8; Woepcke, _Propagation_, states that these numerals were used not for calculation, but very much as we use Roman numerals. These superposed dots are found with both forms of numerals (_Propagation_, pp. 244-246). [255] Gerhardt (_Etudes_, p. 9) from a manuscript in the Bibliotheque Nationale. The numeral forms are [symbols], 20 being indicated by [symbol with dot] and 200 by [symbol with 2 dots]. This scheme of zero dots was also adopted by the Byzantine Greeks, for a manuscript of Planudes in the Bibliotheque Nationale has numbers like [pi alpha with 4 dots] for 8,100,000,000. See Gerhardt, _Etudes_, p. 19. Pihan, _Expose_ etc., p. 208, gives two forms, Asiatic and Maghrebian, of "Ghob[=a]r" numerals. [256] See Chap. IV. [257] Possibly as early as the third century A.D., but probably of the eighth or ninth. See Cantor, I (3), p. 598. [258] Ascribed by the Arabic writer to India. [259] See Woepcke's description of a manuscript in the Chasles library, "Recherches sur l'histoire des sciences mathematiques chez les orientaux," _Journal Asiatique_, IV (5), 1859, p. 358, note. [260] P. 56. [261] Reinaud, _Memoire sur l'Inde_, p. 399. In the fourteenth century one Sih[=a]b al-D[=i]n wrote a work on which, a scholiast to the Bodleian manuscript remarks: "The science is called Algobar because the inventor had the habit of writing the figures on a tablet covered with sand." [Gerhardt, _Etudes, _p. 11, note.] [262] Gerhardt, _Entstehung _etc., p. 20. [263] H. Suter, "Das Rechenbuch des Ab[=u] Zakar[=i]j[=a] el-[H.]a[s.][s.][=a]r," _Bibliotheca Mathematica_, Vol. II (3), p. 15. [264] A. Devoulx, "Les chiffres arabes," _Revue Africaine_, Vol. XVI, pp. 455-458. [265] _Kit[=a]b al-Fihrist_, G. Fluegel, Leipzig, Vol. I, 1871, and Vol. II, 1872. This work was published after Professor Fluegel's death by J. Roediger and A. Mueller. The first volume contains the Arabic text and the second volume contains critical notes upon it. [266] Like those of line 5 in the illustration on page 69. [267] Woepcke, _Recherches sur l'histoire des sciences mathematiques chez les orientaux_, loc. cit.; _Propagation, _p. 57.