ider, "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