[348] "Habebant enim diverse formatos apices vel caracteres." See the reference to Gerbert on p. 117.

[349] C. Henry, "Sur l'origine de quelques notations mathematiques," _Revue Archeologique_, 1879, derives these from the initial letters used as abbreviations for the names of the numerals, a theory that finds few supporters.

[350] E.g., it appears in Schonerus, _Algorithmus Demonstratus_, Nurnberg, 1534, f. A4. In England it appeared in the earliest English arithmetical ma.n.u.script known, _The Crafte of Nombrynge_: "-- fforthermore ye most vndirstonde that in this craft ben vsid teen figurys, as here bene writen for ensampul, [Numerals] ... in the quych we vse teen figurys of Inde.

Questio. -- why ten fyguris of Inde? Solucio. for as I have sayd afore thei were fonde fyrst in Inde of a kynge of that c.u.n.tre, that was called Algor."

See Smith, _An Early English Algorism_, loc. cit.

[351] Friedlein ed., p. 397.

[352] Carlsruhe codex of Gerlando.

[353] Munich codex of Gerlando.

[354] Carlsruhe codex of Bernelinus.

[355] Munich codex of Bernelinus.

[356] Turchill, c. 1200.

[357] Anon. MS., thirteenth century, Alexandrian Library, Rome.

[358] Twelfth-century Boethius, Friedlein, p. 396.

[359] Vatican codex, tenth century, Boethius.

[360] a, h, i, are from the Friedlein ed.; the original in the ma.n.u.script from which a is taken contains a zero symbol, as do all of the six plates given by Friedlein. b-e from the Boncompagni _Bulletino_, Vol. X, p. 596; f ibid., Vol. XV, p. 186; g _Memorie della cla.s.se di sci., Reale Acc. dei Lincei_, An. CCLXXIV (1876-1877), April, 1877. A twelfth-century arithmetician, possibly John of Luna (Hispalensis, of Seville, c. 1150), speaks of the great diversity of these forms even in his day, saying: "Est autem in aliquibus figuram istarum apud multos diuersitas. Quidam enim septimam hanc figuram representant [Symbol] alii autem sic [Symbol], uel sic [Symbol]. Quidam vero quartam sic [Symbol]." [Boncompagni, _Trattati_, Vol. II, p. 28.]

[361] Loc. cit., p. 59.

[362] Ibid., p. 101.

[363] Loc. cit., p. 396.

[364] Khosr[=u] I, who began to reign in 531 A.D. See W. S. W Vaux, _Persia, _London, 1875, p. 169; Th. Noldeke, _Aufsatze zur persichen Geschichte_, Leipzig, 1887, p. 113, and his article in the ninth edition of the _Encyclopaedia Britannica_.

[365] Colebrooke, _Essays_, Vol. II, p. 504, on the authority of Ibn al-Adam[=i], astronomer, in a work published by his continuator Al-Q[=a]sim in 920 A.D.; Al-B[=i]r[=u]n[=i], _India, _Vol. II, p. 15.

[366] H. Suter, _Die Mathematiker_ etc., pp. 4-5, states that Al-Faz[=a]r[=i] died between 796 and 806.

[367] Suter, loc. cit., p. 63.

[368] Suter, loc. cit., p. 74.

[369] Suter, _Das Mathematiker-Verzeichniss im Fihrist_. The references to Suter, unless otherwise stated, are to his later work _Die Mathematiker und Astronomen der Araber_ etc.

[370] Suter, _Fihrist_, p. 37, no date.

[371] Suter, _Fihrist_, p. 38, no date.

[372] Possibly late tenth, since he refers to one arithmetical work which is ent.i.tled _Book of the Cyphers_ in his _Chronology_, English ed., p. 132.

Suter, _Die Mathematiker_ etc., pp. 98-100, does not mention this work; see the _Nachtrage und Berichtigungen_, pp. 170-172.

[373] Suter, pp. 96-97.

[374] Suter, p. 111.

[375] Suter, p. 124. As the name shows, he came from the West.

[376] Suter, p. 138.

[377] Hankel, _Zur Geschichte der Mathematik_, p. 256, refers to him as writing on the Hindu art of reckoning; Suter, p. 162.

[378] [Greek: Psephophoria kat' Indous], Greek ed., C. I. Gerhardt, Halle, 1865; and German translation, _Das Rechenbuch des Maximus Planudes_, H.

Waschke, Halle, 1878.

[379] "Sur une donnee historique relative a l'emploi des chiffres indiens par les Arabes," Tortolini's _Annali di scienze mat. e fis._, 1855.

[380] Suter, p. 80.

[381] Suter, p. 68.

[382] Sprenger also calls attention to this fact, in the _Zeitschrift d.

deutschen morgenland. Gesellschaft_, Vol. XLV, p. 367.

[383] Libri, _Histoire des mathematiques_, Vol. I, p. 147.

[384] "Dictant la paix a l'empereur de Constantinople, l'Arabe victorieux demandait des ma.n.u.scrits et des savans." [Libri, loc. cit., p. 108.]

[385] Persian _bagadata_, "G.o.d-given."

[386] One of the Abba.s.sides, the (at least pretended) descendants of 'Al-Abb[=a]s, uncle and adviser of Mo[h.]ammed.

[387] E. Reclus, _Asia_, American ed., N. Y., 1891, Vol. IV, p. 227.

[388] _Historical Sketches_, Vol. III, chap. iii.

[389] On its prominence at that period see Villicus, p. 70.

[390] See pp. 4-5.

[391] Smith, D. E., in the _Cantor Festschrift_, 1909, note pp. 10-11. See also F. Woepcke, _Propagation_.

[392] Enestrom, in _Bibliotheca Mathematica_, Vol. I (3), p. 499; Cantor, _Geschichte_, Vol. I (3), p. 671.

[393] Cited in Chapter I. It begins: "Dixit algoritmi: laudes deo rectori nostro atque defensori dicamus dignas." It is devoted entirely to the fundamental operations and contains no applications.

[394] M. Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca Mathematica_, Vol. VIII (2), p. 99. See also the reference to this writer in Chapter I.