[477] _The Works of Geoffrey Chaucer_, edited by W. W. Skeat, Vol. IV, Oxford, 1894, p. 92.

[478] Loc. cit., Vol. III, pp. 179 and 180.

[479] In Book II, chap, vii, of _The Testament of Love_, printed with Chaucer's Works, loc. cit., Vol. VII, London, 1897.

[480] _Liber Abacci_, published in Olleris, _Oeuvres de Gerbert_, pp.

357-400.

[481] G. R. Kaye, "The Use of the Abacus in Ancient India," _Journal and Proceedings of the Asiatic Society of Bengal_, 1908, pp. 293-297.

[482] _Liber Abbaci_, by Leonardo Pisano, loc. cit., p. 1.

[483] Friedlein, "Die Entwickelung des Rechnens mit Columnen," _Zeitschrift fur Mathematik und Physik_, Vol. X, p. 247.

[484] The divisor 6 or 16 being increased by the difference 4, to 10 or 20 respectively.

[485] E.g. Cantor, Vol. I, p. 882.

[486] Friedlein, loc. cit.; Friedlein, "Gerbert's Regeln der Division" and "Das Rechnen mit Columnen vor dem 10. Jahrhundert," _Zeitschrift fur Mathematik und Physik_, Vol. IX; Bubnov, loc. cit., pp. 197-245; M.

Chasles, "Histoire de l'arithmetique. Recherches des traces du systeme de l'abacus, apres que cette methode a pris le nom d'Algorisme.--Preuves qu'a toutes les epoques, jusq'au XVI^e siecle, on a su que l'arithmetique vulgaire avait pour origine cette methode ancienne," _Comptes rendus_, Vol.

XVII, pp. 143-154, also "Regles de l'abacus," _Comptes rendus_, Vol. XVI, pp. 218-246, and "a.n.a.lyse et explication du traite de Gerbert," _Comptes rendus_, Vol. XVI, pp. 281-299.

[487] Bubnov, loc. cit., pp. 203-204, "Abbonis abacus."

[488] "Regulae de numerorum abaci rationibus," in Bubnov, loc. cit., pp.

205-225.

[489] P. Treutlein, "Intorno ad alcuni scritti inediti relativi al calcolo dell' abaco," _Bulletino di bibliografia e di storia delle scienze matematiche e fisiche_, Vol. X, pp. 589-647.

[490] "Intorno ad uno scritto inedito di Adelhardo di Bath int.i.tolato 'Regulae Abaci,'" B. Boncompagni, in his _Bulletino_, Vol. XIV, pp. 1-134.

[491] Treutlein, loc. cit.; Boncompagni, "Intorno al Tractatus de Abaco di Gerlando," _Bulletino_, Vol. X, pp. 648-656.

[492] E. Narducci, "Intorno a due trattati inediti d'abaco contenuti in due codici Vaticani del secolo XII," Boncompagni _Bulletino_, Vol. XV, pp.

111-162.

[493] See Molinier, _Les sources de l'histoire de France_, Vol. II, Paris, 1902, pp. 2, 3.

[494] Cantor, _Geschichte_, Vol. I, p. 762. A. Nagl in the _Abhandlungen zur Geschichte der Mathematik_, Vol. V, p. 85.

[495] 1030-1117.

[496] _Abhandlungen zur Geschichte der Mathematik_, Vol. V, pp. 85-133. The work begins "Incipit Liber Radulfi laudunensis de abaco."

[497] _Materialien zur Geschichte der arabischen Zahlzeichen in Frankreich_, loc. cit.

[498] Who died in 1202.

[499] Cantor, _Geschichte_, Vol. I (3), pp. 800-803; Boncompagni, _Trattati_, Part II. M. Steinschneider ("Die Mathematik bei den Juden,"

_Bibliotheca Mathematica_, Vol. X (2), p. 79) ingeniously derives another name by which he is called (Abendeuth) from Ibn Da[=u]d (Son of David). See also _Abhandlungen_, Vol. III, p. 110.

[500] John is said to have died in 1157.

[501] For it says, "Incipit prologus in libro algh.o.a.rismi de practica arismetrice. Qui editus est a magistro Johanne yspalensi." It is published in full in the second part of Boncompagni's _Trattati d'aritmetica_.

[502] Possibly, indeed, the meaning of "libro algh.o.a.rismi" is not "to Al-Khow[=a]razm[=i]'s book," but "to a book of algorism." John of Luna says of it: "Hoc idem est illud etiam quod ... alcorismus dicere videtur."

[_Trattati_, p. 68.]

[503] For a resume, see Cantor, Vol. I (3), pp. 800-803. As to the author, see Enestrom in the _Bibliotheca Mathematica_, Vol. VI (3), p. 114, and Vol. IX (3), p. 2.

[504] Born at Cremona (although some have a.s.serted at Carmona, in Andalusia) in 1114; died at Toledo in 1187. Cantor, loc. cit.; Boncompagni, _Atti d. R. Accad. d. n. Lincei_, 1851.

[505] See _Abhandlungen zur Geschichte der Mathematik_, Vol. XIV, p. 149; _Bibliotheca Mathematica_, Vol. IV (3), p. 206. Boncompagni had a fourteenth-century ma.n.u.script of his work, _Gerardi Cremonensis artis metrice practice_. See also T. L. Heath, _The Thirteen Books of Euclid's Elements_, 3 vols., Cambridge, 1908, Vol. I, pp. 92-94 ; A. A. Bjornbo, "Gerhard von Cremonas ubersetzung von Alkwarizmis Algebra und von Euklids Elementen," _Bibliotheca Mathematica_, Vol. VI (3), pp. 239-248.

[506] Wallis, _Algebra_, 1685, p. 12 seq.

[507] Cantor, _Geschichte_, Vol. I (3), p. 906; A. A. Bjornbo, "Al-Chw[=a]rizm[=i]'s trigonometriske Tavler," _Festskrift til H. G.

Zeuthen_, Copenhagen, 1909, pp. 1-17.

[508] Heath, loc. cit., pp. 93-96.

[509] M. Steinschneider, _Zeitschrift der deutschen morgenlandischen Gesellschaft_, Vol. XXV, 1871, p. 104, and _Zeitschrift fur Mathematik und Physik_, Vol. XVI, 1871, pp. 392-393; M. Curtze, _Centralblatt fur Bibliothekswesen_, 1899, p. 289; E. Wappler, _Zur Geschichte der deutschen Algebra im 15. Jahrhundert_, Programm, Zwickau, 1887; L. C. Karpinski, "Robert of Chester's Translation of the Algebra of Al-Khow[=a]razm[=i],"

_Bibliotheca Mathematica_, Vol. XI (3), p. 125. He is also known as Robertus Retinensis, or Robert of Reading.

[510] Nagl, A., "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und uber die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," in the _Zeitschrift fur Mathematik und Physik, Hist.-lit. Abth._, Vol. x.x.xIV, p. 129. Curtze, _Abhandlungen zur Geschichte der Mathematik_, Vol. VIII, pp. 1-27.

[511] See line _a_ in the plate on p. 143.

[512] _Sefer ha-Mispar, Das Buch der Zahl, ein hebraisch-arithmetisches Werk des R. Abraham ibn Esra_, Moritz Silberberg, Frankfurt a. M., 1895.

[513] Browning's "Rabbi ben Ezra."

[514] "Darum haben auch die Weisen Indiens all ihre Zahlen durch neun bezeichnet und Formen fur die 9 Ziffern gebildet." [_Sefer ha-Mispar_, loc.

cit., p. 2.]

[515] F. Bonaini, "Memoria unica sincrona di Leonardo Fibonacci," Pisa, 1858, republished in 1867, and appearing in the _Giornale Arcadico_, Vol.

CXCVII (N.S. LII); Gaetano Milanesi, _Doc.u.mento inedito e sconosciuto a Lionardo Fibonacci_, Roma, 1867; Guglielmini, _Elogio di Lionardo Pisano_, Bologna, 1812, p. 35; Libri, _Histoire des sciences mathematiques_, Vol.

II, p. 25; D. Martines, _Origine e progressi dell' aritmetica_, Messina, 1865, p. 47; Lucas, in Boncompagni _Bulletino_, Vol. X, pp. 129, 239; Besagne, ibid., Vol. IX, p. 583; Boncompagni, three works as cited in Chap.

I; G. Enestrom, "Ueber zwei angebliche mathematische Schulen im christlichen Mittelalter," _Bibliotheca Mathematica_, Vol. VIII (3), pp.

252-262; Boncompagni, "Della vita e delle opere di Leonardo Pisano," loc.

cit.

[516] The date is purely conjectural. See the _Bibliotheca Mathematica_, Vol. IV (3), p. 215.

[517] An old chronicle relates that in 1063 Pisa fought a great battle with the Saracens at Palermo, capturing six ships, one being "full of wondrous treasure," and this was devoted to building the cathedral.

[518] Heyd, loc. cit., Vol. I, p. 149.

[519] Ibid., p. 211.