[520] J. A. Symonds, _Renaissance in Italy. The Age of Despots._ New York, 1883, p. 62.

[521] Symonds, loc. cit., p. 79.

[522] J. A. Froude, _The Science of History_, London, 1864. "Un brevet d'apothicaire n'empecha pas Dante d'etre le plus grand poete de l'Italie, et ce fut un pet.i.t marchand de Pise qui donna l'algebre aux Chretiens."

[Libri, _Histoire_, Vol. I, p. xvi.]

[523] A doc.u.ment of 1226, found and published in 1858, reads: "Leonardo bigollo quondam Guilielmi."

[524] "Bonaccingo germano suo."

[525] E.g. Libri, Guglielmini, Tiraboschi.

[526] Latin, _Bonaccius_.

[527] Boncompagni and Milanesi.

[528] Reprint, p. 5.

[529] Whence the French name for candle.

[530] Now part of Algiers.

[531] E. Reclus, _Africa_, New York, 1893, Vol. II, p. 253.

[532] "Sed hoc totum et algorismum atque arcus pictagore quasi errorem computavi respectu modi indorum." Woepcke, _Propagation_ etc., regards this as referring to two different systems, but the expression may very well mean algorism as performed upon the Pythagorean arcs (or table).

[533] "Book of the Abacus," this term then being used, and long afterwards in Italy, to mean merely the arithmetic of computation.

[534] "Incipit liber Abaci a Leonardo filio Bonacci compositus anno 1202 et correctus ab eodem anno 1228." Three MSS. of the thirteenth century are known, viz. at Milan, at Siena, and in the Vatican library. The work was first printed by Boncompagni in 1857.

[535] I.e. in relation to the quadrivium. "Non legant in festivis diebus, nisi Philosophos et rhetoricas et quadrivalia et barbarismum et ethicam, si placet." Suter, _Die Mathematik auf den Universitaten des Mittelalters_, Zurich, 1887, p. 56. Roger Bacon gives a still more gloomy view of Oxford in his time in his _Opus minus_, in the _Rerum Britannicarum medii aevi scriptores_, London, 1859, Vol. I, p. 327. For a picture of Cambridge at this time consult F. W. Newman, _The English Universities, translated from the German of V. A. Huber_, London, 1843, Vol. I, p. 61; W. W. R. Ball, _History of Mathematics at Cambridge_, 1889; S. Gunther, _Geschichte des mathematischen Unterrichts im deutschen Mittelalter bis zum Jahre 1525_, Berlin, 1887, being Vol. III of _Monumenta Germaniae paedagogica_.

[536] On the commercial activity of the period, it is known that bills of exchange pa.s.sed between Messina and Constantinople in 1161, and that a bank was founded at Venice in 1170, the Bank of San Marco being established in the following year. The activity of Pisa was very manifest at this time.

Heyd, loc. cit., Vol. II, p. 5; V. Casagrandi, _Storia e cronologia_, 3d ed., Milan, 1901, p. 56.

[537] J. A. Symonds, loc. cit., Vol. II, p. 127.

[538] I. Taylor, _The Alphabet_, London, 1883, Vol. II, p. 263.

[539] Cited by Unger's History, p. 15. The Arabic numerals appear in a Regensburg chronicle of 1167 and in Silesia in 1340. See Schmidt's _Encyclopadie der Erziehung_, Vol. VI, p. 726; A. Kuckuk, "Die Rechenkunst im sechzehnten Jahrhundert," _Festschrift zur dritten Sacularfeier des Berlinischen Gymnasiums zum grauen Kloster_, Berlin, 1874, p. 4.

[540] The text is given in Halliwell, _Rara Mathematica_, London, 1839.

[541] Seven are given in Ashmole's _Catalogue of Ma.n.u.scripts in the Oxford Library_, 1845.

[542] Maximilian Curtze, _Petri Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco commentarius, una c.u.m Algorismo ipso_, Copenhagen, 1897; L. C. Karpinski, "Jorda.n.u.s Nemorarius and John of Halifax," _American Mathematical Monthly_, Vol. XVII, pp. 108-113.

[543] J. Aschbach, _Geschichte der Wiener Universitat im ersten Jahrhunderte ihres Bestehens_, Wien, 1865, p. 93.

[544] Curtze, loc. cit., gives the text.

[545] Curtze, loc. cit., found some forty-five copies of the _Algorismus_ in three libraries of Munich, Venice, and Erfurt (Amploniana). Examination of two ma.n.u.scripts from the Plimpton collection and the Columbia library shows such marked divergence from each other and from the text published by Curtze that the conclusion seems legitimate that these were students'

lecture notes. The shorthand character of the writing further confirms this view, as it shows that they were written largely for the personal use of the writers.

[546] "Quidam philosophus edidit nomine Algus, unde et Algorismus nuncupatur." [Curtze, loc. cit., p. 1.]

[547] "Sinistrorsum autera scribimus in hac arte more arabico sive iudaico, huius scientiae inventorum." [Curtze, loc. cit., p. 7.] The Plimpton ma.n.u.script omits the words "sive iudaico."

[548] "Non enim omnis numerus per quasc.u.mque figuras Indorum repraesentatur, sed tantum determinatus per determinatam, ut 4 non per 5,..." [Curtze, loc. cit., p. 25.]

[549] C. Henry, "Sur les deux plus anciens traites francais d'algorisme et de geometrie," Boncompagni _Bulletino_, Vol. XV, p. 49; Victor Mortet, "Le plus ancien traite francais d'algorisme," loc. cit.

[550] _L'etat des sciences en France, depute la mort du Roy Robert, arrivee en 1031, jusqu'a celle de Philippe le Bel, arrivee en 1314_, Paris, 1741.

[551] _Discours sur l'etat des lettres en France au XIII^e siecle_, Paris, 1824.

[552] _Apercu historique_, Paris, 1876 ed., p. 464.

[553] Ranulf Higden, a native of the west of England, entered St.

Werburgh's monastery at Chester in 1299. He was a Benedictine monk and chronicler, and died in 1364. His _Polychronicon_, a history in seven books, was printed by Caxton in 1480.

[554] Trevisa's translation, Higden having written in Latin.

[555] An ill.u.s.tration of this feeling is seen in the writings of Prosdocimo de' Beldomandi (b. c. 1370-1380, d. 1428): "Inveni in quam pluribus libris algorismi nuncupatis mores circa numeros operandi satis varios atque diversos, qui licet boni existerent atque veri erant, tamen fastidiosi, tum propter ipsarum regularum mult.i.tudinem, tum propter earum deleationes, tum etiam propter ipsarum operationum probationes, utrum si bone fuerint vel ne. Erant et etiam isti modi interim fastidiosi, quod si in aliquo calculo astroloico error contigisset, calculatorem operationem suam a capite incipere oportebat, dato quod error suus adhuc satis propinquus existeret; et hoc propter figuras in sua operatione deletas. Indigebat etiam calculator semper aliquo lapide vel sibi conformi, super quo scribere atque faciliter delere posset figuras c.u.m quibus operabatur in calculo suo. Et quia haec omnia satis fastidiosa atque laboriosa mihi visa sunt, disposui libellum edere in quo omnia ista abicerentur: qui etiam algorismus sive liber de numeris denominari poterit. Scias tamen quod in hoc libello ponere non intendo nisi ea quae ad calculum necessaria sunt, alia quae in aliis libris practice arismetrice tanguntur, ad calculum non necessaria, propter brevitatem dimitendo." [Quoted by A. Nagl, _Zeitschrift fur Mathematik und Physik, Hist.-lit. Abth._, Vol. x.x.xIV, p. 143; Smith, _Rara Arithmetica_, p. 14, in facsimile.]

[556] P. Ewald, loc. cit.; Franz Steffens, _Lateinische Palaographie_, pp.

x.x.xix-xl. We are indebted to Professor J. M. Burnam for a photograph of this rare ma.n.u.script.

[557] See the plate of forms on p. 88.

[558] Karabacek, loc. cit., p. 56; Karpinski, "Hindu Numerals in the Fihrist," _Bibliotheca Mathematica_, Vol. XI (3), p. 121.

[559] Woepcke, "Sur une donnee historique," etc., loc. cit., and "Essai d'une rest.i.tution de travaux perdus d'Apollonius sur les quant.i.tes irrationnelles, d'apres des indications tirees d'un ma.n.u.scrit arabe," _Tome XIV des Memoires presentes par divers savants a l'Academie des sciences_, Paris, 1856, note, pp. 6-14.

[560] _Archeological Report of the Egypt Exploration Fund for 1908-1909_, London, 1910, p. 18.

[561] There was a set of astronomical tables in Boncompagni's library bearing this date: "Nota quod anno d[=n]i [=n]ri ihu x[=p]i. 1264.

perfecto." See Narducci's _Catalogo_, p. 130.

[562] "On the Early use of Arabic Numerals in Europe," read before the Society of Antiquaries April 14, 1910, and published in _Archaeologia_ in the same year.

[563] Ibid., p. 8, n. The date is part of an Arabic inscription.

[564] O. Codrington, _A Manual of Musalman Numismatics_, London, 1904.

[565] See Arbuthnot, _The Mysteries of Chronology_, London, 1900, pp. 75, 78, 98; F. Pichler, _Repertorium der steierischen Munzkunde_, Gratz, 1875, where the claim is made of an Austrian coin of 1458; _Bibliotheca Mathematica_, Vol. X (2), p. 120, and Vol. XII (2), p. 120. There is a Brabant piece of 1478 in the collection of D. E. Smith.

[566] A specimen is in the British Museum. [Arbuthnot, p. 79.]