[405] Cunningham, loc. cit., p. 81.

[406] Putnam, Books, Vol. I, p. 227:

"Non semel externas peregrino tramite terras Jam peragravit ovans, sophiae deductus amore, Si quid forte novi librorum seu studiorum Quod secum ferret, terris reperiret in illis. Hic quoque Romuleum venit devotus ad urbem."

("More than once he has traveled joyfully through remote regions and by strange roads, led on by his zeal for knowledge and seeking to discover in foreign lands novelties in books or in studies which he could take back with him. And this zealous student journeyed to the city of Romulus.")

[407] A. Neander, General History of the Christian Religion and Church, 5th American ed., Boston, 1855, Vol. III, p. 89, note 4; Libri, Histoire, Vol. I, p. 143.

[408] Cunningham, loc. cit., p. 81.

[409] Heyd, loc. cit., Vol. I, p. 4.

[410] Ibid., p. 5.

[411] Ibid., p. 21.

[412] Ibid., p. 23.

[413] Libri, Histoire, Vol. I, p. 167.

[414] Picavet, Gerbert, un pape philosophe, d'apres l'histoire et d'apres la legende, Paris, 1897, p. 19.

[415] Beazley, loc. cit., Vol. I, chap, i, and p. 54 seq.

[416] Ibid., p. 57.

[417] Libri, Histoire, Vol. I, p. 110, n., citing authorities, and p. 152.

[418] Possibly the old tradition, "Prima dedit nautis usum magnetis Amalphis," is true so far as it means the modern form of compass card. See Beazley, loc. cit., Vol. II, p. 398.

[419] R. C. Dutt, loc. cit., Vol. II, p. 312.

[420] E. J. Payne, in The Cambridge Modern History, London, 1902, Vol. I, chap. i.

[421] Geo. Phillips, "The Identity of Marco Polo's Zaitun with Changchau, in T'oung pao," Archives pour servir a l'etude de l'histoire de l'Asie orientale, Leyden, 1890, Vol. I, p. 218. W. Heyd, Geschichte des Levanthandels im Mittelalter, Vol. II, p. 216.

The Palazzo dei Poli, where Marco was born and died, still stands in the Corte del Milione, in Venice. The best description of the Polo travels, and of other travels of the later Middle Ages, is found in C. R. Beazley's Dawn of Modern Geography, Vol. III, chap, ii, and Part II.

[422] Heyd, loc. cit., Vol. II, p. 220; H. Yule, in Encyclopaedia Britannica, 9th (10th) or 11th ed., article "China." The handbook cited is Pegolotti's Libro di divisamenti di paesi, chapters i-ii, where it is implied that $60,000 would be a likely amount for a merchant going to China to invest in his trip.

[423] Cunningham, loc. cit., p. 194.

[424] I.e. a commission house.

[425] Cunningham, loc. cit., p. 186.

[426] J. R. Green, Short History of the English People, New York, 1890, p. 66.

[427] W. Besant, London, New York, 1892, p. 43.

[428] Baldakin, baldekin, baldachino.

[429] Italian Baldacco.

[430] J. K. Mumford, Oriental Rugs, New York, 1901, p. 18.

[431] Or Girbert, the Latin forms Gerbertus and Girbertus appearing indifferently in the documents of his time.

[432] See, for example, J. C. Heilbronner, Historia matheseos universae, p. 740.

[433] "Obscuro loco natum," as an old chronicle of Aurillac has it.

[434] N. Bubnov, Gerberti postea Silvestri II papae opera mathematica, Berlin, 1899, is the most complete and reliable source of information; Picavet, loc. cit., Gerbert etc.; Olleris, Oeuvres de Gerbert, Paris, 1867; Havet, Lettres de Gerbert, Paris, 1889 ; H. Weissenborn, Gerbert; Beitraege zur Kenntnis der Mathematik des Mittelalters, Berlin, 1888, and Zur Geschichte der Einfuehrung der jetzigen Ziffern in Europa durch Gerbert, Berlin, 1892; Buedinger, Ueber Gerberts wissenschaftliche und politische Stellung, Cassel, 1851; Richer, "Historiarum liber III," in Bubnov, loc. cit., pp. 376-381; Nagl, Gerbert und die Rechenkunst des 10. Jahrhunderts, Vienna, 1888.

[435] Richer tells of the visit to Aurillac by Borel, a Spanish nobleman, just as Gerbert was entering into young manhood. He relates how affectionately the abbot received him, asking if there were men in Spain well versed in the arts. Upon Borel's reply in the affirmative, the abbot asked that one of his young men might accompany him upon his return, that he might carry on his studies there.

[436] Vicus Ausona. Hatto also appears as Atton and Hatton.

[437] This is all that we know of his sojourn in Spain, and this comes from his pupil Richer. The stories told by Adhemar of Chabanois, an apparently ignorant and certainly untrustworthy contemporary, of his going to Cordova, are unsupported. (See e.g. Picavet, p. 34.) Nevertheless this testimony is still accepted: K. von Raumer, for example (Geschichte der Paedagogik, 6th ed., 1890, Vol. I, p. 6), says "Mathematik studierte man im Mittelalter bei den Arabern in Spanien. Zu ihnen gieng Gerbert, nachmaliger Pabst Sylvester II."

[438] Thus in a letter to Aldaberon he says: "Quos post repperimus speretis, id est VIII volumina Boeti de astrologia, praeclarissima quoque figurarum geometriae, aliaque non minus admiranda" (Epist. 8). Also in a letter to Rainard (Epist. 130), he says: "Ex tuis sumptibus fac ut michi scribantur M. Manlius (Manilius in one MS.) de astrologia."

[439] Picavet, loc. cit., p. 31.

[440] Picavet, loc. cit., p. 36.

[441] Havet, loc. cit., p. vii.

[442] Picavet, loc. cit., p. 37.

[443] "Con sinistre arti conseguri la dignita del Pontificato.... Lasciato poi l' abito, e 'l monasterio, e datosi tutto in potere del diavolo." [Quoted in Bombelli, L'antica numerazione Italica, Rome, 1876, p. 41 n.]

[444] He writes from Rheims in 984 to one Lupitus, in Barcelona, saying: "Itaque librum de astrologia translatum a te michi petenti dirige," presumably referring to some Arabic treatise. [Epist. no. 24 of the Havet collection, p. 19.]

[445] See Bubnov, loc. cit., p. x.

[446] Olleris, loc. cit., p. 361, l. 15, for Bernelinus; and Bubnov, loc. cit., p. 381, l. 4, for Richer.

[447] Woepcke found this in a Paris MS. of Radulph of Laon, c. 1100. [Propagation, p. 246.] "Et prima quidem trium spaciorum superductio unitatis caractere inscribitur, qui chaldeo nomine dicitur igin." See also Alfred Nagl, "Der arithmetische Tractat des Radulph von Laon" (Abhandlungen zur Geschichte der Mathematik, Vol. V, pp. 85-133), p. 97.

[448] Weissenborn, loc. cit., p. 239. When Olleris (Oeuvres de Gerbert, Paris, 1867, p. cci) says, "C'est a lui et non point aux Arabes, que l'Europe doit son systeme et ses signes de numeration," he exaggerates, since the evidence is all against his knowing the place value. Friedlein emphasizes this in the Zeitschrift fuer Mathematik und Physik, Vol. XII (1867), Literaturzeitung, p. 70: "Fuer das System unserer Numeration ist die Null das wesentlichste Merkmal, und diese kannte Gerbert nicht. Er selbst schrieb alle Zahlen mit den roemischen Zahlzeichen und man kann ihm also nicht verdanken, was er selbst nicht kannte."

[449] E.g., Chasles, Buedinger, Gerhardt, and Richer. So Martin (Recherches nouvelles etc.) believes that Gerbert received them from Boethius or his followers. See Woepcke, Propagation, p. 41.

[450] Buedinger, loc. cit., p. 10. Nevertheless, in Gerbert's time one Al-Manṣūr, governing Spain under the name of Hishām (976-1002), called from the Orient Al-Beġānī to teach his son, so that scholars were recognized. [Picavet, p. 36.]

[451] Weissenborn, loc. cit., p. 235.

[452] Ibid., p. 234.

[453] These letters, of the period 983-997, were edited by Havet, loc. cit., and, less completely, by Olleris, loc. cit. Those touching mathematical topics were edited by Bubnov, loc. cit., pp. 98-106.

[454] He published it in the Monumenta Germaniae historica, "Scriptores," Vol. III, and at least three other editions have since appeared, viz. those by Guadet in 1845, by Poinsignon in 1855, and by Waitz in 1877.

[455] Domino ac beatissimo Patri Gerberto, Remorum archiepiscopo, Richerus Monchus, Gallorum congressibus in volumine regerendis, imperii tui, pater sanctissime Gerberte, auctoritas seminarium dedit.

[456] In epistle 17 (Havet collection) he speaks of the "De multiplicatione et divisione numerorum libellum a Joseph Ispano editum abbas Warnerius" (a person otherwise unknown). In epistle 25 he says: "De multiplicatione et divisione numerorum, Joseph Sapiens sententias quasdam edidit."

[457] H. Suter, "Zur Frage ueber den Josephus Sapiens," Bibliotheca Mathematica, Vol. VIII (2), p. 84; Weissenborn, Einfuehrung, p. 14; also his Gerbert; M. Steinschneider, in Bibliotheca Mathematica, 1893, p. 68. Wallis (Algebra, 1685, chap. 14) went over the list of Spanish Josephs very carefully, but could find nothing save that "Josephus Hispanus seu Josephus sapiens videtur aut Maurus fuisse aut alius quis in Hispania."

[458] P. Ewald, Mittheilungen, Neues Archiv d. Gesellschaft fuer aeltere deutsche Geschichtskunde, Vol. VIII, 1883, pp. 354-364. One of the manuscripts is of 976 A.D. and the other of 992 A.D. See also Franz Steffens, Lateinische Palaeographie, Freiburg (Schweiz), 1903, pp. xxxix-xl. The forms are reproduced in the plate on page 140.

[459] It is entitled Constantino suo Gerbertus scolasticus, because it was addressed to Constantine, a monk of the Abbey of Fleury. The text of the letter to Constantine, preceding the treatise on the Abacus, is given in the Comptes rendus, Vol. XVI (1843), p. 295. This book seems to have been written c. 980 A.D. [Bubnov, loc. cit., p. 6.]

[460] "Histoire de l'Arithmetique," Comptes rendus, Vol. XVI (1843), pp. 156, 281.

[461] Loc. cit., Gerberti Opera etc.

[462] Friedlein thought it spurious. See Zeitschrift fuer Mathematik und Physik, Vol. XII (1867), Hist.-lit. suppl., p. 74. It was discovered in the library of the Benedictine monastry of St. Peter, at Salzburg, and was published by Peter Bernhard Pez in 1721. Doubt was first cast upon it in the Olleris edition (Oeuvres de Gerbert). See Weissenborn, Gerbert, pp. 2, 6, 168, and Picavet, p. 81. Hock, Cantor, and Th. Martin place the composition of the work at c. 996 when Gerbert was in Germany, while Olleris and Picavet refer it to the period when he was at Rheims.

[463] Picavet, loc. cit., p. 182.

[464] Who wrote after Gerbert became pope, for he uses, in his preface, the words, "a domino pape Gerberto." He was quite certainly not later than the eleventh century; we do not have exact information about the time in which he lived.

[465] Picavet, loc. cit., p. 182. Weissenborn, Gerbert, p. 227. In Olleris, Liber Abaci (of Bernelinus), p. 361.

[466] Richer, in Bubnov, loc. cit., p. 381.

[467] Weissenborn, Gerbert, p. 241.

[468] Writers on numismatics are quite uncertain as to their use. See F. Gnecchi, Monete Romane, 2d ed., Milan, 1900, cap. XXXVII. For pictures of old Greek tesserae of Sarmatia, see S. Ambrosoli, Monete Greche, Milan, 1899, p. 202.

[469] Thus Tzwivel's arithmetic of 1507, fol. 2, v., speaks of the ten figures as "characteres sive numerorum apices a diuo Seuerino Boetio."

[470] Weissenborn uses sipos for 0. It is not given by Bernelinus, and appears in Radulph of Laon, in the twelfth century. See Guenther's Geschichte, p. 98, n.; Weissenborn, p. 11; Pihan, Expose etc., pp. xvi-xxii.

In Friedlein's Boetius, p. 396, the plate shows that all of the six important manuscripts from which the illustrations are taken contain the symbol, while four out of five which give the words use the word sipos for 0. The names appear in a twelfth-century anonymous manuscript in the Vatican, in a passage beginning

Ordine primigeno sibi nomen possidet igin. Andras ecce locum mox uendicat ipse secundum Ormis post numeros incompositus sibi primus.

[Boncompagni Buttetino, XV, p. 132.] Turchill (twelfth century) gives the names Igin, andras, hormis, arbas, quimas, caletis, zenis, temenias, celentis, saying: "Has autem figuras, ut donnus [dominus] Gvillelmus Rx testatur, a pytagoricis habemus, nomina uero ab arabibus." (Who the William R. was is not known. Boncompagni Bulletino XV, p. 136.) Radulph of Laon (d. 1131) asserted that they were Chaldean (Propagation, p. 48 n.). A discussion of the whole question is also given in E. C. Bayley, loc. cit. Huet, writing in 1679, asserted that they were of Semitic origin, as did Nesselmann in spite of his despair over ormis, calctis, and celentis; see Woepcke, Propagation, p. 48. The names were used as late as the fifteenth century, without the zero, but with the superscript dot for 10's, two dots for 100's, etc., as among the early Arabs. Gerhardt mentions having seen a fourteenth or fifteenth century manuscript in the Bibliotheca Amploniana with the names "Ingnin, andras, armis, arbas, quinas, calctis, zencis, zemenias, zcelentis," and the statement "Si unum punctum super ingnin ponitur, X significat.... Si duo puncta super ... figuras superponunter, fiet decuplim illius quod cum uno puncto significabatur," in Monatsberichte der K. P. Akad. d. Wiss., Berlin, 1867, p. 40.

[471] A chart of ten numerals in 200 tongues, by Rev. R. Patrick, London, 1812.

[472] "Numeratio figuralis est cuiusuis numeri per notas, et figuras numerates descriptio." [Clichtoveus, edition of c. 1507, fol. C ii, v.] "Aristoteles enim uoces rerum [Greek: sumbola] uocat: id translatum, sonat notas." [Noviomagus, De Numeris Libri II, cap. vi.] "Alphabetum decem notarum." [Schonerus, notes to Ramus, 1586, p. 3 seq.] Richer says: "novem numero notas omnem numerum significantes." [Bubnov, loc. cit., p. 381.]

[473] "Il y a dix Characteres, autrement Figures, Notes, ou Elements." [Peletier, edition of 1607, p. 13.] "Numerorum notas alij figuras, alij signa, alij characteres uocant." [Glareanus, 1545 edition, f. 9, r.] "Per figuras (quas zyphras uocant) assignationem, quales sunt hae notulae, 1. 2. 3. 4...." [Noviomagus, De Numeris Libri II, cap. vi.] Gemma Frisius also uses elementa and Cardan uses literae. In the first arithmetic by an American (Greenwood, 1729) the author speaks of "a few Arabian Charecters or Numeral Figures, called Digits" (p. 1), and as late as 1790, in the third edition of J. J. Blassiere's arithmetic (1st ed. 1769), the name characters is still in use, both for "de Latynsche en de Arabische" (p. 4), as is also the term "Cyfferletters" (p. 6, n.). Ziffer, the modern German form of cipher, was commonly used to designate any of the nine figures, as by Boeschenstein and Riese, although others, like Koebel, used it only for the zero. So zifre appears in the arithmetic by Borgo, 1550 ed. In a Munich codex of the twelfth century, attributed to Gerland, they are called characters only: "Usque ad VIIII. enim porrigitur omnis numerus et qui supercrescit eisdem designator Karacteribus." [Boncompagni Bulletino, Vol. X. p. 607.]

[474] The title of his work is Prologus N. Ocreati in Helceph (Arabic al-qeif, investigation or memoir) ad Adelardum Batensem magistrum suum. The work was made known by C. Henry, in the Zeitschrift fuer Mathematik und Physik, Vol. XXV, p. 129, and in the Abhandlungen zur Geschichte der Mathematik, Vol. III; Weissenborn, Gerbert, p. 188.

[475] The zero is indicated by a vacant column.

[476] Leo Jordan, loc. cit., p. 170. "Chifre en augorisme" is the expression used, while a century later "giffre en argorisme" and "cyffres d'augorisme" are similarly used.

[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 fuer 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 fuer 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 "Analyse 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 intitolato '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 Daūd (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 alghoarismi 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 alghoarismi" is not "to Al-Khowārazmī'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 Enestroem in the Bibliotheca Mathematica, Vol. VI (3), p. 114, and Vol. IX (3), p. 2.

[504] Born at Cremona (although some have asserted 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 manuscript 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. Bjoernbo, "Gerhard von Cremonas Uebersetzung 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. Bjoernbo, "Al-Chwārizmī'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 morgenlaendischen Gesellschaft, Vol. XXV, 1871, p. 104, and Zeitschrift fuer Mathematik und Physik, Vol. XVI, 1871, pp. 392-393; M. Curtze, Centralblatt fuer 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-Khowārazmī," 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 ueber die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," in the Zeitschrift fuer Mathematik und Physik, Hist.-lit. Abth., Vol. XXXIV, 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 hebraeisch-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 fuer 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, Documento 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. Enestroem, "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.

[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 petit marchand de Pise qui donna l'algebre aux Chretiens." [Libri, Histoire, Vol. I, p. xvi.]

[523] A document 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 Universitaeten des Mittelalters, Zuerich, 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. Guenther, 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 passed 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 Encyclopaedie der Erziehung, Vol. VI, p. 726; A. Kuckuk, "Die Rechenkunst im sechzehnten Jahrhundert," Festschrift zur dritten Saecularfeier 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 Manuscripts in the Oxford Library, 1845.

[542] Maximilian Curtze, Petri Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco commentarius, una cum Algorismo ipso, Copenhagen, 1897; L. C. Karpinski, "Jordanus Nemorarius and John of Halifax," American Mathematical Monthly, Vol. XVII, pp. 108-113.

[543] J. Aschbach, Geschichte der Wiener Universitaet 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 manuscripts 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 manuscript omits the words "sive iudaico."

[548] "Non enim omnis numerus per quascumque 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 illustration 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 multitudinem, 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 cum 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 fuer Mathematik und Physik, Hist.-lit. Abth., Vol. XXXIV, p. 143; Smith, Rara Arithmetica, p. 14, in facsimile.]

[556] P. Ewald, loc. cit.; Franz Steffens, Lateinische Palaeographie, pp. xxxix-xl. We are indebted to Professor J. M. Burnam for a photograph of this rare manuscript.

[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 restitution de travaux perdus d'Apollonius sur les quantites irrationnelles, d'apres des indications tirees d'un manuscrit 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 dni nri ihu xpi. 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 Muenzkunde, Graetz, 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.]

[567] Ibid., p. 79.

[568] Liber de Remediis utriusque fortunae Coloniae.

[569] Fr. Walthern et Hans Hurning, Noerdlingen.

[570] Ars Memorandi, one of the oldest European block-books.

[571] Eusebius Caesariensis, De praeparatione evangelica, Venice, Jenson, 1470. The above statement holds for copies in the Astor Library and in the Harvard University Library.

[572] Francisco de Retza, Comestorium vitiorum, Nuernberg, 1470. The copy referred to is in the Astor Library.

[573] See Mauch, "Ueber den Gebrauch arabischer Ziffern und die Veraenderungen derselben," Anzeiger fuer Kunde der deutschen Vorzeit, 1861, columns 46, 81, 116, 151, 189, 229, and 268; Calmet, Recherches sur l'origine des chiffres d'arithmetique, plate, loc. cit.

[574] Guenther, Geschichte, p. 175, n.; Mauch, loc. cit.

[575] These are given by W. R. Lethaby, from drawings by J. T. Irvine, in the Proceedings of the Society of Antiquaries, 1906, p. 200.

[576] There are some ill-tabulated forms to be found in J. Bowring, The Decimal System, London, 1854, pp. 23, 25, and in L. A. Chassant, Dictionnaire des abreviations latines et francaises ... du moyen age, Paris, MDCCCLXVI, p. 113. The best sources we have at present, aside from the Hill monograph, are P. Treutlein, Geschichte unserer Zahlzeichen, Karlsruhe, 1875; Cantor's Geschichte, Vol. I, table; M. Prou, Manuel de paleographie latine et francaise, 2d ed., Paris, 1892, p. 164; A. Cappelli, Dizionario di abbreviature latine ed italiane, Milan, 1899. An interesting early source is found in the rare Caxton work of 1480, The Myrrour of the World. In Chap. X is a cut with the various numerals, the chapter beginning "The fourth scyence is called arsmetrique." Two of the fifteen extant copies of this work are at present in the library of Mr. J. P. Morgan, in New York.

[577] From the twelfth-century manuscript on arithmetic, Curtze, loc. cit., Abhandlungen, and Nagl, loc. cit. The forms are copied from Plate VII in Zeitschrift fuer Mathematik und Physik, Vol. XXXIV.

[578] From the Regensburg chronicle. Plate containing some of these numerals in Monumenta Germaniae historica, "Scriptores" Vol. XVII, plate to p. 184; Wattenbach, Anleitung zur lateinischen Palaeographie, Leipzig, 1886, p. 102; Boehmer, Fontes rerum Germanicarum, Vol. III, Stuttgart, 1852, p. lxv.

[579] French Algorismus of 1275; from an unpublished photograph of the original, in the possession of D. E. Smith. See also p. 135.

[580] From a manuscript of Boethius c. 1294, in Mr. Plimpton's library. Smith, Rara Arithmetica, Plate I.

[581] Numerals in a 1303 manuscript in Sigmaringen, copied from Wattenbach, loc. cit., p. 102.

[582] From a manuscript, Add. Manuscript 27,589, British Museum, 1360 A.D. The work is a computus in which the date 1360 appears, assigned in the British Museum catalogue to the thirteenth century.

[583] From the copy of Sacrabosco's Algorismus in Mr. Plimpton's library. Date c. 1442. See Smith, Rara Arithmetica, p. 450.

[584] See Rara Arithmetica, pp. 446-447.

[585] Ibid., pp. 469-470.

[586] Ibid., pp. 477-478.

[587] The i is used for "one" in the Treviso arithmetic (1478), Clichtoveus (c. 1507 ed., where both i and j are so used), Chiarini (1481), Sacrobosco (1488 ed.), and Tzwivel (1507 ed., where jj and jz are used for 11 and 12). This was not universal, however, for the Algorithmus linealis of c. 1488 has a special type for 1. In a student's notebook of lectures taken at the University of Wuerzburg in 1660, in Mr. Plimpton's library, the ones are all in the form of i.

[588] Thus the date [Numerals 1580], for 1580, appears in a MS. in the Laurentian library at Florence. The second and the following five characters are taken from Cappelli's Dizionario, p. 380, and are from manuscripts of the twelfth, thirteenth, fourteenth, sixteenth, seventeenth, and eighteenth centuries, respectively.

[589] E.g. Chiarini's work of 1481; Clichtoveus (c. 1507).

[590] The first is from an algorismus of the thirteenth century, in the Hannover Library. [See Gerhardt, "Ueber die Entstehung und Ausbreitung des dekadischen Zahlensystems," loc. cit., p. 28.] The second character is from a French algorismus, c. 1275. [Boncompagni Bulletino, Vol. XV, p. 51.] The third and the following sixteen characters are given by Cappelli, loc. cit., and are from manuscripts of the twelfth (1), thirteenth (2), fourteenth (7), fifteenth (3), sixteenth (1), seventeenth (2), and eighteenth (1) centuries, respectively.

[591] Thus Chiarini (1481) has [Symbol] for 23.

[592] The first of these is from a French algorismus, c. 1275. The second and the following eight characters are given by Cappelli, loc. cit., and are from manuscripts of the twelfth (2), thirteenth, fourteenth, fifteenth (3), seventeenth, and eighteenth centuries, respectively.

[593] See Nagl, loc. cit.

[594] Hannover algorismus, thirteenth century.

[595] See the Dagomari manuscript, in Rara Arithmetica, pp. 435, 437-440.

[596] But in the woodcuts of the Margarita Philosophica (1503) the old forms are used, although the new ones appear in the text. In Caxton's Myrrour of the World (1480) the old form is used.

[597] Cappelli, loc. cit. They are partly from manuscripts of the tenth, twelfth, thirteenth (3), fourteenth (7), fifteenth (6), and eighteenth centuries, respectively. Those in the third line are from Chassant's Dictionnaire, p. 113, without mention of dates.

[598] The first is from the Hannover algorismus, thirteenth century. The second is taken from the Rollandus manuscript, 1424. The others in the first two lines are from Cappelli, twelfth (3), fourteenth (6), fifteenth (13) centuries, respectively. The third line is from Chassant, loc. cit., p. 113, no mention of dates.

[599] The first of these forms is from the Hannover algorismus, thirteenth century. The following are from Cappelli, fourteenth (3), fifteenth, sixteenth (2), and eighteenth centuries, respectively.

[600] The first of these is taken from the Hannover algorismus, thirteenth century. The following forms are from Cappelli, twelfth, thirteenth, fourteenth (5), fifteenth (2), seventeenth, and eighteenth centuries, respectively.

[601] All of these are given by Cappelli, thirteenth, fourteenth, fifteenth (2), and sixteenth centuries, respectively.

[602] Smith, Rara Arithmetica, p. 489. This is also seen in several of the Plimpton manuscripts, as in one written at Ancona in 1684. See also Cappelli, loc. cit.

[603] French algorismus, c. 1275, for the first of these forms. Cappelli, thirteenth, fourteenth, fifteenth (3), and seventeenth centuries, respectively. The last three are taken from Byzantinische Analekten, J. L. Heiberg, being forms of the fifteenth century, but not at all common. [Symbol: Qoppa] was the old Greek symbol for 90.

[604] For the first of these the reader is referred to the forms ascribed to Boethius, in the illustration on p. 88; for the second, to Radulph of Laon, see p. 60. The third is used occasionally in the Rollandus (1424) manuscript, in Mr. Plimpton's library. The remaining three are from Cappelli, fourteenth (2) and seventeenth centuries.

[605] Smith, An Early English Algorism.

[606] Kuckuck, p. 5.

[607] A. Cappelli, loc. cit., p. 372.

[608] Smith, Rara Arithmetica, p. 443.

[609] Curtze, Petri Philomeni de Dacia etc., p. IX.

[610] Cappelli, loc. cit., p. 376.

[611] Curtze, loc. cit., pp. VIII-IX, note.

[612] Edition of 1544-1545, f. 52.

[613] De numeris libri II, 1544 ed., cap. XV. Heilbronner, loc. cit., p. 736, also gives them, and compares this with other systems.

[614] Noviomagus says of them: "De quibusdam Astrologicis, sive Chaldaicis numerorum notis.... Sunt & aliae quaedam notae, quibus Chaldaei & Astrologii quemlibet numerum artificiose & argute describunt, scitu periucundae, quas nobis communicauit Rodolphus Paludanus Nouiomagus."

THE END