Ordine primigeno sibi nomen possidet igin.Andras ecce locum mox uendicat ipse secundumOrmis post numeros incompositus sibi primus.[BoncompagniButtetino, 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. BoncompagniBulletinoXV, 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," inMonatsberichte 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 rerumσύμβολαuocat: 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 hæ notulæ, 1. 2. 3. 4...." [Noviomagus,De Numeris Libri II, cap. vi.] Gemma Frisius also useselementaand Cardan usesliterae. In the first arithmetic by an American (Greenwood, 1729) the author speaks of "a few ArabianCharectersor Numeral Figures, calledDigits" (p. 1), and as late as 1790, in the third edition of J. J. Blassière's arithmetic (1st ed. 1769), the namecharactersis 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 Köbel, used it only for the zero. Sozifreappears in the arithmetic by Borgo, 1550 ed. In a Munich codex of the twelfth century, attributed to Gerland, they are calledcharactersonly: "Usque ad VIIII. enim porrigitur omnis numerus et qui supercrescit eisdem designator Karacteribus." [BoncompagniBulletino, Vol. X. p. 607.][474]The title of his work isPrologus N. Ocreati in Helceph(Arabical-qeif, investigation or memoir)ad Adelardum Batensem magistrum suum. The work was made known by C. Henry, in theZeitschrift für Mathematik und Physik, Vol. XXV, p. 129, and in theAbhandlungen 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, ofThe Testament of Love, printed with Chaucer's Works, loc. cit., Vol. VII, London, 1897.[480]Liber Abacci, published in Olleris,Œuvres 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 für 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 für Mathematik und Physik, Vol. IX; Bubnov, loc. cit., pp. 197-245; M. Chasles, "Histoire de l'arithmétique. Recherches des traces du système de l'abacus, après que cette méthode a pris le nom d'Algorisme.—Preuves qu'à toutes les époques, jusq'auXVIesiècle, on a su que l'arithmétique vulgaire avait pour origine cette méthode ancienne,"Comptes rendus, Vol. XVII, pp. 143-154, also "Règles de l'abacus,"Comptes rendus, Vol. XVI, pp. 218-246, and "Analyse et explication du traité 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 hisBulletino, 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," BoncompagniBulletino, 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 theAbhandlungen 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 alsoAbhandlungen, 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'sTrattati 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 résumé, see Cantor, Vol. I (3), pp. 800-803. As to the author, see Eneström in theBibliotheca 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]SeeAbhandlungen 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. Björnbo, "Gerhard von Cremonas Übersetzung 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. Björnbo, "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 morgenländischen Gesellschaft, Vol. XXV, 1871, p. 104, andZeitschrift für Mathematik und Physik, Vol. XVI, 1871, pp. 392-393; M. Curtze,Centralblatt für 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 über die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," in theZeitschrift für Mathematik und Physik, Hist.-lit. Abth., Vol. XXXIV, p. 129. Curtze,Abhandlungen zur Geschichte der Mathematik, Vol. VIII, pp. 1-27.[511]See lineain the plate on p.143.[512]Sefer ha-Mispar, Das Buch der Zahl, ein hebräisch-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 für 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 theGiornale 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 mathématiques, Vol. II, p. 25; D. Martines,Origine e progressi dell' aritmetica, Messina, 1865, p. 47; Lucas, in BoncompagniBulletino, Vol. X, pp. 129, 239; Besagne, ibid., Vol. IX, p. 583; Boncompagni, three works as cited in Chap. I; G. Eneström, "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 theBibliotheca 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'empêcha pas Dante d'être le plus grand poète de l'Italie, et ce fut un petit marchand de Pise qui donna l'algèbre aux Chrétiens." [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,Propagationetc., 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 Universitäten des Mittelalters, Zürich, 1887, p. 56. Roger Bacon gives a still more gloomy view of Oxford in his time in hisOpus minus, in theRerum 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. Günther,Geschichte des mathematischen Unterrichts im deutschen Mittelalter bis zum Jahre 1525, Berlin, 1887, being Vol. III ofMonumenta 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'sEncyclopädie der Erziehung, Vol. VI, p. 726; A. Kuckuk, "Die Rechenkunst im sechzehnten Jahrhundert,"Festschrift zur dritten Säcularfeier 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'sCatalogue 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 Universität 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 theAlgorismusin 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 traités français d'algorisme et de géométrie," BoncompagniBulletino, Vol. XV, p. 49; Victor Mortet, "Le plus ancien traité français d'algorisme," loc. cit.[550]L'État des sciences en France, depute la mort du Roy Robert, arrivée en 1031, jusqu'à celle de Philippe le Bel, arrivée en 1314, Paris, 1741.[551]Discours sur l'état des lettres en France au XIIIesiecle, Paris, 1824.[552]Aperçu 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. HisPolychronicon, 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 für 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 Paläographie, 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 donnée historique," etc., loc. cit., and "Essai d'une restitution de travaux perdus d'Apollonius sur les quantités irrationnelles, d'après des indications tirées d'un manuscrit arabe,"Tome XIV des Mémoires présentés par divers savants à l'Académie 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 dninri ihû xpi. 1264. perfecto." See Narducci'sCatalogo, p. 130.[562]"On the Early use of Arabic Numerals in Europe," read before the Society of Antiquaries April 14, 1910, and published inArchæologiain 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 Münzkunde, Grätz, 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, Nördlingen.[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, Nürnberg, 1470. The copy referred to is in the Astor Library.[573]See Mauch, "Ueber den Gebrauch arabischer Ziffern und die Veränderungen derselben,"Anzeiger für Kunde der deutschen Vorzeit, 1861, columns 46, 81, 116, 151, 189, 229, and 268; Calmet,Recherches sur l'origine des chiffres d'arithmétique, plate, loc. cit.[574]Günther,Geschichte, p. 175, n.; Mauch, loc. cit.[575]These are given by W. R. Lethaby, from drawings by J. T. Irvine, in theProceedings 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 abréviations latines et françaises ... du moyen âge, 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'sGeschichte, Vol. I, table; M. Prou,Manuel de paléographie latine et française, 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 inZeitschrift für Mathematik und Physik, Vol. XXXIV.[578]From the Regensburg chronicle. Plate containing some of these numerals inMonumenta 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, 1360A.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'sAlgorismusin Mr. Plimpton's library. Date c. 1442. See Smith,Rara Arithmetica, p. 450.[584]SeeRara 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 theAlgorithmus linealisof c. 1488 has a special type for 1. In a student's notebook of lectures taken at the University of Würzburg in 1660, in Mr. Plimpton's library, the ones are all in the form of i.[588]Thus the dateNumerals 1580, for 1580, appears in a MS. in the Laurentian library at Florence. The second and the following five characters are taken from Cappelli'sDizionario, 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. [BoncompagniBulletino, 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) hasSymbolfor 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, inRara Arithmetica, pp. 435, 437-440.[596]But in the woodcuts of theMargarita Philosophica(1503) the old forms are used, although the new ones appear in the text. In Caxton'sMyrrour 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'sDictionnaire, 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 fromByzantinische Analekten, J. L. Heiberg, being forms of the fifteenth century, but not at all common.Symbol: Qoppawas 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 Daciaetc., 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 & aliæ quædam notæ, quibus Chaldaei & Astrologii quemlibet numerum artificiose & arguté describunt, scitu periucundae, quas nobis communicauit Rodolphus Paludanus Nouiomagus."
Ordine primigeno sibi nomen possidet igin.Andras ecce locum mox uendicat ipse secundumOrmis post numeros incompositus sibi primus.
Ordine primigeno sibi nomen possidet igin.Andras ecce locum mox uendicat ipse secundumOrmis post numeros incompositus sibi primus.
Ordine primigeno sibi nomen possidet igin.
Andras ecce locum mox uendicat ipse secundum
Ormis post numeros incompositus sibi primus.
[BoncompagniButtetino, 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. BoncompagniBulletinoXV, 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," inMonatsberichte 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 rerumσύμβολαuocat: 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 hæ notulæ, 1. 2. 3. 4...." [Noviomagus,De Numeris Libri II, cap. vi.] Gemma Frisius also useselementaand Cardan usesliterae. In the first arithmetic by an American (Greenwood, 1729) the author speaks of "a few ArabianCharectersor Numeral Figures, calledDigits" (p. 1), and as late as 1790, in the third edition of J. J. Blassière's arithmetic (1st ed. 1769), the namecharactersis 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 Köbel, used it only for the zero. Sozifreappears in the arithmetic by Borgo, 1550 ed. In a Munich codex of the twelfth century, attributed to Gerland, they are calledcharactersonly: "Usque ad VIIII. enim porrigitur omnis numerus et qui supercrescit eisdem designator Karacteribus." [BoncompagniBulletino, Vol. X. p. 607.]
[474]The title of his work isPrologus N. Ocreati in Helceph(Arabical-qeif, investigation or memoir)ad Adelardum Batensem magistrum suum. The work was made known by C. Henry, in theZeitschrift für Mathematik und Physik, Vol. XXV, p. 129, and in theAbhandlungen 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, ofThe Testament of Love, printed with Chaucer's Works, loc. cit., Vol. VII, London, 1897.
[480]Liber Abacci, published in Olleris,Œuvres 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 für 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 für Mathematik und Physik, Vol. IX; Bubnov, loc. cit., pp. 197-245; M. Chasles, "Histoire de l'arithmétique. Recherches des traces du système de l'abacus, après que cette méthode a pris le nom d'Algorisme.—Preuves qu'à toutes les époques, jusq'auXVIesiècle, on a su que l'arithmétique vulgaire avait pour origine cette méthode ancienne,"Comptes rendus, Vol. XVII, pp. 143-154, also "Règles de l'abacus,"Comptes rendus, Vol. XVI, pp. 218-246, and "Analyse et explication du traité 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 hisBulletino, 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," BoncompagniBulletino, 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 theAbhandlungen 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 alsoAbhandlungen, 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'sTrattati 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 résumé, see Cantor, Vol. I (3), pp. 800-803. As to the author, see Eneström in theBibliotheca 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]SeeAbhandlungen 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. Björnbo, "Gerhard von Cremonas Übersetzung 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. Björnbo, "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 morgenländischen Gesellschaft, Vol. XXV, 1871, p. 104, andZeitschrift für Mathematik und Physik, Vol. XVI, 1871, pp. 392-393; M. Curtze,Centralblatt für 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 über die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," in theZeitschrift für Mathematik und Physik, Hist.-lit. Abth., Vol. XXXIV, p. 129. Curtze,Abhandlungen zur Geschichte der Mathematik, Vol. VIII, pp. 1-27.
[511]See lineain the plate on p.143.
[512]Sefer ha-Mispar, Das Buch der Zahl, ein hebräisch-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 für 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 theGiornale 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 mathématiques, Vol. II, p. 25; D. Martines,Origine e progressi dell' aritmetica, Messina, 1865, p. 47; Lucas, in BoncompagniBulletino, Vol. X, pp. 129, 239; Besagne, ibid., Vol. IX, p. 583; Boncompagni, three works as cited in Chap. I; G. Eneström, "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 theBibliotheca 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'empêcha pas Dante d'être le plus grand poète de l'Italie, et ce fut un petit marchand de Pise qui donna l'algèbre aux Chrétiens." [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,Propagationetc., 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 Universitäten des Mittelalters, Zürich, 1887, p. 56. Roger Bacon gives a still more gloomy view of Oxford in his time in hisOpus minus, in theRerum 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. Günther,Geschichte des mathematischen Unterrichts im deutschen Mittelalter bis zum Jahre 1525, Berlin, 1887, being Vol. III ofMonumenta 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'sEncyclopädie der Erziehung, Vol. VI, p. 726; A. Kuckuk, "Die Rechenkunst im sechzehnten Jahrhundert,"Festschrift zur dritten Säcularfeier 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'sCatalogue 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 Universität 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 theAlgorismusin 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 traités français d'algorisme et de géométrie," BoncompagniBulletino, Vol. XV, p. 49; Victor Mortet, "Le plus ancien traité français d'algorisme," loc. cit.
[550]L'État des sciences en France, depute la mort du Roy Robert, arrivée en 1031, jusqu'à celle de Philippe le Bel, arrivée en 1314, Paris, 1741.
[551]Discours sur l'état des lettres en France au XIIIesiecle, Paris, 1824.
[552]Aperçu 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. HisPolychronicon, 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 für 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 Paläographie, 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 donnée historique," etc., loc. cit., and "Essai d'une restitution de travaux perdus d'Apollonius sur les quantités irrationnelles, d'après des indications tirées d'un manuscrit arabe,"Tome XIV des Mémoires présentés par divers savants à l'Académie 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 dninri ihû xpi. 1264. perfecto." See Narducci'sCatalogo, p. 130.
[562]"On the Early use of Arabic Numerals in Europe," read before the Society of Antiquaries April 14, 1910, and published inArchæologiain 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 Münzkunde, Grätz, 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, Nördlingen.
[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, Nürnberg, 1470. The copy referred to is in the Astor Library.
[573]See Mauch, "Ueber den Gebrauch arabischer Ziffern und die Veränderungen derselben,"Anzeiger für Kunde der deutschen Vorzeit, 1861, columns 46, 81, 116, 151, 189, 229, and 268; Calmet,Recherches sur l'origine des chiffres d'arithmétique, plate, loc. cit.
[574]Günther,Geschichte, p. 175, n.; Mauch, loc. cit.
[575]These are given by W. R. Lethaby, from drawings by J. T. Irvine, in theProceedings 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 abréviations latines et françaises ... du moyen âge, 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'sGeschichte, Vol. I, table; M. Prou,Manuel de paléographie latine et française, 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 inZeitschrift für Mathematik und Physik, Vol. XXXIV.
[578]From the Regensburg chronicle. Plate containing some of these numerals inMonumenta 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, 1360A.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'sAlgorismusin Mr. Plimpton's library. Date c. 1442. See Smith,Rara Arithmetica, p. 450.
[584]SeeRara 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 theAlgorithmus linealisof c. 1488 has a special type for 1. In a student's notebook of lectures taken at the University of Würzburg in 1660, in Mr. Plimpton's library, the ones are all in the form of i.
[588]Thus the dateNumerals 1580, for 1580, appears in a MS. in the Laurentian library at Florence. The second and the following five characters are taken from Cappelli'sDizionario, 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. [BoncompagniBulletino, 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) hasSymbolfor 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, inRara Arithmetica, pp. 435, 437-440.
[596]But in the woodcuts of theMargarita Philosophica(1503) the old forms are used, although the new ones appear in the text. In Caxton'sMyrrour 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'sDictionnaire, 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 fromByzantinische Analekten, J. L. Heiberg, being forms of the fifteenth century, but not at all common.Symbol: Qoppawas 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 Daciaetc., 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 & aliæ quædam notæ, quibus Chaldaei & Astrologii quemlibet numerum artificiose & arguté describunt, scitu periucundae, quas nobis communicauit Rodolphus Paludanus Nouiomagus."