Chapter VIII: The Spread of the Numerals in Europe (3)

[294] P. M. Sykes, _Ten Thousand Miles in Persia, or Eight Years in Iran_, London, 1902, p. 167. Sykes was the first European to follow the course of Alexander's army across eastern Persia.

[295] Buehler, _Indian Br[=a]hma Alphabet_, note, p. 27; _Palaeographie_, p. 2; _Herodoti Halicarnassei historia_, Amsterdam, 1763, Bk. IV, p. 300; Isaac Vossius, _Periplus Scylacis Caryandensis_, 1639. It is doubtful whether the work attributed to Scylax was written by him, but in any case the work dates back to the fourth century B.C. See Smith's _Dictionary of Greek and Roman Biography_.

[296] Herodotus, Bk. III.

[297] Rameses II(?), the _Sesoosis_ of Diodorus Siculus.

[298] _Indian Antiquary_, Vol. I, p. 229; F. B. Jevons, _Manual of Greek Antiquities_, London, 1895, p. 386. On the relations, political and commercial, between India and Egypt c. 72 B.C., under Ptolemy Auletes, see the _Journal Asiatique_, 1863, p. 297.

[299] Sikandar, as the name still remains in northern India.

[300] _Harper's Classical Dict._, New York, 1897, Vol. I, p. 724; F. B. Jevons, loc. cit., p. 389; J. C. Marshman, _Abridgment of the History of India_, chaps. i and ii.

[301] Oppert, loc. cit., p. 11. It was at or near this place that the first great Indian mathematician, [=A]ryabha[t.]a, was born in 476 A.D.

[302] Buehler, _Palaeographie_, p. 2, speaks of Greek coins of a period anterior to Alexander, found in northern India. More complete information may be found in _Indian Coins_, by E. J. Rapson, Strassburg, 1898, pp. 3-7.

[303] Oppert, loc. cit., p. 14; and to him is due other similar information.

[304] J. Beloch, _Griechische Geschichte_, Vol. III, Strassburg, 1904, pp. 30-31.

[305] E.g., the denarius, the words for hour and minute ([Greek: hora, lepton]), and possibly the signs of the zodiac. [R. Caldwell, _Comparative Grammar of the Dravidian Languages_, London, 1856, p. 438.] On the probable Chinese origin of the zodiac see Schlegel, loc. cit.

[306] Marie, Vol. II, p. 73; R. Caldwell, loc. cit.

[307] A. Cunningham, loc. cit., p. 50.

[308] C. A. J. Skeel, _Travel_, loc. cit., p. 14.

[309] _Inchiver_, from _inchi_, "the green root." [_Indian Antiquary_, Vol. I, p. 352.]

[310] In China dating only from the second century A.D., however.

[311] The Italian _morra_.

[312] J. Bowring, _The Decimal System_, London, 1854, p. 2.

[313] H. A. Giles, lecture at Columbia University, March 12, 1902, on "China and Ancient Greece."

[314] Giles, loc. cit.

[315] E.g., the names for grape, radish (_la-po_, [Greek: rhaphe]), water-lily (_si-kua_, "west gourds"; [Greek: sikua], "gourds"), are much alike. [Giles, loc. cit.]

[316] _Epistles_, I, 1, 45-46. On the Roman trade routes, see Beazley, loc. cit., Vol. I, p. 179.

[317] _Am. Journ. of Archeol._, Vol. IV, p. 366.

[318] M. Perrot gives this conjectural restoration of his words: "Ad me ex India regum legationes saepe missi sunt numquam antea visae apud quemquam principem Romanorum." [M. Reinaud, "Relations politiques et commerciales de l'empire romain avec l'Asie orientale," _Journ. Asiat._, Vol. I (6), p. 93.]

[319] Reinaud, loc. cit., p. 189. Florus, II, 34 (IV, 12), refers to it: "Seres etiam habitantesque sub ipso sole Indi, cum gemmis et margaritis elephantes quoque inter munera trahentes nihil magis quam longinquitatem viae imputabant." Horace shows his geographical knowledge by saying: "Not those who drink of the deep Danube shall now break the Julian edicts; not the Getae, not the Seres, nor the perfidious Persians, nor those born on the river Tanais." [_Odes_, Bk. IV, Ode 15, 21-24.]

[320] "Qua virtutis moderationisque fama Indos etiam ac Scythas auditu modo cognitos pellexit ad amicitiam suam populique Romani ultro per legatos petendam." [Reinaud, loc. cit., p. 180.]

[321] Reinaud, loc. cit., p. 180.

[322] _Georgics_, II, 170-172. So Propertius (_Elegies_, III, 4):

Arma deus Caesar dites meditatur ad Indos
Et freta gemmiferi findere classe maris.

"The divine Caesar meditated carrying arms against opulent India, and with his ships to cut the gem-bearing seas."

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

[324] Reinaud, loc. cit., p. 393.

[325] The title page of Calandri (1491), for example, represents Pythagoras with these numerals before him. [Smith, _Rara Arithmetica_, p. 46.] Isaacus Vossius, _Observationes ad Pomponium Melam de situ orbis_, 1658, maintained that the Arabs derived these numerals from the west. A learned dissertation to this effect, but deriving them from the Romans instead of the Greeks, was written by Ginanni in 1753 (_Dissertatio mathematica critica de numeralium notarum minuscularum origine_, Venice, 1753). See also Mannert, _De numerorum quos arabicos vocant vera origine Pythagorica_, Nuernberg, 1801. Even as late as 1827 Romagnosi (in his supplement to _Ricerche storiche sull' India_ etc., by Robertson, Vol. II, p. 580, 1827) asserted that Pythagoras originated them. [R. Bombelli, _L'antica numerazione italica_, Rome, 1876, p. 59.] Gow (_Hist. of Greek Math._, p. 98) thinks that Iamblichus must have known a similar system in order to have worked out certain of his theorems, but this is an unwarranted deduction from the passage given.

[326] A. Hillebrandt, _Alt-Indien_, p. 179.

[327] J. C. Marshman, loc. cit., chaps. i and ii.

[328] He reigned 631-579 A.D.; called Nu['s][=i]rw[=a]n, _the holy one_.

[329] J. Keane, _The Evolution of Geography_, London, 1899, p. 38.

[330] The Arabs who lived in and about Mecca.

[331] S. Guyard, in _Encyc. Brit._, 9th ed., Vol. XVI, p. 597.

[332] Oppert, loc. cit., p. 29.

[333] "At non credendum est id in Autographis contigisse, aut vetustioribus Codd. MSS." [Wallis, _Opera omnia_, Vol. II, p. 11.]

[334] In _Observationes ad Pomponium Melam de situ orbis_. The question was next taken up in a large way by Weidler, loc. cit., _De characteribus_ etc., 1727, and in _Spicilegium_ etc., 1755.

[335] The best edition of these works is that of G. Friedlein, _Anicii Manlii Torquati Severini Boetii de institutione arithmetica libri duo, de institutione musica libri quinque. Accedit geometria quae fertur Boetii_.... Leipzig.... MDCCCLXVII.

[336] See also P. Tannery, "Notes sur la pseudo-geometrie de Boece," in _Bibliotheca Mathematica_, Vol. I (3), p. 39. This is not the geometry in two books in which are mentioned the numerals. There is a manuscript of this pseudo-geometry of the ninth century, but the earliest one of the other work is of the eleventh century (Tannery), unless the Vatican codex is of the tenth century as Friedlein (p. 372) asserts.

[337] Friedlein feels that it is partly spurious, but he says: "Eorum librorum, quos Boetius de geometria scripsisse dicitur, investigare veram inscriptionem nihil aliud esset nisi operam et tempus perdere." [Preface, p. v.] N. Bubnov in the Russian _Journal of the Ministry of Public Instruction_, 1907, in an article of which a synopsis is given in the _Jahrbuch ueber die Fortschritte der Mathematik_ for 1907, asserts that the geometry was written in the eleventh century.

[338] The most noteworthy of these was for a long time Cantor (_Geschichte_, Vol. I., 3d ed., pp. 587-588), who in his earlier days even believed that Pythagoras had known them. Cantor says (_Die roemischen Agrimensoren_, Leipzig, 1875, p. 130): "Uns also, wir wiederholen es, ist die Geometrie des Boetius echt, dieselbe Schrift, welche er nach Euklid bearbeitete, von welcher ein Codex bereits in Jahre 821 im Kloster Reichenau vorhanden war, von welcher ein anderes Exemplar im Jahre 982 zu Mantua in die Haende Gerbert's gelangte, von welcher mannigfache Handschriften noch heute vorhanden sind." But against this opinion of the antiquity of MSS. containing these numerals is the important statement of P. Tannery, perhaps the most critical of modern historians of mathematics, that none exists earlier than the eleventh century. See also J. L. Heiberg in _Philologus, Zeitschrift f. d. klass. Altertum_, Vol. XLIII, p. 508.

Of Cantor's predecessors, Th. H. Martin was one of the most prominent, his argument for authenticity appearing in the _Revue Archeologique_ for 1856-1857, and in his treatise _Les signes numeraux_ etc. See also M. Chasles, "De la connaissance qu'ont eu les anciens d'une numeration decimale ecrite qui fait usage de neuf chiffres prenant les valeurs de position," _Comptes rendus_, Vol. VI, pp. 678-680; "Sur l'origine de notre systeme de numeration," _Comptes rendus_, Vol. VIII, pp. 72-81; and note "Sur le passage du premier livre de la geometrie de Boece, relatif a un nouveau systeme de numeration," in his work _Apercu historique sur l'origine et le developpement des methodes en geometrie_, of which the first edition appeared in 1837.

[339] J. L. Heiberg places the book in the eleventh century on philological grounds, _Philologus_, loc. cit.; Woepcke, in _Propagation_, p. 44; Blume, Lachmann, and Rudorff, _Die Schriften der roemischen Feldmesser_, Berlin, 1848; Boeckh, _De abaco graecorum_, Berlin, 1841; Friedlein, in his Leipzig edition of 1867; Weissenborn, _Abhandlungen_, Vol. II, p. 185, his _Gerbert_, pp. 1, 247, and his _Geschichte der Einfuehrung der jetzigen Ziffern in Europa durch Gerbert_, Berlin, 1892, p. 11; Bayley, loc. cit., p. 59; Gerhardt, _Etudes_, p. 17, _Entstehung und Ausbreitung_, p. 14; Nagl, _Gerbert_, p. 57; Bubnov, loc. cit. See also the discussion by Chasles, Halliwell, and Libri, in the _Comptes rendus_, 1839, Vol. IX, p. 447, and in Vols. VIII, XVI, XVII of the same journal.

[340] J. Marquardt, _La vie privee des Romains_, Vol. II (French trans.), p. 505, Paris, 1893.

[341] In a Plimpton manuscript of the arithmetic of Boethius of the thirteenth century, for example, the Roman numerals are all replaced by the Arabic, and the same is true in the first printed edition of the book. (See Smith's _Rara Arithmetica_, pp. 434, 25-27.) D. E. Smith also copied from a manuscript of the arithmetic in the Laurentian library at Florence, of 1370, the following forms, [Forged numerals

[342] Halliwell, in his _Rara Mathematica, _p. 107, states that the disputed passage is not in a manuscript belonging to Mr. Ames, nor in one at Trinity College. See also Woepcke, in _Propagation_, pp. 37 and 42. It was the evident corruption of the texts in such editions of Boethius as those of Venice, 1499, Basel, 1546 and 1570, that led Woepcke to publish his work _Sur l'introduction de l'arithmetique indienne en Occident_.

[343] They are found in none of the very ancient manuscripts, as, for example, in the ninth-century (?) codex in the Laurentian library which one of the authors has examined. It should be said, however, that the disputed passage was written after the arithmetic, for it contains a reference to that work. See the Friedlein ed., p. 397.

[344] Smith, _Rara Arithmetica_, p. 66.

[345] J. L. Heiberg, _Philologus_, Vol. XLIII, p. 507.

[346] "Nosse autem huius artis dispicientem, quid sint digiti, quid articuli, quid compositi, quid incompositi numeri." [Friedlein ed., p. 395.]

[347] _De ratione abaci._ In this he describes "quandam formulam, quam ob honorem sui praeceptoris mensam Pythagoream nominabant ... a posterioribus appellabatur abacus." This, as pictured in the text, is the common Gerbert abacus. In the edition in Migne's _Patrologia Latina_, Vol. LXIII, an ordinary multiplication table (sometimes called Pythagorean abacus) is given in the illustration.

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

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

[350] E.g., it appears in Schonerus, _Algorithmus Demonstratus_, Nuernberg, 1534, f. A4. In England it appeared in the earliest English arithmetical manuscript known, _The Crafte of Nombrynge_: "¶ fforthermore ye most vndirstonde that in this craft ben vsid teen figurys, as here bene writen for ensampul, [Numerals] ... in the quych we vse teen figurys of Inde. Questio. ¶ why ten fyguris of Inde? Solucio. for as I have sayd afore thei were fonde fyrst in Inde of a kynge of that Cuntre, that was called Algor." See Smith, _An Early English Algorism_, loc. cit.

[351] Friedlein ed., p. 397.

[352] Carlsruhe codex of Gerlando.

[353] Munich codex of Gerlando.

[354] Carlsruhe codex of Bernelinus.

[355] Munich codex of Bernelinus.

[356] Turchill, c. 1200.

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

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

[359] Vatican codex, tenth century, Boethius.

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

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

[362] Ibid., p. 101.

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

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

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

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

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

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

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

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

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

[372] Possibly late tenth, since he refers to one arithmetical work which is entitled _Book of the Cyphers_ in his _Chronology_, English ed., p. 132. Suter, _Die Mathematiker_ etc., pp. 98-100, does not mention this work; see the _Nachtraege und Berichtigungen_, pp. 170-172.

[373] Suter, pp. 96-97.

[374] Suter, p. 111.

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

[376] Suter, p. 138.

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

[378] [Greek: Psephophoria kat' Indous], Greek ed., C. I. Gerhardt, Halle, 1865; and German translation, _Das Rechenbuch des Maximus Planudes_, H. Waeschke, Halle, 1878.

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

[380] Suter, p. 80.

[381] Suter, p. 68.

[382] Sprenger also calls attention to this fact, in the _Zeitschrift d. deutschen morgenlaend. Gesellschaft_, Vol. XLV, p. 367.

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

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

[385] Persian _bagadata_, "God-given."

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

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

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

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

[390] See pp. 4-5.

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

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

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

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

[395] Part of this work has been translated from a Leyden MS. by F. Woepcke, _Propagation_, and more recently by H. Suter, _Bibliotheca Mathematica_, Vol. VII (3), pp. 113-119.

[396] A. Neander, _General History of the Christian Religion and Church_, 5th American ed., Boston, 1855, Vol. III, p. 335.

[397] Beazley, loc. cit., Vol. I, p. 49.

[398] Beazley, loc. cit., Vol. I, pp. 50, 460.

[399] See pp. 7-8.

[400] The name also appears as Mo[h.]ammed Ab[=u]'l-Q[=a]sim, and Ibn Hauqal. Beazley, loc. cit., Vol. I, p. 45.

[401] _Kit[=a]b al-mas[=a]lik wa'l-mam[=a]lik._

[402] Reinaud, _Mem. sur l'Inde_; in Gerhardt, _Etudes_, p. 18.

[403] Born at Shiraz in 1193. He himself had traveled from India to Europe.

[404] _Gulistan_ (_Rose Garden_), Gateway the third, XXII. Sir Edwin Arnold's translation, N. Y., 1899, p. 177.

[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-Man[s.][=u]r, governing Spain under the name of Hish[=a]m (976-1002), called from the Orient Al-Be[.g][=a]n[=i] 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 Da[=u]d (Son of David). See also _Abhandlungen_, Vol. III, p. 110.

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

[501] For it says, "Incipit prologus in libro 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-Khow[=a]razm[=i]'s book," but "to a book of algorism." John of Luna says of it: "Hoc idem est illud etiam quod ... alcorismus dicere videtur." [_Trattati_, p. 68.]

[503] For a resume, see Cantor, Vol. I (3), pp. 800-803. As to the author, see 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-Chw[=a]rizm[=i]'s trigonometriske Tavler," _Festskrift til H. G. Zeuthen_, Copenhagen, 1909, pp. 1-17.

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

[509] M. Steinschneider, _Zeitschrift der deutschen 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-Khow[=a]razm[=i]," _Bibliotheca Mathematica_, Vol. XI (3), p. 125. He is also known as Robertus Retinensis, or Robert of Reading.

[510] Nagl, A., "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und 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 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 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."