[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 s.p.a.ciorum 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 fur Mathematik und Physik_, Vol. XII (1867), _Literaturzeitung_, p. 70: "Fur das _System_ unserer Numeration ist die _Null_ das wesentlichste Merkmal, und diese kannte Gerbert nicht. Er selbst schrieb alle Zahlen mit den romischen Zahlzeichen und man kann ihm also nicht verdanken, was er selbst nicht kannte."

[449] E.g., Chasles, Budinger, Gerhardt, and Richer. So Martin (_Recherches nouvelles_ etc.) believes that Gerbert received them from Boethius or his followers. See Woepcke, _Propagation_, p. 41.

[450] Budinger, 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 uber den Josephus Sapiens," _Bibliotheca Mathematica_, Vol. VIII (2), p. 84; Weissenborn, _Einfuhrung_, 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 Hispa.n.u.s seu Josephus sapiens videtur aut Maurus fuisse aut alius quis in Hispania."

[458] P. Ewald, _Mittheilungen, Neues Archiv d. Gesellschaft fur altere deutsche Geschichtskunde_, Vol. VIII, 1883, pp. 354-364. One of the ma.n.u.scripts is of 976 A.D. and the other of 992 A.D. See also Franz Steffens, _Lateinische Palaographie_, Freiburg (Schweiz), 1903, pp.

x.x.xix-xl. The forms are reproduced in the plate on page 140.

[459] It is ent.i.tled _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 fur 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. x.x.xVII. 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 Gunther'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 ma.n.u.scripts from which the ill.u.s.trations 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 ma.n.u.script in the Vatican, in a pa.s.sage beginning

Ordine primigeno sibi nomen possidet igin.

Andras ecce loc.u.m mox uendicat ipse secundum Ormis post numeros incompositus sibi primus.

[Boncompagni _b.u.t.tetino_, 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) a.s.serted 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, a.s.serted 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 ma.n.u.script 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 c.u.m 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." [Glarea.n.u.s, 1545 edition, f. 9, r.] "Per figuras (quas zyphras uocant) a.s.signationem, 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. Bla.s.siere'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 Kobel, 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 t.i.tle 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 fur 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.