"The saintly soul, that showsThe world's deceitfulness, to all who hear him,Is, with the sight of all the good that is,Blest there. The limbs, whence it was driven, lieDown in Cieldauro; and from martyrdomAnd exile came it here."—Paradiso, Canto X.[284]Not, however, in the mercantile schools. The arithmetic of Boethius would have been about the last book to be thought of in such institutions. While referred to by Bæda (672-735) and Hrabanus Maurus (c. 776-856), it was only after Gerbert's time that theBoëtii de institutione arithmetica libri duowas really a common work.[285]Also spelled Cassiodorius.[286]As a matter of fact, Boethius could not have translated any work by Pythagoras on music, because there was no such work, but he did make the theories of the Pythagoreans known. Neither did he translate Nicomachus, although he embodied many of the ideas of the Greek writer in his own arithmetic. Gibbon follows Cassiodorus in these statements in hisDecline and Fall of the Roman Empire, chap. xxxix. Martin pointed out with positiveness the similarity of the first book of Boethius to the first five books of Nicomachus. [Les signes numérauxetc., reprint, p. 4.][287]The general idea goes back to Pythagoras, however.[288]J. C. Scaliger in hisPoëticealso said of him: "Boethii Severini ingenium, eruditio, ars, sapientia facile provocat omnes auctores, sive illi Graeci sint, sive Latini" [Heilbronner,Hist. math. univ., p. 387]. Libri, speaking of the time of Boethius, remarks: "Nous voyons du temps de Théodoric, les lettres reprendre une nouvelle vie en Italie, les écoles florissantes et les savans honorés. Et certes les ouvrages de Boëce, de Cassiodore, de Symmaque, surpassent de beaucoup toutes les productions du siècle précédent." [Histoire des mathématiques, Vol. I, p. 78.][289]Carra de Vaux,Avicenne, Paris, 1900; Woepcke,Sur l'introduction, etc.; Gerhardt,Entstehungetc., p. 20. Avicenna is a corruption from Ibn Sīnā, as pointed out by Wüstenfeld,Geschichte der arabischen Aerzte und Naturforscher, Göttingen, 1840. His full name isAbū ‛Alī al-Ḥosein ibn Sīnā. For notes on Avicenna's arithmetic, see Woepcke,Propagation, p. 502.[290]On the early travel between the East and the West the following works may be consulted: A. Hillebrandt,Alt-Indien, containing "Chinesische Reisende in Indien," Breslau, 1899, p. 179; C. A. Skeel,Travel in the First Century after Christ, Cambridge, 1901, p. 142; M. Reinaud, "Relations politiques et commerciales de l'empire romain avec l'Asie orientale," in theJournal Asiatique, Mars-Avril, 1863, Vol. I (6), p. 93; Beazley,Dawn of Modern Geography, a History of Exploration and Geographical Science from the Conversion of the Roman Empire toA.D.1420, London, 1897-1906, 3 vols.; Heyd,Geschichte des Levanthandels im Mittelalter, Stuttgart, 1897; J. Keane,The Evolution of Geography, London, 1899, p. 38; A. Cunningham,Corpus inscriptionum Indicarum, Calcutta, 1877, Vol. I; A. Neander,General History of the Christian Religion and Church, 5th American ed., Boston, 1855, Vol. III, p. 89; R. C. Dutt,A History of Civilization in Ancient India, Vol. II, Bk. V, chap, ii; E. C. Bayley, loc. cit., p. 28 et seq.; A. C. Burnell, loc. cit., p. 3; J. E. Tennent,Ceylon, London, 1859, Vol. I, p. 159; Geo. Turnour,Epitome of the History of Ceylon, London, n.d., preface; "Philalethes,"History of Ceylon, London, 1816, chap, i; H. C. Sirr,Ceylon and the Cingalese, London, 1850, Vol. I, chap. ix. On the Hindu knowledge of the Nile see F. Wilford,Asiatick Researches, Vol. III, p. 295, Calcutta, 1792.[291]G. Oppert,On the Ancient Commerce of India, Madras, 1879, p. 8.[292]Gerhardt,Étudesetc., pp. 8, 11.[293]See Smith'sDictionary of Greek and Roman Biography and Mythology.[294]P. M. Sykes,Ten Thousand Miles in Persia, or Eight Years in Irán, London, 1902, p. 167. Sykes was the first European to follow the course of Alexander's army across eastern Persia.[295]Bühler,Indian Brāhma 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 centuryB.C.See Smith'sDictionary of Greek and Roman Biography.[296]Herodotus, Bk. III.[297]Rameses II(?), theSesoosisof 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. 72B.C., under Ptolemy Auletes, see theJournal 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,Āryabhaṭa, was born in 476A.D.[302]Bühler,Palaeographie, p. 2, speaks of Greek coins of a period anterior to Alexander, found in northern India. More complete information may be found inIndian 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 (ὥρα, λεπτόν), 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, frominchi, "the green root." [Indian Antiquary, Vol. I, p. 352.][310]In China dating only from the second centuryA.D., however.[311]The Italianmorra.[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,ῥάφη), water-lily (si-kua, "west gourds";σικύα, "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 Tanaïs." [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 IndosEt freta gemmiferi findere classe maris."The divine Cæsar 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, Nürnberg, 1801. Even as late as 1827 Romagnosi (in his supplement toRicerche storiche sull' Indiaetc., 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 631-579A.D.; called Nuśīrwān,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, inEncyc. 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]InObservationes ad Pomponium Melam de situ orbis. The question was next taken up in a large way by Weidler, loc. cit.,De characteribusetc., 1727, and inSpicilegiumetc., 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-géometrie de Boèce," inBibliotheca 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 RussianJournal of the Ministry of Public Instruction, 1907, in an article of which a synopsis is given in theJahrbuch über die Fortschritte der Mathematikfor 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 römischen 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 Hände 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 inPhilologus, 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 theRevue Archéologiquefor 1856-1857, and in his treatiseLes signes numérauxetc. See also M. Chasles, "De la connaissance qu'ont eu les anciens d'une numération décimale écrite qui fait usage de neuf chiffres prenant les valeurs de position,"Comptes rendus, Vol. VI, pp. 678-680; "Sur l'origine de notre système de numération,"Comptes rendus, Vol. VIII, pp. 72-81; and note "Sur le passage du premier livre de la géométrie de Boèce, relatif à un nouveau système de numération," in his workAperçu historique sur l'origine et le devéloppement des méthodes en géométrie, 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, inPropagation, p. 44; Blume, Lachmann, and Rudorff,Die Schriften der römischen Feldmesser, Berlin, 1848; Boeckh,De abaco graecorum, Berlin, 1841; Friedlein, in his Leipzig edition of 1867; Weissenborn,Abhandlungen, Vol. II, p. 185, hisGerbert, pp. 1, 247, and hisGeschichte der Einführung der jetzigen Ziffern in Europa durch Gerbert, Berlin, 1892, p. 11; Bayley, loc. cit., p. 59; Gerhardt,Études, p. 17,Entstehung und Ausbreitung, p. 14; Nagl,Gerbert, p. 57; Bubnov, loc. cit. See also the discussion by Chasles, Halliwell, and Libri, in theComptes rendus, 1839, Vol. IX, p. 447, and in Vols. VIII, XVI, XVII of the same journal.[340]J. Marquardt,La vie privée 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'sRara 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 numeralswhich, of course, are interpolations. An interesting example of a forgery in ecclesiastical matters is in the charter said to have been given by St. Patrick, granting indulgences to the benefactors of Glastonbury, dated "In nomine domini nostri Jhesu Christi Ego Patricius humilis servunculus Dei anno incarnationis ejusdem ccccxxx." Now if the Benedictines are right in saying that Dionysius Exiguus, a Scythian monk, first arranged the Christian chronology c. 532A.D., this can hardly be other than spurious. See Arbuthnot, loc. cit., p. 38.[342]Halliwell, in hisRara 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, inPropagation, 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 workSur l'introduction de l'arithmétique 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'sPatrologia 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 mathématiques,"Revue Archéologique, 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, Nürnberg, 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 BoncompagniBulletino, Vol. X, p. 596; f ibid., Vol. XV, p. 186; gMemorie 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 representantSymbolalii autem sicSymbol, uel sicSymbol. Quidam vero quartam sicSymbol." [Boncompagni,Trattati, Vol. II, p. 28.][361]Loc. cit., p. 59.[362]Ibid., p. 101.[363]Loc. cit., p. 396.[364]Khosrū I, who began to reign in 531A.D.See W. S. W Vaux,Persia,London, 1875, p. 169; Th. Nöldeke,Aufsätze zur persichen Geschichte, Leipzig, 1887, p. 113, and his article in the ninth edition of theEncyclopædia Britannica.[365]Colebrooke,Essays, Vol. II, p. 504, on the authority of Ibn al-Adamī, astronomer, in a work published by his continuator Al-Qāsim in 920A.D.; Al-Bīrūnī,India,Vol. II, p. 15.[366]H. Suter,Die Mathematikeretc., pp. 4-5, states that Al-Fazārī 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 workDie Mathematiker und Astronomen der Araberetc.[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 entitledBook of the Cyphersin hisChronology, English ed., p. 132. Suter,Die Mathematikeretc., pp. 98-100, does not mention this work; see theNachträge 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]Ψηφοφορία κατ' Ἰνδούς, Greek ed., C. I. Gerhardt, Halle, 1865; and German translation,Das Rechenbuch des Maximus Planudes, H. Wäschke, Halle, 1878.[379]"Sur une donnée historique relative à l'emploi des chiffres indiens par les Arabes," Tortolini'sAnnali di scienze mat. e fis., 1855.[380]Suter, p. 80.[381]Suter, p. 68.[382]Sprenger also calls attention to this fact, in theZeitschrift d. deutschen morgenländ. Gesellschaft, Vol. XLV, p. 367.[383]Libri,Histoire des mathématiques, Vol. I, p. 147.[384]"Dictant la paix à l'empereur de Constantinople, l'Arabe victorieux demandait des manuscrits et des savans." [Libri, loc. cit., p. 108.][385]Persianbagadata, "God-given."[386]One of the Abbassides, the (at least pretended) descendants of ‛Al-Abbās, uncle and adviser ofMoḥammed.[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 theCantor Festschrift, 1909, note pp. 10-11. See also F. Woepcke,Propagation.[392]Eneström, inBibliotheca 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 asMoḥammedAbū'l-Qāsim, and Ibn Hauqal. Beazley, loc. cit., Vol. I, p. 45.[401]Kitāb al-masālik wa'l-mamālik.[402]Reinaud,Mém. sur l'Inde; in Gerhardt,Études, 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 terrasJam peragravit ovans, sophiae deductus amore,Si quid forte novi librorum seu studiorumQuod 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'après l'histoire et d'après la légende, 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, inThe 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 à l'étude 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'sDawn of Modern Geography, Vol. III, chap, ii, and Part II.[422]Heyd, loc. cit., Vol. II, p. 220; H. Yule, inEncyclopædia Britannica, 9th (10th) or 11th ed., article "China." The handbook cited is Pegolotti'sLibro 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]ItalianBaldacco.[430]J. K. Mumford,Oriental Rugs, New York, 1901, p. 18.[431]Or Girbert, the Latin formsGerbertusandGirbertusappearing indifferently in the documents of his time.[432]See, for example, J. C. Heilbronner,Historia matheseos universæ, 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.,Gerbertetc.; Olleris,Œuvres de Gerbert, Paris, 1867; Havet,Lettres de Gerbert, Paris, 1889 ; H. Weissenborn,Gerbert; Beiträge zur Kenntnis der Mathematik des Mittelalters, Berlin, 1888, andZur Geschichte der Einführung der jetzigen Ziffern in Europa durch Gerbert, Berlin, 1892; Büdinger,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 Pädagogik, 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 geometriæ, 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 (Œuvres de Gerbert, Paris, 1867, p. cci) says, "C'est à lui et non point aux Arabes, que l'Europe doit son système et ses signes de numération," he exaggerates, since the evidence is all against his knowing the place value. Friedlein emphasizes this in theZeitschrift für Mathematik und Physik, Vol. XII (1867),Literaturzeitung, p. 70: "Für dasSystemunserer Numeration ist dieNulldas wesentlichste Merkmal, und diese kannte Gerbert nicht. Er selbst schrieb alle Zahlen mit den römischen Zahlzeichen und man kann ihm also nicht verdanken, was er selbst nicht kannte."[449]E.g., Chasles, Büdinger, Gerhardt, and Richer. So Martin (Recherches nouvellesetc.) believes that Gerbert received them from Boethius or his followers. See Woepcke,Propagation, p. 41.[450]Büdinger, loc. cit., p. 10. Nevertheless, in Gerbert's time oneAl-Manṣūr, governing Spain under the name of Hishām (976-1002), called from the Orient Al-Beġānī to teach his son, so that scholars were recognized. [Picavet, p. 36.][451]Weissenborn, loc. cit., p. 235.[452]Ibid., p. 234.[453]These letters, of the period 983-997, were edited by Havet, loc. cit., and, less completely, by Olleris, loc. cit. Those touching mathematical topics were edited by Bubnov, loc. cit., pp. 98-106.[454]He published it in theMonumenta 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 über den Josephus Sapiens,"Bibliotheca Mathematica, Vol. VIII (2), p. 84; Weissenborn,Einführung, p. 14; also hisGerbert; M. Steinschneider, inBibliotheca 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 für ältere deutsche Geschichtskunde, Vol. VIII, 1883, pp. 354-364. One of the manuscripts is of 976A.D.and the other of 992A.D.See also Franz Steffens,Lateinische Paläographie, Freiburg (Schweiz), 1903, pp. xxxix-xl. The forms are reproduced in the plate on page 140.[459]It is entitledConstantino 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 theComptes rendus, Vol. XVI (1843), p. 295. This book seems to have been written c. 980A.D.[Bubnov, loc. cit., p. 6.][460]"Histoire de l'Arithmétique,"Comptes rendus, Vol. XVI (1843), pp. 156, 281.[461]Loc. cit.,Gerberti Operaetc.[462]Friedlein thought it spurious. SeeZeitschrift für 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 (Œuvres 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 usessiposfor 0. It is not given by Bernelinus, and appears in Radulph of Laon, in the twelfth century. See Günther'sGeschichte, p. 98, n.; Weissenborn, p. 11; Pihan,Exposéetc., pp. xvi-xxii.In Friedlein'sBoetius, 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 wordsiposfor 0. The names appear in a twelfth-century anonymous manuscript in the Vatican, in a passage beginning
"The saintly soul, that showsThe world's deceitfulness, to all who hear him,Is, with the sight of all the good that is,Blest there. The limbs, whence it was driven, lieDown in Cieldauro; and from martyrdomAnd exile came it here."—Paradiso, Canto X.
"The saintly soul, that showsThe world's deceitfulness, to all who hear him,Is, with the sight of all the good that is,Blest there. The limbs, whence it was driven, lieDown in Cieldauro; and from martyrdomAnd exile came it here."—Paradiso, Canto X.
"The saintly soul, that shows
The world's deceitfulness, to all who hear him,
Is, with the sight of all the good that is,
Blest there. The limbs, whence it was driven, lie
Down in Cieldauro; and from martyrdom
And exile came it here."—Paradiso, Canto X.
[284]Not, however, in the mercantile schools. The arithmetic of Boethius would have been about the last book to be thought of in such institutions. While referred to by Bæda (672-735) and Hrabanus Maurus (c. 776-856), it was only after Gerbert's time that theBoëtii de institutione arithmetica libri duowas really a common work.
[285]Also spelled Cassiodorius.
[286]As a matter of fact, Boethius could not have translated any work by Pythagoras on music, because there was no such work, but he did make the theories of the Pythagoreans known. Neither did he translate Nicomachus, although he embodied many of the ideas of the Greek writer in his own arithmetic. Gibbon follows Cassiodorus in these statements in hisDecline and Fall of the Roman Empire, chap. xxxix. Martin pointed out with positiveness the similarity of the first book of Boethius to the first five books of Nicomachus. [Les signes numérauxetc., reprint, p. 4.]
[287]The general idea goes back to Pythagoras, however.
[288]J. C. Scaliger in hisPoëticealso said of him: "Boethii Severini ingenium, eruditio, ars, sapientia facile provocat omnes auctores, sive illi Graeci sint, sive Latini" [Heilbronner,Hist. math. univ., p. 387]. Libri, speaking of the time of Boethius, remarks: "Nous voyons du temps de Théodoric, les lettres reprendre une nouvelle vie en Italie, les écoles florissantes et les savans honorés. Et certes les ouvrages de Boëce, de Cassiodore, de Symmaque, surpassent de beaucoup toutes les productions du siècle précédent." [Histoire des mathématiques, Vol. I, p. 78.]
[289]Carra de Vaux,Avicenne, Paris, 1900; Woepcke,Sur l'introduction, etc.; Gerhardt,Entstehungetc., p. 20. Avicenna is a corruption from Ibn Sīnā, as pointed out by Wüstenfeld,Geschichte der arabischen Aerzte und Naturforscher, Göttingen, 1840. His full name isAbū ‛Alī al-Ḥosein ibn Sīnā. For notes on Avicenna's arithmetic, see Woepcke,Propagation, p. 502.
[290]On the early travel between the East and the West the following works may be consulted: A. Hillebrandt,Alt-Indien, containing "Chinesische Reisende in Indien," Breslau, 1899, p. 179; C. A. Skeel,Travel in the First Century after Christ, Cambridge, 1901, p. 142; M. Reinaud, "Relations politiques et commerciales de l'empire romain avec l'Asie orientale," in theJournal Asiatique, Mars-Avril, 1863, Vol. I (6), p. 93; Beazley,Dawn of Modern Geography, a History of Exploration and Geographical Science from the Conversion of the Roman Empire toA.D.1420, London, 1897-1906, 3 vols.; Heyd,Geschichte des Levanthandels im Mittelalter, Stuttgart, 1897; J. Keane,The Evolution of Geography, London, 1899, p. 38; A. Cunningham,Corpus inscriptionum Indicarum, Calcutta, 1877, Vol. I; A. Neander,General History of the Christian Religion and Church, 5th American ed., Boston, 1855, Vol. III, p. 89; R. C. Dutt,A History of Civilization in Ancient India, Vol. II, Bk. V, chap, ii; E. C. Bayley, loc. cit., p. 28 et seq.; A. C. Burnell, loc. cit., p. 3; J. E. Tennent,Ceylon, London, 1859, Vol. I, p. 159; Geo. Turnour,Epitome of the History of Ceylon, London, n.d., preface; "Philalethes,"History of Ceylon, London, 1816, chap, i; H. C. Sirr,Ceylon and the Cingalese, London, 1850, Vol. I, chap. ix. On the Hindu knowledge of the Nile see F. Wilford,Asiatick Researches, Vol. III, p. 295, Calcutta, 1792.
[291]G. Oppert,On the Ancient Commerce of India, Madras, 1879, p. 8.
[292]Gerhardt,Étudesetc., pp. 8, 11.
[293]See Smith'sDictionary of Greek and Roman Biography and Mythology.
[294]P. M. Sykes,Ten Thousand Miles in Persia, or Eight Years in Irán, London, 1902, p. 167. Sykes was the first European to follow the course of Alexander's army across eastern Persia.
[295]Bühler,Indian Brāhma 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 centuryB.C.See Smith'sDictionary of Greek and Roman Biography.
[296]Herodotus, Bk. III.
[297]Rameses II(?), theSesoosisof 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. 72B.C., under Ptolemy Auletes, see theJournal 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,Āryabhaṭa, was born in 476A.D.
[302]Bühler,Palaeographie, p. 2, speaks of Greek coins of a period anterior to Alexander, found in northern India. More complete information may be found inIndian 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 (ὥρα, λεπτόν), 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, frominchi, "the green root." [Indian Antiquary, Vol. I, p. 352.]
[310]In China dating only from the second centuryA.D., however.
[311]The Italianmorra.
[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,ῥάφη), water-lily (si-kua, "west gourds";σικύα, "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 Tanaïs." [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 IndosEt freta gemmiferi findere classe maris.
Arma deus Caesar dites meditatur ad IndosEt freta gemmiferi findere classe maris.
Arma deus Caesar dites meditatur ad Indos
Et freta gemmiferi findere classe maris.
"The divine Cæsar 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, Nürnberg, 1801. Even as late as 1827 Romagnosi (in his supplement toRicerche storiche sull' Indiaetc., 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 631-579A.D.; called Nuśīrwān,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, inEncyc. 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]InObservationes ad Pomponium Melam de situ orbis. The question was next taken up in a large way by Weidler, loc. cit.,De characteribusetc., 1727, and inSpicilegiumetc., 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-géometrie de Boèce," inBibliotheca 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 RussianJournal of the Ministry of Public Instruction, 1907, in an article of which a synopsis is given in theJahrbuch über die Fortschritte der Mathematikfor 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 römischen 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 Hände 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 inPhilologus, 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 theRevue Archéologiquefor 1856-1857, and in his treatiseLes signes numérauxetc. See also M. Chasles, "De la connaissance qu'ont eu les anciens d'une numération décimale écrite qui fait usage de neuf chiffres prenant les valeurs de position,"Comptes rendus, Vol. VI, pp. 678-680; "Sur l'origine de notre système de numération,"Comptes rendus, Vol. VIII, pp. 72-81; and note "Sur le passage du premier livre de la géométrie de Boèce, relatif à un nouveau système de numération," in his workAperçu historique sur l'origine et le devéloppement des méthodes en géométrie, 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, inPropagation, p. 44; Blume, Lachmann, and Rudorff,Die Schriften der römischen Feldmesser, Berlin, 1848; Boeckh,De abaco graecorum, Berlin, 1841; Friedlein, in his Leipzig edition of 1867; Weissenborn,Abhandlungen, Vol. II, p. 185, hisGerbert, pp. 1, 247, and hisGeschichte der Einführung der jetzigen Ziffern in Europa durch Gerbert, Berlin, 1892, p. 11; Bayley, loc. cit., p. 59; Gerhardt,Études, p. 17,Entstehung und Ausbreitung, p. 14; Nagl,Gerbert, p. 57; Bubnov, loc. cit. See also the discussion by Chasles, Halliwell, and Libri, in theComptes rendus, 1839, Vol. IX, p. 447, and in Vols. VIII, XVI, XVII of the same journal.
[340]J. Marquardt,La vie privée 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'sRara 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 numeralswhich, of course, are interpolations. An interesting example of a forgery in ecclesiastical matters is in the charter said to have been given by St. Patrick, granting indulgences to the benefactors of Glastonbury, dated "In nomine domini nostri Jhesu Christi Ego Patricius humilis servunculus Dei anno incarnationis ejusdem ccccxxx." Now if the Benedictines are right in saying that Dionysius Exiguus, a Scythian monk, first arranged the Christian chronology c. 532A.D., this can hardly be other than spurious. See Arbuthnot, loc. cit., p. 38.
[342]Halliwell, in hisRara 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, inPropagation, 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 workSur l'introduction de l'arithmétique 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'sPatrologia 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 mathématiques,"Revue Archéologique, 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, Nürnberg, 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 BoncompagniBulletino, Vol. X, p. 596; f ibid., Vol. XV, p. 186; gMemorie 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 representantSymbolalii autem sicSymbol, uel sicSymbol. Quidam vero quartam sicSymbol." [Boncompagni,Trattati, Vol. II, p. 28.]
[361]Loc. cit., p. 59.
[362]Ibid., p. 101.
[363]Loc. cit., p. 396.
[364]Khosrū I, who began to reign in 531A.D.See W. S. W Vaux,Persia,London, 1875, p. 169; Th. Nöldeke,Aufsätze zur persichen Geschichte, Leipzig, 1887, p. 113, and his article in the ninth edition of theEncyclopædia Britannica.
[365]Colebrooke,Essays, Vol. II, p. 504, on the authority of Ibn al-Adamī, astronomer, in a work published by his continuator Al-Qāsim in 920A.D.; Al-Bīrūnī,India,Vol. II, p. 15.
[366]H. Suter,Die Mathematikeretc., pp. 4-5, states that Al-Fazārī 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 workDie Mathematiker und Astronomen der Araberetc.
[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 entitledBook of the Cyphersin hisChronology, English ed., p. 132. Suter,Die Mathematikeretc., pp. 98-100, does not mention this work; see theNachträge 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]Ψηφοφορία κατ' Ἰνδούς, Greek ed., C. I. Gerhardt, Halle, 1865; and German translation,Das Rechenbuch des Maximus Planudes, H. Wäschke, Halle, 1878.
[379]"Sur une donnée historique relative à l'emploi des chiffres indiens par les Arabes," Tortolini'sAnnali di scienze mat. e fis., 1855.
[380]Suter, p. 80.
[381]Suter, p. 68.
[382]Sprenger also calls attention to this fact, in theZeitschrift d. deutschen morgenländ. Gesellschaft, Vol. XLV, p. 367.
[383]Libri,Histoire des mathématiques, Vol. I, p. 147.
[384]"Dictant la paix à l'empereur de Constantinople, l'Arabe victorieux demandait des manuscrits et des savans." [Libri, loc. cit., p. 108.]
[385]Persianbagadata, "God-given."
[386]One of the Abbassides, the (at least pretended) descendants of ‛Al-Abbās, uncle and adviser ofMoḥammed.
[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 theCantor Festschrift, 1909, note pp. 10-11. See also F. Woepcke,Propagation.
[392]Eneström, inBibliotheca 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 asMoḥammedAbū'l-Qāsim, and Ibn Hauqal. Beazley, loc. cit., Vol. I, p. 45.
[401]Kitāb al-masālik wa'l-mamālik.
[402]Reinaud,Mém. sur l'Inde; in Gerhardt,Études, 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 terrasJam peragravit ovans, sophiae deductus amore,Si quid forte novi librorum seu studiorumQuod secum ferret, terris reperiret in illis.Hic quoque Romuleum venit devotus ad urbem."
"Non semel externas peregrino tramite terrasJam peragravit ovans, sophiae deductus amore,Si quid forte novi librorum seu studiorumQuod secum ferret, terris reperiret in illis.Hic quoque Romuleum venit devotus ad urbem."
"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'après l'histoire et d'après la légende, 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, inThe 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 à l'étude 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'sDawn of Modern Geography, Vol. III, chap, ii, and Part II.
[422]Heyd, loc. cit., Vol. II, p. 220; H. Yule, inEncyclopædia Britannica, 9th (10th) or 11th ed., article "China." The handbook cited is Pegolotti'sLibro 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]ItalianBaldacco.
[430]J. K. Mumford,Oriental Rugs, New York, 1901, p. 18.
[431]Or Girbert, the Latin formsGerbertusandGirbertusappearing indifferently in the documents of his time.
[432]See, for example, J. C. Heilbronner,Historia matheseos universæ, 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.,Gerbertetc.; Olleris,Œuvres de Gerbert, Paris, 1867; Havet,Lettres de Gerbert, Paris, 1889 ; H. Weissenborn,Gerbert; Beiträge zur Kenntnis der Mathematik des Mittelalters, Berlin, 1888, andZur Geschichte der Einführung der jetzigen Ziffern in Europa durch Gerbert, Berlin, 1892; Büdinger,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 Pädagogik, 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 geometriæ, 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 (Œuvres de Gerbert, Paris, 1867, p. cci) says, "C'est à lui et non point aux Arabes, que l'Europe doit son système et ses signes de numération," he exaggerates, since the evidence is all against his knowing the place value. Friedlein emphasizes this in theZeitschrift für Mathematik und Physik, Vol. XII (1867),Literaturzeitung, p. 70: "Für dasSystemunserer Numeration ist dieNulldas wesentlichste Merkmal, und diese kannte Gerbert nicht. Er selbst schrieb alle Zahlen mit den römischen Zahlzeichen und man kann ihm also nicht verdanken, was er selbst nicht kannte."
[449]E.g., Chasles, Büdinger, Gerhardt, and Richer. So Martin (Recherches nouvellesetc.) believes that Gerbert received them from Boethius or his followers. See Woepcke,Propagation, p. 41.
[450]Büdinger, loc. cit., p. 10. Nevertheless, in Gerbert's time oneAl-Manṣūr, governing Spain under the name of Hishām (976-1002), called from the Orient Al-Beġānī to teach his son, so that scholars were recognized. [Picavet, p. 36.]
[451]Weissenborn, loc. cit., p. 235.
[452]Ibid., p. 234.
[453]These letters, of the period 983-997, were edited by Havet, loc. cit., and, less completely, by Olleris, loc. cit. Those touching mathematical topics were edited by Bubnov, loc. cit., pp. 98-106.
[454]He published it in theMonumenta 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 über den Josephus Sapiens,"Bibliotheca Mathematica, Vol. VIII (2), p. 84; Weissenborn,Einführung, p. 14; also hisGerbert; M. Steinschneider, inBibliotheca 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 für ältere deutsche Geschichtskunde, Vol. VIII, 1883, pp. 354-364. One of the manuscripts is of 976A.D.and the other of 992A.D.See also Franz Steffens,Lateinische Paläographie, Freiburg (Schweiz), 1903, pp. xxxix-xl. The forms are reproduced in the plate on page 140.
[459]It is entitledConstantino 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 theComptes rendus, Vol. XVI (1843), p. 295. This book seems to have been written c. 980A.D.[Bubnov, loc. cit., p. 6.]
[460]"Histoire de l'Arithmétique,"Comptes rendus, Vol. XVI (1843), pp. 156, 281.
[461]Loc. cit.,Gerberti Operaetc.
[462]Friedlein thought it spurious. SeeZeitschrift für 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 (Œuvres 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 usessiposfor 0. It is not given by Bernelinus, and appears in Radulph of Laon, in the twelfth century. See Günther'sGeschichte, p. 98, n.; Weissenborn, p. 11; Pihan,Exposéetc., pp. xvi-xxii.
In Friedlein'sBoetius, 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 wordsiposfor 0. The names appear in a twelfth-century anonymous manuscript in the Vatican, in a passage beginning