The Hindu-Arabic Numerals c12: Chapter VIII. The Spread of the Numerals in Europe
The Hindu-Arabic Numerals c12
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Chapter VIII. The Spread of the Numerals in Europe
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Of all the medieval writers, probably the one most influential in introducing the new numerals to the scholars of Europe was Leonardo Fibonacci, of Pisa.¹ This remarkable man, the most noteworthy mathematical genius of the Middle Ages, was born at Pisa about 1175.²
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¹ 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 mathématiques*, Vol. II, p. 25; D. Martines, *Origine e progressi dell' aritmetica*, Messina, 1865, p. 47; Lucas, in Boncompagni *Bullettino*, Vol. X, pp. 129, 239; Besagne, ibid., Vol. IX, p. 583; Boncompagni, three works as cited in Chap. I; G. Eneström, "Ueber zwei angebliche mathematische Schulen im christlichen Mittelalter," *Bibliotheca Mathematica*, Vol. VIII (3), pp. 252–262; Boncompagni, "Della vita e delle opere di Leonardo Pisano," loc. cit.
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² The date is purely conjectural. See the *Bibliotheca Mathematica*, Vol. IV (3), p. 215.
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The traveler of to-day may cross the Via Fibonacci on his way to the Campo Santo, and there he may see at the end of the long corridor, across the quadrangle, the statue of Leonardo in scholar's garb. Few towns have honored a mathematician more, and few mathematicians have so distinctly honored their birthplace. Leonardo was born in the golden age of this city, the period of its commercial, religious, and intellectual prosperity.³
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³ 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.
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Situated practically at the mouth of the Arno, Pisa formed with Genoa and Venice the trio of the greatest commercial centers of Italy at the opening of the thirteenth century. Even before Venice had captured the Levantine trade, Pisa had close relations with the East. An old Latin chronicle relates that in 1005 “Pisa was captured by the Saracens,” that in the following year “the Pisans overthrew the Saracens at Reggio,” and that in 1012 “the Saracens came to Pisa and destroyed it.” The city soon recovered, however, sending no fewer than a hundred and twenty ships to Syria in 1099,¹ founding a merchant colony in Constantinople a few years later,² and meanwhile carrying on an interurban warfare in Italy that seemed to stimulate it to great activity.³ A writer of 1114 tells us that at that time there were many heathen people — Turks, Libyans, Parthians, and Chaldeans — to be found in Pisa. It was in the midst of such wars, in a cosmopolitan and commercial town, in a center where literary work was not appreciated,⁴ that the genius of Leonardo appears as one of the surprises of history, warning us again that “we should draw no horoscope; that we should expect little, for what we expect will not come to pass.”⁵
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¹ Heyd, loc. cit., Vol. I, p. 149.
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² Ibid., p. 211.
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³ J. A. Symonds, *Renaissance in Italy. The Age of Despots*. New York, 1883, p. 62.
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⁴ Symonds, loc. cit., p. 79.
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⁵ J. A. Froude, *The Science of History*, London, 1864. “Un brevet d’apothicaire n’empêcha pas Dante d’être le plus grand poète de l’Italie, et ce fut un petit marchand de Pise qui donna l’algèbre aux Chrétiens.” [Libri, *Histoire*, Vol. I, p. xvi.]
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Leonardo’s father was one William,⁶ and he had a brother named Bonaccingus,⁷ but nothing further is known of his family. As to Fibonacci, most writers¹ have assumed that his father's name was Bonaccio,² whence *filius Bonaccii*, or Fibonacci. Others³ believe that the name, even in the Latin form of *filius Bonaccii* as used in Leonardo's work, was simply a general one, like our Johnson or Bronson (Brown's son); and the only contemporary evidence that we have bears out this view. As to the name Bigollo, used by Leonardo, some have thought it a self-assumed one meaning blockhead, a term that had been applied to him by the commercial world or possibly by the university circle, and taken by him that he might prove what a blockhead could do. Milanesi,⁴ however, has shown that the word Bigollo (or Pigollo) was used in Tuscany to mean a traveler, and was naturally assumed by one who had studied, as Leonardo had, in foreign lands.
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⁶ A document of 1226, found and published in 1858, reads: “Leonardo bigollo quondam Guilielmi.”
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⁷ “Bonaccingo germano suo.”
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¹ E.g. Libri, Guglielmini, Tiraboschi.
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² Latin, *Bonaccius*.
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³ Boncompagni and Milanesi.
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⁴ Reprint, p. 5.
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Leonardo's father was a commercial agent at Bugia, the modern Bougie,⁵ the ancient Saldae on the coast of Barbary,⁶ a royal capital under the Vandals and again, a century before Leonardo, under the Beni Hammad. It had one of the best harbors on the coast, sheltered as it is by Mt. Lalla Guraia,⁷ and at the close of the twelfth century it was a center of African commerce. It was here that Leonardo was taken as a child, and here he went to school to a Moorish master. When he reached the years of young manhood he started on a tour of the Mediterranean Sea, and visited Egypt, Syria, Greece, Sicily, and Provence, meeting with scholars as well as with merchants, and imbibing a knowledge of the various systems of numbers in use in the centers of trade. All these systems, however, he says he counted almost as errors compared with that of the Hindus.¹ Returning to Pisa, he wrote his *Liber Abaci*² in 1202, rewriting it in 1228.³ In this work the numerals are explained and are used in the usual computations of business. Such a treatise was not destined to be popular, however, because it was too advanced for the mercantile class, and too novel for the conservative university circles. Indeed, at this time mathematics had only slight place in the newly established universities, as witness the oldest known statute of the Sorbonne at Paris, dated 1215, where the subject is referred to only in an incidental way.⁴ The period was one of great commercial activity, and on this very account such a book would attract even less attention than usual.¹
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⁵ Whence the French name for candle.
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⁶ Now part of Algiers.
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⁷ E. Reclus, *Africa*, New York, 1893, Vol. II, p. 253.
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¹ "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).
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² "Book of the Abacus," this term then being used, and long afterwards in Italy, to mean merely the arithmetic of computation.
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³ "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.
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⁴ I.e. in relation to the quadrivium. "Non legant in festivis diebus, nisi Philosophos et rhetoricas et quadrivalia et barbarismum et ethicam, si placet." Suter, *Die Mathematik auf den Universitäten des Mittelalters*, Zürich, 1887, p. 56. Roger Bacon gives a still more gloomy view of Oxford in his time in 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. Günther, *Geschichte des mathematischen Unterrichts im deutschen Mittelalter bis zum Jahre 1525*, Berlin, 1887, being Vol. III of *Monumenta Germaniae paedagogica*.
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¹ 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.
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It would now be thought that the western world would at once adopt the new numerals which Leonardo had made known, and which were so much superior to anything that had been in use in Christian Europe. The antagonism of the universities would avail but little, it would seem, against such an improvement. It must be remembered, however, that there was great difficulty in spreading knowledge at this time, some two hundred and fifty years before printing was invented. "Popes and princes and even great religious institutions possessed far fewer books than many farmers of the present age. The library belonging to the Cathedral Church of San Martino at Lucca in the ninth century contained only nineteen volumes of abridgments from ecclesiastical commentaries."² Indeed, it was not until the early part of the fifteenth century that Palla degli Strozzi took steps to carry out the project that had been in the mind of Petrarch, the founding of a public library. It was largely by word of mouth, therefore, that this early knowledge had to be transmitted. Fortunately the presence of foreign students in Italy at this time made this transmission feasible. (If human nature was the same then as now, it is not impossible that the very opposition of the faculties to the works of Leonardo led the students to investigate them the more zealously.) At Vicenza in 1209, for example, there were Bohemians, Poles, Frenchmen, Burgundians, Germans, and Spaniards, not to speak of representatives of divers towns of Italy; and what was true there was also true of other intellectual centers. The knowledge could not fail to spread, therefore, and as a matter of fact we find numerous bits of evidence that this was the case. Although the bankers of Florence were forbidden to use these numerals in 1299, and the statutes of the university of Padua required stationers to keep the price lists of books “non per cifras, sed per literas claros,”¹ the numerals really made much headway from about 1275 on.
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² J. A. Symonds, loc. cit., Vol. II, p. 127.
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¹ ^1 I. Taylor, *The Alphabet*, London, 1883, Vol. II, p. 263.
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It was, however, rather exceptional for the common people of Germany to use the Arabic numerals before the sixteenth century, a good witness to this fact being the popular almanacs. Calendars of 1457–1496² have generally the Roman numerals, while Köbel’s calendar of 1518 gives the Arabic forms as subordinate to the Roman. In the register of the Kreuzschule at Dresden the Roman forms were used even until 1539.
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² ^2 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 *Encyclopädie der Erziehung*, Vol. VI, p. 726; A. Kuckuk, “Die Rechenkunst im sechzehnten Jahrhundert,” *Festschrift zur dritten Säcularfeier des Berlinischen Gymnasiums zum grauen Kloster*, Berlin, 1874, p. 4.
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While not minimizing the importance of the scientific work of Leonardo of Pisa, we may note that the more popular treatises by Alexander de Villa Dei (*c.* 1240 a.d.) and John of Halifax (Sacrobosco, *c.* 1250 a.d.) were much more widely used, and doubtless contributed more to the spread of the numerals among the common people.
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The *Carmen de Algorismo*¹ of Alexander de Villa Dei was written in verse, as indeed were many other textbooks of that time. That it was widely used is evidenced by the large number of manuscripts² extant in European libraries. Sacrobosco’s *Algorismus*,³ in which some lines from the *Carmen* are quoted, enjoyed a wide popularity as a textbook for university instruction.⁴ The work was evidently written with this end in view, as numerous commentaries by university lecturers are found. Probably the most widely used of these was that of Petrus de Dacia⁵ written in 1291. These works throw an interesting light upon the method of instruction in mathematics in use in the universities from the thirteenth even to the sixteenth century. Evidently the text was first read and copied by students.⁶ Following this came line by line an exposition of the text, such as is given in Petrus de Dacia’s commentary.
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¹ The text is given in Halliwell, *Rara Mathematica*, London, 1839.
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² Seven are given in Ashmole’s *Catalogue of Manuscripts in the Oxford Library*, 1845.
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³ 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.
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⁴ J. Aschbach, *Geschichte der Wiener Universität im ersten Jahrhunderte ihres Bestehens*, Wien, 1865, p. 93.
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⁵ Curtze, loc. cit., gives the text.
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⁶ 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.
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Sacrobosco’s work is of interest also because it was probably due to the extended use of this work that the term *Arabic numerals* became common. In two places there is mention of the inventors of this system. In the introduction it is stated that this science of reckoning was due to a philosopher named Algus, whence the name *algorismus*,¹ and in the section on numeration reference is made to the Arabs as the inventors of this science.² While some of the commentators, Petrus de Dacia³ among them, knew of the Hindu origin, most of them undoubtedly took the text as it stood; and so the Arabs were credited with the invention of the system.
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¹ “Quidam philosophus edidit nomine Algus, unde et Algorismus nuncupatur.” [Curtze, loc. cit., p. 1.]
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² “Sinistrorsum autem scribimus in hac arte more arabico sive iudaico, huius scientiae inventorum.” [Curtze, loc. cit., p. 7.] The Plimpton manuscript omits the words “sive iudaico.”
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³ “Non enim omnis numerus per quascumque figuras Indorum repraesentatur, sed tantum determinatus per determinatam, ut 4 non per 5, ...” [Curtze, loc. cit., p. 25.]
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The first definite trace that we have of an algorism in the French language is found in a manuscript written about 1275.⁴ This interesting leaf, for the part on algorism consists of a single folio, was noticed by the Abbé Lebœuf as early as 1741,⁵ and by Daunou in 1824.⁶ It then seems to have been lost in the multitude of Paris manuscripts; for although Chasles⁷ relates his vain search for it, it was not rediscovered until 1882. In that year M. Ch. Henry found it, and to his care we owe our knowledge of the interesting manuscript. The work is anonymous and is devoted almost entirely to geometry, only two pages (one folio) relating to arithmetic. In these the forms of the numerals are given, and a very brief statement as to the operations, it being evident that the writer himself had only the slightest understanding of the subject.
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⁴ C. Henry, “Sur les deux plus anciens traités français d’algorisme et de géométrie,” *Boncompagni Bulletino*, Vol. XV, p. 49; Victor Mortet, “Le plus ancien traité français d’algorisme,” loc. cit.
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⁵ *L’État des sciences en France, depuis la mort du Roy Robert, arrivée en 1031, jusqu’à celle de Philippe le Bel, arrivée en 1314*, Paris, 1741.
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⁶ *Discours sur l’état des lettres en France au XIII^e siècle*, Paris, 1824.
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⁷ *Aperçu historique*, Paris, 1875 ed., p. 464.
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Once the new system was known in France, even thus superficially, it would be passed across the Channel to England. Higden,¹ writing soon after the opening of the fourteenth century, speaks of the French influence at that time and for some generations preceding:² “For two hundred years children in scole, agenst the usage and manir of all other nations beeth compelled for to leave hire own language, and for to construe hir lessons and hire thynges in Frensche. . . . Gentilmen children beeth taught to speke Frensche from the tyme that they bith rokked in hir cradell; and uplondissche men will likne himself to gentylmen, and fondeth with greet besynesse for to speke Frensche.”
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¹ 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.
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² Trevisa's translation, Higden having written in Latin.
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The question is often asked, why did not these new numerals attract more immediate attention? Why did they have to wait until the sixteenth century to be generally used in business and in the schools? In reply it may be said that in their elementary work the schools always wait upon the demands of trade. That work which pretends to touch the life of the people must come reasonably near doing so. Now the computations of business until about 1500 did not demand the new figures, for two reasons: First, cheap paper was not known. Paper-making of any kind was not introduced into Europe until the twelfth century, and cheap paper is a product of the nineteenth. Pencils, too, of the modern type, date only from the sixteenth century. In the second place, modern methods of operating, particularly of multiplying and dividing (operations of relatively greater importance when all measures were in compound numbers requiring reductions at every step), were not yet invented. The old plan required the erasing of figures after they had served their purpose, an operation very simple with counters, since they could be removed. The new plan did not as easily permit this. Hence we find the new numerals very tardily admitted to the counting-house, and not welcomed with any enthusiasm by teachers.¹
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¹ An illustration of this feeling is seen in the writings of Prosdocimo de' Beldomandi (b. c. 1370–1380, d. 1428): "Inveni in quam pluribus libris algorismi nuncupatis mores circa numeros operandi satis varios atque diversos, qui licet boni existerent atque veri erant, tamen fastidiosi, tum propter ipsarum regularum multitudinem, tum propter earum deleationes, tum etiam propter ipsarum operationum probationes, utrum si bone fuerint vel ne. Erant et etiam isti modi interim fastidiosi, quod si in aliquo calculo astroloico error contigisset, calculatorem operationem suam a capite incipere oportebat, dato quod error suus adhuc satis propinquus existeret; et hoc propter figuras in sua operatione deletas. Indigebat etiam calculator semper aliquo lapide vel sibi conformi, super quo scribere atque faciliter delere posset figuras cum quibus operabatur in calculo suo. Et quia haec omnia satis fastidiosa atque laboriosa mihi visa sunt, disposui libellum edere in quo omnia ista abicerentur: qui etiam algorismus sive liber de numeris denominari poterit. Scias tamen quod in hoc libello ponere non intendo nisi ea quae ad calculum necessaria sunt, alia quae in aliis libris practice arismetrice tanguntur, ad calculum non necessaria, propter brevitatem dimitendo." [Quoted by A. Nagl, *Zeitschrift für Mathematik und Physik, Hist.-lit. Abth.*, Vol. XXXIV, p. 143; Smith, *Rara Arithmetica*, p. 14, in facsimile.]
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Aside from their use in the early treatises on the new art of reckoning, the numerals appeared from time to time in the dating of manuscripts and upon monuments. The oldest definitely dated European document known to contain the numerals is a Latin manuscript,¹ the *Codex Vigilanus*, written in the Albelda Cloister not far from Logroño in Spain, in 976 a.d. The nine characters (of *ġobār* type), without the zero, are given as an addition to the first chapters of the third book of the *Origines* by Isidorus of Seville, in which the Roman numerals are under discussion. Another Spanish copy of the same work, of 992 a.d., contains the numerals in the corresponding section. The writer ascribes an Indian origin to them in the following words: "Item de figuris arithmeticę. Scire debemus in Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et geometrią et ceteris liberalibus disciplinis concedere. Et hoc manifestum est in nobem figuris, quibus designant unumquemque gradum cuiuslibet gradus. Quarum hec sunt forma." The nine *ġobār* characters follow. Some of the abacus forms² previously given are doubtless also of the tenth century. The earliest Arabic documents containing the numerals are two manuscripts of 874 and 888 a.d.³ They appear about a century later in a work⁴ written at Shiraz in 970 a.d. There is also an early trace of their use on a pillar recently discovered in a church apparently destroyed as early as the tenth century, not far from the Jeremias Monastery, in Egypt.
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¹ P. Ewald, loc. cit.; Franz Steffens, *Lateinische Paläographie*, pp. xxxix–xl. We are indebted to Professor J. M. Burnam for a photograph of this rare manuscript.
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² See the plate of forms on p. 88.
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³ Karabacek, loc. cit., p. 56; Karpinski, "Hindu Numerals in the Fihrist," *Bibliotheca Mathematica*, Vol. XI (3), p. 121.
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⁴ Woepcke, "Sur une donnée historique," etc., loc. cit., and "Essai d'une restitution de travaux perdus d'Apollonius sur les quantités irrationnelles, d'après des indications tirées d'un manuscrit arabe," *Tome XIV des Mémoires présentés par divers savants à l'Académie des sciences*, Paris, 1856, note, pp. 6–14.
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A graffito in Arabic on this pillar has the date 349 a.h., which corresponds to 961 a.d.¹ For the dating of Latin documents the Arabic forms were used as early as the thirteenth century.²
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¹ *Archeological Report of the Egypt Exploration Fund for 1908–1909*, London, 1910, p. 18.
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² There was a set of astronomical tables in Boncompagni's library bearing this date: "Nota quod anno dñi n̂ri ihû xp̂i. 1264. perfecto." See Narducci's *Catalogo*, p. 130.
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On the early use of these numerals in Europe the only scientific study worthy the name is that made by Mr. G. F. Hill of the British Museum.³ From his investigations it appears that the earliest occurrence of a date in these numerals on a coin is found in the reign of Roger of Sicily in 1138.⁴ Until recently it was thought that the earliest such date was 1217 a.d. for an Arabic piece and 1388 for a Turkish one.⁵ Most of the seals and medals containing dates that were at one time thought to be very early have been shown by Mr. Hill to be of relatively late workmanship. There are, however, in European manuscripts, numerous instances of the use of these numerals before the twelfth century. Besides the example in the Codex Vigilanus, another of the tenth century has been found in the St. Gall MS. now in the University Library at Zürich, the forms differing materially from those in the Spanish codex.
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³ "On the Early use of Arabic Numerals in Europe," read before the Society of Antiquaries April 14, 1910, and published in *Archæologia* in the same year.
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⁴ Ibid., p. 8, n. The date is part of an Arabic inscription.
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⁵ O. Codrington, *A Manual of Musalman Numismatics*, London, 1904.
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The third specimen in point of time in Mr. Hill's list is from a Vatican MS. of 1077. The fourth and fifth specimens are from the Erlangen MS. of Boethius, of the same (eleventh) century, and the sixth and seventh are also from an eleventh-century MS. of Boethius at Chartres.
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Earliest Manuscript Forms
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These and other early forms are given by Mr. Hill in this table, which is reproduced with his kind permission.
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This is one of more than fifty tables given in Mr. Hill's valuable paper, and to this monograph students are referred for details as to the development of number-forms in Europe from the tenth to the sixteenth century. It is of interest to add that he has found that among the earliest dates of European coins or medals in these numerals, after the Sicilian one already mentioned, are the following: Austria, 1484; Germany, 1489 (Cologne); Switzerland, 1424 (St. Gall); Netherlands, 1474; France, 1485; Italy, 1390.¹
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¹ See Arbuthnot, *The Mysteries of Chronology*, London, 1900, pp. 75, 78, 98; F. Pichler, *Repertorium der steierischen Münzkunde*, Grätz, 1875, where the claim is made of an Austrian coin of 1458; *Bibliotheca Mathematica*, Vol. X (2), p. 120, and Vol. XII (2), p. 120. There is a Brabant piece of 1478 in the collection of D. E. Smith.
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The earliest English coin dated in these numerals was struck in 1551,² although there is a Scotch piece of 1539.³ In numbering pages of a printed book these numerals were first used in a work of Petrarch's published at Cologne in 1471.⁴ The date is given in the following form in the *Biblia Pauperum*,⁵ a block-book of 1470, while in
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² A specimen is in the British Museum. [Arbuthnot, p. 79.]
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³ Ibid., p. 79.
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⁴ *Liber de Remediis utriusque fortunae Coloniae*.
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⁵ Fr. Walthern et Hans Hurning, Nördlingen.
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another block-book which possibly goes back to *c.* 1430⁶ the numerals appear in several illustrations, with forms as follows:
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⁶ *Ars Memorandi*, one of the oldest European block-books.
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Many printed works anterior to 1471 have pages or chapters numbered by hand, but many of these numerals are of date much later than the printing of the work. Other works were probably numbered directly after printing. Thus the chapters 2, 3, 4, 5, 6 in a book of 1470¹ are numbered as follows: Capitulem zm., . . . 3m., . . . 4m., . . . v, . . . vi, and followed by Roman numerals. This appears in the body of the text, in spaces left by the printer to be filled in by hand. Another book² of 1470 has pages numbered by hand with a mixture of Roman and Hindu numerals, thus,
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¹ Eusebius Caesariensis, *De praeparatione evangelica*, Venice, Jenson, 1470. The above statement holds for copies in the Astor Library and in the Harvard University Library.
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² Francisco de Retza, *Comestorium vitiorum*, Nürnberg, 1470. The copy referred to is in the Astor Library.
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As to monumental inscriptions,³ there was once thought to be a gravestone at Katharein, near Troppau, with the date 1007, and one at Biebrich of 1299. There is no doubt, however, of one at Pforzheim of 1371 and one at Ulm of 1388.⁴ Certain numerals on Wells Cathedral have been assigned to the thirteenth century, but they are undoubtedly considerably later.⁵
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³ See Mauch, "Ueber den Gebrauch arabischer Ziffern und die Veränderungen derselben," *Anzeiger für Kunde der deutschen Vorzeit*, 1861, columns 46, 81, 116, 151, 189, 229, and 268; Calmet, *Recherches sur l'origine des chiffres d'arithmétique*, plate, loc. cit.
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⁴ Günther, *Geschichte*, p. 175, n.; Mauch, loc. cit.
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⁵ These are given by W. R. Lethaby, from drawings by J. T. Irvine, in the *Proceedings of the Society of Antiquaries*, 1906, p. 200.
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The table on page 143 will serve to supplement that from Mr. Hill's work.⁶
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⁶ There are some ill-tabulated forms to be found in J. Bowring, *The Decimal System*, London, 1854, pp. 23, 25, and in L. A. Chassant, *Dictionnaire des abréviations latines et françaises . . . du moyen âge*,
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Early Manuscript Forms
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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 paléographie latine et française*, 2d ed., Paris, 1892, p. 164; A. Cappelli, *Dizionario di abbreviature latine ed italiane*, Milan, 1899. An interesting early source is found in the rare Caxton work of 1480, *The Myrrour of the World*. In Chap. X is a cut with the various numerals, the chapter beginning “The fourth scyence is called arsmetrique.” Two of the fifteen extant copies of this work are at present in the library of Mr. J. P. Morgan, in New York.
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a From the twelfth-century manuscript on arithmetic, Curtze, loc. cit., *Abhandlungen*, and Nagl, loc. cit. The forms are copied from Plate VII in *Zeitschrift für Mathematik und Physik*, Vol. XXXIV.
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b 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.
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c French Algorismus of 1275; from an unpublished photograph of the original, in the possession of D. E. Smith. See also p. 135.
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d From a manuscript of Boethius c. 1294, in Mr. Plimpton's library. Smith, *Rara Arithmetica*, Plate I.
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e Numerals in a 1303 manuscript in Sigmaringen, copied from Wattenbach, loc. cit., p. 102.
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f 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.
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g From the copy of Sacrobosco's *Algorismus* in Mr. Plimpton's library. Date c. 1442. See Smith, *Rara Arithmetica*, p. 450.
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For the sake of further comparison, three illustrations from works in Mr. Plimpton's library, reproduced from the *Rara Arithmetica*, may be considered. The first is from a Latin manuscript on arithmetic,¹ of which the original was written at Paris in 1424 by Rollandus, a Portuguese physician, who prepared the work at the command of John of Lancaster, Duke of Bedford, at one time Protector of England and Regent of France, to whom the work is dedicated. The figures show the successive powers of 2. The second illustration is from Luca da Firenze's *Inprencipio darte dabacho*,² c. 1475, and the third is from an anonymous manuscript³ of about 1500.
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¹ See *Rara Arithmetica*, pp. 446–447.
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² Ibid., pp. 469–470.
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³ Ibid., pp. 477–478.
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As to the forms of the numerals, fashion played a leading part until printing was invented. This tended to fix these forms, although in writing there is still a great variation,
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as witness the French 5 and the German 7 and 9. Even in printing there is not complete uniformity, and it is often difficult for a foreigner to distinguish between the 3 and 5 of the French types.
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As to the particular numerals, the following are some of the forms to be found in the later manuscripts and in the early printed books.
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1. In the early printed books "one" was often i, perhaps to save types, just as some modern typewriters use the same character for l and 1.¹ In the manuscripts the "one" appears in such forms as ²
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¹ 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 Würzburg in 1660, in Mr. Plimpton's library, the ones are all in the form of i.
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² Thus the date [Figure:], 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.
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2. "Two" often appears as z in the early printed books, 12 appearing as iz.³ In the medieval manuscripts the following forms are common :⁴
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³ E. g. Chiarini's work of 1481; Clichtoveus (c. 1507).
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⁴ 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.
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It is evident, from the early traces, that it is merely a cursive form for the primitive =, just as 3 comes from ≡, as in the Nānā Ghāt inscriptions.
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3. "Three" usually had a special type in the first printed books, although occasionally it appears as [Figure:].¹ In the medieval manuscripts it varied rather less than most of the others. The following are common forms:²
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¹ Thus Chiarini (1481) has 13 for 23.
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² 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.
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4. "Four" has changed greatly; and one of the first tests as to the age of a manuscript on arithmetic, and the place where it was written, is the examination of this numeral. Until the time of printing the most common form was [Figure:], although the Florentine manuscript of Leonard of Pisa's work has the form [Figure:];³ but the manuscripts show that the Florentine arithmeticians and astronomers rather early began to straighten the first of these forms up to forms like [Figure:]⁴ and [Figure:]⁴ or [Figure:], ⁵ more closely resembling our own. The first printed books generally used our present form⁶ with the closed top [Figure:], the open top used in writing (4) being purely modern. The following are other forms of the four, from various manuscripts:¹
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³ See Nagl, loc. cit.
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⁴ Hannover algorismus, thirteenth century.
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⁵ See the Dagomari manuscript, in *Rara Arithmetica*, pp. 435, 437-440.
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⁶ 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.
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¹ 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.
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5. "Five" also varied greatly before the time of printing. The following are some of the forms:²
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² 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 (5), fifteenth (13) centuries, respectively. The third line is from Chassant, loc. cit., p. 113, no mention of dates.
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6. "Six" has changed rather less than most of the others. The chief variation has been in the slope of the top, as will be seen in the following:³
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³ 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.
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7. "Seven," like "four," has assumed its present erect form only since the fifteenth century. In medieval times it appeared as follows:⁴ 8. "Eight," like "six," has changed but little. In medieval times there are a few variants of interest as follows:¹
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⁴ 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.
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¹ All of these are given by Cappelli, thirteenth, fourteenth, fifteenth (2), and sixteenth centuries, respectively.
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In the sixteenth century, however, there was manifested a tendency to write it ∞.²
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² 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.*
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9. "Nine" has not varied as much as most of the others. Among the medieval forms are the following:³
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³ 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. Ϙ was the old Greek symbol for 90.
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0. The shape of the zero also had a varied history. The following are common medieval forms:⁴
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⁴ 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.
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The explanation of the place value was a serious matter to most of the early writers. If they had been using an abacus constructed like the Russian chotü, and had placed this before all learners of the positional system, there would have been little trouble. But the medieval line-reckoning, where the lines stood for powers of 10 and the spaces for half of such powers, did not lend itself to this comparison. Accordingly we find such labored explanations as the following, from *The Crafte of Nombrynge*:
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“Euery of these figuris bitokens hym selfe & no more, yf he stonde in the first place of the rewele. . . .
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“If it stonde in the secunde place of the rewle, he betokens ten tymes hym selfe, as this figure 2 here 20 tokens ten tyme hym selfe, that is twenty, for he hym selfe betokens tweyne, & ten tymes twene is twenty. And for he stondis on the lyft side & in the secunde place, he betokens ten tyme hym selfe. And so go forth. . . .
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“Nil cifra significat sed dat signare sequenti. Expone this verse. A cifre tokens noȝt, bot he makes the figure to betoken that comes after hym more than he shuld & he were away, as thus 10. here the figure of one tokens ten, & yf the cifre were away & no figure byfore hym he schuld token bot one, for than he schuld stonde in the first place. . . .”¹
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¹ Smith, *An Early English Algorism*.
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It would seem that a system that was thus used for dating documents, coins, and monuments, would have been generally adopted much earlier than it was, particularly in those countries north of Italy where it did not come into general use until the sixteenth century. This, however, has been the fate of many inventions, as witness our neglect of logarithms and of contracted processes to-day.
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As to Germany, the fifteenth century saw the rise of the new symbolism; the sixteenth century saw it slowly gain the mastery; the seventeenth century saw it finally conquer the system that for two thousand years had dominated the arithmetic of business. Not a little of the success of the new plan was due to Luther's demand that all learning should go into the vernacular.¹
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¹ Kuckuck, p. 5.
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During the transition period from the Roman to the Arabic numerals, various anomalous forms found place. For example, we have in the fourteenth century ca for 104;² 1000.300.80 et 4 for 1384;³ and in a manuscript of the fifteenth century 12901 for 1291.⁴ In the same century m.cccc. 8II appears for 1482,⁵ while M°CCCC°50 (1450) and MCCCCXL6 (1446) are used by Theodoricus Ruffi about the same time.⁶ To the next century belongs the form 1vojj for 1502. Even in Sfortunati's *Nuovo lume*⁷ the use of ordinals is quite confused, the propositions on a single page being numbered "tertia," "4," and "V."
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² A. Cappelli, loc. cit., p. 372.
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³ Smith, *Rara Arithmetica*, p. 443.
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⁴ Curtze, *Petri Philomeni de Dacia etc.*, p. ix.
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⁵ Cappelli, loc. cit., p. 376.
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⁶ Curtze, loc. cit., pp. viii-ix, note.
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⁷ Edition of 1544-1545, f. 52.
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Although not connected with the Arabic numerals in any direct way, the medieval astrological numerals may here be mentioned. These are given by several early writers, but notably by Noviomagus (1539),⁸ as follows⁹:
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⁸ *De numeris libri II*, 1544 ed., cap. xv. Heilbronner, loc. cit., p. 736, also gives them, and compares this with other systems.
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⁹ Noviomagus says of them: "De quibusdam Astrologicis, sive Chaldaicis numerorum notis. . . . Sunt & aliæ quædam notæ, quibus Chaldaei & Astrologii quemlibet numerum artificiose & arguté describunt, scitu periucundae, quas nobis communicauit Rodolphus Paludanus Nouiomagus."
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Thus we find the numerals gradually replacing the Roman forms all over Europe, from the time of Leonardo of Pisa until the seventeenth century. But in the Far East to-day they are quite unknown in many countries, and they still have their way to make. In many parts of India, among the common people of Japan and China, in Siam and generally about the Malay Peninsula, in Tibet, and among the East India islands, the natives still adhere to their own numeral forms. Only as Western civilization is making its way into the commercial life of the East do the numerals as used by us find place, save as the Sanskrit forms appear in parts of India. It is therefore with surprise that the student of mathematics comes to realize how modern are these forms so common in the West, how limited is their use even at the present time, and how slow the world has been and is in adopting such a simple device as the Hindu-Arabic numerals.
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