Chapter 8

THE SPREAD OF THE NUMERALS IN EUROPE

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.[515] This remarkable man, the most noteworthy mathematical genius of the Middle Ages, was born at Pisa about 1175.[516]

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 scholars 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.[517] {129} 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,[518] founding a merchant colony in Constantinople a few years later,[519] and meanwhile carrying on an interurban warfare in Italy that seemed to stimulate it to great activity.[520] 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,[521] 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."[522]

Leonardo's father was one William,[523] and he had a brother named Bonaccingus,[524] but nothing further is {130} known of his family. As to Fibonacci, most writers[525] have assumed that his father's name was Bonaccio,[526] whence _filius Bonaccii_, or Fibonacci. Others[527] 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,[528] 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.

Leonardo's father was a commercial agent at Bugia, the modern Bougie,[529] the ancient Saldae on the coast of Barbary,[530] 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,[531] 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 {131} 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.[532] Returning to Pisa, he wrote his _Liber Abaci_[533] in 1202, rewriting it in 1228.[534] 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.[535] The period was one of great commercial activity, and on this very {132} account such a book would attract even less attention than usual.[536]

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."[537] 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 {133} 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,"[538] the numerals really made much headway from about 1275 on.

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 1457-1496[539] 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.

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.

{134}

The _Carmen de Algorismo_[540] 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 manuscripts[541] extant in European libraries. Sacrobosco's _Algorismus_,[542] in which some lines from the Carmen are quoted, enjoyed a wide popularity as a textbook for university instruction.[543] 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 Dacia[544] 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.[545] Following this came line by line an exposition of the text, such as is given in Petrus de Dacia's commentary.

Sacrobosco's work is of interest also because it was probably due to the extended use of this work that the {135} 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_,[546] and in the section on numeration reference is made to the Arabs as the inventors of this science.[547] While some of the commentators, Petrus de Dacia[548] 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.

The first definite trace that we have of an algorism in the French language is found in a manuscript written about 1275.[549] This interesting leaf, for the part on algorism consists of a single folio, was noticed by the Abbé Leboeuf as early as 1741,[550] and by Daunou in 1824.[551] It then seems to have been lost in the multitude of Paris manuscripts; for although Chasles[552] 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 {136} 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.

Once the new system was known in France, even thus superficially, it would be passed across the Channel to England. Higden,[553] writing soon after the opening of the fourteenth century, speaks of the French influence at that time and for some generations preceding:[554] "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."

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 {137} 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.[555]

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 {138} to contain the numerals is a Latin manuscript,[556] the Codex Vigilanus, written in the Albelda Cloister not far from Logroño in Spain, in 976 A.D. The nine characters (of [.g]ob[=a]r 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 arithmetic[e,]. Scire debemus in Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et geometria et ceteris liberalibus disciplinis concedere. Et hoc manifestum est in nobem figuris, quibus designant unumquemque gradum cuiuslibet gradus. Quarum hec sunt forma." The nine [.g]ob[=a]r characters follow. Some of the abacus forms[557] 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.[558] They appear about a century later in a work[559] 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. {139} A graffito in Arabic on this pillar has the date 349 A.H., which corresponds to 961 A.D.[560] For the dating of Latin documents the Arabic forms were used as early as the thirteenth century.[561]

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.[562] 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.[563] Until recently it was thought that the earliest such date was 1217 A.D. for an Arabic piece and 1388 for a Turkish one.[564] 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.

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 {140} (eleventh) century, and the sixth and seventh are also from an eleventh-century MS. of Boethius at Chartres. These and other early forms are given by Mr. Hill in this table, which is reproduced with his kind permission.

EARLIEST MANUSCRIPT FORMS

This is one of more than fifty tables given in Mr. Hill's valuable paper, and to this monograph students {141} 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.[565]

The earliest English coin dated in these numerals was struck in 1551,[566] although there is a Scotch piece of 1539.[567] In numbering pages of a printed book these numerals were first used in a work of Petrarch's published at Cologne in 1471.[568] The date is given in the following form in the _Biblia Pauperum_,[569] a block-book of 1470,

while in another block-book which possibly goes back to c. 1430[570] the numerals appear in several illustrations, with forms as follows:

Many printed works anterior to 1471 have pages or chapters numbered by hand, but many of these numerals are {142} 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 1470[571] are numbered as follows: Capitulem [Symbol 2]m.,... [Symbol 3]m.,... 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 book[572] of 1470 has pages numbered by hand with a mixture of Roman and Hindu numerals, thus,

The table on page 143 will serve to supplement that from Mr. Hill's work.[576]

{143}

EARLY MANUSCRIPT FORMS

[577] [Illustration] Twelfth century A.D. [578] [Illustration] 1197 A.D. [579] [Illustration] 1275 A.D. [580] [Illustration] c. 1294 A.D. [581] [Illustration] c. 1303 A.D. [582] [Illustration] c. 1360 A.D. [583] [Illustration] c. 1442 A.D.

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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,[584] 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_,[585] c. 1475, and the third is from an anonymous manuscript[586] of about 1500.

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, as witness the French 5 and the German 7 and 9. Even in printing there is not complete uniformity, {145} and it is often difficult for a foreigner to distinguish between the 3 and 5 of the French types.

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.

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.[587] In the manuscripts the "one" appears in such forms as[588]

2. "Two" often appears as z in the early printed books, 12 appearing as iz.[589] In the medieval manuscripts the following forms are common:[590]

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It is evident, from the early traces, that it is merely a cursive form for the primitive [2 horizontal strokes], just as 3 comes from [3 horizontal strokes], as in the N[=a]n[=a] Gh[=a]t inscriptions.

3. "Three" usually had a special type in the first printed books, although occasionally it appears as [Symbol].[591] In the medieval manuscripts it varied rather less than most of the others. The following are common forms:[592]

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 [Symbol], although the Florentine manuscript of Leonard of Pisa's work has the form [Symbol];[593] but the manuscripts show that the Florentine arithmeticians and astronomers rather early began to straighten the first of these forms up to forms like [Symbol][594] and [Symbol][594] or [Symbol],[595] more closely resembling our own. The first printed books generally used our present form[596] with the closed top [Symbol], the open top used in writing ( [Symbol]) being {147} purely modern. The following are other forms of the four, from various manuscripts:[597]

5. "Five" also varied greatly before the time of printing. The following are some of the forms:[598]

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:[599]

7. "Seven," like "four," has assumed its present erect form only since the fifteenth century. In medieval times it appeared as follows:[600]

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8. "Eight," like "six," has changed but little. In medieval times there are a few variants of interest as follows:[601]

In the sixteenth century, however, there was manifested a tendency to write it [Symbol].[602]

9. "Nine" has not varied as much as most of the others. Among the medieval forms are the following:[603]

0. The shape of the zero also had a varied history. The following are common medieval forms:[604]

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 {149} 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_:

"Euery of these figuris bitokens hym selfe & no more, yf he stonde in the first place of the rewele....

"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....

"Nil cifra significat sed dat signare sequenti. Expone this verse. A cifre tokens no[gh]t, 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...."[605]

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.

As to Germany, the fifteenth century saw the rise of the new symbolism; the sixteenth century saw it slowly {150} 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.[606]

During the transition period from the Roman to the Arabic numerals, various anomalous forms found place. For example, we have in the fourteenth century c[alpha] for 104;[607] 1000. 300. 80 et 4 for 1384;[608] and in a manuscript of the fifteenth century 12901 for 1291.[609] In the same century m. cccc. 8II appears for 1482,[610] while M^oCCCC^o50 (1450) and MCCCCXL6 (1446) are used by Theodoricus Ruffi about the same time.[611] To the next century belongs the form 1vojj for 1502. Even in Sfortunati's _Nuovo lume_[612] the use of ordinals is quite confused, the propositions on a single page being numbered "tertia," "4," and "V."

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),[613] as follows[614]:

{151}

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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INDEX

_Transcriber's note: many of the entries refer to footnotes linked from the page numbers given._

Abbo of Fleury, 122 `Abdall[=a]h ibn al-[H.]asan, 92 `Abdallat[=i]f ibn Y[=u]suf, 93 `Abdalq[=a]dir ibn `Al[=i] al-Sakh[=a]w[=i], 6 Abenragel, 34 Abraham ibn Meïr ibn Ezra, _see_ Rabbi ben Ezra Ab[=u] `Al[=i] al-[H.]osein ibn S[=i]n[=a], 74 Ab[=u] 'l-[H.]asan, 93, 100 Ab[=u] 'l-Q[=a]sim, 92 Ab[=u] 'l-[T.]eiyib, 97 Ab[=u] Na[s.]r, 92 Ab[=u] Roshd, 113 Abu Sahl Dunash ibn Tamim, 65, 67 Adelhard of Bath, 5, 55, 97, 119, 123, 126 Adhemar of Chabanois, 111 A[h.]med al-Nasaw[=i], 98 A[h.]med ibn `Abdall[=a]h, 9, 92 A[h.]med ibn Mo[h.]ammed, 94 A[h.]med ibn `Omar, 93 Ak[s.]aras, 32 Alanus ab Insulis, 124 Al-Ba[.g]d[=a]d[=i], 93 Al-Batt[=a]n[=i], 54 Albelda (Albaida) MS., 116 Albert, J., 62 Albert of York, 103 Al-B[=i]r[=u]n[=i], 6, 41, 49, 65, 92, 93 Alcuin, 103 Alexander the Great, 76 Alexander de Villa Dei, 11, 133 Alexandria, 64, 82 Al-Faz[=a]r[=i], 92 Alfred, 103 Algebra, etymology, 5 Algerian numerals, 68 Algorism, 97 Algorismus, 124, 126, 135 Algorismus cifra, 120 Al-[H.]a[s.][s.][=a]r, 65 `Al[=i] ibn Ab[=i] Bekr, 6 `Al[=i] ibn A[h.]med, 93, 98 Al-Kar[=a]b[=i]s[=i], 93 Al-Khow[=a]razm[=i], 4, 9, 10, 92, 97, 98, 125, 126 Al-Kind[=i], 10, 92 Almagest, 54 Al-Ma[.g]reb[=i], 93 Al-Ma[h.]all[=i], 6 Al-M[=a]m[=u]n, 10, 97 Al-Man[s.][=u]r, 96, 97 Al-Mas`[=u]d[=i], 7, 92 Al-Nad[=i]m, 9 Al-Nasaw[=i], 93, 98 Alphabetic numerals, 39, 40, 43 Al-Q[=a]sim, 92 Al-Qass, 94 Al-Sakh[=a]w[=i], 6 Al-[S.]ardaf[=i], 93 Al-Sijz[=i], 94 Al-S[=u]f[=i], 10, 92 Ambrosoli, 118 A[.n]kapalli, 43 Apices, 87, 117, 118 Arabs, 91-98 Arbuthnot, 141 {154} Archimedes, 15, 16 Arcus Pictagore, 122 Arjuna, 15 Arnold, E., 15, 102 Ars memorandi, 141 [=A]ryabha[t.]a, 39, 43, 44 Aryan numerals, 19 Aschbach, 134 Ashmole, 134 A['s]oka, 19, 20, 22, 81 A[s.]-[s.]ifr, 57, 58 Astrological numerals, 150 Atharva-Veda, 48, 49, 55 Augustus, 80 Averroës, 113 Avicenna, 58, 74, 113

Babylonian numerals, 28 Babylonian zero, 51 Bacon, R., 131 Bactrian numerals, 19, 30 Bæda, 2, 72 Bagdad, 4, 96 Bakh[s.][=a]l[=i] manuscript, 43, 49, 52, 53 Ball, C. J., 35 Ball, W. W. R., 36, 131 B[=a][n.]a, 44 Barth, A., 39 Bayang inscriptions, 39 Bayer, 33 Bayley, E. C., 19, 23, 30, 32, 52, 89 Beazley, 75 Bede, _see_ Bæda Beldomandi, 137 Beloch, J., 77 Bendall, 25, 52 Benfey, T., 26 Bernelinus, 88, 112, 117, 121 Besagne, 128 Besant, W., 109 Bettino, 36 Bhandarkar, 18, 47, 49 Bh[=a]skara, 53, 55 Biernatzki, 32 Biot, 32 Björnbo, A. A., 125, 126 Blassière, 119 Bloomfield, 48 Blume, 85 Boeckh, 62 Boehmer, 143 Boeschenstein, 119 Boethius, 63, 70-73, 83-90 Boissière, 63 Bombelli, 81 Bonaini, 128 Boncompagni, 5, 6, 10, 48, 49, 123, 125 Borghi, 59 Borgo, 119 Bougie, 130 Bowring, J., 56 Brahmagupta, 52 Br[=a]hma[n.]as, 12, 13 Br[=a]hm[=i], 19, 20, 31, 83 Brandis, J., 54 B[r.]hat-Sa[m.]hita, 39, 44, 78 Brockhaus, 43 Bubnov, 65, 84, 110, 116 Buddha, education of, 15, 16 Büdinger, 110 Bugia, 130 Bühler, G., 15, 19, 22, 31, 44, 49 Burgess, 25 Bürk, 13 Burmese numerals, 36 Burnell, A. C., 18, 40 Buteo, 61

Calandri, 59, 81 Caldwell, R., 19 Calendars, 133 Calmet, 34 Cantor, M., 5, 13, 30, 43, 84 {155} Capella, 86 Cappelli, 143 Caracteres, 87, 113, 117, 119 Cardan, 119 Carmen de Algorismo, 11, 134 Casagrandi, 132 Casiri, 8, 10 Cassiodorus, 72 Cataldi, 62 Cataneo, 3 Caxton, 143, 146 Ceretti, 32 Ceylon numerals, 36 Chalfont, F. H., 28 Champenois, 60 Characters, _see_ Caracteres Charlemagne, 103 Chasles, 54, 60, 85, 116, 122, 135 Chassant, L. A., 142 Chaucer, 121 Chiarini, 145, 146 Chiffre, 58 Chinese numerals, 28, 56 Chinese zero, 56 Cifra, 120, 124 Cipher, 58 Circulus, 58, 60 Clichtoveus, 61, 119, 145 Codex Vigilanus, 138 Codrington, O., 139 Coins dated, 141 Colebrooke, 8, 26, 46, 53 Constantine, 104, 105 Cosmas, 82 Cossali, 5 Counters, 117 Courteille, 8 Coxe, 59 Crafte of Nombrynge, 11, 87, 149 Crusades, 109 Cunningham, A., 30, 75 Curtze, 55, 59, 126, 134 Cyfra, 55

Dagomari, 146 D'Alviella, 15 Dante, 72 Dasypodius, 33, 67, 63 Daunou, 135 Delambre, 54 Devan[=a]gar[=i], 7 Devoulx, A., 68 Dhruva, 49 Dicæarchus of Messana, 77 Digits, 119 Diodorus Siculus, 76 Du Cange, 62 Dumesnil, 36 Dutt, R. C., 12, 15, 18, 75 Dvived[=i], 44

East and West, relations, 73-81, 100-109 Egyptian numerals, 27 Eisenlohr, 28 Elia Misrachi, 57 Enchiridion Algorismi, 58 Eneström, 5, 48, 59, 97, 125, 128 Europe, numerals in, 63, 99, 128, 136 Eusebius Caesariensis, 142 Euting, 21 Ewald, P., 116

Fazzari, 53, 54 Fibonacci, _see_ Leonardo of Pisa Figura nihili, 58 Figures, 119. _See_ numerals. Fihrist, 67, 68, 93 Finaeus, 57 Firdus[=i], 81 Fitz Stephen, W., 109 Fleet, J. C., 19, 20, 49 {156} Florus, 80 Flügel, G., 68 Francisco de Retza, 142 François, 58 Friedlein, G., 84, 113, 116, 122 Froude, J. A., 129

Gandh[=a]ra, 19 Garbe, 48 Gasbarri, 58 Gautier de Coincy, 120, 124 Gemma Frisius, 2, 3, 119 Gerber, 113 Gerbert, 108, 110-120, 122 Gerhardt, C. I., 43, 56, 93, 118 Gerland, 88, 123 Gherard of Cremona, 125 Gibbon, 72 Giles, H. A., 79 Ginanni, 81 Giovanni di Danti, 58 Glareanus, 4, 119 Gnecchi, 71, 117 [.G]ob[=a]r numerals, 65, 100, 112, 124, 138 Gow, J., 81 Grammateus, 61 Greek origin, 33 Green, J. R., 109 Greenwood, I., 62, 119 Guglielmini, 128 Gulist[=a]n, 102 Günther, S., 131 Guyard, S., 82

[H.]abash, 9, 92 Hager, J. (G.), 28, 32 Halliwell, 59, 85 Hankel, 93 H[=a]r[=u]n al-Rash[=i]d, 97, 106 Havet, 110 Heath, T. L., 125 Hebrew numerals, 127 Hecatæus, 75 Heiberg, J. L., 55, 85, 148 Heilbronner, 5 Henry, C., 5, 31, 55, 87, 120, 135 Heriger, 122 Hermannus Contractus, 123 Herodotus, 76, 78 Heyd, 75 Higden, 136 Hill, G. F., 52, 139, 142 Hillebrandt, A., 15, 74 Hilprecht, H. V., 28 Hindu forms, early, 12 Hindu number names, 42 Hodder, 62 Hoernle, 43, 49 Holywood, _see_ Sacrobosco Hopkins, E. W., 12 Horace, 79, 80 [H.]osein ibn Mo[h.]ammed al-Ma[h.]all[=i], 6 Hostus, M., 56 Howard, H. H., 29 Hrabanus Maurus, 72 Huart, 7 Huet, 33 Hugo, H., 57 Humboldt, A. von, 62 Huswirt, 58

Iamblichus, 81 Ibn Ab[=i] Ya`q[=u]b, 9 Ibn al-Adam[=i], 92 Ibn al-Bann[=a], 93 Ibn Khord[=a][d.]beh, 101, 106 Ibn Wahab, 103 India, history of, 14 writing in, 18 Indicopleustes, 83 Indo-Bactrian numerals, 19 {157} Indr[=a]j[=i], 23 Is[h.][=a]q ibn Y[=u]suf al-[S.]ardaf[=i], 93

Jacob of Florence, 57 Jacquet, E., 38 Jamshid, 56 Jehan Certain, 59 Jetons, 58, 117 Jevons, F. B., 76 Johannes Hispalensis, 48, 88, 124 John of Halifax, _see_ Sacrobosco John of Luna, _see_ Johannes Hispalensis Jordan, L., 58, 124 Joseph Ispanus (Joseph Sapiens), 115 Justinian, 104

Kále, M. R., 26 Karabacek, 56 Karpinski, L. C., 126, 134, 138 K[=a]ty[=a]yana, 39 Kaye, C. R., 6, 16, 43, 46, 121 Keane, J., 75, 82 Keene, H. G., 15 Kern, 44 Kharo[s.][t.]h[=i], 19, 20 Khosr[=u], 82, 91 Kielhorn, F., 46, 47 Kircher, A., 34 Kit[=a]b al-Fihrist, _see_ Fihrist Kleinwächter, 32 K[=l]os, 62 Köbel, 4, 58, 60, 119, 123 Krumbacher, K., 57 Kuckuck, 62, 133 Kugler, F. X., 51

Lachmann, 85 Lacouperie, 33, 35 Lalitavistara, 15, 17 Lami, G., 57 La Roche, 61 Lassen, 39 L[=a][t.]y[=a]yana, 39 Leboeuf, 135 Leonardo of Pisa, 5, 10, 57, 64, 74, 120, 128-133 Lethaby, W. R., 142 Levi, B., 13 Levias, 3 Libri, 73, 85, 95 Light of Asia, 16 Luca da Firenze, 144 Lucas, 128

Mah[=a]bh[=a]rata, 18 Mah[=a]v[=i]r[=a]c[=a]rya, 53 Malabar numerals, 36 Malayalam numerals, 36 Mannert, 81 Margarita Philosophica, 146 Marie, 78 Marquardt, J., 85 Marshman, J. C., 17 Martin, T. H., 30, 62, 85, 113 Martines, D. C., 58 M[=a]sh[=a]ll[=a]h, 3 Maspero, 28 Mauch, 142 Maximus Planudes, 2, 57, 66, 93, 120 Megasthenes, 77 Merchants, 114 Meynard, 8 Migne, 87 Mikami, Y., 56 Milanesi, 128 Mo[h.]ammed ibn `Abdall[=a]h, 92 Mo[h.]ammed ibn A[h.]med, 6 Mo[h.]ammed ibn `Al[=i] `Abd[=i], 8 Mo[h.]ammed ibn M[=u]s[=a], _see_ Al-Khow[=a]razm[=i] Molinier, 123 Monier-Williams, 17 {158} Morley, D., 126 Moroccan numerals, 68, 119 Mortet, V., 11 Moseley, C. B., 33 Mo[t.]ahhar ibn [T.][=a]hir, 7 Mueller, A., 68 Mumford, J. K., 109 Muwaffaq al-D[=i]n, 93

Nabatean forms, 21 Nallino, 4, 54, 55 Nagl, A., 55, 110, 113, 126 N[=a]n[=a] Gh[=a]t inscriptions, 20, 22, 23, 40 Narducci, 123 Nasik cave inscriptions, 24 Na[z.][=i]f ibn Yumn, 94 Neander, A., 75 Neophytos, 57, 62 Neo-Pythagoreans, 64 Nesselmann, 58 Newman, Cardinal, 96 Newman, F. W., 131 Nöldeke, Th., 91 Notation, 61 Note, 61, 119 Noviomagus, 45, 61, 119, 150 Null, 61 Numerals, Algerian, 68 astrological, 150 Br[=a]hm[=i], 19-22, 83 early ideas of origin, 1 Hindu, 26 Hindu, classified, 19, 38 Kharo[s.][t.]h[=i], 19-22 Moroccan, 68 Nabatean, 21 origin, 27, 30, 31, 37 supposed Arabic origin, 2 supposed Babylonian origin, 28 supposed Chaldean and Jewish origin, 3 supposed Chinese origin, 28, 32 supposed Egyptian origin, 27, 30, 69, 70 supposed Greek origin, 33 supposed Phoenician origin, 32 tables of, 22-27, 36, 48, 49, 69, 88, 140, 143, 145-148

O'Creat, 5, 55, 119, 120 Olleris, 110, 113 Oppert, G., 14, 75

Pali, 22 Pañcasiddh[=a]ntik[=a], 44 Paravey, 32, 57 P[=a]tal[=i]pu[t.]ra, 77 Patna, 77 Patrick, R., 119 Payne, E. J., 106 Pegolotti, 107 Peletier, 2, 62 Perrot, 80 Persia, 66, 91, 107 Pertz, 115 Petrus de Dacia, 59, 61, 62 Pez, P. B., 117 "Philalethes," 75 Phillips, G., 107 Picavet, 105 Pichler, F., 141 Pihan, A. P., 36 Pisa, 128 Place value, 26, 42, 46, 48 Planudes, _see_ Maximus Planudes Plimpton, G. A., 56, 59, 85, 143, 144, 145, 148 Pliny, 76 Polo, N. and M., 107 {159} Prändel, J. G., 54 Prinsep, J., 20, 31 Propertius, 80 Prosdocimo de' Beldomandi, 137 Prou, 143 Ptolemy, 54, 78 Putnam, 103 Pythagoras, 63 Pythagorean numbers, 13 Pytheas of Massilia, 76

Rabbi ben Ezra, 60, 127 Radulph of Laon, 60, 113, 118, 124 Raets, 62 Rainer, _see_ Gemma Frisius R[=a]m[=a]yana, 18 Ramus, 2, 41, 60, 61 Raoul Glaber, 123 Rapson, 77 Rauhfuss, _see_ Dasypodius Raumer, K. von, 111 Reclus, E., 14, 96, 130 Recorde, 3, 58 Reinaud, 67, 74, 80 Reveillaud, 36 Richer, 110, 112, 115 Riese, A., 119 Robertson, 81 Robertus Cestrensis, 97, 126 Rodet, 5, 44 Roediger, J., 68 Rollandus, 144 Romagnosi, 81 Rosen, F., 5 Rotula, 60 Rudolff, 85 Rudolph, 62, 67 Ruffi, 150

Sachau, 6 Sacrobosco, 3, 58, 133 Sacy, S. de, 66, 70 Sa`d[=i], 102 ['S]aka inscriptions, 20 Sam[=u]'[=i]l ibn Ya[h.]y[=a], 93 ['S][=a]rad[=a] characters, 55 Savonne, 60 Scaliger, J. C., 73 Scheubel, 62 Schlegel, 12 Schmidt, 133 Schonerus, 87, 119 Schroeder, L. von, 13 Scylax, 75 Sedillot, 8, 34 Senart, 20, 24, 25 Sened ibn `Al[=i], 10, 98 Sfortunati, 62, 150 Shelley, W., 126 Siamese numerals, 36 Siddh[=a]nta, 8, 18 [S.]ifr, 57 Sigsboto, 55 Sih[=a]b al-D[=i]n, 67 Silberberg, 60 Simon, 13 Sin[=a]n ibn al-Fat[h.], 93 Sindbad, 100 Sindhind, 97 Sipos, 60 Sirr, H. C., 75 Skeel, C. A., 74 Smith, D. E., 11, 17, 53, 86, 141, 143 Smith, V. A., 20, 35, 46, 47 Smith, Wm., 75 Sm[r.]ti, 17 Spain, 64, 65, 100 Spitta-Bey, 5 Sprenger, 94 ['S]rautas[=u]tra, 39 Steffens, F., 116 Steinschneider, 5, 57, 65, 66, 98, 126 Stifel, 62 {160} Subandhus, 44 Suetonius, 80 Suleim[=a]n, 100 ['S][=u]nya, 43, 53, 57 Suter, 5, 9, 68, 69, 93, 116, 131 S[=u]tras, 13 Sykes, P. M., 75 Sylvester II, _see_ Gerbert Symonds, J. A., 129

Tannery, P., 62, 84, 85 Tartaglia, 4, 61 Taylor, I., 19, 30 Teca, 55, 61 Tennent, J. E., 75 Texada, 60 Theca, 58, 61 Theophanes, 64 Thibaut, G., 12, 13, 16, 44, 47 Tibetan numerals, 36 Timotheus, 103 Tonstall, C., 3, 61 Trenchant, 60 Treutlein, 5, 63, 123 Trevisa, 136 Treviso arithmetic, 145 Trivium and quadrivium, 73 Tsin, 56 Tunis, 65 Turchill, 88, 118, 123 Turnour, G., 75 Tziphra, 57, 62 [Greek: tziphra], 55, 57, 62 Tzwivel, 61, 118, 145

Ujjain, 32 Unger, 133 Upanishads, 12 Usk, 121

Valla, G., 61 Van der Schuere, 62 Var[=a]ha-Mihira, 39, 44, 78 V[=a]savadatt[=a], 44 Vaux, Carra de, 9, 74 Vaux, W. S. W., 91 Ved[=a][.n]gas, 17 Vedas, 12, 15, 17 Vergil, 80 Vincent, A. J. H., 57 Vogt, 13 Voizot, P., 36 Vossius, 4, 76, 81, 84

Wallis, 3, 62, 84, 116 Wappler, E., 54, 126 Wäschke, H., 2, 93 Wattenbach, 143 Weber, A., 31 Weidler, I. F., 34, 66 Weidler, I. F. and G. I., 63, 66 Weissenborn, 85, 110 Wertheim, G., 57, 61 Whitney, W. D., 13 Wilford, F., 75 Wilkens, 62 Wilkinson, J. G., 70 Willichius, 3 Woepcke, 3, 6, 42, 63, 64, 65, 67, 69, 70, 94, 113, 138 Wolack, G., 54 Woodruff, C. E., 32 Word and letter numerals, 38, 44 Wüstenfeld, 74

Yule, H., 107

Zephirum, 57, 58 Zephyr, 59 Zepiro, 58 Zero, 26, 38, 40, 43, 45, 49, 51-62, 67 Zeuero, 58

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