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] 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 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 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 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 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 K?bel'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.

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 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 Leb?uf 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 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 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 to contain the numerals is a Latin manuscript,[556] the Codex Vigilanus, written in the Albelda Cloister not far from Logro?o in Spain, in 976 A.D. The nine characters (of ?obār 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?. 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 ?obār 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. 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 (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 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 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 m.,... 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,

for 125 for 150

for 147 for 202

As to monumental inscriptions,[573] 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.[574] Certain numerals on Wells Cathedral have been assigned to the thirteenth century, but they are undoubtedly considerably later.[575]

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

Early Manuscript Forms

a [577] Twelfth century A.D.

b [578] 1197 A.D.

c [579] 1275 A.D.

d [580] c. 1294 A.D.

e [581] c. 1303 A.D.

f [582] c. 1360 A.D.

g [583] c. 1442 A.D.

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, 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]

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.

3. "Three" usually had a special type in the first printed books, although occasionally it appears as .[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 , although the Florentine manuscript of Leonard of Pisa's work has the form ;[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 [594] and [594] or ,[595] more closely resembling our own. The first printed books generally used our present form[596] with the closed top , the open top used in writing ( ) being 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]

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 .[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 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?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 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 cα 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 MoCCCCo50 (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]:

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.

* * *

INDEX

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

Abbo of Fleury, 122

'Abdallāh ibn al-?asan, 92

'Abdallatīf ibn Yūsuf, 93

'Abdalqādir ibn 'Alī al-Sakhāwī, 6

Abenragel, 34

Abraham ibn Me?r ibn Ezra, see Rabbi ben Ezra

Abū 'Alī al-?osein ibn Sīnā, 74

Abū 'l-?asan, 93, 100

Abū 'l-Qāsim, 92

Abū 'l-?eiyib, 97

Abū Na?r, 92

Abū Roshd, 113

Abu Sahl Dunash ibn Tamim, 65, 67

Adelhard of Bath, 5, 55, 97, 119, 123, 126

Adhemar of Chabanois, 111

A?med al-Nasawī, 98

A?med ibn 'Abdallāh, 9, 92

A?med ibn Mo?ammed, 94

A?med ibn 'Omar, 93

Ak?aras, 32

Alanus ab Insulis, 124

Al-Ba?dādī, 93

Al-Battānī, 54

Albelda (Albaida) MS., 116

Albert, J., 62

Albert of York, 103

Al-Bīrūnī, 6, 41, 49, 65, 92, 93

Alcuin, 103

Alexander the Great, 76

Alexander de Villa Dei, 11, 133

Alexandria, 64, 82

Al-Fazārī, 92

Alfred, 103

Algebra, etymology, 5

Algerian numerals, 68

Algorism, 97

Algorismus, 124, 126, 135

Algorismus cifra, 120

Al-?a??ār, 65

'Alī ibn Abī Bekr, 6

'Alī ibn A?med, 93, 98

Al-Karābīsī, 93

Al-Khowārazmī, 4, 9, 10, 92, 97, 98, 125, 126

Al-Kindī, 10, 92

Almagest, 54

Al-Ma?rebī, 93

Al-Ma?allī, 6

Al-Māmūn, 10, 97

Al-Man?ūr, 96, 97

Al-Mas'ūdī, 7, 92

Al-Nadīm, 9

Al-Nasawī, 93, 98

Alphabetic numerals, 39, 40, 43

Al-Qāsim, 92

Al-Qass, 94

Al-Sakhāwī, 6

Al-?ardafī, 93

Al-Sijzī, 94

Al-Sūfī, 10, 92

Ambrosoli, 118

A?kapalli, 43

Apices, 87, 117, 118

Arabs, 91-98

Arbuthnot, 141

Archimedes, 15, 16

Arcus Pictagore, 122

Arjuna, 15

Arnold, E., 15, 102

Ars memorandi, 141

āryabha?a, 39, 43, 44

Aryan numerals, 19

Aschbach, 134

Ashmole, 134

A?oka, 19, 20, 22, 81

A?-?ifr, 57, 58

Astrological numerals, 150

Atharva-Veda, 48, 49, 55

Augustus, 80

Averro?s, 113

Avicenna, 58, 74, 113

Babylonian numerals, 28

Babylonian zero, 51

Bacon, R., 131

Bactrian numerals, 19, 30

B?da, 2, 72

Bagdad, 4, 96

Bakh?ālī manuscript, 43, 49, 52, 53

Ball, C. J., 35

Ball, W. W. R., 36, 131

Bā?a, 44

Barth, A., 39

Bayang inscriptions, 39

Bayer, 33

Bayley, E. C., 19, 23, 30, 32, 52, 89

Beazley, 75

Bede, see B?da

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

Bhāskara, 53, 55

Biernatzki, 32

Biot, 32

Bj?rnbo, 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

Brāhma?as, 12, 13

Brāhmī, 19, 20, 31, 83

Brandis, J., 54

B?hat-Sa?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

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

Devanāgarī, 7

Devoulx, A., 68

Dhruva, 49

Dic?archus of Messana, 77

Digits, 119

Diodorus Siculus, 76

Du Cange, 62

Dumesnil, 36

Dutt, R. C., 12, 15, 18, 75

Dvivedī, 44

East and West, relations, 73-81, 100-109

Egyptian numerals, 27

Eisenlohr, 28

Elia Misrachi, 57

Enchiridion Algorismi, 58

Enestr?m, 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

Firdusī, 81

Fitz Stephen, W., 109

Fleet, J. C., 19, 20, 49

Florus, 80

Flügel, G., 68

Francisco de Retza, 142

Fran?ois, 58

Friedlein, G., 84, 113, 116, 122

Froude, J. A., 129

Gandhāra, 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

?obār numerals, 65, 100, 112, 124, 138

Gow, J., 81

Grammateus, 61

Greek origin, 33

Green, J. R., 109

Greenwood, I., 62, 119

Guglielmini, 128

Gulistān, 102

Günther, S., 131

Guyard, S., 82

?abash, 9, 92

Hager, J. (G.), 28, 32

Halliwell, 59, 85

Hankel, 93

Hārūn al-Rashīd, 97, 106

Havet, 110

Heath, T. L., 125

Hebrew numerals, 127

Hecat?us, 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

?osein ibn Mo?ammed al-Ma?allī, 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 Abī Ya'qūb, 9

Ibn al-Adamī, 92

Ibn al-Bannā, 93

Ibn Khordā?beh, 101, 106

Ibn Wahab, 103

India, history of, 14

writing in, 18

Indicopleustes, 83

Indo-Bactrian numerals, 19

Indrājī, 23

Is?āq ibn Yūsuf al-?ardafī, 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

Kātyāyana, 39

Kaye, C. R., 6, 16, 43, 46, 121

Keane, J., 75, 82

Keene, H. G., 15

Kern, 44

Kharo??hī, 19, 20

Khosrū, 82, 91

Kielhorn, F., 46, 47

Kircher, A., 34

Kitāb al-Fihrist, see Fihrist

Kleinw?chter, 32

Klos, 62

K?bel, 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

Lā?yāyana, 39

Leb?uf, 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

Mahābhārata, 18

Mahāvīrācārya, 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

Māshāllāh, 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?ammed ibn 'Abdallāh, 92

Mo?ammed ibn A?med, 6

Mo?ammed ibn 'Alī 'Abdī, 8

Mo?ammed ibn Mūsā, see Al-Khowārazmī

Molinier, 123

Monier-Williams, 17

Morley, D., 126

Moroccan numerals, 68, 119

Mortet, V., 11

Moseley, C. B., 33

Mo?ahhar ibn ?āhir, 7

Mueller, A., 68

Mumford, J. K., 109

Muwaffaq al-Dīn, 93

Nabatean forms, 21

Nallino, 4, 54, 55

Nagl, A., 55, 110, 113, 126

Nānā Ghāt inscriptions, 20, 22, 23, 40

Narducci, 123

Nasik cave inscriptions, 24

Na?īf ibn Yumn, 94

Neander, A., 75

Neophytos, 57, 62

Neo-Pythagoreans, 64

Nesselmann, 58

Newman, Cardinal, 96

Newman, F. W., 131

N?ldeke, Th., 91

Notation, 61

Note, 61, 119

Noviomagus, 45, 61, 119, 150

Null, 61

Numerals,

Algerian, 68

astrological, 150

Brāhmī, 19-22, 83

early ideas of origin, 1

Hindu, 26

Hindu, classified, 19, 38

Kharo??hī, 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 Ph?nician 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

Pa?casiddhāntikā, 44

Paravey, 32, 57

Pātalīpu?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

Pr?ndel, 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

Rāmāyana, 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'dī, 102

?aka inscriptions, 20

Samū'īl ibn Ya?yā, 93

?āradā 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 'Alī, 10, 98

Sfortunati, 62, 150

Shelley, W., 126

Siamese numerals, 36

Siddhānta, 8, 18

?ifr, 57

Sigsboto, 55

Sihāb al-Dīn, 67

Silberberg, 60

Simon, 13

Sinān ibn al-Fat?, 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?ti, 17

Spain, 64, 65, 100

Spitta-Bey, 5

Sprenger, 94

?rautasūtra, 39

Steffens, F., 116

Steinschneider, 5, 57, 65, 66, 98, 126

Stifel, 62

Subandhus, 44

Suetonius, 80

Suleimān, 100

?ūnya, 43, 53, 57

Suter, 5, 9, 68, 69, 93, 116, 131

Sūtras, 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

τζ?φρα, 55, 57, 62

Tzwivel, 61, 118, 145

Ujjain, 32

Unger, 133

Upanishads, 12

Usk, 121

Valla, G., 61

Van der Schuere, 62

Varāha-Mihira, 39, 44, 78

Vāsavadattā, 44

Vaux, Carra de, 9, 74

Vaux, W. S. W., 91

Vedā?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

W?schke, 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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Notes

[1] "Discipulus. Quis primus invenit numerum apud Hebr?os et ?gyptios? Magister. Abraham primus invenit numerum apud Hebr?os, deinde Moses; et Abraham tradidit istam scientiam numeri ad ?gyptios, et docuit eos: deinde Josephus." [Bede, De computo dialogus (doubtfully assigned to him), Opera omnia, Paris, 1862, Vol. I, p. 650.]

"Alii referunt ad Ph?nices inventores arithmetic?, propter eandem commerciorum caussam: Alii ad Indos: Ioannes de Sacrobosco, cujus sepulchrum est Luteti? in comitio Maturinensi, refert ad Arabes." [Ramus, Arithmetic? libri dvo, Basel, 1569, p. 112.]

Similar notes are given by Peletarius in his commentary on the arithmetic of Gemma Frisius (1563 ed., fol. 77), and in his own work (1570 Lyons ed., p. 14): "La valeur des Figures commence au coste dextre tirant vers le coste senestre: au rebours de notre maniere d'escrire par ce que la premiere prattique est venue des Chaldees: ou des Pheniciens, qui ont été les premiers traffiquers de marchandise."

[2] Maximus Planudes (c. 1330) states that "the nine symbols come from the Indians." [W?schke's German translation, Halle, 1878, p. 3.] Willichius speaks of the "Zyphr? Indic?," in his Arithmetic? libri tres (Strasburg, 1540, p. 93), and Cataneo of "le noue figure de gli Indi," in his Le pratiche delle dve prime mathematiche (Venice, 1546, fol. 1). Woepcke is not correct, therefore, in saying ("Mémoire sur la propagation des chiffres indiens," hereafter referred to as Propagation [Journal Asiatique, Vol. I (6), 1863, p. 34]) that Wallis (A Treatise on Algebra, both historical and practical, London, 1685, p. 13, and De algebra tractatus, Latin edition in his Opera omnia, 1693, Vol. II, p. 10) was one of the first to give the Hindu origin.

[3] From the 1558 edition of The Grovnd of Artes, fol. C, 5. Similarly Bishop Tonstall writes: "Qui a Chaldeis primum in finitimos, deinde in omnes pene gentes fluxit.... Numerandi artem a Chaldeis esse profectam: qui dum scribunt, a dextra incipiunt, et in leuam progrediuntur." [De arte supputandi, London, 1522, fol. B, 3.] Gemma Frisius, the great continental rival of Recorde, had the same idea: "Primùm autem appellamus dexterum locum, eo quòd haec ars vel à Chald?is, vel ab Hebr?is ortum habere credatur, qui etiam eo ordine scribunt"; but this refers more evidently to the Arabic numerals. [Arithmetic? practic? methodvs facilis, Antwerp, 1540, fol. 4 of the 1563 ed.] Sacrobosco (c. 1225) mentions the same thing. Even the modern Jewish writers claim that one of their scholars, Māshāllāh (c. 800), introduced them to the Mohammedan world. [C. Levias, The Jewish Encyclopedia, New York, 1905, Vol. IX, p. 348.]

[4] "... & que esto fu trouato di fare da gli Arabi con diece figure." [La prima parte del general trattato di nvmeri, et misvre, Venice, 1556, fol. 9 of the 1592 edition.]

[5] "Vom welchen Arabischen auch disz Kunst entsprungen ist." [Ain nerv geordnet Rechenbiechlin, Augsburg, 1514, fol. 13 of the 1531 edition. The printer used the letters rv for w in "new" in the first edition, as he had no w of the proper font.]

[6] Among them Glareanus: "Characteres simplices sunt nouem significatiui, ab Indis usque, siue Chald?is asciti .1.2.3.4.5.6.7.8.9. Est item unus .0 circulus, qui nihil significat." [De VI. Arithmeticae practicae speciebvs, Paris, 1539, fol. 9 of the 1543 edition.]

[7] "Barbarische oder gemeine Ziffern." [Anonymous, Das Einmahl Eins cum notis variorum, Dresden, 1703, p. 3.] So Vossius (De universae matheseos natura et constitutione liber, Amsterdam, 1650, p. 34) calls them "Barbaras numeri notas." The word at that time was possibly synonymous with Arabic.

[8] His full name was 'Abū 'Abdallāh Mo?ammed ibn Mūsā al-Khowārazmī. He was born in Khowārezm, "the lowlands," the country about the present Khiva and bordering on the Oxus, and lived at Bagdad under the caliph al-Māmūn. He died probably between 220 and 230 of the Mohammedan era, that is, between 835 and 845 A.D., although some put the date as early as 812. The best account of this great scholar may be found in an article by C. Nallino, "Al-?uwārizmī" in the Atti della R. Accad. dei Lincei, Rome, 1896. See also Verhandlungen des 5. Congresses der Orientalisten, Berlin, 1882, Vol. II, p. 19; W. Spitta-Bey in the Zeitschrift der deutschen Morgenl?nd. Gesellschaft, Vol. XXXIII, p. 224; Steinschneider in the Zeitschrift der deutschen Morgenl?nd. Gesellschaft, Vol. L, p. 214; Treutlein in the Abhandlungen zur Geschichte der Mathematik, Vol. I, p. 5; Suter, "Die Mathematiker und Astronomen der Araber und ihre Werke," Abhandlungen zur Geschichte der Mathematik, Vol. X, Leipzig, 1900, p. 10, and "Nachtr?ge," in Vol. XIV, p. 158; Cantor, Geschichte der Mathematik, Vol. I, 3d ed., pp. 712-733 etc.; F. Woepcke in Propagation, p. 489. So recently has he become known that Heilbronner, writing in 1742, merely mentions him as "Ben-Musa, inter Arabes celebris Geometra, scripsit de figuris planis & sphericis." [Historia matheseos univers?, Leipzig, 1742, p. 438.]

In this work most of the Arabic names will be transliterated substantially as laid down by Suter in his work Die Mathematiker etc., except where this violates English pronunciation. The scheme of pronunciation of oriental names is set forth in the preface.

[9] Our word algebra is from the title of one of his works, Al-jabr wa'l-muqābalah, Completion and Comparison. The work was translated into English by F. Rosen, London, 1831, and treated in L'Algèbre d'al-Khārizmi et les méthodes indienne et grecque, Léon Rodet, Paris, 1878, extract from the Journal Asiatique. For the derivation of the word algebra, see Cossali, Scritti Inediti, pp. 381-383, Rome, 1857; Leonardo's Liber Abbaci (1202), p. 410, Rome, 1857; both published by B. Boncompagni. "Almuchabala" also was used as a name for algebra.

[10] This learned scholar, teacher of O'Creat who wrote the Helceph ("Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum"), studied in Toledo, learned Arabic, traveled as far east as Egypt, and brought from the Levant numerous manuscripts for study and translation. See Henry in the Abhandlungen zur Geschichte der Mathematik, Vol. III, p. 131; Woepcke in Propagation, p. 518.

[11] The title is Algoritmi de numero Indorum. That he did not make this translation is asserted by Enestr?m in the Bibliotheca Mathematica, Vol. I (3), p. 520.

[12] Thus he speaks "de numero indorum per .IX. literas," and proceeds: "Dixit algoritmi: Cum uidissem yndos constituisse .IX. literas in uniuerso numero suo, propter dispositionem suam quam posuerunt, uolui patefacere de opera quod fit per eas aliquid quod esset leuius discentibus, si deus uoluerit." [Boncompagni, Trattati d'Aritmetica, Rome, 1857.] Discussed by F. Woepcke, Sur l'introduction de l'arithmétique indienne en Occident, Rome, 1859.

[13] Thus in a commentary by 'Alī ibn Abī Bekr ibn al-Jamāl al-An?ārī al-Mekkī on a treatise on ?obār arithmetic (explained later) called Al-murshidah, found by Woepcke in Paris (Propagation, p. 66), there is mentioned the fact that there are "nine Indian figures" and "a second kind of Indian figures ... although these are the figures of the ?obār writing." So in a commentary by ?osein ibn Mo?ammed al-Ma?allī (died in 1756) on the Mokhta?ar fī'ilm el-?isāb (Extract from Arithmetic) by 'Abdalqādir ibn 'Alī al-Sakhāwī (died c. 1000) it is related that "the preface treats of the forms of the figures of Hindu signs, such as were established by the Hindu nation." [Woepcke, Propagation, p. 63.]

[14] See also Woepcke, Propagation, p. 505. The origin is discussed at much length by G. R. Kaye, "Notes on Indian Mathematics.-Arithmetical Notation," Journ. and Proc. of the Asiatic Soc. of Bengal, Vol. III, 1907, p. 489.

[15] Alberuni's India, Arabic version, London, 1887; English translation, ibid., 1888.

[16] Chronology of Ancient Nations, London, 1879. Arabic and English versions, by C. E. Sachau.

[17] India, Vol. I, chap. xvi.

[18] The Hindu name for the symbols of the decimal place system.

[19] Sachau's English edition of the Chronology, p. 64.

[20] Littérature arabe, Cl. Huart, Paris, 1902.

[21] Huart, History of Arabic Literature, English ed., New York, 1903, p. 182 seq.

[22] Al-Mas'ūdī's Meadows of Gold, translated in part by Aloys Sprenger, London, 1841; Les prairies d'or, trad. par C. Barbier de Meynard et Pavet de Courteille, Vols. I to IX, Paris, 1861-1877.

[23] Les prairies d'or, Vol. VIII, p. 289 seq.

[24] Essays, Vol. II, p. 428.

[25] Loc. cit., p. 504.

[26] Matériaux pour servir à l'histoire comparée des sciences mathématiques chez les Grecs et les Orientaux, 2 vols., Paris, 1845-1849, pp. 438-439.

[27] He made an exception, however, in favor of the numerals, loc. cit., Vol. II, p. 503.

[28] Bibliotheca Arabico-Hispana Escurialensis, Madrid, 1760-1770, pp. 426-427.

[29] The author, Ibn al-Qif?ī, flourished A.D. 1198 [Colebrooke, loc. cit., note Vol. II, p. 510].

[30] "Liber Artis Logisticae à Mohamado Ben Musa Alkhuarezmita exornatus, qui ceteros omnes brevitate methodi ac facilitate praestat, Indorum que in praeclarissimis inventis ingenium & acumen ostendit." [Casiri, loc. cit., p. 427.]

[31] Ma?oudi, Le livre de l'avertissement et de la révision. Translation by B. Carra de Vaux, Paris, 1896.

[32] Verifying the hypothesis of Woepcke, Propagation, that the Sindhind included a treatment of arithmetic.

[33] A?med ibn 'Abdallāh, Suter, Die Mathematiker, etc., p. 12.

[34] India, Vol. II, p. 15.

[35] See H. Suter, "Das Mathematiker-Verzeichniss im Fihrist," Abhandlungen zur Geschichte der Mathematik, Vol. VI, Leipzig, 1892. For further references to early Arabic writers the reader is referred to H. Suter, Die Mathematiker und Astronomen der Araber und ihre Werke. Also "Nachtr?ge und Berichtigungen" to the same (Abhandlungen, Vol. XIV, 1902, pp. 155-186).

[36] Suter, loc. cit., note 165, pp. 62-63.

[37] "Send Ben Ali,... tùm arithmetica scripta maximè celebrata, quae publici juris fecit." [Loc. cit., p. 440.]

[38] Scritti di Leonardo Pisano, Vol. I, Liber Abbaci (1857); Vol. II, Scritti (1862); published by Baldassarre Boncompagni, Rome. Also Tre Scritti Inediti, and Intorno ad Opere di Leonardo Pisano, Rome, 1854.

[39] "Ubi ex mirabili magisterio in arte per novem figuras indorum introductus" etc. In another place, as a heading to a separate division, he writes, "De cognitione novem figurarum yndorum" etc. "Novem figure indorum he sunt 9 8 7 6 5 4 3 2 1."

[40] See An Ancient English Algorism, by David Eugene Smith, in Festschrift Moritz Cantor, Leipzig, 1909. See also Victor Mortet, "Le plus ancien traité francais d'algorisme," Bibliotheca Mathematica, Vol. IX (3), pp. 55-64.

[41] These are the two opening lines of the Carmen de Algorismo that the anonymous author is explaining. They should read as follows:

Haec algorismus ars praesens dicitur, in qua

Talibus Indorum fruimur bis quinque figuris.

What follows is the translation.

[42] Thibaut, Astronomie, Astrologie und Mathematik, Strassburg, 1899.

[43] Gustave Schlegel, Uranographie chinoise ou preuves directes que l'astronomie primitive est originaire de la Chine, et qu'elle a été empruntée par les anciens peuples occidentaux à la sphère chinoise; ouvrage accompagné d'un atlas céleste chinois et grec, The Hague and Leyden, 1875.

[44] E. W. Hopkins, The Religions of India, Boston, 1898, p. 7.

[45] R. C. Dutt, History of India, London, 1906.

[46] W. D. Whitney, Sanskrit Grammar, 3d ed., Leipzig, 1896.

[47] "Das āpastamba-?ulba-Sūtra," Zeitschrift der deutschen Morgenl?ndischen Gesellschaft, Vol. LV, p. 543, and Vol. LVI, p. 327.

[48] Geschichte der Math., Vol. I, 2d ed., p. 595.

[49] L. von Schroeder, Pythagoras und die Inder, Leipzig, 1884; H. Vogt, "Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale gekannt?" Bibliotheca Mathematica, Vol. VII (3), pp. 6-20; A. Bürk, loc. cit.; Max Simon, Geschichte der Mathematik im Altertum, Berlin, 1909, pp. 137-165; three Sūtras are translated in part by Thibaut, Journal of the Asiatic Society of Bengal, 1875, and one appeared in The Pandit, 1875; Beppo Levi, "Osservazioni e congetture sopra la geometria degli indiani," Bibliotheca Mathematica, Vol. IX (3), 1908, pp. 97-105.

[50] Loc. cit.; also Indiens Literatur und Cultur, Leipzig, 1887.

[51] It is generally agreed that the name of the river Sindhu, corrupted by western peoples to Hindhu, Indos, Indus, is the root of Hindustan and of India. Reclus, Asia, English ed., Vol. III, p. 14.

[52] See the comments of Oppert, On the Original Inhabitants of Bharatavar?a or India, London, 1893, p. 1.

[53] A. Hillebrandt, Alt-Indien, Breslau, 1899, p. 111. Fragmentary records relate that Khāravela, king of Kali?ga, learned as a boy lekhā (writing), ga?anā (reckoning), and rūpa (arithmetic applied to monetary affairs and mensuration), probably in the 5th century B.C. [Bühler, Indische Palaeographie, Strassburg, 1896, p. 5.]

[54] R. C. Dutt, A History of Civilization in Ancient India, London, 1893, Vol. I, p. 174.

[55] The Buddha. The date of his birth is uncertain. Sir Edwin Arnold put it c. 620 B.C.

[56] I.e. 100·107.

[57] There is some uncertainty about this limit.

[58] This problem deserves more study than has yet been given it. A beginning may be made with Comte Goblet d'Alviella, Ce que l'Inde doit à la Grèce, Paris, 1897, and H. G. Keene's review, "The Greeks in India," in the Calcutta Review, Vol. CXIV, 1902, p. 1. See also F. Woepeke, Propagation, p. 253; G. R. Kaye, loc. cit., p. 475 seq., and "The Source of Hindu Mathematics," Journal of the Royal Asiatic Society, July, 1910, pp. 749-760; G. Thibaut, Astronomie, Astrologie und Mathematik, pp. 43-50 and 76-79. It will be discussed more fully in Chapter VI.

[59] I.e. to 100,000. The lakh is still the common large unit in India, like the myriad in ancient Greece and the million in the West.

[60] This again suggests the Psammites, or De harenae numero as it is called in the 1544 edition of the Opera of Archimedes, a work in which the great Syracusan proposes to show to the king "by geometric proofs which you can follow, that the numbers which have been named by us ... are sufficient to exceed not only the number of a sand-heap as large as the whole earth, but one as large as the universe." For a list of early editions of this work see D. E. Smith, Rara Arithmetica, Boston, 1909, p. 227.

[61] I.e. the Wise.

[62] Sir Monier Monier-Williams, Indian Wisdom, 4th ed., London, 1893, pp. 144, 177. See also J. C. Marshman, Abridgment of the History of India, London, 1893, p. 2.

[63] For a list and for some description of these works see R. C. Dutt, A History of Civilization in Ancient India, Vol. II, p. 121.

[64] Professor Ramkrishna Gopal Bhandarkar fixes the date as the fifth century B.C. ["Consideration of the Date of the Mahābhārata," in the Journal of the Bombay Branch of the R. A. Soc., Bombay, 1873, Vol. X, p. 2.].

[65] Marshman, loc. cit., p. 2.

[66] A. C. Burnell, South Indian Pal?ography, 2d ed., London, 1878, p. 1, seq.

[67] This extensive subject of palpable arithmetic, essentially the history of the abacus, deserves to be treated in a work by itself.

[68] The following are the leading sources of information upon this subject: G. Bühler, Indische Palaeographie, particularly chap. vi; A. C. Burnell, South Indian Pal?ography, 2d ed., London, 1878, where tables of the various Indian numerals are given in Plate XXIII; E. C. Bayley, "On the Genealogy of Modern Numerals," Journal of the Royal Asiatic Society, Vol. XIV, part 3, and Vol. XV, part 1, and reprint, London, 1882; I. Taylor, in The Academy, January 28, 1882, with a repetition of his argument in his work The Alphabet, London, 1883, Vol. II, p. 265, based on Bayley; G. R. Kaye, loc. cit., in some respects one of the most critical articles thus far published; J. C. Fleet, Corpus inscriptionum Indicarum, London, 1888, Vol. III, with facsimiles of many Indian inscriptions, and Indian Epigraphy, Oxford, 1907, reprinted from the Imperial Gazetteer of India, Vol. II, pp. 1-88, 1907; G. Thibaut, loc. cit., Astronomie etc.; R. Caldwell, Comparative Grammar of the Dravidian Languages, London, 1856, p. 262 seq.; and Epigraphia Indica (official publication of the government of India), Vols. I-IX. Another work of Bühler's, On the Origin of the Indian Brāhma Alphabet, is also of value.

[69] The earliest work on the subject was by James Prinsep, "On the Inscriptions of Piyadasi or A?oka," etc., Journal of the Asiatic Society of Bengal, 1838, following a preliminary suggestion in the same journal in 1837. See also "A?oka Notes," by V. A. Smith, The Indian Antiquary, Vol. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, pp. 151-159, June, 1909; The Early History of India, 2d ed., Oxford, 1908, p. 154; J. F. Fleet, "The Last Words of A?oka," Journal of the Royal Asiatic Society, October, 1909, pp. 981-1016; E. Senart, Les inscriptions de Piyadasi, 2 vols., Paris, 1887.

[70] For a discussion of the minor details of this system, see Bühler, loc. cit., p. 73.

[71] Julius Euting, Nabat?ische Inschriften aus Arabien, Berlin, 1885, pp. 96-97, with a table of numerals.

[72] For the five principal theories see Bühler, loc. cit., p. 10.

[73] Bayley, loc. cit., reprint p. 3.

[74] Bühler, loc. cit.; Epigraphia Indica, Vol. III, p. 134; Indian Antiquary, Vol. VI, p. 155 seq., and Vol. X, p. 107.

[75] Pandit Bhagavānlāl Indrājī, "On Ancient Nāgāri Numeration; from an Inscription at Nāneghāt," Journal of the Bombay Branch of the Royal Asiatic Society, 1876, Vol. XII, p. 404.

[76] Ib., p. 405. He gives also a plate and an interpretation of each numeral.

[77] These may be compared with Bühler's drawings, loc. cit.; with Bayley, loc. cit., p. 337 and plates; and with Bayley's article in the Encyclop?dia Britannica, 9th ed., art. "Numerals."

[78] E. Senart, "The Inscriptions in the Caves at Nasik," Epigraphia Indica, Vol. VIII, pp. 59-96; "The Inscriptions in the Cave at Karle," Epigraphia Indica, Vol. VII, pp. 47-74; Bühler, Palaeographie, Tafel IX.

[79] See Fleet, loc. cit. See also T. Benfey, Sanskrit Grammar, London, 1863, p. 217; M. R. Kále, Higher Sanskrit Grammar, 2d ed., Bombay, 1898, p. 110, and other authorities as cited.

[80] Kharo??hī numerals, A?oka inscriptions, c. 250 B.C. Senart, Notes d'épigraphie indienne. Given by Bühler, loc. cit., Tafel I.

[81] Same, ?aka inscriptions, probably of the first century B.C. Senart, loc. cit.; Bühler, loc. cit.

[82] Brāhmī numerals, A?oka inscriptions, c. 250 B.C. Indian Antiquary, Vol. VI, p. 155 seq.

[83] Same, Nānā Ghāt inscriptions, c. 150 B.C. Bhagavānlāl Indrājī, On Ancient Nāgarī Numeration, loc. cit. Copied from a squeeze of the original.

[84] Same, Nasik inscription, c. 100 B.C. Burgess, Archeological Survey Report, Western India; Senart, Epigraphia Indica, Vol. VII, pp. 47-79, and Vol. VIII, pp. 59-96.

[85] K?atrapa coins, c. 200 A.D. Journal of the Royal Asiatic Society, 1890, p. 639.

[86] Ku?ana inscriptions, c. 150 A.D. Epigraphia Indica, Vol. I, p. 381, and Vol. II, p. 201.

[87] Gupta Inscriptions, c. 300 A.D. to 450 A.D. Fleet, loc. cit., Vol. III.

[88] Valhabī, c. 600 A.D. Corpus, Vol. III.

[89] Bendall's Table of Numerals, in Cat. Sansk. Budd. MSS., British Museum.

[90] Indian Antiquary, Vol. XIII, 120; Epigraphia Indica, Vol. III, 127 ff.

[91] Fleet, loc. cit.

[92] Bayley, loc. cit., p. 335.

[93] From a copper plate of 493 A.D., found at Kārītalāī, Central India. [Fleet, loc. cit., Plate XVI.] It should be stated, however, that many of these copper plates, being deeds of property, have forged dates so as to give the appearance of antiquity of title. On the other hand, as Colebrooke long ago pointed out, a successful forgery has to imitate the writing of the period in question, so that it becomes evidence well worth considering, as shown in Chapter III.

[94] From a copper plate of 510 A.D., found at Majhgawāin, Central India. [Fleet, loc. cit., Plate XIV.]

[95] From an inscription of 588 A.D., found at Bōdh-Gayā, Bengal Presidency. [Fleet, loc. cit., Plate XXIV.]

[96] From a copper plate of 571 A.D., found at Māliyā, Bombay Presidency. [Fleet, loc. cit., Plate XXIV.]

[97] From a Bijayaga?h pillar inscription of 372 A.D. [Fleet, loc. cit., Plate XXXVI, C.]

[98] From a copper plate of 434 A.D. [Indian Antiquary, Vol. I, p. 60.]

[99] Gadhwa inscription, c. 417 A.D. [Fleet, loc. cit., Plate IV, D.]

[100] Kārītalāī plate of 493 A.D., referred to above.

[101] It seems evident that the Chinese four, curiously enough called "eight in the mouth," is only a cursive .

[102] Chalfont, F. H., Memoirs of the Carnegie Museum, Vol. IV, no. 1; J. Hager, An Explanation of the Elementary Characters of the Chinese, London, 1801.

[103] H. V. Hilprecht, Mathematical, Metrological and Chronological Tablets from the Temple Library at Nippur, Vol. XX, part I, of Series A, Cuneiform Texts Published by the Babylonian Expedition of the University of Pennsylvania, 1906; A. Eisenlohr, Ein altbabylonischer Felderplan, Leipzig, 1906; Maspero, Dawn of Civilization, p. 773.

[104] Sir H. H. Howard, "On the Earliest Inscriptions from Chaldea," Proceedings of the Society of Biblical Arch?ology, XXI, p. 301, London, 1899.

[105] For a bibliography of the principal hypotheses of this nature see Bühler, loc. cit., p. 77. Bühler (p. 78) feels that of all these hypotheses that which connects the Brāhmī with the Egyptian numerals is the most plausible, although he does not adduce any convincing proof. Th. Henri Martin, "Les signes numéraux et l'arithmétique chez les peuples de l'antiquité et du moyen age" (being an examination of Cantor's Mathematische Beitr?ge zum Culturleben der V?lker), Annali di matematica pura ed applicata, Vol. V, Rome, 1864, pp. 8, 70. Also, same author, "Recherches nouvelles sur l'origine de notre système de numération écrite," Revue Archéologique, 1857, pp. 36, 55. See also the tables given later in this work.

[106] Journal of the Royal Asiatic Society, Bombay Branch, Vol. XXIII.

[107] Loc. cit., reprint, Part I, pp. 12, 17. Bayley's deductions are generally regarded as unwarranted.

[108] The Alphabet; London, 1883, Vol. II, pp. 265, 266, and The Academy of Jan. 28, 1882.

[109] Taylor, The Alphabet, loc. cit., table on p. 266.

[110] Bühler, On the Origin of the Indian Brāhma Alphabet, Strassburg, 1898, footnote, pp. 52, 53.

[111] Albrecht Weber, History of Indian Literature, English ed., Boston, 1878, p. 256: "The Indian figures from 1-9 are abbreviated forms of the initial letters of the numerals themselves...: the zero, too, has arisen out of the first letter of the word ?unya (empty) (it occurs even in Pi?gala). It is the decimal place value of these figures which gives them significance." C. Henry, "Sur l'origine de quelques notations mathématiques," Revue Archéologique, June and July, 1879, attempts to derive the Boethian forms from the initials of Latin words. See also J. Prinsep, "Examination of the Inscriptions from Girnar in Gujerat, and Dhauli in Cuttach," Journal of the Asiatic Society of Bengal, 1838, especially Plate XX, p. 348; this was the first work on the subject.

[112] Bühler, Palaeographie, p. 75, gives the list, with the list of letters (p. 76) corresponding to the number symbols.

[113] For a general discussion of the connection between the numerals and the different kinds of alphabets, see the articles by U. Ceretti, "Sulla origine delle cifre numerali moderne," Rivista di fisica, matematica e scienze naturali, Pisa and Pavia, 1909, anno X, numbers 114, 118, 119, and 120, and continuation in 1910.

[114] This is one of Bühler's hypotheses. See Bayley, loc. cit., reprint p. 4; a good bibliography of original sources is given in this work, p. 38.

[115] Loc. cit., reprint, part I, pp. 12, 17. See also Burnell, loc. cit., p. 64, and tables in plate XXIII.

[116] This was asserted by G. Hager (Memoria sulle cifre arabiche, Milan, 1813, also published in Fundgruben des Orients, Vienna, 1811, and in Bibliothèque Britannique, Geneva, 1812). See also the recent article by Major Charles E. Woodruff, "The Evolution of Modern Numerals from Tally Marks," American Mathematical Monthly, August-September, 1909. Biernatzki, "Die Arithmetik der Chinesen," Crelle's Journal für die reine und angewandte Mathematik, Vol. LII, 1857, pp. 59-96, also asserts the priority of the Chinese claim for a place system and the zero, but upon the flimsiest authority. Ch. de Paravey, Essai sur l'origine unique et hiéroglyphique des chiffres et des lettres de tous les peuples, Paris, 1826; G. Kleinw?chter, "The Origin of the Arabic Numerals," China Review, Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp. 28-30; Biot, "Note sur la connaissance que les Chinois ont eue de la valeur de position des chiffres," Journal Asiatique, 1839, pp. 497-502. A. Terrien de Lacouperie, "The Old Numerals, the Counting-Rods and the Swan-Pan in China," Numismatic Chronicle, Vol. III (3), pp. 297-340, and Crowder B. Moseley, "Numeral Characters: Theory of Origin and Development," American Antiquarian, Vol. XXII, pp. 279-284, both propose to derive our numerals from Chinese characters, in much the same way as is done by Major Woodruff, in the article above cited.

[117] The Greeks, probably following the Semitic custom, used nine letters of the alphabet for the numerals from 1 to 9, then nine others for 10 to 90, and further letters to represent 100 to 900. As the ordinary Greek alphabet was insufficient, containing only twenty-four letters, an alphabet of twenty-seven letters was used.

[118] Institutiones mathematicae, 2 vols., Strassburg, 1593-1596, a somewhat rare work from which the following quotation is taken:

"Quis est harum Cyphrarum autor?

"A quibus hae usitatae syphrarum notae sint inventae: hactenus incertum fuit: meo tamen iudicio, quod exiguum esse fateor: a graecis librarijs (quorum olim magna fuit copia) literae Graecorum quibus veteres Graeci tamquam numerorum notis sunt usi: fuerunt corruptae. vt ex his licet videre.

"Graecorum Literae corruptae.

"Sed qua ratione graecorum literae ita fuerunt corruptae?

"Finxerunt has corruptas Graecorum literarum notas: vel abiectione vt in nota binarij numeri, vel additione vt in ternarij, vel inuersione vt in septenarij, numeri nota, nostrae notae, quibus hodie utimur: ab his sola differunt elegantia, vt apparet."

See also Bayer, Historia regni Graecorum Bactriani, St. Petersburg, 1788, pp. 129-130, quoted by Martin, Recherches nouvelles, etc., loc. cit.

[119] P. D. Huet, Demonstratio evangelica, Paris, 1769, note to p. 139 on p. 647: "Ab Arabibus vel ab Indis inventas esse, non vulgus eruditorum modo, sed doctissimi quique ad hanc diem arbitrati sunt. Ego vero falsum id esse, merosque esse Graecorum characteres aio; à librariis Graecae linguae ignaris interpolatos, et diuturna scribendi consuetudine corruptos. Nam primum 1 apex fuit, seu virgula, nota μον?δο?. 2, est ipsum β extremis suis truncatum. γ, si in sinistram partem inclinaveris & cauda mutilaveris & sinistrum cornu sinistrorsum flexeris, fiet 3. Res ipsa loquitur 4 ipsissimum esse Δ, cujus crus sinistrum erigitur κατ? κ?θετον, & infra basim descendit; basis vero ipsa ultra crus producta eminet. Vides quam 5 simile sit τ? ; infimo tantum semicirculo, qui sinistrorsum patebat, dextrorsum converso. ?π?σημον βα? quod ita notabatur , rotundato ventre, pede detracto, peperit τ? 6. Ex Ζ basi sua mutilato, ortum est τ? 7. Si Η inflexis introrsum apicibus in rotundiorem & commodiorem formam mutaveris, exurget τ? 8. At 9 ipsissimum est ."

I. Weidler, Spicilegium observationum ad historiam notarum numeralium, Wittenberg, 1755, derives them from the Hebrew letters; Dom Augustin Calmet, "Recherches sur l'origine des chiffres d'arithmétique," Mémoires pour l'histoire des sciences et des beaux arts, Trévoux, 1707 (pp. 1620-1635, with two plates), derives the current symbols from the Romans, stating that they are relics of the ancient "Notae Tironianae." These "notes" were part of a system of shorthand invented, or at least perfected, by Tiro, a slave who was freed by Cicero. L. A. Sedillot, "Sur l'origine de nos chiffres," Atti dell' Accademia pontificia dei nuovi Lincei, Vol. XVIII, 1864-1865, pp. 316-322, derives the Arabic forms from the Roman numerals.

[120] Athanasius Kircher, Arithmologia sive De abditis Numerorum, mysterijs qua origo, antiquitas & fabrica Numerorum exponitur, Rome, 1665.

[121] See Suter, Die Mathematiker und Astronomen der Araber, p. 100.

[122] "Et hi numeri sunt numeri Indiani, a Brachmanis Indiae Sapientibus ex figura circuli secti inuenti."

[123] V. A. Smith, The Early History of India, Oxford, 2d ed., 1908, p. 333.

[124] C. J. Ball, "An Inscribed Limestone Tablet from Sippara," Proceedings of the Society of Biblical Arch?ology, Vol. XX, p. 25 (London, 1898). Terrien de Lacouperie states that the Chinese used the circle for 10 before the beginning of the Christian era. [Catalogue of Chinese Coins, London, 1892, p. xl.]

[125] For a purely fanciful derivation from the corresponding number of strokes, see W. W. R. Ball, A Short Account of the History of Mathematics, 1st ed., London, 1888, p. 147; similarly J. B. Reveillaud, Essai sur les chiffres arabes, Paris, 1883; P. Voizot, "Les chiffres arabes et leur origine," La Nature, 1899, p. 222; G. Dumesnil, "De la forme des chiffres usuels," Annales de l'université de Grenoble, 1907, Vol. XIX, pp. 657-674, also a note in Revue Archéologique, 1890, Vol. XVI (3), pp. 342-348; one of the earliest references to a possible derivation from points is in a work by Bettino entitled Apiaria universae philosophiae mathematicae in quibus paradoxa et noua machinamenta ad usus eximios traducta, et facillimis demonstrationibus confirmata, Bologna, 1545, Vol. II, Apiarium XI, p. 5.

[126] Alphabetum Barmanum, Romae, MDCCLXXVI, p. 50. The 1 is evidently Sanskrit, and the 4, 7, and possibly 9 are from India.

[127] Alphabetum Grandonico-Malabaricum, Romae, MDCCLXXII, p. 90. The zero is not used, but the symbols for 10, 100, and so on, are joined to the units to make the higher numbers.

[128] Alphabetum Tangutanum, Romae, MDCCLXXIII, p. 107. In a Tibetan MS. in the library of Professor Smith, probably of the eighteenth century, substantially these forms are given.

[129] Bayley, loc. cit., plate II. Similar forms to these here shown, and numerous other forms found in India, as well as those of other oriental countries, are given by A. P. Pihan, Exposé des signes de numération usités chez les peuples orientaux anciens et modernes, Paris, 1860.

[130] Bühler, loc. cit., p. 80; J. F. Fleet, Corpus inscriptionum Indicarum, Vol. III, Calcutta, 1888. Lists of such words are given also by Al-Bīrūnī in his work India; by Burnell, loc. cit.; by E. Jacquet, "Mode d'expression symbolique des nombres employé par les Indiens, les Tibétains et les Javanais," Journal Asiatique, Vol. XVI, Paris, 1835.

[131] This date is given by Fleet, loc. cit., Vol. III, p. 73, as the earliest epigraphical instance of this usage in India proper.

[132] Weber, Indische Studien, Vol. VIII, p. 166 seq.

[133] Journal of the Royal Asiatic Society, Vol. I (N.S.), p. 407.

[134] VIII, 20, 21.

[135] Th. H. Martin, Les signes numéraux ..., Rome, 1864; Lassen, Indische Alterthumskunde, Vol. II, 2d ed., Leipzig and London, 1874, p. 1153.

[136] But see Burnell, loc. cit., and Thibaut, Astronomie, Astrologie und Mathematik, p. 71.

[137] A. Barth, "Inscriptions Sanscrites du Cambodge," in the Notices et extraits des Mss. de la Bibliothèque nationale, Vol. XXVII, Part I, pp. 1-180, 1885; see also numerous articles in Journal Asiatique, by Aymonier.

[138] Bühler, loc. cit., p. 82.

[139] Loc. cit., p. 79.

[140] Bühler, loc. cit., p. 83. The Hindu astrologers still use an alphabetical system of numerals. [Burnell, loc. cit., p. 79.]

[141] Well could Ramus say, "Quicunq; autem fuerit inventor decem notarum laudem magnam meruit."

[142] Al-Bīrūnī gives lists.

[143] Propagation, loc. cit., p. 443.

[144] See the quotation from The Light of Asia in Chapter II, p. 16.

[145] The nine ciphers were called a?ka.

[146] "Zur Geschichte des indischen Ziffernsystems," Zeitschrift für die Kunde des Morgenlandes, Vol. IV, 1842, pp. 74-83.

[147] It is found in the Bakh?ālī MS. of an elementary arithmetic which Hoernle placed, at first, about the beginning of our era, but the date is much in question. G. Thibaut, loc. cit., places it between 700 and 900 A.D.; Cantor places the body of the work about the third or fourth century A.D., Geschichte der Mathematik, Vol. I (3), p. 598.

[148] For the opposite side of the case see G. R. Kaye, "Notes on Indian Mathematics, No. 2.-āryabha?a," Journ. and Proc. of the Asiatic Soc. of Bengal, Vol. IV, 1908, pp. 111-141.

[149] He used one of the alphabetic systems explained above. This ran up to 1018 and was not difficult, beginning as follows:

the same letter (ka) appearing in the successive consonant forms, ka, kha, ga, gha, etc. See C. I. Gerhardt, über die Entstehung und Ausbreitung des dekadischen Zahlensystems, Programm, p. 17, Salzwedel, 1853, and études historiques sur l'arithmétique de position, Programm, p. 24, Berlin, 1856; E. Jacquet, Mode d'expression symbolique des nombres, loc. cit., p. 97; L. Rodet, "Sur la véritable signification de la notation numérique inventée par āryabhata," Journal Asiatique, Vol. XVI (7), pp. 440-485. On the two āryabha?as see Kaye, Bibl. Math., Vol. X (3), p. 289.

[150] Using kha, a synonym of ?ūnya. [Bayley, loc. cit., p. 22, and L. Rodet, Journal Asiatique, Vol. XVI (7), p. 443.]

[151] Varāha-Mihira, Pa?casiddhāntikā, translated by G. Thibaut and M. S. Dvivedī, Benares, 1889; see Bühler, loc. cit., p. 78; Bayley, loc. cit., p. 23.

[152] B?hat Sa?hitā, translated by Kern, Journal of the Royal Asiatic Society, 1870-1875.

[153] It is stated by Bühler in a personal letter to Bayley (loc. cit., p. 65) that there are hundreds of instances of this usage in the B?hat Sa?hitā. The system was also used in the Pa?casiddhāntikā as early as 505 A.D. [Bühler, Palaeographie, p. 80, and Fleet, Journal of the Royal Asiatic Society, 1910, p. 819.]

[154] Cantor, Geschichte der Mathematik, Vol. I (3), p. 608.

[155] Bühler, loc. cit., p. 78.

[156] Bayley, p. 38.

[157] Noviomagus, in his De numeris libri duo, Paris, 1539, confesses his ignorance as to the origin of the zero, but says: "D. Henricus Grauius, vir Graecè & Hebraicè eximè doctus, Hebraicam originem ostendit," adding that Valla "Indis Orientalibus gentibus inventionem tribuit."

[158] See Essays, Vol. II, pp. 287 and 288.

[159] Vol. XXX, p. 205 seqq.

[160] Loc. cit., p. 284 seqq.

[161] Colebrooke, loc. cit., p. 288.

[162] Loc. cit., p. 78.

[163] Hereafter, unless expressly stated to the contrary, we shall use the word "numerals" to mean numerals with place value.

[164] "The Gurjaras of Rājputāna and Kanauj," in Journal of the Royal Asiatic Society, January and April, 1909.

[165] Vol. IX, 1908, p. 248.

[166] Epigraphia Indica, Vol. IX, pp. 193 and 198.

[167] Epigraphia Indica, Vol. IX, p. 1.

[168] Loc. cit., p. 71.

[169] Thibaut, p. 71.

[170] "Est autem in aliquibus figurarum istaram apud multos diuersitas. Quidam enim septimam hanc figuram representant," etc. [Boncompagni, Trattati, p. 28.] Enestr?m has shown that very likely this work is incorrectly attributed to Johannes Hispalensis. [Bibliotheca Mathematica, Vol. IX (3), p. 2.]

[171] Indische Palaeographie, Tafel IX.

[172] Edited by Bloomfield and Garbe, Baltimore, 1901, containing photographic reproductions of the manuscript.

[173] Bakh?ālī MS. See page 43; Hoernle, R., The Indian Antiquary, Vol. XVII, pp. 33-48, 1 plate; Hoernle, Verhandlungen des VII. Internationalen Orientalisten-Congresses, Arische Section, Vienna, 1888, "On the Bakshālī Manuscript," pp. 127-147, 3 plates; Bühler, loc. cit.

[174] 3, 4, 6, from H. H. Dhruva, "Three Land-Grants from Sankheda," Epigraphia Indica, Vol. II, pp. 19-24 with plates; date 595 A.D. 7, 1, 5, from Bhandarkar, "Daulatabad Plates," Epigraphia Indica, Vol. IX, part V; date c. 798 A.D.

[175] 8, 7, 2, from "Buckhala Inscription of Nagabhatta," Bhandarkar, Epigraphia Indica, Vol. IX, part V; date 815 A.D. 5 from "The Morbi Copper-Plate," Bhandarkar, The Indian Antiquary, Vol. II, pp. 257-258, with plate; date 804 A.D. See Bühler, loc. cit.

[176] 8 from the above Morbi Copper-Plate. 4, 5, 7, 9, and 0, from "Asni Inscription of Mahipala," The Indian Antiquary, Vol. XVI, pp. 174-175; inscription is on red sandstone, date 917 A.D. See Bühler.

[177] 8, 9, 4, from "Rashtrakuta Grant of Amoghavarsha," J. F. Fleet, The Indian Antiquary, Vol. XII, pp. 263-272; copper-plate grant of date c. 972 A.D. See Bühler. 7, 3, 5, from "Torkhede Copper-Plate Grant of the Time of Govindaraja of Gujerat," Fleet, Epigraphia Indica, Vol. III, pp. 53-58. See Bühler.

[178] From "A Copper-Plate Grant of King Tritochanapala Chanlukya of Lā?ade?a," H.H. Dhruva, Indian Antiquary, Vol. XII, pp. 196-205; date 1050 A.D. See Bühler.

[179] Burnell, A. C., South Indian Pal?ography, plate XXIII, Telugu-Canarese numerals of the eleventh century. See Bühler.

[180] From a manuscript of the second half of the thirteenth century, reproduced in "Della vita e delle opere di Leonardo Pisano," Baldassare Boncompagni, Rome, 1852, in Atti dell' Accademia Pontificia dei nuovi Lincei, anno V.

[181] From a fourteenth-century manuscript, as reproduced in Della vita etc., Boncompagni, loc. cit.

[182] From a Tibetan MS. in the library of D. E. Smith.

[183] From a Tibetan block-book in the library of D. E. Smith.

[184] ?āradā numerals from The Kashmirian Atharva-Veda, reproduced by chromophotography from the manuscript in the University Library at Tübingen, Bloomfield and Garbe, Baltimore, 1901. Somewhat similar forms are given under "Numération Cachemirienne," by Pihan, Exposé etc., p. 84.

[185] Franz X. Kugler, Die Babylonische Mondrechnung, Freiburg i. Br., 1900, in the numerous plates at the end of the book; practically all of these contain the symbol to which reference is made. Cantor, Geschichte, Vol. I, p. 31.

[186] F. X. Kugler, Sternkunde und Sterndienst in Babel, I. Buch, from the beginnings to the time of Christ, Münster i. Westfalen, 1907. It also has numerous tables containing the above zero.

[187] From a letter to D. E. Smith, from G. F. Hill of the British Museum. See also his monograph "On the Early Use of Arabic Numerals in Europe," in Arch?ologia, Vol. LXII (1910), p. 137.

[188] R. Hoernle, "The Bakshālī Manuscript," Indian Antiquary, Vol. XVII, pp. 33-48 and 275-279, 1888; Thibaut, Astronomie, Astrologie und Mathematik, p. 75; Hoernle, Verhandlungen, loc. cit., p. 132.

[189] Bayley, loc. cit., Vol. XV, p. 29. Also Bendall, "On a System of Numerals used in South India," Journal of the Royal Asiatic Society, 1896, pp. 789-792.

[190] V. A. Smith, The Early History of India, 2d ed., Oxford, 1908, p. 14.

[191] Colebrooke, Algebra, with Arithmetic and Mensuration, from the Sanskrit of Brahmegupta and Bháscara, London, 1817, pp. 339-340.

[192] Ibid., p. 138.

[193] D. E. Smith, in the Bibliotheca Mathematica, Vol. IX (3), pp. 106-110.

[194] As when we use three dots (...).

[195] "The Hindus call the nought explicitly ?ūnyabindu 'the dot marking a blank,' and about 500 A.D. they marked it by a simple dot, which latter is commonly used in inscriptions and MSS. in order to mark a blank, and which was later converted into a small circle." [Bühler, On the Origin of the Indian Alphabet, p. 53, note.]

[196] Fazzari, Dell' origine delle parole zero e cifra, Naples, 1903.

[197] E. Wappler, "Zur Geschichte der Mathematik im 15. Jahrhundert," in the Zeitschrift für Mathematik und Physik, Vol. XLV, Hist.-lit. Abt., p. 47. The manuscript is No. C. 80, in the Dresden library.

[198] J. G. Pr?ndel, Algebra nebst ihrer literarischen Geschichte, p. 572, Munich, 1795.

[199] See the table, p. 23. Does the fact that the early European arithmetics, following the Arab custom, always put the 0 after the 9, suggest that the 0 was derived from the old Hindu symbol for 10?

[200] Bayley, loc. cit., p. 48. From this fact Delambre (Histoire de l'astronomie ancienne) inferred that Ptolemy knew the zero, a theory accepted by Chasles, Aper?u historique sur l'origine et le développement des méthodes en géométrie, 1875 ed., p. 476; Nesselmann, however, showed (Algebra der Griechen, 1842, p. 138), that Ptolemy merely used ο for ο?δ?ν, with no notion of zero. See also G. Fazzari, "Dell' origine delle parole zero e cifra," Ateneo, Anno I, No. 11, reprinted at Naples in 1903, where the use of the point and the small cross for zero is also mentioned. Th. H. Martin, Les signes numéraux etc., reprint p. 30, and J. Brandis, Das Münz-, Mass- und Gewichtswesen in Vorderasien bis auf Alexander den Grossen, Berlin, 1866, p. 10, also discuss this usage of ο, without the notion of place value, by the Greeks.

[201] Al-Battānī sive Albatenii opus astronomicum. Ad fidem codicis escurialensis arabice editum, latine versum, adnotationibus instructum a Carolo Alphonso Nallino, 1899-1907. Publicazioni del R. Osservatorio di Brera in Milano, No. XL.

[202] Loc. cit., Vol. II, p. 271.

[203] C. Henry, "Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum," Abhandlungen zur Geschichte der Mathematik, Vol. III, 1880.

[204] Max. Curtze, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts," Abhandlungen zur Geschichte der Mathematik, Vol. VIII, 1898, pp. 1-27; Alfred Nagl, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und über die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," Zeitschrift für Mathematik und Physik, Hist.-lit. Abth., Vol. XXXIV, pp. 129-146 and 161-170, with one plate.

[205] "Byzantinische Analekten," Abhandlungen zur Geschichte der Mathematik, Vol. IX, pp. 161-189.

[206] or for 0. also used for 5. for 13. [Heiberg, loc. cit.]

[207] Gerhardt, études historiques sur l'arithmétique de position, Berlin, 1856, p. 12; J. Bowring, The Decimal System in Numbers, Coins, & Accounts, London, 1854, p. 33.

[208] Karabacek, Wiener Zeitschrift für die Kunde des Morgenlandes, Vol. XI, p. 13; Führer durch die Papyrus-Ausstellung Erzherzog Rainer, Vienna, 1894, p. 216.

[209] In the library of G. A. Plimpton, Esq.

[210] Cantor, Geschichte, Vol. I (3), p. 674; Y. Mikami, "A Remark on the Chinese Mathematics in Cantor's Geschichte der Mathematik," Archiv der Mathematik und Physik, Vol. XV (3), pp. 68-70.

[211] Of course the earlier historians made innumerable guesses as to the origin of the word cipher. E.g. Matthew Hostus, De numeratione emendata, Antwerp, 1582, p. 10, says: "Siphra vox Hebr?am originem sapit refértque: & ut docti arbitrantur, à verbo saphar, quod Ordine numerauit significat. Unde Sephar numerus est: hinc Siphra (vulgo corruptius). Etsi verò gens Iudaica his notis, qu? hodie Siphr? vocantur, usa non fuit: mansit tamen rei appellatio apud multas gentes." Dasypodius, Institutiones mathematicae, Vol. I, 1593, gives a large part of this quotation word for word, without any mention of the source. Hermannus Hugo, De prima scribendi origine, Trajecti ad Rhenum, 1738, pp. 304-305, and note, p. 305; Karl Krumbacher, "Woher stammt das Wort Ziffer (Chiffre)?", études de philologie néo-grecque, Paris, 1892.

[212] Bühler, loc. cit., p. 78 and p. 86.

[213] Fazzari, loc. cit., p. 4. So Elia Misrachi (1455-1526) in his posthumous Book of Number, Constantinople, 1534, explains sifra as being Arabic. See also Steinschneider, Bibliotheca Mathematica, 1893, p. 69, and G. Wertheim, Die Arithmetik des Elia Misrachi, Programm, Frankfurt, 1893.

[214] "Cum his novem figuris, et cum hoc signo 0, quod arabice zephirum appellatur, scribitur quilibet numerus."

[215] τζ?φρα, a form also used by Neophytos (date unknown, probably c. 1330). It is curious that Finaeus (1555 ed., f. 2) used the form tziphra throughout. A. J. H. Vincent ["Sur l'origine de nos chiffres," Notices et Extraits des MSS., Paris, 1847, pp. 143-150] says: "Ce cercle fut nommé par les uns, sipos, rota, galgal ...; par les autres tsiphra (de ???, couronne ou diadème) ou ciphra (de ???, numération)." Ch. de Paravey, Essai sur l'origine unique et hiéroglyphique des chiffres et des lettres de tous les peuples, Paris, 1826, p. 165, a rather fanciful work, gives "vase, vase arrondi et fermé par un couvercle, qui est le symbole de la 10e Heure, ," among the Chinese; also "Tsiphron Zéron, ou tout à fait vide en arabe, τζ?φρα en grec ... d'où chiffre (qui dérive plut?t, suivant nous, de l'Hébreu Sepher, compter.")

[216] "Compilatus a Magistro Jacobo de Florentia apud montem pesalanum," and described by G. Lami in his Catalogus codicum manuscriptorum qui in bibliotheca Riccardiana Florenti? adservantur. See Fazzari, loc. cit., p. 5.

[217] "Et doveto sapere chel zeuero per se solo non significa nulla ma è potentia di fare significare, ... Et decina o centinaia o migliaia non si puote scrivere senza questo segno 0. la quale si chiama zeuero." [Fazzari, loc. cit., p. 5.]

[218] Ibid., p. 6.

[219] Avicenna (980-1036), translation by Gasbarri et Fran?ois, "più il punto (gli Arabi adoperavano il punto in vece dello zero il cui segno 0 in arabo si chiama zepiro donde il vocabolo zero), che per sè stesso non esprime nessun numero." This quotation is taken from D. C. Martines, Origine e progressi dell' aritmetica, Messina, 1865.

[220] Leo Jordan, "Materialien zur Geschichte der arabischen Zahlzeichen in Frankreich," Archiv für Kulturgeschichte, Berlin, 1905, pp. 155-195, gives the following two schemes of derivation, (1) "zefiro, zeviro, zeiro, zero," (2) "zefiro, zefro, zevro, zero."

[221] K?bel (1518 ed., f. A_4) speaks of the numerals in general as "die der gemain man Zyfer nendt." Recorde (Grounde of Artes, 1558 ed., f. B_6) says that the zero is "called priuatly a Cyphar, though all the other sometimes be likewise named."

[222] "Decimo X 0 theca, circul cifra sive figura nihili appelat′." [Enchiridion Algorismi, Cologne, 1501.] Later, "quoniam de integris tam in cifris quam in proiectilibus,"-the word proiectilibus referring to markers "thrown" and used on an abacus, whence the French jetons and the English expression "to cast an account."

[223] "Decima vero o dicitur teca, circulus, vel cyfra vel figura nichili." [Maximilian Curtze, Petri Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco commentarius, una cum Algorismo ipso, Copenhagen, 1897, p. 2.] Curtze cites five manuscripts (fourteenth and fifteenth centuries) of Dacia's commentary in the libraries at Erfurt, Leipzig, and Salzburg, in addition to those given by Enestr?m, ?fversigt af Kongl. Vetenskaps-Akademiens F?rhandlingar, 1885, pp. 15-27, 65-70; 1886, pp. 57-60.

[224] Curtze, loc. cit., p. VI.

[225] Rara Mathematica, London, 1841, chap, i, "Joannis de Sacro-Bosco Tractatus de Arte Numerandi."

[226] Smith, Rara Arithmetica, Boston, 1909.

[227] In the 1484 edition, Borghi uses the form "?efiro: ouero nulla:" while in the 1488 edition he uses "zefiro: ouero nulla," and in the 1540 edition, f. 3, appears "Chiamata zero, ouero nulla." Woepcke asserted that it first appeared in Calandri (1491) in this sentence: "Sono dieci le figure con le quali ciascuno numero si può significare: delle quali n'è una che si chiama zero: et per se sola nulla significa." (f. 4). [See Propagation, p. 522.]

[228] Boncompagni Bulletino, Vol. XVI, pp. 673-685.

[229] Leo Jordan, loc. cit. In the Catalogue of MSS., Bibl. de l'Arsenal, Vol. III, pp. 154-156, this work is No. 2904 (184 S.A.F.), Bibl. Nat., and is also called Petit traicté de algorisme.

[230] Texada (1546) says that there are "nueue letros yvn zero o cifra" (f. 3).

[231] Savonne (1563, 1751 ed., f. 1): "Vne ansi formee (o) qui s'appelle nulle, & entre marchans zero," showing the influence of Italian names on French mercantile customs. Trenchant (Lyons, 1566, 1578 ed., p. 12) also says: "La derniere qui s'apele nulle, ou zero;" but Champenois, his contemporary, writing in Paris in 1577 (although the work was not published until 1578), uses "cipher," the Italian influence showing itself less in this center of university culture than in the commercial atmosphere of Lyons.

[232] Thus Radulph of Laon (c. 1100): "Inscribitur in ultimo ordine et figura sipos nomine, quae, licet numerum nullum signitet, tantum ad alia quaedam utilis, ut insequentibus declarabitur." ["Der Arithmetische Tractat des Radulph von Laon," Abhandlungen zur Geschichte der Mathematik, Vol. V, p. 97, from a manuscript of the thirteenth century.] Chasles (Comptes rendus, t. 16, 1843, pp. 1393, 1408) calls attention to the fact that Radulph did not know how to use the zero, and he doubts if the sipos was really identical with it. Radulph says: "... figuram, cui sipos nomen est in motum rotulae formatam nullius numeri significatione inscribi solere praediximus," and thereafter uses rotula. He uses the sipos simply as a kind of marker on the abacus.

[233] Rabbi ben Ezra (1092-1168) used both ????, galgal (the Hebrew for wheel), and ????, sifra. See M. Steinschneider, "Die Mathematik bei den Juden," in Bibliotheca Mathematica, 1893, p. 69, and Silberberg, Das Buch der Zahl des R. Abraham ibn Esra, Frankfurt a. M., 1895, p. 96, note 23; in this work the Hebrew letters are used for numerals with place value, having the zero.

[234] E.g., in the twelfth-century Liber aligorismi (see Boncompagni's Trattati, II, p. 28). So Ramus (Libri II, 1569 ed., p. 1) says: "Circulus qu? nota est ultima: nil per se significat." (See also the Schonerus ed. of Ramus, 1586, p. 1.)

[235] "Und wirt das ringlein o. die Ziffer genant die nichts bedeut." [K?bel's Rechenbuch, 1549 ed., f. 10, and other editions.]

[236] I.e. "circular figure," our word notation having come from the medieval nota. Thus Tzwivel (1507, f. 2) says: "Nota autem circularis .o. per se sumpta nihil vsus habet. alijs tamen adiuncta earum significantiam et auget et ordinem permutat quantum quo ponit ordinem. vt adiuncta note binarij hoc modo 20 facit eam significare bis decem etc." Also (ibid., f. 4), "figura circularis," "circularis nota." Clichtoveus (1503 ed., f. XXXVII) calls it "nota aut circularis o," "circularis nota," and "figura circularis." Tonstall (1522, f. B_3) says of it: "Decimo uero nota ad formam litter? circulari figura est: quam alij circulum, uulgus cyphram uocat," and later (f. C_4) speaks of the "circulos." Grammateus, in his Algorismus de integris (Erfurt, 1523, f. A_2), speaking of the nine significant figures, remarks: "His autem superadditur decima figura circularis ut 0 existens que ratione sua nihil significat." Noviomagus (De Numeris libri II, Paris, 1539, chap. xvi, "De notis numerorum, quas zyphras vocant") calls it "circularis nota, quam ex his solam, alij sipheram, Georgius Valla zyphram."

[237] Huswirt, as above. Ramus (Scholae mathematicae, 1569 ed., p. 112) discusses the name interestingly, saying: "Circulum appellamus cum multis, quam alii thecam, alii figuram nihili, alii figuram privationis, seu figuram nullam vocant, alii ciphram, cùm tamen hodie omnes h? not? vulgò ciphr? nominentur, & his notis numerare idem sit quod ciphrare." Tartaglia (1592 ed., f. 9) says: "si chiama da alcuni tecca, da alcuni circolo, da altri cifra, da altri zero, & da alcuni altri nulla."

[238] "Quare autem aliis nominibus vocetur, non dicit auctor, quia omnia alia nomina habent rationem suae lineationis sive figurationis. Quia rotunda est, dicitur haec figura teca ad similitudinem tecae. Teca enim est ferrum figurae rotundae, quod ignitum solet in quibusdam regionibus imprimi fronti vel maxillae furis seu latronum." [Loc. cit., p. 26.] But in Greek theca (, θ?κη) is a place to put something, a receptacle. If a vacant column, e.g. in the abacus, was so called, the initial might have given the early forms and for the zero.

[239] Buteo, Logistica, Lyons, 1559. See also Wertheim in the Bibliotheca Mathematica, 1901, p. 214.

[240] "0 est appellee chiffre ou nulle ou figure de nulle valeur." [La Roche, L'arithmétique, Lyons, 1520.]

[241] "Decima autem figura nihil uocata," "figura nihili (quam etiam cifram uocant)." [Stifel, Arithmetica integra, 1544, f. 1.]

[242] "Zifra, & Nulla uel figura Nihili." [Scheubel, 1545, p. 1 of ch. 1.] Nulla is also used by Italian writers. Thus Sfortunati (1545 ed., f. 4) says: "et la decima nulla & e chiamata questa decima zero;" Cataldi (1602, p. 1): "La prima, che è o, si chiama nulla, ouero zero, ouero niente." It also found its way into the Dutch arithmetics, e.g. Raets (1576, 1580 ed., f. A_3): "Nullo dat ist niet;" Van der Schuere (1600, 1624 ed., f. 7); Wilkens (1669 ed., p. 1). In Germany Johann Albert (Wittenberg, 1534) and Rudolff (1526) both adopted the Italian nulla and popularized it. (See also Kuckuck, Die Rechenkunst im sechzehnten Jahrhundert, Berlin, 1874, p. 7; Günther, Geschichte, p. 316.)

[243] "La dixième s'appelle chifre vulgairement: les vns l'appellant zero: nous la pourrons appeller vn Rien." [Peletier, 1607 ed., p. 14.]

[244] It appears in the Polish arithmetic of Klos (1538) as cyfra. "The Ciphra 0 augmenteth places, but of himselfe signifieth not," Digges, 1579, p. 1. Hodder (10th ed., 1672, p. 2) uses only this word (cypher or cipher), and the same is true of the first native American arithmetic, written by Isaac Greenwood (1729, p. 1). Petrus de Dacia derives cyfra from circumference. "Vocatur etiam cyfra, quasi circumfacta vel circumferenda, quod idem est, quod circulus non habito respectu ad centrum." [Loc. cit., p. 26.]

[245] Opera mathematica, 1695, Oxford, Vol. I, chap. ix, Mathesis universalis, "De figuris numeralibus," pp. 46-49; Vol. II, Algebra, p. 10.

[246] Martin, Origine de notre système de numération écrite, note 149, p. 36 of reprint, spells τσ?φρα from Maximus Planudes, citing Wallis as an authority. This is an error, for Wallis gives the correct form as above.

Alexander von Humboldt, "über die bei verschiedenen V?lkern üblichen Systeme von Zahlzeichen und über den Ursprung des Stellenwerthes in den indischen Zahlen," Crelle's Journal für reine und angewandte Mathematik, Vol. IV, 1829, called attention to the work ?ριθμο? ?νδικο? of the monk Neophytos, supposed to be of the fourteenth century. In this work the forms τζ?φρα and τζ?μφρα appear. See also Boeckh, De abaco Graecorum, Berlin, 1841, and Tannery, "Le Scholie du moine Néophytos," Revue Archéologique, 1885, pp. 99-102. Jordan, loc. cit., gives from twelfth and thirteenth century manuscripts the forms cifra, ciffre, chifras, and cifrus. Du Cange, Glossarium mediae et infimae Latinitatis, Paris, 1842, gives also chilerae. Dasypodius, Institutiones Mathematicae, Strassburg, 1593-1596, adds the forms zyphra and syphra. Boissière, L'art d'arythmetique contenant toute dimention, tres-singulier et commode, tant pour l'art militaire que autres calculations, Paris, 1554: "Puis y en a vn autre dict zero lequel ne designe nulle quantité par soy, ains seulement les loges vuides."

[247] Propagation, pp. 27, 234, 442. Treutlein, "Das Rechnen im 16. Jahrhundert," Abhandlungen zur Geschichte der Mathematik, Vol. I, p. 5, favors the same view. It is combated by many writers, e.g. A. C. Burnell, loc. cit., p. 59. Long before Woepcke, I. F. and G. I. Weidler, De characteribus numerorum vulgaribus et eorum aetatibus, Wittenberg, 1727, asserted the possibility of their introduction into Greece by Pythagoras or one of his followers: "Potuerunt autem ex oriente, uel ex phoenicia, ad graecos traduci, uel Pythagorae, uel eius discipulorum auxilio, cum aliquis eo, proficiendi in literis causa, iter faceret, et hoc quoque inuentum addisceret."

[248] E.g., they adopted the Greek numerals in use in Damascus and Syria, and the Coptic in Egypt. Theophanes (758-818 A.D.), Chronographia, Scriptores Historiae Byzantinae, Vol. XXXIX, Bonnae, 1839, p. 575, relates that in 699 A.D. the caliph Walīd forbade the use of the Greek language in the bookkeeping of the treasury of the caliphate, but permitted the use of the Greek alphabetic numerals, since the Arabs had no convenient number notation: κα? ?κ?λυσε γρ?φεσθαι ?λληνιστ? το?? δημοσ?ου? τ?ν λογοθεσ?ων κ?δικα?, ?λλ' ?ραβ?οι? α?τ? παρασημα?νεσθαι, χωρ?? τ?ν ψ?φων, ?πειδ? ?δ?νατον τ? ?κε?νων γλ?σσ? μον?δα ? δυ?δα ? τρι?δα ? ?κτ? ?μισυ ? τρ?α γρ?φεσθαι? δι? κα? ?ω? σ?μερ?ν ε?σιν σ?ν α?το?? νοτ?ριοι Χριστιανο?. The importance of this contemporaneous document was pointed out by Martin, loc. cit. Karabacek, "Die Involutio im arabischen Schriftwesen," Vol. CXXXV of Sitzungsberichte d. phil.-hist. Classe d. k. Akad. d. Wiss., Vienna, 1896, p. 25, gives an Arabic date of 868 A.D. in Greek letters.

[249] The Origin and History of Our Numerals (in Russian), Kiev, 1908; The Independence of European Arithmetic (in Russian), Kiev.

[250] Woepcke, loc. cit., pp. 462, 262.

[251] Woepcke, loc. cit., p. 240. ?isāb-al-?obār, by an anonymous author, probably Abū Sahl Dunash ibn Tamim, is given by Steinschneider, "Die Mathematik bei den Juden," Bibliotheca Mathematica, 1896, p. 26.

[252] Steinschneider in the Abhandlungen, Vol. III, p. 110.

[253] See his Grammaire arabe, Vol. I, Paris, 1810, plate VIII; Gerhardt, études, pp. 9-11, and Entstehung etc., p. 8; I. F. Weidler, Spicilegium observationum ad historiam notarum numeralium pertinentium, Wittenberg, 1755, speaks of the "figura cifrarum Saracenicarum" as being different from that of the "characterum Boethianorum," which are similar to the "vulgar" or common numerals; see also Humboldt, loc. cit.

[254] Gerhardt mentions it in his Entstehung etc., p. 8; Woepcke, Propagation, states that these numerals were used not for calculation, but very much as we use Roman numerals. These superposed dots are found with both forms of numerals (Propagation, pp. 244-246).

[255] Gerhardt (études, p. 9) from a manuscript in the Bibliothèque Nationale. The numeral forms are , 20 being indicated by and 200 by . This scheme of zero dots was also adopted by the Byzantine Greeks, for a manuscript of Planudes in the Bibliothèque Nationale has numbers like for 8,100,000,000. See Gerhardt, études, p. 19. Pihan, Exposé etc., p. 208, gives two forms, Asiatic and Maghrebian, of "Ghobār" numerals.

[256] See Chap. IV.

[257] Possibly as early as the third century A.D., but probably of the eighth or ninth. See Cantor, I (3), p. 598.

[258] Ascribed by the Arabic writer to India.

[259] See Woepcke's description of a manuscript in the Chasles library, "Recherches sur l'histoire des sciences mathématiques chez les orientaux," Journal Asiatique, IV (5), 1859, p. 358, note.

[260] P. 56.

[261] Reinaud, Mémoire sur l'Inde, p. 399. In the fourteenth century one Sihāb al-Dīn wrote a work on which, a scholiast to the Bodleian manuscript remarks: "The science is called Algobar because the inventor had the habit of writing the figures on a tablet covered with sand." [Gerhardt, études, p. 11, note.]

[262] Gerhardt, Entstehung etc., p. 20.

[263] H. Suter, "Das Rechenbuch des Abū Zakarījā el-?a??ār," Bibliotheca Mathematica, Vol. II (3), p. 15.

[264] A. Devoulx, "Les chiffres arabes," Revue Africaine, Vol. XVI, pp. 455-458.

[265] Kitāb al-Fihrist, G. Flügel, Leipzig, Vol. I, 1871, and Vol. II, 1872. This work was published after Professor Flügel's death by J. Roediger and A. Mueller. The first volume contains the Arabic text and the second volume contains critical notes upon it.

[266] Like those of line 5 in the illustration on page 69.

[267] Woepcke, Recherches sur l'histoire des sciences mathématiques chez les orientaux, loc. cit.; Propagation, p. 57.

[268] Al-?a??ār's forms, Suter, Bibliotheca Mathematica, Vol. II (3), p. 15.

[269] Woepcke, Sur une donnée historique, etc., loc. cit. The name ?obār is not used in the text. The manuscript from which these are taken is the oldest (970 A.D.) Arabic document known to contain all of the numerals.

[270] Silvestre de Sacy, loc. cit. He gives the ordinary modern Arabic forms, calling them Indien.

[271] Woepcke, "Introduction au calcul Gobārī et Hawāī," Atti dell' accademia pontificia dei nuovi Lincei, Vol. XIX. The adjective applied to the forms in 5 is gobārī and to those in 6 indienne. This is the direct opposite of Woepcke's use of these adjectives in the Recherches sur l'histoire cited above, in which the ordinary Arabic forms (like those in row 5) are called indiens.

These forms are usually written from right to left.

[272] J. G. Wilkinson, The Manners and Customs of the Ancient Egyptians, revised by S. Birch, London, 1878, Vol. II, p. 493, plate XVI.

[273] There is an extensive literature on this "Boethius-Frage." The reader who cares to go fully into it should consult the various volumes of the Jahrbuch über die Fortschritte der Mathematik.

[274] This title was first applied to Roman emperors in posthumous coins of Julius C?sar. Subsequently the emperors assumed it during their own lifetimes, thus deifying themselves. See F. Gnecchi, Monete romane, 2d ed., Milan, 1900, p. 299.

[275] This is the common spelling of the name, although the more correct Latin form is Bo?tius. See Harper's Dict. of Class. Lit. and Antiq., New York, 1897, Vol. I, p. 213. There is much uncertainty as to his life. A good summary of the evidence is given in the last two editions of the Encyclop?dia Britannica.

[276] His father, Flavius Manlius Boethius, was consul in 487.

[277] There is, however, no good historic evidence of this sojourn in Athens.

[278] His arithmetic is dedicated to Symmachus: "Domino suo patricio Symmacho Boetius." [Friedlein ed., p. 3.]

[279] It was while here that he wrote De consolatione philosophiae.

[280] It is sometimes given as 525.

[281] There was a medieval tradition that he was executed because of a work on the Trinity.

[282] Hence the Divus in his name.

[283] Thus Dante, speaking of his burial place in the monastery of St. Pietro in Ciel d'Oro, at Pavia, says:

"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 B?da (672-735) and Hrabanus Maurus (c. 776-856), it was only after Gerbert's time that the Bo?tii de institutione arithmetica libri duo was 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 his Decline 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 numéraux etc., reprint, p. 4.]

[287] The general idea goes back to Pythagoras, however.

[288] J. C. Scaliger in his Po?tice also 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 Bo?ce, 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, Entstehung etc., p. 20. Avicenna is a corruption from Ibn Sīnā, as pointed out by Wüstenfeld, Geschichte der arabischen Aerzte und Naturforscher, G?ttingen, 1840. His full name is Abū 'Alī al-?osein 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 the Journal 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 to A.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, études etc., pp. 8, 11.

[293] See Smith's Dictionary 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 century B.C. See Smith's Dictionary of Greek and Roman Biography.

[296] Herodotus, Bk. III.

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

[298] Indian Antiquary, Vol. I, p. 229; F. B. Jevons, Manual of Greek Antiquities, London, 1895, p. 386. On the relations, political and commercial, between India and Egypt c. 72 B.C., under Ptolemy Auletes, see the Journal Asiatique, 1863, p. 297.

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

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

[301] Oppert, loc. cit., p. 11. It was at or near this place that the first great Indian mathematician, āryabha?a, was born in 476 A.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 in Indian Coins, by E. J. Rapson, Strassburg, 1898, pp. 3-7.

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

[304] J. Beloch, Griechische Geschichte, Vol. III, Strassburg, 1904, pp. 30-31.

[305] E.g., the denarius, the words for hour and minute (?ρα, λεπτ?ν), and possibly the signs of the zodiac. [R. Caldwell, Comparative Grammar of the Dravidian Languages, London, 1856, p. 438.] On the probable Chinese origin of the zodiac see Schlegel, loc. cit.

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

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

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

[309] Inchiver, from inchi, "the green root." [Indian Antiquary, Vol. I, p. 352.]

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

[311] The Italian morra.

[312] J. Bowring, The Decimal System, London, 1854, p. 2.

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

[314] Giles, loc. cit.

[315] E.g., the names for grape, radish (la-po, ??φη), water-lily (si-kua, "west gourds"; σικ?α, "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 Tana?s." [Odes, Bk. IV, Ode 15, 21-24.]

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

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

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

Arma deus Caesar dites meditatur ad Indos

Et freta gemmiferi findere classe maris.

"The divine C?sar 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 to Ricerche storiche sull' India etc., by Robertson, Vol. II, p. 580, 1827) asserted that Pythagoras originated them. [R. Bombelli, L'antica numerazione italica, Rome, 1876, p. 59.] Gow (Hist. of Greek Math., p. 98) thinks that Iamblichus must have known a similar system in order to have worked out certain of his theorems, but this is an unwarranted deduction from the passage given.

[326] A. Hillebrandt, Alt-Indien, p. 179.

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

[328] He reigned 631-579 A.D.; called Nu?īrwān, the holy one.

[329] J. Keane, The Evolution of Geography, London, 1899, p. 38.

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

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

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

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

[334] In Observationes ad Pomponium Melam de situ orbis. The question was next taken up in a large way by Weidler, loc. cit., De characteribus etc., 1727, and in Spicilegium etc., 1755.

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

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

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

[338] The most noteworthy of these was for a long time Cantor (Geschichte, Vol. I., 3d ed., pp. 587-588), who in his earlier days even believed that Pythagoras had known them. Cantor says (Die r?mischen 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 H?nde Gerbert's gelangte, von welcher mannigfache Handschriften noch heute vorhanden sind." But against this opinion of the antiquity of MSS. containing these numerals is the important statement of P. Tannery, perhaps the most critical of modern historians of mathematics, that none exists earlier than the eleventh century. See also J. L. Heiberg in Philologus, Zeitschrift f. d. klass. Altertum, Vol. XLIII, p. 508.

Of Cantor's predecessors, Th. H. Martin was one of the most prominent, his argument for authenticity appearing in the Revue Archéologique for 1856-1857, and in his treatise Les signes numéraux etc. 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 work Aper?u 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, in Propagation, p. 44; Blume, Lachmann, and Rudorff, Die Schriften der r?mischen Feldmesser, Berlin, 1848; Boeckh, De abaco graecorum, Berlin, 1841; Friedlein, in his Leipzig edition of 1867; Weissenborn, Abhandlungen, Vol. II, p. 185, his Gerbert, pp. 1, 247, and his Geschichte der 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 the Comptes 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's Rara Arithmetica, pp. 434, 25-27.) D. E. Smith also copied from a manuscript of the arithmetic in the Laurentian library at Florence, of 1370, the following forms, which, 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. 532 A.D., this can hardly be other than spurious. See Arbuthnot, loc. cit., p. 38.

[342] Halliwell, in his Rara Mathematica, p. 107, states that the disputed passage is not in a manuscript belonging to Mr. Ames, nor in one at Trinity College. See also Woepcke, in Propagation, pp. 37 and 42. It was the evident corruption of the texts in such editions of Boethius as those of Venice, 1499, Basel, 1546 and 1570, that led Woepcke to publish his work Sur l'introduction de l'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's Patrologia Latina, Vol. LXIII, an ordinary multiplication table (sometimes called Pythagorean abacus) is given in the illustration.

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

[349] C. Henry, "Sur l'origine de quelques notations 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, ... in the quych we vse teen figurys of Inde. Questio. ? why ten fyguris of Inde? Solucio. for as I have sayd afore thei were fonde fyrst in Inde of a kynge of that Cuntre, that was called Algor." See Smith, An Early English Algorism, loc. cit.

[351] Friedlein ed., p. 397.

[352] Carlsruhe codex of Gerlando.

[353] Munich codex of Gerlando.

[354] Carlsruhe codex of Bernelinus.

[355] Munich codex of Bernelinus.

[356] Turchill, c. 1200.

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

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

[359] Vatican codex, tenth century, Boethius.

[360] a, h, i, are from the Friedlein ed.; the original in the manuscript from which a is taken contains a zero symbol, as do all of the six plates given by Friedlein. b-e from the Boncompagni Bulletino, Vol. X, p. 596; f ibid., Vol. XV, p. 186; g Memorie della classe di sci., Reale Acc. dei Lincei, An. CCLXXIV (1876-1877), April, 1877. A twelfth-century arithmetician, possibly John of Luna (Hispalensis, of Seville, c. 1150), speaks of the great diversity of these forms even in his day, saying: "Est autem in aliquibus figuram istarum apud multos diuersitas. Quidam enim septimam hanc figuram representant alii autem sic , uel sic . Quidam vero quartam sic ." [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 531 A.D. See W. S. W Vaux, Persia, London, 1875, p. 169; Th. N?ldeke, Aufs?tze zur persichen Geschichte, Leipzig, 1887, p. 113, and his article in the ninth edition of the Encyclop?dia 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 920 A.D.; Al-Bīrūnī, India, Vol. II, p. 15.

[366] H. Suter, Die Mathematiker etc., 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 work Die Mathematiker und Astronomen der Araber etc.

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

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

[372] Possibly late tenth, since he refers to one arithmetical work which is entitled Book of the Cyphers in his Chronology, English ed., p. 132. Suter, Die Mathematiker etc., pp. 98-100, does not mention this work; see the Nachtr?ge 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] Ψηφοφορ?α κατ' ?νδο??, Greek ed., C. I. Gerhardt, Halle, 1865; and German translation, Das Rechenbuch des Maximus Planudes, H. W?schke, Halle, 1878.

[379] "Sur une donnée historique relative à l'emploi des chiffres indiens par les Arabes," Tortolini's Annali di scienze mat. e fis., 1855.

[380] Suter, p. 80.

[381] Suter, p. 68.

[382] Sprenger also calls attention to this fact, in the Zeitschrift d. deutschen morgenl?nd. 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] Persian bagadata, "God-given."

[386] One of the Abbassides, the (at least pretended) descendants of 'Al-Abbās, uncle and adviser of Mo?ammed.

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

[388] Historical Sketches, Vol. III, chap. iii.

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

[390] See pp. 4-5.

[391] Smith, D. E., in the Cantor Festschrift, 1909, note pp. 10-11. See also F. Woepcke, Propagation.

[392] Enestr?m, in Bibliotheca Mathematica, Vol. I (3), p. 499; Cantor, Geschichte, Vol. I (3), p. 671.

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

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

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

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

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

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

[399] See pp. 7-8.

[400] The name also appears as Mo?ammed Abū'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 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, in The Cambridge Modern History, London, 1902, Vol. I, chap. i.

[421] Geo. Phillips, "The Identity of Marco Polo's Zaitun with Changchau, in T'oung pao," Archives pour servir 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's Dawn of Modern Geography, Vol. III, chap, ii, and Part II.

[422] Heyd, loc. cit., Vol. II, p. 220; H. Yule, in Encyclop?dia Britannica, 9th (10th) or 11th ed., article "China." The handbook cited is Pegolotti's Libro di divisamenti di paesi, chapters i-ii, where it is implied that $60,000 would be a likely amount for a merchant going to China to invest in his trip.

[423] Cunningham, loc. cit., p. 194.

[424] I.e. a commission house.

[425] Cunningham, loc. cit., p. 186.

[426] J. R. Green, Short History of the English People, New York, 1890, p. 66.

[427] W. Besant, London, New York, 1892, p. 43.

[428] Baldakin, baldekin, baldachino.

[429] Italian Baldacco.

[430] J. K. Mumford, Oriental Rugs, New York, 1901, p. 18.

[431] Or Girbert, the Latin forms Gerbertus and Girbertus appearing indifferently in the documents of his time.

[432] See, for example, J. C. Heilbronner, Historia matheseos univers?, p. 740.

[433] "Obscuro loco natum," as an old chronicle of Aurillac has it.

[434] N. Bubnov, Gerberti postea Silvestri II papae opera mathematica, Berlin, 1899, is the most complete and reliable source of information; Picavet, loc. cit., Gerbert etc.; Olleris, ?uvres de Gerbert, Paris, 1867; Havet, Lettres de Gerbert, Paris, 1889 ; H. Weissenborn, Gerbert; Beitr?ge zur Kenntnis der Mathematik des Mittelalters, Berlin, 1888, and Zur 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 P?dagogik, 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 geometri?, 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 (?uvres 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 the Zeitschrift für Mathematik und Physik, Vol. XII (1867), Literaturzeitung, p. 70: "Für das System unserer Numeration ist die Null das wesentlichste Merkmal, und diese kannte Gerbert nicht. Er selbst schrieb alle Zahlen mit den r?mischen Zahlzeichen und man kann ihm also nicht verdanken, was er selbst nicht kannte."

[449] E.g., Chasles, Büdinger, Gerhardt, and Richer. So Martin (Recherches nouvelles etc.) 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 one Al-Man?ūr, governing Spain under the name of Hishām (976-1002), called from the Orient Al-Be?ānī to teach his son, so that scholars were recognized. [Picavet, p. 36.]

[451] Weissenborn, loc. cit., p. 235.

[452] Ibid., p. 234.

[453] These letters, of the period 983-997, were edited by Havet, loc. cit., and, less completely, by Olleris, loc. cit. Those touching mathematical topics were edited by Bubnov, loc. cit., pp. 98-106.

[454] He published it in the Monumenta Germaniae historica, "Scriptores," Vol. III, and at least three other editions have since appeared, viz. those by Guadet in 1845, by Poinsignon in 1855, and by Waitz in 1877.

[455] Domino ac beatissimo Patri Gerberto, Remorum archiepiscopo, Richerus Monchus, Gallorum congressibus in volumine regerendis, imperii tui, pater sanctissime Gerberte, auctoritas seminarium dedit.

[456] In epistle 17 (Havet collection) he speaks of the "De multiplicatione et divisione numerorum libellum a Joseph Ispano editum abbas Warnerius" (a person otherwise unknown). In epistle 25 he says: "De multiplicatione et divisione numerorum, Joseph Sapiens sententias quasdam edidit."

[457] H. Suter, "Zur Frage über den Josephus Sapiens," Bibliotheca Mathematica, Vol. VIII (2), p. 84; Weissenborn, Einführung, p. 14; also his Gerbert; M. Steinschneider, in Bibliotheca Mathematica, 1893, p. 68. Wallis (Algebra, 1685, chap. 14) went over the list of Spanish Josephs very carefully, but could find nothing save that "Josephus Hispanus seu Josephus sapiens videtur aut Maurus fuisse aut alius quis in Hispania."

[458] P. Ewald, Mittheilungen, Neues Archiv d. Gesellschaft für ?ltere deutsche Geschichtskunde, Vol. VIII, 1883, pp. 354-364. One of the manuscripts is of 976 A.D. and the other of 992 A.D. See also Franz Steffens, Lateinische Pal?ographie, Freiburg (Schweiz), 1903, pp. xxxix-xl. The forms are reproduced in the plate on page 140.

[459] It is entitled Constantino suo Gerbertus scolasticus, because it was addressed to Constantine, a monk of the Abbey of Fleury. The text of the letter to Constantine, preceding the treatise on the Abacus, is given in the Comptes rendus, Vol. XVI (1843), p. 295. This book seems to have been written c. 980 A.D. [Bubnov, loc. cit., p. 6.]

[460] "Histoire de l'Arithmétique," Comptes rendus, Vol. XVI (1843), pp. 156, 281.

[461] Loc. cit., Gerberti Opera etc.

[462] Friedlein thought it spurious. See Zeitschrift 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 (?uvres de Gerbert). See Weissenborn, Gerbert, pp. 2, 6, 168, and Picavet, p. 81. Hock, Cantor, and Th. Martin place the composition of the work at c. 996 when Gerbert was in Germany, while Olleris and Picavet refer it to the period when he was at Rheims.

[463] Picavet, loc. cit., p. 182.

[464] Who wrote after Gerbert became pope, for he uses, in his preface, the words, "a domino pape Gerberto." He was quite certainly not later than the eleventh century; we do not have exact information about the time in which he lived.

[465] Picavet, loc. cit., p. 182. Weissenborn, Gerbert, p. 227. In Olleris, Liber Abaci (of Bernelinus), p. 361.

[466] Richer, in Bubnov, loc. cit., p. 381.

[467] Weissenborn, Gerbert, p. 241.

[468] Writers on numismatics are quite uncertain as to their use. See F. Gnecchi, Monete Romane, 2d ed., Milan, 1900, cap. XXXVII. For pictures of old Greek tesserae of Sarmatia, see S. Ambrosoli, Monete Greche, Milan, 1899, p. 202.

[469] Thus Tzwivel's arithmetic of 1507, fol. 2, v., speaks of the ten figures as "characteres sive numerorum apices a diuo Seuerino Boetio."

[470] Weissenborn uses sipos for 0. It is not given by Bernelinus, and appears in Radulph of Laon, in the twelfth century. See Günther's Geschichte, p. 98, n.; Weissenborn, p. 11; Pihan, Exposé etc., pp. xvi-xxii.

In Friedlein's Boetius, p. 396, the plate shows that all of the six important manuscripts from which the illustrations are taken contain the symbol, while four out of five which give the words use the word sipos for 0. The names appear in a twelfth-century anonymous manuscript in the Vatican, in a passage beginning

Ordine primigeno sibi nomen possidet igin.

Andras ecce locum mox uendicat ipse secundum

Ormis post numeros incompositus sibi primus.

[Boncompagni Buttetino, XV, p. 132.] Turchill (twelfth century) gives the names Igin, andras, hormis, arbas, quimas, caletis, zenis, temenias, celentis, saying: "Has autem figuras, ut donnus [dominus] Gvillelmus Rx testatur, a pytagoricis habemus, nomina uero ab arabibus." (Who the William R. was is not known. Boncompagni Bulletino XV, p. 136.) Radulph of Laon (d. 1131) asserted that they were Chaldean (Propagation, p. 48 n.). A discussion of the whole question is also given in E. C. Bayley, loc. cit. Huet, writing in 1679, asserted that they were of Semitic origin, as did Nesselmann in spite of his despair over ormis, calctis, and celentis; see Woepcke, Propagation, p. 48. The names were used as late as the fifteenth century, without the zero, but with the superscript dot for 10's, two dots for 100's, etc., as among the early Arabs. Gerhardt mentions having seen a fourteenth or fifteenth century manuscript in the Bibliotheca Amploniana with the names "Ingnin, andras, armis, arbas, quinas, calctis, zencis, zemenias, zcelentis," and the statement "Si unum punctum super ingnin ponitur, X significat.... Si duo puncta super ... figuras superponunter, fiet decuplim illius quod cum uno puncto significabatur," in Monatsberichte der K. P. Akad. d. Wiss., Berlin, 1867, p. 40.

[471] A chart of ten numerals in 200 tongues, by Rev. R. Patrick, London, 1812.

[472] "Numeratio figuralis est cuiusuis numeri per notas, et figuras numerates descriptio." [Clichtoveus, edition of c. 1507, fol. C ii, v.] "Aristoteles enim uoces rerum σ?μβολα uocat: id translatum, sonat notas." [Noviomagus, De Numeris Libri II, cap. vi.] "Alphabetum decem notarum." [Schonerus, notes to Ramus, 1586, p. 3 seq.] Richer says: "novem numero notas omnem numerum significantes." [Bubnov, loc. cit., p. 381.]

[473] "Il y a dix Characteres, autrement Figures, Notes, ou Elements." [Peletier, edition of 1607, p. 13.] "Numerorum notas alij figuras, alij signa, alij characteres uocant." [Glareanus, 1545 edition, f. 9, r.] "Per figuras (quas zyphras uocant) assignationem, quales sunt h? notul?, 1. 2. 3. 4...." [Noviomagus, De Numeris Libri II, cap. vi.] Gemma Frisius also uses elementa and Cardan uses literae. In the first arithmetic by an American (Greenwood, 1729) the author speaks of "a few Arabian Charecters or Numeral Figures, called Digits" (p. 1), and as late as 1790, in the third edition of J. J. Blassière's arithmetic (1st ed. 1769), the name characters is still in use, both for "de Latynsche en de Arabische" (p. 4), as is also the term "Cyfferletters" (p. 6, n.). Ziffer, the modern German form of cipher, was commonly used to designate any of the nine figures, as by Boeschenstein and Riese, although others, like K?bel, used it only for the zero. So zifre appears in the arithmetic by Borgo, 1550 ed. In a Munich codex of the twelfth century, attributed to Gerland, they are called characters only: "Usque ad VIIII. enim porrigitur omnis numerus et qui supercrescit eisdem designator Karacteribus." [Boncompagni Bulletino, Vol. X. p. 607.]

[474] The title of his work is Prologus N. Ocreati in Helceph (Arabic al-qeif, investigation or memoir) ad Adelardum Batensem magistrum suum. The work was made known by C. Henry, in the Zeitschrift für Mathematik und Physik, Vol. XXV, p. 129, and in the Abhandlungen zur Geschichte der Mathematik, Vol. III; Weissenborn, Gerbert, p. 188.

[475] The zero is indicated by a vacant column.

[476] Leo Jordan, loc. cit., p. 170. "Chifre en augorisme" is the expression used, while a century later "giffre en argorisme" and "cyffres d'augorisme" are similarly used.

[477] The Works of Geoffrey Chaucer, edited by W. W. Skeat, Vol. IV, Oxford, 1894, p. 92.

[478] Loc. cit., Vol. III, pp. 179 and 180.

[479] In Book II, chap, vii, of The Testament of Love, printed with Chaucer's Works, loc. cit., Vol. VII, London, 1897.

[480] Liber Abacci, published in Olleris, ?uvres de Gerbert, pp. 357-400.

[481] G. R. Kaye, "The Use of the Abacus in Ancient India," Journal and Proceedings of the Asiatic Society of Bengal, 1908, pp. 293-297.

[482] Liber Abbaci, by Leonardo Pisano, loc. cit., p. 1.

[483] Friedlein, "Die Entwickelung des Rechnens mit Columnen," Zeitschrift für Mathematik und Physik, Vol. X, p. 247.

[484] The divisor 6 or 16 being increased by the difference 4, to 10 or 20 respectively.

[485] E.g. Cantor, Vol. I, p. 882.

[486] Friedlein, loc. cit.; Friedlein, "Gerbert's Regeln der Division" and "Das Rechnen mit Columnen vor dem 10. Jahrhundert," Zeitschrift für Mathematik und Physik, Vol. IX; Bubnov, loc. cit., pp. 197-245; M. Chasles, "Histoire de l'arithmétique. Recherches des traces du système de l'abacus, après que cette méthode a pris le nom d'Algorisme.-Preuves qu'à toutes les époques, jusq'au XVIe siècle, on a su que l'arithmétique vulgaire avait pour origine cette méthode ancienne," Comptes rendus, Vol. XVII, pp. 143-154, also "Règles de l'abacus," Comptes rendus, Vol. XVI, pp. 218-246, and "Analyse et explication du traité de Gerbert," Comptes rendus, Vol. XVI, pp. 281-299.

[487] Bubnov, loc. cit., pp. 203-204, "Abbonis abacus."

[488] "Regulae de numerorum abaci rationibus," in Bubnov, loc. cit., pp. 205-225.

[489] P. Treutlein, "Intorno ad alcuni scritti inediti relativi al calcolo dell' abaco," Bulletino di bibliografia e di storia delle scienze matematiche e fisiche, Vol. X, pp. 589-647.

[490] "Intorno ad uno scritto inedito di Adelhardo di Bath intitolato 'Regulae Abaci,'" B. Boncompagni, in his Bulletino, Vol. XIV, pp. 1-134.

[491] Treutlein, loc. cit.; Boncompagni, "Intorno al Tractatus de Abaco di Gerlando," Bulletino, Vol. X, pp. 648-656.

[492] E. Narducci, "Intorno a due trattati inediti d'abaco contenuti in due codici Vaticani del secolo XII," Boncompagni Bulletino, Vol. XV, pp. 111-162.

[493] See Molinier, Les sources de l'histoire de France, Vol. II, Paris, 1902, pp. 2, 3.

[494] Cantor, Geschichte, Vol. I, p. 762. A. Nagl in the Abhandlungen zur Geschichte der Mathematik, Vol. V, p. 85.

[495] 1030-1117.

[496] Abhandlungen zur Geschichte der Mathematik, Vol. V, pp. 85-133. The work begins "Incipit Liber Radulfi laudunensis de abaco."

[497] Materialien zur Geschichte der arabischen Zahlzeichen in Frankreich, loc. cit.

[498] Who died in 1202.

[499] Cantor, Geschichte, Vol. I (3), pp. 800-803; Boncompagni, Trattati, Part II. M. Steinschneider ("Die Mathematik bei den Juden," Bibliotheca Mathematica, Vol. X (2), p. 79) ingeniously derives another name by which he is called (Abendeuth) from Ibn Daūd (Son of David). See also Abhandlungen, Vol. III, p. 110.

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

[501] For it says, "Incipit prologus in libro alghoarismi de practica arismetrice. Qui editus est a magistro Johanne yspalensi." It is published in full in the second part of Boncompagni's Trattati d'aritmetica.

[502] Possibly, indeed, the meaning of "libro alghoarismi" is not "to Al-Khowārazmī's book," but "to a book of algorism." John of Luna says of it: "Hoc idem est illud etiam quod ... alcorismus dicere videtur." [Trattati, p. 68.]

[503] For a résumé, see Cantor, Vol. I (3), pp. 800-803. As to the author, see Enestr?m in the Bibliotheca Mathematica, Vol. VI (3), p. 114, and Vol. IX (3), p. 2.

[504] Born at Cremona (although some have asserted at Carmona, in Andalusia) in 1114; died at Toledo in 1187. Cantor, loc. cit.; Boncompagni, Atti d. R. Accad. d. n. Lincei, 1851.

[505] See Abhandlungen zur Geschichte der Mathematik, Vol. XIV, p. 149; Bibliotheca Mathematica, Vol. IV (3), p. 206. Boncompagni had a fourteenth-century manuscript of his work, Gerardi Cremonensis artis metrice practice. See also T. L. Heath, The Thirteen Books of Euclid's Elements, 3 vols., Cambridge, 1908, Vol. I, pp. 92-94 ; A. A. Bj?rnbo, "Gerhard von Cremonas übersetzung von Alkwarizmis Algebra und von Euklids Elementen," Bibliotheca Mathematica, Vol. VI (3), pp. 239-248.

[506] Wallis, Algebra, 1685, p. 12 seq.

[507] Cantor, Geschichte, Vol. I (3), p. 906; A. A. Bj?rnbo, "Al-Chwārizmī's trigonometriske Tavler," Festskrift til H. G. Zeuthen, Copenhagen, 1909, pp. 1-17.

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

[509] M. Steinschneider, Zeitschrift der deutschen morgenl?ndischen Gesellschaft, Vol. XXV, 1871, p. 104, and Zeitschrift für Mathematik und Physik, Vol. XVI, 1871, pp. 392-393; M. Curtze, Centralblatt für Bibliothekswesen, 1899, p. 289; E. Wappler, Zur Geschichte der deutschen Algebra im 15. Jahrhundert, Programm, Zwickau, 1887; L. C. Karpinski, "Robert of Chester's Translation of the Algebra of Al-Khowārazmī," Bibliotheca Mathematica, Vol. XI (3), p. 125. He is also known as Robertus Retinensis, or Robert of Reading.

[510] Nagl, A., "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und über die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im christl. Abendlande," in the Zeitschrift für Mathematik und Physik, Hist.-lit. Abth., Vol. XXXIV, p. 129. Curtze, Abhandlungen zur Geschichte der Mathematik, Vol. VIII, pp. 1-27.

[511] See line a in the plate on p. 143.

[512] Sefer ha-Mispar, Das Buch der Zahl, ein hebr?isch-arithmetisches Werk des R. Abraham ibn Esra, Moritz Silberberg, Frankfurt a. M., 1895.

[513] Browning's "Rabbi ben Ezra."

[514] "Darum haben auch die Weisen Indiens all ihre Zahlen durch neun bezeichnet und Formen für die 9 Ziffern gebildet." [Sefer ha-Mispar, loc. cit., p. 2.]

[515] F. Bonaini, "Memoria unica sincrona di Leonardo Fibonacci," Pisa, 1858, republished in 1867, and appearing in the Giornale Arcadico, Vol. CXCVII (N.S. LII); Gaetano Milanesi, Documento inedito e sconosciuto a Lionardo Fibonacci, Roma, 1867; Guglielmini, Elogio di Lionardo Pisano, Bologna, 1812, p. 35; Libri, Histoire des sciences mathématiques, Vol. II, p. 25; D. Martines, Origine e progressi dell' aritmetica, Messina, 1865, p. 47; Lucas, in Boncompagni Bulletino, Vol. X, pp. 129, 239; Besagne, ibid., Vol. IX, p. 583; Boncompagni, three works as cited in Chap. I; G. Enestr?m, "Ueber zwei angebliche mathematische Schulen im christlichen Mittelalter," Bibliotheca Mathematica, Vol. VIII (3), pp. 252-262; Boncompagni, "Della vita e delle opere di Leonardo Pisano," loc. cit.

[516] The date is purely conjectural. See the Bibliotheca Mathematica, Vol. IV (3), p. 215.

[517] An old chronicle relates that in 1063 Pisa fought a great battle with the Saracens at Palermo, capturing six ships, one being "full of wondrous treasure," and this was devoted to building the cathedral.

[518] Heyd, loc. cit., Vol. I, p. 149.

[519] Ibid., p. 211.

[520] J. A. Symonds, Renaissance in Italy. The Age of Despots. New York, 1883, p. 62.

[521] Symonds, loc. cit., p. 79.

[522] J. A. Froude, The Science of History, London, 1864. "Un brevet d'apothicaire n'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.]

[523] A document of 1226, found and published in 1858, reads: "Leonardo bigollo quondam Guilielmi."

[524] "Bonaccingo germano suo."

[525] E.g. Libri, Guglielmini, Tiraboschi.

[526] Latin, Bonaccius.

[527] Boncompagni and Milanesi.

[528] Reprint, p. 5.

[529] Whence the French name for candle.

[530] Now part of Algiers.

[531] E. Reclus, Africa, New York, 1893, Vol. II, p. 253.

[532] "Sed hoc totum et algorismum atque arcus pictagore quasi errorem computavi respectu modi indorum." Woepcke, Propagation etc., regards this as referring to two different systems, but the expression may very well mean algorism as performed upon the Pythagorean arcs (or table).

[533] "Book of the Abacus," this term then being used, and long afterwards in Italy, to mean merely the arithmetic of computation.

[534] "Incipit liber Abaci a Leonardo filio Bonacci compositus anno 1202 et correctus ab eodem anno 1228." Three MSS. of the thirteenth century are known, viz. at Milan, at Siena, and in the Vatican library. The work was first printed by Boncompagni in 1857.

[535] I.e. in relation to the quadrivium. "Non legant in festivis diebus, nisi Philosophos et rhetoricas et quadrivalia et barbarismum et ethicam, si placet." Suter, Die Mathematik auf den Universit?ten 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.

[536] On the commercial activity of the period, it is known that bills of exchange passed between Messina and Constantinople in 1161, and that a bank was founded at Venice in 1170, the Bank of San Marco being established in the following year. The activity of Pisa was very manifest at this time. Heyd, loc. cit., Vol. II, p. 5; V. Casagrandi, Storia e cronologia, 3d ed., Milan, 1901, p. 56.

[537] J. A. Symonds, loc. cit., Vol. II, p. 127.

[538] I. Taylor, The Alphabet, London, 1883, Vol. II, p. 263.

[539] Cited by Unger's History, p. 15. The Arabic numerals appear in a Regensburg chronicle of 1167 and in Silesia in 1340. See Schmidt's Encyclop?die der Erziehung, Vol. VI, p. 726; A. Kuckuk, "Die Rechenkunst im sechzehnten Jahrhundert," Festschrift zur dritten S?cularfeier des Berlinischen Gymnasiums zum grauen Kloster, Berlin, 1874, p. 4.

[540] The text is given in Halliwell, Rara Mathematica, London, 1839.

[541] Seven are given in Ashmole's Catalogue of Manuscripts in the Oxford Library, 1845.

[542] Maximilian Curtze, Petri Philomeni de Dacia in Algorismum Vulgarem Johannis de Sacrobosco commentarius, una cum Algorismo ipso, Copenhagen, 1897; L. C. Karpinski, "Jordanus Nemorarius and John of Halifax," American Mathematical Monthly, Vol. XVII, pp. 108-113.

[543] J. Aschbach, Geschichte der Wiener Universit?t im ersten Jahrhunderte ihres Bestehens, Wien, 1865, p. 93.

[544] Curtze, loc. cit., gives the text.

[545] Curtze, loc. cit., found some forty-five copies of the Algorismus in three libraries of Munich, Venice, and Erfurt (Amploniana). Examination of two manuscripts from the Plimpton collection and the Columbia library shows such marked divergence from each other and from the text published by Curtze that the conclusion seems legitimate that these were students' lecture notes. The shorthand character of the writing further confirms this view, as it shows that they were written largely for the personal use of the writers.

[546] "Quidam philosophus edidit nomine Algus, unde et Algorismus nuncupatur." [Curtze, loc. cit., p. 1.]

[547] "Sinistrorsum autera scribimus in hac arte more arabico sive iudaico, huius scientiae inventorum." [Curtze, loc. cit., p. 7.] The Plimpton manuscript omits the words "sive iudaico."

[548] "Non enim omnis numerus per quascumque figuras Indorum repraesentatur, sed tantum determinatus per determinatam, ut 4 non per 5,..." [Curtze, loc. cit., p. 25.]

[549] C. Henry, "Sur les deux plus anciens traités fran?ais d'algorisme et de géométrie," Boncompagni Bulletino, Vol. XV, p. 49; Victor Mortet, "Le plus ancien traité fran?ais d'algorisme," loc. cit.

[550] L'état des sciences en France, depute la mort du Roy Robert, arrivée en 1031, jusqu'à celle de Philippe le Bel, arrivée en 1314, Paris, 1741.

[551] Discours sur l'état des lettres en France au XIIIe siecle, Paris, 1824.

[552] Aper?u historique, Paris, 1876 ed., p. 464.

[553] Ranulf Higden, a native of the west of England, entered St. Werburgh's monastery at Chester in 1299. He was a Benedictine monk and chronicler, and died in 1364. His Polychronicon, a history in seven books, was printed by Caxton in 1480.

[554] Trevisa's translation, Higden having written in Latin.

[555] An illustration of this feeling is seen in the writings of Prosdocimo de' Beldomandi (b. c. 1370-1380, d. 1428): "Inveni in quam pluribus libris algorismi nuncupatis mores circa numeros operandi satis varios atque diversos, qui licet boni existerent atque veri erant, tamen fastidiosi, tum propter ipsarum regularum multitudinem, tum propter earum deleationes, tum etiam propter ipsarum operationum probationes, utrum si bone fuerint vel ne. Erant et etiam isti modi interim fastidiosi, quod si in aliquo calculo astroloico error contigisset, calculatorem operationem suam a capite incipere oportebat, dato quod error suus adhuc satis propinquus existeret; et hoc propter figuras in sua operatione deletas. Indigebat etiam calculator semper aliquo lapide vel sibi conformi, super quo scribere atque faciliter delere posset figuras cum quibus operabatur in calculo suo. Et quia haec omnia satis fastidiosa atque laboriosa mihi visa sunt, disposui libellum edere in quo omnia ista abicerentur: qui etiam algorismus sive liber de numeris denominari poterit. Scias tamen quod in hoc libello ponere non intendo nisi ea quae ad calculum necessaria sunt, alia quae in aliis libris practice arismetrice tanguntur, ad calculum non necessaria, propter brevitatem dimitendo." [Quoted by A. Nagl, Zeitschrift für Mathematik und Physik, Hist.-lit. Abth., Vol. XXXIV, p. 143; Smith, Rara Arithmetica, p. 14, in facsimile.]

[556] P. Ewald, loc. cit.; Franz Steffens, Lateinische Pal?ographie, pp. xxxix-xl. We are indebted to Professor J. M. Burnam for a photograph of this rare manuscript.

[557] See the plate of forms on p. 88.

[558] Karabacek, loc. cit., p. 56; Karpinski, "Hindu Numerals in the Fihrist," Bibliotheca Mathematica, Vol. XI (3), p. 121.

[559] Woepcke, "Sur une 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.

[560] Archeological Report of the Egypt Exploration Fund for 1908-1909, London, 1910, p. 18.

[561] There was a set of astronomical tables in Boncompagni's library bearing this date: "Nota quod anno dni nri ih? xpi. 1264. perfecto." See Narducci's Catalogo, p. 130.

[562] "On the Early use of Arabic Numerals in Europe," read before the Society of Antiquaries April 14, 1910, and published in Arch?ologia in the same year.

[563] Ibid., p. 8, n. The date is part of an Arabic inscription.

[564] O. Codrington, A Manual of Musalman Numismatics, London, 1904.

[565] See Arbuthnot, The Mysteries of Chronology, London, 1900, pp. 75, 78, 98; F. Pichler, Repertorium der steierischen Münzkunde, Gr?tz, 1875, where the claim is made of an Austrian coin of 1458; Bibliotheca Mathematica, Vol. X (2), p. 120, and Vol. XII (2), p. 120. There is a Brabant piece of 1478 in the collection of D. E. Smith.

[566] A specimen is in the British Museum. [Arbuthnot, p. 79.]

[567] Ibid., p. 79.

[568] Liber de Remediis utriusque fortunae Coloniae.

[569] Fr. Walthern et Hans Hurning, N?rdlingen.

[570] Ars Memorandi, one of the oldest European block-books.

[571] Eusebius Caesariensis, De praeparatione evangelica, Venice, Jenson, 1470. The above statement holds for copies in the Astor Library and in the Harvard University Library.

[572] Francisco de Retza, Comestorium vitiorum, Nürnberg, 1470. The copy referred to is in the Astor Library.

[573] See Mauch, "Ueber den Gebrauch arabischer Ziffern und die Ver?nderungen 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.

[574] Günther, Geschichte, p. 175, n.; Mauch, loc. cit.

[575] These are given by W. R. Lethaby, from drawings by J. T. Irvine, in the Proceedings of the Society of Antiquaries, 1906, p. 200.

[576] There are some ill-tabulated forms to be found in J. Bowring, The Decimal System, London, 1854, pp. 23, 25, and in L. A. Chassant, Dictionnaire des abréviations latines et fran?aises ... du moyen age, Paris, MDCCCLXVI, p. 113. The best sources we have at present, aside from the Hill monograph, are P. Treutlein, Geschichte unserer Zahlzeichen, Karlsruhe, 1875; Cantor's Geschichte, Vol. I, table; M. Prou, Manuel de paléographie latine et fran?aise, 2d ed., Paris, 1892, p. 164; A. Cappelli, Dizionario di abbreviature latine ed italiane, Milan, 1899. An interesting early source is found in the rare Caxton work of 1480, The Myrrour of the World. In Chap. X is a cut with the various numerals, the chapter beginning "The fourth scyence is called arsmetrique." Two of the fifteen extant copies of this work are at present in the library of Mr. J. P. Morgan, in New York.

[577] From the twelfth-century manuscript on arithmetic, Curtze, loc. cit., Abhandlungen, and Nagl, loc. cit. The forms are copied from Plate VII in Zeitschrift für Mathematik und Physik, Vol. XXXIV.

[578] From the Regensburg chronicle. Plate containing some of these numerals in Monumenta Germaniae historica, "Scriptores" Vol. XVII, plate to p. 184; Wattenbach, Anleitung zur lateinischen Palaeographie, Leipzig, 1886, p. 102; Boehmer, Fontes rerum Germanicarum, Vol. III, Stuttgart, 1852, p. lxv.

[579] French Algorismus of 1275; from an unpublished photograph of the original, in the possession of D. E. Smith. See also p. 135.

[580] From a manuscript of Boethius c. 1294, in Mr. Plimpton's library. Smith, Rara Arithmetica, Plate I.

[581] Numerals in a 1303 manuscript in Sigmaringen, copied from Wattenbach, loc. cit., p. 102.

[582] From a manuscript, Add. Manuscript 27,589, British Museum, 1360 A.D. The work is a computus in which the date 1360 appears, assigned in the British Museum catalogue to the thirteenth century.

[583] From the copy of Sacrabosco's Algorismus in Mr. Plimpton's library. Date c. 1442. See Smith, Rara Arithmetica, p. 450.

[584] See Rara Arithmetica, pp. 446-447.

[585] Ibid., pp. 469-470.

[586] Ibid., pp. 477-478.

[587] The i is used for "one" in the Treviso arithmetic (1478), Clichtoveus (c. 1507 ed., where both i and j are so used), Chiarini (1481), Sacrobosco (1488 ed.), and Tzwivel (1507 ed., where jj and jz are used for 11 and 12). This was not universal, however, for the Algorithmus linealis of c. 1488 has a special type for 1. In a student's notebook of lectures taken at the University of Würzburg in 1660, in Mr. Plimpton's library, the ones are all in the form of i.

[588] Thus the date , for 1580, appears in a MS. in the Laurentian library at Florence. The second and the following five characters are taken from Cappelli's Dizionario, p. 380, and are from manuscripts of the twelfth, thirteenth, fourteenth, sixteenth, seventeenth, and eighteenth centuries, respectively.

[589] E.g. Chiarini's work of 1481; Clichtoveus (c. 1507).

[590] The first is from an algorismus of the thirteenth century, in the Hannover Library. [See Gerhardt, "Ueber die Entstehung und Ausbreitung des dekadischen Zahlensystems," loc. cit., p. 28.] The second character is from a French algorismus, c. 1275. [Boncompagni Bulletino, Vol. XV, p. 51.] The third and the following sixteen characters are given by Cappelli, loc. cit., and are from manuscripts of the twelfth (1), thirteenth (2), fourteenth (7), fifteenth (3), sixteenth (1), seventeenth (2), and eighteenth (1) centuries, respectively.

[591] Thus Chiarini (1481) has for 23.

[592] The first of these is from a French algorismus, c. 1275. The second and the following eight characters are given by Cappelli, loc. cit., and are from manuscripts of the twelfth (2), thirteenth, fourteenth, fifteenth (3), seventeenth, and eighteenth centuries, respectively.

[593] See Nagl, loc. cit.

[594] Hannover algorismus, thirteenth century.

[595] See the Dagomari manuscript, in Rara Arithmetica, pp. 435, 437-440.

[596] But in the woodcuts of the Margarita Philosophica (1503) the old forms are used, although the new ones appear in the text. In Caxton's Myrrour of the World (1480) the old form is used.

[597] Cappelli, loc. cit. They are partly from manuscripts of the tenth, twelfth, thirteenth (3), fourteenth (7), fifteenth (6), and eighteenth centuries, respectively. Those in the third line are from Chassant's Dictionnaire, p. 113, without mention of dates.

[598] The first is from the Hannover algorismus, thirteenth century. The second is taken from the Rollandus manuscript, 1424. The others in the first two lines are from Cappelli, twelfth (3), fourteenth (6), fifteenth (13) centuries, respectively. The third line is from Chassant, loc. cit., p. 113, no mention of dates.

[599] The first of these forms is from the Hannover algorismus, thirteenth century. The following are from Cappelli, fourteenth (3), fifteenth, sixteenth (2), and eighteenth centuries, respectively.

[600] The first of these is taken from the Hannover algorismus, thirteenth century. The following forms are from Cappelli, twelfth, thirteenth, fourteenth (5), fifteenth (2), seventeenth, and eighteenth centuries, respectively.

[601] All of these are given by Cappelli, thirteenth, fourteenth, fifteenth (2), and sixteenth centuries, respectively.

[602] Smith, Rara Arithmetica, p. 489. This is also seen in several of the Plimpton manuscripts, as in one written at Ancona in 1684. See also Cappelli, loc. cit.

[603] French algorismus, c. 1275, for the first of these forms. Cappelli, thirteenth, fourteenth, fifteenth (3), and seventeenth centuries, respectively. The last three are taken from Byzantinische Analekten, J. L. Heiberg, being forms of the fifteenth century, but not at all common. was the old Greek symbol for 90.

[604] For the first of these the reader is referred to the forms ascribed to Boethius, in the illustration on p. 88; for the second, to Radulph of Laon, see p. 60. The third is used occasionally in the Rollandus (1424) manuscript, in Mr. Plimpton's library. The remaining three are from Cappelli, fourteenth (2) and seventeenth centuries.

[605] Smith, An Early English Algorism.

[606] Kuckuck, p. 5.

[607] A. Cappelli, loc. cit., p. 372.

[608] Smith, Rara Arithmetica, p. 443.

[609] Curtze, Petri Philomeni de Dacia etc., p. IX.

[610] Cappelli, loc. cit., p. 376.

[611] Curtze, loc. cit., pp. VIII-IX, note.

[612] Edition of 1544-1545, f. 52.

[613] De numeris libri II, 1544 ed., cap. XV. Heilbronner, loc. cit., p. 736, also gives them, and compares this with other systems.

[614] Noviomagus says of them: "De quibusdam Astrologicis, sive Chaldaicis numerorum notis.... Sunt & ali? qu?dam not?, quibus Chaldaei & Astrologii quemlibet numerum artificiose & arguté describunt, scitu periucundae, quas nobis communicauit Rodolphus Paludanus Nouiomagus."