A century after Scylax, Herodotus showed considerable knowledge of India, speaking of its cotton and its gold,[296] telling how Sesostris[297] fitted out ships to sail to that country, and mentioning the routes to the east. These routes were generally by the Red Sea, and had been followed by the Phoenicians and the Sabaeans, and later were taken by the Greeks and Romans.[298]

In the fourth century B.C. the West and East came into very close relations. As early as 330, Pytheas of Massilia (Marseilles) had explored as far north as the northern end of the British Isles and the coasts of the German Sea, while Macedon, in close touch with southern France, was also sending her armies under Alexander[299] through Afghanistan as far east as the Punjab.[300] Pliny tells us that Alexander the Great employed surveyors to measure {77} the roads of India; and one of the great highways is described by Megasthenes, who in 295 B.C., as the ambassador of Seleucus, resided at Pātalīpuṭra, the present Patna.[301]

The Hindus also learned the art of coining from the Greeks, or possibly from the Chinese, and the stores of Greco-Hindu coins still found in northern India are a constant source of historical information.[302] The Rāmāyana speaks of merchants traveling in great caravans and embarking by sea for foreign lands.[303] Ceylon traded with Malacca and Siam, and Java was colonized by Hindu traders, so that mercantile knowledge was being spread about the Indies during all the formative period of the numerals.

Moreover the results of the early Greek invasion were embodied by Dicaearchus of Messana (about 320 B.C.) in a map that long remained a standard. Furthermore, Alexander did not allow his influence on the East to cease. He divided India into three satrapies,[304] placing Greek governors over two of them and leaving a Hindu ruler in charge of the third, and in Bactriana, a part of Ariana or ancient Persia, he left governors; and in these the western civilization was long in evidence. Some of the Greek and Roman metrical and astronomical terms {78} found their way, doubtless at this time, into the Sanskrit language.[305] Even as late as from the second to the fifth centuries A.D., Indian coins showed the Hellenic influence. The Hindu astronomical terminology reveals the same relationship to western thought, for Varāha-Mihira (6th century A.D.), a contemporary of Āryabhaṭa, entitled a work of his the Bṛhat-Saṃhitā, a literal translation of [Greek: megale suntaxis] of Ptolemy;[306] and in various ways is this interchange of ideas apparent.[307] It could not have been at all unusual for the ancient Greeks to go to India, for Strabo lays down the route, saying that all who make the journey start from Ephesus and traverse Phrygia and Cappadocia before taking the direct road.[308] The products of the East were always finding their way to the West, the Greeks getting their ginger[309] from Malabar, as the Phoenicians had long before brought gold from Malacca.

Greece must also have had early relations with China, for there is a notable similarity between the Greek and Chinese life, as is shown in their houses, their domestic customs, their marriage ceremonies, the public story-tellers, the puppet shows which Herodotus says were introduced from Egypt, the street jugglers, the games of dice,[310] the game of finger-guessing,[311] the water clock, the {79} music system, the use of the myriad,[312] the calendars, and in many other ways.[313] In passing through the suburbs of Peking to-day, on the way to the Great Bell temple, one is constantly reminded of the semi-Greek architecture of Pompeii, so closely does modern China touch the old classical civilization of the Mediterranean. The Chinese historians tell us that about 200 B.C. their arms were successful in the far west, and that in 180 B.C. an ambassador went to Bactria, then a Greek city, and reported that Chinese products were on sale in the markets there.[314] There is also a noteworthy resemblance between certain Greek and Chinese words,[315] showing that in remote times there must have been more or less interchange of thought.

The Romans also exchanged products with the East. Horace says, "A busy trader, you hasten to the farthest Indies, flying from poverty over sea, over crags, over fires."[316] The products of the Orient, spices and jewels from India, frankincense from Persia, and silks from China, being more in demand than the exports from the Mediterranean lands, the balance of trade was against the West, and thus Roman coin found its way eastward. In 1898, for example, a number of Roman coins dating from 114 B.C. to Hadrian's time were found at Paklī, a part of the Hazāra district, sixteen miles north of Abbottābād,[317] and numerous similar discoveries have been made from time to time.

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Augustus speaks of envoys received by him from India, a thing never before known,[318] and it is not improbable that he also received an embassy from China.[319] Suetonius (first century A.D.) speaks in his history of these relations,[320] as do several of his contemporaries,[321] and Vergil[322] tells of Augustus doing battle in Persia. In Pliny's time the trade of the Roman Empire with Asia amounted to a million and a quarter dollars a year, a sum far greater relatively then than now,[323] while by the time of Constantine Europe was in direct communication with the Far East.[324]

In view of these relations it is not beyond the range of possibility that proof may sometime come to light to show that the Greeks and Romans knew something of the {81} number system of India, as several writers have maintained.[325]

Returning to the East, there are many evidences of the spread of knowledge in and about India itself. In the third century B.C. Buddhism began to be a connecting medium of thought. It had already permeated the Himalaya territory, had reached eastern Turkestan, and had probably gone thence to China. Some centuries later (in 62 A.D.) the Chinese emperor sent an ambassador to India, and in 67 A.D. a Buddhist monk was invited to China.[326] Then, too, in India itself Aśoka, whose name has already been mentioned in this work, extended the boundaries of his domains even into Afghanistan, so that it was entirely possible for the numerals of the Punjab to have worked their way north even at that early date.[327]

Furthermore, the influence of Persia must not be forgotten in considering this transmission of knowledge. In the fifth century the Persian medical school at Jondi-Sapur admitted both the Hindu and the Greek doctrines, and Firdusī tells us that during the brilliant reign of {82} Khosrū I,[328] the golden age of Pahlavī literature, the Hindu game of chess was introduced into Persia, at a time when wars with the Greeks were bringing prestige to the Sassanid dynasty.

Again, not far from the time of Boethius, in the sixth century, the Egyptian monk Cosmas, in his earlier years as a trader, made journeys to Abyssinia and even to India and Ceylon, receiving the name Indicopleustes (the Indian traveler). His map (547 A.D.) shows some knowledge of the earth from the Atlantic to India. Such a man would, with hardly a doubt, have observed every numeral system used by the people with whom he sojourned,[329] and whether or not he recorded his studies in permanent form he would have transmitted such scraps of knowledge by word of mouth.

As to the Arabs, it is a mistake to feel that their activities began with Mohammed. Commerce had always been held in honor by them, and the Qoreish[330] had annually for many generations sent caravans bearing the spices and textiles of Yemen to the shores of the Mediterranean. In the fifth century they traded by sea with India and even with China, and Ḥira was an emporium for the wares of the East,[331] so that any numeral system of any part of the trading world could hardly have remained isolated.

Long before the warlike activity of the Arabs, Alexandria had become the great market-place of the world. From this center caravans traversed Arabia to Hadramaut, where they met ships from India. Others went north to Damascus, while still others made their way {83} along the southern shores of the Mediterranean. Ships sailed from the isthmus of Suez to all the commercial ports of Southern Europe and up into the Black Sea. Hindus were found among the merchants[332] who frequented the bazaars of Alexandria, and Brahmins were reported even in Byzantium.

Such is a very brief resume of the evidence showing that the numerals of the Punjab and of other parts of India as well, and indeed those of China and farther Persia, of Ceylon and the Malay peninsula, might well have been known to the merchants of Alexandria, and even to those of any other seaport of the Mediterranean, in the time of Boethius. The Brāhmī numerals would not have attracted the attention of scholars, for they had no zero so far as we know, and therefore they were no better and no worse than those of dozens of other systems. If Boethius was attracted to them it was probably exactly as any one is naturally attracted to the bizarre or the mystic, and he would have mentioned them in his works only incidentally, as indeed they are mentioned in the manuscripts in which they occur.

In answer therefore to the second question, Could Boethius have known the Hindu numerals? the reply must be, without the slightest doubt, that he could easily have known them, and that it would have been strange if a man of his inquiring mind did not pick up many curious bits of information of this kind even though he never thought of making use of them.

Let us now consider the third question, Is there any positive or strong circumstantial evidence that Boethius did know these numerals? The question is not new, {84} nor is it much nearer being answered than it was over two centuries ago when Wallis (1693) expressed his doubts about it[333] soon after Vossius (1658) had called attention to the matter.[334] Stated briefly, there are three works on mathematics attributed to Boethius:[335] (1) the arithmetic, (2) a work on music, and (3) the geometry.[336]

The genuineness of the arithmetic and the treatise on music is generally recognized, but the geometry, which contains the Hindu numerals with the zero, is under suspicion.[337] There are plenty of supporters of the idea that Boethius knew the numerals and included them in this book,[338] and on the other hand there are as many who {85} feel that the geometry, or at least the part mentioning the numerals, is spurious.[339] The argument of those who deny the authenticity of the particular passage in question may briefly be stated thus:

1. The falsification of texts has always been the subject of complaint. It was so with the Romans,[340] it was common in the Middle Ages,[341] and it is much more prevalent {86} to-day than we commonly think. We have but to see how every hymn-book compiler feels himself authorized to change at will the classics of our language, and how unknown editors have mutilated Shakespeare, to see how much more easy it was for medieval scribes to insert or eliminate paragraphs without any protest from critics.[342]

2. If Boethius had known these numerals he would have mentioned them in his arithmetic, but he does not do so.[343]

3. If he had known them, and had mentioned them in any of his works, his contemporaries, disciples, and successors would have known and mentioned them. But neither Capella (c. 475)[344] nor any of the numerous medieval writers who knew the works of Boethius makes any reference to the system.[345]

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4. The passage in question has all the appearance of an interpolation by some scribe. Boethius is speaking of angles, in his work on geometry, when the text suddenly changes to a discussion of classes of numbers.[346] This is followed by a chapter in explanation of the abacus,[347] in which are described those numeral forms which are called apices or caracteres.[348] The forms[349] of these characters vary in different manuscripts, but in general are about as shown on page 88. They are commonly written with the 9 at the left, decreasing to the unit at the right, numerous writers stating that this was because they were derived from Semitic sources in which the direction of writing is the opposite of our own. This practice continued until the sixteenth century.[350] The writer then leaves the subject entirely, using the Roman numerals for the rest of his discussion, a proceeding so foreign to the method of Boethius as to be inexplicable on the hypothesis of authenticity. Why should such a scholarly writer have given them with no mention of their origin or use? Either he would have mentioned some historical interest attaching to them, or he would have used them in some discussion; he certainly would not have left the passage as it is.

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FORMS OF THE NUMERALS, LARGELY FROM WORKS ON THE ABACUS[351]

a[352] b[353] c[354] d[355] e[356] f[357] g[358] h[359] i[360]

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Sir E. Clive Bayley has added[361] a further reason for believing them spurious, namely that the 4 is not of the Nānā Ghāt type, but of the Kabul form which the Arabs did not receive until 776;[362] so that it is not likely, even if the characters were known in Europe in the time of Boethius, that this particular form was recognized. It is worthy of mention, also, that in the six abacus forms from the chief manuscripts as given by Friedlein,[363] each contains some form of zero, which symbol probably originated in India about this time or later. It could hardly have reached Europe so soon.

As to the fourth question, Did Boethius probably know the numerals? It seems to be a fair conclusion, according to our present evidence, that (1) Boethius might very easily have known these numerals without the zero, but, (2) there is no reliable evidence that he did know them. And just as Boethius might have come in contact with them, so any other inquiring mind might have done so either in his time or at any time before they definitely appeared in the tenth century. These centuries, five in number, represented the darkest of the Dark Ages, and even if these numerals were occasionally met and studied, no trace of them would be likely to show itself in the {90} literature of the period, unless by chance it should get into the writings of some man like Alcuin. As a matter of fact, it was not until the ninth or tenth century that there is any tangible evidence of their presence in Christendom. They were probably known to merchants here and there, but in their incomplete state they were not of sufficient importance to attract any considerable attention.

As a result of this brief survey of the evidence several conclusions seem reasonable: (1) commerce, and travel for travel's sake, never died out between the East and the West; (2) merchants had every opportunity of knowing, and would have been unreasonably stupid if they had not known, the elementary number systems of the peoples with whom they were trading, but they would not have put this knowledge in permanent written form; (3) wandering scholars would have known many and strange things about the peoples they met, but they too were not, as a class, writers; (4) there is every reason a priori for believing that the ġobār numerals would have been known to merchants, and probably to some of the wandering scholars, long before the Arabs conquered northern Africa; (5) the wonder is not that the Hindu-Arabic numerals were known about 1000 A.D., and that they were the subject of an elaborate work in 1202 by Fibonacci, but rather that more extended manuscript evidence of their appearance before that time has not been found. That they were more or less known early in the Middle Ages, certainly to many merchants of Christian Europe, and probably to several scholars, but without the zero, is hardly to be doubted. The lack of documentary evidence is not at all strange, in view of all of the circumstances.

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CHAPTER VI

THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS

If the numerals had their origin in India, as seems most probable, when did the Arabs come to know of them? It is customary to say that it was due to the influence of Mohammedanism that learning spread through Persia and Arabia; and so it was, in part. But learning was already respected in these countries long before Mohammed appeared, and commerce flourished all through this region. In Persia, for example, the reign of Khosrū Nuśīrwān,[364] the great contemporary of Justinian the law-maker, was characterized not only by an improvement in social and economic conditions, but by the cultivation of letters. Khosrū fostered learning, inviting to his court scholars from Greece, and encouraging the introduction of culture from the West as well as from the East. At this time Aristotle and Plato were translated, and portions of the Hito-padēśa, or Fables of Pilpay, were rendered from the Sanskrit into Persian. All this means that some three centuries before the great intellectual ascendancy of Bagdad a similar fostering of learning was taking place in Persia, and under pre-Mohammedan influences.

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The first definite trace that we have of the introduction of the Hindu system into Arabia dates from 773 A.D.,[365] when an Indian astronomer visited the court of the caliph, bringing with him astronomical tables which at the caliph's command were translated into Arabic by Al-Fazārī.[366] Al-Khowārazmī and Ḥabash (Aḥmed ibn 'Abdallāh, died c. 870) based their well-known tables upon the work of Al-Fāzarī. It may be asserted as highly probable that the numerals came at the same time as the tables. They were certainly known a few decades later, and before 825 A.D., about which time the original of the Algoritmi de numero Indorum was written, as that work makes no pretense of being the first work to treat of the Hindu numerals.

The three writers mentioned cover the period from the end of the eighth to the end of the ninth century. While the historians Al-Maś'ūdī and Al-Bīrūnī follow quite closely upon the men mentioned, it is well to note again the Arab writers on Hindu arithmetic, contemporary with Al-Khowārazmī, who were mentioned in chapter I, viz. Al-Kindī, Sened ibn 'Alī, and Al-Ṣūfī.

For over five hundred years Arabic writers and others continued to apply to works on arithmetic the name "Indian." In the tenth century such writers are 'Abdallāh ibn al-Ḥasan, Abū 'l-Qāsim[367] (died 987 A.D.) of Antioch, and Moḥammed ibn 'Abdallāh, Abū Naṣr[368] (c. 982), of Kalwādā near Bagdad. Others of the same period or {93} earlier (since they are mentioned in the Fihrist,[369] 987 A.D.), who explicitly use the word "Hindu" or "Indian," are Sinān ibn al-Fatḥ[370] of Ḥarrān, and Ahmed ibn 'Omar, al-Karābīsī.[371] In the eleventh century come Al-Bīrūnī[372] (973-1048) and 'Ali ibn Aḥmed, Abū 'l-Ḥasan, Al-Nasawī[373] (c. 1030). The following century brings similar works by Ishāq ibn Yūsuf al-Ṣardafī[374] and Samū'īl ibn Yaḥyā ibn 'Abbās al-Maġrebī al-Andalusī[375] (c. 1174), and in the thirteenth century are 'Abdallatīf ibn Yūsuf ibn Moḥammed, Muwaffaq al-Dīn Abū Moḥammed al-Baġdādī[376] (c. 1231), and Ibn al-Bannā.[377]

The Greek monk Maximus Planudes, writing in the first half of the fourteenth century, followed the Arabic usage in calling his work Indian Arithmetic.[378] There were numerous other Arabic writers upon arithmetic, as that subject occupied one of the high places among the sciences, but most of them did not feel it necessary to refer to the origin of the symbols, the knowledge of which might well have been taken for granted.

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One document, cited by Woepcke,[379] is of special interest since it shows at an early period, 970 A.D., the use of the ordinary Arabic forms alongside the ġobār. The title of the work is Interesting and Beautiful Problems on Numbers copied by Aḥmed ibn Moḥammed ibn 'Abdaljalīl, Abū Sa'īd, al-Sijzī,[380] (951-1024) from a work by a priest and physician, Naẓīf ibn Yumn,[381] al-Qass (died c. 990). Suter does not mention this work of Naẓīf.

The second reason for not ascribing too much credit to the purely Arab influence is that the Arab by himself never showed any intellectual strength. What took place after Moḥammed had lighted the fire in the hearts of his people was just what always takes place when different types of strong races blend,—a great renaissance in divers lines. It was seen in the blending of such types at Miletus in the time of Thales, at Rome in the days of the early invaders, at Alexandria when the Greek set firm foot on Egyptian soil, and we see it now when all the nations mingle their vitality in the New World. So when the Arab culture joined with the Persian, a new civilization rose and flourished.[382] The Arab influence came not from its purity, but from its intermingling with an influence more cultured if less virile.

As a result of this interactivity among peoples of diverse interests and powers, Mohammedanism was to the world from the eighth to the thirteenth century what Rome and Athens and the Italo-Hellenic influence generally had {95} been to the ancient civilization. "If they did not possess the spirit of invention which distinguished the Greeks and the Hindus, if they did not show the perseverance in their observations that characterized the Chinese astronomers, they at least possessed the virility of a new and victorious people, with a desire to understand what others had accomplished, and a taste which led them with equal ardor to the study of algebra and of poetry, of philosophy and of language."[383]

It was in 622 A.D. that Moḥammed fled from Mecca, and within a century from that time the crescent had replaced the cross in Christian Asia, in Northern Africa, and in a goodly portion of Spain. The Arab empire was an ellipse of learning with its foci at Bagdad and Cordova, and its rulers not infrequently took pride in demanding intellectual rather than commercial treasure as the result of conquest.[384]

It was under these influences, either pre-Mohammedan or later, that the Hindu numerals found their way to the North. If they were known before Moḥammed's time, the proof of this fact is now lost. This much, however, is known, that in the eighth century they were taken to Bagdad. It was early in that century that the Mohammedans obtained their first foothold in northern India, thus foreshadowing an epoch of supremacy that endured with varied fortunes until after the golden age of Akbar the Great (1542-1605) and Shah Jehan. They also conquered Khorassan and Afghanistan, so that the learning and the commercial customs of India at once found easy {96} access to the newly-established schools and the bazaars of Mesopotamia and western Asia. The particular paths of conquest and of commerce were either by way of the Khyber Pass and through Kabul, Herat and Khorassan, or by sea through the strait of Ormuz to Basra (Busra) at the head of the Persian Gulf, and thence to Bagdad. As a matter of fact, one form of Arabic numerals, the one now in use by the Arabs, is attributed to the influence of Kabul, while the other, which eventually became our numerals, may very likely have reached Arabia by the other route. It is in Bagdad,[385] Dār al-Salām—"the Abode of Peace," that our special interest in the introduction of the numerals centers. Built upon the ruins of an ancient town by Al-Manṣūr[386] in the second half of the eighth century, it lies in one of those regions where the converging routes of trade give rise to large cities.[387] Quite as well of Bagdad as of Athens might Cardinal Newman have said:[388]

"What it lost in conveniences of approach, it gained in its neighborhood to the traditions of the mysterious East, and in the loveliness of the region in which it lay. Hither, then, as to a sort of ideal land, where all archetypes of the great and the fair were found in substantial being, and all departments of truth explored, and all diversities of intellectual power exhibited, where taste and philosophy were majestically enthroned as in a royal court, where there was no sovereignty but that of mind, and no nobility but that of genius, where professors were {97} rulers, and princes did homage, thither flocked continually from the very corners of the orbis terrarum the many-tongued generation, just rising, or just risen into manhood, in order to gain wisdom." For here it was that Al-Manṣūr and Al-Māmūn and Hārūn al-Rashīd (Aaron the Just) made for a time the world's center of intellectual activity in general and in the domain of mathematics in particular.[389] It was just after the Sindhind was brought to Bagdad that Moḥammed ibn Mūsā al-Khowārazmī, whose name has already been mentioned,[390] was called to that city. He was the most celebrated mathematician of his time, either in the East or West, writing treatises on arithmetic, the sundial, the astrolabe, chronology, geometry, and algebra, and giving through the Latin transliteration of his name, algoritmi, the name of algorism to the early arithmetics using the new Hindu numerals.[391] Appreciating at once the value of the position system so recently brought from India, he wrote an arithmetic based upon these numerals, and this was translated into Latin in the time of Adelhard of Bath (c. 1180), although possibly by his contemporary countryman Robert Cestrensis.[392] This translation was found in Cambridge and was published by Boncompagni in 1857.[393]

Contemporary with Al-Khowārazmī, and working also under Al-Māmūn, was a Jewish astronomer, Abū 'l-Ṭeiyib, {98} Sened ibn 'Alī, who is said to have adopted the Mohammedan religion at the caliph's request. He also wrote a work on Hindu arithmetic,[394] so that the subject must have been attracting considerable attention at that time. Indeed, the struggle to have the Hindu numerals replace the Arabic did not cease for a long time thereafter. 'Alī ibn Aḥmed al-Nasawī, in his arithmetic of c. 1025, tells us that the symbolism of number was still unsettled in his day, although most people preferred the strictly Arabic forms.[395]

We thus have the numerals in Arabia, in two forms: one the form now used there, and the other the one used by Al-Khowārazmī. The question then remains, how did this second form find its way into Europe? and this question will be considered in the next chapter.

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CHAPTER VII

THE DEFINITE INTRODUCTION OF THE NUMERALS INTO EUROPE

It being doubtful whether Boethius ever knew the Hindu numeral forms, certainly without the zero in any case, it becomes necessary now to consider the question of their definite introduction into Europe. From what has been said of the trade relations between the East and the West, and of the probability that it was the trader rather than the scholar who carried these numerals from their original habitat to various commercial centers, it is evident that we shall never know when they first made their inconspicuous entrance into Europe. Curious customs from the East and from the tropics,—concerning games, social peculiarities, oddities of dress, and the like,—are continually being related by sailors and traders in their resorts in New York, London, Hamburg, and Rotterdam to-day, customs that no scholar has yet described in print and that may not become known for many years, if ever. And if this be so now, how much more would it have been true a thousand years before the invention of printing, when learning was at its lowest ebb. It was at this period of low esteem of culture that the Hindu numerals undoubtedly made their first appearance in Europe.

There were many opportunities for such knowledge to reach Spain and Italy. In the first place the Moors went into Spain as helpers of a claimant of the throne, and {100} remained as conquerors. The power of the Goths, who had held Spain for three centuries, was shattered at the battle of Jerez de la Frontera in 711, and almost immediately the Moors became masters of Spain and so remained for five hundred years, and masters of Granada for a much longer period. Until 850 the Christians were absolutely free as to religion and as to holding political office, so that priests and monks were not infrequently skilled both in Latin and Arabic, acting as official translators, and naturally reporting directly or indirectly to Rome. There was indeed at this time a complaint that Christian youths cultivated too assiduously a love for the literature of the Saracen, and married too frequently the daughters of the infidel.[396] It is true that this happy state of affairs was not permanent, but while it lasted the learning and the customs of the East must have become more or less the property of Christian Spain. At this time the ġobār numerals were probably in that country, and these may well have made their way into Europe from the schools of Cordova, Granada, and Toledo.

Furthermore, there was abundant opportunity for the numerals of the East to reach Europe through the journeys of travelers and ambassadors. It was from the records of Suleimān the Merchant, a well-known Arab trader of the ninth century, that part of the story of Sindbād the Sailor was taken.[397] Such a merchant would have been particularly likely to know the numerals of the people whom he met, and he is a type of man that may well have taken such symbols to European markets. A little later, {101} Abū 'l-Ḥasan 'Alī al-Mas'ūdī (d. 956) of Bagdad traveled to the China Sea on the east, at least as far south as Zanzibar, and to the Atlantic on the west,[398] and he speaks of the nine figures with which the Hindus reckoned.[399]

There was also a Bagdad merchant, one Abū 'l-Qāsim 'Obeidallāh ibn Aḥmed, better known by his Persian name Ibn Khordāḍbeh,[400] who wrote about 850 A.D. a work entitled Book of Roads and Provinces[401] in which the following graphic account appears:[402] "The Jewish merchants speak Persian, Roman (Greek and Latin), Arabic, French, Spanish, and Slavic. They travel from the West to the East, and from the East to the West, sometimes by land, sometimes by sea. They take ship from France on the Western Sea, and they voyage to Farama (near the ruins of the ancient Pelusium); there they transfer their goods to caravans and go by land to Colzom (on the Red Sea). They there reembark on the Oriental (Red) Sea and go to Hejaz and to Jiddah, and thence to the Sind, India, and China. Returning, they bring back the products of the oriental lands.... These journeys are also made by land. The merchants, leaving France and Spain, cross to Tangier and thence pass through the African provinces and Egypt. They then go to Ramleh, visit Damascus, Kufa, Bagdad, and Basra, penetrate into Ahwaz, Fars, Kerman, Sind, and thus reach India and China." Such travelers, about 900 A.D., must necessarily have spread abroad a knowledge of all number {102} systems used in recording prices or in the computations of the market. There is an interesting witness to this movement, a cruciform brooch now in the British Museum. It is English, certainly as early as the eleventh century, but it is inlaid with a piece of paste on which is the Mohammedan inscription, in Kufic characters, "There is no God but God." How did such an inscription find its way, perhaps in the time of Alcuin of York, to England? And if these Kufic characters reached there, then why not the numeral forms as well?

Even in literature of the better class there appears now and then some stray proof of the important fact that the great trade routes to the far East were never closed for long, and that the customs and marks of trade endured from generation to generation. The Gulistān of the Persian poet Sa'dī[403] contains such a passage:

"I met a merchant who owned one hundred and forty camels, and fifty slaves and porters.... He answered to me: 'I want to carry sulphur of Persia to China, which in that country, as I hear, bears a high price; and thence to take Chinese ware to Roum; and from Roum to load up with brocades for Hind; and so to trade Indian steel (pulab) to Halib. From Halib I will convey its glass to Yeman, and carry the painted cloths of Yeman back to Persia.'"[404] On the other hand, these men were not of the learned class, nor would they preserve in treatises any knowledge that they might have, although this knowledge would occasionally reach the ears of the learned as bits of curious information.

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There were also ambassadors passing back and forth from time to time, between the East and the West, and in particular during the period when these numerals probably began to enter Europe. Thus Charlemagne (c. 800) sent emissaries to Bagdad just at the time of the opening of the mathematical activity there.[405] And with such ambassadors must have gone the adventurous scholar, inspired, as Alcuin says of Archbishop Albert of York (766-780),[406] to seek the learning of other lands. Furthermore, the Nestorian communities, established in Eastern Asia and in India at this time, were favored both by the Persians and by their Mohammedan conquerors. The Nestorian Patriarch of Syria, Timotheus (778-820), sent missionaries both to India and to China, and a bishop was appointed for the latter field. Ibn Wahab, who traveled to China in the ninth century, found images of Christ and the apostles in the Emperor's court.[407] Such a learned body of men, knowing intimately the countries in which they labored, could hardly have failed to make strange customs known as they returned to their home stations. Then, too, in Alfred's time (849-901) emissaries went {104} from England as far as India,[408] and generally in the Middle Ages groceries came to Europe from Asia as now they come from the colonies and from America. Syria, Asia Minor, and Cyprus furnished sugar and wool, and India yielded her perfumes and spices, while rich tapestries for the courts and the wealthy burghers came from Persia and from China.[409] Even in the time of Justinian (c. 550) there seems to have been a silk trade with China, which country in turn carried on commerce with Ceylon,[410] and reached out to Turkestan where other merchants transmitted the Eastern products westward. In the seventh century there was a well-defined commerce between Persia and India, as well as between Persia and Constantinople.[411] The Byzantine commerciarii were stationed at the outposts not merely as customs officers but as government purchasing agents.[412]

Occasionally there went along these routes of trade men of real learning, and such would surely have carried the knowledge of many customs back and forth. Thus at a period when the numerals are known to have been partly understood in Italy, at the opening of the eleventh century, one Constantine, an African, traveled from Italy through a great part of Africa and Asia, even on to India, for the purpose of learning the sciences of the Orient. He spent thirty-nine years in travel, having been hospitably received in Babylon, and upon his return he was welcomed with great honor at Salerno.[413]

A very interesting illustration of this intercourse also appears in the tenth century, when the son of Otto I {105} (936-973) married a princess from Constantinople. This monarch was in touch with the Moors of Spain and invited to his court numerous scholars from abroad,[414] and his intercourse with the East as well as the West must have brought together much of the learning of each.

Another powerful means for the circulation of mysticism and philosophy, and more or less of culture, took its start just before the conversion of Constantine (c. 312), in the form of Christian pilgrim travel. This was a feature peculiar to the zealots of early Christianity, found in only a slight degree among their Jewish predecessors in the annual pilgrimage to Jerusalem, and almost wholly wanting in other pre-Christian peoples. Chief among these early pilgrims were the two Placentians, John and Antonine the Elder (c. 303), who, in their wanderings to Jerusalem, seem to have started a movement which culminated centuries later in the crusades.[415] In 333 a Bordeaux pilgrim compiled the first Christian guide-book, the Itinerary from Bordeaux to Jerusalem,[416] and from this time on the holy pilgrimage never entirely ceased.

Still another certain route for the entrance of the numerals into Christian Europe was through the pillaging and trading carried on by the Arabs on the northern shores of the Mediterranean. As early as 652 A.D., in the thirtieth year of the Hejira, the Mohammedans descended upon the shores of Sicily and took much spoil. Hardly had the wretched Constans given place to the {106} young Constantine IV when they again attacked the island and plundered ancient Syracuse. Again in 827, under Asad, they ravaged the coasts. Although at this time they failed to conquer Syracuse, they soon held a good part of the island, and a little later they successfully besieged the city. Before Syracuse fell, however, they had plundered the shores of Italy, even to the walls of Rome itself; and had not Leo IV, in 849, repaired the neglected fortifications, the effects of the Moslem raid of that year might have been very far-reaching. Ibn Khordāḍbeh, who left Bagdad in the latter part of the ninth century, gives a picture of the great commercial activity at that time in the Saracen city of Palermo. In this same century they had established themselves in Piedmont, and in 906 they pillaged Turin.[417] On the Sorrento peninsula the traveler who climbs the hill to the beautiful Ravello sees still several traces of the Arab architecture, reminding him of the fact that about 900 A.D. Amalfi was a commercial center of the Moors.[418] Not only at this time, but even a century earlier, the artists of northern India sold their wares at such centers, and in the courts both of Hārūn al-Rashīd and of Charlemagne.[419] Thus the Arabs dominated the Mediterranean Sea long before Venice

"held the gorgeous East in fee And was the safeguard of the West,"

and long before Genoa had become her powerful rival.[420]

{107}

Only a little later than this the brothers Nicolo and Maffeo Polo entered upon their famous wanderings.[421] Leaving Constantinople in 1260, they went by the Sea of Azov to Bokhara, and thence to the court of Kublai Khan, penetrating China, and returning by way of Acre in 1269 with a commission which required them to go back to China two years later. This time they took with them Nicolo's son Marco, the historian of the journey, and went across the plateau of Pamir; they spent about twenty years in China, and came back by sea from China to Persia.

The ventures of the Poli were not long unique, however: the thirteenth century had not closed before Roman missionaries and the merchant Petrus de Lucolongo had penetrated China. Before 1350 the company of missionaries was large, converts were numerous, churches and Franciscan convents had been organized in the East, travelers were appealing for the truth of their accounts to the "many" persons in Venice who had been in China, Tsuan-chau-fu had a European merchant community, and Italian trade and travel to China was a thing that occupied two chapters of a commercial handbook.[422]

{108}

It is therefore reasonable to conclude that in the Middle Ages, as in the time of Boethius, it was a simple matter for any inquiring scholar to become acquainted with such numerals of the Orient as merchants may have used for warehouse or price marks. And the fact that Gerbert seems to have known only the forms of the simplest of these, not comprehending their full significance, seems to prove that he picked them up in just this way.

Even if Gerbert did not bring his knowledge of the Oriental numerals from Spain, he may easily have obtained them from the marks on merchant's goods, had he been so inclined. Such knowledge was probably obtainable in various parts of Italy, though as parts of mere mercantile knowledge the forms might soon have been lost, it needing the pen of the scholar to preserve them. Trade at this time was not stagnant. During the eleventh and twelfth centuries the Slavs, for example, had very great commercial interests, their trade reaching to Kiev and Novgorod, and thence to the East. Constantinople was a great clearing-house of commerce with the Orient,[423] and the Byzantine merchants must have been entirely familiar with the various numerals of the Eastern peoples. In the eleventh century the Italian town of Amalfi established a factory[424] in Constantinople, and had trade relations with Antioch and Egypt. Venice, as early as the ninth century, had a valuable trade with Syria and Cairo.[425] Fifty years after Gerbert died, in the time of Cnut, the Dane and the Norwegian pushed their commerce far beyond the northern seas, both by caravans through Russia to the Orient, and by their venturesome barks which {109} sailed through the Strait of Gibraltar into the Mediterranean.[426] Only a little later, probably before 1200 A.D., a clerk in the service of Thomas a Becket, present at the latter's death, wrote a life of the martyr, to which (fortunately for our purposes) he prefixed a brief eulogy of the city of London.[427] This clerk, William Fitz Stephen by name, thus speaks of the British capital:

Aurum mittit Arabs: species et thura Sabaeus: Arma Sythes: oleum palmarum divite sylva Pingue solum Babylon: Nilus lapides pretiosos: Norwegi, Russi, varium grisum, sabdinas: Seres, purpureas vestes: Galli, sua vina.

Although, as a matter of fact, the Arabs had no gold to send, and the Scythians no arms, and Egypt no precious stones save only the turquoise, the Chinese (Seres) may have sent their purple vestments, and the north her sables and other furs, and France her wines. At any rate the verses show very clearly an extensive foreign trade.

Then there were the Crusades, which in these times brought the East in touch with the West. The spirit of the Orient showed itself in the songs of the troubadours, and the baudekin,[428] the canopy of Bagdad,[429] became common in the churches of Italy. In Sicily and in Venice the textile industries of the East found place, and made their way even to the Scandinavian peninsula.[430]

We therefore have this state of affairs: There was abundant intercourse between the East and West for {110} some centuries before the Hindu numerals appear in any manuscripts in Christian Europe. The numerals must of necessity have been known to many traders in a country like Italy at least as early as the ninth century, and probably even earlier, but there was no reason for preserving them in treatises. Therefore when a man like Gerbert made them known to the scholarly circles, he was merely describing what had been familiar in a small way to many people in a different walk of life.

Since Gerbert[431] was for a long time thought to have been the one to introduce the numerals into Italy,[432] a brief sketch of this unique character is proper. Born of humble parents,[433] this remarkable man became the counselor and companion of kings, and finally wore the papal tiara as Sylvester II, from 999 until his death in 1003.[434] He was early brought under the influence of the monks at Aurillac, and particularly of Raimund, who had been a pupil of Odo of Cluny, and there in due time he himself took holy orders. He visited Spain in about 967 in company with Count Borel,[435] remaining there three years, {111} and studying under Bishop Hatto of Vich,[436] a city in the province of Barcelona,[437] then entirely under Christian rule. Indeed, all of Gerbert's testimony is as to the influence of the Christian civilization upon his education. Thus he speaks often of his study of Boethius,[438] so that if the latter knew the numerals Gerbert would have learned them from him.[439] If Gerbert had studied in any Moorish schools he would, under the decree of the emir Hishām (787-822), have been obliged to know Arabic, which would have taken most of his three years in Spain, and of which study we have not the slightest hint in any of his letters.[440] On the other hand, Barcelona was the only Christian province in immediate touch with the Moorish civilization at that time.[441] Furthermore we know that earlier in the same century King Alonzo of Asturias (d. 910) confided the education of his son Ordono to the Arab scholars of the court of the {112} wālī of Saragossa,[442] so that there was more or less of friendly relation between Christian and Moor.

After his three years in Spain, Gerbert went to Italy, about 970, where he met Pope John XIII, being by him presented to the emperor Otto I. Two years later (972), at the emperor's request, he went to Rheims, where he studied philosophy, assisting to make of that place an educational center; and in 983 he became abbot at Bobbio. The next year he returned to Rheims, and became archbishop of that diocese in 991. For political reasons he returned to Italy in 996, became archbishop of Ravenna in 998, and the following year was elected to the papal chair. Far ahead of his age in wisdom, he suffered as many such scholars have even in times not so remote by being accused of heresy and witchcraft. As late as 1522, in a biography published at Venice, it is related that by black art he attained the papacy, after having given his soul to the devil.[443] Gerbert was, however, interested in astrology,[444] although this was merely the astronomy of that time and was such a science as any learned man would wish to know, even as to-day we wish to be reasonably familiar with physics and chemistry.

That Gerbert and his pupils knew the ġobār numerals is a fact no longer open to controversy.[445] Bernelinus and Richer[446] call them by the well-known name of {113} "caracteres," a word used by Radulph of Laon in the same sense a century later.[447] It is probable that Gerbert was the first to describe these ġobār numerals in any scientific way in Christian Europe, but without the zero. If he knew the latter he certainly did not understand its use.[448]

The question still to be settled is as to where he found these numerals. That he did not bring them from Spain is the opinion of a number of careful investigators.[449] This is thought to be the more probable because most of the men who made Spain famous for learning lived after Gerbert was there. Such were Ibn Sīnā (Avicenna) who lived at the beginning, and Gerber of Seville who flourished in the middle, of the eleventh century, and Abū Roshd (Averroes) who lived at the end of the twelfth.[450] Others hold that his proximity to {114} the Arabs for three years makes it probable that he assimilated some of their learning, in spite of the fact that the lines between Christian and Moor at that time were sharply drawn.[451] Writers fail, however, to recognize that a commercial numeral system would have been more likely to be made known by merchants than by scholars. The itinerant peddler knew no forbidden pale in Spain, any more than he has known one in other lands. If the ġobār numerals were used for marking wares or keeping simple accounts, it was he who would have known them, and who would have been the one rather than any Arab scholar to bring them to the inquiring mind of the young French monk. The facts that Gerbert knew them only imperfectly, that he used them solely for calculations, and that the forms are evidently like the Spanish ġobār, make it all the more probable that it was through the small tradesman of the Moors that this versatile scholar derived his knowledge. Moreover the part of the geometry bearing his name, and that seems unquestionably his, shows the Arab influence, proving that he at least came into contact with the transplanted Oriental learning, even though imperfectly.[452] There was also the persistent Jewish merchant trading with both peoples then as now, always alive to the acquiring of useful knowledge, and it would be very natural for a man like Gerbert to welcome learning from such a source.

On the other hand, the two leading sources of information as to the life of Gerbert reveal practically nothing to show that he came within the Moorish sphere of influence during his sojourn in Spain. These sources {115} are his letters and the history written by Richer. Gerbert was a master of the epistolary art, and his exalted position led to the preservation of his letters to a degree that would not have been vouchsafed even by their classic excellence.[453] Richer was a monk at St. Remi de Rheims, and was doubtless a pupil of Gerbert. The latter, when archbishop of Rheims, asked Richer to write a history of his times, and this was done. The work lay in manuscript, entirely forgotten until Pertz discovered it at Bamberg in 1833.[454] The work is dedicated to Gerbert as archbishop of Rheims,[455] and would assuredly have testified to such efforts as he may have made to secure the learning of the Moors.

Now it is a fact that neither the letters nor this history makes any statement as to Gerbert's contact with the Saracens. The letters do not speak of the Moors, of the Arab numerals, nor of Cordova. Spain is not referred to by that name, and only one Spanish scholar is mentioned. In one of his letters he speaks of Joseph Ispanus,[456] or Joseph Sapiens, but who this Joseph the Wise of Spain may have been we do not know. Possibly {116} it was he who contributed the morsel of knowledge so imperfectly assimilated by the young French monk.[457] Within a few years after Gerbert's visit two young Spanish monks of lesser fame, and doubtless with not that keen interest in mathematical matters which Gerbert had, regarded the apparently slight knowledge which they had of the Hindu numeral forms as worthy of somewhat permanent record[458] in manuscripts which they were transcribing. The fact that such knowledge had penetrated to their modest cloisters in northern Spain—the one Albelda or Albaida—indicates that it was rather widely diffused.

Gerbert's treatise Libellus de numerorum divisione[459] is characterized by Chasles as "one of the most obscure documents in the history of science."[460] The most complete information in regard to this and the other mathematical works of Gerbert is given by Bubnov,[461] who considers this work to be genuine.[462]

{117}

So little did Gerbert appreciate these numerals that in his works known as the Regula de abaco computi and the Libellus he makes no use of them at all, employing only the Roman forms.[463] Nevertheless Bernelinus[464] refers to the nine ġobār characters.[465] These Gerbert had marked on a thousand jetons or counters,[466] using the latter on an abacus which he had a sign-maker prepare for him.[467] Instead of putting eight counters in say the tens' column, Gerbert would put a single counter marked 8, and so for the other places, leaving the column empty where we would place a zero, but where he, lacking the zero, had no counter to place. These counters he possibly called caracteres, a name which adhered also to the figures themselves. It is an interesting speculation to consider whether these apices, as they are called in the Boethius interpolations, were in any way suggested by those Roman jetons generally known in numismatics as tesserae, and bearing the figures I-XVI, the sixteen referring to the number of assi in a sestertius.[468] The {118} name apices adhered to the Hindu-Arabic numerals until the sixteenth century.[469]

To the figures on the apices were given the names Igin, andras, ormis, arbas, quimas, calctis or caltis, zenis, temenias, celentis, sipos,[470] the origin and meaning of which still remain a mystery. The Semitic origin of several of the words seems probable. Wahud, thaneine, {119} thalata, arba, kumsa, setta, sebba, timinia, taseud are given by the Rev. R. Patrick[471] as the names, in an Arabic dialect used in Morocco, for the numerals from one to nine. Of these the words for four, five, and eight are strikingly like those given above.

The name apices was not, however, a common one in later times. Notae was more often used, and it finally gave the name to notation.[472] Still more common were the names figures, ciphers, signs, elements, and characters.[473]

So little effect did the teachings of Gerbert have in making known the new numerals, that O'Creat, who lived a century later, a friend and pupil of Adelhard {120} of Bath, used the zero with the Roman characters, in contrast to Gerbert's use of the ġobār forms without the zero.[474] O'Creat uses three forms for zero, o, ō, and [Greek: t], as in Maximus Planudes. With this use of the zero goes, naturally, a place value, for he writes III III for 33, ICCOO and I. II. [tau]. [tau] for 1200, I. O. VIII. IX for 1089, and I. IIII. IIII. [tau][tau][tau][tau] for the square of 1200.

The period from the time of Gerbert until after the appearance of Leonardo's monumental work may be called the period of the abacists. Even for many years after the appearance early in the twelfth century of the books explaining the Hindu art of reckoning, there was strife between the abacists, the advocates of the abacus, and the algorists, those who favored the new numerals. The words cifra and algorismus cifra were used with a somewhat derisive significance, indicative of absolute uselessness, as indeed the zero is useless on an abacus in which the value of any unit is given by the column which it occupies.[475] So Gautier de Coincy (1177-1236) in a work on the miracles of Mary says:

A horned beast, a sheep, An algorismus-cipher, Is a priest, who on such a feast day Does not celebrate the holy Mother.[476]

So the abacus held the field for a long time, even against the new algorism employing the new numerals. {121} Geoffrey Chaucer[477] describes in The Miller's Tale the clerk with

"His Almageste and bokes grete and smale, His astrelabie, longinge for his art, His augrim-stones layen faire apart On shelves couched at his beddes heed."

So, too, in Chaucer's explanation of the astrolabe,[478] written for his son Lewis, the number of degrees is expressed on the instrument in Hindu-Arabic numerals: "Over the whiche degrees ther ben noumbres of augrim, that devyden thilke same degrees fro fyve to fyve," and "... the nombres ... ben writen in augrim," meaning in the way of the algorism. Thomas Usk about 1387 writes:[479] "a sypher in augrim have no might in signification of it-selve, yet he yeveth power in signification to other." So slow and so painful is the assimilation of new ideas.

Bernelinus[480] states that the abacus is a well-polished board (or table), which is covered with blue sand and used by geometers in drawing geometrical figures. We have previously mentioned the fact that the Hindus also performed mathematical computations in the sand, although there is no evidence to show that they had any column abacus.[481] For the purposes of computation, Bernelinus continues, the board is divided into thirty vertical columns, three of which are reserved for fractions. Beginning with the units columns, each set of {122} three columns (lineae is the word which Bernelinus uses) is grouped together by a semicircular arc placed above them, while a smaller arc is placed over the units column and another joins the tens and hundreds columns. Thus arose the designation arcus pictagore[482] or sometimes simply arcus.[483] The operations of addition, subtraction, and multiplication upon this form of the abacus required little explanation, although they were rather extensively treated, especially the multiplication of different orders of numbers. But the operation of division was effected with some difficulty. For the explanation of the method of division by the use of the complementary difference,[484] long the stumbling-block in the way of the medieval arithmetician, the reader is referred to works on the history of mathematics[485] and to works relating particularly to the abacus.[486]

Among the writers on the subject may be mentioned Abbo[487] of Fleury (c. 970), Heriger[488] of Lobbes or Laubach {123} (c. 950-1007), and Hermannus Contractus[489] (1013-1054), all of whom employed only the Roman numerals. Similarly Adelhard of Bath (c. 1130), in his work Regulae Abaci,[490] gives no reference to the new numerals, although it is certain that he knew them. Other writers on the abacus who used some form of Hindu numerals were Gerland[491] (first half of twelfth century) and Turchill[492] (c. 1200). For the forms used at this period the reader is referred to the plate on page 88.

After Gerbert's death, little by little the scholars of Europe came to know the new figures, chiefly through the introduction of Arab learning. The Dark Ages had passed, although arithmetic did not find another advocate as prominent as Gerbert for two centuries. Speaking of this great revival, Raoul Glaber[493] (985-c. 1046), a monk of the great Benedictine abbey of Cluny, of the eleventh century, says: "It was as though the world had arisen and tossed aside the worn-out garments of ancient time, and wished to apparel itself in a white robe of churches." And with this activity in religion came a corresponding interest in other lines. Algorisms began to appear, and knowledge from the outside world found {124} interested listeners. Another Raoul, or Radulph, to whom we have referred as Radulph of Laon,[494] a teacher in the cloister school of his city, and the brother of Anselm of Laon[495] the celebrated theologian, wrote a treatise on music, extant but unpublished, and an arithmetic which Nagl first published in 1890.[496] The latter work, preserved to us in a parchment manuscript of seventy-seven leaves, contains a curious mixture of Roman and ġobār numerals, the former for expressing large results, the latter for practical calculation. These ġobār "caracteres" include the sipos (zero), [Symbol], of which, however, Radulph did not know the full significance; showing that at the opening of the twelfth century the system was still uncertain in its status in the church schools of central France.

At the same time the words algorismus and cifra were coming into general use even in non-mathematical literature. Jordan [497] cites numerous instances of such use from the works of Alanus ab Insulis[498] (Alain de Lille), Gautier de Coincy (1177-1236), and others.

Another contributor to arithmetic during this interesting period was a prominent Spanish Jew called variously John of Luna, John of Seville, Johannes Hispalensis, Johannes Toletanus, and Johannes Hispanensis de Luna.[499] {125} His date is rather closely fixed by the fact that he dedicated a work to Raimund who was archbishop of Toledo between 1130 and 1150.[500] His interests were chiefly in the translation of Arabic works, especially such as bore upon the Aristotelian philosophy. From the standpoint of arithmetic, however, the chief interest centers about a manuscript entitled Joannis Hispalensis liber Algorismi de Practica Arismetrice which Boncompagni found in what is now the Bibliotheque nationale at Paris. Although this distinctly lays claim to being Al-Khowārazmī's work,[501] the evidence is altogether against the statement,[502] but the book is quite as valuable, since it represents the knowledge of the time in which it was written. It relates to the operations with integers and sexagesimal fractions, including roots, and contains no applications.[503]

Contemporary with John of Luna, and also living in Toledo, was Gherard of Cremona,[504] who has sometimes been identified, but erroneously, with Gernardus,[505] the {126} author of a work on algorism. He was a physician, an astronomer, and a mathematician, translating from the Arabic both in Italy and in Spain. In arithmetic he was influential in spreading the ideas of algorism.

Four Englishmen—Adelhard of Bath (c. 1130), Robert of Chester (Robertus Cestrensis, c. 1143), William Shelley, and Daniel Morley (1180)—are known[506] to have journeyed to Spain in the twelfth century for the purpose of studying mathematics and Arabic. Adelhard of Bath made translations from Arabic into Latin of Al-Khowārazmī's astronomical tables[507] and of Euclid's Elements,[508] while Robert of Chester is known as the translator of Al-Khowārazmī's algebra.[509] There is no reason to doubt that all of these men, and others, were familiar with the numerals which the Arabs were using.

The earliest trace we have of computation with Hindu numerals in Germany is in an Algorismus of 1143, now in the Hofbibliothek in Vienna.[510] It is bound in with a {127} Computus by the same author and bearing the date given. It contains chapters "De additione," "De diminutione," "De mediatione," "De divisione," and part of a chapter on multiplication. The numerals are in the usual medieval forms except the 2 which, as will be seen from the illustration,[511] is somewhat different, and the 3, which takes the peculiar shape [Symbol], a form characteristic of the twelfth century.

It was about the same time that the Sefer ha-Mispar,[512] the Book of Number, appeared in the Hebrew language. The author, Rabbi Abraham ibn Meir ibn Ezra,[513] was born in Toledo (c. 1092). In 1139 he went to Egypt, Palestine, and the Orient, spending also some years in Italy. Later he lived in southern France and in England. He died in 1167. The probability is that he acquired his knowledge of the Hindu arithmetic[514] in his native town of Toledo, but it is also likely that the knowledge of other systems which he acquired on travels increased his appreciation of this one. We have mentioned the fact that he used the first letters of the Hebrew alphabet, [Hebrew: A B G D H W Z CH T'], for the numerals 9 8 7 6 5 4 3 2 1, and a circle for the zero. The quotation in the note given below shows that he knew of the Hindu origin; but in his manuscript, although he set down the Hindu forms, he used the above nine Hebrew letters with place value for all computations.

* * * * *

{128}

CHAPTER VIII

THE SPREAD OF THE NUMERALS IN EUROPE

Of all the medieval writers, probably the one most influential in introducing the new numerals to the scholars of Europe was Leonardo Fibonacci, of Pisa.[515] This remarkable man, the most noteworthy mathematical genius of the Middle Ages, was born at Pisa about 1175.[516]

The traveler of to-day may cross the Via Fibonacci on his way to the Campo Santo, and there he may see at the end of the long corridor, across the quadrangle, the statue of Leonardo in scholars garb. Few towns have honored a mathematician more, and few mathematicians have so distinctly honored their birthplace. Leonardo was born in the golden age of this city, the period of its commercial, religious, and intellectual prosperity.[517] {129} Situated practically at the mouth of the Arno, Pisa formed with Genoa and Venice the trio of the greatest commercial centers of Italy at the opening of the thirteenth century. Even before Venice had captured the Levantine trade, Pisa had close relations with the East. An old Latin chronicle relates that in 1005 "Pisa was captured by the Saracens," that in the following year "the Pisans overthrew the Saracens at Reggio," and that in 1012 "the Saracens came to Pisa and destroyed it." The city soon recovered, however, sending no fewer than a hundred and twenty ships to Syria in 1099,[518] founding a merchant colony in Constantinople a few years later,[519] and meanwhile carrying on an interurban warfare in Italy that seemed to stimulate it to great activity.[520] A writer of 1114 tells us that at that time there were many heathen people—Turks, Libyans, Parthians, and Chaldeans—to be found in Pisa. It was in the midst of such wars, in a cosmopolitan and commercial town, in a center where literary work was not appreciated,[521] that the genius of Leonardo appears as one of the surprises of history, warning us again that "we should draw no horoscope; that we should expect little, for what we expect will not come to pass."[522]

Leonardo's father was one William,[523] and he had a brother named Bonaccingus,[524] but nothing further is {130} known of his family. As to Fibonacci, most writers[525] have assumed that his father's name was Bonaccio,[526] whence filius Bonaccii, or Fibonacci. Others[527] believe that the name, even in the Latin form of filius Bonaccii as used in Leonardo's work, was simply a general one, like our Johnson or Bronson (Brown's son); and the only contemporary evidence that we have bears out this view. As to the name Bigollo, used by Leonardo, some have thought it a self-assumed one meaning blockhead, a term that had been applied to him by the commercial world or possibly by the university circle, and taken by him that he might prove what a blockhead could do. Milanesi,[528] however, has shown that the word Bigollo (or Pigollo) was used in Tuscany to mean a traveler, and was naturally assumed by one who had studied, as Leonardo had, in foreign lands.

Leonardo's father was a commercial agent at Bugia, the modern Bougie,[529] the ancient Saldae on the coast of Barbary,[530] a royal capital under the Vandals and again, a century before Leonardo, under the Beni Hammad. It had one of the best harbors on the coast, sheltered as it is by Mt. Lalla Guraia,[531] and at the close of the twelfth century it was a center of African commerce. It was here that Leonardo was taken as a child, and here he went to school to a Moorish master. When he reached the years of young manhood he started on a tour of the Mediterranean Sea, and visited Egypt, Syria, Greece, Sicily, and Provence, meeting with scholars as well as with {131} merchants, and imbibing a knowledge of the various systems of numbers in use in the centers of trade. All these systems, however, he says he counted almost as errors compared with that of the Hindus.[532] Returning to Pisa, he wrote his Liber Abaci[533] in 1202, rewriting it in 1228.[534] In this work the numerals are explained and are used in the usual computations of business. Such a treatise was not destined to be popular, however, because it was too advanced for the mercantile class, and too novel for the conservative university circles. Indeed, at this time mathematics had only slight place in the newly established universities, as witness the oldest known statute of the Sorbonne at Paris, dated 1215, where the subject is referred to only in an incidental way.[535] The period was one of great commercial activity, and on this very {132} account such a book would attract even less attention than usual.[536]

It would now be thought that the western world would at once adopt the new numerals which Leonardo had made known, and which were so much superior to anything that had been in use in Christian Europe. The antagonism of the universities would avail but little, it would seem, against such an improvement. It must be remembered, however, that there was great difficulty in spreading knowledge at this time, some two hundred and fifty years before printing was invented. "Popes and princes and even great religious institutions possessed far fewer books than many farmers of the present age. The library belonging to the Cathedral Church of San Martino at Lucca in the ninth century contained only nineteen volumes of abridgments from ecclesiastical commentaries."[537] Indeed, it was not until the early part of the fifteenth century that Palla degli Strozzi took steps to carry out the project that had been in the mind of Petrarch, the founding of a public library. It was largely by word of mouth, therefore, that this early knowledge had to be transmitted. Fortunately the presence of foreign students in Italy at this time made this transmission feasible. (If human nature was the same then as now, it is not impossible that the very opposition of the faculties to the works of Leonardo led the students to investigate {133} them the more zealously.) At Vicenza in 1209, for example, there were Bohemians, Poles, Frenchmen, Burgundians, Germans, and Spaniards, not to speak of representatives of divers towns of Italy; and what was true there was also true of other intellectual centers. The knowledge could not fail to spread, therefore, and as a matter of fact we find numerous bits of evidence that this was the case. Although the bankers of Florence were forbidden to use these numerals in 1299, and the statutes of the university of Padua required stationers to keep the price lists of books "non per cifras, sed per literas claros,"[538] the numerals really made much headway from about 1275 on.

It was, however, rather exceptional for the common people of Germany to use the Arabic numerals before the sixteenth century, a good witness to this fact being the popular almanacs. Calendars of 1457-1496[539] have generally the Roman numerals, while Koebel's calendar of 1518 gives the Arabic forms as subordinate to the Roman. In the register of the Kreuzschule at Dresden the Roman forms were used even until 1539.

While not minimizing the importance of the scientific work of Leonardo of Pisa, we may note that the more popular treatises by Alexander de Villa Dei (c. 1240 A.D.) and John of Halifax (Sacrobosco, c. 1250 A.D.) were much more widely used, and doubtless contributed more to the spread of the numerals among the common people.

{134}

The Carmen de Algorismo[540] of Alexander de Villa Dei was written in verse, as indeed were many other textbooks of that time. That it was widely used is evidenced by the large number of manuscripts[541] extant in European libraries. Sacrobosco's Algorismus,[542] in which some lines from the Carmen are quoted, enjoyed a wide popularity as a textbook for university instruction.[543] The work was evidently written with this end in view, as numerous commentaries by university lecturers are found. Probably the most widely used of these was that of Petrus de Dacia[544] written in 1291. These works throw an interesting light upon the method of instruction in mathematics in use in the universities from the thirteenth even to the sixteenth century. Evidently the text was first read and copied by students.[545] Following this came line by line an exposition of the text, such as is given in Petrus de Dacia's commentary.

Sacrobosco's work is of interest also because it was probably due to the extended use of this work that the {135} term Arabic numerals became common. In two places there is mention of the inventors of this system. In the introduction it is stated that this science of reckoning was due to a philosopher named Algus, whence the name algorismus,[546] and in the section on numeration reference is made to the Arabs as the inventors of this science.[547] While some of the commentators, Petrus de Dacia[548] among them, knew of the Hindu origin, most of them undoubtedly took the text as it stood; and so the Arabs were credited with the invention of the system.

The first definite trace that we have of an algorism in the French language is found in a manuscript written about 1275.[549] This interesting leaf, for the part on algorism consists of a single folio, was noticed by the Abbe Leboeuf as early as 1741,[550] and by Daunou in 1824.[551] It then seems to have been lost in the multitude of Paris manuscripts; for although Chasles[552] relates his vain search for it, it was not rediscovered until 1882. In that year M. Ch. Henry found it, and to his care we owe our knowledge of the interesting manuscript. The work is anonymous and is devoted almost entirely to geometry, only {136} two pages (one folio) relating to arithmetic. In these the forms of the numerals are given, and a very brief statement as to the operations, it being evident that the writer himself had only the slightest understanding of the subject.

Once the new system was known in France, even thus superficially, it would be passed across the Channel to England. Higden,[553] writing soon after the opening of the fourteenth century, speaks of the French influence at that time and for some generations preceding:[554] "For two hundred years children in scole, agenst the usage and manir of all other nations beeth compelled for to leave hire own language, and for to construe hir lessons and hire thynges in Frensche.... Gentilmen children beeth taught to speke Frensche from the tyme that they bith rokked in hir cradell; and uplondissche men will likne himself to gentylmen, and fondeth with greet besynesse for to speke Frensche."

The question is often asked, why did not these new numerals attract more immediate attention? Why did they have to wait until the sixteenth century to be generally used in business and in the schools? In reply it may be said that in their elementary work the schools always wait upon the demands of trade. That work which pretends to touch the life of the people must come reasonably near doing so. Now the computations of business until about 1500 did not demand the new figures, for two reasons: First, cheap paper was not known. Paper-making of any kind was not introduced into Europe until {137} the twelfth century, and cheap paper is a product of the nineteenth. Pencils, too, of the modern type, date only from the sixteenth century. In the second place, modern methods of operating, particularly of multiplying and dividing (operations of relatively greater importance when all measures were in compound numbers requiring reductions at every step), were not yet invented. The old plan required the erasing of figures after they had served their purpose, an operation very simple with counters, since they could be removed. The new plan did not as easily permit this. Hence we find the new numerals very tardily admitted to the counting-house, and not welcomed with any enthusiasm by teachers.[555]

Aside from their use in the early treatises on the new art of reckoning, the numerals appeared from time to time in the dating of manuscripts and upon monuments. The oldest definitely dated European document known {138} to contain the numerals is a Latin manuscript,[556] the Codex Vigilanus, written in the Albelda Cloister not far from Logrono in Spain, in 976 A.D. The nine characters (of ġobār type), without the zero, are given as an addition to the first chapters of the third book of the Origines by Isidorus of Seville, in which the Roman numerals are under discussion. Another Spanish copy of the same work, of 992 A.D., contains the numerals in the corresponding section. The writer ascribes an Indian origin to them in the following words: "Item de figuris arithmeticȩ. 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 ġobār characters follow. Some of the abacus forms[557] previously given are doubtless also of the tenth century. The earliest Arabic documents containing the numerals are two manuscripts of 874 and 888 A.D.[558] They appear about a century later in a work[559] written at Shiraz in 970 A.D. There is also an early trace of their use on a pillar recently discovered in a church apparently destroyed as early as the tenth century, not far from the Jeremias Monastery, in Egypt. {139} A graffito in Arabic on this pillar has the date 349 A.H., which corresponds to 961 A.D.[560] For the dating of Latin documents the Arabic forms were used as early as the thirteenth century.[561]

On the early use of these numerals in Europe the only scientific study worthy the name is that made by Mr. G. F. Hill of the British Museum.[562] From his investigations it appears that the earliest occurrence of a date in these numerals on a coin is found in the reign of Roger of Sicily in 1138.[563] Until recently it was thought that the earliest such date was 1217 A.D. for an Arabic piece and 1388 for a Turkish one.[564] Most of the seals and medals containing dates that were at one time thought to be very early have been shown by Mr. Hill to be of relatively late workmanship. There are, however, in European manuscripts, numerous instances of the use of these numerals before the twelfth century. Besides the example in the Codex Vigilanus, another of the tenth century has been found in the St. Gall MS. now in the University Library at Zuerich, the forms differing materially from those in the Spanish codex.

The third specimen in point of time in Mr. Hill's list is from a Vatican MS. of 1077. The fourth and fifth specimens are from the Erlangen MS. of Boethius, of the same {140} (eleventh) century, and the sixth and seventh are also from an eleventh-century MS. of Boethius at Chartres. These and other early forms are given by Mr. Hill in this table, which is reproduced with his kind permission.

EARLIEST MANUSCRIPT FORMS



This is one of more than fifty tables given in Mr. Hill's valuable paper, and to this monograph students {141} are referred for details as to the development of number-forms in Europe from the tenth to the sixteenth century. It is of interest to add that he has found that among the earliest dates of European coins or medals in these numerals, after the Sicilian one already mentioned, are the following: Austria, 1484; Germany, 1489 (Cologne); Switzerland, 1424 (St. Gall); Netherlands, 1474; France, 1485; Italy, 1390.[565]

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



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



Many printed works anterior to 1471 have pages or chapters numbered by hand, but many of these numerals are {142} of date much later than the printing of the work. Other works were probably numbered directly after printing. Thus the chapters 2, 3, 4, 5, 6 in a book of 1470[571] are numbered as follows: Capitulem [Symbol 2]m.,... [Symbol 3]m.,... 4m.,... v,... vi, and followed by Roman numerals. This appears in the body of the text, in spaces left by the printer to be filled in by hand. Another book[572] of 1470 has pages numbered by hand with a mixture of Roman and Hindu numerals, thus,

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]

{143}

EARLY MANUSCRIPT FORMS

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

{144}



For the sake of further comparison, three illustrations from works in Mr. Plimpton's library, reproduced from the Rara Arithmetica, may be considered. The first is from a Latin manuscript on arithmetic,[584] of which the original was written at Paris in 1424 by Rollandus, a Portuguese physician, who prepared the work at the command of John of Lancaster, Duke of Bedford, at one time Protector of England and Regent of France, to whom the work is dedicated. The figures show the successive powers of 2. The second illustration is from Luca da Firenze's Inprencipio darte dabacho,[585] c. 1475, and the third is from an anonymous manuscript[586] of about 1500.



As to the forms of the numerals, fashion played a leading part until printing was invented. This tended to fix these forms, although in writing there is still a great variation, as witness the French 5 and the German 7 and 9. Even in printing there is not complete uniformity, {145} and it is often difficult for a foreigner to distinguish between the 3 and 5 of the French types.



As to the particular numerals, the following are some of the forms to be found in the later manuscripts and in the early printed books.

1. In the early printed books "one" was often i, perhaps to save types, just as some modern typewriters use the same character for l and 1.[587] In the manuscripts the "one" appears in such forms as[588]



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



{146}

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

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



4. "Four" has changed greatly; and one of the first tests as to the age of a manuscript on arithmetic, and the place where it was written, is the examination of this numeral. Until the time of printing the most common form was [Symbol], although the Florentine manuscript of Leonard of Pisa's work has the form [Symbol];[593] but the manuscripts show that the Florentine arithmeticians and astronomers rather early began to straighten the first of these forms up to forms like [Symbol][594] and [Symbol][594] or [Symbol],[595] more closely resembling our own. The first printed books generally used our present form[596] with the closed top [Symbol], the open top used in writing ( [Symbol]) being {147} purely modern. The following are other forms of the four, from various manuscripts:[597]



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



6. "Six" has changed rather less than most of the others. The chief variation has been in the slope of the top, as will be seen in the following:[599]



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



{148}

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



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

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



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



The explanation of the place value was a serious matter to most of the early writers. If they had been using an abacus constructed like the Russian chotue, and had placed this before all learners of the positional system, there would have been little trouble. But the medieval {149} line-reckoning, where the lines stood for powers of 10 and the spaces for half of such powers, did not lend itself to this comparison. Accordingly we find such labored explanations as the following, from The Crafte of Nombrynge:

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

"If it stonde in the secunde place of the rewle, he betokens ten tymes hym selfe, as this figure 2 here 20 tokens ten tyme hym selfe, that is twenty, for he hym selfe betokens tweyne, & ten tymes twene is twenty. And for he stondis on the lyft side & in the secunde place, he betokens ten tyme hym selfe. And so go forth....

"Nil cifra significat sed dat signare sequenti. Expone this verse. A cifre tokens no[gh]t, bot he makes the figure to betoken that comes after hym more than he shuld & he were away, as thus 10. here the figure of one tokens ten, & yf the cifre were away & no figure byfore hym he schuld token bot one, for than he schuld stonde in the first place...."[605]

It would seem that a system that was thus used for dating documents, coins, and monuments, would have been generally adopted much earlier than it was, particularly in those countries north of Italy where it did not come into general use until the sixteenth century. This, however, has been the fate of many inventions, as witness our neglect of logarithms and of contracted processes to-day.

As to Germany, the fifteenth century saw the rise of the new symbolism; the sixteenth century saw it slowly {150} gain the mastery; the seventeenth century saw it finally conquer the system that for two thousand years had dominated the arithmetic of business. Not a little of the success of the new plan was due to Luther's demand that all learning should go into the vernacular.[606]

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

Although not connected with the Arabic numerals in any direct way, the medieval astrological numerals may here be mentioned. These are given by several early writers, but notably by Noviomagus (1539),[613] as follows[614]:



{151}

Thus we find the numerals gradually replacing the Roman forms all over Europe, from the time of Leonardo of Pisa until the seventeenth century. But in the Far East to-day they are quite unknown in many countries, and they still have their way to make. In many parts of India, among the common people of Japan and China, in Siam and generally about the Malay Peninsula, in Tibet, and among the East India islands, the natives still adhere to their own numeral forms. Only as Western civilization is making its way into the commercial life of the East do the numerals as used by us find place, save as the Sanskrit forms appear in parts of India. It is therefore with surprise that the student of mathematics comes to realize how modern are these forms so common in the West, how limited is their use even at the present time, and how slow the world has been and is in adopting such a simple device as the Hindu-Arabic numerals.

* * * * *

{153}

INDEX

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

Abbo of Fleury, 122 'Abdallāh ibn al-Ḥasan, 92 'Abdallatīf ibn Yūsuf, 93 'Abdalqādir ibn 'Alī al-Sakhāwī, 6 Abenragel, 34 Abraham ibn Meir ibn Ezra, see Rabbi ben Ezra Abū 'Alī al-Ḥosein ibn Sīnā, 74 Abū 'l-Ḥasan, 93, 100 Abū 'l-Qāsim, 92 Abū 'l-Ṭeiyib, 97 Abū Naṣr, 92 Abū Roshd, 113 Abu Sahl Dunash ibn Tamim, 65, 67 Adelhard of Bath, 5, 55, 97, 119, 123, 126 Adhemar of Chabanois, 111 Aḥmed al-Nasawī, 98 Aḥmed ibn 'Abdallāh, 9, 92 Aḥmed ibn Moḥammed, 94 Aḥmed ibn 'Omar, 93 Akṣaras, 32 Alanus ab Insulis, 124 Al-Baġdādī, 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-Ḥaṣṣār, 65 'Alī ibn Abī Bekr, 6 'Alī ibn Aḥmed, 93, 98 Al-Karābīsī, 93 Al-Khowārazmī, 4, 9, 10, 92, 97, 98, 125, 126 Al-Kindī, 10, 92 Almagest, 54 Al-Maġrebī, 93 Al-Maḥallī, 6 Al-Māmūn, 10, 97 Al-Manṣūr, 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-Ṣardafī, 93 Al-Sijzī, 94 Al-Sūfī, 10, 92 Ambrosoli, 118 Aṅkapalli, 43 Apices, 87, 117, 118 Arabs, 91-98 Arbuthnot, 141 {154} Archimedes, 15, 16 Arcus Pictagore, 122 Arjuna, 15 Arnold, E., 15, 102 Ars memorandi, 141 Āryabhaṭa, 39, 43, 44 Aryan numerals, 19 Aschbach, 134 Ashmole, 134 Aśoka, 19, 20, 22, 81 Aṣ-ṣifr, 57, 58 Astrological numerals, 150 Atharva-Veda, 48, 49, 55 Augustus, 80 Averroes, 113 Avicenna, 58, 74, 113

Babylonian numerals, 28 Babylonian zero, 51 Bacon, R., 131 Bactrian numerals, 19, 30 Baeda, 2, 72 Bagdad, 4, 96 Bakhṣālī manuscript, 43, 49, 52, 53 Ball, C. J., 35 Ball, W. W. R., 36, 131 Bāṇa, 44 Barth, A., 39 Bayang inscriptions, 39 Bayer, 33 Bayley, E. C., 19, 23, 30, 32, 52, 89 Beazley, 75 Bede, see Baeda 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 Bjoernbo, A. A., 125, 126 Blassiere, 119 Bloomfield, 48 Blume, 85 Boeckh, 62 Boehmer, 143 Boeschenstein, 119 Boethius, 63, 70-73, 83-90 Boissiere, 63 Bombelli, 81 Bonaini, 128 Boncompagni, 5, 6, 10, 48, 49, 123, 125 Borghi, 59 Borgo, 119 Bougie, 130 Bowring, J., 56 Brahmagupta, 52 Brāhmaṇas, 12, 13 Brāhmī, 19, 20, 31, 83 Brandis, J., 54 Bṛhat-Saṃhita, 39, 44, 78 Brockhaus, 43 Bubnov, 65, 84, 110, 116 Buddha, education of, 15, 16 Buedinger, 110 Bugia, 130 Buehler, G., 15, 19, 22, 31, 44, 49 Burgess, 25 Buerk, 13 Burmese numerals, 36 Burnell, A. C., 18, 40 Buteo, 61

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

Dagomari, 146 D'Alviella, 15 Dante, 72 Dasypodius, 33, 67, 63 Daunou, 135 Delambre, 54 Devanāgarī, 7 Devoulx, A., 68 Dhruva, 49 Dicaearchus 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 Enestroem, 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 {156} Florus, 80 Fluegel, G., 68 Francisco de Retza, 142 Francois, 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 Ġobār 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 Guenther, S., 131 Guyard, S., 82

Ḥabash, 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 Hecataeus, 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 Ḥosein ibn Moḥammed al-Maḥallī, 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 Khordāḍbeh, 101, 106 Ibn Wahab, 103 India, history of, 14 writing in, 18 Indicopleustes, 83 Indo-Bactrian numerals, 19 {157} Indrājī, 23 Isḥāq ibn Yūsuf al-Ṣardafī, 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

Kale, 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 Kharoṣṭhī, 19, 20 Khosrū, 82, 91 Kielhorn, F., 46, 47 Kircher, A., 34 Kitāb al-Fihrist, see Fihrist Kleinwaechter, 32 Klos, 62 Koebel, 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 Lāṭyāyana, 39 Leboeuf, 135 Leonardo of Pisa, 5, 10, 57, 64, 74, 120, 128-133 Lethaby, W. R., 142 Levi, B., 13 Levias, 3 Libri, 73, 85, 95 Light of Asia, 16 Luca da Firenze, 144 Lucas, 128

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 Moḥammed ibn 'Abdallāh, 92 Moḥammed ibn Aḥmed, 6 Moḥammed ibn 'Alī 'Abdī, 8 Moḥammed ibn Mūsā, see Al-Khowārazmī Molinier, 123 Monier-Williams, 17 {158} Morley, D., 126 Moroccan numerals, 68, 119 Mortet, V., 11 Moseley, C. B., 33 Moṭahhar ibn Ṭāhir, 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 Naẓīf ibn Yumn, 94 Neander, A., 75 Neophytos, 57, 62 Neo-Pythagoreans, 64 Nesselmann, 58 Newman, Cardinal, 96 Newman, F. W., 131 Noeldeke, 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 Kharoṣṭhī, 19-22 Moroccan, 68 Nabatean, 21 origin, 27, 30, 31, 37 supposed Arabic origin, 2 supposed Babylonian origin, 28 supposed Chaldean and Jewish origin, 3 supposed Chinese origin, 28, 32 supposed Egyptian origin, 27, 30, 69, 70 supposed Greek origin, 33 supposed Phoenician origin, 32 tables of, 22-27, 36, 48, 49, 69, 88, 140, 143, 145-148

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

Pali, 22 Pancasiddhāntikā, 44 Paravey, 32, 57 Pātalīpuṭra, 77 Patna, 77 Patrick, R., 119 Payne, E. J., 106 Pegolotti, 107 Peletier, 2, 62 Perrot, 80 Persia, 66, 91, 107 Pertz, 115 Petrus de Dacia, 59, 61, 62 Pez, P. B., 117 "Philalethes," 75 Phillips, G., 107 Picavet, 105 Pichler, F., 141 Pihan, A. P., 36 Pisa, 128 Place value, 26, 42, 46, 48 Planudes, see Maximus Planudes Plimpton, G. A., 56, 59, 85, 143, 144, 145, 148 Pliny, 76 Polo, N. and M., 107 {159} Praendel, 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 Śaka inscriptions, 20 Samū'īl ibn Yaḥyā, 93 Śāradā 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 Ṣifr, 57 Sigsboto, 55 Sihāb al-Dīn, 67 Silberberg, 60 Simon, 13 Sinān ibn al-Fatḥ, 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 Smṛti, 17 Spain, 64, 65, 100 Spitta-Bey, 5 Sprenger, 94 Śrautasūtra, 39 Steffens, F., 116 Steinschneider, 5, 57, 65, 66, 98, 126 Stifel, 62 {160} Subandhus, 44 Suetonius, 80 Suleimān, 100 Śūnya, 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