Contemporary with Al-Khowārazmī, and working also under Al-Māmūn, was a Jewish astronomer,Abū 'l-Ṭeiyib,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 ibnAḥmedal-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.
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, andremained 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,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 ibnAḥmed, better known by his Persian nameIbn Khordāḍbeh,[400]who wrote about 850A.D.a work entitledBook 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 reëmbark 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 900A.D., must necessarily have spread abroad a knowledge of all numbersystems 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. TheGulistānof 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 (pûlab) 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.
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 wentfrom 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 Byzantinecommerciariiwere 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(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, theItinerary 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 652A.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 theyoung 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 900A.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 feeAnd was the safeguard of the West,"
"held the gorgeous East in feeAnd was the safeguard of the West,"
"held the gorgeous East in fee
And was the safeguard of the West,"
and long before Genoa had become her powerful rival.[420]
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]
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 whichsailed through the Strait of Gibraltar into the Mediterranean.[426]Only a little later, probably before 1200A.D., a clerk in the service of Thomas à 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 Sabæus:Arma Sythes: oleum palmarum divite sylvaPingue solum Babylon: Nilus lapides pretiosos:Norwegi, Russi, varium grisum, sabdinas:Seres, purpureas vestes: Galli, sua vina.
Aurum mittit Arabs: species et thura Sabæus:Arma Sythes: oleum palmarum divite sylvaPingue solum Babylon: Nilus lapides pretiosos:Norwegi, Russi, varium grisum, sabdinas:Seres, purpureas vestes: Galli, sua vina.
Aurum mittit Arabs: species et thura Sabæus:
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 thebaudekin,[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 forsome 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,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 Ordoño to the Arab scholars of the court of thewālī 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"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 (Averroës) who lived at the end of the twelfth.[450]Others hold that his proximity tothe 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 sourcesare 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. Possiblyit 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 treatiseLibellus 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]
So little did Gerbert appreciate these numerals that in his works known as theRegula de abaco computiand theLibellushe 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 thousandjetonsor 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 calledcaracteres, a name which adhered also to the figures themselves. It is an interesting speculation to consider whether theseapices, as they are called in the Boethius interpolations, were in any way suggested by those Roman jetons generally known in numismatics astesserae, and bearing the figures I-XVI, the sixteen referring to the number ofassiin asestertius.[468]Thenameapicesadhered to the Hindu-Arabic numerals until the sixteenth century.[469]
To the figures on theapiceswere 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,thalata,arba,kumsa,setta,sebba,timinia,taseudare 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 nameapiceswas not, however, a common one in later times.Notaewas more often used, and it finally gave the name to notation.[472]Still more common were the namesfigures,ciphers,signs,elements, andcharacters.[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 Adelhardof 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, andτ, 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.τ.τfor 1200, I. O. VIII. IX for 1089, and I. IIII. IIII.ττττfor 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 wordscifraandalgorismus cifrawere 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 dayDoes not celebrate the holy Mother.[476]
A horned beast, a sheep,An algorismus-cipher,Is a priest, who on such a feast dayDoes not celebrate the holy Mother.[476]
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.Geoffrey Chaucer[477]describes inThe Miller's Talethe clerk with
"His Almageste and bokes grete and smale,His astrelabie, longinge for his art,His augrim-stones layen faire apartOn shelves couched at his beddes heed."
"His Almageste and bokes grete and smale,His astrelabie, longinge for his art,His augrim-stones layen faire apartOn shelves couched at his beddes heed."
"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 ofthree columns (lineaeis 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 designationarcus pictagore[482]or sometimes simplyarcus.[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(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 workRegulae 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 page88.
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 foundinterested 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 wordsalgorismusandcifrawere 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]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 entitledJoannis Hispalensis liber Algorismi de Practica Arismetricewhich Boncompagni found in what is now theBibliothèque nationaleat 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]theauthor 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 aComputusby 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 shapeSymbol, a form characteristic of the twelfth century.
It was about the same time that theSefer ha-Mispar,[512]the Book of Number, appeared in the Hebrew language. The author, Rabbi Abraham ibn Meïr 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,א ב ג ד ה ו ז ח ט, 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.
THE SPREAD OF THE NUMERALS IN EUROPE
Of all the medieval writers, probably the one most influential in introducing the new numerals to the scholars of Europe was Leonardo Fibonacci, of Pisa.[515]This remarkable man, the most noteworthy mathematical genius of the Middle Ages, was born at Pisa about 1175.[516]
The traveler of to-day may cross the Via Fibonacci on his way to the Campo Santo, and there he may see at the end of the long corridor, across the quadrangle, the statue of Leonardo in scholars garb. Few towns have honored a mathematician more, and few mathematicians have so distinctly honored their birthplace. Leonardo was born in the golden age of this city, the period of its commercial, religious, and intellectual prosperity.[517]Situated practically at the mouth of the Arno, Pisa formed with Genoa and Venice the trio of the greatest commercial centers of Italy at the opening of the thirteenth century. Even before Venice had captured the Levantine trade, Pisa had close relations with the East. An old Latin chronicle relates that in 1005 "Pisa was captured by the Saracens," that in the following year "the Pisans overthrew the Saracens at Reggio," and that in 1012 "the Saracens came to Pisa and destroyed it." The city soon recovered, however, sending no fewer than a hundred and twenty ships to Syria in 1099,[518]founding a merchant colony in Constantinople a few years later,[519]and meanwhile carrying on an interurban warfare in Italy that seemed to stimulate it to great activity.[520]A writer of 1114 tells us that at that time there were many heathen people—Turks, Libyans, Parthians, and Chaldeans—to be found in Pisa. It was in the midst of such wars, in a cosmopolitan and commercial town, in a center where literary work was not appreciated,[521]that the genius of Leonardo appears as one of the surprises of history, warning us again that "we should draw no horoscope; that we should expect little, for what we expect will not come to pass."[522]
Leonardo's father was one William,[523]and he had a brother named Bonaccingus,[524]but nothing further isknown of his family. As to Fibonacci, most writers[525]have assumed that his father's name was Bonaccio,[526]whencefilius Bonaccii, or Fibonacci. Others[527]believe that the name, even in the Latin form offilius Bonacciias 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 withmerchants, 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 hisLiber 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 veryaccount 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 investigatethem the more zealously.) At Vicenza in 1209, for example, there were Bohemians, Poles, Frenchmen, Burgundians, Germans, and Spaniards, not to speak of representatives of divers towns of Italy; and what was true there was also true of other intellectual centers. The knowledge could not fail to spread, therefore, and as a matter of fact we find numerous bits of evidence that this was the case. Although the bankers of Florence were forbidden to use these numerals in 1299, and the statutes of the university of Padua required stationers to keep the price lists of books "non per cifras, sed per literas claros,"[538]the numerals really made much headway from about 1275 on.
It was, however, rather exceptional for the common people of Germany to use the Arabic numerals before the sixteenth century, a good witness to this fact being the popular almanacs. Calendars of 1457-1496[539]have generally the Roman numerals, while Köbel's calendar of 1518 gives the Arabic forms as subordinate to the Roman. In the register of the Kreuzschule at Dresden the Roman forms were used even until 1539.
While not minimizing the importance of the scientific work of Leonardo of Pisa, we may note that the more popular treatises by Alexander de Villa Dei (c. 1240A.D.) and John of Halifax (Sacrobosco, c. 1250A.D.) were much more widely used, and doubtless contributed more to the spread of the numerals among the common people.
TheCarmen 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'sAlgorismus,[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 thetermArabic numeralsbecame 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 namealgorismus,[546]and in the section on numeration reference is made to the Arabs as the inventors of this science.[547]While some of the commentators, Petrus de Dacia[548]among them, knew of the Hindu origin, most of them undoubtedly took the text as it stood; and so the Arabs were credited with the invention of the system.
The first definite trace that we have of an algorism in the French language is found in a manuscript written about 1275.[549]This interesting leaf, for the part on algorism consists of a single folio, was noticed by the Abbé Lebœuf 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, onlytwo 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 untilthe 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 knownto contain the numerals is a Latin manuscript,[556]the Codex Vigilanus, written in the Albelda Cloister not far from Logroño in Spain, in 976A.D.The nine characters (of ġobār type), without the zero, are given as an addition to the first chapters of the third book of theOriginesby Isidorus of Seville, in which the Roman numerals are under discussion. Another Spanish copy of the same work, of 992A.D., contains the numerals in the corresponding section. The writer ascribes an Indian origin to them in the following words: "Item de figuris arithmeticę. Scire debemus in Indos subtilissimum ingenium habere et ceteras gentes eis in arithmetica et 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 888A.D.[558]They appear about a century later in a work[559]written at Shiraz in 970A.D.There is also an early trace of their use on a pillar recently discovered in a church apparently destroyed as early as the tenth century, not far from the Jeremias Monastery, in Egypt.A graffito in Arabic on this pillar has the date 349A.H., which corresponds to 961A.D.[560]For the dating of Latin documents the Arabic forms were used as early as the thirteenth century.[561]