Even in literature of the better cla.s.s 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 _Gulist[=a]n_ of the Persian poet Sa'd[=i][403] contains such a pa.s.sage:

"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 gla.s.s to Yeman, and carry the painted cloths of Yeman back to Persia.'"[404] On the other hand, these men were not of the learned cla.s.s, 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.

{103}

There were also amba.s.sadors pa.s.sing 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 amba.s.sadors 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 ill.u.s.tration 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 sh.o.r.es of the Mediterranean. As early as 652 A.D., in the thirtieth year of the Hejira, the Mohammedans descended upon the sh.o.r.es 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 sh.o.r.es 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 Khord[=a][d.]beh, 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 H[=a]r[=u]n al-Rash[=i]d 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 c.n.u.t, 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 mitt.i.t 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 ma.n.u.scripts 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 Hish[=a]m (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} w[=a]l[=i] 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, a.s.sisting 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 [.g]ob[=a]r 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 [.g]ob[=a]r 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 S[=i]n[=a] (Avicenna) who lived at the beginning, and Gerber of Seville who flourished in the middle, of the eleventh century, and Ab[=u] 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 a.s.similated 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 [.g]ob[=a]r 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 [.g]ob[=a]r, 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 cla.s.sic 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 ma.n.u.script, entirely forgotten until Pertz discovered it at Bamberg in 1833.[454] The work is dedicated to Gerbert as archbishop of Rheims,[455]

and would a.s.suredly 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 Ispa.n.u.s,[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 a.s.similated 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 ma.n.u.scripts 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 doc.u.ments 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 [.g]ob[=a]r 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 _a.s.si_ 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_, _k.u.msa_, _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 [.g]ob[=a]r forms without the zero.[474]

O'Creat uses three forms for zero, o, [=o], 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 a.s.similation 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 pa.s.sed, 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 ma.n.u.script of seventy-seven leaves, contains a curious mixture of Roman and [.g]ob[=a]r numerals, the former for expressing large results, the latter for practical calculation. These [.g]ob[=a]r "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 Ala.n.u.s ab Insulis[498]

(Alain de Lille), Gautier de Coincy (1177-1236), and others.