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 [.g]ob[=a]r 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 Suleim[=a]n the Merchant, a well-known Arab trader of the
ninth century, that part of the story of Sindb[=a]d 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} Ab[=u]
'l-[H.]asan `Al[=i] al-Mas`[=u]d[=i] (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 Ab[=u] 'l-Q[=a]sim `Obeidall[=a]h ibn
A[h.]med, better known by his Persian name Ibn Khord[=a][d.]beh,[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 _Gulist[=a]n_ of the Persian
poet Sa`d[=i][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.
{103}
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 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 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
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, 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 [.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 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 [.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
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 [.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 _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 [.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 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 [.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 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-Khow[=a]razm[=i]'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-Khow[=a]razm[=i]'s astronomical
tables[507] and of Euclid's Elements,[508] while Robert of Chester is known
as the translator of Al-Khow[=a]razm[=i]'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}