THE QUESTION OF THE INTRODUCTION OF THE NUMERALS INTO EUROPE BY BOETHIUS
Just as we were quite uncertain as to the origin of the numeral forms, so
too are we uncertain as to the time and place of their introduction into
Europe. There are two general theories as to this introduction. The first
is that they were carried by the Moors to Spain in the eighth or ninth
century, and thence were transmitted to Christian Europe, a theory which
will be considered later. The second, advanced by Woepcke,[247] is that
they were not brought to Spain by the Moors, but that they were already in
Spain when the Arabs arrived there, having reached the West through the
Neo-Pythagoreans. There are two facts to support this second theory: (1)
the forms of these numerals are characteristic, differing materially from
those which were brought by Leonardo of Pisa from Northern Africa early in
the thirteenth century (before 1202 A.D.); (2) they are essentially those
which {64} tradition has so persistently assigned to Boethius (c. 500
A.D.), and which he would naturally have received, if at all, from these
same Neo-Pythagoreans or from the sources from which they derived them.
Furthermore, Woepcke points out that the Arabs on entering Spain (711 A.D.)
would naturally have followed their custom of adopting for the computation
of taxes the numerical systems of the countries they conquered,[248] so
that the numerals brought from Spain to Italy, not having undergone the
same modifications as those of the Eastern Arab empire, would have
differed, as they certainly did, from those that came through Bagdad. The
theory is that the Hindu system, without the zero, early reached Alexandria
(say 450 A.D.), and that the Neo-Pythagorean love for the mysterious and
especially for the Oriental led to its use as something bizarre and
cabalistic; that it was then passed along the Mediterranean, reaching
Boethius in Athens or in Rome, and to the schools of Spain, being
discovered in Africa and Spain by the Arabs even before they themselves
knew the improved system with the place value.
{65}
A recent theory set forth by Bubnov[249] also deserves mention, chiefly
because of the seriousness of purpose shown by this well-known writer.
Bubnov holds that the forms first found in Europe are derived from ancient
symbols used on the abacus, but that the zero is of Hindu origin. This
theory does not seem tenable, however, in the light of the evidence already
set forth.
Two questions are presented by Woepcke's theory: (1) What was the nature of
these Spanish numerals, and how were they made known to Italy? (2) Did
Boethius know them?
The Spanish forms of the numerals were called the _[h.]ur[=u]f
al-[.g]ob[=a]r_, the [.g]ob[=a]r or dust numerals, as distinguished from
the _[h.]ur[=u]f al-jumal_ or alphabetic numerals. Probably the latter,
under the influence of the Syrians or Jews,[250] were also used by the
Arabs. The significance of the term [.g]ob[=a]r is doubtless that these
numerals were written on the dust abacus, this plan being distinct from the
counter method of representing numbers. It is also worthy of note that
Al-B[=i]r[=u]n[=i] states that the Hindus often performed numerical
computations in the sand. The term is found as early as c. 950, in the
verses of an anonymous writer of Kairw[=a]n, in Tunis, in which the author
speaks of one of his works on [.g]ob[=a]r calculation;[251] and, much
later, the Arab writer Ab[=u] Bekr Mo[h.]ammed ibn `Abdall[=a]h, surnamed
al-[H.]a[s.][s.][=a]r {66} (the arithmetician), wrote a work of which the
second chapter was "On the dust figures."[252]
The [.g]ob[=a]r numerals themselves were first made known to modern
scholars by Silvestre de Sacy, who discovered them in an Arabic manuscript
from the library of the ancient abbey of St.-Germain-des-Pres.[253] The
system has nine characters, but no zero. A dot above a character indicates
tens, two dots hundreds, and so on, [5 with dot] meaning 50, and [5 with 3
dots] meaning 5000. It has been suggested that possibly these dots,
sprinkled like dust above the numerals, gave rise to the word
_[.g]ob[=a]r_,[254] but this is not at all probable. This system of dots is
found in Persia at a much later date with numerals quite like the modern
Arabic;[255] but that it was used at all is significant, for it is hardly
likely that the western system would go back to Persia, when the perfected
Hindu one was near at hand.
At first sight there would seem to be some reason for believing that this
feature of the [.g]ob[=a]r system was of {67} Arabic origin, and that the
present zero of these people,[256] the dot, was derived from it. It was
entirely natural that the Semitic people generally should have adopted such
a scheme, since their diacritical marks would suggest it, not to speak of
the possible influence of the Greek accents in the Hellenic number system.
When we consider, however, that the dot is found for zero in the
Bakh[s.][=a]l[=i] manuscript,[257] and that it was used in subscript form
in the _Kit[=a]b al-Fihrist_[258] in the tenth century, and as late as the
sixteenth century,[259] although in this case probably under Arabic
influence, we are forced to believe that this form may also have been of
Hindu origin.
The fact seems to be that, as already stated,[260] the Arabs did not
immediately adopt the Hindu zero, because it resembled their 5; they used
the superscript dot as serving their purposes fairly well; they may,
indeed, have carried this to the west and have added it to the [.g]ob[=a]r
forms already there, just as they transmitted it to the Persians.
Furthermore, the Arab and Hebrew scholars of Northern Africa in the tenth
century knew these numerals as Indian forms, for a commentary on the
_S[=e]fer Ye[s.][=i]r[=a]h_ by Ab[=u] Sahl ibn Tamim (probably composed at
Kairw[=a]n, c. 950) speaks of "the Indian arithmetic known under the name
of _[.g]ob[=a]r_ or dust calculation."[261] All this suggests that the
Arabs may very {68} likely have known the [.g]ob[=a]r forms before the
numerals reached them again in 773.[262] The term "[.g]ob[=a]r numerals"
was also used without any reference to the peculiar use of dots.[263] In
this connection it is worthy of mention that the Algerians employed two
different forms of numerals in manuscripts even of the fourteenth
century,[264] and that the Moroccans of to-day employ the European forms
instead of the present Arabic.
The Indian use of subscript dots to indicate the tens, hundreds, thousands,
etc., is established by a passage in the _Kit[=a]b al-Fihrist_[265] (987
A.D.) in which the writer discusses the written language of the people of
India. Notwithstanding the importance of this reference for the early
history of the numerals, it has not been mentioned by previous writers on
this subject. The numeral forms given are those which have usually been
called Indian,[266] in opposition to [.g]ob[=a]r. In this document the dots
are placed below the characters, instead of being superposed as described
above. The significance was the same.
In form these [.g]ob[=a]r numerals resemble our own much more closely than
the Arab numerals do. They varied more or less, but were substantially as
follows:
{69}
1[267][Illustration]
2[268][Illustration]
3[269][Illustration]
4[270][Illustration]
5[271][Illustration]
6[271][Illustration]
The question of the possible influence of the Egyptian demotic and hieratic
ordinal forms has been so often suggested that it seems well to introduce
them at this point, for comparison with the [.g]ob[=a]r forms. They would
as appropriately be used in connection with the Hindu forms, and the
evidence of a relation of the first three with all these systems is
apparent. The only further resemblance is in the Demotic 4 and in the 9, so
that the statement that the Hindu forms in general came from {70} this
source has no foundation. The first four Egyptian cardinal numerals[272]
resemble more the modern Arabic.
[Illustration: DEMOTIC AND HIERATIC ORDINALS]
This theory of the very early introduction of the numerals into Europe
fails in several points. In the first place the early Western forms are not
known; in the second place some early Eastern forms are like the
[.g]ob[=a]r, as is seen in the third line on p. 69, where the forms are
from a manuscript written at Shiraz about 970 A.D., and in which some
western Arabic forms, e.g. [symbol] for 2, are also used. Probably most
significant of all is the fact that the [.g]ob[=a]r numerals as given by
Sacy are all, with the exception of the symbol for eight, either single
Arabic letters or combinations of letters. So much for the Woepcke theory
and the meaning of the [.g]ob[=a]r numerals. We now have to consider the
question as to whether Boethius knew these [.g]ob[=a]r forms, or forms akin
to them.
This large question[273] suggests several minor ones: (1) Who was Boethius?
(2) Could he have known these numerals? (3) Is there any positive or strong
circumstantial evidence that he did know them? (4) What are the
probabilities in the case?
{71}
First, who was Boethius,--Divus[274] Boethius as he was called in the
Middle Ages? Anicius Manlius Severinus Boethius[275] was born at Rome c.
475. He was a member of the distinguished family of the Anicii,[276] which
had for some time before his birth been Christian. Early left an orphan,
the tradition is that he was taken to Athens at about the age of ten, and
that he remained there eighteen years.[277] He married Rusticiana, daughter
of the senator Symmachus, and this union of two such powerful families
allowed him to move in the highest circles.[278] Standing strictly for the
right, and against all iniquity at court, he became the object of hatred on
the part of all the unscrupulous element near the throne, and his bold
defense of the ex-consul Albinus, unjustly accused of treason, led to his
imprisonment at Pavia[279] and his execution in 524.[280] Not many
generations after his death, the period being one in which historical
criticism was at its lowest ebb, the church found it profitable to look
upon his execution as a martyrdom.[281] He was {72} accordingly looked upon
as a saint,[282] his bones were enshrined,[283] and as a natural
consequence his books were among the classics in the church schools for a
thousand years.[284] It is pathetic, however, to think of the medieval
student trying to extract mental nourishment from a work so abstract, so
meaningless, so unnecessarily complicated, as the arithmetic of Boethius.
He was looked upon by his contemporaries and immediate successors as a
master, for Cassiodorus[285] (c. 490-c. 585 A.D.) says to him: "Through
your translations the music of Pythagoras and the astronomy of Ptolemy are
read by those of Italy, and the arithmetic of Nicomachus and the geometry
of Euclid are known to those of the West."[286] Founder of the medieval
scholasticism, {73} distinguishing the trivium and quadrivium,[287] writing
the only classics of his time, Gibbon well called him "the last of the
Romans whom Cato or Tully could have acknowledged for their
countryman."[288]
The second question relating to Boethius is this: Could he possibly have
known the Hindu numerals? In view of the relations that will be shown to
have existed between the East and the West, there can only be an
affirmative answer to this question. The numerals had existed, without the
zero, for several centuries; they had been well known in India; there had
been a continued interchange of thought between the East and West; and
warriors, ambassadors, scholars, and the restless trader, all had gone back
and forth, by land or more frequently by sea, between the Mediterranean
lands and the centers of Indian commerce and culture. Boethius could very
well have learned one or more forms of Hindu numerals from some traveler or
merchant.
To justify this statement it is necessary to speak more fully of these
relations between the Far East and Europe. It is true that we have no
records of the interchange of learning, in any large way, between eastern
Asia and central Europe in the century preceding the time of Boethius. But
it is one of the mistakes of scholars to believe that they are the sole
transmitters of knowledge. {74} As a matter of fact there is abundant
reason for believing that Hindu numerals would naturally have been known to
the Arabs, and even along every trade route to the remote west, long before
the zero entered to make their place-value possible, and that the
characters, the methods of calculating, the improvements that took place
from time to time, the zero when it appeared, and the customs as to solving
business problems, would all have been made known from generation to
generation along these same trade routes from the Orient to the Occident.
It must always be kept in mind that it was to the tradesman and the
wandering scholar that the spread of such learning was due, rather than to
the school man. Indeed, Avicenna[289] (980-1037 A.D.) in a short biography
of himself relates that when his people were living at Bokh[=a]ra his
father sent him to the house of a grocer to learn the Hindu art of
reckoning, in which this grocer (oil dealer, possibly) was expert. Leonardo
of Pisa, too, had a similar training.
The whole question of this spread of mercantile knowledge along the trade
routes is so connected with the [.g]ob[=a]r numerals, the Boethius
question, Gerbert, Leonardo of Pisa, and other names and events, that a
digression for its consideration now becomes necessary.[290]
{75}
Even in very remote times, before the Hindu numerals were sculptured in the
cave of N[=a]n[=a] Gh[=a]t, there were trade relations between Arabia and
India. Indeed, long before the Aryans went to India the great Turanian race
had spread its civilization from the Mediterranean to the Indus.[291] At a
much later period the Arabs were the intermediaries between Egypt and Syria
on the west, and the farther Orient.[292] In the sixth century B.C.,
Hecataeus,[293] the father of geography, was acquainted not only with the
Mediterranean lands but with the countries as far as the Indus,[294] and in
Biblical times there were regular triennial voyages to India. Indeed, the
story of Joseph bears witness to the caravan trade from India, across
Arabia, and on to the banks of the Nile. About the same time as Hecataeus,
Scylax, a Persian admiral under Darius, from Caryanda on the coast of Asia
Minor, traveled to {76} northwest India and wrote upon his ventures.[295]
He induced the nations along the Indus to acknowledge the Persian
supremacy, and such number systems as there were in these lands would
naturally have been known to a man of his attainments.
A century after Scylax, Herodotus showed considerable knowledge of India,
speaking of its cotton and its gold,[296] telling how Sesostris[297] fitted
out ships to sail to that country, and mentioning the routes to the east.
These routes were generally by the Red Sea, and had been followed by the
Phoenicians and the Sabaeans, and later were taken by the Greeks and
Romans.[298]
In the fourth century B.C. the West and East came into very close
relations. As early as 330, Pytheas of Massilia (Marseilles) had explored
as far north as the northern end of the British Isles and the coasts of the
German Sea, while Macedon, in close touch with southern France, was also
sending her armies under Alexander[299] through Afghanistan as far east as
the Punjab.[300] Pliny tells us that Alexander the Great employed surveyors
to measure {77} the roads of India; and one of the great highways is
described by Megasthenes, who in 295 B.C., as the ambassador of Seleucus,
resided at P[=a]tal[=i]pu[t.]ra, the present Patna.[301]
The Hindus also learned the art of coining from the Greeks, or possibly
from the Chinese, and the stores of Greco-Hindu coins still found in
northern India are a constant source of historical information.[302] The
R[=a]m[=a]yana speaks of merchants traveling in great caravans and
embarking by sea for foreign lands.[303] Ceylon traded with Malacca and
Siam, and Java was colonized by Hindu traders, so that mercantile knowledge
was being spread about the Indies during all the formative period of the
numerals.
Moreover the results of the early Greek invasion were embodied by
Dicaearchus of Messana (about 320 B.C.) in a map that long remained a
standard. Furthermore, Alexander did not allow his influence on the East to
cease. He divided India into three satrapies,[304] placing Greek governors
over two of them and leaving a Hindu ruler in charge of the third, and in
Bactriana, a part of Ariana or ancient Persia, he left governors; and in
these the western civilization was long in evidence. Some of the Greek and
Roman metrical and astronomical terms {78} found their way, doubtless at
this time, into the Sanskrit language.[305] Even as late as from the second
to the fifth centuries A.D., Indian coins showed the Hellenic influence.
The Hindu astronomical terminology reveals the same relationship to western
thought, for Var[=a]ha-Mihira (6th century A.D.), a contemporary of
[=A]ryabha[t.]a, entitled a work of his the _B[r.]hat-Sa[m.]hit[=a]_, a
literal translation of [Greek: megale suntaxis] of Ptolemy;[306] and in
various ways is this interchange of ideas apparent.[307] It could not have
been at all unusual for the ancient Greeks to go to India, for Strabo lays
down the route, saying that all who make the journey start from Ephesus and
traverse Phrygia and Cappadocia before taking the direct road.[308] The
products of the East were always finding their way to the West, the Greeks
getting their ginger[309] from Malabar, as the Phoenicians had long before
brought gold from Malacca.
Greece must also have had early relations with China, for there is a
notable similarity between the Greek and Chinese life, as is shown in their
houses, their domestic customs, their marriage ceremonies, the public
story-tellers, the puppet shows which Herodotus says were introduced from
Egypt, the street jugglers, the games of dice,[310] the game of
finger-guessing,[311] the water clock, the {79} music system, the use of
the myriad,[312] the calendars, and in many other ways.[313] In passing
through the suburbs of Peking to-day, on the way to the Great Bell temple,
one is constantly reminded of the semi-Greek architecture of Pompeii, so
closely does modern China touch the old classical civilization of the
Mediterranean. The Chinese historians tell us that about 200 B.C. their
arms were successful in the far west, and that in 180 B.C. an ambassador
went to Bactria, then a Greek city, and reported that Chinese products were
on sale in the markets there.[314] There is also a noteworthy resemblance
between certain Greek and Chinese words,[315] showing that in remote times
there must have been more or less interchange of thought.
The Romans also exchanged products with the East. Horace says, "A busy
trader, you hasten to the farthest Indies, flying from poverty over sea,
over crags, over fires."[316] The products of the Orient, spices and jewels
from India, frankincense from Persia, and silks from China, being more in
demand than the exports from the Mediterranean lands, the balance of trade
was against the West, and thus Roman coin found its way eastward. In 1898,
for example, a number of Roman coins dating from 114 B.C. to Hadrian's time
were found at Pakl[=i], a part of the Haz[=a]ra district, sixteen miles
north of Abbott[=a]b[=a]d,[317] and numerous similar discoveries have been
made from time to time.
{80}
Augustus speaks of envoys received by him from India, a thing never before
known,[318] and it is not improbable that he also received an embassy from
China.[319] Suetonius (first century A.D.) speaks in his history of these
relations,[320] as do several of his contemporaries,[321] and Vergil[322]
tells of Augustus doing battle in Persia. In Pliny's time the trade of the
Roman Empire with Asia amounted to a million and a quarter dollars a year,
a sum far greater relatively then than now,[323] while by the time of
Constantine Europe was in direct communication with the Far East.[324]
In view of these relations it is not beyond the range of possibility that
proof may sometime come to light to show that the Greeks and Romans knew
something of the {81} number system of India, as several writers have
maintained.[325]
Returning to the East, there are many evidences of the spread of knowledge
in and about India itself. In the third century B.C. Buddhism began to be a
connecting medium of thought. It had already permeated the Himalaya
territory, had reached eastern Turkestan, and had probably gone thence to
China. Some centuries later (in 62 A.D.) the Chinese emperor sent an
ambassador to India, and in 67 A.D. a Buddhist monk was invited to
China.[326] Then, too, in India itself A['s]oka, whose name has already
been mentioned in this work, extended the boundaries of his domains even
into Afghanistan, so that it was entirely possible for the numerals of the
Punjab to have worked their way north even at that early date.[327]
Furthermore, the influence of Persia must not be forgotten in considering
this transmission of knowledge. In the fifth century the Persian medical
school at Jondi-Sapur admitted both the Hindu and the Greek doctrines, and
Firdus[=i] tells us that during the brilliant reign of {82} Khosr[=u]
I,[328] the golden age of Pahlav[=i] literature, the Hindu game of chess
was introduced into Persia, at a time when wars with the Greeks were
bringing prestige to the Sassanid dynasty.
Again, not far from the time of Boethius, in the sixth century, the
Egyptian monk Cosmas, in his earlier years as a trader, made journeys to
Abyssinia and even to India and Ceylon, receiving the name _Indicopleustes_
(the Indian traveler). His map (547 A.D.) shows some knowledge of the earth
from the Atlantic to India. Such a man would, with hardly a doubt, have
observed every numeral system used by the people with whom he
sojourned,[329] and whether or not he recorded his studies in permanent
form he would have transmitted such scraps of knowledge by word of mouth.
As to the Arabs, it is a mistake to feel that their activities began with
Mohammed. Commerce had always been held in honor by them, and the
Qoreish[330] had annually for many generations sent caravans bearing the
spices and textiles of Yemen to the shores of the Mediterranean. In the
fifth century they traded by sea with India and even with China, and
[H.]ira was an emporium for the wares of the East,[331] so that any numeral
system of any part of the trading world could hardly have remained
isolated.
Long before the warlike activity of the Arabs, Alexandria had become the
great market-place of the world. From this center caravans traversed Arabia
to Hadramaut, where they met ships from India. Others went north to
Damascus, while still others made their way {83} along the southern shores
of the Mediterranean. Ships sailed from the isthmus of Suez to all the
commercial ports of Southern Europe and up into the Black Sea. Hindus were
found among the merchants[332] who frequented the bazaars of Alexandria,
and Brahmins were reported even in Byzantium.
Such is a very brief resume of the evidence showing that the numerals of
the Punjab and of other parts of India as well, and indeed those of China
and farther Persia, of Ceylon and the Malay peninsula, might well have been
known to the merchants of Alexandria, and even to those of any other
seaport of the Mediterranean, in the time of Boethius. The Br[=a]hm[=i]
numerals would not have attracted the attention of scholars, for they had
no zero so far as we know, and therefore they were no better and no worse
than those of dozens of other systems. If Boethius was attracted to them it
was probably exactly as any one is naturally attracted to the bizarre or
the mystic, and he would have mentioned them in his works only
incidentally, as indeed they are mentioned in the manuscripts in which they
occur.
In answer therefore to the second question, Could Boethius have known the
Hindu numerals? the reply must be, without the slightest doubt, that he
could easily have known them, and that it would have been strange if a man
of his inquiring mind did not pick up many curious bits of information of
this kind even though he never thought of making use of them.
Let us now consider the third question, Is there any positive or strong
circumstantial evidence that Boethius did know these numerals? The question
is not new, {84} nor is it much nearer being answered than it was over two
centuries ago when Wallis (1693) expressed his doubts about it[333] soon
after Vossius (1658) had called attention to the matter.[334] Stated
briefly, there are three works on mathematics attributed to Boethius:[335]
(1) the arithmetic, (2) a work on music, and (3) the geometry.[336]
The genuineness of the arithmetic and the treatise on music is generally
recognized, but the geometry, which contains the Hindu numerals with the
zero, is under suspicion.[337] There are plenty of supporters of the idea
that Boethius knew the numerals and included them in this book,[338] and on
the other hand there are as many who {85} feel that the geometry, or at
least the part mentioning the numerals, is spurious.[339] The argument of
those who deny the authenticity of the particular passage in question may
briefly be stated thus:
1. The falsification of texts has always been the subject of complaint. It
was so with the Romans,[340] it was common in the Middle Ages,[341] and it
is much more prevalent {86} to-day than we commonly think. We have but to
see how every hymn-book compiler feels himself authorized to change at will
the classics of our language, and how unknown editors have mutilated
Shakespeare, to see how much more easy it was for medieval scribes to
insert or eliminate paragraphs without any protest from critics.[342]
2. If Boethius had known these numerals he would have mentioned them in his
arithmetic, but he does not do so.[343]
3. If he had known them, and had mentioned them in any of his works, his
contemporaries, disciples, and successors would have known and mentioned
them. But neither Capella (c. 475)[344] nor any of the numerous medieval
writers who knew the works of Boethius makes any reference to the
system.[345]
{87}
4. The passage in question has all the appearance of an interpolation by
some scribe. Boethius is speaking of angles, in his work on geometry, when
the text suddenly changes to a discussion of classes of numbers.[346] This
is followed by a chapter in explanation of the abacus,[347] in which are
described those numeral forms which are called _apices_ or
_caracteres_.[348] The forms[349] of these characters vary in different
manuscripts, but in general are about as shown on page 88. They are
commonly written with the 9 at the left, decreasing to the unit at the
right, numerous writers stating that this was because they were derived
from Semitic sources in which the direction of writing is the opposite of
our own. This practice continued until the sixteenth century.[350] The
writer then leaves the subject entirely, using the Roman numerals for the
rest of his discussion, a proceeding so foreign to the method of Boethius
as to be inexplicable on the hypothesis of authenticity. Why should such a
scholarly writer have given them with no mention of their origin or use?
Either he would have mentioned some historical interest attaching to them,
or he would have used them in some discussion; he certainly would not have
left the passage as it is.
{88}
FORMS OF THE NUMERALS, LARGELY FROM WORKS ON THE ABACUS[351]
a[352] [Illustration]
b[353] [Illustration]
c[354] [Illustration]
d[355] [Illustration]
e[356] [Illustration]
f[357] [Illustration]
g[358] [Illustration]
h[359] [Illustration]
i[360] [Illustration]
{89}
Sir E. Clive Bayley has added[361] a further reason for believing them
spurious, namely that the 4 is not of the N[=a]n[=a] Gh[=a]t type, but of
the Kabul form which the Arabs did not receive until 776;[362] so that it
is not likely, even if the characters were known in Europe in the time of
Boethius, that this particular form was recognized. It is worthy of
mention, also, that in the six abacus forms from the chief manuscripts as
given by Friedlein,[363] each contains some form of zero, which symbol
probably originated in India about this time or later. It could hardly have
reached Europe so soon.
As to the fourth question, Did Boethius probably know the numerals? It
seems to be a fair conclusion, according to our present evidence, that (1)
Boethius might very easily have known these numerals without the zero, but,
(2) there is no reliable evidence that he did know them. And just as
Boethius might have come in contact with them, so any other inquiring mind
might have done so either in his time or at any time before they definitely
appeared in the tenth century. These centuries, five in number, represented
the darkest of the Dark Ages, and even if these numerals were occasionally
met and studied, no trace of them would be likely to show itself in the
{90} literature of the period, unless by chance it should get into the
writings of some man like Alcuin. As a matter of fact, it was not until the
ninth or tenth century that there is any tangible evidence of their
presence in Christendom. They were probably known to merchants here and
there, but in their incomplete state they were not of sufficient importance
to attract any considerable attention.
As a result of this brief survey of the evidence several conclusions seem
reasonable: (1) commerce, and travel for travel's sake, never died out
between the East and the West; (2) merchants had every opportunity of
knowing, and would have been unreasonably stupid if they had not known, the
elementary number systems of the peoples with whom they were trading, but
they would not have put this knowledge in permanent written form; (3)
wandering scholars would have known many and strange things about the
peoples they met, but they too were not, as a class, writers; (4) there is
every reason a priori for believing that the [.g]ob[=a]r numerals would
have been known to merchants, and probably to some of the wandering
scholars, long before the Arabs conquered northern Africa; (5) the wonder
is not that the Hindu-Arabic numerals were known about 1000 A.D., and that
they were the subject of an elaborate work in 1202 by Fibonacci, but rather
that more extended manuscript evidence of their appearance before that time
has not been found. That they were more or less known early in the Middle
Ages, certainly to many merchants of Christian Europe, and probably to
several scholars, but without the zero, is hardly to be doubted. The lack
of documentary evidence is not at all strange, in view of all of the
circumstances.
* * * * *
{91}