THE

HINDU-ARABIC NUMERALS

BY
DAVID EUGENE SMITH
AND
LOUIS CHARLES KARPINSKI

BOSTON AND LONDON
GINN AND COMPANY, PUBLISHERS
1911

COPYRIGHT, 1911, BY DAVID EUGENE SMITH
AND LOUIS CHARLES KARPINSKI
ALL RIGHTS RESERVED
811.7

The Athenæum Press
GINN AND COMPANY · PROPRIETORS
BOSTON · U.S.A.

[iii]

PREFACE

So familiar are we with the numerals that bear the misleading name of
Arabic, and so extensive is their use in Europe and the Americas, that it
is difficult for us to realize that their general acceptance in the
transactions of commerce is a matter of only the last four centuries, and
that they are unknown to a very large part of the human race to-day. It
seems strange that such a labor-saving device should have struggled for
nearly a thousand years after its system of place value was perfected
before it replaced such crude notations as the one that the Roman
conqueror made substantially universal in Europe. Such, however, is the
case, and there is probably no one who has not at least some slight
passing interest in the story of this struggle. To the mathematician and
the student of civilization the interest is generally a deep one; to the
teacher of the elements of knowledge the interest may be less marked, but
nevertheless it is real; and even the business man who makes daily use of
the curious symbols by which we express the numbers of commerce, cannot
fail to have some appreciation for the story of the rise and progress of
these tools of his trade.

This story has often been told in part, but it is a long time since
any effort has been made to bring together the fragmentary narrations and
to set forth the general problem of the origin and development of these
[iv]numerals. In this little work we have
attempted to state the history of these forms in small compass, to place
before the student materials for the investigation of the problems
involved, and to express as clearly as possible the results of the labors
of scholars who have studied the subject in different parts of the world.
We have had no theory to exploit, for the history of mathematics has seen
too much of this tendency already, but as far as possible we have weighed
the testimony and have set forth what seem to be the reasonable
conclusions from the evidence at hand.

To facilitate the work of students an index has been prepared which we
hope may be serviceable. In this the names of authors appear only when
some use has been made of their opinions or when their works are first
mentioned in full in a footnote.

If this work shall show more clearly the value of our number system,
and shall make the study of mathematics seem more real to the teacher and
student, and shall offer material for interesting some pupil more fully
in his work with numbers, the authors will feel that the considerable
labor involved in its preparation has not been in vain.

We desire to acknowledge our especial indebtedness to Professor
Alexander Ziwet for reading all the proof, as well as for the digest of a
Russian work, to Professor Clarence L. Meader for Sanskrit
transliterations, and to Mr. Steven T. Byington for Arabic
transliterations and the scheme of pronunciation of Oriental names, and
also our indebtedness to other scholars in Oriental learning for
information.

DAVID EUGENE SMITH

LOUIS CHARLES KARPINSKI

[v]

CONTENTS

[vi]

PRONUNCIATION OF ORIENTAL NAMES

(S) = in Sanskrit names and words; (A) = in Arabic names and
words.

b, d, f, g, h, j, l,
m, n, p, sh (A), t, th (A),
v, w, x, z, as in English.

a, (S) like u in but: thus pandit,
pronounced pundit. (A) like a in ask or in
man. ā, as in father.

c, (S) like ch in church (Italian c in
cento).

ḍ,
ṇ,
ṣ,
ṭ, (S)
d, n, sh, t, made with the tip of the tongue
turned up and back into the dome of the palate. ḍ, ṣ, ṭ, ẓ, (A) d, s,
t, z, made with the tongue spread so that the sounds are
produced largely against the side teeth. Europeans commonly pronounce
ḍ,
ṇ,
ṣ,
ṭ,
ẓ, both
(S) and (A), as simple d, n, sh (S) or s (A),
t, z. ḏ (A), like th in
this.

e, (S) as in they. (A) as in bed.

ġ, (A) a voiced consonant formed below the vocal cords;
its sound is compared by some to a g, by others to a guttural
r; in Arabic words adopted into English it is represented by
gh (e.g. ghoul), less often r (e.g.
razzia).

h preceded by b, c, t, ṭ, etc. does not
form a single sound with these letters, but is a more or less distinct
h sound following them; cf. the sounds in abhor, boathook,
etc., or, more accurately for (S), the “bhoys” etc. of Irish brogue.
h (A) retains its consonant sound at the end of a word. ḥ, (A) an unvoiced
consonant formed below the vocal cords; its sound is sometimes compared
to German hard ch, and may be represented by an h as strong
as possible. In Arabic words adopted into English it is represented by
h, e.g. in sahib, hakeem. ḥ (S) is final consonant h,
like final h (A).

i, as in pin. ī, as in pique.

k, as in kick.

kh, (A) the hard ch of Scotch loch, German
ach, especially of German as pronounced by the Swiss.

ṁ,
ṅ, (S)
like French final m or n, nasalizing the preceding
vowel.

ṇ, see
ḍ.
ñ, like ng in singing.

o, (S) as in so. (A) as in obey.

q, (A) like k (or c) in cook; further back
in the mouth than in kick.

r, (S) English r, smooth and untrilled. (A) stronger.
ṛ, (S) r
used as vowel, as in apron when pronounced aprn and not
apern; modern Hindus say ri, hence our amrita,
Krishna, for a-mṛta, Kṛṣṇa.

s, as in same. ṣ, see ḍ. ś, (S) English
sh (German sch).

ṭ, see
ḍ.

u, as in put. ū, as in rule.

y, as in you.

ẓ, see
ḍ.

‛, (A) a sound kindred to the spiritus lenis (that is,
to our ears, the mere distinct separation of a vowel from the preceding
sound, as at the beginning of a word in German) and to ḥ. The ‛ is
a very distinct sound in Arabic, but is more nearly represented by the
spiritus lenis than by any sound that we can produce without much special
training. That is, it should be treated as silent, but the sounds that
precede and follow it should not run together. In Arabic words adopted
into English it is treated as silent, e.g. in Arab, amber,
Caaba (‛Arab, ‛anbar,
ka‛abah).

(A) A final long vowel is shortened before al (‘l) or
ibn (whose i is then silent).

Accent: (S) as if Latin; in determining the place of the accent
ṁ and
ṅ count as
consonants, but h after another consonant does not. (A), on the
last syllable that contains a long vowel or a vowel followed by two
consonants, except that a final long vowel is not ordinarily accented; if
there is no long vowel nor two consecutive consonants, the accent falls
on the first syllable. The words al and ibn are never
accented.

[1]

THE HINDU-ARABIC NUMERALS

CHAPTER I

EARLY IDEAS OF THEIR ORIGIN

It has long been recognized that the common numerals used in daily
life are of comparatively recent origin. The number of systems of
notation employed before the Christian era was about the same as the
number of written languages, and in some cases a single language had
several systems. The Egyptians, for example, had three systems of
writing, with a numerical notation for each; the Greeks had two
well-defined sets of numerals, and the Roman symbols for number changed
more or less from century to century. Even to-day the number of methods
of expressing numerical concepts is much greater than one would believe
before making a study of the subject, for the idea that our common
numerals are universal is far from being correct. It will be well, then,
to think of the numerals that we still commonly call Arabic, as only one
of many systems in use just before the Christian era. As it then existed
the system was no better than many others, it was of late origin, it
contained no zero, it was cumbersome and little used, [2]and it had no
particular promise. Not until centuries later did the system have any
standing in the world of business and science; and had the place value
which now characterizes it, and which requires a zero, been worked out in
Greece, we might have been using Greek numerals to-day instead of the
ones with which we are familiar.

Of the first number forms that the world used this is not the place to
speak. Many of them are interesting, but none had much scientific value.
In Europe the invention of notation was generally assigned to the eastern
shores of the Mediterranean until the critical period of about a century
ago,—sometimes to the Hebrews, sometimes to the Egyptians, but more
often to the early trading Phœnicians.^[1]

The idea that our common numerals are Arabic in origin is not an old
one. The mediæval and Renaissance writers generally recognized them as
Indian, and many of them expressly stated that they were of Hindu
origin.^[2] [3]Others argued that
they were probably invented by the Chaldeans or the Jews because they
increased in value from right to left, an argument that would apply quite
as well to the Roman and Greek systems, or to any other. It was, indeed,
to the general idea of notation that many of these writers referred, as
is evident from the words of England’s earliest arithmetical
textbook-maker, Robert Recorde (c. 1542): “In that thinge all men do
agree, that the Chaldays, whiche fyrste inuented thys arte, did set these
figures as thei set all their letters. for they wryte backwarde as you
tearme it, and so doo they reade. And that may appeare in all Hebrewe,
Chaldaye and Arabike bookes … where as the Greekes, Latines, and all
nations of Europe, do wryte and reade from the lefte hand towarde the
ryghte.”^[3] Others, and [4]among them
such influential writers as Tartaglia^[4] in Italy and Köbel^[5] in Germany, asserted the Arabic origin of
the numerals, while still others left the matter undecided^[6] or simply dismissed them as “barbaric.”^[7] Of course the Arabs
themselves never laid claim to the invention, always recognizing their
indebtedness to the Hindus both for the numeral forms and for the
distinguishing feature of place value. Foremost among these writers was
the great master of the golden age of Bagdad, one of the first of the
Arab writers to collect the mathematical classics of both the East and
the West, preserving them and finally passing them on to awakening
Europe. This man was Moḥammed the Son of Moses, from
Khowārezm, or, more after the manner of the Arab, Moḥammed ibn
Mūsā al-Khowārazmī,^[8] a man of great [5]learning and one to whom
the world is much indebted for its present knowledge of algebra^[9] and of arithmetic. Of him
there will often be occasion to speak; and in the arithmetic which he
wrote, and of which Adelhard of Bath^[10] (c. 1130) may have made the translation
or paraphrase,^[11] he
stated distinctly that the numerals were due to the Hindus.^[12] This is as plainly
asserted by later Arab [6]writers, even to the present day.^[13] Indeed the phrase
‛ilm hindī, “Indian science,” is used by them for
arithmetic, as also the adjective hindī alone.^[14]

Probably the most striking testimony from Arabic sources is that given
by the Arabic traveler and scholar Mohammed ibn Aḥmed, Abū ‘l-Rīḥān al-Bīrūnī
(973-1048), who spent many years in Hindustan. He wrote a large work on
India,^[15] one on ancient
chronology,^[16] the “Book
of the Ciphers,” unfortunately lost, which treated doubtless of the Hindu
art of calculating, and was the author of numerous other works.
Al-Bīrūnī was a man of unusual attainments, being
versed in Arabic, Persian, Sanskrit, Hebrew, and Syriac, as well as in
astronomy, chronology, and mathematics. In his work on India he gives
detailed information concerning the language and [7]customs of the people of
that country, and states explicitly^[17] that the Hindus of his time did not use
the letters of their alphabet for numerical notation, as the Arabs did.
He also states that the numeral signs called aṅka^[18] had different shapes in various parts
of India, as was the case with the letters. In his Chronology of
Ancient Nations he gives the sum of a geometric progression and shows
how, in order to avoid any possibility of error, the number may be
expressed in three different systems: with Indian symbols, in sexagesimal
notation, and by an alphabet system which will be touched upon later. He
also speaks^[19] of “179,
876, 755, expressed in Indian ciphers,” thus again attributing these
forms to Hindu sources.

Preceding Al-Bīrūnī there was another Arabic writer
of the tenth century, Moṭahhar ibn Ṭāhir,^[20] author of the Book of the Creation
and of History, who gave as a curiosity, in Indian
(Nāgarī) symbols, a large number asserted by the people of
India to represent the duration of the world. Huart feels positive that
in Moṭahhar’s time
the present Arabic symbols had not yet come into use, and that the Indian
symbols, although known to scholars, were not current. Unless this were
the case, neither the author nor his readers would have found anything
extraordinary in the appearance of the number which he cites.

Mention should also be made of a widely-traveled student,
Al-Mas‛ūdī (885?-956), whose journeys carried him from
Bagdad to Persia, India, Ceylon, and even [8]across the China sea, and
at other times to Madagascar, Syria, and Palestine.^[21] He seems to have neglected no
accessible sources of information, examining also the history of the
Persians, the Hindus, and the Romans. Touching the period of the Caliphs
his work entitled Meadows of Gold furnishes a most entertaining
fund of information. He states^[22] that the wise men of India, assembled
by the king, composed the Sindhind. Further on^[23] he states, upon the authority of the
historian Moḥammed
ibn ‛Alī ‛Abdī, that by order of Al-Manṣūr many
works of science and astrology were translated into Arabic, notably the
Sindhind (Siddhānta). Concerning the meaning and
spelling of this name there is considerable diversity of opinion.
Colebrooke^[24] first
pointed out the connection between Siddhānta and
Sindhind. He ascribes to the word the meaning “the revolving
ages.”^[25] Similar
designations are collected by Sédillot,^[26] who inclined to the Greek origin of the
sciences commonly attributed to the Hindus.^[27] Casiri,^[28] citing the Tārīkh
al-ḥokamā or Chronicles of the Learned,^[29] refers to the work [9]as the
Sindum-Indum with the meaning “perpetuum æternumque.” The
reference^[30] in this
ancient Arabic work to Al-Khowārazmī is worthy of note.

This Sindhind is the book, says Mas‛ūdī,^[31] which gives all that the
Hindus know of the spheres, the stars, arithmetic,^[32] and the other branches of science. He
mentions also Al-Khowārazmī and Ḥabash^[33] as translators of the tables of the
Sindhind. Al-Bīrūnī^[34] refers to two other translations from a
work furnished by a Hindu who came to Bagdad as a member of the political
mission which Sindh sent to the caliph Al-Manṣūr, in the year of the
Hejira 154 (A.D. 771).

The oldest work, in any sense complete, on the history of Arabic
literature and history is the Kitāb al-Fihrist, written in
the year 987 A.D., by Ibn Abī
Ya‛qūb al-Nadīm. It is of fundamental importance for
the history of Arabic culture. Of the ten chief divisions of the work,
the seventh demands attention in this discussion for the reason that its
second subdivision treats of mathematicians and astronomers.^[35]

[10]

The first of the Arabic writers mentioned is Al-Kindī (800-870
A.D.), who wrote five books on arithmetic and
four books on the use of the Indian method of reckoning. Sened ibn
‛Alī, the Jew, who was converted to Islam under the caliph
Al-Māmūn, is also given as the author of a work on the Hindu
method of reckoning. Nevertheless, there is a possibility^[36] that some of the works ascribed to
Sened ibn ‛Alī are really works of
Al-Khowārazmī, whose name immediately precedes his. However,
it is to be noted in this connection that Casiri^[37] also mentions the same writer as the
author of a most celebrated work on arithmetic.

To Al-Ṣūfī, who died in 986 A.D., is also credited a large work on the same
subject, and similar treatises by other writers are mentioned. We are
therefore forced to the conclusion that the Arabs from the early ninth
century on fully recognized the Hindu origin of the new numerals.

Leonard of Pisa, of whom we shall speak at length in the chapter on
the Introduction of the Numerals into Europe, wrote his Liber
Abbaci^[38] in 1202. In
this work he refers frequently to the nine Indian figures,^[39] thus showing again the
general consensus of opinion in the Middle Ages that the numerals were of
Hindu origin.

Some interest also attaches to the oldest documents on arithmetic in
our own language. One of the earliest [11]treatises on algorism is
a commentary^[40] on a set
of verses called the Carmen de Algorismo, written by Alexander de
Villa Dei (Alexandra de Ville-Dieu), a Minorite monk of about 1240 A.D. The text of the first few lines is as
follows:

“This boke is called the boke of algorim or augrym after lewder use.
And this boke tretys of the Craft of Nombryng, the quych crafte is called
also Algorym. Ther was a kyng of Inde the quich heyth Algor & he made
this craft…. Algorisms, in the quych we use teen figurys of Inde.”

[12]

CHAPTER II

EARLY HINDU FORMS WITH NO PLACE VALUE

While it is generally conceded that the scientific development of
astronomy among the Hindus towards the beginning of the Christian era
rested upon Greek^[42] or
Chinese^[43] sources, yet
their ancient literature testifies to a high state of civilization, and
to a considerable advance in sciences, in philosophy, and along literary
lines, long before the golden age of Greece. From the earliest times even
up to the present day the Hindu has been wont to put his thought into
rhythmic form. The first of this poetry—it well deserves this name,
being also worthy from a metaphysical point of view^[44]—consists of the Vedas, hymns of
praise and poems of worship, collected during the Vedic period which
dates from approximately 2000 B.C. to 1400
B.C.^[45] Following this work, or possibly
contemporary with it, is the Brahmanic literature, which is partly
ritualistic (the Brāhmaṇas), and partly
philosophical (the Upanishads). Our especial interest is [13]in the
Sūtras, versified abridgments of the ritual and of ceremonial
rules, which contain considerable geometric material used in connection
with altar construction, and also numerous examples of rational numbers
the sum of whose squares is also a square, i.e. “Pythagorean numbers,”
although this was long before Pythagoras lived. Whitney^[46] places the whole of the Veda
literature, including the Vedas, the Brāhmaṇas, and the Sūtras,
between 1500 B.C. and 800 B.C., thus agreeing with Bürk^[47] who holds that the knowledge of the
Pythagorean theorem revealed in the Sūtras goes back to the eighth
century B.C.

The importance of the Sūtras as showing an independent origin of
Hindu geometry, contrary to the opinion long held by Cantor^[48] of a Greek origin, has
been repeatedly emphasized in recent literature,^[49] especially since the appearance of the
important work of Von Schroeder.^[50] Further fundamental mathematical
notions such as the conception of irrationals and the use of gnomons, as
well as the philosophical doctrine of the transmigration of
souls,—all of these having long been attributed to the
Greeks,—are shown in these works to be native to India. Although
this discussion does not bear directly upon the [14]origin of our numerals,
yet it is highly pertinent as showing the aptitude of the Hindu for
mathematical and mental work, a fact further attested by the independent
development of the drama and of epic and lyric poetry.

It should be stated definitely at the outset, however, that we are not
at all sure that the most ancient forms of the numerals commonly known as
Arabic had their origin in India. As will presently be seen, their forms
may have been suggested by those used in Egypt, or in Eastern Persia, or
in China, or on the plains of Mesopotamia. We are quite in the dark as to
these early steps; but as to their development in India, the approximate
period of the rise of their essential feature of place value, their
introduction into the Arab civilization, and their spread to the West, we
have more or less definite information. When, therefore, we consider the
rise of the numerals in the land of the Sindhu,^[51] it must be understood that it is only
the large movement that is meant, and that there must further be
considered the numerous possible sources outside of India itself and long
anterior to the first prominent appearance of the number symbols.

No one attempts to examine any detail in the history of ancient India
without being struck with the great dearth of reliable material.^[52] So little sympathy have
the people with any save those of their own caste that a general
literature is wholly lacking, and it is only in the observations of
strangers that any all-round view of scientific progress is to be found.
There is evidence that primary schools [15]existed in earliest
times, and of the seventy-two recognized sciences writing and arithmetic
were the most prized.^[53]
In the Vedic period, say from 2000 to 1400 B.C., there was the same attention to astronomy that
was found in the earlier civilizations of Babylon, China, and Egypt, a
fact attested by the Vedas themselves.^[54] Such advance in science presupposes a
fair knowledge of calculation, but of the manner of calculating we are
quite ignorant and probably always shall be. One of the Buddhist sacred
books, the Lalitavistara, relates that when the
Bōdhisattva^[55] was
of age to marry, the father of Gopa, his intended bride, demanded an
examination of the five hundred suitors, the subjects including
arithmetic, writing, the lute, and archery. Having vanquished his rivals
in all else, he is matched against Arjuna the great arithmetician and is
asked to express numbers greater than 100 kotis.^[56] In reply he gave a scheme of number
names as high as 10⁵³, adding that he could proceed as far as
10⁴²¹,^[57] all
of which suggests the system of Archimedes and the unsettled question of
the indebtedness of the West to the East in the realm of ancient
mathematics.^[58] Sir Edwin
Arnold, [16]in The Light of Asia, does not
mention this part of the contest, but he speaks of Buddha’s training at
the hands of the learned Viṣvamitra:

[17]

Thereupon Viṣvamitra Ācārya^[61] expresses his approval of the task, and
asks to hear the “measure of the line” as far as yōjana, the
longest measure bearing name. This given, Buddha adds:

It is needless to say that this is far from being history. And yet it
puts in charming rhythm only what the ancient Lalitavistara
relates of the number-series of the Buddha’s time. While it extends
beyond all reason, nevertheless it reveals a condition that would have
been impossible unless arithmetic had attained a considerable degree of
advancement.

To this pre-Christian period belong also the Vedāṅgas, or “limbs for
supporting the Veda,” part of that great branch of Hindu literature known
as Smṛiti
(recollection), that which was to be handed down by tradition. Of these
the sixth is known as Jyotiṣa (astronomy), a short treatise
of only thirty-six verses, written not earlier than 300 B.C., and affording us some knowledge of the extent
of number work in that period.^[62] The Hindus [18]also speak of eighteen
ancient Siddhāntas or astronomical works, which, though mostly
lost, confirm this evidence.^[63]

As to authentic histories, however, there exist in India none relating
to the period before the Mohammedan era (622 A.D.). About all that we know of the earlier
civilization is what we glean from the two great epics, the
Mahābhārata^[64]
and the Rāmāyana, from coins, and from a few inscriptions.^[65]

It is with this unsatisfactory material, then, that we have to deal in
searching for the early history of the Hindu-Arabic numerals, and the
fact that many unsolved problems exist and will continue to exist is no
longer strange when we consider the conditions. It is rather surprising
that so much has been discovered within a century, than that we are so
uncertain as to origins and dates and the early spread of the system. The
probability being that writing was not introduced into India before the
close of the fourth century B.C., and
literature existing only in spoken form prior to that period,^[66] the number work was
doubtless that of all primitive peoples, palpable, merely a matter of
placing sticks or cowries or pebbles on the ground, of marking a
sand-covered board, or of cutting notches or tying cords as is still done
in parts of Southern India to-day.^[67]

[19]

The early Hindu numerals^[68] may be classified into three great
groups, (1) the Kharoṣṭhī, (2) the
Brāhmī, and (3) the word and letter forms; and these will be
considered in order.

The Kharoṣṭhī numerals are found
in inscriptions formerly known as Bactrian, Indo-Bactrian, and Aryan, and
appearing in ancient Gandhāra, now eastern Afghanistan and northern
Punjab. The alphabet of the language is found in inscriptions dating from
the fourth century B.C. to the third century
A.D., and from the fact that the words are
written from right to left it is assumed to be of Semitic origin. No
numerals, however, have been found in the earliest of these inscriptions,
number-names probably having been written out in words as was the custom
with many ancient peoples. Not until the time of the powerful King
Aśoka, in the third century B.C., do
numerals appear in any inscriptions thus far discovered; and then only in
the primitive form of marks, quite as they would be found in Egypt,
Greece, Rome, or in [20]various other parts of the world. These
Aśoka^[69]
inscriptions, some thirty in all, are found in widely separated parts of
India, often on columns, and are in the various vernaculars that were
familiar to the people. Two are in the Kharoṣṭhī characters, and
the rest in some form of Brāhmī. In the Kharoṣṭhī inscriptions only
four numerals have been found, and these are merely vertical marks for
one, two, four, and five, thus:

In the so-called Śaka inscriptions, possibly of the first
century B.C., more numerals are found, and in
more highly developed form, the right-to-left system appearing, together
with evidences of three different scales of counting,—four, ten,
and twenty. The numerals of this period are as follows:

There are several noteworthy points to be observed in studying this
system. In the first place, it is probably not as early as that shown in
the Nānā Ghāt forms hereafter given, although the
inscriptions themselves at Nānā Ghāt are later than
those of the Aśoka period. The [21]four is to this system
what the X was to the Roman, probably a canceling of three marks as a
workman does to-day for five, or a laying of one stick across three
others. The ten has never been satisfactorily explained. It is similar to
the A of the Kharoṣṭhī alphabet, but we
have no knowledge as to why it was chosen. The twenty is evidently a
ligature of two tens, and this in turn suggested a kind of radix, so that
ninety was probably written in a way reminding one of the
quatre-vingt-dix of the French. The hundred is unexplained, although it
resembles the letter ta or tra of the Brāhmī
alphabet with 1 before (to the right of) it. The two hundred is only a
variant of the symbol for hundred, with two vertical marks.^[70]

This system has many points of similarity with the Nabatean numerals^[71] in use in the first
centuries of the Christian era. The cross is here used for four, and the
Kharoṣṭhī form is employed
for twenty. In addition to this there is a trace of an analogous use of a
scale of twenty. While the symbol for 100 is quite different, the method
of forming the other hundreds is the same. The correspondence seems to be
too marked to be wholly accidental.

It is not in the Kharoṣṭhī numerals,
therefore, that we can hope to find the origin of those used by us, and
we turn to the second of the Indian types, the Brāhmī
characters. The alphabet attributed to Brahmā is the oldest of the
several known in India, and was used from the earliest historic times.
There are various theories of its origin, [22]none of which has as yet
any wide acceptance,^[72]
although the problem offers hope of solution in due time. The numerals
are not as old as the alphabet, or at least they have not as yet been
found in inscriptions earlier than those in which the edicts of
Aśoka appear, some of these having been incised in
Brāhmī as well as Kharoṣṭhī. As already
stated, the older writers probably wrote the numbers in words, as seems
to have been the case in the earliest Pali writings of Ceylon.^[73]

The following numerals are, as far as known, the only ones to appear
in the Aśoka edicts:^[74]

These fragments from the third century B.C.,
crude and unsatisfactory as they are, are the undoubted early forms from
which our present system developed. They next appear in the second
century B.C. in some inscriptions in the cave
on the top of the Nānā Ghāt hill, about seventy-five
miles from Poona in central India. These inscriptions may be memorials of
the early Andhra dynasty of southern India, but their chief interest lies
in the numerals which they contain.

The cave was made as a resting-place for travelers ascending the hill,
which lies on the road from Kalyāna to Junar. It seems to have been
cut out by a descendant [23]of King Śātavāhana,^[75] for inside the wall
opposite the entrance are representations of the members of his family,
much defaced, but with the names still legible. It would seem that the
excavation was made by order of a king named Vedisiri, and “the
inscription contains a list of gifts made on the occasion of the
performance of several yagnas or religious sacrifices,” and
numerals are to be seen in no less than thirty places.^[76]

There is considerable dispute as to what numerals are really found in
these inscriptions, owing to the difficulty of deciphering them; but the
following, which have been copied from a rubbing, are probably number
forms:^[77]

The inscription itself, so important as containing the earliest
considerable Hindu numeral system connected with our own, is of
sufficient interest to warrant reproducing part of it in facsimile, as is
done on page 24.

[24]

The next very noteworthy evidence of the numerals, and this quite
complete as will be seen, is found in certain other cave inscriptions
dating back to the first or second century A.D.
In these, the Nasik^[78]
cave inscriptions, the forms are as follows:

From this time on, until the decimal system finally adopted the first
nine characters and replaced the rest of the Brāhmī notation
by adding the zero, the progress of these forms is well marked. It is
therefore well to present synoptically the best-known specimens that have
come down to us, and this is done in the table on page 25.^[79]

[25]

Table showing the Progress of Number Forms in India

Numerals
Aśoka[80]
Śaka[81]
Aśoka[82]
Nāgarī[83]
Nasik[84]
Kṣatrapa[85]
Kuṣana[86]
Gupta[87]
Valhabī[88]
Nepal[89]
Kaliṅga[90]
Vākāṭaka[91]

[Most of these numerals are given by Bühler, loc. cit., Tafel IX.]

[26]

With respect to these numerals it should first be noted that no zero
appears in the table, and as a matter of fact none existed in any of the
cases cited. It was therefore impossible to have any place value, and the
numbers like twenty, thirty, and other multiples of ten, one hundred, and
so on, required separate symbols except where they were written out in
words. The ancient Hindus had no less than twenty of these symbols,^[92] a number that was
afterward greatly increased. The following are examples of their method
of indicating certain numbers between one hundred and one thousand:

[27]

To these may be added the following numerals below one hundred,
similar to those in the table:

We have thus far spoken of the Kharoṣṭhī and
Brāhmī numerals, and it remains to mention the third type,
the word and letter forms. These are, however, so closely connected with
the perfecting of the system by the invention of the zero that they are
more appropriately considered in the next chapter, particularly as they
have little relation to the problem of the origin of the forms known as
the Arabic.

Having now examined types of the early forms it is appropriate to turn
our attention to the question of their origin. As to the first three
there is no question. The or is
simply one stroke, or one stick laid down by the computer. The or represents two
strokes or two sticks, and so for the and .
From some primitive came
the two of Egypt, of Rome, of early Greece, and of various other
civilizations. It appears in the three Egyptian numeral systems in the
following forms:

The last of these is merely a cursive form as in the Arabic , which becomes our 2 if tipped
through a right angle. From some primitive came the Chinese [28]symbol, which is
practically identical with the symbols found commonly in India from 150
B.C. to 700 A.D. In
the cursive form it becomes , and this was frequently used for two
in Germany until the 18th century. It finally went into the modern form
2, and the in
the same way became our 3.

There is, however, considerable ground for interesting speculation
with respect to these first three numerals. The earliest Hindu forms were
perpendicular. In the Nānā Ghāt inscriptions they are
vertical. But long before either the Aśoka or the Nānā
Ghāt inscriptions the Chinese were using the horizontal forms for
the first three numerals, but a vertical arrangement for four.^[101] Now where did China
get these forms? Surely not from India, for she had them, as her
monuments and literature^[102] show, long before the Hindus knew
them. The tradition is that China brought her civilization around the
north of Tibet, from Mongolia, the primitive habitat being Mesopotamia,
or possibly the oases of Turkestan. Now what numerals did Mesopotamia
use? The Babylonian system, simple in its general principles but very
complicated in many of its details, is now well known.^[103] In particular, one, two, and three
were represented by vertical arrow-heads. Why, then, did the Chinese
write [29]theirs horizontally? The problem now takes a
new interest when we find that these Babylonian forms were not the
primitive ones of this region, but that the early Sumerian forms were
horizontal.^[104]

What interpretation shall be given to these facts? Shall we say that
it was mere accident that one people wrote “one” vertically and that
another wrote it horizontally? This may be the case; but it may also be
the case that the tribal migrations that ended in the Mongol invasion of
China started from the Euphrates while yet the Sumerian civilization was
prominent, or from some common source in Turkestan, and that they carried
to the East the primitive numerals of their ancient home, the first
three, these being all that the people as a whole knew or needed. It is
equally possible that these three horizontal forms represent primitive
stick-laying, the most natural position of a stick placed in front of a
calculator being the horizontal one. When, however, the cuneiform writing
developed more fully, the vertical form may have been proved the easier
to make, so that by the time the migrations to the West began these were
in use, and from them came the upright forms of Egypt, Greece, Rome, and
other Mediterranean lands, and those of Aśoka’s time in India.
After Aśoka, and perhaps among the merchants of earlier centuries,
the horizontal forms may have come down into India from China, thus
giving those of the Nānā Ghāt cave and of later
inscriptions. This is in the realm of speculation, but it is not
improbable that further epigraphical studies may confirm the
hypothesis.

[30]

As to the numerals above three there have been very many conjectures.
The figure one of the Demotic looks like the one of the Sanskrit, the two
(reversed) like that of the Arabic, the four has some resemblance to that
in the Nasik caves, the five (reversed) to that on the Kṣatrapa coins, the nine
to that of the Kuṣana
inscriptions, and other points of similarity have been imagined. Some
have traced resemblance between the Hieratic five and seven and those of
the Indian inscriptions. There have not, therefore, been wanting those
who asserted an Egyptian origin for these numerals.^[105] There has already been mentioned the
fact that the Kharoṣṭhī numerals were
formerly known as Bactrian, Indo-Bactrian, and Aryan. Cunningham^[106] was the first to
suggest that these numerals were derived from the alphabet of the
Bactrian civilization of Eastern Persia, perhaps a thousand years before
our era, and in this he was supported by the scholarly work of Sir E.
Clive Bayley,^[107] who in
turn was followed by Canon Taylor.^[108] The resemblance has not proved
convincing, however, and Bayley’s drawings [31]have been criticized as
being affected by his theory. The following is part of the hypothesis:^[109]

Numeral	Hindu	Bactrian	Sanskrit
4		= ch	chatur, Lat. quattuor
5		= p	pancha, Gk. πέντε
6		= s	ṣaṣ
7		= ṣ	sapta
(the s and ṣ are interchanged as occasionally in N. W. India)

Bühler^[110] rejects
this hypothesis, stating that in four cases (four, six, seven, and ten)
the facts are absolutely against it.

While the relation to ancient Bactrian forms has been generally
doubted, it is agreed that most of the numerals resemble
Brāhmī letters, and we would naturally expect them to be
initials.^[111] But,
knowing the ancient pronunciation of most of the number names,^[112] we find this not to be
the case. We next fall back upon the hypothesis [32]that they represent the
order of letters^[113] in
the ancient alphabet. From what we know of this order, however, there
seems also no basis for this assumption. We have, therefore, to confess
that we are not certain that the numerals were alphabetic at all, and if
they were alphabetic we have no evidence at present as to the basis of
selection. The later forms may possibly have been alphabetical
expressions of certain syllables called akṣaras, which possessed in Sanskrit
fixed numerical values,^[114] but this is equally uncertain with
the rest. Bayley also thought^[115] that some of the forms were
Phœnician, as notably the use of a circle for twenty, but the
resemblance is in general too remote to be convincing.

There is also some slight possibility that Chinese influence is to be
seen in certain of the early forms of Hindu numerals.^[116]

[33]

More absurd is the hypothesis of a Greek origin, supposedly supported
by derivation of the current symbols from the first nine letters of the
Greek alphabet.^[117] This
difficult feat is accomplished by twisting some of the letters, cutting
off, adding on, and effecting other changes to make the letters fit the
theory. This peculiar theory was first set up by Dasypodius^[118] (Conrad Rauhfuss), and
was later elaborated by Huet.^[119]

[34]

A bizarre derivation based upon early Arabic (c. 1040 A.D.) sources is given by Kircher in his work^[120] on number mysticism.
He quotes from Abenragel,^[121] giving the Arabic and a Latin
translation^[122] and
stating that the ordinary Arabic forms are derived from sectors of a
circle, .

Out of all these conflicting theories, and from all the resemblances
seen or imagined between the numerals of the West and those of the East,
what conclusions are we prepared to draw as the evidence now stands?
Probably none that is satisfactory. Indeed, upon the evidence at [35]hand we
might properly feel that everything points to the numerals as being
substantially indigenous to India. And why should this not be the case?
If the king Srong-tsan-Gampo (639 A.D.), the
founder of Lhāsa,^[123] could have set about to devise a new
alphabet for Tibet, and if the Siamese, and the Singhalese, and the
Burmese, and other peoples in the East, could have created alphabets of
their own, why should not the numerals also have been fashioned by some
temple school, or some king, or some merchant guild? By way of
illustration, there are shown in the table on page 36 certain systems of
the East, and while a few resemblances are evident, it is also evident
that the creators of each system endeavored to find original forms that
should not be found in other systems. This, then, would seem to be a fair
interpretation of the evidence. A human mind cannot readily create simple
forms that are absolutely new; what it fashions will naturally resemble
what other minds have fashioned, or what it has known through hearsay or
through sight. A circle is one of the world’s common stock of figures,
and that it should mean twenty in Phœnicia and in India is hardly
more surprising than that it signified ten at one time in Babylon.^[124] It is therefore quite
probable that an extraneous origin cannot be found for the very
sufficient reason that none exists.

Of absolute nonsense about the origin of the symbols which we use much
has been written. Conjectures, [36]however, without any historical evidence for
support, have no place in a serious discussion of the gradual evolution
of the present numeral forms.^[125]

Table of Certain Eastern Systems

Siam
Burma[126]
Malabar[127]
Tibet[128]
Ceylon[129]
Malayalam[129]

[37]

We may summarize this chapter by saying that no one knows what
suggested certain of the early numeral forms used in India. The origin of
some is evident, but the origin of others will probably never be known.
There is no reason why they should not have been invented by some priest
or teacher or guild, by the order of some king, or as part of the
mysticism of some temple. Whatever the origin, they were no better than
scores of other ancient systems and no better than the present Chinese
system when written without the zero, and there would never have been any
chance of their triumphal progress westward had it not been for this
relatively late symbol. There could hardly be demanded a stronger proof
of the Hindu origin of the character for zero than this, and to it
further reference will be made in Chapter IV.

[38]

CHAPTER III

LATER HINDU FORMS, WITH A PLACE VALUE

Before speaking of the perfected Hindu numerals with the zero and the
place value, it is necessary to consider the third system mentioned on
page 19,—the word and letter forms. The use of words with place
value began at least as early as the 6th century of the Christian era. In
many of the manuals of astronomy and mathematics, and often in other
works in mentioning dates, numbers are represented by the names of
certain objects or ideas. For example, zero is represented by “the void”
(śūnya), or “heaven-space” (ambara
ākāśa); one by “stick” (rupa), “moon”
(indu śaśin), “earth” (bhū), “beginning”
(ādi), “Brahma,” or, in general, by anything markedly
unique; two by “the twins” (yama), “hands” (kara), “eyes”
(nayana), etc.; four by “oceans,” five by “senses” (viṣaya) or “arrows” (the
five arrows of Kāmadēva); six by “seasons” or “flavors”;
seven by “mountain” (aga), and so on.^[130] These names, accommodating themselves
to the verse in which scientific works were written, had the additional
advantage of not admitting, as did the figures, easy alteration, since
any change would tend to disturb the meter.

[39]

As an example of this system, the date “Śaka Saṃvat, 867″ (A.D. 945 or 946), is given by “giri-raṣa-vasu,”
meaning “the mountains” (seven), “the flavors” (six), and the gods
“Vasu” of which there were eight. In reading the date these are
read from right to left.^[131] The period of invention of this
system is uncertain. The first trace seems to be in the
Śrautasūtra of Kātyāyana and Lāṭyāyana.^[132] It was certainly known
to Varāha-Mihira (d. 587),^[133] for he used it in the Bṛhat-Saṃhitā.^[134] It has also been
asserted^[135] that Āryabhaṭa (c. 500
A.D.) was familiar with this system, but there
is nothing to prove the statement.^[136] The earliest epigraphical examples of
the system are found in the Bayang (Cambodia) inscriptions of 604 and 624
A.D.^[137]

Mention should also be made, in this connection, of a curious system
of alphabetic numerals that sprang up in southern India. In this we have
the numerals represented by the letters as given in the following
table:

[40]

By this plan a numeral might be represented by any one of several
letters, as shown in the preceding table, and thus it could the more
easily be formed into a word for mnemonic purposes. For example, the
word

has the value 1,565,132, reading from right to left.^[138] This, the oldest specimen (1184 A.D.) known of this notation, is given in a
commentary on the Rigveda, representing the number of days that had
elapsed from the beginning of the Kaliyuga. Burnell^[139] states that this system is even yet
in use for remembering rules to calculate horoscopes, and for
astronomical tables.

A second system of this kind is still used in the pagination of
manuscripts in Ceylon, Siam, and Burma, having also had its rise in
southern India. In this the thirty-four consonants when followed by
a (as ka … la) designate the numbers 1-34; by
ā (as kā … lā), those from 35 to
68; by i (ki … li), those from 69 to 102,
inclusive; and so on.^[140]

As already stated, however, the Hindu system as thus far described was
no improvement upon many others of the ancients, such as those used by
the Greeks and the Hebrews. Having no zero, it was impracticable to
designate the tens, hundreds, and other units of higher order by the same
symbols used for the units from one to nine. In other words, there was no
possibility of place value without some further improvement. So the
Nānā Ghāt [41]symbols required the writing of “thousand
seven twenty-four” about like T 7, tw, 4 in modern symbols, instead of
7024, in which the seven of the thousands, the two of the tens (concealed
in the word twenty, being originally “twain of tens,” the -ty
signifying ten), and the four of the units are given as spoken and the
order of the unit (tens, hundreds, etc.) is given by the place. To
complete the system only the zero was needed; but it was probably eight
centuries after the Nānā Ghāt inscriptions were cut,
before this important symbol appeared; and not until a considerably later
period did it become well known. Who it was to whom the invention is due,
or where he lived, or even in what century, will probably always remain a
mystery.^[141] It is
possible that one of the forms of ancient abacus suggested to some Hindu
astronomer or mathematician the use of a symbol to stand for the vacant
line when the counters were removed. It is well established that in
different parts of India the names of the higher powers took different
forms, even the order being interchanged.^[142] Nevertheless, as the significance of
the name of the unit was given by the order in reading, these variations
did not lead to error. Indeed the variation itself may have necessitated
the introduction of a word to signify a vacant place or lacking unit,
with the ultimate introduction of a zero symbol for this word.

To enable us to appreciate the force of this argument a large number,
8,443,682,155, may be considered as the Hindus wrote and read it, and
then, by way of contrast, as the Greeks and Arabs would have read it.

[42]

Modern American reading, 8 billion, 443 million, 682 thousand,
155.

Hindu, 8 padmas, 4 vyarbudas, 4 kōṭis, 3 prayutas, 6 lakṣas, 8 ayutas, 2 sahasra,
1 śata, 5 daśan, 5.

Arabic and early German, eight thousand thousand thousand and
four hundred thousand thousand and forty-three thousand thousand, and six
hundred thousand and eighty-two thousand and one hundred fifty-five (or
five and fifty).

Greek, eighty-four myriads of myriads and four thousand three
hundred sixty-eight myriads and two thousand and one hundred
fifty-five.

As Woepcke^[143]
pointed out, the reading of numbers of this kind shows that the notation
adopted by the Hindus tended to bring out the place idea. No other
language than the Sanskrit has made such consistent application, in
numeration, of the decimal system of numbers. The introduction of myriads
as in the Greek, and thousands as in Arabic and in modern numeration, is
really a step away from a decimal scheme. So in the numbers below one
hundred, in English, eleven and twelve are out of harmony with the rest
of the -teens, while the naming of all the numbers between ten and twenty
is not analogous to the naming of the numbers above twenty. To conform to
our written system we should have ten-one, ten-two, ten-three, and so on,
as we have twenty-one, twenty-two, and the like. The Sanskrit is
consistent, the units, however, preceding the tens and hundreds. Nor did
any other ancient people carry the numeration as far as did the Hindus.^[144]

[43]

When the aṅkapalli,^[145] the decimal-place system of writing
numbers, was perfected, the tenth symbol was called the
śūnyabindu, generally shortened to
śūnya (the void). Brockhaus^[146] has well said that if there was any
invention for which the Hindus, by all their philosophy and religion,
were well fitted, it was the invention of a symbol for zero. This making
of nothingness the crux of a tremendous achievement was a step in
complete harmony with the genius of the Hindu.

It is generally thought that this śūnya as a symbol
was not used before about 500 A.D., although
some writers have placed it earlier.^[147] Since Āryabhaṭa gives our common method
of extracting roots, it would seem that he may have known a decimal
notation,^[148] although
he did not use the characters from which our numerals are derived.^[149] Moreover, he
frequently speaks of the [44]void.^[150] If he refers to a symbol this would
put the zero as far back as 500 A.D., but of
course he may have referred merely to the concept of nothingness.

A little later, but also in the sixth century, Varāha-Mihira^[151] wrote a work entitled
Bṛhat
Saṃhitā^[152] in which he frequently uses
śūnya in speaking of numerals, so that it has been
thought that he was referring to a definite symbol. This, of course,
would add to the probability that Āryabhaṭa was doing the same.

It should also be mentioned as a matter of interest, and somewhat
related to the question at issue, that Varāha-Mihira used the
word-system with place value^[153] as explained above.

The first kind of alphabetic numerals and also the word-system (in
both of which the place value is used) are plays upon, or variations of,
position arithmetic, which would be most likely to occur in the country
of its origin.^[154]

At the opening of the next century (c. 620 A.D.) Bāṇa^[155] wrote of Subandhus’s
Vāsavadattā as a celebrated work, [45]and mentioned
that the stars dotting the sky are here compared with zeros, these being
points as in the modern Arabic system. On the other hand, a strong
argument against any Hindu knowledge of the symbol zero at this time is
the fact that about 700 A.D. the Arabs overran
the province of Sind and thus had an opportunity of knowing the common
methods used there for writing numbers. And yet, when they received the
complete system in 776 they looked upon it as something new.^[156] Such evidence is not
conclusive, but it tends to show that the complete system was probably
not in common use in India at the beginning of the eighth century. On the
other hand, we must bear in mind the fact that a traveler in Germany in
the year 1700 would probably have heard or seen nothing of decimal
fractions, although these were perfected a century before that date. The
élite of the mathematicians may have known the zero even in Āryabhaṭa‘s time,
while the merchants and the common people may not have grasped the
significance of the novelty until a long time after. On the whole, the
evidence seems to point to the west coast of India as the region where
the complete system was first seen.^[157] As mentioned above, traces of the
numeral words with place value, which do not, however, absolutely require
a decimal place-system of symbols, are found very early in Cambodia, as
well as in India.

Concerning the earliest epigraphical instances of the use of the nine
symbols, plus the zero, with place value, there [46]is some question.
Colebrooke^[158] in 1807
warned against the possibility of forgery in many of the ancient
copper-plate land grants. On this account Fleet, in the Indian
Antiquary,^[159]
discusses at length this phase of the work of the epigraphists in India,
holding that many of these forgeries were made about the end of the
eleventh century. Colebrooke^[160] takes a more rational view of these
forgeries than does Kaye, who seems to hold that they tend to invalidate
the whole Indian hypothesis. “But even where that may be suspected, the
historical uses of a monument fabricated so much nearer to the times to
which it assumes to belong, will not be entirely superseded. The
necessity of rendering the forged grant credible would compel a
fabricator to adhere to history, and conform to established notions: and
the tradition, which prevailed in his time, and by which he must be
guided, would probably be so much nearer to the truth, as it was less
remote from the period which it concerned.”^[161] Bühler^[162] gives the copper-plate Gurjara
inscription of Cedi-saṃvat 346 (595 A.D.) as the oldest epigraphical use of the
numerals^[163] “in which
the symbols correspond to the alphabet numerals of the period and the
place.” Vincent A. Smith^[164] quotes a stone inscription of 815
A.D., dated Saṃvat 872. So F. Kielhorn in the
Epigraphia Indica^[165] gives a Pathari pillar inscription of
Parabala, dated Vikrama-saṃvat 917, which
corresponds to 861 A.D., [47]and refers also to
another copper-plate inscription dated Vikrama-saṃvat 813 (756 A.D.). The inscription quoted by V. A. Smith above is
that given by D. R. Bhandarkar,^[166] and another is given by the same
writer as of date Saka-saṃvat 715 (798 A.D.), being incised on a pilaster. Kielhorn^[167] also gives two
copper-plate inscriptions of the time of Mahendrapala of Kanauj, Valhabī-saṃvat
574 (893 A.D.) and Vikrama-saṃvat 956 (899 A.D.). That there should be any inscriptions of date
as early even as 750 A.D., would tend to show
that the system was at least a century older. As will be shown in the
further development, it was more than two centuries after the
introduction of the numerals into Europe that they appeared there upon
coins and inscriptions. While Thibaut^[168] does not consider it necessary to
quote any specific instances of the use of the numerals, he states that
traces are found from 590 A.D. on. “That the
system now in use by all civilized nations is of Hindu origin cannot be
doubted; no other nation has any claim upon its discovery, especially
since the references to the origin of the system which are found in the
nations of western Asia point unanimously towards India.”^[169]

The testimony and opinions of men like Bühler, Kielhorn, V. A. Smith,
Bhandarkar, and Thibaut are entitled to the most serious consideration.
As authorities on ancient Indian epigraphy no others rank higher. Their
work is accepted by Indian scholars the world over, and their united
judgment as to the rise of the system with a place value—that it
took place in India as early as the [48]sixth century A.D.—must stand unless new evidence of great
weight can be submitted to the contrary.

Many early writers remarked upon the diversity of Indian numeral
forms. Al-Bīrūnī was probably the first; noteworthy is
also Johannes Hispalensis,^[170] who gives the variant forms for seven
and four. We insert on p. 49 a table of numerals used with place value.
While the chief authority for this is Bühler,^[171] several specimens are given which are
not found in his work and which are of unusual interest.

The Śāradā forms given in the table use the circle
as a symbol for 1 and the dot for zero. They are taken from the paging
and text of The Kashmirian Atharva-Veda^[172], of which the manuscript used is
certainly four hundred years old. Similar forms are found in a manuscript
belonging to the University of Tübingen. Two other series presented are
from Tibetan books in the library of one of the authors.

For purposes of comparison the modern Sanskrit and Arabic numeral
forms are added.

Sanskrit,
Arabic,

[49]

Numerals used with Place Value

a [173]
b [174]
c [175]
d [176]
e [177]
f [178]
g [179]
h [180]
i [180]
j [181]
k [181]
l [182]
m [183]
n [184]

[51]

CHAPTER IV

THE SYMBOL ZERO

What has been said of the improved Hindu system with a place value
does not touch directly the origin of a symbol for zero, although it
assumes that such a symbol exists. The importance of such a sign, the
fact that it is a prerequisite to a place-value system, and the further
fact that without it the Hindu-Arabic numerals would never have dominated
the computation system of the western world, make it proper to devote a
chapter to its origin and history.

It was some centuries after the primitive Brāhmī and Kharoṣṭhī
numerals had made their appearance in India that the zero first appeared
there, although such a character was used by the Babylonians^[185] in the centuries
immediately preceding the Christian era. The symbol is or , and apparently it
was not used in calculation. Nor does it always occur when units of any
order are lacking; thus 180 is written with the meaning three sixties and no units, since
181 immediately following is , three sixties and one unit.^[186] The main [52]use of this Babylonian
symbol seems to have been in the fractions, 60ths, 3600ths, etc., and
somewhat similar to the Greek use of ο, for οὐδέν, with the meaning
vacant.

“The earliest undoubted occurrence of a zero in India is an
inscription at Gwalior, dated Samvat 933 (876 A.D.). Where 50 garlands are mentioned (line 20), 50
is written . 270
(line 4) is written .”^[187] The Bakhṣālī
Manuscript^[188] probably
antedates this, using the point or dot as a zero symbol. Bayley mentions
a grant of Jaika Rashtrakúta of Bharuj, found at Okamandel, of date 738
A.D., which contains a zero, and also a coin
with indistinct Gupta date 707 (897 A.D.), but
the reliability of Bayley’s work is questioned. As has been noted, the
appearance of the numerals in inscriptions and on coins would be of much
later occurrence than the origin and written exposition of the system.
From the period mentioned the spread was rapid over all of India, save
the southern part, where the Tamil and Malayalam people retain the old
system even to the present day.^[189]

Aside from its appearance in early inscriptions, there is still
another indication of the Hindu origin of the symbol in the special
treatment of the concept zero in the early works on arithmetic.
Brahmagupta, who lived in Ujjain, the center of Indian astronomy,^[190] in the early part [53]of the
seventh century, gives in his arithmetic^[191] a distinct treatment of the
properties of zero. He does not discuss a symbol, but he shows by his
treatment that in some way zero had acquired a special significance not
found in the Greek or other ancient arithmetics. A still more scientific
treatment is given by Bhāskara,^[192] although in one place he permits
himself an unallowed liberty in dividing by zero. The most recently
discovered work of ancient Indian mathematical lore, the Ganita-Sāra-Saṅgraha^[193] of
Mahāvīrācārya (c. 830 A.D.), while it does not use the numerals with place
value, has a similar discussion of the calculation with zero.

What suggested the form for the zero is, of course, purely a matter of
conjecture. The dot, which the Hindus used to fill up lacunæ in their
manuscripts, much as we indicate a break in a sentence,^[194] would have been a more natural
symbol; and this is the one which the Hindus first used^[195] and which most Arabs use to-day.
There was also used for this purpose a cross, like our X, and this is
occasionally found as a zero symbol.^[196] In the Bakhṣālī manuscript above
mentioned, the word śūnya, with the dot as its symbol,
is used to denote the unknown quantity, as well as to denote zero. An
analogous use of the [54]zero, for the unknown quantity in a
proportion, appears in a Latin manuscript of some lectures by Gottfried
Wolack in the University of Erfurt in 1467 and 1468.^[197] The usage was noted even as early as
the eighteenth century.^[198]

The small circle was possibly suggested by the spurred circle which
was used for ten.^[199] It
has also been thought that the omicron used by Ptolemy in his
Almagest, to mark accidental blanks in the sexagesimal system
which he employed, may have influenced the Indian writers.^[200] This symbol was used
quite generally in Europe and Asia, and the Arabic astronomer
Al-Battānī^[201] (died 929 A.D.) used a similar symbol in connection with the
alphabetic system of numerals. The occasional use by
Al-Battānī of the Arabic negative, lā, to
indicate the absence of minutes [55](or seconds), is noted by Nallino.^[202] Noteworthy is also the
use of the for unity in the
Śāradā characters of the Kashmirian Atharva-Veda, the
writing being at least 400 years old. Bhāskara (c. 1150) used a
small circle above a number to indicate subtraction, and in the Tartar
writing a redundant word is removed by drawing an oval around it. It
would be interesting to know whether our score mark , read “four in the hole,”
could trace its pedigree to the same sources. O’Creat^[203] (c. 1130), in a letter to his
teacher, Adelhard of Bath, uses τ
for zero, being an abbreviation for the word teca which we shall
see was one of the names used for zero, although it could quite as well
be from τζίφρα. More rarely O’Creat uses , applying the name
cyfra to both forms. Frater Sigsboto^[204] (c. 1150) uses the same symbol. Other
peculiar forms are noted by Heiberg^[205] as being in use among the Byzantine
Greeks in the fifteenth century. It is evident from the text that some of
these writers did not understand the import of the new system.^[206]

Although the dot was used at first in India, as noted above, the small
circle later replaced it and continues in use to this day. The Arabs,
however, did not adopt the [56]circle, since it bore some resemblance to
the letter which expressed the number five in the alphabet system.^[207] The earliest Arabic
zero known is the dot, used in a manuscript of 873 A.D.^[208] Sometimes both the dot and the circle
are used in the same work, having the same meaning, which is the case in
an Arabic MS., an abridged arithmetic of Jamshid,^[209] 982 A.H. (1575 A.D.). As given in this work the numerals are . The form for 5 varies, in some
works becoming or ; is found in Egypt and appears in some fonts of type. To-day the Arabs use the 0 only
when, under European influence, they adopt the ordinary system. Among the
Chinese the first definite trace of zero is in the work of Tsin^[210] of 1247 A.D. The form is the circular one of the Hindus, and
undoubtedly was brought to China by some traveler.

The name of this all-important symbol also demands some attention,
especially as we are even yet quite undecided as to what to call it. We
speak of it to-day as zero, naught, and even cipher; the
telephone operator often calls it O, and the illiterate or
careless person calls it aught. In view of all this uncertainty we
may well inquire what it has been called in the past.^[211]

[57]

As already stated, the Hindus called it śūnya,
“void.”^[212] This passed
over into the Arabic as aṣ-ṣifr or ṣifr.^[213] When Leonard of Pisa (1202) wrote
upon the Hindu numerals he spoke of this character as zephirum.^[214] Maximus Planudes
(1330), writing under both the Greek and the Arabic influence, called it
tziphra.^[215] In a
treatise on arithmetic written in the Italian language by Jacob of
Florence^[216] [58](1307) it is
called zeuero,^[217] while in an arithmetic of Giovanni di
Danti of Arezzo (1370) the word appears as çeuero.^[218] Another form is
zepiro,^[219] which
was also a step from zephirum to zero.^[220]

Of course the English cipher, French chiffre, is derived
from the same Arabic word, aṣ-ṣifr, but in several
languages it has come to mean the numeral figures in general. A trace of
this appears in our word ciphering, meaning figuring or
computing.^[221] Johann
Huswirt^[222] uses the
word with both meanings; he gives for the tenth character the four names
theca, circulus, cifra, and figura nihili. In this
statement Huswirt probably follows, as did many writers of that period,
the Algorismus of Johannes de Sacrobosco (c. 1250 A.D.), who was also known as John of Halifax or John
of Holywood. The commentary of [59]Petrus de Dacia^[223] (c. 1291 A.D.) on the Algorismus vulgaris of Sacrobosco
was also widely used. The widespread use of this Englishman’s work on
arithmetic in the universities of that time is attested by the large
number^[224] of MSS. from
the thirteenth to the seventeenth century still extant, twenty in Munich,
twelve in Vienna, thirteen in Erfurt, several in England given by
Halliwell,^[225] ten
listed in Coxe’s Catalogue of the Oxford College Library, one in
the Plimpton collection,^[226] one in the Columbia University
Library, and, of course, many others.

From aṣ-ṣifr has come zephyr,
cipher, and finally the abridged form zero. The earliest
printed work in which is found this final form appears to be Calandri’s
arithmetic of 1491,^[227]
while in manuscript it appears at least as early as the middle of the
fourteenth century.^[228]
It also appears in a work, Le Kadran des marchans, by Jehan [60]Certain,^[229] written in 1485. This word soon
became fairly well known in Spain^[230] and France.^[231] The medieval writers also spoke of it
as the sipos,^[232]
and occasionally as the wheel,^[233] circulus^[234] (in German das Ringlein^[235]), circular [61]note,^[236] theca,^[237] long supposed to be from its
resemblance to the Greek theta, but explained by Petrus de Dacia as being
derived from the name of the iron^[238] used to brand thieves and robbers
with a circular mark placed on the forehead or on the cheek. It was also
called omicron^[239] (the Greek o), being sometimes
written õ or φ to distinguish it
from the letter o. It also went by the name null^[240] (in the Latin books
[62]nihil^[241] or nulla,^[242] and in the French rien^[243]), and very commonly by
the name cipher.^[244] Wallis^[245] gives one of the earliest extended
discussions of the various forms of the word, giving certain other
variations worthy of note, as ziphra, zifera,
siphra, ciphra, tsiphra, tziphra, and the
Greek τζίφρα.^[246]

[63]

CHAPTER V

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 ḥurūf
al-ġobār, the ġobār or dust numerals,
as distinguished from the ḥurūf 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 ġobār 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-Bīrūnī 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 Kairwān, in Tunis, in which
the author speaks of one of his works on ġobār calculation;^[251] and, much later, the
Arab writer Abū Bekr Moḥammed ibn ‛Abdallāh,
surnamed al-Ḥaṣṣār [66](the
arithmetician), wrote a work of which the second chapter was “On the dust
figures.”^[252]

The ġobār 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-Prés.^[253] The system has nine characters, but
no zero. A dot above a character indicates tens, two dots hundreds, and
so on, meaning 50, and
meaning 5000. It has been
suggested that possibly these dots, sprinkled like dust above the
numerals, gave rise to the word ġobār,^[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 ġobār 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 Bakhṣālī manuscript,^[257] and that it was used
in subscript form in the Kitāb 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 ġobār 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 Sēfer
Yeṣīrāh by Abū Sahl ibn Tamim
(probably composed at Kairwān, c. 950) speaks of “the Indian
arithmetic known under the name of ġobār or dust
calculation.”^[261] All
this suggests that the Arabs may very [68]likely have known the
ġobār forms before the numerals reached them again in 773.^[262] The term
“ġobār 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 Kitāb
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 ġobār.
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 ġobār 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]
2 [268]
3 [269]
4 [270]
5 [271]
6 [271]

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 ġobār
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.

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
ġobār, 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. for 2, are also used. Probably most significant of all is the fact
that the ġobār 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 ġobār numerals. We now have to consider the question
as to whether Boethius knew these ġobār 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 Bokhāra 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 ġobār 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 Nānā Ghāt, 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., Hecatæus,^[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 Hecatæus, 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
Phœnicians and the Sabæans, and later were taken by the Greeks and
Romans.^[298]

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

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

Moreover the results of the early Greek invasion were embodied by
Dicæarchus of Messana (about 320 B.C.) in a map
that long remained a standard. Furthermore, Alexander did not allow his
influence on the East to cease. He divided India into three satrapies,^[304] placing Greek
governors over two of them and leaving a Hindu ruler in charge of the
third, and in Bactriana, a part of Ariana or ancient Persia, he left
governors; and in these the western civilization was long in evidence.
Some of the Greek and Roman metrical and astronomical terms [78]found their way,
doubtless at this time, into the Sanskrit language.^[305] Even as late as from the second to
the fifth centuries A.D., Indian coins showed
the Hellenic influence. The Hindu astronomical terminology reveals the
same relationship to western thought, for Varāha-Mihira (6th
century A.D.), a contemporary of Āryabhaṭa, entitled
a work of his the Bṛhat-Saṃhitā, a
literal translation of μεγάλη
σύνταξις 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
Phœnicians had long before brought gold from Malacca.

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

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

[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
Aśoka, whose name has already been mentioned in this work, extended
the boundaries of his domains even into Afghanistan, so that it was
entirely possible for the numerals of the Punjab to have worked their way
north even at that early date.^[327]

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

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

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

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

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

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

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

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

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

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

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

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

[89]

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

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

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

[91]

CHAPTER VI

THE DEVELOPMENT OF THE NUMERALS
AMONG THE ARABS

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

[92]

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

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

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

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

[94]

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

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

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

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

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

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

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

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

[99]

CHAPTER VII

THE DEFINITE INTRODUCTION OF THE NUMERALS
INTO EUROPE

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

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

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

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

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

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

[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 Khordāḍbeh, who left Bagdad in the latter part
of the ninth century, gives a picture of the great commercial activity at
that time in the Saracen city of Palermo. In this same century they had
established themselves in Piedmont, and in 906 they pillaged Turin.^[417] On the Sorrento
peninsula the traveler who climbs the hill to the beautiful Ravello sees
still several traces of the Arab architecture, reminding him of the fact
that about 900 A.D. Amalfi was a commercial
center of the Moors.^[418]
Not only at this time, but even a century earlier, the artists of
northern India sold their wares at such centers, and in the courts both
of Hārūn al-Rashīd and of Charlemagne.^[419] Thus the Arabs dominated the
Mediterranean Sea long before Venice

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 à
Becket, present at the latter’s death, wrote a life of the martyr, to
which (fortunately for our purposes) he prefixed a brief eulogy of the
city of London.^[427] This
clerk, William Fitz Stephen by name, thus speaks of the British
capital:

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

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

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

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

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

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

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

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

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

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

[117]

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

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

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

So little effect did the teachings of Gerbert have in making known the
new numerals, that O’Creat, who lived a century later, a friend and pupil
of Adelhard [120]of Bath, used the zero with the Roman
characters, in contrast to Gerbert’s use of the ġobār forms
without the zero.^[474]
O’Creat uses three forms for zero, o, ō, and τ, as in Maximus Planudes. With this use of the zero goes,
naturally, a place value, for he writes III III for 33, ICCOO and
I. II. τ. τ for
1200, I. O. VIII. IX for 1089, and I. IIII. IIII. ττττ 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:

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

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

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

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

After Gerbert’s death, little by little the scholars of Europe came to
know the new figures, chiefly through the introduction of Arab learning.
The Dark Ages had passed, although arithmetic did not find another
advocate as prominent as Gerbert for two centuries. Speaking of this
great revival, Raoul Glaber^[493] (985-c. 1046), a monk of the great
Benedictine abbey of Cluny, of the eleventh century, says: “It was as
though the world had arisen and tossed aside the worn-out garments of
ancient time, and wished to apparel itself in a white robe of churches.”
And with this activity in religion came a corresponding interest in other
lines. Algorisms began to appear, and knowledge from the outside world
found [124]interested listeners. Another Raoul, or
Radulph, to whom we have referred as Radulph of Laon,^[494] a teacher in the cloister school of
his city, and the brother of Anselm of Laon^[495] the celebrated theologian, wrote a
treatise on music, extant but unpublished, and an arithmetic which Nagl
first published in 1890.^[496] The latter work, preserved to us in a
parchment manuscript of seventy-seven leaves, contains a curious mixture
of Roman and ġobār numerals, the former for expressing large
results, the latter for practical calculation. These ġobār
“caracteres” include the sipos (zero), , 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 Bibliothèque nationale at Paris. Although this
distinctly lays claim to being Al-Khowārazmī’s work,^[501] the evidence is
altogether against the statement,^[502] but the book is quite as valuable,
since it represents the knowledge of the time in which it was written. It
relates to the operations with integers and sexagesimal fractions,
including roots, and contains no applications.^[503]

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

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

The earliest trace we have of computation with Hindu numerals in
Germany is in an Algorismus of 1143, now in the Hofbibliothek in
Vienna.^[510] It is bound
in with a [127]Computus by the same author and
bearing the date given. It contains chapters “De additione,” “De
diminutione,” “De mediatione,” “De divisione,” and part of a chapter on
multiplication. The numerals are in the usual medieval forms except the 2
which, as will be seen from the illustration,^[511] is somewhat different, and the 3,
which takes the peculiar shape , 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 Meïr ibn
Ezra,^[513] was born in
Toledo (c. 1092). In 1139 he went to Egypt, Palestine, and the Orient,
spending also some years in Italy. Later he lived in southern France and
in England. He died in 1167. The probability is that he acquired his
knowledge of the Hindu arithmetic^[514] in his native town of Toledo, but it
is also likely that the knowledge of other systems which he acquired on
travels increased his appreciation of this one. We have mentioned the
fact that he used the first letters of the Hebrew alphabet, א
ב ג ד ה ו ז ח
ט, for the numerals 9 8 7 6 5 4 3 2 1, and a circle
for the zero. The quotation in the note given below shows that he knew of
the Hindu origin; but in his manuscript, although he set down the Hindu
forms, he used the above nine Hebrew letters with place value for all
computations.

[128]

CHAPTER VIII

THE SPREAD OF THE NUMERALS IN EUROPE

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

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

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

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

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

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

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

[134]

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

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

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

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

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

Aside from their use in the early treatises on the new art of
reckoning, the numerals appeared from time to time in the dating of
manuscripts and upon monuments. The oldest definitely dated European
document known [138]to contain the numerals is a Latin
manuscript,^[556] the
Codex Vigilanus, written in the Albelda Cloister not far from Logroño in
Spain, in 976 A.D. The nine characters (of
ġobār type), without the zero, are given as an addition to
the first chapters of the third book of the Origines by Isidorus
of Seville, in which the Roman numerals are under discussion. Another
Spanish copy of the same work, of 992 A.D.,
contains the numerals in the corresponding section. The writer ascribes
an Indian origin to them in the following words: “Item de figuris
arithmeticę. Scire debemus in Indos subtilissimum ingenium habere
et ceteras gentes eis in arithmetica et geometria et ceteris liberalibus
disciplinis concedere. Et hoc manifestum est in nobem figuris, quibus
designant unumquemque gradum cuiuslibet gradus. Quarum hec sunt forma.”
The nine ġobār characters follow. Some of the abacus forms^[557] previously given are
doubtless also of the tenth century. The earliest Arabic documents
containing the numerals are two manuscripts of 874 and 888 A.D.^[558] They appear about a century later in
a work^[559] written at
Shiraz in 970 A.D. There is also an early trace
of their use on a pillar recently discovered in a church apparently
destroyed as early as the tenth century, not far from the Jeremias
Monastery, in Egypt. [139]A graffito in Arabic on this pillar has
the date 349 A.H., which corresponds to 961
A.D.^[560] For the dating of Latin documents the
Arabic forms were used as early as the thirteenth century.^[561]

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

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

Earliest Manuscript Forms

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

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

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

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

	for 125		for 150
	for 147		for 202

As to monumental inscriptions,^[573] there was once thought to be a
gravestone at Katharein, near Troppau, with the date 1007, and one at
Biebrich of 1299. There is no doubt, however, of one at Pforzheim of 1371
and one at Ulm of 1388.^[574] Certain numerals on Wells Cathedral
have been assigned to the thirteenth century, but they are undoubtedly
considerably later.^[575]

The table on page 143 will serve to supplement that from Mr. Hill’s
work.^[576]

[143]

Early Manuscript Forms

a [577]		Twelfth century A.D.
b [578]		1197 A.D.
c [579]		1275 A.D.
d [580]		c. 1294 A.D.
e [581]		c. 1303 A.D.
f [582]		c. 1360 A.D.
g [583]		c. 1442 A.D.

[144]

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

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

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

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

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

[146]

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

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

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

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

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

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

[148]

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

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

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

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

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

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

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

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

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

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

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

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

[151]

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

[153]

INDEX

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

ANNOUNCEMENTS

WENTWORTH’S

COLLEGE ALGEBRA

REVISED EDITION

12mo. Half morocco. 530 pages. List price, $1.50; mailing price, $1.65

This book is a thorough revision of the author’s “College Algebra.”
Some chapters of the old edition have been wholly rewritten, and the
other chapters have been rewritten in part and greatly improved. The
order of topics has been changed to a certain extent; the plan is to have
each chapter as complete in itself as possible, so that the teacher may
vary the order of succession at his discretion.

As the name implies, the work is intended for colleges and scientific
schools. The first part is simply a review of the principles of algebra
preceding Quadratic Equations, with just enough examples to illustrate
and enforce these principles. By this brief treatment of the first
chapters sufficient space is allowed, without making the book cumbersome,
for a full discussion of Quadratic Equations, The Binomial Theorem,
Choice Chance, Series, Determinants, and the General Properties of
Equations.

Every effort has been made to present in the clearest light each
subject discussed, and to give in matter and methods the best training in
algebraic analysis at present attainable.

ADDITIONAL PUBLICATIONS

By G. A. WENTWORTH

A list of the various editions of Wentworth’s Trigonometries will be sent on request.

GINN AND COMPANY Publishers

ALGEBRA FOR BEGINNERS

By DAVID EUGENE SMITH,

Professor of Mathematics in Teachers College, Columbia University

12mo, cloth, 154 pages, 50 cents

This work is intended to serve as an introduction to the study of
algebra, and is adapted to the needs of the seventh or eighth school
year. It is arranged in harmony with the leading courses of study that
include algebra in the curriculum of the grades.

The relation of algebra to arithmetic is emphasized, the subject is
treated topically, and each important point is touched at least twice.
The book begins by showing the uses of algebra, employing such practical
applications as are within the pupil’s range of knowledge. When an
interest has thus been awakened in the subject, the fundamental
operations are presented with the simple explanations necessary to make
the student independent of dogmatic rules. Throughout the book abundant
oral and written drill exercises are provided. The work includes linear
equations with two unknown quantities, and easy quadratics.

The leading features may be summarized as follows: (1) an arrangement
in harmony with existing courses of study; (2) a presentation designed to
awaken the interest of the pupils; (3) a topical arrangement for each
half year, every important topic being repeated; (4) simplicity of
explanations; (5) development of the relation of algebra to arithmetic
both in theory and in applications; (6) emphasis laid on the importance
of oral as well as written algebra.

GINN & COMPANY Publishers

BOOKS FOR TEACHERS

FOR CLASS RECORDS

GINN AND COMPANY Publishers

TEXTBOOKS ON MATHEMATICS

FOR HIGHER SCHOOLS AND COLLEGES

GINN AND COMPANY Publishers

Notes