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Transcriber's Note:

The following codes are used for characters that are not present in the
character set used for this version of the book.

  [=a] a with macron (etc.)
  [.g] g with dot above (etc.)
  ['s] s with acute accent
  [d.] d with dot below (etc.)
  [d=] d with line below
  [H)] H with breve below





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

  CHAPTER

        PRONUNCIATION OF ORIENTAL NAMES                              vi

  I.    EARLY IDEAS OF THEIR ORIGIN                                   1

  II.   EARLY HINDU FORMS WITH NO PLACE VALUE                        12

  III.  LATER HINDU FORMS, WITH A PLACE VALUE                        38

  IV.   THE SYMBOL ZERO                                              51

  V.    THE QUESTION OF THE INTRODUCTION OF THE
        NUMERALS INTO EUROPE BY BOETHIUS                             63

  VI.   THE DEVELOPMENT OF THE NUMERALS AMONG THE ARABS              91

  VII.  THE DEFINITE INTRODUCTION OF THE NUMERALS INTO EUROPE        99

  VIII. THE SPREAD OF THE NUMERALS IN EUROPE                        128

  INDEX                                                             153

       *       *       *       *       *


{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_. [=A], as in _father_.

C, (S) like _ch_ in _church_ (Italian _c_ in _cento_).

[D.], [N.], [S.], [T.], (S) _d_, _n_, _sh_, _t_, made with the tip of the
tongue turned up and back into the dome of the palate. [D.], [S.], [T.],
[Z.], (A) _d_, _s_, _t_, _z_, made with the tongue spread so that the
sounds are produced largely against the side teeth. Europeans commonly
pronounce [D.], [N.], [S.], [T.], [Z.], both (S) and (A), as simple _d_,
_n_, _sh_ (S) or _s_ (A), _t_, _z_. [D=] (A), like _th_ in _this_.

E, (S) as in _they_. (A) as in _bed_.

[.G], (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_, _[t.]_, 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. [H.], (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_. [H.] (S) is final consonant
_h_, like final _h_ (A).

I, as in _pin_. [=I], 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.

[.M], [.N], (S) like French final _m_ or _n_, nasalizing the preceding
vowel.

[N.], see [D.]. 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. [R.], (S) r used as
vowel, as in _apron_ when pronounced _aprn_ and not _apern_; modern Hindus
say _ri_, hence our _amrita_, _Krishna_, for _a-m[r.]ta, K[r.][s.][n.]a_.

S, as in _same_. [S.], see [D.]. ['S], (S) English _sh_ (German _sch_).

[T.], see [D.].

U, as in _put_. [=U], as in _rule_.

Y, as in _you_.

[Z.], see [D.].

`, (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 _[h.]_. 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 _[.m]_ and
_[.n]_ 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 Phoenicians.[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 Mo[h.]ammed the Son of Moses, from
Khow[=a]rezm, or, more after the manner of the Arab, Mo[h.]ammed ibn
M[=u]s[=a] al-Khow[=a]razm[=i],[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
hind[=i]_, "Indian science," is used by them for arithmetic, as also the
adjective _hind[=i]_ alone.[14]

Probably the most striking testimony from Arabic sources is that given by
the Arabic traveler and scholar Mohammed ibn A[h.]med, Ab[=u]
'l-R[=i][h.][=a]n al-B[=i]r[=u]n[=i] (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-B[=i]r[=u]n[=i] 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 _a[.n]ka_[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-B[=i]r[=u]n[=i] there was another Arabic writer of the tenth
century, Mo[t.]ahhar ibn [T.][=a]hir,[20] author of the _Book of the
Creation and of History_, who gave as a curiosity, in Indian (N[=a]gar[=i])
symbols, a large number asserted by the people of India to represent the
duration of the world. Huart feels positive that in Mo[t.]ahhar'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`[=u]d[=i]
(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 Mo[h.]ammed ibn `Al[=i] `Abd[=i], that by order of
Al-Man[s.][=u]r many works of science and astrology were translated into
Arabic, notably the _Sindhind_ (_Siddh[=a]nta_). Concerning the meaning and
spelling of this name there is considerable diversity of opinion.
Colebrooke[24] first pointed out the connection between _Siddh[=a]nta_ 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 _T[=a]r[=i]kh al-[h.]okam[=a]_ 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-Khow[=a]razm[=i] is worthy of note.

This _Sindhind_ is the book, says Mas`[=u]d[=i],[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-Khow[=a]razm[=i] and [H.]abash[33]
as translators of the tables of the _Sindhind_. Al-B[=i]r[=u]n[=i][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-Man[s.][=u]r, 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 _Kit[=a]b al-Fihrist_, written in the year 987 A.D., by
Ibn Ab[=i] Ya`q[=u]b al-Nad[=i]m. 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-Kind[=i] (800-870 A.D.),
who wrote five books on arithmetic and four books on the use of the Indian
method of reckoning. Sened ibn `Al[=i], the Jew, who was converted to Islam
under the caliph Al-M[=a]m[=u]n, 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 `Al[=i] are really works of
Al-Khow[=a]razm[=i], 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-[S.][=u]f[=i], 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:

 "Hec algorism' ars p'sens dicit' in qua
  Talib; indor[um] fruim bis quinq; figuris.[41]

"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
Br[=a]hma[n.]as), and partly philosophical (the Upanishads). Our especial
interest is {13} in the S[=u]tras, 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
Br[=a]hma[n.]as, and the S[=u]tras, 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 S[=u]tras goes back to the eighth century B.C.

The importance of the S[=u]tras 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 B[=o]dhisattva[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 10^{53}, adding that
he could proceed as far as 10^{421},[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 Vi[s.]vamitra:

  "And Viswamitra said, 'It is enough,
  Let us to numbers. After me repeat
  Your numeration till we reach the lakh,[59]
  One, two, three, four, to ten, and then by tens
  To hundreds, thousands.' After him the child
  Named digits, decads, centuries, nor paused,
  The round lakh reached, but softly murmured on,
  Then comes the k[=o]ti, nahut, ninnahut,
  Khamba, viskhamba, abab, attata,
  To kumuds, gundhikas, and utpalas,
  By pundar[=i]kas into padumas,
  Which last is how you count the utmost grains
  Of Hastagiri ground to finest dust;[60]
  But beyond that a numeration is,
  The K[=a]tha, used to count the stars of night,
  The K[=o]ti-K[=a]tha, for the ocean drops;
  Ingga, the calculus of circulars;
  Sarvanikchepa, by the which you deal
  With all the sands of Gunga, till we come
  To Antah-Kalpas, where the unit is
  The sands of the ten crore Gungas.  If one seeks
  More comprehensive scale, th' arithmic mounts
  By the Asankya, which is the tale
  Of all the drops that in ten thousand years
  Would fall on all the worlds by daily rain;
  Thence unto Maha Kalpas, by the which
  The gods compute their future and their past.'"

{17}

Thereupon Vi[s.]vamitra [=A]c[=a]rya[61] expresses his approval of the
task, and asks to hear the "measure of the line" as far as y[=o]jana, the
longest measure bearing name. This given, Buddha adds:

  ... "'And master! if it please,
  I shall recite how many sun-motes lie
  From end to end within a y[=o]jana.'
  Thereat, with instant skill, the little prince
  Pronounced the total of the atoms true.
  But Viswamitra heard it on his face
  Prostrate before the boy; 'For thou,' he cried,
  'Art Teacher of thy teachers--thou, not I,
  Art G[=u]r[=u].'"

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 _Ved[=a][.n]gas_, or "limbs
for supporting the Veda," part of that great branch of Hindu literature
known as _Sm[r.]iti_ (recollection), that which was to be handed down by
tradition. Of these the sixth is known as _Jyoti[s.]a_ (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 Siddh[=a]ntas 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
Mah[=a]bh[=a]rata[64] and the R[=a]m[=a]yana, 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 Kharo[s.][t.]h[=i], (2) the Br[=a]hm[=i], and (3) the word and letter
forms; and these will be considered in order.

The Kharo[s.][t.]h[=i] numerals are found in inscriptions formerly known as
Bactrian, Indo-Bactrian, and Aryan, and appearing in ancient Gandh[=a]ra,
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 A['s]oka, 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 A['s]oka[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 Kharo[s.][t.]h[=i] characters, and
the rest in some form of Br[=a]hm[=i]. In the Kharo[s.][t.]h[=i]
inscriptions only four numerals have been found, and these are merely
vertical marks for one, two, four, and five, thus:

  |     ||     |||     ||||

In the so-called ['S]aka 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:

[Illustration]

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
N[=a]n[=a] Gh[=a]t forms hereafter given, although the inscriptions
themselves at N[=a]n[=a] Gh[=a]t are later than those of the A['s]oka
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 Kharo[s.][t.]h[=i]
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 Br[=a]hm[=i] 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 Kharo[s.][t.]h[=i] 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 Kharo[s.][t.]h[=i] 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 Br[=a]hm[=i] characters. The alphabet attributed to
Brahm[=a] 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 A['s]oka appear, some of these
having been incised in Br[=a]hm[=i] as well as Kharo[s.][t.]h[=i]. 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
A['s]oka edicts:[74]

[Illustration]

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 N[=a]n[=a] Gh[=a]t 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 Kaly[=a]na to Junar. It seems to have been cut
out by a descendant {23} of King ['S][=a]tav[=a]hana,[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]

[Illustration]

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}

[Illustration]

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:

[Illustration]

From this time on, until the decimal system finally adopted the first nine
characters and replaced the rest of the Br[=a]hm[=i] 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 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 1000
  A['s]oka[80]                [Illustration]
  ['S]aka[81]                 [Illustration]
  A['s]oka[82]                [Illustration]
  N[=a]gar[=i][83]            [Illustration]
  Nasik[84]                   [Illustration]
  K[s.]atrapa[85]             [Illustration]
  Ku[s.]ana [86]              [Illustration]
  Gupta[87]                   [Illustration]
  Valhab[=i][88]              [Illustration]
  Nepal [89]                  [Illustration]
  Kali[.n]ga[90]              [Illustration]
  V[=a]k[=a][t.]aka[91]       [Illustration]

[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:

  [93] [Numerals] for 174
  [94] [Numerals] for 191
  [95] [Numerals] for 269
  [96] [Numerals] for 252
  [97] [Numerals] for 400
  [98] [Numerals] for 356

{27}

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

  [Numerals][99]    for 90
  [Numerals][100]   for 70

We have thus far spoken of the Kharo[s.][t.]h[=i] and Br[=a]hm[=i]
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 [1 vertical stroke] or [1 horizontal stroke] is simply one
stroke, or one stick laid down by the computer. The [2 vertical strokes] or
[2 horizontal strokes] represents two strokes or two sticks, and so for the
[3 vertical strokes] and [3 horizontal strokes]. From some primitive [2
vertical strokes] 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:

  Hieroglyphic [2 vertical strokes]
  Hieratic     [Hieratic 2]
  Demotic      [Demotic 2]

The last of these is merely a cursive form as in the Arabic [Arabic 2],
which becomes our 2 if tipped through a right angle. From some primitive [2
horizontal strokes] 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 [2 horizontal strokes joined], and this
was frequently used for two in Germany until the 18th century. It finally
went into the modern form 2, and the [3 horizontal strokes] 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 N[=a]n[=a] Gh[=a]t inscriptions they are vertical.
But long before either the A['s]oka or the N[=a]n[=a] Gh[=a]t 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 A['s]oka's time in India. After A['s]oka, and perhaps among
the merchants of earlier centuries, the horizontal forms may have come down
into India from China, thus giving those of the N[=a]n[=a] Gh[=a]t 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 K[s.]atrapa coins,
the nine to that of the Ku[s.]ana 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
Kharo[s.][t.]h[=i] 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           [Symbol]      [Symbol] = ch    chatur, Lat. quattuor
     5           [Symbol]      [Symbol] = p     pancha, Gk. [Greek:p/ente]
     6           [Symbol]      [Symbol] = s     [s.]a[s.]
     7           [Symbol]      [Symbol] = [s.]  sapta
     ( the s and [s.] 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 Br[=a]hm[=i] 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 _ak[s.]aras_,
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 Phoenician, 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,
[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 Lh[=a]sa,[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 Phoenicia 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            [Illustration: numerals]
  Burma[126]      [Illustration: numerals]
  Malabar[127]    [Illustration: numerals]
  Tibet[128]      [Illustration: numerals]
  Ceylon[129]     [Illustration: numerals]
  Malayalam[129]  [Illustration: numerals]

{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" (_['s][=u]nya_),
or "heaven-space" (_ambara [=a]k[=a]['s]a_); one by "stick" (_rupa_),
"moon" (_indu ['s]a['s]in_), "earth" (_bh[=u]_), "beginning" (_[=a]di_),
"Brahma," or, in general, by anything markedly unique; two by "the twins"
(_yama_), "hands" (_kara_), "eyes" (_nayana_), etc.; four by "oceans," five
by "senses" (_vi[s.]aya_) or "arrows" (the five arrows of K[=a]mad[=e]va);
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 "['S]aka Sa[m.]vat, 867" (A.D. 945
or 946), is given by "_giri-ra[s.]a-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 _['S]rautas[=u]tra_ of K[=a]ty[=a]yana and
L[=a][t.]y[=a]yana.[132] It was certainly known to Var[=a]ha-Mihira (d.
587),[133] for he used it in the _B[r.]hat-Sa[m.]hit[=a]._[134] It has also
been asserted[135] that [=A]ryabha[t.]a (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:

  1        2       3       4       5        6       7      8     9      0
  k        kh      g       gh      [.n]     c       ch     j     jh     ñ
  [t.]     [t.]h   [d.]    [d.]h   [n.]     t       th     d     th     n
  p        ph      b       bh      m
  y        r       l       v       ['s]     [s.]    s      h      l

{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

    2     3      1     5    6       5       1
  _kha_ _gont_ _yan_ _me_ _[s.]a_ _m[=a]_ _pa_

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 _[=a]_ (as _k[=a]_ ... _l[=a]_), 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
N[=a]n[=a] Gh[=a]t {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
N[=a]n[=a] Gh[=a]t 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 k[=o][t.]is, 3 prayutas, 6 lak[s.]as, 8
ayutas, 2 sahasra, 1 ['s]ata, 5 da['s]an, 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 _a[.n]kapalli_,[145] the decimal-place system of writing numbers,
was perfected, the tenth symbol was called the _['s][=u]nyabindu_,
generally shortened to _['s][=u]nya_ (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 _['s][=u]nya_ as a symbol was not used
before about 500 A.D., although some writers have placed it earlier.[147]
Since [=A]ryabha[t.]a 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, Var[=a]ha-Mihira[151] wrote
a work entitled _B[r.]hat Sa[m.]hit[=a]_[152] in which he frequently uses
_['s][=u]nya_ 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 [=A]ryabha[t.]a was doing the same.

It should also be mentioned as a matter of interest, and somewhat related
to the question at issue, that Var[=a]ha-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.) B[=a][n.]a[155] wrote of
Subandhus's _V[=a]savadatt[=a]_ 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 [=A]ryabha[t.]a'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-sa[m.]vat 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 Sa[m.]vat 872. So F. Kielhorn
in the _Epigraphia Indica_[165] gives a Pathari pillar inscription of
Parabala, dated Vikrama-sa[m.]vat 917, which corresponds to 861 A.D., {47}
and refers also to another copper-plate inscription dated Vikrama-sa[m.]vat
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-sa[m.]vat 715 (798 A.D.), being incised on a pilaster.
Kielhorn[167] also gives two copper-plate inscriptions of the time of
Mahendrapala of Kanauj, Valhab[=i]-sa[m.]vat 574 (893 A.D.) and
Vikrama-sa[m.]vat 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-B[=i]r[=u]n[=i] 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 ['S][=a]rad[=a] 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, [Illustration]
  Arabic, [Illustration]

{49}

NUMERALS USED WITH PLACE VALUE

      1  2  3  4  5  6  7  8  9  0
  a[173] [Illustration]
  b[174] [Illustration]
  c[175] [Illustration]
  d[176] [Illustration]
  e[177] [Illustration]
  f[178] [Illustration]
  g[179] [Illustration]
  h[180] [Illustration]
  i[180] [Illustration]
  j[181] [Illustration]
  k[181] [Illustration]
  l[182] [Illustration]
  m[183] [Illustration]
  n[184] [Illustration]

       *       *       *       *       *


{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 Br[=a]hm[=i] and
Kharo[s.][t.]h[=i] 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 [Babylonian zero symbol] or [Babylonian zero symbol], and
apparently it was not used in calculation. Nor does it always occur when
units of any order are lacking; thus 180 is written [Babylonian numerals
180] with the meaning three sixties and no units, since 181 immediately
following is [Babylonian numerals 181], 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
[Greek: o], for [Greek: ouden], 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 [Gwalior numerals 50]. 270 (line 4) is written [Gwalior
numerals 270]."[187] The Bakh[s.][=a]l[=i] 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 Bh[=a]skara,[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-S[=a]ra-Sa[.n]graha[193] of Mah[=a]v[=i]r[=a]c[=a]rya (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 Bakh[s.][=a]l[=i] manuscript above mentioned, the word
_['s][=u]nya_, 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-Batt[=a]n[=i][201] (died 929 A.D.) used a similar symbol in
connection with the alphabetic system of numerals. The occasional use by
Al-Batt[=a]n[=i] of the Arabic negative, _l[=a]_, to indicate the absence
of minutes {55} (or seconds), is noted by Nallino.[202] Noteworthy is also
the use of the [Circle] for unity in the ['S][=a]rad[=a] characters of the
Kashmirian Atharva-Veda, the writing being at least 400 years old.
Bh[=a]skara (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 [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 [Greek: t] 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 [Greek: tziphra]. More rarely O'Creat uses
[circle with bar], 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 [symbols]. The form for 5 varies, in some works becoming
[symbol] or [symbol]; [symbol] is found in Egypt and [symbol] 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 _['s][=u]nya_, "void."[212] This
passed over into the Arabic as _a[s.]-[s.]ifr_ or _[s.]ifr_.[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, _a[s.]-[s.]ifr_, 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 _a[s.]-[s.]ifr _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 [Greek: ph] 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 [Greek: tziphra].[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 _[h.]ur[=u]f
al-[.g]ob[=a]r_, the [.g]ob[=a]r or dust numerals, as distinguished from
the _[h.]ur[=u]f al-jumal_ or alphabetic numerals. Probably the latter,
under the influence of the Syrians or Jews,[250] were also used by the
Arabs. The significance of the term [.g]ob[=a]r is doubtless that these
numerals were written on the dust abacus, this plan being distinct from the
counter method of representing numbers. It is also worthy of note that
Al-B[=i]r[=u]n[=i] states that the Hindus often performed numerical
computations in the sand. The term is found as early as c. 950, in the
verses of an anonymous writer of Kairw[=a]n, in Tunis, in which the author
speaks of one of his works on [.g]ob[=a]r calculation;[251] and, much
later, the Arab writer Ab[=u] Bekr Mo[h.]ammed ibn `Abdall[=a]h, surnamed
al-[H.]a[s.][s.][=a]r {66} (the arithmetician), wrote a work of which the
second chapter was "On the dust figures."[252]

The [.g]ob[=a]r numerals themselves were first made known to modern
scholars by Silvestre de Sacy, who discovered them in an Arabic manuscript
from the library of the ancient abbey of St.-Germain-des-Prés.[253] The
system has nine characters, but no zero. A dot above a character indicates
tens, two dots hundreds, and so on, [5 with dot] meaning 50, and [5 with 3
dots] meaning 5000. It has been suggested that possibly these dots,
sprinkled like dust above the numerals, gave rise to the word
_[.g]ob[=a]r_,[254] but this is not at all probable. This system of dots is
found in Persia at a much later date with numerals quite like the modern
Arabic;[255] but that it was used at all is significant, for it is hardly
likely that the western system would go back to Persia, when the perfected
Hindu one was near at hand.

At first sight there would seem to be some reason for believing that this
feature of the [.g]ob[=a]r system was of {67} Arabic origin, and that the
present zero of these people,[256] the dot, was derived from it. It was
entirely natural that the Semitic people generally should have adopted such
a scheme, since their diacritical marks would suggest it, not to speak of
the possible influence of the Greek accents in the Hellenic number system.
When we consider, however, that the dot is found for zero in the
Bakh[s.][=a]l[=i] manuscript,[257] and that it was used in subscript form
in the _Kit[=a]b al-Fihrist_[258] in the tenth century, and as late as the
sixteenth century,[259] although in this case probably under Arabic
influence, we are forced to believe that this form may also have been of
Hindu origin.

The fact seems to be that, as already stated,[260] the Arabs did not
immediately adopt the Hindu zero, because it resembled their 5; they used
the superscript dot as serving their purposes fairly well; they may,
indeed, have carried this to the west and have added it to the [.g]ob[=a]r
forms already there, just as they transmitted it to the Persians.
Furthermore, the Arab and Hebrew scholars of Northern Africa in the tenth
century knew these numerals as Indian forms, for a commentary on the
_S[=e]fer Ye[s.][=i]r[=a]h_ by Ab[=u] Sahl ibn Tamim (probably composed at
Kairw[=a]n, c. 950) speaks of "the Indian arithmetic known under the name
of _[.g]ob[=a]r_ or dust calculation."[261] All this suggests that the
Arabs may very {68} likely have known the [.g]ob[=a]r forms before the
numerals reached them again in 773.[262] The term "[.g]ob[=a]r numerals"
was also used without any reference to the peculiar use of dots.[263] In
this connection it is worthy of mention that the Algerians employed two
different forms of numerals in manuscripts even of the fourteenth
century,[264] and that the Moroccans of to-day employ the European forms
instead of the present Arabic.

The Indian use of subscript dots to indicate the tens, hundreds, thousands,
etc., is established by a passage in the _Kit[=a]b al-Fihrist_[265] (987
A.D.) in which the writer discusses the written language of the people of
India. Notwithstanding the importance of this reference for the early
history of the numerals, it has not been mentioned by previous writers on
this subject. The numeral forms given are those which have usually been
called Indian,[266] in opposition to [.g]ob[=a]r. In this document the dots
are placed below the characters, instead of being superposed as described
above. The significance was the same.

In form these [.g]ob[=a]r numerals resemble our own much more closely than
the Arab numerals do. They varied more or less, but were substantially as
follows:

{69}

  1[267][Illustration]
  2[268][Illustration]
  3[269][Illustration]
  4[270][Illustration]
  5[271][Illustration]
  6[271][Illustration]

The question of the possible influence of the Egyptian demotic and hieratic
ordinal forms has been so often suggested that it seems well to introduce
them at this point, for comparison with the [.g]ob[=a]r forms. They would
as appropriately be used in connection with the Hindu forms, and the
evidence of a relation of the first three with all these systems is
apparent. The only further resemblance is in the Demotic 4 and in the 9, so
that the statement that the Hindu forms in general came from {70} this
source has no foundation. The first four Egyptian cardinal numerals[272]
resemble more the modern Arabic.

[Illustration: DEMOTIC AND HIERATIC ORDINALS]

This theory of the very early introduction of the numerals into Europe
fails in several points. In the first place the early Western forms are not
known; in the second place some early Eastern forms are like the
[.g]ob[=a]r, as is seen in the third line on p. 69, where the forms are
from a manuscript written at Shiraz about 970 A.D., and in which some
western Arabic forms, e.g. [symbol] for 2, are also used. Probably most
significant of all is the fact that the [.g]ob[=a]r numerals as given by
Sacy are all, with the exception of the symbol for eight, either single
Arabic letters or combinations of letters. So much for the Woepcke theory
and the meaning of the [.g]ob[=a]r numerals. We now have to consider the
question as to whether Boethius knew these [.g]ob[=a]r forms, or forms akin
to them.

This large question[273] suggests several minor ones: (1) Who was Boethius?
(2) Could he have known these numerals? (3) Is there any positive or strong
circumstantial evidence that he did know them? (4) What are the
probabilities in the case?

{71}

First, who was Boethius,--Divus[274] Boethius as he was called in the
Middle Ages? Anicius Manlius Severinus Boethius[275] was born at Rome c.
475. He was a member of the distinguished family of the Anicii,[276] which
had for some time before his birth been Christian. Early left an orphan,
the tradition is that he was taken to Athens at about the age of ten, and
that he remained there eighteen years.[277] He married Rusticiana, daughter
of the senator Symmachus, and this union of two such powerful families
allowed him to move in the highest circles.[278] Standing strictly for the
right, and against all iniquity at court, he became the object of hatred on
the part of all the unscrupulous element near the throne, and his bold
defense of the ex-consul Albinus, unjustly accused of treason, led to his
imprisonment at Pavia[279] and his execution in 524.[280] Not many
generations after his death, the period being one in which historical
criticism was at its lowest ebb, the church found it profitable to look
upon his execution as a martyrdom.[281] He was {72} accordingly looked upon
as a saint,[282] his bones were enshrined,[283] and as a natural
consequence his books were among the classics in the church schools for a
thousand years.[284] It is pathetic, however, to think of the medieval
student trying to extract mental nourishment from a work so abstract, so
meaningless, so unnecessarily complicated, as the arithmetic of Boethius.

He was looked upon by his contemporaries and immediate successors as a
master, for Cassiodorus[285] (c. 490-c. 585 A.D.) says to him: "Through
your translations the music of Pythagoras and the astronomy of Ptolemy are
read by those of Italy, and the arithmetic of Nicomachus and the geometry
of Euclid are known to those of the West."[286] Founder of the medieval
scholasticism, {73} distinguishing the trivium and quadrivium,[287] writing
the only classics of his time, Gibbon well called him "the last of the
Romans whom Cato or Tully could have acknowledged for their
countryman."[288]

The second question relating to Boethius is this: Could he possibly have
known the Hindu numerals? In view of the relations that will be shown to
have existed between the East and the West, there can only be an
affirmative answer to this question. The numerals had existed, without the
zero, for several centuries; they had been well known in India; there had
been a continued interchange of thought between the East and West; and
warriors, ambassadors, scholars, and the restless trader, all had gone back
and forth, by land or more frequently by sea, between the Mediterranean
lands and the centers of Indian commerce and culture. Boethius could very
well have learned one or more forms of Hindu numerals from some traveler or
merchant.

To justify this statement it is necessary to speak more fully of these
relations between the Far East and Europe. It is true that we have no
records of the interchange of learning, in any large way, between eastern
Asia and central Europe in the century preceding the time of Boethius. But
it is one of the mistakes of scholars to believe that they are the sole
transmitters of knowledge. {74} As a matter of fact there is abundant
reason for believing that Hindu numerals would naturally have been known to
the Arabs, and even along every trade route to the remote west, long before
the zero entered to make their place-value possible, and that the
characters, the methods of calculating, the improvements that took place
from time to time, the zero when it appeared, and the customs as to solving
business problems, would all have been made known from generation to
generation along these same trade routes from the Orient to the Occident.
It must always be kept in mind that it was to the tradesman and the
wandering scholar that the spread of such learning was due, rather than to
the school man. Indeed, Avicenna[289] (980-1037 A.D.) in a short biography
of himself relates that when his people were living at Bokh[=a]ra his
father sent him to the house of a grocer to learn the Hindu art of
reckoning, in which this grocer (oil dealer, possibly) was expert. Leonardo
of Pisa, too, had a similar training.

The whole question of this spread of mercantile knowledge along the trade
routes is so connected with the [.g]ob[=a]r numerals, the Boethius
question, Gerbert, Leonardo of Pisa, and other names and events, that a
digression for its consideration now becomes necessary.[290]

{75}

Even in very remote times, before the Hindu numerals were sculptured in the
cave of N[=a]n[=a] Gh[=a]t, there were trade relations between Arabia and
India. Indeed, long before the Aryans went to India the great Turanian race
had spread its civilization from the Mediterranean to the Indus.[291] At a
much later period the Arabs were the intermediaries between Egypt and Syria
on the west, and the farther Orient.[292] In the sixth century B.C.,
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
Phoenicians 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 P[=a]tal[=i]pu[t.]ra, the present Patna.[301]

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

Moreover the results of the early Greek invasion were embodied by
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 Var[=a]ha-Mihira (6th century A.D.), a contemporary of
[=A]ryabha[t.]a, entitled a work of his the _B[r.]hat-Sa[m.]hit[=a]_, a
literal translation of [Greek: megalê suntaxis] of Ptolemy;[306] and in
various ways is this interchange of ideas apparent.[307] It could not have
been at all unusual for the ancient Greeks to go to India, for Strabo lays
down the route, saying that all who make the journey start from Ephesus and
traverse Phrygia and Cappadocia before taking the direct road.[308] The
products of the East were always finding their way to the West, the Greeks
getting their ginger[309] from Malabar, as the Phoenicians had long before
brought gold from Malacca.

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

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

{80}

Augustus speaks of envoys received by him from India, a thing never before
known,[318] and it is not improbable that he also received an embassy from
China.[319] Suetonius (first century A.D.) speaks in his history of these
relations,[320] as do several of his contemporaries,[321] and Vergil[322]
tells of Augustus doing battle in Persia. In Pliny's time the trade of the
Roman Empire with Asia amounted to a million and a quarter dollars a year,
a sum far greater relatively then than now,[323] while by the time of
Constantine Europe was in direct communication with the Far East.[324]

In view of these relations it is not beyond the range of possibility that
proof may sometime come to light to show that the Greeks and Romans knew
something of the {81} number system of India, as several writers have
maintained.[325]

Returning to the East, there are many evidences of the spread of knowledge
in and about India itself. In the third century B.C. Buddhism began to be a
connecting medium of thought. It had already permeated the Himalaya
territory, had reached eastern Turkestan, and had probably gone thence to
China. Some centuries later (in 62 A.D.) the Chinese emperor sent an
ambassador to India, and in 67 A.D. a Buddhist monk was invited to
China.[326] Then, too, in India itself A['s]oka, whose name has already
been mentioned in this work, extended the boundaries of his domains even
into Afghanistan, so that it was entirely possible for the numerals of the
Punjab to have worked their way north even at that early date.[327]

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

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

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

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

Such is a very brief 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 Br[=a]hm[=i]
numerals would not have attracted the attention of scholars, for they had
no zero so far as we know, and therefore they were no better and no worse
than those of dozens of other systems. If Boethius was attracted to them it
was probably exactly as any one is naturally attracted to the bizarre or
the mystic, and he would have mentioned them in his works only
incidentally, as indeed they are mentioned in the manuscripts in which they
occur.

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

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

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

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

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

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

{87}

4. The passage in question has all the appearance of an interpolation by
some scribe. Boethius is speaking of angles, in his work on geometry, when
the text suddenly changes to a discussion of classes of numbers.[346] This
is followed by a chapter in explanation of the abacus,[347] in which are
described those numeral forms which are called _apices_ or
_caracteres_.[348] The forms[349] of these characters vary in different
manuscripts, but in general are about as shown on page 88. They are
commonly written with the 9 at the left, decreasing to the unit at the
right, numerous writers stating that this was because they were derived
from Semitic sources in which the direction of writing is the opposite of
our own. This practice continued until the sixteenth century.[350] The
writer then leaves the subject entirely, using the Roman numerals for the
rest of his discussion, a proceeding so foreign to the method of Boethius
as to be inexplicable on the hypothesis of authenticity. Why should such a
scholarly writer have given them with no mention of their origin or use?
Either he would have mentioned some historical interest attaching to them,
or he would have used them in some discussion; he certainly would not have
left the passage as it is.

{88}

FORMS OF THE NUMERALS, LARGELY FROM WORKS ON THE ABACUS[351]

  a[352] [Illustration]
  b[353] [Illustration]
  c[354] [Illustration]
  d[355] [Illustration]
  e[356] [Illustration]
  f[357] [Illustration]
  g[358] [Illustration]
  h[359] [Illustration]
  i[360] [Illustration]

{89}

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

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

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

       *       *       *       *       *


{91}

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 Khosr[=u]
Nu['s][=i]rw[=a]n,[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. Khosr[=u] 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-pad[=e]['s]a_, 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-Faz[=a]r[=i].[366] Al-Khow[=a]razm[=i] and [H.]abash (A[h.]med ibn
`Abdall[=a]h, died c. 870) based their well-known tables upon the work of
Al-F[=a]zar[=i]. 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-Ma['s]`[=u]d[=i] and
Al-B[=i]r[=u]n[=i] follow quite closely upon the men mentioned, it is well
to note again the Arab writers on Hindu arithmetic, contemporary with
Al-Khow[=a]razm[=i], who were mentioned in chapter I, viz. Al-Kind[=i],
Sened ibn `Al[=i], and Al-[S.][=u]f[=i].

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 `Abdall[=a]h ibn al-[H.]asan, Ab[=u] 'l-Q[=a]sim[367] (died 987 A.D.)
of Antioch, and Mo[h.]ammed ibn `Abdall[=a]h, Ab[=u] Na[s.]r[368] (c. 982),
of Kalw[=a]d[=a] 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 Sin[=a]n ibn al-Fat[h.][370] of
[H.]arr[=a]n, and Ahmed ibn `Omar, al-Kar[=a]b[=i]s[=i].[371] In the
eleventh century come Al-B[=i]r[=u]n[=i][372] (973-1048) and `Ali ibn
A[h.]med, Ab[=u] 'l-[H.]asan, Al-Nasaw[=i][373] (c. 1030). The following
century brings similar works by Ish[=a]q ibn Y[=u]suf al-[S.]ardaf[=i][374]
and Sam[=u]'[=i]l ibn Ya[h.]y[=a] ibn `Abb[=a]s al-Ma[.g]reb[=i]
al-Andalus[=i][375] (c. 1174), and in the thirteenth century are
`Abdallat[=i]f ibn Y[=u]suf ibn Mo[h.]ammed, Muwaffaq al-D[=i]n Ab[=u]
Mo[h.]ammed al-Ba[.g]d[=a]d[=i][376] (c. 1231), and Ibn al-Bann[=a].[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 [.g]ob[=a]r. The title of the work is _Interesting and
Beautiful Problems on Numbers_ copied by A[h.]med ibn Mo[h.]ammed ibn
`Abdaljal[=i]l, Ab[=u] Sa`[=i]d, al-Sijz[=i],[380] (951-1024) from a work
by a priest and physician, Na[z.][=i]f ibn Yumn,[381] al-Qass (died c.
990). Suter does not mention this work of Na[z.][=i]f.

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 Mo[h.]ammed 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 Mo[h.]ammed 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
Mo[h.]ammed'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] D[=a]r
al-Sal[=a]m--"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-Man[s.][=u]r[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-Man[s.][=u]r and
Al-M[=a]m[=u]n and H[=a]r[=u]n al-Rash[=i]d (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 Mo[h.]ammed ibn M[=u]s[=a] al-Khow[=a]razm[=i],
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-Khow[=a]razm[=i], and working also under
Al-M[=a]m[=u]n, was a Jewish astronomer, Ab[=u] 'l-[T.]eiyib, {98} Sened
ibn `Al[=i], 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. `Al[=i] ibn A[h.]med al-Nasaw[=i], 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-Khow[=a]razm[=i]. 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 [.g]ob[=a]r numerals were probably in
that country, and these may well have made their way into Europe from the
schools of Cordova, Granada, and Toledo.

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

There was also a Bagdad merchant, one Ab[=u] 'l-Q[=a]sim `Obeidall[=a]h ibn
A[h.]med, better known by his Persian name Ibn Khord[=a][d.]beh,[400] who
wrote about 850 A.D. a work entitled _Book of Roads and Provinces_[401] in
which the following graphic account appears:[402] "The Jewish merchants
speak Persian, Roman (Greek and Latin), Arabic, French, Spanish, and
Slavic. They travel from the West to the East, and from the East to the
West, sometimes by land, sometimes by sea. They take ship from France on
the Western Sea, and they voyage to Farama (near the ruins of the ancient
Pelusium); there they transfer their goods to caravans and go by land to
Colzom (on the Red Sea). They there 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 _Gulist[=a]n_ of the Persian
poet Sa`d[=i][403] contains such a passage:

"I met a merchant who owned one hundred and forty camels, and fifty slaves
and porters.... He answered to me: 'I want to carry sulphur of Persia to
China, which in that country, as I hear, bears a high price; and thence to
take Chinese ware to Roum; and from Roum to load up with brocades for Hind;
and so to trade Indian steel (_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 Khord[=a][d.]beh, who
left Bagdad in the latter part of the ninth century, gives a picture of the
great commercial activity at that time in the Saracen city of Palermo. In
this same century they had established themselves in Piedmont, and in 906
they pillaged Turin.[417] On the Sorrento peninsula the traveler who climbs
the hill to the beautiful Ravello sees still several traces of the Arab
architecture, reminding him of the fact that about 900 A.D. Amalfi was a
commercial center of the Moors.[418] Not only at this time, but even a
century earlier, the artists of northern India sold their wares at such
centers, and in the courts both of H[=a]r[=u]n al-Rash[=i]d and of
Charlemagne.[419] Thus the Arabs dominated the Mediterranean Sea long
before Venice

              "held the gorgeous East in fee
  And was the safeguard of the West,"

and long before Genoa had become her powerful rival.[420]

{107}

Only a little later than this the brothers Nicolo and Maffeo Polo entered
upon their famous wanderings.[421] Leaving Constantinople in 1260, they
went by the Sea of Azov to Bokhara, and thence to the court of Kublai Khan,
penetrating China, and returning by way of Acre in 1269 with a commission
which required them to go back to China two years later. This time they
took with them Nicolo's son Marco, the historian of the journey, and went
across the plateau of Pamir; they spent about twenty years in China, and
came back by sea from China to Persia.

The ventures of the Poli were not long unique, however: the thirteenth
century had not closed before Roman missionaries and the merchant Petrus de
Lucolongo had penetrated China. Before 1350 the company of missionaries was
large, converts were numerous, churches and Franciscan convents had been
organized in the East, travelers were appealing for the truth of their
accounts to the "many" persons in Venice who had been in China,
Tsuan-chau-fu had a European merchant community, and Italian trade and
travel to China was a thing that occupied two chapters of a commercial
handbook.[422]

{108}

It is therefore reasonable to conclude that in the Middle Ages, as in the
time of Boethius, it was a simple matter for any inquiring scholar to
become acquainted with such numerals of the Orient as merchants may have
used for warehouse or price marks. And the fact that Gerbert seems to have
known only the forms of the simplest of these, not comprehending their full
significance, seems to prove that he picked them up in just this way.

Even if Gerbert did not bring his knowledge of the Oriental numerals from
Spain, he may easily have obtained them from the marks on merchant's goods,
had he been so inclined. Such knowledge was probably obtainable in various
parts of Italy, though as parts of mere mercantile knowledge the forms
might soon have been lost, it needing the pen of the scholar to preserve
them. Trade at this time was not stagnant. During the eleventh and twelfth
centuries the Slavs, for example, had very great commercial interests,
their trade reaching to Kiev and Novgorod, and thence to the East.
Constantinople was a great clearing-house of commerce with the Orient,[423]
and the Byzantine merchants must have been entirely familiar with the
various numerals of the Eastern peoples. In the eleventh century the
Italian town of Amalfi established a factory[424] in Constantinople, and
had trade relations with Antioch and Egypt. Venice, as early as the ninth
century, had a valuable trade with Syria and Cairo.[425] Fifty years after
Gerbert died, in the time of Cnut, the Dane and the Norwegian pushed their
commerce far beyond the northern seas, both by caravans through Russia to
the Orient, and by their venturesome barks which {109} sailed through the
Strait of Gibraltar into the Mediterranean.[426] Only a little later,
probably before 1200 A.D., a clerk in the service of Thomas à Becket,
present at the latter's death, wrote a life of the martyr, to which
(fortunately for our purposes) he prefixed a brief eulogy of the city of
London.[427] This clerk, William Fitz Stephen by name, thus speaks of the
British capital:

  Aurum mittit Arabs: species et thura Sabæus:
  Arma Sythes: oleum palmarum divite sylva
  Pingue solum Babylon: Nilus lapides pretiosos:
  Norwegi, Russi, varium grisum, sabdinas:
  Seres, purpureas vestes: Galli, sua vina.

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

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

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

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

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

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

The question still to be settled is as to where he found these numerals.
That he did not bring them from Spain is the opinion of a number of careful
investigators.[449] This is thought to be the more probable because most of
the men who made Spain famous for learning lived after Gerbert was there.
Such were Ibn S[=i]n[=a] (Avicenna) who lived at the beginning, and Gerber
of Seville who flourished in the middle, of the eleventh century, and
Ab[=u] Roshd (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 [.g]ob[=a]r numerals were used
for marking wares or keeping simple accounts, it was he who would have
known them, and who would have been the one rather than any Arab scholar to
bring them to the inquiring mind of the young French monk. The facts that
Gerbert knew them only imperfectly, that he used them solely for
calculations, and that the forms are evidently like the Spanish
[.g]ob[=a]r, make it all the more probable that it was through the small
tradesman of the Moors that this versatile scholar derived his knowledge.
Moreover the part of the geometry bearing his name, and that seems
unquestionably his, shows the Arab influence, proving that he at least came
into contact with the transplanted Oriental learning, even though
imperfectly.[452] There was also the persistent Jewish merchant trading
with both peoples then as now, always alive to the acquiring of useful
knowledge, and it would be very natural for a man like Gerbert to welcome
learning from such a source.

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

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

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

{117}

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

To the figures on the _apices_ were given the names Igin, andras, ormis,
arbas, quimas, calctis or caltis, zenis, temenias, celentis, sipos,[470]
the origin and meaning of which still remain a mystery. The Semitic origin
of several of the words seems probable. _Wahud_, _thaneine_, {119}
_thalata_, _arba_, _kumsa_, _setta_, _sebba_, _timinia_, _taseud_ are given
by the Rev. R. Patrick[471] as the names, in an Arabic dialect used in
Morocco, for the numerals from one to nine. Of these the words for four,
five, and eight are strikingly like those given above.

The name _apices_ was not, however, a common one in later times. _Notae_
was more often used, and it finally gave the name to notation.[472] Still
more common were the names _figures_, _ciphers_, _signs_, _elements_, and
_characters_.[473]

So little effect did the teachings of Gerbert have in making known the new
numerals, that O'Creat, who lived a century later, a friend and pupil of
Adelhard {120} of Bath, used the zero with the Roman characters, in
contrast to Gerbert's use of the [.g]ob[=a]r forms without the zero.[474]
O'Creat uses three forms for zero, o, [=o], and [Greek: t], as in Maximus
Planudes. With this use of the zero goes, naturally, a place value, for he
writes III III for 33, ICCOO and I. II. [tau]. [tau] for 1200,
I. O. VIII. IX for 1089, and I. IIII. IIII. [tau][tau][tau][tau] for the
square of 1200.

The period from the time of Gerbert until after the appearance of
Leonardo's monumental work may be called the period of the abacists. Even
for many years after the appearance early in the twelfth century of the
books explaining the Hindu art of reckoning, there was strife between the
abacists, the advocates of the abacus, and the algorists, those who favored
the new numerals. The words _cifra_ and _algorismus cifra_ were used with a
somewhat derisive significance, indicative of absolute uselessness, as
indeed the zero is useless on an abacus in which the value of any unit is
given by the column which it occupies.[475] So Gautier de Coincy
(1177-1236) in a work on the miracles of Mary says:

  A horned beast, a sheep,
  An algorismus-cipher,
  Is a priest, who on such a feast day
  Does not celebrate the holy Mother.[476]

So the abacus held the field for a long time, even against the new algorism
employing the new numerals. {121} Geoffrey Chaucer[477] describes in _The
Miller's Tale_ the clerk with

 "His Almageste and bokes grete and smale,
  His astrelabie, longinge for his art,
  His augrim-stones layen faire apart
  On shelves couched at his beddes heed."

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

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

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

After Gerbert's death, little by little the scholars of Europe came to know
the new figures, chiefly through the introduction of Arab learning. The
Dark Ages had passed, although arithmetic did not find another advocate as
prominent as Gerbert for two centuries. Speaking of this great revival,
Raoul Glaber[493] (985-c. 1046), a monk of the great Benedictine abbey of
Cluny, of the eleventh century, says: "It was as though the world had
arisen and tossed aside the worn-out garments of ancient time, and wished
to apparel itself in a white robe of churches." And with this activity in
religion came a corresponding interest in other lines. Algorisms began to
appear, and knowledge from the outside world found {124} interested
listeners. Another Raoul, or Radulph, to whom we have referred as Radulph
of Laon,[494] a teacher in the cloister school of his city, and the brother
of Anselm of Laon[495] the celebrated theologian, wrote a treatise on
music, extant but unpublished, and an arithmetic which Nagl first published
in 1890.[496] The latter work, preserved to us in a parchment manuscript of
seventy-seven leaves, contains a curious mixture of Roman and [.g]ob[=a]r
numerals, the former for expressing large results, the latter for practical
calculation. These [.g]ob[=a]r "caracteres" include the sipos (zero),
[Symbol], of which, however, Radulph did not know the full significance;
showing that at the opening of the twelfth century the system was still
uncertain in its status in the church schools of central France.

At the same time the words _algorismus_ and _cifra_ were coming into
general use even in non-mathematical literature. Jordan [497] cites
numerous instances of such use from the works of Alanus ab Insulis[498]
(Alain de Lille), Gautier de Coincy (1177-1236), and others.

Another contributor to arithmetic during this interesting period was a
prominent Spanish Jew called variously John of Luna, John of Seville,
Johannes Hispalensis, Johannes Toletanus, and Johannes Hispanensis de
Luna.[499] {125} His date is rather closely fixed by the fact that he
dedicated a work to Raimund who was archbishop of Toledo between 1130 and
1150.[500] His interests were chiefly in the translation of Arabic works,
especially such as bore upon the Aristotelian philosophy. From the
standpoint of arithmetic, however, the chief interest centers about a
manuscript entitled _Joannis Hispalensis liber Algorismi de Practica
Arismetrice_ which Boncompagni found in what is now the _Bibliothèque
nationale_ at Paris. Although this distinctly lays claim to being
Al-Khow[=a]razm[=i]'s work,[501] the evidence is altogether against the
statement,[502] but the book is quite as valuable, since it represents the
knowledge of the time in which it was written. It relates to the operations
with integers and sexagesimal fractions, including roots, and contains no
applications.[503]

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

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

The earliest trace we have of computation with Hindu numerals in Germany is
in an Algorismus of 1143, now in the Hofbibliothek in Vienna.[510] It is
bound in with a {127} _Computus_ by the same author and bearing the date
given. It contains chapters "De additione," "De diminutione," "De
mediatione," "De divisione," and part of a chapter on multiplication. The
numerals are in the usual medieval forms except the 2 which, as will be
seen from the illustration,[511] is somewhat different, and the 3, which
takes the peculiar shape [Symbol], a form characteristic of the twelfth
century.

It was about the same time that the _Sefer ha-Mispar_,[512] the Book of
Number, appeared in the Hebrew language. The author, Rabbi Abraham ibn 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, [Hebrew: A B G D H W Z CH T`], for the numerals 9 8 7 6 5
4 3 2 1, and a circle for the zero. The quotation in the note given below
shows that he knew of the Hindu origin; but in his manuscript, although he
set down the Hindu forms, he used the above nine Hebrew letters with place
value for all computations.

       *       *       *       *       *


{128}

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 Leboeuf 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 [.g]ob[=a]r 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 arithmetic[e,]. 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 [.g]ob[=a]r 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

[Illustration]

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,

[Illustration]

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

[Illustration]

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 [Symbol 2]m.,... [Symbol 3]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,

  [Illustration] for 125      [Illustration] for 150
  [Illustration] for 147      [Illustration] 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

  [577] [Illustration] Twelfth century A.D.
  [578] [Illustration] 1197 A.D.
  [579] [Illustration] 1275 A.D.
  [580] [Illustration] c. 1294 A.D.
  [581] [Illustration] c. 1303 A.D.
  [582] [Illustration] c. 1360 A.D.
  [583] [Illustration] c. 1442 A.D.

{144}

[Illustration]

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.

[Illustration]

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.

[Illustration]

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]

[Illustration]

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]

[Illustration]

{146}

It is evident, from the early traces, that it is merely a cursive form for
the primitive [2 horizontal strokes], just as 3 comes from [3 horizontal
strokes], as in the N[=a]n[=a] Gh[=a]t inscriptions.

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

[Illustration]

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 [Symbol], although the Florentine manuscript of Leonard of Pisa's
work has the form [Symbol];[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 [Symbol][594] and [Symbol][594]
or [Symbol],[595] more closely resembling our own. The first printed books
generally used our present form[596] with the closed top [Symbol], the open
top used in writing ( [Symbol]) being {147} purely modern. The following
are other forms of the four, from various manuscripts:[597]

[Illustration]

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

[Illustration]

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]

[Illustration]

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

[Illustration]

{148}

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

[Illustration]

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

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

[Illustration]

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

[Illustration]

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 no[gh]t, 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
c[alpha] 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]:

[Illustration]

{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._

  Abbo of Fleury, 122
  `Abdall[=a]h ibn al-[H.]asan, 92
  `Abdallat[=i]f ibn Y[=u]suf, 93
  `Abdalq[=a]dir ibn `Al[=i] al-Sakh[=a]w[=i], 6
  Abenragel, 34
  Abraham ibn Meïr ibn Ezra, _see_ Rabbi ben Ezra
  Ab[=u] `Al[=i] al-[H.]osein ibn S[=i]n[=a], 74
  Ab[=u] 'l-[H.]asan, 93, 100
  Ab[=u] 'l-Q[=a]sim, 92
  Ab[=u] 'l-[T.]eiyib, 97
  Ab[=u] Na[s.]r, 92
  Ab[=u] Roshd, 113
  Abu Sahl Dunash ibn Tamim, 65, 67
  Adelhard of Bath, 5, 55, 97, 119, 123, 126
  Adhemar of Chabanois, 111
  A[h.]med al-Nasaw[=i], 98
  A[h.]med ibn `Abdall[=a]h, 9, 92
  A[h.]med ibn Mo[h.]ammed, 94
  A[h.]med ibn `Omar, 93
  Ak[s.]aras, 32
  Alanus ab Insulis, 124
  Al-Ba[.g]d[=a]d[=i], 93
  Al-Batt[=a]n[=i], 54
  Albelda (Albaida) MS., 116
  Albert, J., 62
  Albert of York, 103
  Al-B[=i]r[=u]n[=i], 6, 41, 49, 65, 92, 93
  Alcuin, 103
  Alexander the Great, 76
  Alexander de Villa Dei, 11, 133
  Alexandria, 64, 82
  Al-Faz[=a]r[=i], 92
  Alfred, 103
  Algebra, etymology, 5
  Algerian numerals, 68
  Algorism, 97
  Algorismus, 124, 126, 135
  Algorismus cifra, 120
  Al-[H.]a[s.][s.][=a]r, 65
  `Al[=i] ibn Ab[=i] Bekr, 6
  `Al[=i] ibn A[h.]med, 93, 98
  Al-Kar[=a]b[=i]s[=i], 93
  Al-Khow[=a]razm[=i], 4, 9, 10, 92, 97, 98, 125, 126
  Al-Kind[=i], 10, 92
  Almagest, 54
  Al-Ma[.g]reb[=i], 93
  Al-Ma[h.]all[=i], 6
  Al-M[=a]m[=u]n, 10, 97
  Al-Man[s.][=u]r, 96, 97
  Al-Mas`[=u]d[=i], 7, 92
  Al-Nad[=i]m, 9
  Al-Nasaw[=i], 93, 98
  Alphabetic numerals, 39, 40, 43
  Al-Q[=a]sim, 92
  Al-Qass, 94
  Al-Sakh[=a]w[=i], 6
  Al-[S.]ardaf[=i], 93
  Al-Sijz[=i], 94
  Al-S[=u]f[=i], 10, 92
  Ambrosoli, 118
  A[.n]kapalli, 43
  Apices, 87, 117, 118
  Arabs, 91-98
  Arbuthnot, 141
  {154}
  Archimedes, 15, 16
  Arcus Pictagore, 122
  Arjuna, 15
  Arnold, E., 15, 102
  Ars memorandi, 141
  [=A]ryabha[t.]a, 39, 43, 44
  Aryan numerals, 19
  Aschbach, 134
  Ashmole, 134
  A['s]oka, 19, 20, 22, 81
  A[s.]-[s.]ifr, 57, 58
  Astrological numerals, 150
  Atharva-Veda, 48, 49, 55
  Augustus, 80
  Averroës, 113
  Avicenna, 58, 74, 113

  Babylonian numerals, 28
  Babylonian zero, 51
  Bacon, R., 131
  Bactrian numerals, 19, 30
  Bæda, 2, 72
  Bagdad, 4, 96
  Bakh[s.][=a]l[=i] manuscript, 43, 49, 52, 53
  Ball, C. J., 35
  Ball, W. W. R., 36, 131
  B[=a][n.]a, 44
  Barth, A., 39
  Bayang inscriptions, 39
  Bayer, 33
  Bayley, E. C., 19, 23, 30, 32, 52, 89
  Beazley, 75
  Bede, _see_ Bæda
  Beldomandi, 137
  Beloch, J., 77
  Bendall, 25, 52
  Benfey, T., 26
  Bernelinus, 88, 112, 117, 121
  Besagne, 128
  Besant, W., 109
  Bettino, 36
  Bhandarkar, 18, 47, 49
  Bh[=a]skara, 53, 55
  Biernatzki, 32
  Biot, 32
  Björnbo, A. A., 125, 126
  Blassière, 119
  Bloomfield, 48
  Blume, 85
  Boeckh, 62
  Boehmer, 143
  Boeschenstein, 119
  Boethius, 63, 70-73, 83-90
  Boissière, 63
  Bombelli, 81
  Bonaini, 128
  Boncompagni, 5, 6, 10, 48, 49, 123, 125
  Borghi, 59
  Borgo, 119
  Bougie, 130
  Bowring, J., 56
  Brahmagupta, 52
  Br[=a]hma[n.]as, 12, 13
  Br[=a]hm[=i], 19, 20, 31, 83
  Brandis, J., 54
  B[r.]hat-Sa[m.]hita, 39, 44, 78
  Brockhaus, 43
  Bubnov, 65, 84, 110, 116
  Buddha, education of, 15, 16
  Büdinger, 110
  Bugia, 130
  Bühler, G., 15, 19, 22, 31, 44, 49
  Burgess, 25
  Bürk, 13
  Burmese numerals, 36
  Burnell, A. C., 18, 40
  Buteo, 61

  Calandri, 59, 81
  Caldwell, R., 19
  Calendars, 133
  Calmet, 34
  Cantor, M., 5, 13, 30, 43, 84
  {155}
  Capella, 86
  Cappelli, 143
  Caracteres, 87, 113, 117, 119
  Cardan, 119
  Carmen de Algorismo, 11, 134
  Casagrandi, 132
  Casiri, 8, 10
  Cassiodorus, 72
  Cataldi, 62
  Cataneo, 3
  Caxton, 143, 146
  Ceretti, 32
  Ceylon numerals, 36
  Chalfont, F. H., 28
  Champenois, 60
  Characters, _see_ Caracteres
  Charlemagne, 103
  Chasles, 54, 60, 85, 116, 122, 135
  Chassant, L. A., 142
  Chaucer, 121
  Chiarini, 145, 146
  Chiffre, 58
  Chinese numerals, 28, 56
  Chinese zero, 56
  Cifra, 120, 124
  Cipher, 58
  Circulus, 58, 60
  Clichtoveus, 61, 119, 145
  Codex Vigilanus, 138
  Codrington, O., 139
  Coins dated, 141
  Colebrooke, 8, 26, 46, 53
  Constantine, 104, 105
  Cosmas, 82
  Cossali, 5
  Counters, 117
  Courteille, 8
  Coxe, 59
  Crafte of Nombrynge, 11, 87, 149
  Crusades, 109
  Cunningham, A., 30, 75
  Curtze, 55, 59, 126, 134
  Cyfra, 55

  Dagomari, 146
  D'Alviella, 15
  Dante, 72
  Dasypodius, 33, 67, 63
  Daunou, 135
  Delambre, 54
  Devan[=a]gar[=i], 7
  Devoulx, A., 68
  Dhruva, 49
  Dicæarchus of Messana, 77
  Digits, 119
  Diodorus Siculus, 76
  Du Cange, 62
  Dumesnil, 36
  Dutt, R. C., 12, 15, 18, 75
  Dvived[=i], 44

  East and West, relations, 73-81, 100-109
  Egyptian numerals, 27
  Eisenlohr, 28
  Elia Misrachi, 57
  Enchiridion Algorismi, 58
  Eneström, 5, 48, 59, 97, 125, 128
  Europe, numerals in, 63, 99, 128, 136
  Eusebius Caesariensis, 142
  Euting, 21
  Ewald, P., 116

  Fazzari, 53, 54
  Fibonacci, _see_ Leonardo of Pisa
  Figura nihili, 58
  Figures, 119. _See_ numerals.
  Fihrist, 67, 68, 93
  Finaeus, 57
  Firdus[=i], 81
  Fitz Stephen, W., 109
  Fleet, J. C., 19, 20, 49
  {156}
  Florus, 80
  Flügel, G., 68
  Francisco de Retza, 142
  François, 58
  Friedlein, G., 84, 113, 116, 122
  Froude, J. A., 129

  Gandh[=a]ra, 19
  Garbe, 48
  Gasbarri, 58
  Gautier de Coincy, 120, 124
  Gemma Frisius, 2, 3, 119
  Gerber, 113
  Gerbert, 108, 110-120, 122
  Gerhardt, C. I., 43, 56, 93, 118
  Gerland, 88, 123
  Gherard of Cremona, 125
  Gibbon, 72
  Giles, H. A., 79
  Ginanni, 81
  Giovanni di Danti, 58
  Glareanus, 4, 119
  Gnecchi, 71, 117
  [.G]ob[=a]r numerals, 65, 100, 112, 124, 138
  Gow, J., 81
  Grammateus, 61
  Greek origin, 33
  Green, J. R., 109
  Greenwood, I., 62, 119
  Guglielmini, 128
  Gulist[=a]n, 102
  Günther, S., 131
  Guyard, S., 82

  [H.]abash, 9, 92
  Hager, J. (G.), 28, 32
  Halliwell, 59, 85
  Hankel, 93
  H[=a]r[=u]n al-Rash[=i]d, 97, 106
  Havet, 110
  Heath, T. L., 125
  Hebrew numerals, 127
  Hecatæus, 75
  Heiberg, J. L., 55, 85, 148
  Heilbronner, 5
  Henry, C., 5, 31, 55, 87, 120, 135
  Heriger, 122
  Hermannus Contractus, 123
  Herodotus, 76, 78
  Heyd, 75
  Higden, 136
  Hill, G. F., 52, 139, 142
  Hillebrandt, A., 15, 74
  Hilprecht, H. V., 28
  Hindu forms, early, 12
  Hindu number names, 42
  Hodder, 62
  Hoernle, 43, 49
  Holywood, _see_ Sacrobosco
  Hopkins, E. W., 12
  Horace, 79, 80
  [H.]osein ibn Mo[h.]ammed al-Ma[h.]all[=i], 6
  Hostus, M., 56
  Howard, H. H., 29
  Hrabanus Maurus, 72
  Huart, 7
  Huet, 33
  Hugo, H., 57
  Humboldt, A. von, 62
  Huswirt, 58

  Iamblichus, 81
  Ibn Ab[=i] Ya`q[=u]b, 9
  Ibn al-Adam[=i], 92
  Ibn al-Bann[=a], 93
  Ibn Khord[=a][d.]beh, 101, 106
  Ibn Wahab, 103
  India, history of, 14
    writing in, 18
  Indicopleustes, 83
  Indo-Bactrian numerals, 19
  {157}
  Indr[=a]j[=i], 23
  Is[h.][=a]q ibn Y[=u]suf al-[S.]ardaf[=i], 93

  Jacob of Florence, 57
  Jacquet, E., 38
  Jamshid, 56
  Jehan Certain, 59
  Jetons, 58, 117
  Jevons, F. B., 76
  Johannes Hispalensis, 48, 88, 124
  John of Halifax, _see_ Sacrobosco
  John of Luna, _see_ Johannes Hispalensis
  Jordan, L., 58, 124
  Joseph Ispanus (Joseph Sapiens), 115
  Justinian, 104

  Kále, M. R., 26
  Karabacek, 56
  Karpinski, L. C., 126, 134, 138
  K[=a]ty[=a]yana, 39
  Kaye, C. R., 6, 16, 43, 46, 121
  Keane, J., 75, 82
  Keene, H. G., 15
  Kern, 44
  Kharo[s.][t.]h[=i], 19, 20
  Khosr[=u], 82, 91
  Kielhorn, F., 46, 47
  Kircher, A., 34
  Kit[=a]b al-Fihrist, _see_ Fihrist
  Kleinwächter, 32
  K[=l]os, 62
  Köbel, 4, 58, 60, 119, 123
  Krumbacher, K., 57
  Kuckuck, 62, 133
  Kugler, F. X., 51

  Lachmann, 85
  Lacouperie, 33, 35
  Lalitavistara, 15, 17
  Lami, G., 57
  La Roche, 61
  Lassen, 39
  L[=a][t.]y[=a]yana, 39
  Leboeuf, 135
  Leonardo of Pisa, 5, 10, 57, 64, 74, 120, 128-133
  Lethaby, W. R., 142
  Levi, B., 13
  Levias, 3
  Libri, 73, 85, 95
  Light of Asia, 16
  Luca da Firenze, 144
  Lucas, 128

  Mah[=a]bh[=a]rata, 18
  Mah[=a]v[=i]r[=a]c[=a]rya, 53
  Malabar numerals, 36
  Malayalam numerals, 36
  Mannert, 81
  Margarita Philosophica, 146
  Marie, 78
  Marquardt, J., 85
  Marshman, J. C., 17
  Martin, T. H., 30, 62, 85, 113
  Martines, D. C., 58
  M[=a]sh[=a]ll[=a]h, 3
  Maspero, 28
  Mauch, 142
  Maximus Planudes, 2, 57, 66, 93, 120
  Megasthenes, 77
  Merchants, 114
  Meynard, 8
  Migne, 87
  Mikami, Y., 56
  Milanesi, 128
  Mo[h.]ammed ibn `Abdall[=a]h, 92
  Mo[h.]ammed ibn A[h.]med, 6
  Mo[h.]ammed ibn `Al[=i] `Abd[=i], 8
  Mo[h.]ammed ibn M[=u]s[=a], _see_ Al-Khow[=a]razm[=i]
  Molinier, 123
  Monier-Williams, 17
  {158}
  Morley, D., 126
  Moroccan numerals, 68, 119
  Mortet, V., 11
  Moseley, C. B., 33
  Mo[t.]ahhar ibn [T.][=a]hir, 7
  Mueller, A., 68
  Mumford, J. K., 109
  Muwaffaq al-D[=i]n, 93

  Nabatean forms, 21
  Nallino, 4, 54, 55
  Nagl, A., 55, 110, 113, 126
  N[=a]n[=a] Gh[=a]t inscriptions, 20, 22, 23, 40
  Narducci, 123
  Nasik cave inscriptions, 24
  Na[z.][=i]f ibn Yumn, 94
  Neander, A., 75
  Neophytos, 57, 62
  Neo-Pythagoreans, 64
  Nesselmann, 58
  Newman, Cardinal, 96
  Newman, F. W., 131
  Nöldeke, Th., 91
  Notation, 61
  Note, 61, 119
  Noviomagus, 45, 61, 119, 150
  Null, 61
  Numerals,
    Algerian, 68
    astrological, 150
    Br[=a]hm[=i], 19-22, 83
    early ideas of origin, 1
    Hindu, 26
    Hindu, classified, 19, 38
    Kharo[s.][t.]h[=i], 19-22
    Moroccan, 68
    Nabatean, 21
    origin, 27, 30, 31, 37
    supposed Arabic origin, 2
    supposed Babylonian origin, 28
    supposed Chaldean and Jewish origin, 3
    supposed Chinese origin, 28, 32
    supposed Egyptian origin, 27, 30, 69, 70
    supposed Greek origin, 33
    supposed Phoenician origin, 32
    tables of, 22-27, 36, 48, 49, 69, 88, 140, 143, 145-148

  O'Creat, 5, 55, 119, 120
  Olleris, 110, 113
  Oppert, G., 14, 75

  Pali, 22
  Pañcasiddh[=a]ntik[=a], 44
  Paravey, 32, 57
  P[=a]tal[=i]pu[t.]ra, 77
  Patna, 77
  Patrick, R., 119
  Payne, E. J., 106
  Pegolotti, 107
  Peletier, 2, 62
  Perrot, 80
  Persia, 66, 91, 107
  Pertz, 115
  Petrus de Dacia, 59, 61, 62
  Pez, P. B., 117
  "Philalethes," 75
  Phillips, G., 107
  Picavet, 105
  Pichler, F., 141
  Pihan, A. P., 36
  Pisa, 128
  Place value, 26, 42, 46, 48
  Planudes, _see_ Maximus Planudes
  Plimpton, G. A., 56, 59, 85, 143, 144, 145, 148
  Pliny, 76
  Polo, N. and M., 107
  {159}
  Prändel, J. G., 54
  Prinsep, J., 20, 31
  Propertius, 80
  Prosdocimo de' Beldomandi, 137
  Prou, 143
  Ptolemy, 54, 78
  Putnam, 103
  Pythagoras, 63
  Pythagorean numbers, 13
  Pytheas of Massilia, 76

  Rabbi ben Ezra, 60, 127
  Radulph of Laon, 60, 113, 118, 124
  Raets, 62
  Rainer, _see_ Gemma Frisius
  R[=a]m[=a]yana, 18
  Ramus, 2, 41, 60, 61
  Raoul Glaber, 123
  Rapson, 77
  Rauhfuss, _see_ Dasypodius
  Raumer, K. von, 111
  Reclus, E., 14, 96, 130
  Recorde, 3, 58
  Reinaud, 67, 74, 80
  Reveillaud, 36
  Richer, 110, 112, 115
  Riese, A., 119
  Robertson, 81
  Robertus Cestrensis, 97, 126
  Rodet, 5, 44
  Roediger, J., 68
  Rollandus, 144
  Romagnosi, 81
  Rosen, F., 5
  Rotula, 60
  Rudolff, 85
  Rudolph, 62, 67
  Ruffi, 150

  Sachau, 6
  Sacrobosco, 3, 58, 133
  Sacy, S. de, 66, 70
  Sa`d[=i], 102
  ['S]aka inscriptions, 20
  Sam[=u]'[=i]l ibn Ya[h.]y[=a], 93
  ['S][=a]rad[=a] characters, 55
  Savonne, 60
  Scaliger, J. C., 73
  Scheubel, 62
  Schlegel, 12
  Schmidt, 133
  Schonerus, 87, 119
  Schroeder, L. von, 13
  Scylax, 75
  Sedillot, 8, 34
  Senart, 20, 24, 25
  Sened ibn `Al[=i], 10, 98
  Sfortunati, 62, 150
  Shelley, W., 126
  Siamese numerals, 36
  Siddh[=a]nta, 8, 18
  [S.]ifr, 57
  Sigsboto, 55
  Sih[=a]b al-D[=i]n, 67
  Silberberg, 60
  Simon, 13
  Sin[=a]n ibn al-Fat[h.], 93
  Sindbad, 100
  Sindhind, 97
  Sipos, 60
  Sirr, H. C., 75
  Skeel, C. A., 74
  Smith, D. E., 11, 17, 53, 86, 141, 143
  Smith, V. A., 20, 35, 46, 47
  Smith, Wm., 75
  Sm[r.]ti, 17
  Spain, 64, 65, 100
  Spitta-Bey, 5
  Sprenger, 94
  ['S]rautas[=u]tra, 39
  Steffens, F., 116
  Steinschneider, 5, 57, 65, 66, 98, 126
  Stifel, 62
  {160}
  Subandhus, 44
  Suetonius, 80
  Suleim[=a]n, 100
  ['S][=u]nya, 43, 53, 57
  Suter, 5, 9, 68, 69, 93, 116, 131
  S[=u]tras, 13
  Sykes, P. M., 75
  Sylvester II, _see_ Gerbert
  Symonds, J. A., 129

  Tannery, P., 62, 84, 85
  Tartaglia, 4, 61
  Taylor, I., 19, 30
  Teca, 55, 61
  Tennent, J. E., 75
  Texada, 60
  Theca, 58, 61
  Theophanes, 64
  Thibaut, G., 12, 13, 16, 44, 47
  Tibetan numerals, 36
  Timotheus, 103
  Tonstall, C., 3, 61
  Trenchant, 60
  Treutlein, 5, 63, 123
  Trevisa, 136
  Treviso arithmetic, 145
  Trivium and quadrivium, 73
  Tsin, 56
  Tunis, 65
  Turchill, 88, 118, 123
  Turnour, G., 75
  Tziphra, 57, 62
  [Greek: tziphra], 55, 57, 62
  Tzwivel, 61, 118, 145

  Ujjain, 32
  Unger, 133
  Upanishads, 12
  Usk, 121

  Valla, G., 61
  Van der Schuere, 62
  Var[=a]ha-Mihira, 39, 44, 78
  V[=a]savadatt[=a], 44
  Vaux, Carra de, 9, 74
  Vaux, W. S. W., 91
  Ved[=a][.n]gas, 17
  Vedas, 12, 15, 17
  Vergil, 80
  Vincent, A. J. H., 57
  Vogt, 13
  Voizot, P., 36
  Vossius, 4, 76, 81, 84

  Wallis, 3, 62, 84, 116
  Wappler, E., 54, 126
  Wäschke, H., 2, 93
  Wattenbach, 143
  Weber, A., 31
  Weidler, I. F., 34, 66
  Weidler, I. F. and G. I., 63, 66
  Weissenborn, 85, 110
  Wertheim, G., 57, 61
  Whitney, W. D., 13
  Wilford, F., 75
  Wilkens, 62
  Wilkinson, J. G., 70
  Willichius, 3
  Woepcke, 3, 6, 42, 63, 64, 65, 67, 69, 70, 94, 113, 138
  Wolack, G., 54
  Woodruff, C. E., 32
  Word and letter numerals, 38, 44
  Wüstenfeld, 74

  Yule, H., 107

  Zephirum, 57, 58
  Zephyr, 59
  Zepiro, 58
  Zero, 26, 38, 40, 43, 45, 49, 51-62, 67
  Zeuero, 58

       *       *       *       *       *


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ALGEBRA FOR BEGINNERS

By DAVID EUGENE SMITH,

Professor of Mathematics in Teachers College, Columbia University

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Notes

al-Mekk[=i] on a treatise on [.g]ob[=a]r arithmetic (explained later)
called _Al-murshidah_, found by Woepcke in Paris (_Propagation_, p. 66),
there is mentioned the fact that there are "nine Indian figures" and "a
second kind of Indian figures ... although these are the figures of the
[.g]ob[=a]r writing." So in a commentary by [H.]osein ibn Mo[h.]ammed
al-Ma[h.]all[=i] (died in 1756) on the _Mokhta[s.]ar f[=i]`ilm
el-[h.]is[=a]b_ (Extract from Arithmetic) by `Abdalq[=a]dir ibn `Al[=i]
al-Sakh[=a]w[=i] (died c. 1000) it is related that "the preface treats of
the forms of the figures of Hindu signs, such as were established by the
Hindu nation." [Woepcke, _Propagation_, p. 63.]]

which, of course, are interpolations. An interesting example of a forgery
in ecclesiastical matters is in the charter said to have been given by St.
Patrick, granting indulgences to the benefactors of Glastonbury, dated "In
nomine domini nostri Jhesu Christi Ego Patricius humilis servunculus Dei
anno incarnationis ejusdem ccccxxx." Now if the Benedictines are right in
saying that Dionysius Exiguus, a Scythian monk, first arranged the
Christian chronology c. 532 A.D., this can hardly be other than spurious.
See Arbuthnot, loc. cit., p. 38.

[1] "_Discipulus._ Quis primus invenit numerum apud Hebræos et Ægyptios?
_Magister._ Abraham primus invenit numerum apud Hebræos, deinde Moses; et
Abraham tradidit istam scientiam numeri ad Ægyptios, et docuit eos: deinde
Josephus." [Bede, _De computo dialogus_ (doubtfully assigned to him),
_Opera omnia_, Paris, 1862, Vol. I, p. 650.]

"Alii referunt ad Phoenices inventores arithmeticæ, propter eandem
commerciorum caussam: Alii ad Indos: Ioannes de Sacrobosco, cujus
sepulchrum est Lutetiæ in comitio Maturinensi, refert ad Arabes." [Ramus,
_Arithmeticæ libri dvo_, Basel, 1569, p. 112.]

Similar notes are given by Peletarius in his commentary on the arithmetic
of Gemma Frisius (1563 ed., fol. 77), and in his own work (1570 Lyons ed.,
p. 14): "La valeur des Figures commence au coste dextre tirant vers le
coste senestre: au rebours de notre maniere d'escrire par ce que la
premiere prattique est venue des Chaldees: ou des Pheniciens, qui ont été
les premiers traffiquers de marchandise."

[2] Maximus Planudes (c. 1330) states that "the nine symbols come from the
Indians." [Wäschke's German translation, Halle, 1878, p. 3.] Willichius
speaks of the "Zyphræ Indicæ," in his _Arithmeticæ libri tres_ (Strasburg,
1540, p. 93), and Cataneo of "le noue figure de gli Indi," in his _Le
pratiche delle dve prime mathematiche_ (Venice, 1546, fol. 1). Woepcke is
not correct, therefore, in saying ("Mémoire sur la propagation des chiffres
indiens," hereafter referred to as _Propagation_ [_Journal Asiatique_, Vol.
I (6), 1863, p. 34]) that Wallis (_A Treatise on Algebra, both historical
and practical_, London, 1685, p. 13, and _De algebra tractatus_, Latin
edition in his _Opera omnia_, 1693, Vol. II, p. 10) was one of the first to
give the Hindu origin.

[3] From the 1558 edition of _The Grovnd of Artes_, fol. C, 5. Similarly
Bishop Tonstall writes: "Qui a Chaldeis primum in finitimos, deinde in
omnes pene gentes fluxit.... Numerandi artem a Chaldeis esse profectam: qui
dum scribunt, a dextra incipiunt, et in leuam progrediuntur." [_De arte
supputandi_, London, 1522, fol. B, 3.] Gemma Frisius, the great continental
rival of Recorde, had the same idea: "Primùm autem appellamus dexterum
locum, eo quòd haec ars vel à Chaldæis, vel ab Hebræis ortum habere
credatur, qui etiam eo ordine scribunt"; but this refers more evidently to
the Arabic numerals. [_Arithmeticæ practicæ methodvs facilis_, Antwerp,
1540, fol. 4 of the 1563 ed.] Sacrobosco (c. 1225) mentions the same thing.
Even the modern Jewish writers claim that one of their scholars,
M[=a]sh[=a]ll[=a]h (c. 800), introduced them to the Mohammedan world. [C.
Levias, _The Jewish Encyclopedia_, New York, 1905, Vol. IX, p. 348.]

[4] "... & que esto fu trouato di fare da gli Arabi con diece figure." [_La
prima parte del general trattato di nvmeri, et misvre_, Venice, 1556, fol.
9 of the 1592 edition.]

[5] "Vom welchen Arabischen auch disz Kunst entsprungen ist." [_Ain nerv
geordnet Rechenbiechlin_, Augsburg, 1514, fol. 13 of the 1531 edition. The
printer used the letters _rv_ for _w_ in "new" in the first edition, as he
had no _w_ of the proper font.]

[6] Among them Glareanus: "Characteres simplices sunt nouem significatiui,
ab Indis usque, siue Chaldæis asciti .1.2.3.4.5.6.7.8.9. Est item unus .0
circulus, qui nihil significat." [_De VI. Arithmeticae practicae
speciebvs_, Paris, 1539, fol. 9 of the 1543 edition.]

[7] "Barbarische oder gemeine Ziffern." [Anonymous, _Das Einmahl Eins cum
notis variorum_, Dresden, 1703, p. 3.] So Vossius (_De universae matheseos
natura et constitutione liber_, Amsterdam, 1650, p. 34) calls them
"Barbaras numeri notas." The word at that time was possibly synonymous with
Arabic.

[8] His full name was `Ab[=u] `Abdall[=a]h Mo[h.]ammed ibn M[=u]s[=a]
al-Khow[=a]razm[=i]. He was born in Khow[=a]rezm, "the lowlands," the
country about the present Khiva and bordering on the Oxus, and lived at
Bagdad under the caliph al-M[=a]m[=u]n. He died probably between 220 and
230 of the Mohammedan era, that is, between 835 and 845 A.D., although some
put the date as early as 812. The best account of this great scholar may be
found in an article by C. Nallino, "Al-[H)]uw[=a]rizm[=i]" in the _Atti
della R. Accad. dei Lincei_, Rome, 1896. See also _Verhandlungen des 5.
Congresses der Orientalisten_, Berlin, 1882, Vol. II, p. 19; W. Spitta-Bey
in the _Zeitschrift der deutschen Morgenländ. Gesellschaft_, Vol. XXXIII,
p. 224; Steinschneider in the _Zeitschrift der deutschen Morgenländ.
Gesellschaft_, Vol. L, p. 214; Treutlein in the _Abhandlungen zur
Geschichte der Mathematik_, Vol. I, p. 5; Suter, "Die Mathematiker und
Astronomen der Araber und ihre Werke," _Abhandlungen zur Geschichte der
Mathematik_, Vol. X, Leipzig, 1900, p. 10, and "Nachträge," in Vol. XIV, p.
158; Cantor, _Geschichte der Mathematik_, Vol. I, 3d ed., pp. 712-733 etc.;
F. Woepcke in _Propagation_, p. 489. So recently has he become known that
Heilbronner, writing in 1742, merely mentions him as "Ben-Musa, inter
Arabes celebris Geometra, scripsit de figuris planis & sphericis."
[_Historia matheseos universæ_, Leipzig, 1742, p. 438.]

In this work most of the Arabic names will be transliterated substantially
as laid down by Suter in his work _Die Mathematiker_ etc., except where
this violates English pronunciation. The scheme of pronunciation of
oriental names is set forth in the preface.

[9] Our word _algebra_ is from the title of one of his works, Al-jabr
wa'l-muq[=a]balah, Completion and Comparison. The work was translated into
English by F. Rosen, London, 1831, and treated in _L'Algèbre
d'al-Kh[=a]rizmi et les méthodes indienne et grecque_, Léon Rodet, Paris,
1878, extract from the _Journal Asiatique_. For the derivation of the word
_algebra_, see Cossali, _Scritti Inediti_, pp. 381-383, Rome, 1857;
Leonardo's _Liber Abbaci_ (1202), p. 410, Rome, 1857; both published by B.
Boncompagni. "Almuchabala" also was used as a name for algebra.

[10] This learned scholar, teacher of O'Creat who wrote the _Helceph_
("_Prologus N. Ocreati in Helceph ad Adelardum Batensem magistrum suum_"),
studied in Toledo, learned Arabic, traveled as far east as Egypt, and
brought from the Levant numerous manuscripts for study and translation. See
Henry in the _Abhandlungen zur Geschichte der Mathematik_, Vol. III, p.
131; Woepcke in _Propagation_, p. 518.

[11] The title is _Algoritmi de numero Indorum_. That he did not make this
translation is asserted by Eneström in the _Bibliotheca Mathematica_, Vol.
I (3), p. 520.

[12] Thus he speaks "de numero indorum per .IX. literas," and proceeds:
"Dixit algoritmi: Cum uidissem yndos constituisse .IX. literas in uniuerso
numero suo, propter dispositionem suam quam posuerunt, uolui patefacere de
opera quod fit per eas aliquid quod esset leuius discentibus, si deus
uoluerit." [Boncompagni, _Trattati d'Aritmetica_,  Rome, 1857.] Discussed
by F. Woepcke, _Sur l'introduction de l'arithmétique indienne en Occident_,
Rome, 1859.

[13] Thus in a commentary by `Al[=i] ibn Ab[=i] Bekr ibn al-Jam[=a]l
al-An[s.][=a]r[=i

[14] See also Woepcke, _Propagation_, p. 505. The origin is discussed at
much length by G. R. Kaye, "Notes on Indian Mathematics.--Arithmetical
Notation," _Journ. and Proc. of the Asiatic Soc. of Bengal_, Vol. III,
1907, p. 489.

[15] _Alberuni's India_, Arabic version, London, 1887; English translation,
ibid., 1888.

[16] _Chronology of Ancient Nations_, London, 1879. Arabic and English
versions, by C. E. Sachau.

[17] _India_, Vol. I, chap. xvi.

[18] The Hindu name for the symbols of the decimal place system.

[19] Sachau's English edition of the _Chronology_, p. 64.

[20] _Littérature arabe_, Cl. Huart, Paris, 1902.

[21] Huart, _History of Arabic Literature_, English ed., New York, 1903, p.
182 seq.

[22] Al-Mas`[=u]d[=i]'s _Meadows of Gold_, translated in part by Aloys
Sprenger, London, 1841; _Les prairies d'or_, trad. par C. Barbier de
Meynard et Pavet de Courteille, Vols. I to IX, Paris, 1861-1877.

[23] _Les prairies d'or_, Vol. VIII, p. 289 seq.

[24] _Essays_, Vol. II, p. 428.

[25] Loc. cit., p. 504.

[26] _Matériaux pour servir à l'histoire comparée des sciences
mathématiques chez les Grecs et les Orientaux_, 2 vols., Paris, 1845-1849,
pp. 438-439.

[27] He made an exception, however, in favor of the numerals, loc. cit.,
Vol. II, p. 503.

[28] _Bibliotheca Arabico-Hispana Escurialensis_, Madrid, 1760-1770, pp.
426-427.

[29] The author, Ibn al-Qif[t.][=i], flourished A.D. 1198 [Colebrooke, loc.
cit., note Vol. II, p. 510].

[30] "Liber Artis Logisticae à Mohamado Ben Musa _Alkhuarezmita_ exornatus,
qui ceteros omnes brevitate methodi ac facilitate praestat, Indorum que in
praeclarissimis inventis ingenium & acumen ostendit." [Casiri, loc. cit.,
p. 427.]

[31] Maçoudi, _Le livre de l'avertissement et de la révision_. Translation
by B. Carra de Vaux, Paris, 1896.

[32] Verifying the hypothesis of Woepcke, _Propagation_, that the Sindhind
included a treatment of arithmetic.

[33] A[h.]med ibn `Abdall[=a]h, Suter, _Die Mathematiker_, etc., p. 12.

[34] _India_, Vol. II, p. 15.

[35] See H. Suter, "Das Mathematiker-Verzeichniss im Fihrist,"
_Abhandlungen zur Geschichte der Mathematik_, Vol. VI, Leipzig, 1892. For
further references to early Arabic writers the reader is referred to H.
Suter, _Die Mathematiker und Astronomen der Araber und ihre Werke_. Also
"Nachträge und Berichtigungen" to the same (_Abhandlungen_,  Vol. XIV,
1902, pp. 155-186).

[36] Suter, loc. cit., note 165, pp. 62-63.

[37] "Send Ben Ali,... tùm arithmetica scripta maximè celebrata, quae
publici juris fecit." [Loc. cit., p. 440.]

[38] _Scritti di Leonardo Pisano_, Vol. I, _Liber Abbaci_ (1857); Vol. II,
_Scritti_ (1862); published by Baldassarre Boncompagni, Rome. Also _Tre
Scritti Inediti_, and _Intorno ad Opere di Leonardo Pisano_, Rome, 1854.

[39] "Ubi ex mirabili magisterio in arte per novem figuras indorum
introductus" etc. In another place, as a heading to a separate division, he
writes, "De cognitione novem figurarum yndorum" etc. "Novem figure indorum
he sunt 9 8 7 6 5 4 3 2 1."

[40] See _An Ancient English Algorism_, by David Eugene Smith, in
_Festschrift Moritz Cantor_, Leipzig, 1909. See also Victor Mortet, "Le
plus ancien traité francais d'algorisme," _Bibliotheca Mathematica_, Vol.
IX (3), pp. 55-64.

[41] These are the two opening lines of the _Carmen de Algorismo_ that the
anonymous author is explaining. They should read as follows:

  Haec algorismus ars praesens dicitur, in qua
  Talibus Indorum fruimur bis quinque figuris.

What follows is the translation.

[42] Thibaut, _Astronomie, Astrologie und Mathematik_, Strassburg, 1899.

[43] Gustave Schlegel, _Uranographie chinoise ou preuves directes que
l'astronomie primitive est originaire de la Chine, et qu'elle a été
empruntée par les anciens peuples occidentaux à la sphère chinoise; ouvrage
accompagné d'un atlas céleste chinois et grec_, The Hague and Leyden, 1875.

[44] E. W. Hopkins, _The Religions of India_, Boston, 1898, p. 7.

[45] R. C. Dutt, _History of India_, London, 1906.

[46] W. D. Whitney, _Sanskrit Grammar_, 3d ed., Leipzig, 1896.

[47] "Das [=A]pastamba-['S]ulba-S[=u]tra," _Zeitschrift der deutschen
Morgenländischen Gesellschaft_, Vol. LV, p. 543, and Vol. LVI, p. 327.

[48] _Geschichte der Math._, Vol. I, 2d ed., p. 595.

[49] L. von Schroeder, _Pythagoras und die Inder_, Leipzig, 1884; H. Vogt,
"Haben die alten Inder den Pythagoreischen Lehrsatz und das Irrationale
gekannt?" _Bibliotheca Mathematica_, Vol. VII (3), pp. 6-20; A. Bürk, loc.
cit.; Max Simon, _Geschichte der Mathematik im Altertum_, Berlin, 1909, pp.
137-165; three S[=u]tras are translated in part by Thibaut, _Journal of the
Asiatic Society of Bengal_, 1875, and one appeared in _The Pandit_, 1875;
Beppo Levi, "Osservazioni e congetture sopra la geometria degli indiani,"
_Bibliotheca Mathematica_, Vol. IX (3), 1908, pp. 97-105.

[50] Loc. cit.; also _Indiens Literatur und Cultur_, Leipzig, 1887.

[51] It is generally agreed that the name of the river Sindhu, corrupted by
western peoples to Hindhu, Indos, Indus, is the root of Hindustan and of
India. Reclus, _Asia_, English ed., Vol. III, p. 14.

[52] See the comments of Oppert, _On the Original Inhabitants of
Bharatavar[s.]a or India_, London, 1893, p. 1.

[53] A. Hillebrandt, _Alt-Indien_, Breslau, 1899, p. 111. Fragmentary
records relate that Kh[=a]ravela, king of Kali[.n]ga, learned as a boy
_lekh[=a]_ (writing), _ga[n.]an[=a]_ (reckoning), and _r[=u]pa_ (arithmetic
applied to monetary affairs and mensuration), probably in the 5th century
B.C. [Bühler, _Indische Palaeographie_, Strassburg, 1896, p. 5.]

[54] R. C. Dutt, _A History of Civilization in Ancient India_, London,
1893, Vol. I, p. 174.

[55] The Buddha. The date of his birth is uncertain. Sir Edwin Arnold put
it c. 620 B.C.

[56] I.e. 100·10^7.

[57] There is some uncertainty about this limit.

[58] This problem deserves more study than has yet been given it. A
beginning may be made with Comte Goblet d'Alviella, _Ce que l'Inde doit à
la Grèce_, Paris, 1897, and H. G. Keene's review, "The Greeks in India," in
the _Calcutta Review_, Vol. CXIV, 1902, p. 1. See also F. Woepeke,
_Propagation_, p. 253; G. R. Kaye, loc. cit., p. 475 seq., and "The Source
of Hindu Mathematics," _Journal of the Royal Asiatic Society_, July, 1910,
pp. 749-760; G. Thibaut, _Astronomie, Astrologie und Mathematik_, pp. 43-50
and 76-79. It will be discussed more fully in Chapter VI.

[59] I.e. to 100,000. The lakh is still the common large unit in India,
like the myriad in ancient Greece and the million in the West.

[60] This again suggests the _Psammites_, or _De harenae numero_ as it is
called in the 1544 edition of the _Opera_ of Archimedes, a work in which
the great Syracusan proposes to show to the king "by geometric proofs which
you can follow, that the numbers which have been named by us ... are
sufficient to exceed not only the number of a sand-heap as large as the
whole earth, but one as large as the universe." For a list of early
editions of this work see D. E. Smith, _Rara Arithmetica_, Boston, 1909, p.
227.

[61] I.e. the Wise.

[62] Sir Monier Monier-Williams, _Indian Wisdom_, 4th ed., London, 1893,
pp. 144, 177. See also J. C. Marshman, _Abridgment of the History of
India_, London, 1893, p. 2.

[63] For a list and for some description of these works see R. C. Dutt, _A
History of Civilization in Ancient India_, Vol. II, p. 121.

[64] Professor Ramkrishna Gopal Bhandarkar fixes the date as the fifth
century B.C. ["Consideration of the Date of the Mah[=a]bh[=a]rata," in the
_Journal of the Bombay Branch of the R. A. Soc._, Bombay, 1873, Vol. X, p.
2.].

[65] Marshman, loc. cit., p. 2.

[66] A. C. Burnell, _South Indian Palæography_, 2d ed., London, 1878, p. 1,
seq.

[67] This extensive subject of palpable arithmetic, essentially the history
of the abacus, deserves to be treated in a work by itself.

[68] The following are the leading sources of information upon this
subject: G. Bühler, _Indische Palaeographie_, particularly chap. vi; A. C.
Burnell, _South Indian Palæography_, 2d ed., London, 1878, where tables of
the various Indian numerals are given in Plate XXIII; E. C. Bayley, "On the
Genealogy of Modern Numerals," _Journal of the Royal Asiatic Society_, Vol.
XIV, part 3, and Vol. XV, part 1, and reprint, London, 1882; I. Taylor, in
_The Academy_, January 28, 1882, with a repetition of his argument in his
work _The Alphabet_, London, 1883, Vol. II, p. 265, based on Bayley; G. R.
Kaye, loc. cit., in some respects one of the most critical articles thus
far published; J. C. Fleet, _Corpus inscriptionum Indicarum_, London, 1888,
Vol. III, with facsimiles of many Indian inscriptions, and _Indian
Epigraphy_, Oxford, 1907, reprinted from the _Imperial Gazetteer of India_,
Vol. II, pp. 1-88, 1907; G. Thibaut, loc. cit., _Astronomie_ etc.; R.
Caldwell, _Comparative Grammar of the Dravidian Languages_, London, 1856,
p. 262 seq.; and _Epigraphia Indica_ (official publication of the
government of India), Vols. I-IX. Another work of Bühler's, _On the Origin
of the Indian Br[=a]hma Alphabet_, is also of value.

[69] The earliest work on the subject was by James Prinsep, "On the
Inscriptions of Piyadasi or A['s]oka," etc., _Journal of the Asiatic
Society of Bengal_, 1838, following a preliminary suggestion in the same
journal in 1837. See also "A['s]oka Notes," by V. A. Smith, _The Indian
Antiquary_, Vol. XXXVII, 1908, p. 24 seq., Vol. XXXVIII, pp. 151-159, June,
1909; _The Early History of India_, 2d ed., Oxford, 1908, p. 154; J. F.
Fleet, "The Last Words of A['s]oka," _Journal of the Royal Asiatic
Society_, October, 1909, pp. 981-1016; E. Senart, _Les inscriptions de
Piyadasi_, 2 vols., Paris, 1887.

[70] For a discussion of the minor details of this system, see Bühler, loc.
cit., p. 73.

[71] Julius Euting, _Nabatäische Inschriften aus Arabien_, Berlin, 1885,
pp. 96-97, with a table of numerals.

[72] For the five principal theories see Bühler, loc. cit., p. 10.

[73] Bayley, loc. cit., reprint p. 3.

[74] Bühler, loc. cit.; _Epigraphia Indica_, Vol. III, p. 134; _Indian
Antiquary_, Vol. VI, p. 155 seq., and Vol. X, p. 107.

[75] Pandit Bhagav[=a]nl[=a]l Indr[=a]j[=i], "On Ancient N[=a]g[=a]ri
Numeration; from an Inscription at N[=a]negh[=a]t," _Journal of the Bombay
Branch of the Royal Asiatic Society_, 1876, Vol. XII, p. 404.

[76] Ib., p. 405. He gives also a plate and an interpretation of each
numeral.

[77] These may be compared with Bühler's drawings, loc. cit.; with Bayley,
loc. cit., p. 337 and plates; and with Bayley's article in the
_Encyclopædia Britannica_, 9th ed., art. "Numerals."

[78] E. Senart, "The Inscriptions in the Caves at Nasik," _Epigraphia
Indica_, Vol. VIII, pp. 59-96; "The Inscriptions in the Cave at Karle,"
_Epigraphia Indica_, Vol. VII, pp. 47-74; Bühler, _Palaeographie_, Tafel
IX.

[79] See Fleet, loc. cit. See also T. Benfey, _Sanskrit Grammar_, London,
1863, p. 217; M. R. Kále, _Higher Sanskrit Grammar_, 2d ed., Bombay, 1898,
p. 110, and other authorities as cited.

[80] Kharo[s.][t.]h[=i] numerals, A['s]oka inscriptions, c. 250 B.C.
Senart, _Notes d'épigraphie indienne_. Given by Bühler, loc. cit., Tafel I.

[81] Same, ['S]aka inscriptions, probably of the first century B.C. Senart,
loc. cit.; Bühler, loc. cit.

[82] Br[=a]hm[=i] numerals, A['s]oka inscriptions, c. 250 B.C. _Indian
Antiquary_, Vol. VI, p. 155 seq.

[83] Same, N[=a]n[=a] Gh[=a]t inscriptions, c. 150 B.C. Bhagav[=a]nl[=a]l
Indr[=a]j[=i], _On Ancient N[=a]gar[=i] Numeration_, loc. cit. Copied from
a squeeze of the original.

[84] Same, Nasik inscription, c. 100 B.C. Burgess, _Archeological Survey
Report, Western India_; Senart, _Epigraphia Indica_, Vol. VII, pp. 47-79,
and Vol. VIII, pp. 59-96.

[85] K[s.]atrapa coins, c. 200 A.D. _Journal of the Royal Asiatic Society_,
1890, p. 639.

[86] Ku[s.]ana inscriptions, c. 150 A.D. _Epigraphia Indica_, Vol. I, p.
381, and Vol. II, p. 201.

[87] Gupta Inscriptions, c. 300 A.D. to 450 A.D. Fleet, loc. cit., Vol.
III.

[88] Valhab[=i], c. 600 A.D. _Corpus_, Vol. III.

[89] Bendall's Table of Numerals, in _Cat. Sansk. Budd. MSS._, British
Museum.

[90] _Indian Antiquary_, Vol. XIII, 120; _Epigraphia Indica_, Vol. III, 127
ff.

[91] Fleet, loc. cit.

[92] Bayley, loc. cit., p. 335.

[93] From a copper plate of 493 A.D., found at K[=a]r[=i]tal[=a][=i],
Central India. [Fleet, loc. cit., Plate XVI.] It should be stated, however,
that many of these copper plates, being deeds of property, have forged
dates so as to give the appearance of antiquity of title. On the other
hand, as Colebrooke long ago pointed out, a successful forgery has to
imitate the writing of the period in question, so that it becomes evidence
well worth considering, as shown in Chapter III.

[94] From a copper plate of 510 A.D., found at Majhgaw[=a]in, Central
India. [Fleet, loc. cit., Plate XIV.]

[95] From an inscription of 588 A.D., found at B[=o]dh-Gay[=a], Bengal
Presidency. [Fleet, loc. cit., Plate XXIV.]

[96] From a copper plate of 571 A.D., found at M[=a]liy[=a], Bombay
Presidency. [Fleet, loc. cit., Plate XXIV.]

[97] From a Bijayaga[d.]h pillar inscription of 372 A.D. [Fleet, loc. cit.,
Plate XXXVI, C.]

[98] From a copper plate of 434 A.D. [_Indian Antiquary_, Vol. I, p. 60.]

[99] Gadhwa inscription, c. 417 A.D. [Fleet, loc. cit., Plate IV, D.]

[100] K[=a]r[=i]tal[=a][=i] plate of 493 A.D., referred to above.

[101] It seems evident that the Chinese four, curiously enough called
"eight in the mouth," is only a cursive [4 vertical strokes].

[102] Chalfont, F. H., _Memoirs of the Carnegie Museum_, Vol. IV, no. 1; J.
Hager, _An Explanation of the Elementary Characters of the Chinese_,
London, 1801.

[103] H. V. Hilprecht, _Mathematical, Metrological and Chronological
Tablets from the Temple Library at Nippur_, Vol. XX, part I, of Series A,
Cuneiform Texts Published by the Babylonian Expedition of the University of
Pennsylvania, 1906; A. Eisenlohr, _Ein altbabylonischer Felderplan_,
Leipzig, 1906; Maspero, _Dawn of Civilization_, p. 773.

[104] Sir H. H. Howard, "On the Earliest Inscriptions from Chaldea,"
_Proceedings of the Society of Biblical Archæology_, XXI, p. 301, London,
1899.

[105] For a bibliography of the principal hypotheses of this nature see
Bühler, loc. cit., p. 77. Bühler (p. 78) feels that of all these hypotheses
that which connects the Br[=a]hm[=i] with the Egyptian numerals is the most
plausible, although he does not adduce any convincing proof. Th. Henri
Martin, "Les signes numéraux et l'arithmétique chez les peuples de
l'antiquité et du moyen âge" (being an examination of Cantor's
_Mathematische Beiträge zum Culturleben der Völker_), _Annali di matematica
pura ed applicata_, Vol. V, Rome, 1864, pp. 8, 70. Also, same author,
"Recherches nouvelles sur l'origine de notre système de numération écrite,"
_Revue Archéologique_, 1857, pp. 36, 55. See also the tables given later in
this work.

[106] _Journal of the Royal Asiatic Society, Bombay Branch_, Vol. XXIII.

[107] Loc. cit., reprint, Part I, pp. 12, 17. Bayley's deductions are
generally regarded as unwarranted.

[108] _The Alphabet_; London, 1883, Vol. II, pp. 265, 266, and _The
Academy_ of Jan. 28, 1882.

[109] Taylor, _The Alphabet_, loc. cit., table on p. 266.

[110] Bühler, _On the Origin of the Indian Br[=a]hma Alphabet_, Strassburg,
1898, footnote, pp. 52, 53.

[111] Albrecht Weber, _History of Indian Literature_, English ed., Boston,
1878, p. 256: "The Indian figures from 1-9 are abbreviated forms of the
initial letters of the numerals themselves...: the zero, too, has arisen
out of the first letter of the word _[s.]unya_ (empty) (it occurs even in
Piñgala). It is the decimal place value of these figures which gives them
significance." C. Henry, "Sur l'origine de quelques notations
mathématiques," _Revue Archéologique_, June and July, 1879, attempts to
derive the Boethian forms from the initials of Latin words. See also J.
Prinsep, "Examination of the Inscriptions from Girnar in Gujerat, and
Dhauli in Cuttach," _Journal of the Asiatic Society of Bengal_, 1838,
especially Plate XX, p. 348; this was the first work on the subject.

[112] Bühler, _Palaeographie_, p. 75, gives the list, with the list of
letters (p. 76) corresponding to the number symbols.

[113] For a general discussion of the connection between the numerals and
the different kinds of alphabets, see the articles by U. Ceretti, "Sulla
origine delle cifre numerali moderne," _Rivista di fisica, matematica e
scienze naturali_, Pisa and Pavia, 1909, anno X, numbers 114, 118, 119, and
120, and continuation in 1910.

[114] This is one of Bühler's hypotheses. See Bayley, loc. cit., reprint p.
4; a good bibliography of original sources is given in this work, p. 38.

[115] Loc. cit., reprint, part I, pp. 12, 17. See also Burnell, loc. cit.,
p. 64, and tables in plate XXIII.

[116] This was asserted by G. Hager (_Memoria sulle cifre arabiche_, Milan,
1813, also published in _Fundgruben des Orients_, Vienna, 1811, and in
_Bibliothèque Britannique_, Geneva, 1812). See also the recent article by
Major Charles E. Woodruff, "The Evolution of Modern Numerals from Tally
Marks," _American Mathematical Monthly_, August-September, 1909.
Biernatzki, "Die Arithmetik der Chinesen," _Crelle's Journal für die reine
und angewandte Mathematik_, Vol. LII, 1857, pp. 59-96, also asserts the
priority of the Chinese claim for a place system and the zero, but upon the
flimsiest authority. Ch. de Paravey, _Essai sur l'origine unique et
hiéroglyphique des chiffres et des lettres de tous les peuples_, Paris,
1826; G. Kleinwächter, "The Origin of the Arabic Numerals," _China Review_,
Vol. XI, 1882-1883, pp. 379-381, Vol. XII, pp. 28-30; Biot, "Note sur la
connaissance que les Chinois ont eue de la valeur de position des
chiffres," _Journal Asiatique_, 1839, pp. 497-502. A. Terrien de
Lacouperie, "The Old Numerals, the Counting-Rods and the Swan-Pan in
China," _Numismatic Chronicle_, Vol. III (3), pp. 297-340, and Crowder B.
Moseley, "Numeral Characters: Theory of Origin and Development," _American
Antiquarian_, Vol. XXII, pp. 279-284, both propose to derive our numerals
from Chinese characters, in much the same way as is done by Major Woodruff,
in the article above cited.

[117] The Greeks, probably following the Semitic custom, used nine letters
of the alphabet for the numerals from 1 to 9, then nine others for 10 to
90, and further letters to represent 100 to 900. As the ordinary Greek
alphabet was insufficient, containing only twenty-four letters, an alphabet
of twenty-seven letters was used.

[118] _Institutiones mathematicae_, 2 vols., Strassburg, 1593-1596, a
somewhat rare work from which the following quotation is taken:

"_Quis est harum Cyphrarum autor?_

"A quibus hae usitatae syphrarum notae sint inventae: hactenus incertum
fuit: meo tamen iudicio, quod exiguum esse fateor: a graecis librarijs
(quorum olim magna fuit copia) literae Graecorum quibus veteres Graeci
tamquam numerorum notis sunt usi: fuerunt corruptae. vt ex his licet
videre.

"Graecorum Literae corruptae.

[Illustration]

_"Sed qua ratione graecorum literae ita fuerunt corruptae?_

"Finxerunt has corruptas Graecorum literarum notas: vel abiectione vt in
nota binarij numeri, vel additione vt in ternarij, vel inuersione vt in
septenarij, numeri nota, nostrae notae, quibus hodie utimur: ab his sola
differunt elegantia, vt apparet."

See also Bayer, _Historia regni Graecorum Bactriani_, St. Petersburg, 1788,
pp. 129-130, quoted by Martin, _Recherches nouvelles_, etc., loc. cit.

[119] P. D. Huet, _Demonstratio evangelica_, Paris, 1769, note to p. 139 on
p. 647: "Ab Arabibus vel ab Indis inventas esse, non vulgus eruditorum
modo, sed doctissimi quique ad hanc diem arbitrati sunt. Ego vero falsum id
esse, merosque esse Graecorum characteres aio; à librariis Graecae linguae
ignaris interpolatos, et diuturna scribendi consuetudine corruptos. Nam
primum 1 apex fuit, seu virgula, nota [Greek: monados]. 2, est ipsum [beta]
extremis suis truncatum. [gamma], si in sinistram partem inclinaveris &
cauda mutilaveris & sinistrum cornu sinistrorsum flexeris, fiet 3. Res ipsa
loquitur 4 ipsissimum esse [Delta], cujus crus sinistrum erigitur [Greek:
kata katheton], & infra basim descendit; basis vero ipsa ultra crus
producta eminet. Vides quam 5 simile sit [Greek: tôi] [epsilon]; infimo
tantum semicirculo, qui sinistrorsum patebat, dextrorsum converso. [Greek:
episêmon bau] quod ita notabatur [digamma], rotundato ventre, pede
detracto, peperit [Greek: to] 6. Ex [Zeta] basi sua mutilato, ortum est
[Greek: to] 7. Si [Eta] inflexis introrsum apicibus in rotundiorem &
commodiorem formam mutaveris, exurget [Greek: to] 8. At 9 ipsissimum est
[alt theta]."

I. Weidler, _Spicilegium observationum ad historiam notarum numeralium_,
Wittenberg, 1755, derives them from the Hebrew letters; Dom Augustin
Calmet, "Recherches sur l'origine des chiffres d'arithmétique," _Mémoires
pour l'histoire des sciences et des beaux arts_, Trévoux, 1707 (pp.
1620-1635, with two plates), derives the current symbols from the Romans,
stating that they are relics of the ancient "Notae Tironianae." These
"notes" were part of a system of shorthand invented, or at least perfected,
by Tiro, a slave who was freed by Cicero. L. A. Sedillot, "Sur l'origine de
nos chiffres," _Atti dell' Accademia pontificia dei nuovi Lincei_, Vol.
XVIII, 1864-1865, pp. 316-322, derives the Arabic forms from the Roman
numerals.

[120] Athanasius Kircher, _Arithmologia sive De abditis Numerorum,
mysterijs qua origo, antiquitas & fabrica Numerorum exponitur_, Rome, 1665.

[121] See Suter, _Die Mathematiker und Astronomen der Araber_, p. 100.

[122] "Et hi numeri sunt numeri Indiani, a Brachmanis Indiae Sapientibus ex
figura circuli secti inuenti."

[123] V. A. Smith, _The Early History of India_, Oxford, 2d ed., 1908, p.
333.

[124] C. J. Ball, "An Inscribed Limestone Tablet from Sippara,"
_Proceedings of the Society of Biblical Archæology_, Vol. XX, p. 25
(London, 1898). Terrien de Lacouperie states that the Chinese used the
circle for 10 before the beginning of the Christian era. [_Catalogue of
Chinese Coins_, London, 1892, p. xl.]

[125] For a purely fanciful derivation from the corresponding number of
strokes, see W. W. R. Ball, _A Short Account of the History of
Mathematics_, 1st ed., London, 1888, p. 147; similarly J. B. Reveillaud,
_Essai sur les chiffres arabes_, Paris, 1883; P. Voizot, "Les chiffres
arabes et leur origine," _La Nature_, 1899, p. 222; G. Dumesnil, "De la
forme des chiffres usuels," _Annales de l'université de Grenoble_, 1907,
Vol. XIX, pp. 657-674, also a note in _Revue Archéologique_, 1890, Vol. XVI
(3), pp. 342-348; one of the earliest references to a possible derivation
from points is in a work by Bettino entitled _Apiaria universae
philosophiae mathematicae in quibus paradoxa et noua machinamenta ad usus
eximios traducta, et facillimis demonstrationibus confirmata_, Bologna,
1545, Vol. II, Apiarium XI, p. 5.

[126] _Alphabetum Barmanum_, Romae, MDCCLXXVI, p. 50. The 1 is evidently
Sanskrit, and the 4, 7, and possibly 9 are from India.

[127] _Alphabetum Grandonico-Malabaricum_, Romae, MDCCLXXII, p. 90. The
zero is not used, but the symbols for 10, 100, and so on, are joined to the
units to make the higher numbers.

[128] _Alphabetum Tangutanum_, Romae, MDCCLXXIII, p. 107. In a Tibetan MS.
in the library of Professor Smith, probably of the eighteenth century,
substantially these forms are given.

[129] Bayley, loc. cit., plate II. Similar forms to these here shown, and
numerous other forms found in India, as well as those of other oriental
countries, are given by A. P. Pihan, _Exposé des signes de numération
usités chez les peuples orientaux anciens et modernes_, Paris, 1860.

[130] Bühler, loc. cit., p. 80; J. F. Fleet, _Corpus inscriptionum
Indicarum_, Vol. III, Calcutta, 1888. Lists of such words are given also by
Al-B[=i]r[=u]n[=i] in his work _India_; by Burnell, loc. cit.; by E.
Jacquet, "Mode d'expression symbolique des nombres employé par les Indiens,
les Tibétains et les Javanais," _Journal Asiatique_, Vol. XVI, Paris, 1835.

[131] This date is given by Fleet, loc. cit., Vol. III, p. 73, as the
earliest epigraphical instance of this usage in India proper.

[132] Weber, _Indische Studien_, Vol. VIII, p. 166 seq.

[133] _Journal of the Royal Asiatic Society_, Vol. I (N.S.), p. 407.

[134] VIII, 20, 21.

[135] Th. H. Martin, _Les signes numéraux_ ..., Rome, 1864; Lassen,
_Indische Alterthumskunde_, Vol. II, 2d ed., Leipzig and London, 1874, p.
1153.

[136] But see Burnell, loc. cit., and Thibaut, _Astronomie, Astrologie und
Mathematik_, p. 71.

[137] A. Barth, "Inscriptions Sanscrites du Cambodge," in the _Notices et
extraits des Mss. de la Bibliothèque nationale_, Vol. XXVII, Part I, pp.
1-180, 1885; see also numerous articles in _Journal Asiatique_, by
Aymonier.

[138] Bühler, loc. cit., p. 82.

[139] Loc. cit., p. 79.

[140] Bühler, loc. cit., p. 83. The Hindu astrologers still use an
alphabetical system of numerals. [Burnell, loc. cit., p. 79.]

[141] Well could Ramus say, "Quicunq; autem fuerit inventor decem notarum
laudem magnam meruit."

[142] Al-B[=i]r[=u]n[=i] gives lists.

[143] _Propagation_, loc. cit., p. 443.

[144] See the quotation from _The Light of Asia_ in Chapter II, p. 16.

[145] The nine ciphers were called _a[.n]ka_.

[146] "Zur Geschichte des indischen Ziffernsystems," _Zeitschrift für die
Kunde des Morgenlandes_, Vol. IV, 1842, pp. 74-83.

[147] It is found in the Bakh[s.][=a]l[=i] MS. of an elementary arithmetic
which Hoernle placed, at first, about the beginning of our era, but the
date is much in question. G. Thibaut, loc. cit., places it between 700 and
900 A.D.; Cantor places the body of the work about the third or fourth
century A.D., _Geschichte der Mathematik_, Vol. I (3), p. 598.

[148] For the opposite side of the case see G. R. Kaye, "Notes on Indian
Mathematics, No. 2.--[=A]ryabha[t.]a," _Journ. and Proc. of the Asiatic
Soc. of Bengal_, Vol. IV, 1908, pp. 111-141.

[149] He used one of the alphabetic systems explained above. This ran up to
10^{18} and was not difficult, beginning as follows:

[Illustration]

the same letter (_ka_) appearing in the successive consonant forms, _ka_,
_kha_, _ga_, _gha_, etc. See C. I. Gerhardt, _Über die Entstehung und
Ausbreitung des dekadischen Zahlensystems_, Programm, p. 17, Salzwedel,
1853, and _Études historiques sur l'arithmétique de position_, Programm, p.
24, Berlin, 1856; E. Jacquet, _Mode d'expression symbolique des nombres_,
loc. cit., p. 97; L. Rodet, "Sur la véritable signification de la notation
numérique inventée par [=A]ryabhata," _Journal Asiatique_, Vol. XVI (7),
pp. 440-485. On the two [=A]ryabha[t.]as see Kaye, _Bibl. Math._, Vol. X
(3), p. 289.

[150] Using _kha_, a synonym of _['s][=u]nya_. [Bayley, loc. cit., p. 22,
and L. Rodet, _Journal Asiatique_, Vol. XVI (7), p. 443.]

[151] Var[=a]ha-Mihira, _Pañcasiddh[=a]ntik[=a]_, translated by G. Thibaut
and M. S. Dvived[=i], Benares, 1889; see Bühler, loc. cit., p. 78; Bayley,
loc. cit., p. 23.

[152] _B[r.]hat Sa[m.]hit[=a]_, translated by Kern, _Journal of the Royal
Asiatic Society_, 1870-1875.

[153] It is stated by Bühler in a personal letter to Bayley (loc. cit., p.
65) that there are hundreds of instances of this usage in the _B[r.]hat
Sa[m.]hit[=a]_. The system was also used in the _Pañcasiddh[=a]ntik[=a]_ as
early as 505 A.D. [Bühler, _Palaeographie_, p. 80, and Fleet, _Journal of
the Royal Asiatic Society_, 1910, p. 819.]

[154] Cantor, _Geschichte der Mathematik_, Vol. I (3), p. 608.

[155] Bühler, loc. cit., p. 78.

[156] Bayley, p. 38.

[157] Noviomagus, in his _De numeris libri duo_, Paris, 1539, confesses his
ignorance as to the origin of the zero, but says: "D. Henricus Grauius, vir
Graecè & Hebraicè eximè doctus, Hebraicam originem ostendit," adding that
Valla "Indis Orientalibus gentibus inventionem tribuit."

[158] See _Essays_, Vol. II, pp. 287 and 288.

[159] Vol. XXX, p. 205 seqq.

[160] Loc. cit., p. 284 seqq.

[161] Colebrooke, loc. cit., p. 288.

[162] Loc. cit., p. 78.

[163] Hereafter, unless expressly stated to the contrary, we shall use the
word "numerals" to mean numerals with place value.

[164] "The Gurjaras of R[=a]jput[=a]na and Kanauj," in _Journal of the
Royal Asiatic Society_, January and April, 1909.

[165] Vol. IX, 1908, p. 248.

[166] _Epigraphia Indica_, Vol. IX, pp. 193 and 198.

[167] _Epigraphia Indica_, Vol. IX, p. 1.

[168] Loc. cit., p. 71.

[169] Thibaut, p. 71.

[170] "Est autem in aliquibus figurarum istaram apud multos diuersitas.
Quidam enim septimam hanc figuram representant," etc. [Boncompagni,
_Trattati_, p. 28.] Eneström has shown that very likely this work is
incorrectly attributed to Johannes Hispalensis. [_Bibliotheca Mathematica_,
Vol. IX (3), p. 2.]

[171] _Indische Palaeographie_, Tafel IX.

[172] Edited by Bloomfield and Garbe, Baltimore, 1901, containing
photographic reproductions of the manuscript.

[173] Bakh[s.][=a]l[=i] MS. See page 43; Hoernle, R., _The Indian
Antiquary_, Vol. XVII, pp. 33-48, 1 plate; Hoernle, _Verhandlungen des VII.
Internationalen Orientalisten-Congresses, Arische Section_, Vienna, 1888,
"On the Baksh[=a]l[=i] Manuscript," pp. 127-147, 3 plates; Bühler, loc.
cit.

[174] 3, 4, 6, from H. H. Dhruva, "Three Land-Grants from Sankheda,"
_Epigraphia Indica_, Vol. II, pp. 19-24 with plates; date 595 A.D. 7, 1, 5,
from Bhandarkar, "Daulatabad Plates," _Epigraphia Indica_, Vol. IX, part V;
date c. 798 A.D.

[175] 8, 7, 2, from "Buckhala Inscription of Nagabhatta," Bhandarkar,
_Epigraphia Indica_, Vol. IX, part V; date 815 A.D. 5 from "The Morbi
Copper-Plate," Bhandarkar, _The Indian Antiquary_, Vol. II, pp. 257-258,
with plate; date 804 A.D. See Bühler, loc. cit.

[176] 8 from the above Morbi Copper-Plate. 4, 5, 7, 9, and 0, from "Asni
Inscription of Mahipala," _The Indian Antiquary_, Vol. XVI, pp. 174-175;
inscription is on red sandstone, date 917 A.D. See Bühler.

[177] 8, 9, 4, from "Rashtrakuta Grant of Amoghavarsha," J. F. Fleet, _The
Indian Antiquary_, Vol. XII, pp. 263-272; copper-plate grant of date c. 972
A.D. See Bühler. 7, 3, 5, from "Torkhede Copper-Plate Grant of the Time of
Govindaraja of Gujerat," Fleet, _Epigraphia Indica_, Vol. III, pp. 53-58.
See Bühler.

[178] From "A Copper-Plate Grant of King Tritochanapâla Chanlukya of
L[=a][t.]ade['s]a," H.H. Dhruva, _Indian Antiquary_, Vol. XII, pp. 196-205;
date 1050 A.D. See Bühler.

[179] Burnell, A. C., _South Indian Palæography_, plate XXIII,
Telugu-Canarese numerals of the eleventh century. See Bühler.

[180] From a manuscript of the second half of the thirteenth century,
reproduced in "Della vita e delle opere di Leonardo Pisano," Baldassare
Boncompagni, Rome, 1852, in _Atti dell' Accademia Pontificia dei nuovi
Lincei_, anno V.

[181] From a fourteenth-century manuscript, as reproduced in _Della vita_
etc., Boncompagni, loc. cit.

[182] From a Tibetan MS. in the library of D. E. Smith.

[183] From a Tibetan block-book in the library of D. E. Smith.

[184] ['S][=a]rad[=a] numerals from _The Kashmirian Atharva-Veda,
reproduced by chromophotography from the manuscript in the University
Library at Tübingen_, Bloomfield and Garbe, Baltimore, 1901. Somewhat
similar forms are given under "Numération Cachemirienne," by Pihan,
_Exposé_ etc., p. 84.

[185] Franz X. Kugler, _Die Babylonische Mondrechnung_, Freiburg i. Br.,
1900, in the numerous plates at the end of the book; practically all of
these contain the symbol to which reference is made. Cantor, _Geschichte_,
Vol. I, p. 31.

[186] F. X. Kugler, _Sternkunde und Sterndienst in Babel_, I. Buch, from
the beginnings to the time of Christ, Münster i. Westfalen, 1907. It also
has numerous tables containing the above zero.

[187] From a letter to D. E. Smith, from G. F. Hill of the British Museum.
See also his monograph "On the Early Use of Arabic Numerals in Europe," in
_Archæologia_, Vol. LXII (1910), p. 137.

[188] R. Hoernle, "The Baksh[=a]l[=i] Manuscript," _Indian Antiquary_, Vol.
XVII, pp. 33-48 and 275-279, 1888; Thibaut, _Astronomie, Astrologie und
Mathematik_, p. 75; Hoernle, _Verhandlungen_, loc. cit., p. 132.

[189] Bayley, loc. cit., Vol. XV, p. 29. Also Bendall, "On a System of
Numerals used in South India," _Journal of the Royal Asiatic Society_,
1896, pp. 789-792.

[190] V. A. Smith, _The Early History of India_, 2d ed., Oxford, 1908, p.
14.

[191] Colebrooke, _Algebra, with Arithmetic and Mensuration, from the
Sanskrit of Brahmegupta and Bháscara_, London, 1817, pp. 339-340.

[192] Ibid., p. 138.

[193] D. E. Smith, in the _Bibliotheca Mathematica_, Vol. IX (3), pp.
106-110.

[194] As when we use three dots (...).

[195] "The Hindus call the nought explicitly _['s][=u]nyabindu_ 'the dot
marking a blank,' and about 500 A.D. they marked it by a simple dot, which
latter is commonly used in inscriptions and MSS. in order to mark a blank,
and which was later converted into a small circle." [Bühler, _On the Origin
of the Indian Alphabet_, p. 53, note.]

[196] Fazzari, _Dell' origine delle parole zero e cifra_, Naples, 1903.

[197] E. Wappler, "Zur Geschichte der Mathematik im 15. Jahrhundert," in
the _Zeitschrift für Mathematik und Physik_, Vol. XLV, _Hist.-lit. Abt._,
p. 47. The manuscript is No. C. 80, in the Dresden library.

[198] J. G. Prändel, _Algebra nebst ihrer literarischen Geschichte_, p.
572, Munich, 1795.

[199] See the table, p. 23. Does the fact that the early European
arithmetics, following the Arab custom, always put the 0 after the 9,
suggest that the 0 was derived from the old Hindu symbol for 10?

[200] Bayley, loc. cit., p. 48. From this fact Delambre (_Histoire de
l'astronomie ancienne_) inferred that Ptolemy knew the zero, a theory
accepted by Chasles, _Aperçu historique sur l'origine et le développement
des méthodes en géométrie_, 1875 ed., p. 476; Nesselmann, however, showed
(_Algebra der Griechen_, 1842, p. 138), that Ptolemy merely used [Greek: o]
for [Greek: ouden], with no notion of zero. See also G. Fazzari, "Dell'
origine delle parole zero e cifra," _Ateneo_, Anno I, No. 11, reprinted at
Naples in 1903, where the use of the point and the small cross for zero is
also mentioned. Th. H. Martin, _Les signes numéraux_ etc., reprint p. 30,
and J. Brandis, _Das Münz-, Mass- und Gewichtswesen in Vorderasien bis auf
Alexander den Grossen_, Berlin, 1866, p. 10, also discuss this usage of
[Greek: o], without the notion of place value, by the Greeks.

[201] _Al-Batt[=a]n[=i] sive Albatenii opus astronomicum_. Ad fidem codicis
escurialensis arabice editum, latine versum, adnotationibus instructum a
Carolo Alphonso Nallino, 1899-1907. Publicazioni del R. Osservatorio di
Brera in Milano, No. XL.

[202] Loc. cit., Vol. II, p. 271.

[203] C. Henry, "Prologus N. Ocreati in Helceph ad Adelardum Batensem
magistrum suum," _Abhandlungen zur Geschichte der Mathematik_, Vol. III,
1880.

[204] Max. Curtze, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts,"
_Abhandlungen zur Geschichte der Mathematik_, Vol. VIII, 1898, pp. 1-27;
Alfred Nagl, "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und über
die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im
christl. Abendlande," _Zeitschrift für Mathematik und Physik, Hist.-lit.
Abth._, Vol. XXXIV, pp. 129-146 and 161-170, with one plate.

[205] "Byzantinische Analekten," _Abhandlungen zur Geschichte der
Mathematik_, Vol. IX, pp. 161-189.

[206] [symbol] or [symbol] for 0. [symbol] also used for 5. [symbols] for
13. [Heiberg, loc. cit.]

[207] Gerhardt, _Études historiques sur l'arithmétique de position_,
Berlin, 1856, p. 12; J. Bowring, _The Decimal System in Numbers, Coins, &
Accounts_, London, 1854, p. 33.

[208] Karabacek, _Wiener Zeitschrift für die Kunde des Morgenlandes_, Vol.
XI, p. 13; _Führer durch die Papyrus-Ausstellung Erzherzog Rainer_, Vienna,
1894, p. 216.

[209] In the library of G. A. Plimpton, Esq.

[210] Cantor, _Geschichte_, Vol. I (3), p. 674; Y. Mikami, "A Remark on the
Chinese Mathematics in Cantor's Geschichte der Mathematik," _Archiv der
Mathematik und Physik_, Vol. XV (3), pp. 68-70.

[211] Of course the earlier historians made innumerable guesses as to the
origin of the word _cipher_. E.g. Matthew Hostus, _De numeratione
emendata_, Antwerp, 1582, p. 10, says: "Siphra vox Hebræam originem sapit
refértque: & ut docti arbitrantur, à verbo saphar, quod Ordine numerauit
significat. Unde Sephar numerus est: hinc Siphra (vulgo corruptius). Etsi
verò gens Iudaica his notis, quæ hodie Siphræ vocantur, usa non fuit:
mansit tamen rei appellatio apud multas gentes." Dasypodius, _Institutiones
mathematicae_, Vol. I, 1593, gives a large part of this quotation word for
word, without any mention of the source. Hermannus Hugo, _De prima
scribendi origine_, Trajecti ad Rhenum, 1738, pp. 304-305, and note, p.
305; Karl Krumbacher, "Woher stammt das Wort Ziffer (Chiffre)?", _Études de
philologie néo-grecque_, Paris, 1892.

[212] Bühler, loc. cit., p. 78 and p. 86.

[213] Fazzari, loc. cit., p. 4. So Elia Misrachi (1455-1526) in his
posthumous _Book of Number_, Constantinople, 1534, explains _sifra_ as
being Arabic. See also Steinschneider, _Bibliotheca Mathematica_, 1893, p.
69, and G. Wertheim, _Die Arithmetik des Elia Misrachi_, Programm,
Frankfurt, 1893.

[214] "Cum his novem figuris, et cum hoc signo 0, quod arabice zephirum
appellatur, scribitur quilibet numerus."

[215] [Greek: tziphra], a form also used by Neophytos (date unknown,
probably c. 1330). It is curious that Finaeus (1555 ed., f. 2) used the
form _tziphra_ throughout. A. J. H. Vincent ["Sur l'origine de nos
chiffres," _Notices et Extraits des MSS._, Paris, 1847, pp. 143-150] says:
"Ce cercle fut nommé par les uns, _sipos, rota, galgal_ ...; par les autres
_tsiphra_ (de [Hebrew: TSPR], _couronne_ ou _diadème_) ou _ciphra_ (de
[Hebrew: SPR], _numération_)." Ch. de Paravey, _Essai sur l'origine unique
et hiéroglyphique des chiffres et des lettres de tous les peuples_, Paris,
1826, p. 165, a rather fanciful work, gives "vase, vase arrondi et fermé
par un couvercle, qui est le symbole de la 10^e Heure, [symbol]," among the
Chinese; also "Tsiphron Zéron, ou tout à fait vide en arabe, [Greek:
tziphra] en grec ... d'où chiffre (qui dérive plutôt, suivant nous, de
l'Hébreu _Sepher_, compter.")

[216] "Compilatus a Magistro Jacobo de Florentia apud montem pesalanum,"
and described by G. Lami in his _Catalogus codicum manuscriptorum qui in
bibliotheca Riccardiana Florentiæ adservantur_. See Fazzari, loc. cit., p.
5.

[217] "Et doveto sapere chel zeuero per se solo non significa nulla ma è
potentia di fare significare, ... Et decina o centinaia o migliaia non si
puote scrivere senza questo segno 0. la quale si chiama zeuero." [Fazzari,
loc. cit., p. 5.]

[218] Ibid., p. 6.

[219] Avicenna (980-1036), translation by Gasbarri et François, "più il
punto (gli Arabi adoperavano il punto in vece dello zero il cui segno 0 in
arabo si chiama _zepiro_ donde il vocabolo zero), che per sè stesso non
esprime nessun numero." This quotation is taken from D. C. Martines,
_Origine e progressi dell' aritmetica_, Messina, 1865.

[220] Leo Jordan, "Materialien zur Geschichte der arabischen Zahlzeichen in
Frankreich," _Archiv für Kulturgeschichte_, Berlin, 1905, pp. 155-195,
gives the following two schemes of derivation, (1) "zefiro, zeviro, zeiro,
zero," (2) "zefiro, zefro, zevro, zero."

[221] Köbel (1518 ed., f. A_4) speaks of the numerals in general as "die
der gemain man Zyfer nendt." Recorde (_Grounde of Artes_, 1558 ed., f. B_6)
says that the zero is "called priuatly a Cyphar, though all the other
sometimes be likewise named."

[222] "Decimo X 0 theca, circul[us] cifra sive figura nihili appelat'."
[_Enchiridion Algorismi_, Cologne, 1501.] Later, "quoniam de integris tam
in cifris quam in proiectilibus,"--the word _proiectilibus_ referring to
markers "thrown" and used on an abacus, whence the French _jetons_ and the
English expression "to _cast_ an account."

[223] "Decima vero o dicitur teca, circulus, vel cyfra vel figura nichili."
[Maximilian Curtze, _Petri Philomeni de Dacia in Algorismum Vulgarem
Johannis de Sacrobosco commentarius, una cum Algorismo ipso_, Copenhagen,
1897, p. 2.] Curtze cites five manuscripts (fourteenth and fifteenth
centuries) of Dacia's commentary in the libraries at Erfurt, Leipzig, and
Salzburg, in addition to those given by Eneström, _Öfversigt af Kongl.
Vetenskaps-Akademiens Förhandlingar_, 1885, pp. 15-27, 65-70; 1886, pp.
57-60.

[224] Curtze, loc. cit., p. VI.

[225] _Rara Mathematica_, London, 1841, chap, i, "Joannis de Sacro-Bosco
Tractatus de Arte Numerandi."

[226] Smith, _Rara Arithmetica_, Boston, 1909.

[227] In the 1484 edition, Borghi uses the form "çefiro: ouero nulla:"
while in the 1488 edition he uses "zefiro: ouero nulla," and in the 1540
edition, f. 3, appears "Chiamata zero, ouero nulla." Woepcke asserted that
it first appeared in Calandri (1491) in this sentence: "Sono dieci le
figure con le quali ciascuno numero si può significare: delle quali n'è una
che si chiama zero: et per se sola nulla significa." (f. 4). [See
_Propagation_, p. 522.]

[228] Boncompagni _Bulletino_, Vol. XVI, pp. 673-685.

[229] Leo Jordan, loc. cit. In the _Catalogue of MSS., Bibl. de l'Arsenal_,
Vol. III, pp. 154-156, this work is No. 2904 (184 S.A.F.), Bibl. Nat., and
is also called _Petit traicté de algorisme_.

[230] Texada (1546) says that there are "nueue letros yvn zero o cifra" (f.
3).

[231] Savonne (1563, 1751 ed., f. 1): "Vne ansi formee (o) qui s'appelle
nulle, & entre marchans zero," showing the influence of Italian names on
French mercantile customs. Trenchant (Lyons, 1566, 1578 ed., p. 12) also
says: "La derniere qui s'apele nulle, ou zero;" but Champenois, his
contemporary, writing in Paris in 1577 (although the work was not published
until 1578), uses "cipher," the Italian influence showing itself less in
this center of university culture than in the commercial atmosphere of
Lyons.

[232] Thus Radulph of Laon (c. 1100): "Inscribitur in ultimo ordine et
figura [symbol] sipos nomine, quae, licet numerum nullum signitet, tantum
ad alia quaedam utilis, ut insequentibus declarabitur." ["Der Arithmetische
Tractat des Radulph von Laon," _Abhandlungen zur Geschichte der
Mathematik_, Vol. V, p. 97, from a manuscript of the thirteenth century.]
Chasles (_Comptes rendus_, t. 16, 1843, pp. 1393, 1408) calls attention to
the fact that Radulph did not know how to use the zero, and he doubts if
the sipos was really identical with it. Radulph says: "... figuram, cui
sipos nomen est [symbol] in motum rotulae formatam nullius numeri
significatione inscribi solere praediximus," and thereafter uses _rotula_.
He uses the sipos simply as a kind of marker on the abacus.

[233] Rabbi ben Ezra (1092-1168) used both [Hebrew: GLGL], _galgal_ (the
Hebrew for _wheel_), and [Hebrew: SPR'], _sifra_. See M. Steinschneider,
"Die Mathematik bei den Juden," in _Bibliotheca Mathematica_, 1893, p. 69,
and Silberberg, _Das Buch der Zahl des R. Abraham ibn Esra_, Frankfurt a.
M., 1895, p. 96, note 23; in this work the Hebrew letters are used for
numerals with place value, having the zero.

[234] E.g., in the twelfth-century _Liber aligorismi_ (see Boncompagni's
_Trattati_, II, p. 28). So Ramus (_Libri II_, 1569 ed., p. 1) says:
"Circulus quæ nota est ultima: nil per se significat." (See also the
Schonerus ed. of Ramus, 1586, p. 1.)

[235] "Und wirt das ringlein o. die Ziffer genant die nichts bedeut."
[Köbel's _Rechenbuch_, 1549 ed., f. 10, and other editions.]

[236] I.e. "circular figure," our word _notation_ having come from the
medieval _nota_. Thus Tzwivel (1507, f. 2) says: "Nota autem circularis .o.
per se sumpta nihil vsus habet. alijs tamen adiuncta earum significantiam
et auget et ordinem permutat quantum quo ponit ordinem. vt adiuncta note
binarij hoc modo 20 facit eam significare bis decem etc." Also (ibid., f.
4), "figura circularis," "circularis nota." Clichtoveus (1503 ed., f.
XXXVII) calls it "nota aut circularis o," "circularis nota," and "figura
circularis." Tonstall (1522, f. B_3) says of it: "Decimo uero nota ad
formam [symbol] litteræ circulari figura est: quam alij circulum, uulgus
cyphram uocat," and later (f. C_4) speaks of the "circulos." Grammateus, in
his _Algorismus de integris_ (Erfurt, 1523, f. A_2), speaking of the nine
significant figures, remarks: "His autem superadditur decima figura
circularis ut 0 existens que ratione sua nihil significat." Noviomagus (_De
Numeris libri II_, Paris, 1539, chap. xvi, "De notis numerorum, quas
zyphras vocant") calls it "circularis nota, quam ex his solam, alij
sipheram, Georgius Valla zyphram."

[237] Huswirt, as above. Ramus (_Scholae mathematicae_, 1569 ed., p. 112)
discusses the name interestingly, saying: "Circulum appellamus cum multis,
quam alii thecam, alii figuram nihili, alii figuram privationis, seu
figuram nullam vocant, alii ciphram, cùm tamen hodie omnes hæ notæ vulgò
ciphræ nominentur, & his notis numerare idem sit quod ciphrare." Tartaglia
(1592 ed., f. 9) says: "si chiama da alcuni tecca, da alcuni circolo, da
altri cifra, da altri zero, & da alcuni altri nulla."

[238] "Quare autem aliis nominibus vocetur, non dicit auctor, quia omnia
alia nomina habent rationem suae lineationis sive figurationis. Quia
rotunda est, dicitur haec figura teca ad similitudinem tecae. Teca enim est
ferrum figurae rotundae, quod ignitum solet in quibusdam regionibus imprimi
fronti vel maxillae furis seu latronum." [Loc. cit., p. 26.] But in Greek
_theca_ ([THEKE], [Greek: thêkê]) is a place to put something, a
receptacle. If a vacant column, e.g. in the abacus, was so called, the
initial might have given the early forms [symbol] and [symbol] for the
zero.

[239] Buteo, _Logistica_, Lyons, 1559. See also Wertheim in the
_Bibliotheca Mathematica_, 1901, p. 214.

[240] "0 est appellee chiffre ou nulle ou figure de nulle valeur." [La
Roche, _L'arithmétique_, Lyons, 1520.]

[241] "Decima autem figura nihil uocata," "figura nihili (quam etiam cifram
uocant)." [Stifel, _Arithmetica integra_, 1544, f. 1.]

[242] "Zifra, & Nulla uel figura Nihili." [Scheubel, 1545, p. 1 of ch. 1.]
_Nulla_ is also used by Italian writers. Thus Sfortunati (1545 ed., f. 4)
says: "et la decima nulla & e chiamata questa decima zero;" Cataldi (1602,
p. 1): "La prima, che è o, si chiama nulla, ouero zero, ouero niente." It
also found its way into the Dutch arithmetics, e.g. Raets (1576, 1580 ed.,
f. A_3): "Nullo dat ist niet;" Van der Schuere (1600, 1624 ed., f. 7);
Wilkens (1669 ed., p. 1). In Germany Johann Albert (Wittenberg, 1534) and
Rudolff (1526) both adopted the Italian _nulla_ and popularized it. (See
also Kuckuck, _Die Rechenkunst im sechzehnten Jahrhundert_, Berlin, 1874,
p. 7; Günther, _Geschichte_, p. 316.)

[243] "La dixième s'appelle chifre vulgairement: les vns l'appellant zero:
nous la pourrons appeller vn Rien." [Peletier, 1607 ed., p. 14.]

[244] It appears in the Polish arithmetic of K[=l]os (1538) as _cyfra_.
"The Ciphra 0 augmenteth places, but of himselfe signifieth not," Digges,
1579, p. 1. Hodder (10th ed., 1672, p. 2) uses only this word (cypher or
cipher), and the same is true of the first native American arithmetic,
written by Isaac Greenwood (1729, p. 1). Petrus de Dacia derives _cyfra_
from circumference. "Vocatur etiam cyfra, quasi circumfacta vel
circumferenda, quod idem est, quod circulus non habito respectu ad
centrum." [Loc. cit., p. 26.]

[245] _Opera mathematica_, 1695, Oxford, Vol. I, chap. ix, _Mathesis
universalis_, "De figuris numeralibus," pp. 46-49; Vol. II, _Algebra_, p.
10.

[246] Martin, _Origine de notre système de numération écrite_, note 149, p.
36 of reprint, spells [Greek: tsiphra] from Maximus Planudes, citing Wallis
as an authority. This is an error, for Wallis gives the correct form as
above.

Alexander von Humboldt, "Über die bei verschiedenen Völkern üblichen
Systeme von Zahlzeichen und über den Ursprung des Stellenwerthes in den
indischen Zahlen," Crelle's _Journal für reine und angewandte Mathematik_,
Vol. IV, 1829, called attention to the work [Greek: arithmoi Indikoi] of
the monk Neophytos, supposed to be of the fourteenth century. In this work
the forms [Greek: tzuphra] and [Greek: tzumphra] appear. See also Boeckh,
_De abaco Graecorum_, Berlin, 1841, and Tannery, "Le Scholie du moine
Néophytos," _Revue Archéologique_, 1885, pp. 99-102. Jordan, loc. cit.,
gives from twelfth and thirteenth century manuscripts the forms _cifra_,
_ciffre_, _chifras_, and _cifrus_. Du Cange, _Glossarium mediae et infimae
Latinitatis_, Paris, 1842, gives also _chilerae_. Dasypodius,
_Institutiones Mathematicae_, Strassburg, 1593-1596, adds the forms
_zyphra_ and _syphra_. Boissière, _L'art d'arythmetique contenant toute
dimention, tres-singulier et commode, tant pour l'art militaire que autres
calculations_, Paris, 1554: "Puis y en a vn autre dict zero lequel ne
designe nulle quantité par soy, ains seulement les loges vuides."

[247] _Propagation_, pp. 27, 234, 442. Treutlein, "Das Rechnen im 16.
Jahrhundert," _Abhandlungen zur Geschichte der Mathematik_, Vol. I, p. 5,
favors the same view. It is combated by many writers, e.g. A. C. Burnell,
loc. cit., p. 59. Long before Woepcke, I. F. and G. I. Weidler, _De
characteribus numerorum vulgaribus et eorum aetatibus_, Wittenberg, 1727,
asserted the possibility of their introduction into Greece by Pythagoras or
one of his followers: "Potuerunt autem ex oriente, uel ex phoenicia, ad
graecos traduci, uel Pythagorae, uel eius discipulorum auxilio, cum aliquis
eo, proficiendi in literis causa, iter faceret, et hoc quoque inuentum
addisceret."

[248] E.g., they adopted the Greek numerals in use in Damascus and Syria,
and the Coptic in Egypt. Theophanes (758-818 A.D.), _Chronographia_,
Scriptores Historiae Byzantinae, Vol. XXXIX, Bonnae, 1839, p. 575, relates
that in 699 A.D. the caliph Wal[=i]d forbade the use of the Greek language
in the bookkeeping of the treasury of the caliphate, but permitted the use
of the Greek alphabetic numerals, since the Arabs had no convenient number
notation: [Greek: kai ekôluse graphesthai Hellênisti tous dêmosious tôn
logothesiôn kôdikas, all' Arabiois auta parasêmainesthai, chôris tôn
psêphôn, epeidê adunaton têi ekeinôn glôssêi monada ê duada ê triada ê oktô
hêmisu ê tria graphesthai; dio kai heôs sêmeron eisin sun autois notarioi
Christianoi.] The importance of this contemporaneous document was pointed
out by Martin, loc. cit. Karabacek, "Die Involutio im arabischen
Schriftwesen," Vol. CXXXV of _Sitzungsberichte d. phil.-hist. Classe d. k.
Akad. d. Wiss._, Vienna, 1896, p. 25, gives an Arabic date of 868 A.D. in
Greek letters.

[249] _The Origin and History of Our Numerals_ (in Russian), Kiev, 1908;
_The Independence of European Arithmetic_ (in Russian), Kiev.

[250] Woepcke, loc. cit., pp. 462, 262.

[251] Woepcke, loc. cit., p. 240. _[H.]is[=a]b-al-[.G]ob[=a]r_, by an
anonymous author, probably Ab[=u] Sahl Dunash ibn Tamim, is given by
Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca Mathematica_,
1896, p. 26.

[252] Steinschneider in the _Abhandlungen_, Vol. III, p. 110.

[253] See his _Grammaire arabe_, Vol. I, Paris, 1810, plate VIII; Gerhardt,
_Études_, pp. 9-11, and _Entstehung_ etc., p. 8; I. F. Weidler,
_Spicilegium observationum ad historiam notarum numeralium pertinentium_,
Wittenberg, 1755, speaks of the "figura cifrarum Saracenicarum" as being
different from that of the "characterum Boethianorum," which are similar to
the "vulgar" or common numerals; see also Humboldt, loc. cit.

[254] Gerhardt mentions it in his _Entstehung_ etc., p. 8; Woepcke,
_Propagation_, states that these numerals were used not for calculation,
but very much as we use Roman numerals. These superposed dots are found
with both forms of numerals (_Propagation_, pp. 244-246).

[255] Gerhardt (_Études_, p. 9) from a manuscript in the Bibliothèque
Nationale. The numeral forms are [symbols], 20 being indicated by [symbol
with dot] and 200 by [symbol with 2 dots]. This scheme of zero dots was
also adopted by the Byzantine Greeks, for a manuscript of Planudes in the
Bibliothèque Nationale has numbers like [pi alpha with 4 dots] for
8,100,000,000. See Gerhardt, _Études_, p. 19. Pihan, _Exposé_ etc., p. 208,
gives two forms, Asiatic and Maghrebian, of "Ghob[=a]r" numerals.

[256] See Chap. IV.

[257] Possibly as early as the third century A.D., but probably of the
eighth or ninth. See Cantor, I (3), p. 598.

[258] Ascribed by the Arabic writer to India.

[259] See Woepcke's description of a manuscript in the Chasles library,
"Recherches sur l'histoire des sciences mathématiques chez les orientaux,"
_Journal Asiatique_, IV (5), 1859, p. 358, note.

[260] P. 56.

[261] Reinaud, _Mémoire sur l'Inde_, p. 399. In the fourteenth century one
Sih[=a]b al-D[=i]n wrote a work on which, a scholiast to the Bodleian
manuscript remarks: "The science is called Algobar because the inventor had
the habit of writing the figures on a tablet covered with sand." [Gerhardt,
_Études, _p. 11, note.]

[262] Gerhardt, _Entstehung _etc., p. 20.

[263] H. Suter, "Das Rechenbuch des Ab[=u] Zakar[=i]j[=a]
el-[H.]a[s.][s.][=a]r," _Bibliotheca Mathematica_, Vol. II (3), p. 15.

[264] A. Devoulx, "Les chiffres arabes," _Revue Africaine_, Vol. XVI, pp.
455-458.

[265] _Kit[=a]b al-Fihrist_, G. Flügel, Leipzig, Vol. I, 1871, and Vol. II,
1872. This work was published after Professor Flügel's death by J. Roediger
and A. Mueller. The first volume contains the Arabic text and the second
volume contains critical notes upon it.

[266] Like those of line 5 in the illustration on page 69.

[267] Woepcke, _Recherches sur l'histoire des sciences mathématiques chez
les orientaux_, loc. cit.; _Propagation, _p. 57.

[268] Al-[H.]a[s.][s.][=a]r's forms, Suter, _Bibliotheca Mathematica_, Vol.
II (3), p. 15.

[269] Woepcke, _Sur une donnée historique_, etc., loc. cit. The name
_[.g]ob[=a]r_ is not used in the text. The manuscript from which these are
taken is the oldest (970 A.D.) Arabic document known to contain all of the
numerals.

[270] Silvestre de Sacy, loc. cit. He gives the ordinary modern Arabic
forms, calling them _Indien_.

[271] Woepcke, "Introduction au calcul Gob[=a]r[=i] et Haw[=a][=i]," _Atti
dell' accademia pontificia dei nuovi Lincei_, Vol. XIX. The adjective
applied to the forms in 5 is _gob[=a]r[=i]_ and to those in 6 _indienne_.
This is the direct opposite of Woepcke's use of these adjectives in the
_Recherches sur l'histoire_ cited above, in which the ordinary Arabic forms
(like those in row 5) are called _indiens_.

These forms are usually written from right to left.

[272] J. G. Wilkinson, _The Manners and Customs of the Ancient Egyptians_,
revised by S. Birch, London, 1878, Vol. II, p. 493, plate XVI.

[273] There is an extensive literature on this "Boethius-Frage." The reader
who cares to go fully into it should consult the various volumes of the
_Jahrbuch über die Fortschritte der Mathematik_.

[274] This title was first applied to Roman emperors in posthumous coins of
Julius Cæsar. Subsequently the emperors assumed it during their own
lifetimes, thus deifying themselves. See F. Gnecchi, _Monete romane_, 2d
ed., Milan, 1900, p. 299.

[275] This is the common spelling of the name, although the more correct
Latin form is Boëtius. See Harper's _Dict. of Class. Lit. and Antiq._, New
York, 1897, Vol. I, p. 213. There is much uncertainty as to his life. A
good summary of the evidence is given in the last two editions of the
_Encyclopædia Britannica_.

[276] His father, Flavius Manlius Boethius, was consul in 487.

[277] There is, however, no good historic evidence of this sojourn in
Athens.

[278] His arithmetic is dedicated to Symmachus: "Domino suo patricio
Symmacho Boetius." [Friedlein ed., p. 3.]

[279] It was while here that he wrote _De consolatione philosophiae_.

[280] It is sometimes given as 525.

[281] There was a medieval tradition that he was executed because of a work
on the Trinity.

[282] Hence the _Divus_ in his name.

[283] Thus Dante, speaking of his burial place in the monastery of St.
Pietro in Ciel d'Oro, at Pavia, says:

      "The saintly soul, that shows
  The world's deceitfulness, to all who hear him,
  Is, with the sight of all the good that is,
  Blest there. The limbs, whence it was driven, lie
  Down in Cieldauro; and from martyrdom
  And exile came it here."--_Paradiso_, Canto X.

[284] Not, however, in the mercantile schools. The arithmetic of Boethius
would have been about the last book to be thought of in such institutions.
While referred to by Bæda (672-735) and Hrabanus Maurus (c. 776-856), it
was only after Gerbert's time that the _Boëtii de institutione arithmetica
libri duo_ was really a common work.

[285] Also spelled Cassiodorius.

[286] As a matter of fact, Boethius could not have translated any work by
Pythagoras on music, because there was no such work, but he did make the
theories of the Pythagoreans known. Neither did he translate Nicomachus,
although he embodied many of the ideas of the Greek writer in his own
arithmetic. Gibbon follows Cassiodorus in these statements in his _Decline
and Fall of the Roman Empire_, chap. xxxix. Martin pointed out with
positiveness the similarity of the first book of Boethius to the first five
books of Nicomachus. [_Les signes numéraux_ etc., reprint, p. 4.]

[287] The general idea goes back to Pythagoras, however.

[288] J. C. Scaliger in his _Poëtice_ also said of him: "Boethii Severini
ingenium, eruditio, ars, sapientia facile provocat omnes auctores, sive
illi Graeci sint, sive Latini" [Heilbronner, _Hist. math. univ._, p. 387].
Libri, speaking of the time of Boethius, remarks: "Nous voyons du temps de
Théodoric, les lettres reprendre une nouvelle vie en Italie, les écoles
florissantes et les savans honorés. Et certes les ouvrages de Boëce, de
Cassiodore, de Symmaque, surpassent de beaucoup toutes les productions du
siècle précédent." [_Histoire des mathématiques_, Vol. I, p. 78.]

[289] Carra de Vaux, _Avicenne_, Paris, 1900; Woepcke, _Sur
l'introduction_, etc.; Gerhardt, _Entstehung_ etc., p. 20. Avicenna is a
corruption from Ibn S[=i]n[=a], as pointed out by Wüstenfeld, _Geschichte
der arabischen Aerzte und Naturforscher_, Göttingen, 1840. His full name is
Ab[=u] `Al[=i] al-[H.]osein ibn S[=i]n[=a]. For notes on Avicenna's
arithmetic, see Woepcke, _Propagation_, p. 502.

[290] On the early travel between the East and the West the following works
may be consulted: A. Hillebrandt, _Alt-Indien_, containing "Chinesische
Reisende in Indien," Breslau, 1899, p. 179; C. A. Skeel, _Travel in the
First Century after Christ_, Cambridge, 1901, p. 142; M. Reinaud,
"Relations politiques et commerciales de l'empire romain avec l'Asie
orientale," in the _Journal Asiatique_, Mars-Avril, 1863, Vol. I (6), p.
93; Beazley, _Dawn of Modern Geography, a History of Exploration and
Geographical Science from the Conversion of the Roman Empire to A.D. 1420_,
London, 1897-1906, 3 vols.; Heyd, _Geschichte des Levanthandels im
Mittelalter_, Stuttgart, 1897; J. Keane, _The Evolution of Geography_,
London, 1899, p. 38; A. Cunningham, _Corpus inscriptionum Indicarum_,
Calcutta, 1877, Vol. I; A. Neander, _General History of the Christian
Religion and Church_, 5th American ed., Boston, 1855, Vol. III, p. 89; R.
C. Dutt, _A History of Civilization in Ancient India_, Vol. II, Bk. V,
chap, ii; E. C. Bayley, loc. cit., p. 28 et seq.; A. C. Burnell, loc. cit.,
p. 3; J. E. Tennent, _Ceylon_, London, 1859, Vol. I, p. 159; Geo. Turnour,
_Epitome of the History of Ceylon_, London, n.d., preface; "Philalethes,"
_History of Ceylon_, London, 1816, chap, i; H. C. Sirr, _Ceylon and the
Cingalese_, London, 1850, Vol. I, chap. ix. On the Hindu knowledge of the
Nile see F. Wilford, _Asiatick Researches_, Vol. III, p. 295, Calcutta,
1792.

[291] G. Oppert, _On the Ancient Commerce of India_, Madras, 1879, p. 8.

[292] Gerhardt, _Études_ etc., pp. 8, 11.

[293] See Smith's _Dictionary of Greek and Roman Biography and Mythology_.

[294] P. M. Sykes, _Ten Thousand Miles in Persia, or Eight Years in Irán_,
London, 1902, p. 167. Sykes was the first European to follow the course of
Alexander's army across eastern Persia.

[295] Bühler, _Indian Br[=a]hma Alphabet_, note, p. 27; _Palaeographie_, p.
2; _Herodoti Halicarnassei historia_, Amsterdam, 1763, Bk. IV, p. 300;
Isaac Vossius, _Periplus Scylacis Caryandensis_, 1639. It is doubtful
whether the work attributed to Scylax was written by him, but in any case
the work dates back to the fourth century B.C. See Smith's _Dictionary of
Greek and Roman Biography_.

[296] Herodotus, Bk. III.

[297] Rameses II(?), the _Sesoosis_ of Diodorus Siculus.

[298] _Indian Antiquary_, Vol. I, p. 229; F. B. Jevons, _Manual of Greek
Antiquities_, London, 1895, p. 386. On the relations, political and
commercial, between India and Egypt c. 72 B.C., under Ptolemy Auletes, see
the _Journal Asiatique_, 1863, p. 297.

[299] Sikandar, as the name still remains in northern India.

[300] _Harper's Classical Dict._, New York, 1897, Vol. I, p. 724; F. B.
Jevons, loc. cit., p. 389; J. C. Marshman, _Abridgment of the History of
India_, chaps. i and ii.

[301] Oppert, loc. cit., p. 11. It was at or near this place that the first
great Indian mathematician, [=A]ryabha[t.]a, was born in 476 A.D.

[302] Bühler, _Palaeographie_, p. 2, speaks of Greek coins of a period
anterior to Alexander, found in northern India. More complete information
may be found in _Indian Coins_, by E. J. Rapson, Strassburg, 1898, pp. 3-7.

[303] Oppert, loc. cit., p. 14; and to him is due other similar
information.

[304] J. Beloch, _Griechische Geschichte_, Vol. III, Strassburg, 1904, pp.
30-31.

[305] E.g., the denarius, the words for hour and minute ([Greek: hôra,
lepton]), and possibly the signs of the zodiac. [R. Caldwell, _Comparative
Grammar of the Dravidian Languages_, London, 1856, p. 438.] On the probable
Chinese origin of the zodiac see Schlegel, loc. cit.

[306] Marie, Vol. II, p. 73; R. Caldwell, loc. cit.

[307] A. Cunningham, loc. cit., p. 50.

[308] C. A. J. Skeel, _Travel_, loc. cit., p. 14.

[309] _Inchiver_, from _inchi_, "the green root." [_Indian Antiquary_, Vol.
I, p. 352.]

[310] In China dating only from the second century A.D., however.

[311] The Italian _morra_.

[312] J. Bowring, _The Decimal System_, London, 1854, p. 2.

[313] H. A. Giles, lecture at Columbia University, March 12, 1902, on
"China and Ancient Greece."

[314] Giles, loc. cit.

[315] E.g., the names for grape, radish (_la-po_, [Greek: rhaphê]),
water-lily (_si-kua_,  "west gourds"; [Greek: sikua], "gourds"), are much
alike. [Giles, loc. cit.]

[316] _Epistles_, I, 1, 45-46. On the Roman trade routes, see Beazley, loc.
cit., Vol. I, p. 179.

[317] _Am. Journ. of Archeol._, Vol. IV, p. 366.

[318] M. Perrot gives this conjectural restoration of his words: "Ad me ex
India regum legationes saepe missi sunt numquam antea visae apud quemquam
principem Romanorum." [M. Reinaud, "Relations politiques et commerciales de
l'empire romain avec l'Asie orientale," _Journ. Asiat._, Vol. I (6), p.
93.]

[319] Reinaud, loc. cit., p. 189. Florus, II, 34 (IV, 12), refers to it:
"Seres etiam habitantesque sub ipso sole Indi, cum gemmis et margaritis
elephantes quoque inter munera trahentes nihil magis quam longinquitatem
viae imputabant." Horace shows his geographical knowledge by saying: "Not
those who drink of the deep Danube shall now break the Julian edicts; not
the Getae, not the Seres, nor the perfidious Persians, nor those born on
the river Tanaïs." [_Odes_, Bk. IV, Ode 15, 21-24.]

[320] "Qua virtutis moderationisque fama Indos etiam ac Scythas auditu modo
cognitos pellexit ad amicitiam suam populique Romani ultro per legatos
petendam." [Reinaud, loc. cit., p. 180.]

[321] Reinaud, loc. cit., p. 180.

[322] _Georgics_, II, 170-172. So Propertius (_Elegies_, III, 4):

  Arma deus Caesar dites meditatur ad Indos
    Et freta gemmiferi findere classe maris.

"The divine Cæsar meditated carrying arms against opulent India, and with
his ships to cut the gem-bearing seas."

[323] Heyd, loc. cit., Vol. I, p. 4.

[324] Reinaud, loc. cit., p. 393.

[325] The title page of Calandri (1491), for example, represents Pythagoras
with these numerals before him. [Smith, _Rara Arithmetica_, p. 46.] Isaacus
Vossius, _Observationes ad Pomponium Melam de situ orbis_, 1658, maintained
that the Arabs derived these numerals from the west. A learned dissertation
to this effect, but deriving them from the Romans instead of the Greeks,
was written by Ginanni in 1753 (_Dissertatio mathematica critica de
numeralium notarum minuscularum origine_, Venice, 1753). See also Mannert,
_De numerorum quos arabicos vocant vera origine Pythagorica_, Nürnberg,
1801. Even as late as 1827 Romagnosi (in his supplement to _Ricerche
storiche sull' India_ etc., by Robertson, Vol. II, p. 580, 1827) asserted
that Pythagoras originated them. [R. Bombelli, _L'antica numerazione
italica_, Rome, 1876, p. 59.] Gow (_Hist. of Greek Math._, p. 98) thinks
that Iamblichus must have known a similar system in order to have worked
out certain of his theorems, but this is an unwarranted deduction from the
passage given.

[326] A. Hillebrandt, _Alt-Indien_, p. 179.

[327] J. C. Marshman, loc. cit., chaps. i and ii.

[328] He reigned 631-579 A.D.; called Nu['s][=i]rw[=a]n, _the holy one_.

[329] J. Keane, _The Evolution of Geography_, London, 1899, p. 38.

[330] The Arabs who lived in and about Mecca.

[331] S. Guyard, in _Encyc. Brit._, 9th ed., Vol. XVI, p. 597.

[332] Oppert, loc. cit., p. 29.

[333] "At non credendum est id in Autographis contigisse, aut vetustioribus
Codd. MSS." [Wallis, _Opera omnia_, Vol. II, p. 11.]

[334] In _Observationes ad Pomponium Melam de situ orbis_. The question was
next taken up in a large way by Weidler, loc. cit., _De characteribus_
etc., 1727, and in _Spicilegium_ etc., 1755.

[335] The best edition of these works is that of G. Friedlein, _Anicii
Manlii Torquati Severini Boetii de institutione arithmetica libri duo, de
institutione musica libri quinque. Accedit geometria quae fertur
Boetii_.... Leipzig.... MDCCCLXVII.

[336] See also P. Tannery, "Notes sur la pseudo-géometrie de Boèce," in
_Bibliotheca Mathematica_, Vol. I (3), p. 39. This is not the geometry in
two books in which are mentioned the numerals. There is a manuscript of
this pseudo-geometry of the ninth century, but the earliest one of the
other work is of the eleventh century (Tannery), unless the Vatican codex
is of the tenth century as Friedlein (p. 372) asserts.

[337] Friedlein feels that it is partly spurious, but he says: "Eorum
librorum, quos Boetius de geometria scripsisse dicitur, investigare veram
inscriptionem nihil aliud esset nisi operam et tempus perdere." [Preface,
p. v.] N. Bubnov in the Russian _Journal of the Ministry of Public
Instruction_, 1907, in an article of which a synopsis is given in the
_Jahrbuch über die Fortschritte der Mathematik_ for 1907, asserts that the
geometry was written in the eleventh century.

[338] The most noteworthy of these was for a long time Cantor
(_Geschichte_, Vol. I., 3d ed., pp. 587-588), who in his earlier days even
believed that Pythagoras had known them. Cantor says (_Die römischen
Agrimensoren_, Leipzig, 1875, p. 130): "Uns also, wir wiederholen es, ist
die Geometrie des Boetius echt, dieselbe Schrift, welche er nach Euklid
bearbeitete, von welcher ein Codex bereits in Jahre 821 im  Kloster
Reichenau vorhanden war, von welcher ein anderes Exemplar im Jahre 982 zu
Mantua in die Hände Gerbert's gelangte, von welcher mannigfache
Handschriften noch heute vorhanden sind." But against this opinion of the
antiquity of MSS. containing these numerals is the important statement of
P. Tannery, perhaps the most critical of modern historians of mathematics,
that none exists earlier than the eleventh century. See also J. L. Heiberg
in _Philologus, Zeitschrift f. d. klass. Altertum_, Vol. XLIII, p. 508.

Of Cantor's predecessors, Th. H. Martin was one of the most prominent, his
argument for authenticity appearing in the _Revue Archéologique_  for
1856-1857, and in his treatise _Les signes numéraux_ etc. See also M.
Chasles, "De la connaissance qu'ont eu les anciens d'une numération
décimale écrite qui fait usage de neuf chiffres prenant les valeurs de
position," _Comptes rendus_, Vol. VI, pp. 678-680; "Sur l'origine de notre
système de numération," _Comptes rendus_, Vol. VIII, pp. 72-81; and note
"Sur le passage du premier livre de la géométrie de Boèce, relatif à un
nouveau système de numération," in his work _Aperçu historique sur
l'origine et le devéloppement des méthodes en géométrie_, of which the
first edition appeared in 1837.

[339] J. L. Heiberg places the book in the eleventh century on philological
grounds, _Philologus_, loc. cit.; Woepcke, in _Propagation_, p. 44; Blume,
Lachmann, and Rudorff, _Die Schriften der römischen Feldmesser_, Berlin,
1848; Boeckh, _De abaco graecorum_, Berlin, 1841; Friedlein, in his Leipzig
edition of 1867; Weissenborn, _Abhandlungen_, Vol. II, p. 185, his
_Gerbert_, pp. 1, 247, and his _Geschichte der Einführung der jetzigen
Ziffern in Europa durch Gerbert_, Berlin, 1892, p. 11; Bayley, loc. cit.,
p. 59; Gerhardt, _Études_, p. 17, _Entstehung und Ausbreitung_, p. 14;
Nagl, _Gerbert_, p. 57; Bubnov, loc. cit. See also the discussion by
Chasles, Halliwell, and Libri, in the _Comptes rendus_, 1839, Vol. IX, p.
447, and in Vols. VIII, XVI, XVII of the same journal.

[340] J. Marquardt, _La vie privée des Romains_, Vol. II (French trans.),
p. 505, Paris, 1893.

[341] In a Plimpton manuscript of the arithmetic of Boethius of the
thirteenth century, for example, the Roman numerals are all replaced by the
Arabic, and the same is true in the first printed edition of the book. (See
Smith's _Rara Arithmetica_, pp. 434, 25-27.) D. E. Smith also copied from a
manuscript of the arithmetic in the Laurentian library at Florence, of
1370, the following forms, [Forged numerals

[342] Halliwell, in his _Rara Mathematica, _p. 107, states that the
disputed passage is not in a manuscript belonging to Mr. Ames, nor in one
at Trinity College. See also Woepcke, in _Propagation_, pp. 37 and 42. It
was the evident corruption of the texts in such editions of Boethius as
those of Venice, 1499, Basel, 1546 and 1570, that led Woepcke to publish
his work _Sur l'introduction de l'arithmétique indienne en Occident_.

[343] They are found in none of the very ancient manuscripts, as, for
example, in the ninth-century (?) codex in the Laurentian library which one
of the authors has examined. It should be said, however, that the disputed
passage was written after the arithmetic, for it contains a reference to
that work. See the Friedlein ed., p. 397.

[344] Smith, _Rara Arithmetica_, p. 66.

[345] J. L. Heiberg, _Philologus_, Vol. XLIII, p. 507.

[346] "Nosse autem huius artis dispicientem, quid sint digiti, quid
articuli, quid compositi, quid incompositi numeri." [Friedlein ed., p.
395.]

[347] _De ratione abaci._ In this he describes "quandam formulam, quam ob
honorem sui praeceptoris mensam Pythagoream nominabant ... a posterioribus
appellabatur abacus." This, as pictured in the text, is the common Gerbert
abacus. In the edition in Migne's _Patrologia Latina_, Vol. LXIII, an
ordinary multiplication table (sometimes called Pythagorean abacus) is
given in the illustration.

[348] "Habebant enim diverse formatos apices vel caracteres." See the
reference to Gerbert on p. 117.

[349] C. Henry, "Sur l'origine de quelques notations mathématiques," _Revue
Archéologique_, 1879, derives these from the initial letters used as
abbreviations for the names of the numerals, a theory that finds few
supporters.

[350] E.g., it appears in Schonerus, _Algorithmus Demonstratus_, Nürnberg,
1534, f. A4. In England it appeared in the earliest English arithmetical
manuscript known, _The Crafte of Nombrynge_: "¶ fforthermore ye most
vndirstonde that in this craft ben vsid teen figurys, as here bene writen
for ensampul, [Numerals] ... in the quych we vse teen figurys of Inde.
Questio. ¶ why ten fyguris of Inde? Solucio. for as I have sayd afore thei
were fonde fyrst in Inde of a kynge of that Cuntre, that was called Algor."
See Smith, _An Early English Algorism_, loc. cit.

[351] Friedlein ed., p. 397.

[352] Carlsruhe codex of Gerlando.

[353] Munich codex of Gerlando.

[354] Carlsruhe codex of Bernelinus.

[355] Munich codex of Bernelinus.

[356] Turchill, c. 1200.

[357] Anon. MS., thirteenth century, Alexandrian Library, Rome.

[358] Twelfth-century Boethius, Friedlein, p. 396.

[359] Vatican codex, tenth century, Boethius.

[360] a, h, i, are from the Friedlein ed.; the original in the manuscript
from which a is taken contains a zero symbol, as do all of the six plates
given by Friedlein. b-e from the Boncompagni _Bulletino_, Vol. X, p. 596; f
ibid., Vol. XV, p. 186; g _Memorie della classe di sci., Reale Acc. dei
Lincei_, An. CCLXXIV (1876-1877), April, 1877. A twelfth-century
arithmetician, possibly John of Luna (Hispalensis, of Seville, c. 1150),
speaks of the great diversity of these forms even in his day, saying: "Est
autem in aliquibus figuram istarum apud multos diuersitas. Quidam enim
septimam hanc figuram representant [Symbol] alii autem sic [Symbol], uel
sic [Symbol]. Quidam vero quartam sic [Symbol]." [Boncompagni, _Trattati_,
Vol. II, p. 28.]

[361] Loc. cit., p. 59.

[362] Ibid., p. 101.

[363] Loc. cit., p. 396.

[364] Khosr[=u] I, who began to reign in 531 A.D. See W. S. W Vaux,
_Persia, _London, 1875, p. 169; Th. Nöldeke, _Aufsätze zur persichen
Geschichte_, Leipzig, 1887, p. 113, and his article in the ninth edition of
the _Encyclopædia Britannica_.

[365] Colebrooke, _Essays_, Vol. II, p. 504, on the authority of Ibn
al-Adam[=i], astronomer, in a work published by his continuator Al-Q[=a]sim
in 920 A.D.; Al-B[=i]r[=u]n[=i], _India, _Vol. II, p. 15.

[366] H. Suter, _Die Mathematiker_ etc., pp. 4-5, states that
Al-Faz[=a]r[=i] died between 796 and 806.

[367] Suter, loc. cit., p. 63.

[368] Suter, loc. cit., p. 74.

[369] Suter, _Das Mathematiker-Verzeichniss im Fihrist_. The references to
Suter, unless otherwise stated, are to his later work _Die Mathematiker und
Astronomen der Araber_ etc.

[370] Suter, _Fihrist_, p. 37, no date.

[371] Suter, _Fihrist_, p. 38, no date.

[372] Possibly late tenth, since he refers to one arithmetical work which
is entitled _Book of the Cyphers_ in his _Chronology_, English ed., p. 132.
Suter, _Die Mathematiker_ etc., pp. 98-100, does not mention this work; see
the _Nachträge und Berichtigungen_, pp. 170-172.

[373] Suter, pp. 96-97.

[374] Suter, p. 111.

[375] Suter, p. 124. As the name shows, he came from the West.

[376] Suter, p. 138.

[377] Hankel, _Zur Geschichte der Mathematik_, p. 256, refers to him as
writing on the Hindu art of reckoning; Suter, p. 162.

[378] [Greek: Psêphophoria kat' Indous], Greek ed., C. I. Gerhardt, Halle,
1865; and German translation, _Das Rechenbuch des Maximus Planudes_, H.
Wäschke, Halle, 1878.

[379] "Sur une donnée historique relative à l'emploi des chiffres indiens
par les Arabes," Tortolini's _Annali di scienze mat. e fis._, 1855.

[380] Suter, p. 80.

[381] Suter, p. 68.

[382] Sprenger also calls attention to this fact, in the _Zeitschrift d.
deutschen morgenländ. Gesellschaft_, Vol. XLV, p. 367.

[383] Libri, _Histoire des mathématiques_, Vol. I, p. 147.

[384] "Dictant la paix à l'empereur de Constantinople, l'Arabe victorieux
demandait des manuscrits et des savans." [Libri, loc. cit., p. 108.]

[385] Persian _bagadata_, "God-given."

[386] One of the Abbassides, the (at least pretended) descendants of
`Al-Abb[=a]s, uncle and adviser of Mo[h.]ammed.

[387] E. Reclus, _Asia_, American ed., N. Y., 1891, Vol. IV, p. 227.

[388] _Historical Sketches_, Vol. III, chap. iii.

[389] On its prominence at that period see Villicus, p. 70.

[390] See pp. 4-5.

[391] Smith, D. E., in the _Cantor Festschrift_, 1909, note pp. 10-11. See
also F. Woepcke, _Propagation_.

[392] Eneström, in _Bibliotheca Mathematica_, Vol. I (3), p. 499; Cantor,
_Geschichte_, Vol. I (3), p. 671.

[393] Cited in Chapter I. It begins: "Dixit algoritmi: laudes deo rectori
nostro atque defensori dicamus dignas." It is devoted entirely to the
fundamental operations and contains no applications.

[394] M. Steinschneider, "Die Mathematik bei den Juden," _Bibliotheca
Mathematica_, Vol. VIII (2), p. 99. See also the reference to this writer
in Chapter I.

[395] Part of this work has been translated from a Leyden MS. by F.
Woepcke, _Propagation_, and more recently by H. Suter, _Bibliotheca
Mathematica_, Vol. VII (3), pp. 113-119.

[396] A. Neander, _General History of the Christian Religion and Church_,
5th American ed., Boston, 1855, Vol. III, p. 335.

[397] Beazley, loc. cit., Vol. I, p. 49.

[398] Beazley, loc. cit., Vol. I, pp. 50, 460.

[399] See pp. 7-8.

[400] The name also appears as Mo[h.]ammed Ab[=u]'l-Q[=a]sim, and Ibn
Hauqal. Beazley, loc. cit., Vol. I, p. 45.

[401] _Kit[=a]b al-mas[=a]lik wa'l-mam[=a]lik._

[402] Reinaud, _Mém. sur l'Inde_; in Gerhardt, _Études_, p. 18.

[403] Born at Shiraz in 1193. He himself had traveled from India to Europe.

[404] _Gulistan_ (_Rose Garden_), Gateway the third, XXII. Sir Edwin
Arnold's translation, N. Y., 1899, p. 177.

[405] Cunningham, loc. cit., p. 81.

[406] Putnam, _Books_, Vol. I, p. 227:

 "Non semel externas peregrino tramite terras
  Jam peragravit ovans, sophiae deductus amore,
  Si quid forte novi librorum seu studiorum
  Quod secum ferret, terris reperiret in illis.
  Hic quoque Romuleum venit devotus ad urbem."

("More than once he has traveled joyfully through remote regions and by
strange roads, led on by his zeal for knowledge and seeking to discover in
foreign lands novelties in books or in studies which he could take back
with him. And this zealous student journeyed to the city of Romulus.")

[407] A. Neander, _General History of the Christian Religion and Church_,
5th American ed., Boston, 1855, Vol. III, p. 89, note 4; Libri, _Histoire_,
Vol. I, p. 143.

[408] Cunningham, loc. cit., p. 81.

[409] Heyd, loc. cit., Vol. I, p. 4.

[410] Ibid., p. 5.

[411] Ibid., p. 21.

[412] Ibid., p. 23.

[413] Libri, _Histoire_, Vol. I, p. 167.

[414] Picavet, _Gerbert, un pape philosophe, d'après l'histoire et d'après
la légende_, Paris, 1897, p. 19.

[415] Beazley, loc. cit., Vol. I, chap, i, and p. 54 seq.

[416] Ibid., p. 57.

[417] Libri, _Histoire_, Vol. I, p. 110, n., citing authorities, and p.
152.

[418] Possibly the old tradition, "Prima dedit nautis usum magnetis
Amalphis," is true so far as it means the modern form of compass card. See
Beazley, loc. cit., Vol. II, p. 398.

[419] R. C. Dutt, loc. cit., Vol. II, p. 312.

[420] E. J. Payne, in _The Cambridge Modern History_, London, 1902, Vol. I,
chap. i.

[421] Geo. Phillips, "The Identity of Marco Polo's Zaitun with Changchau,
in T'oung pao," _Archives pour servir à l'étude de l'histoire de l'Asie
orientale_, Leyden, 1890, Vol. I, p. 218. W. Heyd, _Geschichte des
Levanthandels im Mittelalter_, Vol. II, p. 216.

The Palazzo dei Poli, where Marco was born and died, still stands in the
Corte del Milione, in Venice. The best description of the Polo travels, and
of other travels of the later Middle Ages, is found in C. R. Beazley's
_Dawn of Modern Geography_, Vol. III, chap, ii, and Part II.

[422] Heyd, loc. cit., Vol. II, p. 220; H. Yule, in _Encyclopædia
Britannica_, 9th (10th) or 11th ed., article "China." The handbook cited is
Pegolotti's _Libro di divisamenti di paesi_, chapters i-ii, where it is
implied that $60,000 would be a likely amount for a merchant going to China
to invest in his trip.

[423] Cunningham, loc. cit., p. 194.

[424] I.e. a commission house.

[425] Cunningham, loc. cit., p. 186.

[426] J. R. Green, _Short History of the English People_, New York, 1890,
p. 66.

[427] W. Besant, _London_, New York, 1892, p. 43.

[428] _Baldakin_, _baldekin_, _baldachino_.

[429] Italian _Baldacco_.

[430] J. K. Mumford, _Oriental Rugs_, New York, 1901, p. 18.

[431] Or Girbert, the Latin forms _Gerbertus_ and _Girbertus_ appearing
indifferently in the documents of his time.

[432] See, for example, J. C. Heilbronner, _Historia matheseos universæ_,
p. 740.

[433] "Obscuro loco natum," as an old chronicle of Aurillac has it.

[434] N. Bubnov, _Gerberti postea Silvestri II papae opera mathematica_,
Berlin, 1899, is the most complete and reliable source of information;
Picavet, loc. cit., _Gerbert_ etc.; Olleris, _Oeuvres de Gerbert_, Paris,
1867; Havet, _Lettres de Gerbert_, Paris, 1889 ; H. Weissenborn, _Gerbert;
Beiträge zur Kenntnis der Mathematik des Mittelalters_, Berlin, 1888, and
_Zur Geschichte der Einführung der jetzigen Ziffern in Europa durch
Gerbert_, Berlin, 1892; Büdinger, _Ueber Gerberts wissenschaftliche und
politische Stellung_, Cassel, 1851; Richer, "Historiarum liber III," in
Bubnov, loc. cit., pp. 376-381; Nagl, _Gerbert und die Rechenkunst des 10.
Jahrhunderts_, Vienna, 1888.

[435] Richer tells of the visit to Aurillac by Borel, a Spanish nobleman,
just as Gerbert was entering into young manhood. He relates how
affectionately the abbot received him, asking if there were men in Spain
well versed in the arts. Upon Borel's reply in the affirmative, the abbot
asked that one of his young men might accompany him upon his return, that
he might carry on his studies there.

[436] Vicus Ausona. Hatto also appears as Atton and Hatton.

[437] This is all that we know of his sojourn in Spain, and this comes from
his pupil Richer. The stories told by Adhemar of Chabanois, an apparently
ignorant and certainly untrustworthy contemporary, of his going to Cordova,
are unsupported. (See e.g. Picavet, p. 34.) Nevertheless this testimony is
still accepted: K. von Raumer, for example (_Geschichte der Pädagogik_, 6th
ed., 1890, Vol. I, p. 6), says "Mathematik studierte man im Mittelalter bei
den Arabern in Spanien. Zu ihnen gieng Gerbert, nachmaliger Pabst Sylvester
II."

[438] Thus in a letter to Aldaberon he says: "Quos post repperimus
speretis, id est VIII volumina Boeti de astrologia, praeclarissima quoque
figurarum geometriæ, aliaque non minus admiranda" (Epist. 8). Also in a
letter to Rainard (Epist. 130), he says: "Ex tuis sumptibus fac ut michi
scribantur M. Manlius (Manilius in one MS.) de astrologia."

[439] Picavet, loc. cit., p. 31.

[440] Picavet, loc. cit., p. 36.

[441] Havet, loc. cit., p. vii.

[442] Picavet, loc. cit., p. 37.

[443] "Con sinistre arti conseguri la dignita del Pontificato.... Lasciato
poi l' abito, e 'l monasterio, e datosi tutto in potere del diavolo."
[Quoted in Bombelli, _L'antica numerazione Italica_, Rome, 1876, p. 41 n.]

[444] He writes from Rheims in 984 to one Lupitus, in Barcelona, saying:
"Itaque librum de astrologia translatum a te michi petenti dirige,"
presumably referring to some Arabic treatise. [Epist. no. 24 of the Havet
collection, p. 19.]

[445] See Bubnov, loc. cit., p. x.

[446] Olleris, loc. cit., p. 361, l. 15, for Bernelinus; and Bubnov, loc.
cit., p. 381, l. 4, for Richer.

[447] Woepcke found this in a Paris MS. of Radulph of Laon, c. 1100.
[_Propagation_, p. 246.] "Et prima quidem trium spaciorum superductio
unitatis caractere inscribitur, qui chaldeo nomine dicitur igin." See also
Alfred Nagl, "Der arithmetische Tractat des Radulph von Laon"
(_Abhandlungen zur Geschichte der Mathematik_, Vol. V, pp. 85-133), p. 97.

[448] Weissenborn, loc. cit., p. 239. When Olleris (_Oeuvres de Gerbert_,
Paris, 1867, p. cci) says, "C'est à lui et non point aux Arabes, que
l'Europe doit son système et ses signes de numération," he exaggerates,
since the evidence is all against his knowing the place value. Friedlein
emphasizes this in the _Zeitschrift für Mathematik und Physik_, Vol. XII
(1867), _Literaturzeitung_, p. 70: "Für das _System_ unserer Numeration ist
die _Null_ das wesentlichste Merkmal, und diese kannte Gerbert nicht. Er
selbst schrieb alle Zahlen mit den römischen Zahlzeichen und man kann ihm
also nicht verdanken, was er selbst nicht kannte."

[449] E.g., Chasles, Büdinger, Gerhardt, and Richer. So Martin (_Recherches
nouvelles_ etc.) believes that Gerbert received them from Boethius or his
followers. See Woepcke, _Propagation_, p. 41.

[450] Büdinger, loc. cit., p. 10. Nevertheless, in Gerbert's time one
Al-Man[s.][=u]r, governing Spain under the name of Hish[=a]m (976-1002),
called from the Orient Al-Be[.g][=a]n[=i] to teach his son, so that
scholars were recognized. [Picavet, p. 36.]

[451] Weissenborn, loc. cit., p. 235.

[452] Ibid., p. 234.

[453] These letters, of the period 983-997, were edited by Havet, loc.
cit., and, less completely, by Olleris, loc. cit. Those touching
mathematical topics were edited by Bubnov, loc. cit., pp. 98-106.

[454] He published it in the _Monumenta Germaniae historica_, "Scriptores,"
Vol. III, and at least three other editions have since appeared, viz. those
by Guadet in 1845, by Poinsignon in 1855, and by Waitz in 1877.

[455] Domino ac beatissimo Patri Gerberto, Remorum archiepiscopo, Richerus
Monchus, Gallorum congressibus in volumine regerendis, imperii tui, pater
sanctissime Gerberte, auctoritas seminarium dedit.

[456] In epistle 17 (Havet collection) he speaks of the "De multiplicatione
et divisione numerorum libellum a Joseph Ispano editum abbas Warnerius" (a
person otherwise unknown). In epistle 25 he says: "De multiplicatione et
divisione numerorum, Joseph Sapiens sententias quasdam edidit."

[457] H. Suter, "Zur Frage über den Josephus Sapiens," _Bibliotheca
Mathematica_, Vol. VIII (2), p. 84; Weissenborn, _Einführung_, p. 14; also
his _Gerbert_; M. Steinschneider, in _Bibliotheca Mathematica_, 1893, p.
68. Wallis (_Algebra_, 1685, chap. 14) went over the list of Spanish
Josephs very carefully, but could find nothing save that "Josephus Hispanus
seu Josephus sapiens videtur aut Maurus fuisse aut alius quis in Hispania."

[458] P. Ewald, _Mittheilungen, Neues Archiv d. Gesellschaft für ältere
deutsche Geschichtskunde_, Vol. VIII, 1883, pp. 354-364. One of the
manuscripts is of 976 A.D. and the other of 992 A.D. See also Franz
Steffens, _Lateinische Paläographie_, Freiburg (Schweiz), 1903, pp.
xxxix-xl. The forms are reproduced in the plate on page 140.

[459] It is entitled _Constantino suo Gerbertus scolasticus_, because it
was addressed to Constantine, a monk of the Abbey of Fleury. The text of
the letter to Constantine, preceding the treatise on the Abacus, is given
in the _Comptes rendus_, Vol. XVI (1843), p. 295. This book seems to have
been written c. 980 A.D. [Bubnov, loc. cit., p. 6.]

[460] "Histoire de l'Arithmétique," _Comptes rendus_, Vol. XVI (1843), pp.
156, 281.

[461] Loc. cit., _Gerberti Opera_ etc.

[462] Friedlein thought it spurious. See _Zeitschrift für Mathematik und
Physik_, Vol. XII (1867), Hist.-lit. suppl., p. 74. It was discovered in
the library of the Benedictine monastry of St. Peter, at Salzburg, and was
published by Peter Bernhard Pez in 1721. Doubt was first cast upon it in
the Olleris edition (_Oeuvres de Gerbert_). See Weissenborn, _Gerbert_, pp.
2, 6, 168, and Picavet, p. 81. Hock, Cantor, and Th. Martin place the
composition of the work at c. 996 when Gerbert was in Germany, while
Olleris and Picavet refer it to the period when he was at Rheims.

[463] Picavet, loc. cit., p. 182.

[464] Who wrote after Gerbert became pope, for he uses, in his preface, the
words, "a domino pape Gerberto." He was quite certainly not later than the
eleventh century; we do not have exact information about the time in which
he lived.

[465] Picavet, loc. cit., p. 182. Weissenborn, _Gerbert_, p. 227. In
Olleris, _Liber Abaci_ (of Bernelinus), p. 361.

[466] Richer, in Bubnov, loc. cit., p. 381.

[467] Weissenborn, _Gerbert_, p. 241.

[468] Writers on numismatics are quite uncertain as to their use. See F.
Gnecchi, _Monete Romane_, 2d ed., Milan, 1900, cap. XXXVII. For pictures of
old Greek tesserae of Sarmatia, see S. Ambrosoli, _Monete Greche_, Milan,
1899, p. 202.

[469] Thus Tzwivel's arithmetic of 1507, fol. 2, v., speaks of the ten
figures as "characteres sive numerorum apices a diuo Seuerino Boetio."

[470] Weissenborn uses _sipos_ for 0. It is not given by Bernelinus, and
appears in Radulph of Laon, in the twelfth century. See Günther's
_Geschichte_, p. 98, n.; Weissenborn, p. 11; Pihan, _Exposé_ etc., pp.
xvi-xxii.

In Friedlein's _Boetius_, p. 396, the plate shows that all of the six
important manuscripts from which the illustrations are taken contain the
symbol, while four out of five which give the words use the word _sipos_
for 0. The names appear in a twelfth-century anonymous manuscript in the
Vatican, in a passage beginning

  Ordine primigeno sibi nomen possidet igin.
  Andras ecce locum mox uendicat ipse secundum
  Ormis post numeros incompositus sibi primus.

[Boncompagni _Buttetino_, XV, p. 132.] Turchill (twelfth century) gives the
names Igin, andras, hormis, arbas, quimas, caletis, zenis, temenias,
celentis, saying: "Has autem figuras, ut donnus [dominus] Gvillelmus Rx
testatur, a pytagoricis habemus, nomina uero ab arabibus." (Who the William
R. was is not known. Boncompagni _Bulletino_ XV, p. 136.) Radulph of Laon
(d. 1131) asserted that they were Chaldean (_Propagation_, p. 48 n.). A
discussion of the whole question is also given in E. C. Bayley, loc. cit.
Huet, writing in 1679, asserted that they were of Semitic origin, as did
Nesselmann in spite of his despair over ormis, calctis, and celentis; see
Woepcke, _Propagation_, p. 48. The names were used as late as the fifteenth
century, without the zero, but with the superscript dot for 10's, two dots
for 100's, etc., as among the early Arabs. Gerhardt mentions having seen a
fourteenth or fifteenth century manuscript in the Bibliotheca Amploniana
with the names "Ingnin, andras, armis, arbas, quinas, calctis, zencis,
zemenias, zcelentis," and the statement "Si unum punctum super ingnin
ponitur, X significat.... Si duo puncta super ... figuras superponunter,
fiet decuplim illius quod cum uno puncto significabatur," in
_Monatsberichte der K. P. Akad. d. Wiss._, Berlin, 1867, p. 40.

[471] _A chart of ten numerals in 200 tongues_, by Rev. R. Patrick, London,
1812.

[472] "Numeratio figuralis est cuiusuis numeri per notas, et figuras
numerates descriptio." [Clichtoveus, edition of c. 1507, fol. C ii, v.]
"Aristoteles enim uoces rerum [Greek: sumbola] uocat: id translatum, sonat
notas." [Noviomagus, _De Numeris Libri II_, cap. vi.] "Alphabetum decem
notarum." [Schonerus, notes to Ramus, 1586, p. 3 seq.] Richer says: "novem
numero notas omnem numerum significantes." [Bubnov, loc. cit., p. 381.]

[473] "Il y a dix Characteres, autrement Figures, Notes, ou Elements."
[Peletier, edition of 1607, p. 13.] "Numerorum notas alij figuras, alij
signa, alij characteres uocant." [Glareanus, 1545 edition, f. 9, r.] "Per
figuras (quas zyphras uocant) assignationem, quales sunt hæ notulæ, 1. 2.
3. 4...." [Noviomagus, _De Numeris Libri II_, cap. vi.] Gemma Frisius also
uses _elementa_ and Cardan uses _literae_. In the first arithmetic by an
American (Greenwood, 1729) the author speaks of "a few Arabian _Charecters_
or Numeral Figures, called _Digits_" (p. 1), and as late as 1790, in the
third edition of J. J. Blassière's arithmetic (1st ed. 1769), the name
_characters_ is still in use, both for "de Latynsche en de Arabische" (p.
4), as is also the term "Cyfferletters" (p. 6, n.). _Ziffer_, the modern
German form of cipher, was commonly used to designate any of the nine
figures, as by Boeschenstein and Riese, although others, like Köbel, used
it only for the zero. So _zifre_ appears in the arithmetic by Borgo, 1550
ed. In a Munich codex of the twelfth century, attributed to Gerland, they
are called _characters_ only: "Usque ad VIIII. enim porrigitur omnis
numerus et qui supercrescit eisdem designator Karacteribus." [Boncompagni
_Bulletino_, Vol. X. p. 607.]

[474] The title of his work is _Prologus N. Ocreati in Helceph_ (Arabic
_al-qeif_, investigation or memoir) _ad Adelardum Batensem magistrum suum_.
The work was made known by C. Henry, in the _Zeitschrift für Mathematik und
Physik_, Vol. XXV, p. 129, and in the _Abhandlungen zur Geschichte der
Mathematik_, Vol. III; Weissenborn, _Gerbert_, p. 188.

[475] The zero is indicated by a vacant column.

[476] Leo Jordan, loc. cit., p. 170. "Chifre en augorisme" is the
expression used, while a century later "giffre en argorisme" and "cyffres
d'augorisme" are similarly used.

[477] _The Works of Geoffrey Chaucer_, edited by W. W. Skeat, Vol. IV,
Oxford, 1894, p. 92.

[478] Loc. cit., Vol. III, pp. 179 and 180.

[479] In Book II, chap, vii, of _The Testament of Love_, printed with
Chaucer's Works, loc. cit., Vol. VII, London, 1897.

[480] _Liber Abacci_, published in Olleris, _Oeuvres de Gerbert_, pp.
357-400.

[481] G. R. Kaye, "The Use of the Abacus in Ancient India," _Journal and
Proceedings of the Asiatic Society of Bengal_, 1908, pp. 293-297.

[482] _Liber Abbaci_, by Leonardo Pisano, loc. cit., p. 1.

[483] Friedlein, "Die Entwickelung des Rechnens mit Columnen," _Zeitschrift
für Mathematik und Physik_, Vol. X, p. 247.

[484] The divisor 6 or 16 being increased by the difference 4, to 10 or 20
respectively.

[485] E.g. Cantor, Vol. I, p. 882.

[486] Friedlein, loc. cit.; Friedlein, "Gerbert's Regeln der Division" and
"Das Rechnen mit Columnen vor dem 10. Jahrhundert," _Zeitschrift für
Mathematik und Physik_, Vol. IX; Bubnov, loc. cit., pp. 197-245; M.
Chasles, "Histoire de l'arithmétique. Recherches des traces du système de
l'abacus, après que cette méthode a pris le nom d'Algorisme.--Preuves qu'à
toutes les époques, jusq'au XVI^e siècle, on a su que l'arithmétique
vulgaire avait pour origine cette méthode ancienne," _Comptes rendus_, Vol.
XVII, pp. 143-154, also "Règles de l'abacus," _Comptes rendus_, Vol. XVI,
pp. 218-246, and "Analyse et explication du traité de Gerbert," _Comptes
rendus_, Vol. XVI, pp. 281-299.

[487] Bubnov, loc. cit., pp. 203-204, "Abbonis abacus."

[488] "Regulae de numerorum abaci rationibus," in Bubnov, loc. cit., pp.
205-225.

[489] P. Treutlein, "Intorno ad alcuni scritti inediti relativi al calcolo
dell' abaco," _Bulletino di bibliografia e di storia delle scienze
matematiche e fisiche_, Vol. X, pp. 589-647.

[490] "Intorno ad uno scritto inedito di Adelhardo di Bath intitolato
'Regulae Abaci,'" B. Boncompagni, in his _Bulletino_, Vol. XIV, pp. 1-134.

[491] Treutlein, loc. cit.; Boncompagni, "Intorno al Tractatus de Abaco di
Gerlando," _Bulletino_, Vol. X, pp. 648-656.

[492] E. Narducci, "Intorno a due trattati inediti d'abaco contenuti in due
codici Vaticani del secolo XII," Boncompagni _Bulletino_, Vol. XV, pp.
111-162.

[493] See Molinier, _Les sources de l'histoire de France_, Vol. II, Paris,
1902, pp. 2, 3.

[494] Cantor, _Geschichte_, Vol. I, p. 762. A. Nagl in the _Abhandlungen
zur Geschichte der Mathematik_, Vol. V, p. 85.

[495] 1030-1117.

[496] _Abhandlungen zur Geschichte der Mathematik_, Vol. V, pp. 85-133. The
work begins "Incipit Liber Radulfi laudunensis de abaco."

[497] _Materialien zur Geschichte der arabischen Zahlzeichen in
Frankreich_, loc. cit.

[498] Who died in 1202.

[499] Cantor, _Geschichte_, Vol. I (3), pp. 800-803; Boncompagni,
_Trattati_, Part II. M. Steinschneider ("Die Mathematik bei den Juden,"
_Bibliotheca Mathematica_, Vol. X (2), p. 79) ingeniously derives another
name by which he is called (Abendeuth) from Ibn Da[=u]d (Son of David). See
also _Abhandlungen_, Vol. III, p. 110.

[500] John is said to have died in 1157.

[501] For it says, "Incipit prologus in libro alghoarismi de practica
arismetrice. Qui editus est a magistro Johanne yspalensi." It is published
in full in the second part of Boncompagni's _Trattati d'aritmetica_.

[502] Possibly, indeed, the meaning of "libro alghoarismi" is not "to
Al-Khow[=a]razm[=i]'s book," but "to a book of algorism." John of Luna says
of it: "Hoc idem est illud etiam quod ... alcorismus dicere videtur."
[_Trattati_, p. 68.]

[503] For a résumé, see Cantor, Vol. I (3), pp. 800-803. As to the author,
see Eneström in the _Bibliotheca Mathematica_, Vol. VI (3), p. 114, and
Vol. IX (3), p. 2.

[504] Born at Cremona (although some have asserted at Carmona, in
Andalusia) in 1114; died at Toledo in 1187. Cantor, loc. cit.; Boncompagni,
_Atti d. R. Accad. d. n. Lincei_, 1851.

[505] See _Abhandlungen zur Geschichte der Mathematik_, Vol. XIV, p. 149;
_Bibliotheca Mathematica_, Vol. IV (3), p. 206. Boncompagni had a
fourteenth-century manuscript of his work, _Gerardi Cremonensis artis
metrice practice_. See also T. L. Heath, _The Thirteen Books of Euclid's
Elements_, 3 vols., Cambridge, 1908, Vol. I, pp. 92-94 ; A. A. Björnbo,
"Gerhard von Cremonas Übersetzung von Alkwarizmis Algebra und von Euklids
Elementen," _Bibliotheca Mathematica_, Vol. VI (3), pp. 239-248.

[506] Wallis, _Algebra_, 1685, p. 12 seq.

[507] Cantor, _Geschichte_, Vol. I (3), p. 906; A. A. Björnbo,
"Al-Chw[=a]rizm[=i]'s trigonometriske Tavler," _Festskrift til H. G.
Zeuthen_, Copenhagen, 1909, pp. 1-17.

[508] Heath, loc. cit., pp. 93-96.

[509] M. Steinschneider, _Zeitschrift der deutschen morgenländischen
Gesellschaft_, Vol. XXV, 1871, p. 104, and _Zeitschrift für Mathematik und
Physik_, Vol. XVI, 1871, pp. 392-393; M. Curtze, _Centralblatt für
Bibliothekswesen_, 1899, p. 289; E. Wappler, _Zur Geschichte der deutschen
Algebra im 15. Jahrhundert_, Programm, Zwickau, 1887; L. C. Karpinski,
"Robert of Chester's Translation of the Algebra of Al-Khow[=a]razm[=i],"
_Bibliotheca Mathematica_, Vol. XI (3), p. 125. He is also known as
Robertus Retinensis, or Robert of Reading.

[510] Nagl, A., "Ueber eine Algorismus-Schrift des XII. Jahrhunderts und
über die Verbreitung der indisch-arabischen Rechenkunst und Zahlzeichen im
christl. Abendlande," in the _Zeitschrift für Mathematik und Physik,
Hist.-lit. Abth._, Vol. XXXIV, p. 129. Curtze, _Abhandlungen zur Geschichte
der Mathematik_, Vol. VIII, pp. 1-27.

[511] See line _a_ in the plate on p. 143.

[512] _Sefer ha-Mispar, Das Buch der Zahl, ein hebräisch-arithmetisches
Werk des R. Abraham ibn Esra_, Moritz Silberberg, Frankfurt a. M., 1895.

[513] Browning's "Rabbi ben Ezra."

[514] "Darum haben auch die Weisen Indiens all ihre Zahlen durch neun
bezeichnet und Formen für die 9 Ziffern gebildet." [_Sefer ha-Mispar_, loc.
cit., p. 2.]

[515] F. Bonaini, "Memoria unica sincrona di Leonardo Fibonacci," Pisa,
1858, republished in 1867, and appearing in the _Giornale Arcadico_, Vol.
CXCVII (N.S. LII); Gaetano Milanesi, _Documento inedito e sconosciuto a
Lionardo Fibonacci_, Roma, 1867; Guglielmini, _Elogio di Lionardo Pisano_,
Bologna, 1812, p. 35; Libri, _Histoire des sciences mathématiques_, Vol.
II, p. 25; D. Martines, _Origine e progressi dell' aritmetica_, Messina,
1865, p. 47; Lucas, in Boncompagni _Bulletino_, Vol. X, pp. 129, 239;
Besagne, ibid., Vol. IX, p. 583; Boncompagni, three works as cited in Chap.
I; G. Eneström, "Ueber zwei angebliche mathematische Schulen im
christlichen Mittelalter," _Bibliotheca Mathematica_, Vol. VIII (3), pp.
252-262; Boncompagni, "Della vita e delle opere di Leonardo Pisano," loc.
cit.

[516] The date is purely conjectural. See the _Bibliotheca Mathematica_,
Vol. IV (3), p. 215.

[517] An old chronicle relates that in 1063 Pisa fought a great battle with
the Saracens at Palermo, capturing six ships, one being "full of wondrous
treasure," and this was devoted to building the cathedral.

[518] Heyd, loc. cit., Vol. I, p. 149.

[519] Ibid., p. 211.

[520] J. A. Symonds, _Renaissance in Italy. The Age of Despots._ New York,
1883, p. 62.

[521] Symonds, loc. cit., p. 79.

[522] J. A. Froude, _The Science of History_, London, 1864. "Un brevet
d'apothicaire n'empêcha pas Dante d'être le plus grand poète de l'Italie,
et ce fut un petit marchand de Pise qui donna l'algèbre aux Chrétiens."
[Libri, _Histoire_, Vol. I, p. xvi.]

[523] A document of 1226, found and published in 1858, reads: "Leonardo
bigollo quondam Guilielmi."

[524] "Bonaccingo germano suo."

[525] E.g. Libri, Guglielmini, Tiraboschi.

[526] Latin, _Bonaccius_.

[527] Boncompagni and Milanesi.

[528] Reprint, p. 5.

[529] Whence the French name for candle.

[530] Now part of Algiers.

[531] E. Reclus, _Africa_, New York, 1893, Vol. II, p. 253.

[532] "Sed hoc totum et algorismum atque arcus pictagore quasi errorem
computavi respectu modi indorum." Woepcke, _Propagation_ etc., regards this
as referring to two different systems, but the expression may very well
mean algorism as performed upon the Pythagorean arcs (or table).

[533] "Book of the Abacus," this term then being used, and long afterwards
in Italy, to mean merely the arithmetic of computation.

[534] "Incipit liber Abaci a Leonardo filio Bonacci compositus anno 1202 et
correctus ab eodem anno 1228." Three MSS. of the thirteenth century are
known, viz. at Milan, at Siena, and in the Vatican library. The work was
first printed by Boncompagni in 1857.

[535] I.e. in relation to the quadrivium. "Non legant in festivis diebus,
nisi Philosophos et rhetoricas et quadrivalia et barbarismum et ethicam, si
placet." Suter, _Die Mathematik auf den Universitäten des Mittelalters_,
Zürich, 1887, p. 56. Roger Bacon gives a still more gloomy view of Oxford
in his time in his _Opus minus_, in the _Rerum Britannicarum medii aevi
scriptores_, London, 1859, Vol. I, p. 327. For a picture of Cambridge at
this time consult F. W. Newman, _The English Universities, translated from
the German of V. A. Huber_, London, 1843, Vol. I, p. 61; W. W. R. Ball,
_History of Mathematics at Cambridge_, 1889; S. Günther, _Geschichte des
mathematischen Unterrichts im deutschen Mittelalter bis zum Jahre 1525_,
Berlin, 1887, being Vol. III of _Monumenta Germaniae paedagogica_.

[536] On the commercial activity of the period, it is known that bills of
exchange passed between Messina and Constantinople in 1161, and that a bank
was founded at Venice in 1170, the Bank of San Marco being established in
the following year. The activity of Pisa was very manifest at this time.
Heyd, loc. cit., Vol. II, p. 5; V. Casagrandi, _Storia e cronologia_, 3d
ed., Milan, 1901, p. 56.

[537] J. A. Symonds, loc. cit., Vol. II, p. 127.

[538] I. Taylor, _The Alphabet_, London, 1883, Vol. II, p. 263.

[539] Cited by Unger's History, p. 15. The Arabic numerals appear in a
Regensburg chronicle of 1167 and in Silesia in 1340. See Schmidt's
_Encyclopädie der Erziehung_, Vol. VI, p. 726; A. Kuckuk, "Die Rechenkunst
im sechzehnten Jahrhundert," _Festschrift zur dritten Säcularfeier des
Berlinischen Gymnasiums zum grauen Kloster_, Berlin, 1874, p. 4.

[540] The text is given in Halliwell, _Rara Mathematica_, London, 1839.

[541] Seven are given in Ashmole's _Catalogue of Manuscripts in the Oxford
Library_, 1845.

[542] Maximilian Curtze, _Petri Philomeni de Dacia in Algorismum Vulgarem
Johannis de Sacrobosco commentarius, una cum Algorismo ipso_, Copenhagen,
1897; L. C. Karpinski, "Jordanus Nemorarius and John of Halifax," _American
Mathematical Monthly_, Vol. XVII, pp. 108-113.

[543] J. Aschbach, _Geschichte der Wiener Universität im ersten
Jahrhunderte ihres Bestehens_, Wien, 1865, p. 93.

[544] Curtze, loc. cit., gives the text.

[545] Curtze, loc. cit., found some forty-five copies of the _Algorismus_
in three libraries of Munich, Venice, and Erfurt (Amploniana). Examination
of two manuscripts from the Plimpton collection and the Columbia library
shows such marked divergence from each other and from the text published by
Curtze that the conclusion seems legitimate that these were students'
lecture notes. The shorthand character of the writing further confirms this
view, as it shows that they were written largely for the personal use of
the writers.

[546] "Quidam philosophus edidit nomine Algus, unde et Algorismus
nuncupatur." [Curtze, loc. cit., p. 1.]

[547] "Sinistrorsum autera scribimus in hac arte more arabico sive iudaico,
huius scientiae inventorum." [Curtze, loc. cit., p. 7.] The Plimpton
manuscript omits the words "sive iudaico."

[548] "Non enim omnis numerus per quascumque figuras Indorum
repraesentatur, sed tantum determinatus per determinatam, ut 4 non per
5,..." [Curtze, loc. cit., p. 25.]

[549] C. Henry, "Sur les deux plus anciens traités français d'algorisme et
de géométrie," Boncompagni _Bulletino_, Vol. XV, p. 49; Victor Mortet, "Le
plus ancien traité français d'algorisme," loc. cit.

[550] _L'État des sciences en France, depute la mort du Roy Robert, arrivée
en 1031, jusqu'à celle de Philippe le Bel, arrivée en 1314_, Paris, 1741.

[551] _Discours sur l'état des lettres en France au XIII^e siecle_, Paris,
1824.

[552] _Aperçu historique_, Paris, 1876 ed., p. 464.

[553] Ranulf Higden, a native of the west of England, entered St.
Werburgh's monastery at Chester in 1299. He was a Benedictine monk and
chronicler, and died in 1364. His _Polychronicon_, a history in seven
books, was printed by Caxton in 1480.

[554] Trevisa's translation, Higden having written in Latin.

[555] An illustration of this feeling is seen in the writings of Prosdocimo
de' Beldomandi (b. c. 1370-1380, d. 1428): "Inveni in quam pluribus libris
algorismi nuncupatis mores circa numeros operandi satis varios atque
diversos, qui licet boni existerent atque veri erant, tamen fastidiosi, tum
propter ipsarum regularum multitudinem, tum propter earum deleationes, tum
etiam propter ipsarum operationum probationes, utrum si bone fuerint vel
ne. Erant et etiam isti modi interim fastidiosi, quod si in aliquo calculo
astroloico error contigisset, calculatorem operationem suam a capite
incipere oportebat, dato quod error suus adhuc satis propinquus existeret;
et hoc propter figuras in sua operatione deletas. Indigebat etiam
calculator semper aliquo lapide vel sibi conformi, super quo scribere atque
faciliter delere posset figuras cum quibus operabatur in calculo suo. Et
quia haec omnia satis fastidiosa atque laboriosa mihi visa sunt, disposui
libellum edere in quo omnia ista abicerentur: qui etiam algorismus sive
liber de numeris denominari poterit. Scias tamen quod in hoc libello ponere
non intendo nisi ea quae ad calculum necessaria sunt, alia quae in aliis
libris practice arismetrice tanguntur, ad calculum non necessaria, propter
brevitatem dimitendo." [Quoted by A. Nagl, _Zeitschrift für Mathematik und
Physik, Hist.-lit. Abth._, Vol. XXXIV, p. 143; Smith, _Rara Arithmetica_,
p. 14, in facsimile.]

[556] P. Ewald, loc. cit.; Franz Steffens, _Lateinische Paläographie_, pp.
xxxix-xl. We are indebted to Professor J. M. Burnam for a photograph of
this rare manuscript.

[557] See the plate of forms on p. 88.

[558] Karabacek, loc. cit., p. 56; Karpinski, "Hindu Numerals in the
Fihrist," _Bibliotheca Mathematica_, Vol. XI (3), p. 121.

[559] Woepcke, "Sur une donnée historique," etc., loc. cit., and "Essai
d'une restitution de travaux perdus d'Apollonius sur les quantités
irrationnelles, d'après des indications tirées d'un manuscrit arabe," _Tome
XIV des Mémoires présentés par divers savants à l'Académie des sciences_,
Paris, 1856, note, pp. 6-14.

[560] _Archeological Report of the Egypt Exploration Fund for 1908-1909_,
London, 1910, p. 18.

[561] There was a set of astronomical tables in Boncompagni's library
bearing this date: "Nota quod anno d[=n]i [=n]ri ihû x[=p]i. 1264.
perfecto." See Narducci's _Catalogo_, p. 130.

[562] "On the Early use of Arabic Numerals in Europe," read before the
Society of Antiquaries April 14, 1910, and published in _Archæologia_ in
the same year.

[563] Ibid., p. 8, n. The date is part of an Arabic inscription.

[564] O. Codrington, _A Manual of Musalman Numismatics_, London, 1904.

[565] See Arbuthnot, _The Mysteries of Chronology_, London, 1900, pp. 75,
78, 98; F. Pichler, _Repertorium der steierischen Münzkunde_, Grätz, 1875,
where the claim is made of an Austrian coin of 1458; _Bibliotheca
Mathematica_, Vol. X (2), p. 120, and Vol. XII (2), p. 120. There is a
Brabant piece of 1478 in the collection of D. E. Smith.

[566] A specimen is in the British Museum. [Arbuthnot, p. 79.]

[567] Ibid., p. 79.

[568] _Liber de Remediis utriusque fortunae Coloniae._

[569] Fr. Walthern et Hans Hurning, Nördlingen.

[570] _Ars Memorandi_, one of the oldest European block-books.

[571] Eusebius Caesariensis, _De praeparatione evangelica_, Venice, Jenson,
1470. The above statement holds for copies in the Astor Library and in the
Harvard University Library.

[572] Francisco de Retza, _Comestorium vitiorum_, Nürnberg, 1470. The copy
referred to is in the Astor Library.

[573] See Mauch, "Ueber den Gebrauch arabischer Ziffern und die
Veränderungen derselben," _Anzeiger für Kunde der deutschen Vorzeit_, 1861,
columns 46, 81, 116, 151, 189, 229, and 268; Calmet, _Recherches sur
l'origine des chiffres d'arithmétique_, plate, loc. cit.

[574] Günther, _Geschichte_, p. 175, n.; Mauch, loc. cit.

[575] These are given by W. R. Lethaby, from drawings by J. T. Irvine, in
the _Proceedings of the Society of Antiquaries_, 1906, p. 200.

[576] There are some ill-tabulated forms to be found in J. Bowring, _The
Decimal System_, London, 1854, pp. 23, 25, and in L. A. Chassant,
_Dictionnaire des abréviations latines et françaises ... du moyen âge_,
Paris, MDCCCLXVI, p. 113. The best sources we have at present, aside from
the Hill monograph, are P. Treutlein, _Geschichte unserer Zahlzeichen_,
Karlsruhe, 1875; Cantor's _Geschichte_, Vol. I, table; M. Prou, _Manuel de
paléographie latine et française_, 2d ed., Paris, 1892, p. 164; A.
Cappelli, _Dizionario di abbreviature latine ed italiane_, Milan, 1899. An
interesting early source is found in the rare Caxton work of 1480, _The
Myrrour of the World_. In Chap. X is a cut with the various numerals, the
chapter beginning "The fourth scyence is called arsmetrique." Two of the
fifteen extant copies of this work are at present in the library of Mr. J.
P. Morgan, in New York.

[577] From the twelfth-century manuscript on arithmetic, Curtze, loc. cit.,
_Abhandlungen_, and Nagl, loc. cit. The forms are copied from Plate VII in
_Zeitschrift für Mathematik und Physik_, Vol. XXXIV.

[578] From the Regensburg chronicle. Plate containing some of these
numerals in _Monumenta Germaniae historica_, "Scriptores" Vol. XVII, plate
to p. 184; Wattenbach, _Anleitung zur lateinischen Palaeographie_, Leipzig,
1886, p. 102; Boehmer, _Fontes rerum Germanicarum_, Vol. III, Stuttgart,
1852, p. lxv.

[579] French Algorismus of 1275; from an unpublished photograph of the
original, in the possession of D. E. Smith. See also p. 135.

[580] From a manuscript of Boethius c. 1294, in Mr. Plimpton's library.
Smith, _Rara Arithmetica_, Plate I.

[581] Numerals in a 1303 manuscript in Sigmaringen, copied from Wattenbach,
loc. cit., p. 102.

[582] From a manuscript, Add. Manuscript 27,589, British Museum, 1360 A.D.
The work is a computus in which the date 1360 appears, assigned in the
British Museum catalogue to the thirteenth century.

[583] From the copy of Sacrabosco's _Algorismus_ in Mr. Plimpton's library.
Date c. 1442. See Smith, _Rara Arithmetica_, p. 450.

[584] See _Rara Arithmetica_, pp. 446-447.

[585] Ibid., pp. 469-470.

[586] Ibid., pp. 477-478.

[587] The i is used for "one" in the Treviso arithmetic (1478), Clichtoveus
(c. 1507 ed., where both i and j are so used), Chiarini (1481), Sacrobosco
(1488 ed.), and Tzwivel (1507 ed., where jj and jz are used for 11 and 12).
This was not universal, however, for the _Algorithmus linealis_ of c. 1488
has a special type for 1. In a student's notebook of lectures taken at the
University of Würzburg in 1660, in Mr. Plimpton's library, the ones are all
in the form of i.

[588] Thus the date [Numerals 1580], for 1580, appears in a MS. in the
Laurentian library at Florence. The second and the following five
characters are taken from Cappelli's _Dizionario_, p. 380, and are from
manuscripts of the twelfth, thirteenth, fourteenth, sixteenth, seventeenth,
and eighteenth centuries, respectively.

[589] E.g. Chiarini's work of 1481; Clichtoveus (c. 1507).

[590] The first is from an algorismus of the thirteenth century, in the
Hannover Library. [See Gerhardt, "Ueber die Entstehung und Ausbreitung des
dekadischen Zahlensystems," loc. cit., p. 28.] The second character is from
a French algorismus, c. 1275. [Boncompagni _Bulletino_, Vol. XV, p. 51.]
The third and the following sixteen characters are given by Cappelli, loc.
cit., and are from manuscripts of the twelfth (1), thirteenth (2),
fourteenth (7), fifteenth (3), sixteenth (1), seventeenth (2), and
eighteenth (1) centuries, respectively.

[591] Thus Chiarini (1481) has [Symbol] for 23.

[592] The first of these is from a French algorismus, c. 1275. The second
and the following eight characters are given by Cappelli, loc. cit., and
are from manuscripts of the twelfth (2), thirteenth, fourteenth, fifteenth
(3), seventeenth, and eighteenth centuries, respectively.

[593] See Nagl, loc. cit.

[594] Hannover algorismus, thirteenth century.

[595] See the Dagomari manuscript, in _Rara Arithmetica_, pp. 435, 437-440.

[596] But in the woodcuts of the _Margarita Philosophica_ (1503) the old
forms are used, although the new ones appear in the text. In Caxton's
_Myrrour of the World_ (1480) the old form is used.

[597] Cappelli, loc. cit. They are partly from manuscripts of the tenth,
twelfth, thirteenth (3), fourteenth (7), fifteenth (6), and eighteenth
centuries, respectively. Those in the third line are from Chassant's
_Dictionnaire_, p. 113, without mention of dates.

[598] The first is from the Hannover algorismus, thirteenth century. The
second is taken from the Rollandus manuscript, 1424. The others in the
first two lines are from Cappelli, twelfth (3), fourteenth (6), fifteenth
(13) centuries, respectively. The third line is from Chassant, loc. cit.,
p. 113, no mention of dates.

[599] The first of these forms is from the Hannover algorismus, thirteenth
century. The following are from Cappelli, fourteenth (3), fifteenth,
sixteenth (2), and eighteenth centuries, respectively.

[600] The first of these is taken from the Hannover algorismus, thirteenth
century. The following forms are from Cappelli, twelfth, thirteenth,
fourteenth (5), fifteenth (2), seventeenth, and eighteenth centuries,
respectively.

[601] All of these are given by Cappelli, thirteenth, fourteenth, fifteenth
(2), and sixteenth centuries, respectively.

[602] Smith, _Rara Arithmetica_, p. 489. This is also seen in several of
the Plimpton manuscripts, as in one written at Ancona in 1684. See also
Cappelli, loc. cit.

[603] French algorismus, c. 1275, for the first of these forms. Cappelli,
thirteenth, fourteenth, fifteenth (3), and seventeenth centuries,
respectively. The last three are taken from _Byzantinische Analekten_, J.
L. Heiberg, being forms of the fifteenth century, but not at all common.
[Symbol: Qoppa] was the old Greek symbol for 90.

[604] For the first of these the reader is referred to the forms ascribed
to Boethius, in the illustration on p. 88; for the second, to Radulph of
Laon, see p. 60. The third is used occasionally in the Rollandus (1424)
manuscript, in Mr. Plimpton's library. The remaining three are from
Cappelli, fourteenth (2) and seventeenth centuries.

[605] Smith, _An Early English Algorism_.

[606] Kuckuck, p. 5.

[607] A. Cappelli, loc. cit., p. 372.

[608] Smith, _Rara Arithmetica_, p. 443.

[609] Curtze, _Petri Philomeni de Dacia_ etc., p. IX.

[610] Cappelli, loc. cit., p. 376.

[611] Curtze, loc. cit., pp. VIII-IX, note.

[612] Edition of 1544-1545, f. 52.

[613] _De numeris libri II_, 1544 ed., cap. XV. Heilbronner, loc. cit., p.
736, also gives them, and compares this with other systems.

[614] Noviomagus says of them: "De quibusdam Astrologicis, sive Chaldaicis
numerorum notis.... Sunt & aliæ quædam notæ, quibus Chaldaei & Astrologii
quemlibet numerum artificiose & arguté describunt, scitu periucundae, quas
nobis communicauit Rodolphus Paludanus Nouiomagus."