Produced by David Newman, Chuck Greif, Keith Edkins and
the Online Distributed Proofreading Team at
https://www.pgdp.net (This file was produced from images
from the Cornell University Library: Historical Mathematics
Monographs collection.)
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 ATHENAEUM 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.]. N, 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 mediaeval 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 Koebel[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 Sedillot,[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 aeternumque." 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}