Chapter I: 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

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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

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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.

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THE HINDU-ARABIC NUMERALS