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_.