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. LATE