The Hindu-Arabic Numerals

The Hindu-Arabic Numerals, by

David Eugene Smith and Louis Charles Karpinski This eBook is for the
use of anyone anywhere at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms
of the Project Gutenberg License included with this eBook or online at
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Title: The Hindu-Arabic Numerals
Author: David Eugene Smith Louis Charles Karpinski
Release Date: September 14, 2007 [EBook #22599]
Language: English
Character set encoding: ISO-8859-1
*** START OF THIS PROJECT GUTENBERG EBOOK THE
HINDU-ARABIC NUMERALS ***

Produced by David Newman, Chuck Greif, Keith Edkins and the
Online Distributed Proofreading Team at http://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 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.