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.