Produced by Jonathan Ingram, Hagen von Eitzen and the Online Distributed Proofreading Team at https://www.pgdp.net [*Transcriber's Note: The following errors found in the original have been left as is. Chapter I, 14th paragraph: drop double quote before 'It is said'; Chapter IV, 1st paragraph: 'so similar than' read 'so similar that'; Chapter IV, table of Hebrew numerals (near footnote 144): insert comma after 'shemoneh'; Chapter V, table of Tahuatan numerals (near footnote 201): 'tahi,' read 'tahi.'; Same table: ' 20,000. tufa' read '200,000. tufa'; Chapter VI, table of Bagrimma numerals (near footnote 259): 'marta = 5 + 2' read 'marta = 5 + 3'; Same table: 'do-so = [5] + 3' read 'do-so = [5] + 4'; Chapter VII, table of Nahuatl numerals (near footnote 365): '90-10' read '80-10'; In paragraph following that table: '+ (15 + 4) x 400 x 800' read '(15 + 4) x 20 x 400 x 8000 + (15 + 4) x 400 x 8000'; In text of footnote 297: 'II. I. p. 179' read 'II. i. p. 179'; *] THE MACMILLAN COMPANY NEW YORK . BOSTON . CHICAGO . DALLAS ATLANTA . SAN FRANCISCO MACMILLAN & CO., LIMITED LONDON . BOMBAY . CALCUTTA MELBOURNE THE MACMILLAN COMPANY OF CANADA, LIMITED TORONTO THE NUMBER CONCEPT ITS ORIGIN AND DEVELOPMENT BY LEVI LEONARD CONANT, PH.D. ASSOCIATE PROFESSOR OF MATHEMATICS IN THE WORCESTER POLYTECHNIC INSTITUTE New York MACMILLAN AND CO. AND LONDON 1931 COPYRIGHT, 1896, BY THE MACMILLAN COMPANY. COPYRIGHT, 1924, BY EMMA B. CONANT. All rights reserved--no part of this book may be reproduced in any form without permission in writing from the publisher. Set up and electrotyped. Published July, 1896. Norwood Press J.S. Cushing Co.--Berwick & Smith Co. Norwood, Mass., U.S.A. PREFACE. In the selection of authorities which have been consulted in the preparation of this work, and to which reference is made in the following pages, great care has been taken. Original sources have been drawn upon in the majority of cases, and nearly all of these are the most recent attainable. Whenever it has not been possible to cite original and recent works, the author has quoted only such as are most standard and trustworthy. In the choice of orthography of proper names and numeral words, the forms have, in almost all cases, been written as they were found, with no attempt to reduce them to a systematic English basis. In many instances this would have been quite impossible; and, even if possible, it would have been altogether unimportant. Hence the forms, whether German, French, Italian, Spanish, or Danish in their transcription, are left unchanged. Diacritical marks are omitted, however, since the proper key could hardly be furnished in a work of this kind.