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PREFACE TO THE FIRST EDITION THE education of the child must accord both in mode and arrangement with the education of mankind as consid ered historically or, in other words, the genesis of knowledge in the individual must follow the same course as the genesis of knowledge in the race. To M. Comte we believe society owes the enunciation of this doctrine a doctrine which we may accept without committing ourselves to his theory of the genesis of knowledge, either in its causes or its order. 1 If this principle, held also by Pestalozzi and Proebel, be correct, then it would seem as if the knowledge of the history of a science must be an effectual aid in teaching that science. Be this doctrine true or false, certainly the experience of many instructors establishes the importance of mathematical history in teaching. 2 With the hope of being of some assistance to niy fellow-teachers, I have pre pared this book and have interlined my narrative with occasional remarks and suggestions on methods of teaching. No doubt, the thoughtful reader will draw many useful 1 HJBRBEHT SFBNCEK, Education Intellectual, Moral, and Physical New York, 1894, p. 122. See also B. H. QUICK, Educational Beformers, 1879, p. 191. 2 See G. HEFFEL, The Use of History in Teaching Mathematics, Nature, Vol. 48, 18S08, J 1175497 M 21 946 lessons from f the . study of matliematical history which, are not directly pctiitfed out in the text. In the preparation of this history, I have made extensive use of the works of Cantor, Hankel, linger, De Morgan, Pea cock, Gow, Allman, Loria, and of other prominent writers on the history of mathematics. Original sources have been consulted, whenever opportunity has presented itself. Itgives me much pleasure to acknowledge the assistance ren dered by the United States Bureau of Education, in for warding for examination a number of old text-books which otherwise would have been inaccessible to me. It should also be said that a large number of passages in this book are taken, with only slight alteration, from my History of Mathematics, Macmillan Co., 1895. . . . ELOBIAN CAJORI. COLORADO COLLEGE, COLORADO SPRINGS, July, 1896. PREFACE TO THE SECOND EDITION IN the endeavor to bring this history down to date, nume ous alterations and additions have been made. FLORIAN CAJOBI. COLORADO COLLEGE, December, 1916. CONTENTS PA S ANTIQUITY 1 NUMBER-SYSTEMS AND NUMERALS 1 ARITHMETIC AND ALGEBRA 19 Egypt,19 Greece 26 Some .37 GEOMETRY AND TRIGONOMETRY 43 Egypt and Babylonia 43 Greece . 46 Home 89 MIDDLE AGES .93 ARITHMETIC AND ALGEBRA 93 Hindus 93 Arabs 103 Europe during the Middle Ages Ill Introduction of Roman Arithmetic Ill Translation of Arabic Manuscripts ..... 118 The First Awakening 119 GEOMETRY AND TRIGONOMETRY 122 Hindus 122 Arabs, 12 Europe during the Middle Ages 131 Introduction of Roman Geometry ..... 131 Translation of Arabic Manuscripts 132 The First Awakening 134 ttt Till CONTENTS MODERN TIMES ARITHMETIC ....... ... 139 Its Development as a Science and Art ..... 139 English Weights and Measures ...... 157 Rise of the Commercial School in England . . . .179 Causes which Checked the Growth of Demonstrative Arith metic in England . . ...... 204 Reforms in Arithmetical Teaching ..... 211 Arithmetic in the United States ...... 215 Pleasant and Diverting Questions .......