;-NRLF I 1 UNIVERSITY OF CALIFORNIA PEFARTMENT OF CIVIL ENGINEERING BERKELEY, CALIFORNIA Engineering Library A HISTORY OF ELEMENTARY MATHEMATICS THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO DALLAS ATLANTA SAN FRANCISCO MACMILLAN & CO., LIMITED LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN CO. OF CANADA, LTD. TORONTO A HISTORY OF ELEMENTARY MATHEMATICS WITH HINTS ON METHODS OF TEACHING BY FLORIAN CAJORI, PH.D. PROFESSOR OF MATHEMATICS IN COLORADO COLLEGE REVISED AND ENLARGED EDITION THE MACMILLAN COMPANY LONDON : MACMILLAN & CO., LTD. 1917 All rights reserved Engineering Library COPYRIGHT, 1896 AND 1917, BY THE MACMILLAN COMPANY. Set up and electrotyped September, 1896. Reprinted August, 1897; March, 1905; October, 1907; August, 1910; February, 1914. Revised and enlarged edition, February, 1917. o ^ 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 in other the of historically ; or, words, genesis 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 1 the genesis of knowledge, either in its causes or its order." If this principle, held also by Pestalozzi and Froebel, 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 2 of mathematical history in teaching. With the hope of being of some assistance to my 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 HERBERT SPENCER, Education : Intellectual, Moral, and Physical New York, 1894, p. 122. See also R. H. QUICK, Educational Reformers, 1879, p. 191. 2 See G. HEPPEL, "The Use of History in Teaching Mathematics," Nature, Vol. 48, 1893, pp. 16-18. v OOOOOO Vi PREFACE lessons from the study of mathematical history which are not directly pointed 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. It gives 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. FLORIAN CAJORI. COLORADO COLLEGE, COLORADO SPRINGS, July, 1896. PREFACE TO THE SECOND EDITION IN the endeavor to bring this history down to date, numer- ous alterations and additions have been made. FLORIAN CAJORI. COLORADO COLLEGE, December, 1916. CONTENTS PAQ ANTIQUITY 1 NUMBER-SYSTEMS AND NUMERALS 1 ARITHMETIC AND ALGEBRA 19 ' Egypt . 19 Greece 26 Home 37 GEOMETRY AND TRIGONOMETRY 43 Egypt and Babylonia . 43 Greece 46 Borne 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 . 125 Europe during the Middle Ages . 131 Introduction of Roman Geometry ..... 131 Translation of Arabic Manuscripts . 132 The First Awakening 134 vii Vlll CONTENTS PAGE MODERN TIMES 139 ARITHMETIC . * . 139 Its Development as a Science and Art ..... 139 English Weights and Measures ...... 167 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" 219 ALGEBRA 224 The Renaissance 224 The Last Three Centuries 234 GEOMETRY AND TRIGONOMETRY 245 Editions of Euclid. Early Researches ..... 245 The Beginning of Modern Synthetic Geometry . 252 Modern Elementary Geometry 256 Modern Synthetic Geometry 257 Modern Geometry of the Triangle and Circle . 259 Non-Euclidean Geometry 266 Text-books on Elementary Geometry .... 275 RECENT MOVEMENTS IN TEACHING 290 Tfie Perry Movement 291 International Commission 297 American Associations ........ 301 Attacks upon the Study of Mathematics as a Training of the Mind . 304 A HISTORY OF MATHEMATICS ANTIQUITY NUMBER-SYSTEMS AND NUMERALS NEARLY all number-systems, both ancient and modern, are based on the scale of 5, 10, or 20. The reason for this it is not difficult to see. When a child learns to count, he makes use of his fingers and perhaps of his toes. In the same way the savages of prehistoric times unquestionably counted on their fingers and in some cases also on their toes. Such is indeed the practice of the African, the Eskimo, and the South 1 Sea Islander of to-day. This recourse to the fingers has often resulted in the development of a more or less extended pantomime number-system, in which the fingers were used as 1 in. a deaf and dumb alphabet. Evidence of the prevalence of finger symbolisms is found among the ancient Egyptians, Babylonians, Greeks, and Romans, as also among the Euro- peans of the middle ages : even now nearly all Eastern nations use finger symbolisms. The Chinese express on the left hand 1 L. L. CONANT, "Primitive Number-Systems," in Smithsonian lie- port, 1892, p. 584. *, . * 2 '*^' A fr!ST?ORY-OF MATHEMATICS " less the thumb nail of the all numbers than 100,000 ; right hand touches each joint of the little finger, passing first up the external side, then down the middle, and afterwards up of in order to the nine the the other side it, express digits ; tens are denoted in the same way, on the second finger; the third the thousands on the fourth and ten- hundreds on the ; ; thousands on the thumb. It would be merely necessary to proceed to the right hand in order to be able to extend this l system of numeration." So common is the use of this finger- symbolism that traders are said to communicate to one another the price at which they are willing to buy or sell by touching hands, the act being concealed by their cloaks from observa- tion of by-standers. Had the number of fingers and toes been different in man, then the prevalent number-systems of the world would have been different also. We are safe in saying that had one more finger sprouted from each human hand, making twelve fingers in all, then the numerical scale adopted by civilized nations would not be the decimal, but the duodecimal. Two more symbols would be necessary to represent 10 and 11, respec- tively. As far as arithmetic is concerned, it is certainly to be regretted that a sixth finger did not appear. Except for the necessity of using two more signs or numerals and of being obliged to learn the multiplication table as far as 12 x 12, the duodecimal system is decidedly superior to the decimal. The number twelve has for its exact divisors 2, 3, 4, 6, while ten has 2 and 5. In business the fractions only ordinary affairs, -J-, , J, are used extensively, and it is very convenient to have a base which is an exact multiple of 2, 3, and 4. Among the most zealous advocates of the duodecimal scale was Charles XII. 1 GEORGE PEACOCK, article "Arithmetic," in Encyclopaedia Metropoli- tana (The Encyclopedia of Pure Mathematics}, p. 394. Hereafter we shall cite this very valuable article as PEACOCK. NUMBER-SYSTEMS AND NUMERALS 3 of Sweden, who, at the time of his death, was contemplating the change for his dominions from the decimal to the duo- 1 decimal. But it is not likely that the change will ever be brought about. So deeply rooted is the decimal system that when the storm of the French Devolution swept out of exist- ence other old institutions, the decimal system not only remained unshaken, but was more firmly established than ever. The advantages of twelve as a base were not recognized until arithmetic was so far developed as to make a change impossible. "The case is the not uncommon one of high civilization bearing evident traces of the rudeness of its origin in ancient barbaric life." 2 Of the notations based on human anatomy, the quinary and vigesimal systems are frequent among the lower races, while the higher nations have usually avoided the one as too scanty and the other as too cumbrous, preferring the intermediate 3 decimal system. Peoples have not always consistently adhered to any one scale. In the quinary system, 5, 25, 125, 625, etc., should be the values of the successive higher units, but a quinary system thus carried out was never in actual use : whenever it was extended to higher numbers it invariably ran " either into the decimal or into the vigesimal system. The home par excellence of the quinary, or rather of the quinary- vigesimal scale, is America. It is practically universal among the Eskimo tribes of the Arctic regions. It prevailed among a considerable portion of the North American Indian tribes, and was almost universal with the native races of Central and 1 CONANT, op. cit., p. 589. 2 E. B. TYLOR, Primitive Culture, New York, 1889, Vol. I., p. 272. In some respects a scale having for its base a power of 2 the base 8 or 16, for instance, is superior to the duodecimal, but it has the disadvantage " of not being divisible by 3.