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A History of Japanese Mathematics

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A History of Japanese Mathematics A HISTORY OF JAPANESE MATHEMATICS BY DAVID EUGENE SMITH AND YOSHIO MIKAMI CHICAGO THE OPEN COURT PUBLISHING COMPANY 1914 , Printed by W. Drugulin, Leipzig PREFACE Although for nearly a century the greatest mathematical classics of India have been known to western scholars, and several of the more important works of the Arabs for even longer, the mathematics of China and Japan has been closed to all European and American students until very recently. Even now we have not a single translation of a Chinese treatise upon the subject, and it is only within the last dozen years that the contributions of the native Japanese school have become known in the West even by name. At the second International Congress of Mathematicians, held at Paris in 1900, Professor Fujisawa of the Imperial University of Tokio gave a brief address upon Mathematics of the old Japanese School, and this may be taken as the first contribution to the history of mathematics made by a native of that country in a European language. The next effort of this kind showed itself in occasional articles by Baron Kikuchi, as in the Niemv Archief voor Wiskunde, some of which were based upon his contributions in Japanese to one of the scientific journals of Tokio. But -the only serious attempt made up to the present time to present a well-ordered history of the subject in a European language is to be found in the very commendable papers by T. Hayashi, of the Imperial University at Sendai. The most important of these have appeared in the Nieuw Archief voor Wiskunde, and to them the authors are much indebted. Having made an extensive collection of mathematical manu- scripts, early printed works, and early instruments, and having IV Preface. brought together most of the European literature upon the subject and embodied it in a series of lectures for my classes in the history of mathematics, I welcomed the suggestion of Dr. Cams that I join with Mr. Mikami in the preparation of the present work. Mr. Mikami has already made for himself an enviable reputation as an authority upon the wasan or native Japanese mathematics, and his contributions to the BibliotJieca Mathcmatica have attracted the attention of western scholars. He has also published, as a volume of the Abhand- lungen zur GeschicJite der Mathematik, a work entitled Mathe- matical Papers from the Far East. Moreover his labors with the learned T. Endo, the greatest of the historians of Japanese mathematics, and his consequent familiarity with the classics of his country, eminently fit him for a work of this nature. Our labors nave been divided in the manner that the cir- cumstances would suggest. For the European literature, the general planning of the work, and the final writing of the text, the responsibility has naturally fallen to a considerable extent upon me. For the furnishing of the Japanese material, the initial translations, the scholarly search through the excellent library of the Academy of Sciences of Tokio, where Mr. Endo is librarian, and the further examination of the large amount of native secondary material, the responsibility has been Mr. Mi- kami's. To his scholarship and indefatigable labors I am in- debted for more material than could be used in this work, and whatever praise our efforts may merit should be awarded in large measure to him. The aim in writing this work has been to give a brief survey of the leading features in the development of the wasan. It has not seemed best to enter very fully into the details of demonstration or into the methods of solution employed by the great writers whose works are described. This would not be done in a general history of European mathematics, and there is no reason why it should be done here, save in cases where some peculiar feature is under discussion. Undoubtedly several names of importance have been omitted, and at least a score of names that might properly have had mention have Preface. V been the subject of correspondence between the authors for the past year. But on the whole it may be said that most of those writers in whose works European scholars are likely to have much interest have been mentioned. It is the hope of the authors that this brief history may serve to show to the West the nature of the mathematics that was indigenous to Japan, and to strengthen the bonds that unite the scholars of the world through an increase in knowledge of and respect for the scientific attainments of a people whose progress in the past four centuries has been one of the marvels of history. It is only just to mention at this time the generous assistance rendered by Mr. Leslie Leland Locke, one of my graduate students 'in the history of mathematics, who made in my library the photographs for all of the illustrations used in this work. His intelligent and painstaking efforts to carry out the wishes of the authors have resulted in a series of illustrations that not merely elucidate the text, but give a visual idea of the genius of the Japanese mathematics that words alone cannot give. To him I take pleasure in ascribing the credit for this arduous labor, and in expressing the thanks of the authors. Teachers College, Columbia University, David Eugene Smith. New York City, December I, 1913. VOCABULARY FOR REFERENCE The following brief vocabulary will be convenient for reference in con- sidering some of the Japanese titles: ho, method or theory. Synonym Q{ jutsu. It is found in expressions like shosa ho (method of differences). hyo, table. jutsu, method or theory. Synonym of ho, It is found in words like kaku jutsu (polygonal theory) and tatsnjittsu (method of expanding a root of a literal equation). ki, a treatise. roku, a treatise. Synonym of ki. sampo, mathematical treatise, or mathematical lules. sangi, rods used in computing, and as numerical coefficients in equations soroban, the Japanese abacus. tengen, celestial element. The Japanese name for the Chinese algebra. tenzcin, the algebra of the Seki school. ivasati, the native Japanese mathematics as distinguished from the yosan, the European mathematics. yenri, circle principle. A term applied to the native calculus of Japan. In Japanese proper names the surname is placed first in accordance with the native custom, excepting in the cases of persons now living and who follow the European custom of placing the surname last. CONTENTS CHAPTER PAGE I. The Earliest Period I II. The Second Period 7 III. The Development of the Soroban 18 IV. The Sangi applied to Algebra . 47 V. The Third Period 59 VI. Seki Kowa 91 VII. Seki's Contemporaries and possible Western Influences .... 128 VIII. The Yenri or Circle Principle 143 IX. The Eighteenth Century 163 X. Ajirna Chokuyen 195 ' XI. The Opening of the Nineteenth Century 206 XII. Wada Nei 220 XIII. The Close of the Old Wasan 230 XIV. The Introduction of Occidental Mathematics 252 Index . .... 281 CHAPTER I. The Earliest Period. The history of Japanese mathematics, from the most remote times to the present, may be divided into six fairly distinct periods. Of these the first extended from the earliest ages to T a that was influenced 552 , period only indirectly if at all by Chinese mathematics. The second period of approximately a thousand years (552 1600) was characterized by the influx of Chinese learning, first through Korea and then direct from China itself, by some resulting native development, and by a season of stagnation comparable to the Dark Ages of Europe. The third period was less than a century in duration, extend- ing from about 1600 to the beginning of Seki's influence (about 1675). This may be called the Renaissance period of Japanese mathematics, since it saw a new and vigorous importation of Chinese science, the revival of native interest through the efforts of the immediate predecessors of Seki, and some slight intro- duction of European learning through the early Dutch traders and through the Jesuits. The fourth period, also about a century in length (1675 to 1775) may be compared to the synchro- nous period in Europe. Just as the initiative of Descartes, Newton, and Leibnitz prepared the way for the labors of the Bernoullis, Euler, Laplace, D'Alembert, and their contemporaries of the eighteenth century, so the work of the great Japanese teacher, Seki, and of his pupil Takebe, made possible a note- * worthy development of the wasan of Japan during the same 1 All dates are expressed according to the Christian calendar and are to be taken as after Christ unless the contrary is stated. 2 The native mathematics, from Wa (Japan) and son (mathematics). The word is modern, having been employed to distinguish the native theory from the western mathematics, the yosan. I 2 I. The Earliest Period. century. The fifth period, which might indeed be joined with the fourth, but which differs from it much as the nineteenth century of European mathematics differs from the eighteenth, extended from 1775 to 1868, the date of the opening of Japan to the Western World. This is the period of the culmination of native Japanese mathematics, as influenced more or less by the European learning that managed to find some entrance through the Dutch trading station at Nagasaki and through the first Christian missionaries. The sixth and final period begins with the opening of Japan to intercourse with other countries and extends to the present time, a period of marvelous change in government, in ideals, in art, in industry, in edu- cation, in mathematics and the sciences generally, and in all that makes a nation great.
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