David Eugene Smith

“ Thus passes and closes a period of a thousand years, with not a single book of any merit, and without advancing the science of mathematics a single pace. Europe was backward enough, but Japan was worse. China was doing a little, India was doing more, but the Arab was accomplishing still more through his restlessness of spirit if not through his mathematical genius. The world's rebirth was approaching, and this Renaissance came to Japan at about the time that it came to Europe, accompanied in both cases by a grafting of foreign learning upon native stock. ”

“ This list includes all of the important Chinese classics in mathematics that had appeared before it was made, and it shows a serious attempt to introduce the best material available into the schools of Japan at the opening of the 8th century. It seemed that the country had entered upon an era of great intellectual prosperity, but it was like the period of Charlemagne, so nearly synchronous with it—a temporary beacon in a dark night. ”

“ The second period in the history of Japanese mathematics (552—1600) corresponds both in time and in nature with the Dark Ages of Europe. Just as the Northern European lands came in contact with the South, and imbibed some slight draught of classical learning, and then lapsed into a state of indifference except for the influence of an occasional great soul like that of Charlemagne or of certain noble minds in the Church, so Japan, subject to the same Zeitgeist, drank lightly at the Chinese fountain and then lapsed again into semi-barbarism. ”

“ When, therefore, we find the myriad as the base of the Greek system, [12] and find it more or less in use in India, [13] and know that it still persists in China, [14] and see systematically used in the ancient Japanese system as well as in the modern number names, we are convinced that there must have been a considerable intercourse of scholars at an early date. [15] Of the rest of Japanese mathematics in this early period we are wholly ignorant, save that we know a little of the ancient system of measures and that a calendar existed. ”

“ 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. ”

“ A crude theology, a purposeless logic, a feeble literature,—these had some standing; but mathematicians save for calendar purposes was ever an outcast in the temple and the church, save as it occasionally found some eccentric individual to befriend it. In the period of the Ashikaga shoguns it is asserted that there hardly could be found in all Japan a man who was versed in the art of division. ”

“ Moreover his labors with the learned T. Endō, 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 have been divided in the manner that the circumstances 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. ”

“ The general expression of wonder that comes from the study of number, of forms, and of the arrangements of words and objects, is indicated in the language and the traditions of Japan as in the language and traditions of all other peoples. ”

“ Knowledge of numbers was indispensable in the mastering of forces as well as in the distribution of benefices and fiefs, but the counting of money was left to meaner hands.” Only in the Buddhist temples in Japan, as in the Christian church schools in Europe, was the lamp of learning kept burning. [34] In each case, however, mathematics was not a subject that appealed to the religious body. ”

“ VOCABULARY FOR REFERENCE The following brief vocabulary will be convenient for reference in considering some of the Japanese titles:
hō, method or theory. Synonym of jutsu. It is found in, expressions like shōta hō (method of differences) . ”

“ This, however, may be a meaning that came with the influx of Chinese learning, and we are not at all certain that in ancient Japanese it stood for the Greek myriad [7] . The use of yorozu for 10000 was adopted in later times when the number names came to be based upon Chinese roots, and it may possibly have preceded the entry of Chinses learning in historic times. Thus 10 [8] was not “hundred thousand” [9] in this later period, but “ten myriads”, [10] and our million [11] is a hundred myriads. ”

“ If these problems were in the original text, and that text has the antiquity usually assigned to it, concerning neither of which we are at all certain, then they contain the oldest known quadratic equation. The interrelation of ancient mathematics is seen in two problems in this chapter. One is that of the reed growing 1 foot above the surface in the center of a pond 10 feet square, which just reaches the surface when drawn to the edge of the pond, it being required to find the depth of the water. ”

“ Now this system of numeration by myriads is one of the frequently observed evidences of early intercourse between the scholars of the East and the West. Trades people and the populace at large did not need such large numbers, but to the scholar they were significant. ”

“ Another interesting form of counting is still in use in Japan, and is more closely connected with the ancient one that is the common form above given. It is as follows: (1) hitotsu, (2) futatsu, (3) mittsu, (4) yottsu, (5) itsutsu, (6) muttsu, (7) nanatsu, (8) yattsu, (9) kokonotsu, (10) tō. Still another form at present in use, and also related to the ancient one, is as follows: (1) hi, (2) fu, (3) mi, (4) yō, (5) itsu, (6) mū, (7) nana, (8) ya, (9) kono, (10) tō. ”

“ Chow is thought to be the name of a certain scholar who died in 1105 B. C., but it may have been simply the name of the dynasty. This scholar is sometimes spoken of as Chow Kung, [15] and is said to have had a discussion with a nobleman named Kaou, or Shang Kao, [16] which is set forth in this book in the form of a dialogue. ”

“ Mikami's. To his scholarship and indefatigable labors I am indebted 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 topic is our so-called Pythagorean theorem, and the time is over five hundred years before Pythagoras gave what was probably the first scientific proof of the proposition. The work relates to geometric measures and to astronomy. [17]
2. Sun-tsu Suan-ching. This treatise consists of three books, and is commonly known in China as the Swan-king (Arithmetical classic) of Sun-tsu (Sun-tsze, or Swen-tse) , a writer who lived probably in the 3d century A. D., but possibly much earlier. ”

“ It should be added that the modern forms given above are from Chinese roots. ↑ Μυρίοι, 10 000. ↑ 8.0 8.1 Mono yorozu, or, in modern Japanese, hyaku man. ↑ Which would, if so considered, appear as momo chi, or in modern Japanese as hyaku sen. ↑ So yorozu, a softened form of tō yorozu. In modern Japanese, jiu man, man being the myriad. ↑ Mille + on, “big thousand”, just as saloon is salle + on, a big hall, and gallon is gill + on, a big gill. ↑ See, for example, Gow, J., History of Greek Mathematics, Cambridge 1884, and similar works. ”

“ Some writers give the date much earlier. ↑ Or Li-shou. ↑ Wylie, A., Jottings of the Science of Chinese Arithmetic, North China Herald for 1852, Shanghai Almanac for 1853, Chinese Researches, Shanghai 1897, Part III, page 159; Biernatzki, Die Arithmetik der Chinesen, Crelle's Journal for 1856, vol. 52. ↑ For this information we are indebted to the testimony of Liu Hui, whose commentary was written in 263. ↑ Also known as the Western Han. ↑ Legge, J., The Chinese Classics. Oxford 1893, 2nd edition, vol. I, p. 4. ↑ The Regula falsi or Regula positionis of the Middle Ages in Europe. ”