Chinese and Japanese Mathematical Studies of the 1700s

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Chinese and Japanese Mathematical Studies of the 1700s

Overview

The 1700s were a time of transition for China and Japan, in mathematics as in every other aspect of society and culture. Barriers to Western influence and information, in place for centuries, were beginning to crumble. Scholars and scientists in both countries, eager to learn more of Western science and technology, undertook ambitious programs of translation, aimed at making Western science—and the mathematics that underlay it—available to their countrymen. More importantly, the exposure to Western mathematics served as a spur to Chinese and Japanese mathematics, not in imitation or competition, but rather as a catalyst for rediscovering the roots and resources of their own mathematical traditions; both countries took long looks back at those roots, both extending traditional mathematics and melding those traditions with new skills acquired from the West. The West as well learned from Asian mathematical traditions, which had anticipated many Western discoveries. Mathematical education began to spread throughout both countries, becoming an important aspect of basic literacy.

Background

The Chinese mathematical tradition stretches back to the dawn of recorded history: oracle bones from the fourteenth century b.c. are etched with counting symbols. By the first century a.d. (to use Western notation) the Chinese were employing a decimal system, according to Nine Chapters on the Mathematical Procedures, the classic Chinese text of the time. There is a rich tradition of pictographic symbol-driven (rather than numerical) mathematics in China, with the unique addition of an auditory, or sound-based, mathematical component. Fundamentally algebraic—geometry seems not to have developed far there—Chinese math was at one time a powerful and sophisticated tool for the exploration of patterns, positions, and matrices of numbers. Indeed, Chinese mathematics anticipated some of Blaise Pascal's (1623-1662) findings by more than four centuries. Chinese mathematics also exerted a large influence over the development of Japanese mathematics.

By the 1700s, though, Chinese mathematics had largely abandoned theory to focus on practical matters, used for finance and agricultural and other records. The abacus had come into use in China during the 1600s, and found application in the counting of inventories and products. Also during the 1600s, Western approaches to mathematics, the calendar, and astronomy began to be imported and translated, along with the works of Euclid (c. 330-260 b.c.) and others.

This increasing contact with Western priests (particularly Jesuits) and traders helped re-ignite Chinese interest in pure mathematics, even as practical math was undergoing its own refinements and improvements. Scholars and mathematicians were delving into their country's mathematical past, much of the research and re-discovery prodded by K'ang-hsi (1654-1722), fourth emperor of the Manchu (Ch'ing) dynasty and a ruler devoted to scholarship and learning. At K'ang-hsi's request a vast encyclopedia, the Ku-chin t'u-shu chi-ch'eng (Synthesis of Books and Illustrations Past and Present or The Imperial Encyclopedia) was compiled, although not completed until after K'ang-hsi's death. With more than 10,000 entries, the first of the encyclopedia's main headings was "Calendar, Astronomy, Mathematics." Also at the emperor's behest, and with the assistance of Father Antoine Thomas, the basic Chinese unit of measurement, the li, was recalculated as a function of the meridian, nearly a century before such an approach was undertaken in the West.

Mathematical scholarship flowered during the 1700s, guided by the example of Mei Wenting (1632-1721), who assembled a large comparative study of Chinese and Western mathematics. The great Chinese scholar of historical mathematics during the period was Tai Chen (1723-1777), who applied himself to the study of the Chinese tradition of counting with rods. Tai Chen wrote books on the ability to use counting rods to solve equations with unknown quantities and variables, as well as a 1755 volume on the measurement of circles; he was also responsible for overseeing the republication of classical Chinese mathematical treatises.

The flow of mathematical knowledge traveled in both directions, and many European mathematicians and scholars were fascinated and influenced by Chinese mathematics. Notable among these was German philosopherGottfried Wilhelm Leibniz (1646-1716), who spent much of his final years applying himself to Chinese studies, particularly the ways in which Chinese mathematics either anticipated or independently duplicated Western findings. Much of Leibniz's energy was devoted to studies of early Chinese use of binary notation, although his scholarship was somewhat tainted by his determination to fit his findings into already existing philosophical conclusions. Leibniz also found superiorities in the Chinese symbology of mathematics over the Arabic numerical system. Some scholars feel that Leibniz's studies of Chinese mathematics stimulated the growth of mathematical logic in Europe.

While social disruption and cultural clashes would dominate the next three centuries of Chinese/Western relations, the 1700s were a period of sometimes wary exchange, with mathematics one of the "universal" languages of that exchange. China and the West learned from and were influenced by each other. In one area, astronomy, it should be noted that the Chinese (and the Japanese) did not suffer the social dislocation Europe had faced with the Copernican revolution; Chinese mathematics and astronomy, indeed Chinese philosophy overall, viewed the universe as a whole, with our world and its inhabitants merely a part of that whole. To the Chinese, Copernicus's removal of Earth from the center of the universe was a revelation, not a revolution: in Chinese science, math, and philosophy, humans had never been the absolute center.

The 1700s were a time of transition for Japanese mathematics as well, although the transition may have been more fitful, as Japan's barriers against Western influence were stronger and more rigid than those in China. Still, Jesuits and Dutch and Portuguese traders did establish a presence in Japan, and, slowly, the exchange of mathematical ideas began to flow, much as many Japanese mathematical traditions had flowed from China in the past. In fact, hewing to the prohibition against purely Western texts, many of the imported treatises were first translated into Chinese.

As in China under K'ang-hsi, Japan benefited from the scholarly interests of its leader, or shogun, Tokugawa Yoshimune (1684-1751.) Yoshimune was a great reformer who overruled—without overturning—longstanding edicts against the importation of foreign knowledge, particularly the publication of foreign texts. Yoshimune had a particular fascination with mathematics and its practical applications. Concerned over the accuracy of the Japanese calendar—whose social and religious importance was vast—and aware of new astronomical findings, Yoshimune encouraged the publication of European astronomical texts.

The importation of foreign mathematics resulted in a division in Japanese math studies: traditional Japanese methods of calculation came to be known as wasan, while mathematical tools and techniques imported from the West were referred to as yosan.

Perhaps the most influential, but in some ways least known, Japanese mathematician of the period was Seki Kowa (1642-1708), whose influence lasted throughout the century. A child of a samurai family, Seki Kowa made large contributions to Japanese calculus, as well as creating a theory of determinants that anticipated the work, shortly thereafter, of Leibniz.

Practical mathematics outweighed the theoretical in Japan at the time. Application of math to navigation and cartography took precedence over pure math.

Much of the mathematical focus of the period was on education, with mathematical training being extended throughout Japanese (male) society. The terakoya, or one-room school, common to Japanese villages included instruction in the use of the abacus, as well as reading and writing. Japanese primers, or orai, included volumes devoted to calculation, particularly useful for the tedai, or clerk class. As commerce became more and more central to Japanese daily life, so did mathematics.

Scholars of the time noted the increasing importance of math for all citizens, at least male citizens. Confucian scholar Ito Jinsai (1627-1705) had written at the beginning of the century that "It goes without saying that those of low status should also learn writing and arithmetic." That insight continued to guide the spread of mathematics education throughout eighteenth-century Japan.

Impact

As the eighteenth century closed, both China and Japan stood on the brink of full discourse—sometimes violent—with the West. Barriers would be raised and lowered more than once in the decades to come, but the process of cultural and intellectual exchange had begun, and could not be stopped.

The increasing Westernization of Japanese and Chinese mathematics was a cause of some concern for scholars and philosophers in both countries. There was a tendency among scientists, bedazzled by the scientific and technical achievements of the West, to belittle the equally vast achievements of China and Japan. Gradually, Western approaches to mathematics, particularly the introduction of geometry, began to exert more sway than traditional methods.

In practical matters as well, Western mathematics moved to the forefront as China and, particularly, Japan, began the race to technological and military modernity.

Yet those local traditions possessed great inherent as well as mathematical value, and the challenge for scholars, historians, and mathematicians was to strike a balance between the two, to find, as it were, the best of both mathematical worlds.

KEITH FERRELL