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Murray's China (Vol. III., Chap. iv.) contains a fair account of the attainments of the Chinese in mathematics and astronomy. The notation of the Chinese is based on the decimal principle, but their figures are not changed in value by position, and it is difficult therefore to write out clearly the solution of a question. Arithmetical calculations are performed with the assistance of an abacus, swanpan or counting-board, which is simply a shallow case divided longitudinally by a bar, and crossed by several wires; on one side of this bar the wires bear five balls and two on the other side. The five balls on any wire stand for units, and the two balls are each worth five units. When the balls on any wire are taken for units, those next to the right stand for tens, the third for hundreds, and so on ; while the balls on the left denote tenths, hundredths, &c. Simple calculations are done on this board with great accuracy and rapidity, but the machine is more a convenient index for the progress and result of a calculation performed in the head, than a regular notation of it in detail; consequently, if an error be made, the whole must be performed again, since the result only appears when the sum is finished. There are three sorts of figures, partly answering to the English, Roman, and Arabic forms, as Seven, VII, and 7, the most common of which are given on page 495 of vol. I. ; the complicated form is used in drafts and bills to prevent their alteration, and the abbreviated in common operations, accounts, &c., and in setting down large amounts in a more compact form than can be done by the other characters. This mode of notation is employed by the Japanese and Cochinchinese, and possesses some advantages over the method of using letters practised by the Greeks and Romans, and over the counters once employed in England, but falls far behind the Arabic system now in general use in the west.

There are several treatises on arithmetic, in which the simple rules are clearly explained and illustrated by examples and questions. One of the best is the Swan-fah Tung Tsung, or General Comprehensive Arithmetic, in 5 vols. 8vo. ; the author, Ching Yu-sz', lived in the Ming dynasty. The Tsuimi-shan Fang Sho Hioh, or Mathematics of the Lagerstraemia Hill Institution, in 36 vols. 8vo., contains a complete course of mathematical instruction in geometry, trigonometry, mensuration, &c., together with a table of natural sines and tangents, and one of

MATHEMATICAL KNOWLEDGE OF THE CHINESE. 147

logarithmic sines, tangents, secants, &c., for every degree and minute. Both these compilations derive most of their value from the mathematical writings of the Romish missionaries; it is stated in the latter work that “the western scholar, John Napier, made logarithms.” The knowledge of mathematics even among learned men is very small, and the common people study it only as far as their business requires; the cumbersome notation, and the little aid such studies give in the examinations, doubtless discourage men from pursuing what they seem to have no taste for as a people. A curious fact regarding the existence of six errors in these tables discovered by Babbage to have been perpetuated in most of the European logarithmic tables since the publication of the Trigonometria Artificialis of Vlacq in 1633, proves the source whence the Chinese derived them, and their imitative fidelity in copying them. Mathematical treatises on plane and spherical trigonometry and geometry exist in the language, based upon the works of foreigners, but mixed up with some crude notions of the compilers. Chinese authors readily acknowledge the superiority of western mathematicians, and generally ascribe their advances in the exact sciences to them. The attainments made by the ancient Chinese in astronomy are not easily understood from their records, for the mere notice of an eclipse is a very different thing from its calculation or description. The Book of Records contains some remarkable notices of the orders given by Yau to his astronomers Hi and Ho to ascertain the solstices and equinoxes, and employ intercalary months, and fix the four seasons, in order that the husbandman might know when to commit his seed to the ground. If the time of the deluge be reckoned according to Hales at B.c. 3155, there will be an interval of about eight centuries to the days of Yau, B.c. 2357, which would be ample time for the observation that the primitive sacred year of 360 days in Noah's time was wrong, and that the lunar year of about 354 days was also wrong, and required additional correction, which this ancient monarch is said to have rectified by an intercalation of seven lunar months in nineteen years, like the metonic cycle of the Greeks. It is also remarkable that the time given as the date of the commencement of the astronomical observations sent to Aristotle from Babylon by command of Alexander should be B. c. 2233, or only a few years after the death of Yau; at that time, the five additional days to complete the solar year were intercalated by the Chaldeans, and celebrated with great mirth as days of festivity. Dr. Hales, who mentions this, says that many ancient nations had the same custom, among whom he enumerates the Mexicans, but there are no traces of any particular observance of them by the Chinese, who, indeed, would hardly notice them in a lunar year. The intercalation made by Yau has continued with little variation to this day, the Romish missionaries having rectified the calendar as much as it needed on their arrival in the country, and continued its preparation since that time. The adoption of the Julian solar year of 365+ days at this remote period is not certain, though it is mentioned by the commentator upon the Book of Records, who flourished A.D. 1200. The attention the Chinese paid to the lunar year, and the very small difference their seven intercalations left between the true harmonizing of the lunar and solar years (only 1h. 27m. 32s.), would not derange the calculations to a degree to attract their notice. The chronological cycle of sixty years, called luh-shih hwa kiah-tsz', contrived nearly three centuries before the time of Yau, B.c. 2637, seems to have been perfectly arbitrary, or at least no authentic account exists of the reasons which induced its inventor, Hwangti, or his minister Nau the Great, to select this number. The years of the cycle have each of them a separate name, formed by taking ten characters called shih kan or ten stems, and joining to them twelve other characters, called the shih-'rh chi or twelve branches, five times repeated. These horary characters are also applied to minutes and seconds, hours, days and months, signs of the zodiac, points of the compass, &c. By giving the twelve branches the names of as many animals, and apportioning the ten stems in couplets among the five elements, they are also made to play an important part in divination and astrology. The present year (1847) is the 44th year of the 75th cycle, or the 4484th since its institution, being the longest chronological era known. The year is lunar, but its commencement is regulated by the sun, and newyear falls on the first new moon after the sun enters Aquarius; which makes it come not before the 21st of January nor after the 19th of February. Besides the division into lunar months, the year is apportioned into twenty-four tsieh,

DiVISIONS OF TiMe AND OF THE ZODIAC. 149

periods or terms, of about fifteen days each, depending upon the position of the sun; these are continued on from year to year, irrespective of the intercalations, the first one commencing about the 6th of February, when the sun is 15° in Aquarius. Their names have reference to the season of the year and obvious changes in nature at the time they come round, as rain-water, vernal-equinor, spiked-grain, little-heat, &c. The Chinese divide the zodiac, huang tau or yellow road, into twenty-eight kung, constellations or lunar mansions, but instead of an equable allotment, the signs occupy from 18 up to 31°; quite unlike the Hindus, who have the same number, but arrange them nearly in spaces of 13° each. The first two mansions correspond to Virgo, and the succeeding ones follow in a very irregular manner through the signs and constellations, entirely omitting Leo, and ending with the Raven. The zodiac is further divided into twelve signs or palaces, varying from 25° to 38° in length, named after the twelve branches or the animals representing them, commencing with Aquarius or the rat, followed by the cow, tiger, rabbit, dragon, snake, horse, sheep, monkey, cock, dog, and bear. The name of one of the twentyeight mansions is given to every day in the year in perpetual rotation, and of course the same day of our week in every fourth week has the same character applied to it. The days are numbered numerically from the first to the last day of the month, and the months from one to twelve through the year, except the intercalary month called jun yueh; and there is also a trine division of the month into decades.” Astronomy has been studied by the Chinese chiefly for astrological purposes, and their recorded observations of eclipses, comets, &c., are of small value to European astronomers. Mailla has collected the notices of 460 solar eclipses (no lunar ones are recorded), extending from B. c. 2159 to A. D. 1699, but this proves nothing of the science of the observers; his summary of their observations and calculations shows, that although they came very near the truth in their calculations respecting the length of the lunar and solar year, the obliquity of the ecliptic, and the general motions of the plants, yet “on the whole, their astronomy contains only the most vague notions, and has no pre

• Chinese Repository, Vol. IX., pp. 573–584. De Guignes' Voyages, Vol. II., p. 414. Chinese Chrestomathy.

cepts which could be of any use, or throw the least light on the science. Indeed, it may be confidently said, that its true principles were never established in China, notwithstanding the very long period during which it was cultivated by a succession of learned men, supported by the state. The Chinese appear never to have had any fixed rules for determining eclipses, and all their tables are entirely empirical. The precession of the equinoxes was estimated very differently at different times, varying between from fifty to one hundred and eighty years to a degree.” Even with all the aid they have derived from Europeans, the Chinese seem to be unable to advance in this science when left to themselves, and to cling to their superstitions against every evidence. Some clouds having on one occasion covered the sky, so that an eclipse could not be seen, the courtiers joyfully repaired to the emperor to felicitate him, that heaven, touched by his virtues, had spared him the pain of witnessing the “eating of the sun.” A native writer on astronomy, called Tsinglai, who published several works under the patronage of Yuen Yuen, the liberal minded governor of Kwangtung in 1820, even at that late day, “makes the heavens to consist of ten concentric hollow spheres or envelops: the first contains the moon’s orbit; the second that of Mercury; those of Venus, the Sun, Mars, Jupiter, Saturn, and the twenty-eight constellations, follow ; the ninth envelops and binds together the eight interior ones, and revolves daily; while the tenth is the abode of the celestial Sovereign, the great Ruler, with all the gods and sages, where they enjoy eternal tranquillity.” He further says, “there are two north and two south poles, those of the equator and those of the ecliptic. The poles of the ecliptic regulate the varied machinery of the heavenly revolutions, and turn round unceasingly. The poles of the equator are the pivots of the primitive celestial body, and remain permanently unmoved. What are called the two poles, therefore, are really not stars, but two immovable points in the north and in the south.”f The author of this astute cosmogony studied under Europeans, and published these remarks as the fruit of his researches. The names given to the five principal planets, Mercury, Venus, Mars, Jupiter, and Saturn, are water, metal, fire, wood,

* Murray's China, Vol. III., pages 225–265. f Chinese Chrestomathy, page 397.

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