A Short Treatise

數者 乃兩間之一巨物也

夫天自可為天 地自可為地也 萬物自可為萬物 不可使之相同 而猶不可使之相限

數則不然 以之參天 天不失其度 以之兩地 地因之而制 萬物限於大小者則量之 限於長短者則度之 限於輕重者則權之 限於聲音者 以律呂升降之 限於世代者 以歲月紀載之 無往而不有數也 此其有形之數也 至於陰陽五行氣化遷流 屈伸交錯 此其無形之數也 其數也 所統者大 所該者切 不亦於兩間為一巨物乎

聖人因物格理辨形 因形設數 不期而承河圖洛書之用矣 所以黃帝命隸首作九章 別理所在施之於法 而周公與商高尚論周髀 展從橫高下以廣其用 而古經濟聖帝明王之所必需 今其學也

日趨於亡 學士大夫不精心以究其理 庶人即究心於理 而見聞不廣 因而斯道幾至於息

幸逢我聖祖仁皇帝道學本宗深通物理 萬幾之暇 留心於數 如此四十餘年 緒前聖將絕之學 闡數理未宣之秘 橫幾何原本申明九章 句股 少廣 方田之義 探算法原本播揚差分粟布均輸之該以借根方 較方程 盈朒 衰分之微 辨假數乃約差分之層級以假形真之用 於是數學自命日復明於天下後世矣 因而於大內開館 召天下知數之士 增廣其傳註書百卷極盡精微

我皇上善繼善述將其書頒布中外 使天下後世士子身通六藝以弘其道 但初學者望之浩大 難窺其奧 今余謹將有形可見 有繩可循 及有通率之數者 每錄一二條特為初學之一階云爾

雍正十三年歲在乙卯五月

廣寧年希堯序

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Number may be reckoned among the greatest entities that subsist between Heaven and Earth.

Heaven remains Heaven, Earth remains Earth, and the myriad things remain the myriad things. They cannot be made identical with one another, nor may they be mutually confined.

Numbers are otherwise. By means of numbers Heaven may be examined, and Heaven does not lose its proper measure. By means of numbers Earth may be surveyed, and Earth may thereby be duly regulated. That which is limited by magnitude may be measured; that which is limited by length may be determined; that which is limited by weight may be weighed. That which is limited by sound may be adjusted according to the rise and fall of the musical pitches. That which is limited by generations may be recorded by the reckoning of years and months. There is nowhere that number does not extend. These are the numbers that belong to things possessed of form. As for the circulation and transformations of the vital influences of Yin and Yang and of the Five Elements, their contractions and expansions, and their interweaving operations — these constitute the numbers of that which has no form. Numbers comprehend and regulate all things; they embrace what they govern and extend everywhere. Are they not therefore among the greatest entities between Heaven and Earth?

The sages, observing the principles inherent in things, distinguished their forms and accordingly established numbers. Thus they inherited the uses of the Ho-t’u and the Lo-shu. Wherefore the Yellow Emperor commanded Li-shou to compose the Chiu-chang, wherein the principles therein contained were distinguished and reduced to method. Afterwards Duke Chou and Shang Kao discoursed upon the Chou-pi, extending its applications from the horizontal and the vertical to the high and the low. Hence in antiquity the canonical learning required by sage emperors and enlightened kings was dependent upon these studies.

In later ages, however, the art gradually declined. Scholars and great officials no longer applied themselves diligently to investigate its principles. The common people, though sometimes inclined to reason upon such matters, possessed but narrow learning and limited knowledge; thus this Way came almost to cessation.

Fortunately our Sacred Ancestor the Emperor Jen, the Emperor K’ang-hsi, penetrated profoundly the fundamental principles of learning and comprehended the laws of things. Amid the intervals of his manifold affairs he devoted his attention to numbers for more than forty years. Before his departure he transmitted the ultimate learning and revealed the hidden principles of number which had not yet been made manifest. The original meanings of the Chiu-chang, the Kou-ku rule, the Shao-kuang and Fang-t’ien methods; the principles of calculation, the propagation of differences, and the equal distribution of transport in grain and cloth; the comparison of equations, the increase and decrease of the binomial, and the refinements of residual fractions — by establishing provisional numbers and arranging the gradations of differences, the use of hypothetical forms whereby the real might be approached was rendered clear. Thus the learning of mathematics once more shone forth for the world of later ages. Accordingly a hall was established within the Inner Court, and those throughout the empire who possessed knowledge of numbers were summoned. The transmission and annotation of these writings were enlarged into many hundreds of volumes, in which the utmost subtlety was exhausted.

His Imperial Majesty graciously continued and perfected these teachings, and caused the books to be promulgated both within and beyond the court, so that scholars of later generations throughout the empire might become versed in the Six Arts and thereby extend this Way. Yet beginners, beholding the vastness of this learning, find it difficult to penetrate its essentials. Therefore I have respectfully selected those matters which possess visible form, those which may be followed by definite rules, and those numbers that may readily be communicated. Of these I have recorded one or two items, as a first step for those who would enter upon the study.

Written in the fifth month of the year I-mao, being the thirteenth year of the Yung-cheng reign.

Preface by Nien Hsi-yao of Kuang-ning.