(fl. China, 1280–1303),
Chu Shih-chieh (literary name, Han-ch’ing; appellation, Sung-t’ing) lived in Yen-shan (near modern Peking). George Sarton describes him, along with Ch’in Chiu-shao, as “one of the greatest mathematicians of his race, of his time, and indeed of all times.” However, except for the preface of his mathematical work, the Ssu-yüan yü-chien (“Precious Mirror of the Four Elements”), there is no record of his personal life. The preface says that for over twenty years he traveled extensively in China as a renowned mathematician; thereafter he also visited Kuang-ling, where pupils flocked to study under him. We can deduce from this that Chu Shih-chieh flourished as a mathematician and teacher of mathematics during the last two decades of the thirteenth century, a situation possible only after the reunification of China through the Mongol conquest of the Sung dynasty in 1279.
Chu Shih-chieh wrote the Suan-hsüeh ch’i-meng (“Introduction to Mathematical Studies”) in 1299 and the Ssu-yüan yü-chien in 1303. The former was meant essentially as a textbook for beginners, and the latter contained the so-called “method of the four elements” invented by Chu. In the Ssu-yüan yü-chien, Chinese algebra reached its peak of development, but this work also marked the end of the golden age of Chinese mathematics, which began with the works of Liu I, Chia Hsien, and others in the eleventh and the twelfth centuries, and continued in the following century with the writings of Ch’in Chiu-shao, Li Chih, Yang Hui, and Chu Shih-chieh himself.
It appears that the Suan-hsüeh ch’i-meng was lost for some time in China. However, it and the works of Yang Hui were adopted as textbooks in Korea during the fifteenth century. An edition now preserved in Tokyo is believed to have been printed in 1433 in Korea, during the reign of King Sejo. In Japan a punctuated edition of the book (Chinese texts were then not punctuated) under the title Sangaku keimo kunten, appeared in 1658; and an edition annotated by Sanenori Hoshino, entitled Sangaku keimo chūkai, was printed in 1672. In 1690 there was an extensive commentary by Katahiro Takebe, entitled Sangaku keimō genkai, that ran to seven volumes. Several abridged versions of Takebe’s commentary also appeared. The Suan-hsüeh ch’i-meng reappeared in China in the nineteenth century, when Lo Shih-lin discovered a 1660 Korean edition of the text in Peking. The book was reprinted in 1839 at Yangchow with a preface by Juan Yuan and a colophon by Lo Shih-lin. Other editions appeared in 1882 and in 1895. It was also included in the ts’e-hai-shan-fang chung-hsisuan-hsüeh ts’ung-shu collection. Wang Chien wrote a commentary entitled Suan-hsüeh ch’i-meng shu i in 1884 abd Hsu Feng-k’ao produced another, Suan-hsüeh ch’i-meng t’ung-shih, in 1887.
The Ssu-yüan yü-chien also disappeared from China for some time, probably during the later part of the eighteenth century. It was last quoted by Mei Kuch’eng in 1761, but it did not appear in the vast imperial library collection, the Ssu-k’u ch’üan shu, of 1772; and it was not found by Juan Yuan when he compiled the Ch’ou-jen chuan in 1799. In the early part of the nineteenth century, however, Juan Yuan found a copy of the text in Chekiang province and was instrumental in having the book made part of the Ssu-k’u ch’üan-shu. He sent a handwritten copy to Li Jui for editing, but Li Jui died before the task was completed. This handwritten copy was subsequently printed by Ho Yüan-shih. The rediscovery of the Ssu-yüan yü-chien attracted the attention of many Chinese mathematicians besides Li Jui, Hsü Yu-jen, Lo Shih-lin, and Tai Hsü. A preface to the Ssu-yüan yü-chien was written by Shen Ch’in-p’ei in 1829. In his work entitled Ssu yüan yü-chien hsi ts’ao (1834), Lo Shih-lin included the methods of solving the problems after making many changes. Shen Ch’in-p’ei also wrote a so-called hsi ts’ao (“detailed workings”) for this text, but hsi work has not been printed and is not as well known as that by Lo Shih-lin. Ting Ch’ü-chung included Lo’s Ssu-yüan yü-chien hsi ts’ao in his Pai-fu-t’ang suan hsüeh ts’ung shu (1876). According to Tu Shih-jan, Li Yen had a complete handwritten copy of Shen’s version, which in many respects is far superior to Lo’s.
Following the publication of Lo Shih-lin’s Ssu-yüan yü-chien hsi-ts’ao, the “method of the four elements” began to receive much attention from Chinese mathematicians. I Chih-han wrote the K’ai-fang shih-li (“Illustrations of the Method of Root Extraction”), which has since been appended to Lo’s work. Li Shan-lan wrote the Ssu-yüan chieh (“Explanation of the Four Elements”) ans included it in his anthology of mathematical texts, the Tse-ku-shih-chai suan-hsüeh, first published in Peking in 1867. Wu Chia-shan wrote the Ssu-yüan ming-shih shih-li (“Examples Illustrating the Terms and Forms in the Four Elements Method”), the Ssu-yüan ts’ao (“Workings in the Four Elements Method”), and the Ssu-yüan ch’ien-shih (“Simplified Explanations of the Four Elements Method”), and incorporated them in his Pai-fu-t’ang suan-hsüeh ch’u chi (1862). In his Hsüeh-suan pi-t’an (“Jottings in the Study of Mathematics”), Hua Heng-fang also discussed the “method of the four elements” in great detail.
A French translation of the Ssu-yüan yü-chien was made by L. van Hée. Both George Sarton and Joseph Needham refer to an English translation of the text by Ch’en Tsai-hsin. Tu Shih-jan reported in 1966 that the manuscript of this work was still in the Institute of the History of the Natural Sciences, Academia Sinica, Peking.
In the Ssu-yüan yü-chien the “method of the celestial element” (t’ien-yuan shu) was extended for the first time to express four unknown quantities in the same algebraic equation. Thus used, the method became known as the “method of the four elements” (su-yüan shu)—these four elements were t’ien (heaven), ti (earth), jen (man), and wu (things or matter). An epilogue written by Tsu I says that the “method of the celestial element” was first mentioned in Chiang Chou’s I-ku-chi, Li Wen-i’s Chao-tan, Shih Hsin-tao’s Ch’ien-ching, and Liu Yu-chieh’s Ju-chi shih-so, and that a detailed explanation of the solutions was given by Yuan Hao-wen. Tsu I goes on to say that the “earth element” was first used by Li Te-tsai in his Liang-i ch’un-ying chi-chen while the “man element” was introduced by Liu Ta-chien (literary name, Liu Junfu), the author of the Ch’ien-k’un kua-nang; it was his friend Chu Shih-chieh, however, who invented the “method of the four elements.” “Except for Chu Shih-chieh and Yüan Hao-wen, a close friend of Li Chih, wer know nothing else about Tsu I and all the mathematicians he lists. None of the books he mentions has survived. It is also significant that none of the three great Chinese mathematicians of of the thirteenth century—Ch’in Chiu-shao, Li Chih, and Yang Hui—is mentioned in Chu Shih-chieh’s works. It is thought that the “method of the celestial element” was known in China before their time and that Li Chih’s I-ku yen-tuan was a later but expanded version of Chiang Chou’s I-ku-chi.
Tsu I also explains the “method of the four elements,” as does Mo Jo in his preface to the Ssu-yüan yü-chien. Each of the “four elements” represents an unkown quantity—u, v, w, and x, respectively. Heaven (u) is placed below the constant, which is denoted by t’ai, so that the power of u increases as it moves downward; earth (v) is placed to the left of the constant so that the power of v increases as it moves toward the left; man (w) is placed to the right of the constant so that the power of w increases as it moves toward the right; and matter (x) is placed above the constant so that the power of x increases as it moves upward. For example, u + v + w + x = 0 is represented in Fig. 1.
Chu Shih-chieh could also represent the products of any two of these unknowns by using the space (on the countingboard) between them rather as it is used in Cartesian geometry. For example, the square of
(u + v + w + x) = 0,
u2 + v2 + w2 + x2 + 2ux + 2vw + 2ux + 2wx = 0,
can be represented as shown in Fig. 2 (below). Obviously, this was as far as Chu Shih-chieh could go, for he was limited by the two-dimensional space of the countingboard. The method cannot be used to represent more than four unknowns or the cross product of more than two unknowns.
Numerical equations of higher degree, even up to the power fourteen, are dealt with in the Suan-hsüeh ch’i-meng as well as the Ssu-yüan yü-chien. Sometimes a transformation method (fan fa) is employed. Although there is no description of this transformation method, Chu Shih-chieh could arrive at the transformation only after having used a method similar to that independently rediscovered in the early nineteenth century by Horner and Ruffini for the solution of cubic equations. Using his method of fan fa, Chu Shih-chieh changed the quartic equation.
x4 – 1496x2 – x + 558236 = 0
to the form
y4 – 80y3 + 904y2 – 27841y – 119816 = 0.
Employing Horner’s method in finding the first approximate figure, 20, for the root, one can derive the coefficients of the second equation as follows:
Eigher Chu Shih-chieh was not very particular about the signs for the coefficients shown in the above example, or there are printer’s errors. This can be seen in another example, where the equation x2 – 17x – 3120 = 0 became y2 + 103y + 540 = 0 by the fan fa method. In other cases, however, all the signs in the second equations are correct. For example,
109x2 – 2288x – 348432 = 0
gives rise to
109y2 + 10792y – 93312 = 0
9x4 – 2736x2 – 48x + 207936 = 0
gives rise to
9y4 + 360y3 + 2664y2 – 18768y + 23856 = 0.
Where the root of an equation was not a whole number, Chu Shih-chieh sometimes found the next approximation by using the coefficients obtained after applying Horner’s method to find the root. For example, for the equation x2 + 252x – 5292 = 0, the approximate value x1 = 19 was obtained; and, by the method of fan fa, the equation y2 + 290y – 143 = 0. Chu Shih-chieh then gave the root as x = 19(143/1 + 290). In the case of the cubic equation x3 – 574 = 0, the equation obtained by the fan fa method after finding the first approximate root, x1 = 8, becomes y3 + 24y2 + 192y – 62 = 0. In this case the root is given as x = 8(62/1 + 24 + 192) = 8 2/7. The above was not the only method adopted by Chu Shih-chieh in cases where exact roots were not found. Sometimes he would find the next decimal place for the root by continuing the process of root extraction. For example, the answer x = 19.2 was obtained in this fashion in the case of the equation
135x2 + 4608x – 138240 = 0.
For finding square roots, there are the following examples in the Ssu-yüan yü-chien:
Like Ch’in Chiu-shao, Chu Shih-chieh also employed a method of substitution to give the next approximate number. For example, in solving the equation –8x2 + 578x – 3419 = 0, he let x = y/8. Through substitution, the equation became –y2 + 578y – 3419 × 8 = 0. Hence, y = 526 and x = 526/8 = 65–3/4. In another example, 24649x2 – 1562500 = 0, letting x = y/157, leads to y2 – 1562500 = 0, from which y = 1250 and x = 1250/157 = 7 151/157. Sometimes there is a combination of two of the above-mentioned methods. For example, in the equation 63x2 – 740x – 432000 = 0, the root to the nearest whole number, 88, is found by using Horner’s method. The equation 63y2 + 10348y – 9248 = 0 results when the fan fa method is applied. Then, using the substitution method, y = z/63 and the equation becomes z2 + 10348z – 582624 = 0, giving z = 56 and y = 56/63 = 7/8. Hence, x = 88 7/8.
The Ssu-yüan yü-chien begins with a diagram showing the so-called Pascal triangle (shown in modern form in Fig. 3), in which
(x + 1)4 = x4 + 4x3 + 6x2 + 4x + 1.
Although the Pascal triangle was used by Yang Hui in the thirteenth century and by Chia Hsien in the twelfth, the diagram drawn by Chu Shih-chieh differs
from those of his predecessors by having parallel oblique lines drawn across the numbers. On top of the triangle are the words pen chi (“the absolute term”). Along the left side of the triangle are the values of the absolute terms for (x + 1)n from n = 1 to n = 8, while along the right side of the triangle are the values of the coefficient of the highest power of x. To the left, away from the top of the triangle, is the explanation that the numbers in the triangle should be used horizontally when (x + 1) is to be raised to the power n. Opposite this is an explanation that the numbers inside the triangle give the lien, i.e., all coefficients of x from x2 to xn-1. Below the triangle are the technical terms of all the coefficients in the polynomial. It is interesting that Chu Shih-chieh refers to this diagram as the ku-fa (“old method”).
The interest of Chinese mathematicians in problems involving series and progressions is indicated in the earliest Chinese mathematical texts extant, the Choupei suan-ching (ca. fourth century b.c.) and Liu Hui’s commentary on the Chiu-chang suan-shu. Although arithmetical and geometrical series were subsequently handled by a number of Chinese mathematicians, it was not until the time of Chu Shih-chieh that the study of higher series was raised to a more advanced level. In his Ssu-yüan yü-chien Chu Shih-chieh dealt with bundles of arrows of various cross sections, such as circular or square, and with piles of balls arranged so that they formed a triangle, a pyramid, a cone, and so on. Although no theoretical proofs are given, among the series found in the Ssu-yüan yü-chien are the following:
After Chu Shih-chieh, Chinese mathemathicians made almost no progress in the study of higher series. It was only after arrival of the Jesuits that interest in his work was revived. Wang Lai, for example, showed in his Heng-chai suan hsüeh that the first five series above can be represented in the generalized form
where r is a positive integer.
Further contributions to the study of finite integral series were made during the nineteenth century by such Chines mathematicians as Tung Yu-ch’eng, Li Shan-lan, and Lo Shih-lin. They attempted to express Chu Shih-chieh’s series in more generalized and modern forms. Tu Shih-jan has recently stated that the following relationship, often erroneously attributed to Chu Shih-chieh, can be traced only as far as the work of Li Shan-lan.
If , where r and p are positive integre, then
with the examples
where q is any other positive integer.
Another significant contribution by Chu Shih-chieh is his study of the methods of chao ch’a (“finite differences”). Quadratic expression had been used by Chinese astronomers in the process of finding arbitrary constants in formulas for celestial motions. We know that his methods was used by Li Shun-feng when he computed the Lin Te calender in a.d. 665. It is believed that Liu Ch’uo invented the chao ch’a method when he made the Huang Chi calender in a.d. 604, for he established the earliest terms used to denote the differences in the expression
S = U1 + U2 + U3… + Un,
calling Δ = U1shang ch’a (“upper difference”),
Δ2 = U2 – U1erh ch’a (“second difference”),
Δ3 = U3 – (2Δ2 + Δ) san ch’a (“third difference”),
Δ4 = U4 – [3(Δ3 + Δ2) + Δ] hsia ch’a (“lower difference”).
Chu-Shih-chieh illustrated how the method of finite differences could be applied in the last five problems on the subject in chapter 2 of Ssu-yüan yü-chien:
If the cube law is applied to [the rate of] recruiting soldiers, [it is found that on the first day] the ch’u chao [Δ] is equal to the number given by a cube with a side of three feet and the tz’u chao [U2 – U1] is a cube with a side one foot longer, such that on each succeeding day the difference is given by an cube with a side one foot longer that that of the preceding day. Find the total recruitment after fifteen days.
Writing down Δ, Δ2, Δ3, and Δ4 for the given number we have what is shown is Fig. 4 Employing the Conventions of Liu Ch’uo, Chu Shih-chieh gave shang ch’a (Δ)= 27 erh ch’a (Δ2) = 37; san ch’a (Δ3) = 24;
and hsia ch’a (Δ4) = 6. He then proceeded to find the number of recruits on the nth day, as follows:
Take the number of day [n] as the shang chi. Subtracting unity from the shang chi [n – 1], one gets the last term of a chiao ts’ao to [a pile of balls of triangular cross section, or S = 1 + 2 + 3 +… + (n – 1)]. The sum [of the series] is taken as the erh chi. Subtracting two from the shang chi [n – 2], one gets the last term of a san chiao to [a pile of balls of pyramidal cross section, or S = 1 + 3 + 6 +… + n(n – 1)/2]. The sum [of this series] is taken as the san chi. Subtracting three from the shang chi [n – 3], one gets the last term of a san chio lo i to series
The sum [of this series] is taken as the hsia chi. By multiplying the differences [ch’a] by their respective sums [chi] and adding the four results, the total recruitment is obtained.
From the above we have:
Shang chi = n
Multiplying these by the shang ch’a erh ch’a san ch’a, and hsia ch’a respectively, and adding the four terms, we get.
The following results are given in the same section of the Ssu yüan yü-chien:
The chai ch’a method was also employed by Chu’s contemporary, the great Yuan astronomer, mathematician, and hydraulic engineer Kuo Shou-ching, for the summation of power progressions. After them the chao ch’a method was not taken up seriously again in China until the eighteenth century, when Mei Wen-ting fully expounded the theory. Known as shōsa in Japan, the study of finite differences also received considerable attention from Japanese mathematicians, such as Seki Takakazu (or Seki Kōwa) in the seventeenth century.
For further information on Chu Shih-chieh and his work, consult Ch’ien Pao-tsung, Ku-suan k’ao-yüan (“Origin of Ancient Chinese Mathematics”) (Shanghai, 1935), pp. 67–80; and Chung kuo shu hsüeh-shih (“History of Chinese Mathematics”) (Peking, 1964), 179–205; Ch’ien Pao-tsung et al., Sung yuan shu-hsüeh-shih lun-wen-chi (“Collected Essays of Sung and Yuan Chinese Mathematics”) (Peking, 1966), pp. 166–209; L. van Hée, “Le précieux miroir des quatre éléments,” Asia Major, 7 (1932), 242, Hsü Shunfang, Chung-suan-chia ti tai-shu-hsüeh yen-chiu (“Study of Algebra by Chinese Mathematicians”) (Peking, 1952), pp. 34–55; E. L. Konantz, “The Precious Mirror of the Four Elements,” in China Journal of Science and Arts, 2 (1924), 304; Li Yen, Chung-Kuo shu-hsüeh ta-kang (“Outline of Chinese Mathematics”), I (Shanghai, 1931), 184–211; “Chiuchang suan-shu pu-chu” Chuug-suan-shih lun-ts’ung (German trans.), in Gesammelte Abhandlungen über die Geschichte der chinesischen Mathematik, III (Shanghai, 1935), 1–9; Chung-kuo Suan-hsüeh-shih (“History of Chinese Mathematics”) (Shanghai, 1937; repr. 1955), pp. 105–109, 121–128, 132–133; and Chung Suan-chia ti nei-ch’a fa yen-chiu (Investigation of the Interpolation Formulas in Chinese Mathematics”) (Peking, 1957), of which an English trans. and abridagement is “The Interpolation Formulas of Early Chinese Mathematicians,” in Proceedings of the Eighth International Congress of the History of Science (Florence, 1956), pp. 70–72; Li Yen and Tu Shih-jan, Chung-kuo ku-tai shu-hsüeh chien-shih (“A Short History of Ancient Chinese Mathematics”), II (Peking, 1964), 183–193, 203–216; Lo Shih-lin, Supplement to the Ch’ou-jen chuan (1840, repr. Shanghai, 1935), pp. 614–620; Yoshio Mikami, The Development of Mathematics in China and Japan (Leipzig, 1913; repr. New York), 89–98; Joseph Needham, Science and Civilisation in China, III (Cambridge, 1959), 41, 46–47, 125, 129–133, 134–139; George Sarton, Introduction to the Hisṭory of Science, III (Baltimore, 1947), 701–703; and Alexander Wylie, Chinese Researches (Shanghai, 1897; repr. Peking, 1936; Taipei, 1966), pp. 186–188.
fl. c. 1280-1303
The career of Chu Shih-chieh advanced studies in arithmetic and geometric series, as well as finite differences. He produced two notable written works, both of which were destined to have an impact on mathematical studies in East Asia for centuries.
Chu (Zhu Shiejie in the modern spelling) was also known by the literary name of Hanch'ing. He appears to have come from the area of Yen'shan, near Beijing, and spent much of his career as a teacher travelling throughout China. As his reputation spread, more students came to him requesting instruction.
The first of his notable works, which appeared in 1299, was Suan-hsueh ch'i-meng, or Introduction to Mathematical Studies. As its name indicates, this was a book written for novices. The manuscript disappeared from China some time after his death, and was only recovered in the nineteenth century; in the meantime, it had spread to Japan and Korea, where it came into wide use as a textbook beginning in the 1400s.
By contrast to the earlier work, Ssu-yuan yuchien (Precious mirror of the four elements 1303;) was a book to challenge the thinking of mathematical scholars. The four elements referred to in the title were four variables in a single algebraic equation, which Chu expressed using what he called the "method of the celestial element."
Chu's transformation method for solving equations, which he developed up to the degree of 14, would not be equaled by European mathematicians until the nineteenth century, with the Ruffini-Horner procedure of Paolo Ruffini (1765-1822) and William George Horner (1786-1837). The book also discussed what came to be known as the arithmetic or Pascal triangle, actually discovered earlier by other Chinese mathematicians.
Chu's era was one of those periods of turmoil that have traditionally punctuated Chinese history. In 1279, the Sung Dynasty had been overthrown by the Mongols, whose Yüan Dynasty was destined to last fewer than 90 years before it too was replaced by the Ming, China's last native-born ruling house. Given the instability in the country at that time, it is perhaps no surprise that it would be many years before Chinese mathematicians made new advances.