Chu ShihChieh
Chu ShihChieh
(fl. China, 1280–1303),
mathematics.
Chu Shihchieh (literary name, Hanch’ing; appellation, Sungt’ing) lived in Yenshan (near modern Peking). George Sarton describes him, along with Ch’in Chiushao, 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 Ssuyüan yüchien (“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 Kuangling, where pupils flocked to study under him. We can deduce from this that Chu Shihchieh 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 Shihchieh wrote the Suanhsüeh ch’imeng (“Introduction to Mathematical Studies”) in 1299 and the Ssuyüan yüchien in 1303. The former was meant essentially as a textbook for beginners, and the latter contained the socalled “method of the four elements” invented by Chu. In the Ssuyüan yüchien, 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 Chiushao, Li Chih, Yang Hui, and Chu Shihchieh himself.
It appears that the Suanhsüeh ch’imeng 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 Suanhsüeh ch’imeng reappeared in China in the nineteenth century, when Lo Shihlin 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 Shihlin. Other editions appeared in 1882 and in 1895. It was also included in the ts’ehaishanfang chunghsisuanhsüeh ts’ungshu collection. Wang Chien wrote a commentary entitled Suanhsüeh ch’imeng shu i in 1884 abd Hsu Fengk’ao produced another, Suanhsüeh ch’imeng t’ungshih, in 1887.
The Ssuyüan yüchien 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 Ssuk’u ch’üan shu, of 1772; and it was not found by Juan Yuan when he compiled the Ch’oujen 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 Ssuk’u ch’üanshu. 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 Yüanshih. The rediscovery of the Ssuyüan yüchien attracted the attention of many Chinese mathematicians besides Li Jui, Hsü Yujen, Lo Shihlin, and Tai Hsü. A preface to the Ssuyüan yüchien was written by Shen Ch’inp’ei in 1829. In his work entitled Ssu yüan yüchien hsi ts’ao (1834), Lo Shihlin included the methods of solving the problems after making many changes. Shen Ch’inp’ei also wrote a socalled 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 Shihlin. Ting Ch’üchung included Lo’s Ssuyüan yüchien hsi ts’ao in his Paifut’ang suan hsüeh ts’ung shu (1876). According to Tu Shihjan, 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 Shihlin’s Ssuyüan yüchien hsits’ao, the “method of the four elements” began to receive much attention from Chinese mathematicians. I Chihhan wrote the K’aifang shihli (“Illustrations of the Method of Root Extraction”), which has since been appended to Lo’s work. Li Shanlan wrote the Ssuyüan chieh (“Explanation of the Four Elements”) ans included it in his anthology of mathematical texts, the Tsekushihchai suanhsüeh, first published in Peking in 1867. Wu Chiashan wrote the Ssuyüan mingshih shihli (“Examples Illustrating the Terms and Forms in the Four Elements Method”), the Ssuyüan ts’ao (“Workings in the Four Elements Method”), and the Ssuyüan ch’ienshih (“Simplified Explanations of the Four Elements Method”), and incorporated them in his Paifut’ang suanhsüeh ch’u chi (1862). In his Hsüehsuan pit’an (“Jottings in the Study of Mathematics”), Hua Hengfang also discussed the “method of the four elements” in great detail.
A French translation of the Ssuyüan yüchien was made by L. van Hée. Both George Sarton and Joseph Needham refer to an English translation of the text by Ch’en Tsaihsin. Tu Shihjan 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 Ssuyüan yüchien the “method of the celestial element” (t’ienyuan 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” (suyüan 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 Ikuchi, Li Weni’s Chaotan, Shih Hsintao’s Ch’ienching, and Liu Yuchieh’s Juchi shihso, and that a detailed explanation of the solutions was given by Yuan Haowen. Tsu I goes on to say that the “earth element” was first used by Li Tetsai in his Liangi ch’unying chichen while the “man element” was introduced by Liu Tachien (literary name, Liu Junfu), the author of the Ch’ienk’un kuanang; it was his friend Chu Shihchieh, however, who invented the “method of the four elements.” “Except for Chu Shihchieh and Yüan Haowen, 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 Chiushao, Li Chih, and Yang Hui—is mentioned in Chu Shihchieh’s works. It is thought that the “method of the celestial element” was known in China before their time and that Li Chih’s Iku yentuan was a later but expanded version of Chiang Chou’s Ikuchi.
Tsu I also explains the “method of the four elements,” as does Mo Jo in his preface to the Ssuyüan yüchien. 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 Shihchieh 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,
i.e.,
u^{2} + v^{2} + w^{2} + x^{2} + 2ux + 2vw + 2ux + 2wx = 0,
can be represented as shown in Fig. 2 (below). Obviously, this was as far as Chu Shihchieh could go, for he was limited by the twodimensional 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 Suanhsüeh ch’imeng as well as the Ssuyüan yüchien. Sometimes a transformation method (fan fa) is employed. Although there is no description of this transformation method, Chu Shihchieh 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 Shihchieh changed the quartic equation.
x^{4} – 1496x^{2} – x + 558236 = 0
to the form
y^{4} – 80y^{3} + 904y^{2} – 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 Shihchieh 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 x^{2} – 17x – 3120 = 0 became y^{2} + 103y + 540 = 0 by the fan fa method. In other cases, however, all the signs in the second equations are correct. For example,
109x^{2} – 2288x – 348432 = 0
gives rise to
109y^{2} + 10792y – 93312 = 0
and
9x^{4} – 2736x^{2} – 48x + 207936 = 0
gives rise to
9y^{4} + 360y^{3} + 2664y^{2} – 18768y + 23856 = 0.
Where the root of an equation was not a whole number, Chu Shihchieh sometimes found the next approximation by using the coefficients obtained after applying Horner’s method to find the root. For example, for the equation x^{2} + 252x – 5292 = 0, the approximate value x_{1} = 19 was obtained; and, by the method of fan fa, the equation y^{2} + 290y – 143 = 0. Chu Shihchieh then gave the root as x = 19(143/1 + 290). In the case of the cubic equation x^{3} – 574 = 0, the equation obtained by the fan fa method after finding the first approximate root, x_{1} = 8, becomes y^{3} + 24y^{2} + 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 Shihchieh 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
135x^{2} + 4608x – 138240 = 0.
For finding square roots, there are the following examples in the Ssuyüan yüchien:
Like Ch’in Chiushao, Chu Shihchieh also employed a method of substitution to give the next approximate number. For example, in solving the equation –8x^{2} + 578x – 3419 = 0, he let x = y/8. Through substitution, the equation became –y^{2} + 578y – 3419 × 8 = 0. Hence, y = 526 and x = 526/8 = 65–3/4. In another example, 24649x^{2} – 1562500 = 0, letting x = y/157, leads to y^{2} – 1562500 = 0, from which y = 1250 and x = 1250/157 = 7 151/157. Sometimes there is a combination of two of the abovementioned methods. For example, in the equation 63x^{2} – 740x – 432000 = 0, the root to the nearest whole number, 88, is found by using Horner’s method. The equation 63y^{2} + 10348y – 9248 = 0 results when the fan fa method is applied. Then, using the substitution method, y = z/63 and the equation becomes z^{2} + 10348z – 582624 = 0, giving z = 56 and y = 56/63 = 7/8. Hence, x = 88 7/8.
The Ssuyüan yüchien begins with a diagram showing the socalled Pascal triangle (shown in modern form in Fig. 3), in which
(x + 1)^{4} = x^{4} + 4x^{3} + 6x^{2} + 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 Shihchieh 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 x^{2} to x^{n1}. Below the triangle are the technical terms of all the coefficients in the polynomial. It is interesting that Chu Shihchieh refers to this diagram as the kufa (“old method”).
The interest of Chinese mathematicians in problems involving series and progressions is indicated in the earliest Chinese mathematical texts extant, the Choupei suanching (ca. fourth century b.c.) and Liu Hui’s commentary on the Chiuchang suanshu. Although arithmetical and geometrical series were subsequently handled by a number of Chinese mathematicians, it was not until the time of Chu Shihchieh that the study of higher series was raised to a more advanced level. In his Ssuyüan yüchien Chu Shihchieh 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 Ssuyüan yüchien are the following:
After Chu Shihchieh, 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 Hengchai 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 Yuch’eng, Li Shanlan, and Lo Shihlin. They attempted to express Chu Shihchieh’s series in more generalized and modern forms. Tu Shihjan has recently stated that the following relationship, often erroneously attributed to Chu Shihchieh, can be traced only as far as the work of Li Shanlan.
If , where r and p are positive integre, then
(a)
with the examples
and
(b)
where q is any other positive integer.
Another significant contribution by Chu Shihchieh 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 Shunfeng 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 = U_{1} + U_{2} + U_{3}… + U_{n},
calling Δ = U_{1}shang ch’a (“upper difference”),
Δ^{2} = U_{2} – U_{1}erh ch’a (“second difference”),
Δ^{3} = U_{3} – (2Δ^{2} + Δ) san ch’a (“third difference”),
Δ^{4} = U_{4} – [3(Δ^{3} + Δ^{2}) + Δ] hsia ch’a (“lower difference”).
ChuShihchieh illustrated how the method of finite differences could be applied in the last five problems on the subject in chapter 2 of Ssuyüan yüchien:
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 [U_{2} – U_{1}] 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, Δ^{2}, Δ^{3}, and Δ^{4} for the given number we have what is shown is Fig. 4 Employing the Conventions of Liu Ch’uo, Chu Shihchieh gave shang ch’a (Δ)= 27 erh ch’a (Δ^{2}) = 37; san ch’a (Δ^{3}) = 24;
and hsia ch’a (Δ^{4}) = 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 yüchien:
The chai ch’a method was also employed by Chu’s contemporary, the great Yuan astronomer, mathematician, and hydraulic engineer Kuo Shouching, 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 Wenting 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.
BIBLIOGRAPHY
For further information on Chu Shihchieh and his work, consult Ch’ien Paotsung, Kusuan k’aoyüan (“Origin of Ancient Chinese Mathematics”) (Shanghai, 1935), pp. 67–80; and Chung kuo shu hsüehshih (“History of Chinese Mathematics”) (Peking, 1964), 179–205; Ch’ien Paotsung et al., Sung yuan shuhsüehshih lunwenchi (“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, Chungsuanchia ti taishuhsüeh yenchiu (“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, ChungKuo shuhsüeh takang (“Outline of Chinese Mathematics”), I (Shanghai, 1931), 184–211; “Chiuchang suanshu puchu” Chuugsuanshih lunts’ung (German trans.), in Gesammelte Abhandlungen über die Geschichte der chinesischen Mathematik, III (Shanghai, 1935), 1–9; Chungkuo Suanhsüehshih (“History of Chinese Mathematics”) (Shanghai, 1937; repr. 1955), pp. 105–109, 121–128, 132–133; and Chung Suanchia ti neich’a fa yenchiu (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 Shihjan, Chungkuo kutai shuhsüeh chienshih (“A Short History of Ancient Chinese Mathematics”), II (Peking, 1964), 183–193, 203–216; Lo Shihlin, Supplement to the Ch’oujen 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.
Ho PengYoke
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Chu Shihchieh
Chu Shihchieh (jōō shŭjĕ), fl. 1280–1303, Chinese mathematician. He contributed to the study of arithmetic and geometric series and to that of finite differences. His two mathematical works, Introduction to Mathematical Studies and Precious Mirror of the Four Elements, were lost for a time in China and were recovered only in the 19th cent.
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