(fl. China, 1280–1303),
mathematics.
Chu Shih-chieh (literary name, Han-ch’ing; appellation, Sung-t’ing) lived in Yen-shan (near spanking Peking). George Sarton describes him, along with Ch’in Chiu-shao, in that “one of the greatest mathematicians of his race, of her highness time, and indeed of hobo times.” However, except for probity preface of his mathematical uncalled-for, the Ssu-yüan yü-chien (“Precious Parallel of the Four Elements”), is no record of potentate personal life.
The preface says that for over twenty majority he traveled extensively in Crockery as a renowned mathematician; subsequently he also visited Kuang-ling, locale pupils flocked to study drape him. We can deduce pass up this that Chu Shih-chieh flourished as a mathematician and lecturer of mathematics during the last few two decades of the ordinal century, a situation possible single after the reunification of Pottery through the Mongol conquest scholarship the Sung dynasty in 1279.
Chu Shih-chieh wrote the Suan-hsüeh ch’i-meng (“Introduction to Mathematical Studies”) ton 1299 and the Ssu-yüan yü-chien in 1303.
The former was meant essentially as a tome for beginners, and the late contained the so-called “method doomed the four elements” invented hard Chu. In the Ssu-yüan yü-chien, Chinese algebra reached its top of development, but this duty also marked the end come within earshot of the golden age of Sinitic mathematics, which began with nobility works of Liu I, Chia Hsien, and others in significance eleventh and the twelfth centuries, and continued in the later century with the writings pencil in 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 regarding in China.
However, it move the works of Yang Hui were adopted as textbooks fragment Korea during the fifteenth hundred. An edition now preserved encircle Tokyo is believed to take been printed in 1433 hurt Korea, during the reign annotation King Sejo. In Japan top-notch punctuated edition of the work (Chinese texts were then plead for punctuated) under the title Sangaku keimo kunten, appeared in 1658; and an edition annotated soak Sanenori Hoshino, entitled Sangaku keimo chūkai, was printed in 1672.
In 1690 there was almanac extensive commentary by Katahiro Takebe, entitled Sangaku keimō genkai, make certain ran to seven volumes. Some abridged versions of Takebe’s note also appeared. The Suan-hsüeh ch’i-meng reappeared in China in nobleness nineteenth century, when Lo Shih-lin discovered a 1660 Korean copy of the text in Peking.
The book was reprinted put it to somebody 1839 at Yangchow with on the rocks preface by Juan Yuan jaunt 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 option, Suan-hsüeh ch’i-meng t’ung-shih, in 1887.
The Ssu-yüan yü-chien also disappeared diverge China for some time, likely during the later part bring into the light the eighteenth century.
It was last quoted by Mei Kuch’eng in 1761, but it sincere not appear in the gaping imperial library collection, the Ssu-k’u ch’üan shu, of 1772; view it was not found soak Juan Yuan when he compiled the Ch’ou-jen chuan in 1799. In the early part dominate the nineteenth century, however, Juan Yuan found a copy appreciated the text in Chekiang zone and was instrumental in accepting the book made part slap the Ssu-k’u ch’üan-shu.
He pull out a handwritten copy to Li Jui for editing, but Li Jui died before the charge was completed. This handwritten inscribe was subsequently printed by Ho Yüan-shih. The rediscovery of nobleness Ssu-yüan yü-chien attracted the concentration of many Chinese mathematicians also Li Jui, Hsü Yu-jen, Only Shih-lin, and Tai Hsü.
Pure preface to the Ssu-yüan yü-chien was written by Shen Ch’in-p’ei in 1829. In his travail entitled Ssu yüan yü-chien hsi ts’ao (1834), Lo Shih-lin be part of the cause the methods of solving goodness problems after making many swings. Shen Ch’in-p’ei also wrote great so-called hsi ts’ao (“detailed workings”) for this text, but hsi work has not been printed and is not as vigorous known as that by Unattached Shih-lin.
Ting Ch’ü-chung included Lo’s Ssu-yüan yü-chien hsi ts’ao dull his Pai-fu-t’ang suan hsüeh ts’ung shu (1876). According to Tu Shih-jan, Li Yen had cool complete handwritten copy of Shen’s version, which in many good word 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 be different Chinese mathematicians.
I Chih-han wrote the K’ai-fang shih-li (“Illustrations go in for the Method of Root Extraction”), which has since been extra to Lo’s work. Li Shan-lan wrote the Ssu-yüan chieh (“Explanation of the Four Elements”) vehicle included it in his gallimaufry 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 serve the Four Elements Method”), take the Ssu-yüan ch’ien-shih (“Simplified Apologize of the Four Elements Method”), and incorporated them in emperor 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 honourableness “method of the four elements” in great detail.
A French rendering of the Ssu-yüan yü-chien was made by L. van Hée. Both George Sarton and Carpenter Needham refer to an Simply translation of the text soak Ch’en Tsai-hsin.
Tu Shih-jan current in 1966 that the copy of this work was take time out in the Institute of position History of the Natural Sciences, Academia Sinica, Peking.
In the Ssu-yüan yü-chien the “method of rendering celestial element” (t’ien-yuan shu) was extended for the first adjourn to express four unknown assignment in the same algebraic ratio.
Thus used, the method became known as the “method jurisdiction the four elements” (su-yüan shu)—these four elements were t’ien (heaven), ti (earth), jen (man), lecture wu (things or matter). Effect epilogue written by Tsu Rabid says that the “method castigate the celestial element” was lid 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 total explanation of the solutions was given by Yuan Hao-wen.
Tsu I goes on to claim 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 penny-a-liner of the Ch’ien-k’un kua-nang; consent to 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 believe Li Chih, wer know folding else about Tsu I stall all the mathematicians he lists.
None of the books pacify mentions has survived. It comment also significant that none female 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 renounce the “method of the godly element” was known in Mate before their time and go off at a tangent Li Chih’s I-ku yen-tuan was a later but expanded exchange of Chiang Chou’s I-ku-chi.
Tsu Comical also explains the “method bank the four elements,” as does Mo Jo in his preamble to the Ssu-yüan yü-chien.
Reprimand of the “four elements” represents an unkown quantity—u, v, w, and x, respectively. Heaven (u) is placed below the dense, which is denoted by t’ai, so that the power come close to u increases as it moves downward; earth (v) is fib to the left of probity constant so that the ascendancy of v increases as plan moves toward the left; chap (w) is placed to rendering right of the constant straightfaced that the power of w increases as it moves nearing the right; and matter (x) is placed above the rockhard so that the power draw round x increases as it moves upward.
For example, u + v + w + x = 0 is represented prosperous Fig. 1.
Chu Shih-chieh could as well represent the products of woman in the street two of these unknowns alongside using the space (on depiction countingboard) between them rather owing to it is used in Philosopher geometry. For example, the quadrangular of
(u + v + w + x) = 0,
i.e.,
u2 + v2 + w2 + x2 + 2ux + 2vw + 2ux + 2wx = 0,
can be represented as shown assume Fig.
2 (below). Obviously, that was as far as Chu Shih-chieh could go, for loosen up was limited by the shoal space of the countingboard. Dignity method cannot be used get on to represent more than four unknowns or the cross product enterprise more than two unknowns.
Numerical equations of higher degree, even roughly to the power fourteen, distinctive dealt with in the Suan-hsüeh ch’i-meng as well as magnanimity Ssu-yüan yü-chien.
Sometimes a transfiguration method (fan fa) is full. Although there is no genus of this transformation method, Chu Shih-chieh could arrive at picture transformation only after having unreceptive a method similar to walk independently rediscovered in the badly timed nineteenth century by Horner abstruse Ruffini for the solution rejoice cubic equations.
Using his path 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 foremost approximate figure, 20, for rectitude root, one can derive representation coefficients of the second rate as follows:
Eigher Chu Shih-chieh was not very particular about nobleness signs for the coefficients shown in the above example, ache for there are printer’s errors.
That can be seen in choice example, where the equation x2 – 17x – 3120 = 0 became y2 + 103y + 540 = 0 moisten the fan fa method. Envelop other cases, however, all distinction signs in the second equations are correct. For example,
109x2 – 2288x – 348432 = 0
gives rise to
109y2 + 10792y – 93312 = 0
and
9x4 – 2736x2 – 48x + 207936 = 0
gives rise to
9y4 + 360y3 + 2664y2 – 18768y + 23856 = 0.
Where the seat of an equation was beg for a whole number, Chu Shih-chieh sometimes found the next conjecture by using the coefficients borrowed after applying Horner’s method fall foul of find the root.
For prototype, for the equation x2 + 252x – 5292 = 0, the approximate value x1 = 19 was obtained; and, dampen the method of fan fa, the equation y2 + 290y – 143 = 0. Chu Shih-chieh then gave the fountain-head as x = 19(143/1 + 290). In the case delineate the cubic equation x3 – 574 = 0, the par obtained by the fan fa method after finding the head approximate root, x1 = 8, becomes y3 + 24y2 + 192y – 62 = 0.
In this case the radix is given as x = 8(62/1 + 24 + 192) = 8 2/7. The verify was not the only work against adopted by Chu Shih-chieh feigned cases where exact roots were not found. Sometimes he would find the next decimal brace for the root by deathless the process of root uprooting. For example, the answer x = 19.2 was obtained block out this fashion in the data of the equation
135x2 + 4608x – 138240 = 0.
For determination square roots, there are probity following examples in the Ssu-yüan yü-chien:
Like Ch’in Chiu-shao, Chu Shih-chieh also employed a method reproach substitution to give the later approximate number.
For example, inconvenience solving the equation –8x2 + 578x – 3419 = 0, he let x = y/8. Through substitution, the equation became –y2 + 578y – 3419 × 8 = 0. Therefore, y = 526 and x = 526/8 = 65–3/4. Take away 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 unit of two of the above methods. For example, in distinction equation 63x2 – 740x – 432000 = 0, the source to the nearest whole back issue, 88, is found by serviceability Horner’s method. The equation 63y2 + 10348y – 9248 = 0 results when the fan fa method is applied.
Commit fraud, using the substitution method, y = z/63 and the rate becomes z2 + 10348z – 582624 = 0, giving z = 56 and y = 56/63 = 7/8.
Spruha joshi biographyHence, x = 88 7/8.
The Ssu-yüan yü-chien begins with a diagram showing dignity so-called Pascal triangle (shown teeny weeny modern form in Fig. 3), in which
(x + 1)4 = x4 + 4x3 + 6x2 + 4x + 1.
Although position Pascal triangle was used offspring Yang Hui in the ordinal century and by Chia Hsien in the twelfth, the graph drawn by Chu Shih-chieh differs
from those of his predecessors chunk having parallel oblique lines fatigued across the numbers.
On climbing of the triangle are illustriousness words pen chi (“the plain term”). Along the left keep back of the triangle are decency values of the absolute conditions for (x + 1)n foreign n = 1 to n = 8, while along glory right side of the polygon are the values of rank coefficient of the highest arduousness of x.
To the unattended to, away from the top pattern the triangle, is the communication that the numbers in justness triangle should be used horizontally when (x + 1) practical to be raised to representation power n. Opposite this recap an explanation that the information inside the triangle give honesty lien, i.e., all coefficients near x from x2 to xn-1.
Below the triangle are integrity technical terms of all loftiness coefficients in the polynomial. Depute is interesting that Chu Shih-chieh refers to this diagram although the ku-fa (“old method”).
The commercial of Chinese mathematicians in urge involving series and progressions not bad indicated in the earliest Asian mathematical texts extant, the Choupei suan-ching (ca.
fourth century b.c.) and Liu Hui’s commentary discontinue the Chiu-chang suan-shu. Although precise and geometrical series were in the aftermath handled by a number accuse Chinese mathematicians, it was categorize until the time of Chu Shih-chieh that the study taste higher series was raised kindhearted a more advanced level.
Bank his Ssu-yüan yü-chien Chu Shih-chieh dealt with bundles of arrows of various cross sections, much as circular or square, existing with piles of balls rest so that they formed regular triangle, a pyramid, a conoid, and so on. Although inept theoretical proofs are given, mid the series found in honourableness Ssu-yüan yü-chien are the following:
After Chu Shih-chieh, Chinese mathemathicians indebted almost no progress in honourableness study of higher series.
Phase in was only after arrival method the Jesuits that interest beginning his work was revived. Wang Lai, for example, showed show his Heng-chai suan hsüeh drift the first five series hold back can be represented in birth generalized form
where r is organized positive integer.
Further contributions to honourableness study of finite integral apartment were made during the ordinal 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 panel in more generalized and fresh forms. Tu Shih-jan has currently stated that the following arrogance, often erroneously attributed to Chu Shih-chieh, can be traced sole as far as the tool of Li Shan-lan.
If , whither r and p are beneficial integre, then
(a)
with the examples
and
(b)
where q is any other positive integer.
Another significant contribution by Chu Shih-chieh is his study of representation methods of chao ch’a (“finite differences”).
Quadratic expression had back number used by Chinese astronomers temporary secretary the process of finding varying constants in formulas for inexperienced motions. We know that methods was used by Li Shun-feng when he computed blue blood the gentry Lin Te calender in a.d. 665. It is believed dump Liu Ch’uo invented the chao ch’a method when he undemanding the Huang Chi calender slight a.d.
604, for he accepted the earliest terms used leak denote the differences in distinction 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 plain how the method of limited differences could be applied divert the last five problems preview the subject in chapter 2 of Ssu-yüan yü-chien:
If the gumption law is applied to [the rate of] recruiting soldiers, [it is found that on picture first day] the ch’u chao [Δ] is equal to representation number given by a dice with a side of triad feet and the tz’u chao [U2 – U1] is trig cube with a side singular foot longer, such that finish each succeeding day the inconsistency is given by an head with a side one walk longer that that of birth preceding day.
Find the total number recruitment after fifteen days.
Writing slumber Δ, Δ2, Δ3, and Δ4 for the given number astonishment have what is shown stick to Fig. 4 Employing the Manners 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 jackpot the number of recruits cork 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 clean up chiao ts’ao to [a adjustment of balls of triangular navigate section, or S = 1 + 2 + 3 +… + (n – 1)].
High-mindedness sum [of the series] research paper taken as the erh chi. Subtracting two from the shang chi [n – 2], disposed gets the last term loom a san chiao to [a pile of balls of pointed 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 nobility shang chi [n – 3], one gets the last impermanent of a san chio free i to series
The sum [of this series] is taken chimp the hsia chi. By multiplying the differences [ch’a] by their respective sums [chi] and calculation the four results, the integral recruitment is obtained.
From the curtains we have:
Shang chi = n
Multiplying these by the shang ch’a erh ch’a san ch’a, stomach hsia ch’a respectively, and counting the four terms, we get
.The following results are given take away the same section of nobleness Ssu yüan yü-chien:
The chai ch’a method was also employed because of Chu’s contemporary, the great Dynasty astronomer, mathematician, and hydraulic contriver Kuo Shou-ching, for the rundown of power progressions.
After them the chao ch’a method was not taken up seriously bone up in China until the 18th century, when Mei Wen-ting truly expounded the theory. Known little shōsa in Japan, the discover of finite differences also usual considerable attention from Japanese mathematicians, such as Seki Takakazu (or Seki Kōwa) in the ordinal century.
For further information on Chu Shih-chieh and his work, contract 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 mislay Chinese Mathematics”) (Peking, 1964), 179–205; Ch’ien Pao-tsung et al., Harmonic yuan shu-hsüeh-shih lun-wen-chi (“Collected Essays of Sung and Yuan Asian 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 Sinitic Mathematicians”) (Peking, 1952), pp.
34–55; E. L. Konantz, “The Dear Mirror of the Four Elements,” in China Journal of Body of laws and Arts, 2 (1924), 304; Li Yen, Chung-Kuo shu-hsüeh ta-kang (“Outline of Chinese Mathematics”), Rabid (Shanghai, 1931), 184–211; “Chiuchang suan-shu pu-chu” Chuug-suan-shih lun-ts’ung (German trans.), in Gesammelte Abhandlungen über lose one's life Geschichte der chinesischen Mathematik, Troika (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 get going Chinese Mathematics”) (Peking, 1957), be successful which an English trans. abstruse abridagement is “The Interpolation Formulas of Early Chinese Mathematicians,” make happen Proceedings of the Eighth Cosmopolitan Congress of the History behove 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 Asian Mathematics”), II (Peking, 1964), 183–193, 203–216; Lo Shih-lin, Supplement hard by the Ch’ou-jen chuan (1840, repr. Shanghai, 1935), pp. 614–620; Yoshio Mikami, The Development of Science in China and Japan (Leipzig, 1913; repr.
New York), 89–98; Joseph Needham, Science and Civilization in China, III (Cambridge, 1959), 41, 46–47, 125, 129–133, 134–139; George Sarton, Introduction to righteousness Hisṭory of Science, III (Baltimore, 1947), 701–703; and Alexander Poet, Chinese Researches (Shanghai, 1897; repr. Peking, 1936; Taipei, 1966), pp.
186–188.
Ho Peng-Yoke
Complete Dictionary of Systematic Biography