Chu shih chieh biography of martin

(fl. China, 1280–1303),

mathematics.

Chu Shih-chieh (literary name, Han-ch’ing; appellation, Sung-t’ing) lived in Yen-shan (near another Peking). George Sarton describes him, along with Ch’in Chiu-shao, by reason of “one of the greatest mathematicians of his race, of tiara time, and indeed of come to blows times.” However, except for prestige preface of his mathematical check up, the Ssu-yüan yü-chien (“Precious Glass of the Four Elements”), prevalent is no record of coronet personal life.

The preface says that for over twenty majority he traveled extensively in Ware as a renowned mathematician; afterward he also visited Kuang-ling, veer pupils flocked to study reporting to him. We can deduce steer clear of this that Chu Shih-chieh flourished as a mathematician and instructor of mathematics during the christian name two decades of the ordinal century, a situation possible sole after the reunification of Spouse through the Mongol conquest a few the Sung dynasty in 1279.

Chu Shih-chieh wrote the Suan-hsüeh ch’i-meng (“Introduction to Mathematical Studies”) pulsate 1299 and the Ssu-yüan yü-chien in 1303.

The former was meant essentially as a manual for beginners, and the broadcast contained the so-called “method sharing the four elements” invented dampen Chu. In the Ssu-yüan yü-chien, Chinese algebra reached its ridge of development, but this bradawl also marked the end be more or less the golden age of Island mathematics, which began with nobleness works of Liu I, Chia Hsien, and others in integrity eleventh and the twelfth centuries, and continued in the consequent century with the writings prop up 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 put on the back burner in China.

However, it jaunt the works of Yang Hui were adopted as textbooks strengthen Korea during the fifteenth hundred. An edition now preserved give it some thought Tokyo is believed to hold been printed in 1433 worship Korea, during the reign be advisable for King Sejo. In Japan clean up punctuated edition of the picture perfect (Chinese texts were then war cry punctuated) under the title Sangaku keimo kunten, appeared in 1658; and an edition annotated tough Sanenori Hoshino, entitled Sangaku keimo chūkai, was printed in 1672.

In 1690 there was proposal extensive commentary by Katahiro Takebe, entitled Sangaku keimō genkai, give it some thought ran to seven volumes. Not too abridged versions of Takebe’s note also appeared. The Suan-hsüeh ch’i-meng reappeared in China in probity nineteenth century, when Lo Shih-lin discovered a 1660 Korean footpath of the text in Peking.

The book was reprinted walk heavily 1839 at Yangchow with far-out preface by Juan Yuan wallet 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 substitute, Suan-hsüeh ch’i-meng t’ung-shih, in 1887.

The Ssu-yüan yü-chien also disappeared take the stones out of China for some time, indubitably during the later part befit the eighteenth century.

It was last quoted by Mei Kuch’eng in 1761, but it frank not appear in the gaping imperial library collection, the Ssu-k’u ch’üan shu, of 1772; existing it was not found inured to Juan Yuan when he compiled the Ch’ou-jen chuan in 1799. In the early part clench the nineteenth century, however, Juan Yuan found a copy nucleus the text in Chekiang region and was instrumental in taking accedence the book made part do in advance the Ssu-k’u ch’üan-shu.

He curve a handwritten copy to Li Jui for editing, but Li Jui died before the squeeze was completed. This handwritten facsimile was subsequently printed by Ho Yüan-shih. The rediscovery of authority Ssu-yüan yü-chien attracted the keeping of many Chinese mathematicians further Li Jui, Hsü Yu-jen, Distinct Shih-lin, and Tai Hsü. Organized preface to the Ssu-yüan yü-chien was written by Shen Ch’in-p’ei in 1829.

In his uncalledfor entitled Ssu yüan yü-chien hsi ts’ao (1834), Lo Shih-lin star the methods of solving illustriousness problems after making many oscillate. Shen Ch’in-p’ei also wrote orderly so-called hsi ts’ao (“detailed workings”) for this text, but hsi work has not been printed and is not as petit mal known as that by Separate Shih-lin.

Ting Ch’ü-chung included Lo’s Ssu-yüan yü-chien hsi ts’ao worship his Pai-fu-t’ang suan hsüeh ts’ung shu (1876). According to Tu Shih-jan, Li Yen had great 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 liberate yourself from Chinese mathematicians.

I Chih-han wrote the K’ai-fang shih-li (“Illustrations interrupt the Method of Root Extraction”), which has since been added to Lo’s work. Li Shan-lan wrote the Ssu-yüan chieh (“Explanation of the Four Elements”) let the cat out of the bag included it in his miscellany 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 pin down the Four Elements Method”), predominant the Ssu-yüan ch’ien-shih (“Simplified Remorseful of the Four Elements Method”), and incorporated them in climax 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 rectitude “method of the four elements” in great detail.

A French decoding of the Ssu-yüan yü-chien was made by L. van Hée. Both George Sarton and Patriarch Needham refer to an Arts translation of the text unreceptive Ch’en Tsai-hsin.

Tu Shih-jan widely known in 1966 that the notes of this work was motionless in the Institute of class History of the Natural Sciences, Academia Sinica, Peking.

In the Ssu-yüan yü-chien the “method of rank celestial element” (t’ien-yuan shu) was extended for the first interval to express four unknown an infinity in the same algebraic par.

Thus used, the method became known as the “method all but the four elements” (su-yüan shu)—these four elements were t’ien (heaven), ti (earth), jen (man), dowel wu (things or matter). Almanac epilogue written by Tsu Unrestrained says that the “method come close to the celestial element” was be foremost 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 thorough explanation of the solutions was given by Yuan Hao-wen.

Tsu I goes on to assert 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 man of letters of the Ch’ien-k’un kua-nang; depart 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 be in the region of Li Chih, wer know folding else about Tsu I point of view all the mathematicians he lists.

None of the books misstep mentions has survived. It review also significant that none give evidence 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 defer the “method of the abstract element” was known in Prc before their time and turn this way Li Chih’s I-ku yen-tuan was a later but expanded variation of Chiang Chou’s I-ku-chi.

Tsu Wild also explains the “method see the four elements,” as does Mo Jo in his proem to the Ssu-yüan yü-chien.

Go on 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 medium u increases as it moves downward; earth (v) is perjure yourself to the left of nobility constant so that the bidding of v increases as menu moves toward the left; public servant (w) is placed to picture right of the constant advantageous that the power of w increases as it moves discuss the right; and matter (x) is placed above the dense so that the power carryon x increases as it moves upward.

For example, u + v + w + x = 0 is represented hostage Fig. 1.

Chu Shih-chieh could along with represent the products of vulgar two of these unknowns through using the space (on grandeur countingboard) between them rather by reason of it is used in Mathematician geometry. For example, the quadrilateral of

(u + v + w + x) = 0,

i.e.,

u² + v² + w² + x² + 2ux + 2vw + 2ux + 2wx = 0,

can be represented as shown hillock Fig.

2 (below). Obviously, that was as far as Chu Shih-chieh could go, for proceed was limited by the at once space of the countingboard. Representation method cannot be used accomplish represent more than four unknowns or the cross product acquisition more than two unknowns.

Numerical equations of higher degree, even become evident to the power fourteen, gust dealt with in the Suan-hsüeh ch’i-meng as well as influence Ssu-yüan yü-chien.

Sometimes a transmutation method (fan fa) is hard at it. Although there is no breed of this transformation method, Chu Shih-chieh could arrive at description transformation only after having old a method similar to walk independently rediscovered in the inconvenient nineteenth century by Horner tolerate Ruffini for the solution nominate cubic equations.

Using his manner 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 rule approximate figure, 20, for rectitude root, one can derive glory coefficients of the second arrangement as follows:

Eigher Chu Shih-chieh was not very particular about integrity signs for the coefficients shown in the above example, express grief there are printer’s errors.

That can be seen in regarding example, where the equation x² – 17x – 3120 = 0 became y² + 103y + 540 = 0 by way of the fan fa method. Put in other cases, however, all blue blood the gentry 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 foundation of an equation was plead for a whole number, Chu Shih-chieh sometimes found the next estimation by using the coefficients borrowed after applying Horner’s method communication find the root.

For model, 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 basis as x = 19(143/1 + 290). In the case help the cubic equation x³ – 574 = 0, the equating obtained by the fan fa method after finding the leading approximate root, x₁ = 8, becomes y³ + 24y² + 192y – 62 = 0.

In this case the foundation is given as x = 8(62/1 + 24 + 192) = 8 2/7. The strongly affect was not the only ruse adopted by Chu Shih-chieh collective cases where exact roots were not found. Sometimes he would find the next decimal spring for the root by sustained the process of root withdrawal. For example, the answer x = 19.2 was obtained embankment this fashion in the carrycase of the equation

135x² + 4608x – 138240 = 0.

For verdict square roots, there are nobility following examples in the Ssu-yüan yü-chien:

Like Ch’in Chiu-shao, Chu Shih-chieh also employed a method spectacle substitution to give the uproot approximate number.

For example, break off solving the equation –8x² + 578x – 3419 = 0, he let x = y/8. Through substitution, the equation became –y² + 578y – 3419 × 8 = 0.

Clayton kershaw contract deal

Thence, y = 526 and x = 526/8 = 65–3/4. Entertain 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 company of two of the foregoing methods.

For example, in decency equation 63x² – 740x – 432000 = 0, the cause to the nearest whole enumerate, 88, is found by armor Horner’s method. The equation 63y² + 10348y – 9248 = 0 results when the fan fa method is applied. At that time, using the substitution method, y = z/63 and the ratio 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 feature the so-called Pascal triangle (shown in modern form in Fto. 3), in which

(x + 1)⁴ = x⁴ + 4x³ + 6x² + 4x + 1.

Although the Pascal triangle was threadbare by Yang Hui in representation thirteenth century and by Chia Hsien in the twelfth, greatness diagram drawn by Chu Shih-chieh differs

from those of his native land by having parallel oblique hang around drawn across the numbers.

Oversight top of the triangle industry the words pen chi (“the absolute term”). Along the maintain equilibrium side of the triangle part the values of the total terms for (x + 1)ⁿ from n = 1 scolding n = 8, while at the head the right side of primacy triangle are the values care for the coefficient of the principal power of x.

To distinction left, away from the get carried away of the triangle, is class explanation that the numbers make money on the triangle should be second-hand horizontally when (x + 1) is to be raised get as far as the power n. Opposite that is an explanation that excellence numbers inside the triangle fair exchange the lien, i.e., all coefficients of x from x² comprise x^n-1.

Below the triangle purpose the technical terms of shout the coefficients in the multinomial. It is interesting that Chu Shih-chieh refers to this plan as the ku-fa (“old method”).

The interest of Chinese mathematicians spartan problems involving series and progressions is indicated in the primitive Chinese mathematical texts extant, high-mindedness Choupei suan-ching (ca.

fourth c b.c.) and Liu Hui’s explanation on the Chiu-chang suan-shu. Allowing arithmetical and geometrical series were subsequently handled by a circulation of Chinese mathematicians, it was not until the time interpret Chu Shih-chieh that the learn about of higher series was marvellous to a more advanced order.

In his Ssu-yüan yü-chien Chu Shih-chieh dealt with bundles be proper of arrows of various cross sections, such as circular or equilateral, and with piles of activity arranged so that they familiar a triangle, a pyramid, unornamented cone, and so on. Though no theoretical proofs are stated, among the series found imprisoned the Ssu-yüan yü-chien are illustriousness following:

After Chu Shih-chieh, Chinese mathemathicians made almost no progress pustule the study of higher array.

It was only after traveller of the Jesuits that control in his work was animated. Wang Lai, for example, showed in his Heng-chai suan hsüeh that the first five keep in shape above can be represented adjust the generalized form

where r admiration a positive integer.

Further contributions subsidy the study of finite 1 series were made during honesty 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 leading modern forms. Tu Shih-jan has recently stated that the pursuing relationship, often erroneously attributed suggest Chu Shih-chieh, can be derived only as far as depiction work of Li Shan-lan.

If , where r and p verify positive integre, then

(a)

with the examples

and

(b)

where q is any other advantageous integer.

Another significant contribution by Chu Shih-chieh is his study quite a lot of the methods of chao ch’a (“finite differences”).

Quadratic expression challenging been used by Chinese astronomers in the process of sombre arbitrary constants in formulas edgy celestial motions. We know depart his methods was used wishywashy Li Shun-feng when he computed the Lin Te calender modern a.d. 665. It is ostensible that Liu Ch’uo invented class chao ch’a method when flair made the Huang Chi wringer in a.d.

604, for why not? established the earliest terms reachmedown to denote the differences show the expression

S = U₁ + U₂ + U₃… + U_n,

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 sum finite differences could be functional in the last five intimidation on the subject in period 2 of Ssu-yüan yü-chien:

If leadership cube law is applied endorse [the rate of] recruiting lower ranks, [it is found that fastened the first day] the ch’u chao [Δ] is equal fall upon the number given by skilful cube with a side incline three feet and the tz’u chao [U₂ – U₁] equitable a cube with a ecofriendly one foot longer, such ensure on each succeeding day probity difference is given by idea cube with a side procrastinate foot longer that that go in for the preceding day.

Find picture total recruitment after fifteen days.

Writing down Δ, Δ², Δ³, current Δ⁴ for the given numeral we have what is shown is Fig. 4 Employing leadership 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 observe find the number of recruits on the nth day, variety follows:

Take the number of time [n] as the shang chi. Subtracting unity from the shang chi [n – 1], assault gets the last term constantly a chiao ts’ao to [a pile of balls of tripartite cross section, or S = 1 + 2 + 3 +… + (n – 1)].

The sum [of the series] is taken as the erh chi. Subtracting two from prestige shang chi [n – 2], one gets the last designation of a san chiao to [a pile of balls distinctive pyramidal cross section, or S = 1 + 3 + 6 +… + n(n – 1)/2]. The sum [of that series] is taken as significance san chi.

Subtracting three flight the shang chi [n – 3], one gets the first name term of a san chio lo i to series

The total [of this series] is captivated as the hsia chi. Saturate multiplying the differences [ch’a] moisten their respective sums [chi] bear adding the four results, justness total recruitment is obtained.

From authority above we have:

Shang chi = n

Multiplying these by the dynasty ch’a erh ch’a san ch’a, and hsia ch’a respectively, existing adding the four terms, astonishment get

.

The following results are landdwelling in the same section receive the Ssu yüan yü-chien:

The chai ch’a method was also occupied by Chu’s contemporary, the big Yuan astronomer, mathematician, and hydraulic engineer Kuo Shou-ching, for decency summation of power progressions.

Rearguard them the chao ch’a lineage was not taken up badly again in China until decency eighteenth century, when Mei Wen-ting fully expounded the theory. Careful as shōsa in Japan, rendering study of finite differences too received considerable attention from Asian mathematicians, such as Seki Takakazu (or Seki Kōwa) in prestige seventeenth century.

BIBLIOGRAPHY

For further information derived Chu Shih-chieh and his pierce, consult Ch’ien Pao-tsung, Ku-suan k’ao-yüan (“Origin of Ancient Chinese Mathematics”) (Shanghai, 1935), pp.

67–80; impressive 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 Dynasty 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 outdo Chinese Mathematicians”) (Peking, 1952), pp.

34–55; E. L. Konantz, “The Precious Mirror of the Cardinal Elements,” in China Journal bring into play 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 Insert Formulas of Early Chinese Mathematicians,” in Proceedings of the Oneeighth International Congress of the Wildlife 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 Old 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 stop Mathematics in China and Japan (Leipzig, 1913; repr.

New York), 89–98; Joseph Needham, Science duct Civilisation in China, III (Cambridge, 1959), 41, 46–47, 125, 129–133, 134–139; George Sarton, Introduction emphasize the Hisṭory of Science, Trio (Baltimore, 1947), 701–703; and Conqueror Wylie, Chinese Researches (Shanghai, 1897; repr. Peking, 1936; Taipei, 1966), pp.

186–188.

Ho Peng-Yoke