Chu shih chieh identity evropa

(fl. China, –),

mathematics.

Chu Shih-chieh (literary name, Han-ch’ing; appellation, Sung-t’ing) lived in Yen-shan (near extra Peking). George Sarton describes him, along with Ch’in Chiu-shao, whilst “one of the greatest mathematicians of his race, of coronate time, and indeed of approach times.” However, except for high-mindedness preface of his mathematical industry, the Ssu-yüan yü-chien (“Precious Speculum of the Four Elements”), to is no record of authority personal life.

The preface says that for over twenty ripen he traveled extensively in Better half as a renowned mathematician; next he also visited Kuang-ling, at pupils flocked to study go under the surface him. We can deduce distance from this that Chu Shih-chieh flourished as a mathematician and don of mathematics during the dense two decades of the ordinal century, a situation possible after the reunification of Chum through the Mongol conquest boss the Sung dynasty in

Chu Shih-chieh wrote the Suan-hsüeh ch’i-meng (“Introduction to Mathematical Studies”) trim and the Ssu-yüan yü-chien crucial The former was meant fundamentally as a textbook for beginners, and the latter contained class so-called “method of the link elements” invented by Chu.

Be next to the Ssu-yüan yü-chien, Chinese algebra reached its peak of expansion, but this work also forcible the end of the flourishing age of Chinese mathematics, which began with the works personal Liu I, Chia Hsien, increase in intensity others in the eleventh boss the twelfth centuries, and extended in the following century handle the writings of Ch’in Chiu-shao, Li Chih, Yang Hui, leading Chu Shih-chieh himself.

It appears cruise the Suan-hsüeh ch’i-meng was missing for some time in Wife buddy.

However, it and the output of Yang Hui were adoptive as textbooks in Korea via the fifteenth century. An run riot now preserved in Tokyo evolution believed to have been printed in in Korea, during greatness reign of King Sejo. Well-off Japan a punctuated edition exempt the book (Chinese texts were then not punctuated) under decency title Sangaku keimo kunten, arrived in ; and an demonstration annotated by Sanenori Hoshino, privileged Sangaku keimo chūkai, was printed in In there was fleece extensive commentary by Katahiro Takebe, entitled Sangaku keimō genkai, think about it ran to seven volumes.

Diverse abridged versions of Takebe’s annotation also appeared. The Suan-hsüeh ch’i-meng reappeared in China in prestige nineteenth century, when Lo Shih-lin discovered a Korean edition interrupt the text in Peking. Nobleness book was reprinted in associate with Yangchow with a preface tough Juan Yuan and a legend pleasure by Lo Shih-lin. Other editions appeared in and in Get underway was also included in picture ts’e-hai-shan-fang chung-hsisuan-hsüeh ts’ung-shu collection.

Wang Chien wrote a commentary advantaged Suan-hsüeh ch’i-meng shu i beget abd Hsu Feng-k’ao produced all over the place, Suan-hsüeh ch’i-meng t’ung-shih, in

The Ssu-yüan yü-chien also disappeared shun China for some time, unquestionably during the later part have available the eighteenth century. It was last quoted by Mei Kuch’eng in , but it blunt not appear in the endless imperial library collection, the Ssu-k’u ch’üan shu, of ; brook it was not found brush aside Juan Yuan when he compiled the Ch’ou-jen chuan in Flash the early part of blue blood the gentry nineteenth century, however, Juan Kwai found a copy of leadership text in Chekiang province favour was instrumental in having character book made part of rendering Ssu-k’u ch’üan-shu.

He sent grand 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 presumption many Chinese mathematicians besides Li Jui, Hsü Yu-jen, Lo Shih-lin, and Tai Hsü.

A preliminary to the Ssu-yüan yü-chien was written by Shen Ch’in-p’ei clear up In his work entitled Ssu yüan yü-chien hsi ts’ao (), Lo Shih-lin included the approachs of solving the problems tail making many changes. Shen Ch’in-p’ei also wrote a so-called hsi ts’ao (“detailed workings”) for that text, but hsi work has not been printed and assay not as well known slightly that by Lo Shih-lin.

Thought-provoking Ch’ü-chung included Lo’s Ssu-yüan yü-chien hsi ts’ao in his Pai-fu-t’ang suan hsüeh ts’ung shu (). According to Tu Shih-jan, Li Yen had a complete handwritten copy of Shen’s version, which in many respects is distant superior to Lo’s.

Following the check over of Lo Shih-lin’s Ssu-yüan yü-chien hsi-ts’ao, the “method of description four elements” began to catch much attention from Chinese mathematicians.

I Chih-han wrote the K’ai-fang shih-li (“Illustrations of the Schematic of Root Extraction”), which has since been appended to Lo’s work. Li Shan-lan wrote goodness Ssu-yüan chieh (“Explanation of integrity Four Elements”) ans included performance in his anthology of systematic texts, the Tse-ku-shih-chai suan-hsüeh, precede published in Peking in Wu Chia-shan wrote the Ssu-yüan ming-shih shih-li (“Examples Illustrating the Price and Forms in the Pair Elements Method”), the Ssu-yüan ts’ao (“Workings in the Four Bit Method”), and the Ssu-yüan ch’ien-shih (“Simplified Explanations of the Yoke Elements Method”), and incorporated them in his Pai-fu-t’ang suan-hsüeh ch’u chi ().

In his Hsüeh-suan pi-t’an (“Jottings in the Bone up on of Mathematics”), Hua Heng-fang as well discussed the “method of integrity four elements” in great detail.

A French translation of the Ssu-yüan yü-chien was made by Plaudits. van Hée. Both George Sarton and Joseph Needham refer run into an English translation of birth text by Ch’en Tsai-hsin.

Tu Shih-jan reported in that depiction manuscript of this work was still in the Institute neat as a new pin the History of the Leader Sciences, Academia Sinica, Peking.

In honesty Ssu-yüan yü-chien the “method censure the celestial element” (t’ien-yuan shu) was extended for the precede time to express four nameless quantities in the same algebraical equation.

Thus used, the family 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-okay detailed explanation of the solutions was given by Yuan Hao-wen.

Tsu I goes on castigate 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), blue blood the gentry author of the Ch’ien-k’un kua-nang; it was his friend Chu Shih-chieh, however, who invented rectitude “method of the four elements.” “Except for Chu Shih-chieh near Yüan Hao-wen, a close reviewer of Li Chih, wer be versed nothing else about Tsu Funny and all the mathematicians inaccuracy lists.

None of the books he mentions has survived. Arouse is also significant that nobody of the three great Asian mathematicians of of the ordinal century—Ch’in Chiu-shao, Li Chih, endure Yang Hui—is mentioned in Chu Shih-chieh’s works. It is design that the “method of magnanimity celestial element” was known score China before their time beginning that Li Chih’s I-ku yen-tuan was a later but encyclopedic version of Chiang Chou’s I-ku-chi.

Tsu I also explains the “method of the four elements,” slightly does Mo Jo in climax preface to the Ssu-yüan yü-chien.

Each of the “four elements” represents an unkown quantity—u, altogether, w, and x, respectively. Olympus (u) is placed below nobleness constant, which is denoted insensitive to t’ai, so that the independence of u increases as rush moves downward; earth (v) equitable placed to the left help the constant so that honesty power of v increases though it moves toward the left; man (w) is placed attack the right of the frozen so that the power govern w increases as it moves toward the right; and stuff (x) is placed above character constant so that the administrate of x increases as drenching moves upward.

For example, u + v + w + x = 0 is insubstantial in Fig. 1.

Chu Shih-chieh could also represent the products translate any two of these unknowns by using the space (on the countingboard) between them relatively as it is used get Cartesian geometry. For example, probity square of

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

i.e.,

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

can be represented as shown in Fig.

2 (below). Evidently, this was as far bit Chu Shih-chieh could go, collaboration he was limited by rendering two-dimensional space of the countingboard. The method cannot be cast-off to represent more than join unknowns or the cross artefact of more than two unknowns.

Numerical equations of higher degree, smooth up to the power 14, are dealt with in justness Suan-hsüeh ch’i-meng as well bring in the Ssu-yüan yü-chien.

Sometimes fine transformation method (fan fa) level-headed employed. Although there is thumb description of this transformation lineage, Chu Shih-chieh could arrive usage the transformation only after obtaining used a method similar perfect that independently rediscovered in class early nineteenth century by Horner and Ruffini for the fiddle of cubic equations.

Using reward method of fan fa, Chu Shih-chieh changed the quartic equation.

x⁴ – x² – x + = 0

to the form

y⁴ – 80y³ + y² – y – = 0.

Employing Horner’s representation in finding the first relate figure, 20, for the seat, one can derive the coefficients of the second equation likewise follows:

Eigher Chu Shih-chieh was troupe very particular about the script for the coefficients shown surround the above example, or in are printer’s errors.

This stool be seen in another instance, where the equation x² – 17x – = 0 became y² + y + = 0 by the fan fa method. In other cases, notwithstanding, all the signs in greatness second equations are correct. En route for example,

x² – x – = 0

gives rise to

y² + y – = 0

and

9x⁴ – x² – 48x + = 0

gives rise to

9y⁴ + y³ + y² – y + = 0.

Where the root of uncorrupted equation was not a full number, Chu Shih-chieh sometimes make ineffective the next approximation by service the coefficients obtained after levying Horner’s method to find blue blood the gentry root.

For example, for picture equation x² + x – = 0, the approximate mean x₁ = 19 was obtained; and, by the method commemorate fan fa, the equation y² + y – = 0. Chu Shih-chieh then gave high-mindedness root as x = 19(/1 + ). In the argue of the cubic equation x³ – = 0, the equalization obtained by the fan fa method after finding the foremost approximate root, x₁ = 8, becomes y³ + 24y² + y – 62 = 0.

In this case the core is given as x = 8(62/1 + 24 + ) = 8 2/7. The curtains was not the only pathway adopted by Chu Shih-chieh importance cases where exact roots were not found. Sometimes he would find the next decimal resource for the root by in progress the process of root rescission. For example, the answer x = was obtained in that fashion in the case replica the equation

x² + x – = 0.

For finding square citizenship, there are the following examples in the Ssu-yüan yü-chien:

Like Ch’in Chiu-shao, Chu Shih-chieh also hard at it a method of substitution communication give the next approximate enumerate.

For example, in solving influence equation –8x² + x – = 0, he let x = y/8. Through substitution, probity equation became –y² + y – × 8 = 0. Hence, y = and x = /8 = 65–3/4. Affix another example, x² – = 0, letting x = y/, leads to y² – = 0, from which y = and x = / = 7 / Sometimes there keep to a combination of two near the above-mentioned methods.

For give, in the equation 63x² – x – = 0, dignity root to the nearest undivided faultless number, 88, is found strong using Horner’s method. The correspondence 63y² + y – = 0 results when the fan fa method is applied. Next, using the substitution method, y = z/63 and the ratio becomes z² + z – = 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 ethics so-called Pascal triangle (shown donation modern form in Fig. 3), in which

(x + 1)⁴ = x⁴ + 4x³ + 6x² + 4x + 1.

Although excellence Pascal triangle was used close to Yang Hui in the ordinal century and by Chia Hsien in the twelfth, the map drawn by Chu Shih-chieh differs

from those of his predecessors alongside having parallel oblique lines ragged across the numbers.

On refrain from of the triangle are position words pen chi (“the shady term”). Along the left portrayal of the triangle are position values of the absolute cost for (x + 1)ⁿ implant n = 1 to n = 8, while along integrity right side of the polygon are the values of honesty coefficient of the highest force of x.

To the not done, away from the top work at the triangle, is the communication that the numbers in excellence triangle should be used horizontally when (x + 1) assessment to be raised to grandeur power n. Opposite this practical an explanation that the everywhere inside the triangle give integrity lien, i.e., all coefficients get the message x from x² to x^n-1.

Below the triangle are birth technical terms of all glory coefficients in the polynomial. Overtake is interesting that Chu Shih-chieh refers to this diagram pass for the ku-fa (“old method”).

The worried of Chinese mathematicians in demands involving series and progressions assay indicated in the earliest Asian mathematical texts extant, the Choupei suan-ching (ca.

fourth century b.c.) and Liu Hui’s commentary incidence the Chiu-chang suan-shu. Although arithmetic and geometrical series were afterward handled by a number manage Chinese mathematicians, it was shed tears until the time of Chu Shih-chieh that the study reminiscent of higher series was raised adopt a more advanced level.

Sky his Ssu-yüan yü-chien Chu Shih-chieh dealt with bundles of arrows of various cross sections, specified as circular or square, cope with with piles of balls fit so that they formed cool triangle, a pyramid, a strobile, and so on. Although thumb theoretical proofs are given, middle the series found in rank Ssu-yüan yü-chien are the following:

After Chu Shih-chieh, Chinese mathemathicians beholden almost no progress in representation study of higher series.

Do business was only after arrival cut into the Jesuits that interest in vogue his work was revived. Wang Lai, for example, showed middle his Heng-chai suan hsüeh renounce the first five series hold back can be represented in dignity generalized form

where r is a-okay positive integer.

Further contributions to character study of finite integral mound 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 convoy in more generalized and latest forms. Tu Shih-jan has currently stated that the following affinity, often erroneously attributed to Chu Shih-chieh, can be traced solitary as far as the be anxious of Li Shan-lan.

If , disc r and p are definite 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 nobility methods of chao ch’a (“finite differences”).

Quadratic expression had back number used by Chinese astronomers send down the process of finding uncertain constants in formulas for inexperienced motions. We know that fulfil methods was used by Li Shun-feng when he computed goodness Lin Te calender in a.d. It is believed that Liu Ch’uo invented the chao ch’a method when he made representation Huang Chi calender in a.d.

, for he established leadership earliest terms used to give up the differences in 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 accumulate the method of finite differences could be applied in influence last five problems on goodness subject in chapter 2 produce Ssu-yüan yü-chien:

If the cube knock about is applied to [the not up to scratch of] recruiting soldiers, [it go over found that on the be foremost day] the ch’u chao [Δ] is equal to the digit given by a cube come to mind a side of three frontier fingers and the tz’u chao [U₂ – U₁] is a cake with a side one pier longer, such that on each one succeeding day the difference stick to given by an cube work stoppage a side one foot individual that that of the previous day.

Find the total achievement after fifteen days.

Writing down Δ, Δ², Δ³, and Δ⁴ mind the given number we take what is shown is Illustration. 4 Employing the Conventions look up to 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.

Without fear then proceeded to find probity number of recruits on ethics nth day, as follows:

Take interpretation number of day [n] importation the shang chi. Subtracting singleness from the shang chi [n – 1], one gets authority last term of a chiao ts’ao to [a pile confront balls of triangular cross division, or S = 1 + 2 + 3 +… + (n – 1)].

The supplement [of the series] is in use as the erh chi. Subtracting two from the shang chi [n – 2], one gets the last term of pure san chiao to [a mountain of balls of pyramidal bad-tempered section, or S = 1 + 3 + 6 +… + n(n – 1)/2]. High-mindedness sum [of this series] laboratory analysis taken as the san chi.

Subtracting three from the shang chi [n – 3], make sure of gets the last term allround a san chio lo wild to series

The sum [of that series] is taken as depiction hsia chi. By multiplying justness differences [ch’a] by their particular sums [chi] and adding character four results, the total engagement is obtained.

From the above amazement have:

Shang chi = n

Multiplying these by the shang ch’a erh ch’a san ch’a, and hsia ch’a respectively, and adding honourableness four terms, we get

.

The followers results are given in illustriousness same section of the Ssu yüan yü-chien:

The chai ch’a administer was also employed by Chu’s contemporary, the great Yuan uranologist, mathematician, and hydraulic engineer Kuo Shou-ching, for the summation footnote power progressions.

After them loftiness chao ch’a method was mass taken up seriously again exertion China until the eighteenth 100, when Mei Wen-ting fully expounded the theory. Known as shōsa in Japan, the study deduction finite differences also received cumbersome attention from Japanese mathematicians, specified as Seki Takakazu (or Seki Kōwa) in the seventeenth century.

BIBLIOGRAPHY

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

67–80; and Chung kuo shu hsüeh-shih (“History of Sinitic Mathematics”) (Peking, ), –; Ch’ien Pao-tsung et al., Sung kwai shu-hsüeh-shih lun-wen-chi (“Collected Essays line of attack Sung and Yuan Chinese Mathematics”) (Peking, ), pp. –; Acclaim. van Hée, “Le précieux miroir des quatre éléments,” Asia Major, 7 (), , Hsü Shunfang, Chung-suan-chia ti tai-shu-hsüeh yen-chiu (“Study of Algebra by Chinese Mathematicians”) (Peking, ), pp.

34–55; Liken. L. Konantz, “The Precious Speculum of the Four Elements,” burst China Journal of Science illustrious Arts, 2 (), ; Li Yen, Chung-Kuo shu-hsüeh ta-kang (“Outline of Chinese Mathematics”), I (Shanghai, ), –; “Chiuchang suan-shu pu-chu” Chuug-suan-shih lun-ts’ung (German trans.), elation Gesammelte Abhandlungen über die Geschichte der chinesischen Mathematik, III (Shanghai, ), 1–9; Chung-kuo Suan-hsüeh-shih (“History of Chinese Mathematics”) (Shanghai, ; repr.

), pp. –, –, –; and Chung Suan-chia ti nei-ch’a fa yen-chiu (Investigation a choice of the Interpolation Formulas in Island Mathematics”) (Peking, ), of which an English trans. and abridagement is “The Interpolation Formulas be in possession of Early Chinese Mathematicians,” in Proceedings of the Eighth International Get-together of the History of Science (Florence, ), pp.

70–72; Li Yen and Tu Shih-jan, Chung-kuo ku-tai shu-hsüeh chien-shih (“A Diminutive History of Ancient Chinese Mathematics”), II (Peking, ), –, –; Lo Shih-lin, Supplement to representation Ch’ou-jen chuan (, repr. Impress, ), pp. –; Yoshio Mikami, The Development of Mathematics do China and Japan (Leipzig, ; repr. New York), 89–98; Carpenter Needham, Science and Civilisation revel in China, III (Cambridge, ), 41, 46–47, , –, –; Martyr Sarton, Introduction to the Hisṭory of Science, III (Baltimore, ), –; and Alexander Wylie, Chinese Researches (Shanghai, ; repr.

Peking, ; Taipei, ), pp. –

Ho Peng-Yoke