Knowledge of Mathemattesand Science in

I. Introduction

Ching-Hua-Yan

by Yu Wang Luen

(Kuala Lumpur)

Mathematics and science seldom come into Chinese novels as they dit not often have much appeal to the writers and still less to the general public. Although some casual references are made elsewhere, such as in the 18th-century novel Yeh-sou p'u-yenl11 of Hsia Ching-ch'ul11 (1'105-1'18'1), to one or two aspects of Chinese mathematics, we cannot find a novel that embodies a corpus of k.nowledge of mathematics and science, however limited it may be, than the early 19th-century novel Ching-hua-yan 1 (a) of Li Ju-chen 1 11 (c. 1763-1830).

In critisizing the novel Ching-hua-yan, Lu Hsn 181 has, among commen-dation and censure, made the following remarks:

The author obviously considered this book a storehause of know-ledge and art, but in this sense it is more like an encyclopaedia than a novel 8

1 Ching-hua-yan was first printed in 1828. 1t was reprinted in Kwangtung in 1829, since then many editions have been puhlished. The best edition now availa"ole is the one published by Jen-min dl'u-pan-she, Peking 1957. There are two English translations: 1. Gladys Yang, A Joumey into Strange Lands (Translations of Li Ju-dlen's Ching-

hua-yan, dlapters 7-40), Chinese Literature Peking 1958. vol. 1, 76-122. 2. Flowers in the Mirrow by Li Ju-dlen. Translated and edited by Lin Tai-yi, Berke-

ley, University of Califomia Press 1965. 2 Li Ju-dlen (c. 1763--1830), a novelist and pbonetician, was a native of Ta-hsing.

He went to Hai-dlou, Kiangsu, in 1782, accompanying his elder brother Li Ju-huang 151, who was then appointed an. official in Hai-dlou. There he became a pupil of a famous sdlolar Ling T'ing-k'an [J, ~d came in close contact with a number of s~uthern sc:holars. According to himself, he was greatly benefited by Ling's instruc-tlon. . As he. despised the fashionable Eight-legged essay, he was not quite successful m the c1vil service examination, so he directed his energies to other studies. He wr~te L_i-shih yln-dlien [7) (Li's system of phonetics) and Chlng-hua-yan, a novel wh1c:h d1splays the extensiveness of his knowledge.

1 See Lu Hsn, Chung-kuo hsiao-shuo shlh leh 111, Hongkong, Hsin-yi dl'u-pan-sh.e, 1967 ed., 267; English translation by Yang Hsien-yi and Gladys Yang, A Briet History ol Chinese Flction, Peking, Foreign Langnage Press 1964.334.

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While it is true that the novel touched on a large variety of subjects of academic interest, the opinion of some critics that the author did it purposely to show offhispersonal achievements 4 is subject to qualification.

During the time of Li Ju-chen 1231, classical studies in sinology were in vogue in China, involving the methodology of research by comparison and verification. In the study of any subject, knowledge of revelant disciplines was required. Hence an encyclopaedic knowledge was not a rare qualifica-tion among scholars, or at least, among the scholars of classical schools.

On the other hand, except for some reference made to the author' s in-vention in phonetics 5, the information given in Ching-hua-yan on various branches of knowledge were by no means original or profound. They were rather superficial and of common interest to contemporary scholars. It is, therefore, more sensible to regard the insertion of material on various sub-jects in the noveal as playful writings, rather than a parade of personal achievements. This was also the author's own opinion of the novel. Li Ju-chensays:

4 Exemples may be given as follows: Liu Ta-chieh 1101, Chung-kuo wen-hseh Ja-chan shih 1111 (A history of the develop-

ment of Chinese literature), vol. 2, 349, says, "Li Ju-chen was born in a time when classical studies in sinology reached its peak. Deeply influenced by the literary thoughts of that period, the author of Ching-hua-yan made a display of his achieve-ments in verification on classics and etymology in the novel".

Meng Yao 1121, Chung-kuo hsiao-shuo shih (13) Taiwan, Shih-chieh shu-ch 1966, vol. 4, 584, says, "The author, being disappointed of his career, attempted to direct his talents and ideals to th.e writing of this novel. Purposed to make a parade of his rieb erudition and great ability, he only jembled and confused the contents of the book".

See also: Kuo Chen-yi 1141 Chung-kuo hsiao-shuo shih 1151; Taipei, Shang-wu yin-shu kuan 1967, 435--439; Ch'ien Ching-fang (ts) Hsiao-shuo ts'ung k'ao 117J (Textura! critique and verification on novels), Shanghai, Ku-tien wen-hseh cb'u-pan-she, 1957 reprint, 53; Li Kuo 1181 Ching-hua-yan chien-lun [101, Singapore, Ch'ing-nien shu-ch, 2; Wu Hsiao-ju [201, Chung-kuo hsiao-shuo chiang-hua chi ch'i-t'a 1211 Ku-tien wen-hseh cb'u-pan-she, 54-55.

Also refer to E. C. S. Adkins, "Ching-hua-yan - China's Gulliver's Travels", China Society Annual, Singapore 1954, 34-37 & 50, quote: "Though I have called it a satirical work, it is classified in the Chiness divison as a novel which parades the author's leaming (hsien-ts'ai hsiao-shuo [22)) and, in this case the range of the author's learning was very wide".

5 See Hu Shih 1241, "A Chinese Declaration of the Rights of Women", CSPSR, Peking 1924, 8 : 2, 100--109.

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Though meant to be playful writings, it cherishes the object of ex-horting to moral reformation. It contains subjects such as writings of various philosophers; references to men and things, birds and plants, skills in calligraphy, painting, lyre-playing, chess; and knowledge of medicine, divination, astrology, phonology, mathemaUcs, etc. It also contains various kinds of riddles, drinking games, as weil as sports and games .... All may dispel drowsiness and excite laughter 8.

To the readers of today, the lengthy discussions on these subjects may Iook dull and verbose. But to the scholars of the time of Li Ju-chen, there would be plenty of fun.

In this article, we shall endeavour to verify this conclusion through a study of the material on mathematics and science contained in the novel.

li. Topics oJ Mathematics and Science contained in the Novel:

The subjects of academic interest touched on in Ching-hua-yan include mathemaUes and science which were presented in the form of chats and quizes by the women scholars at the parUes held to celebrate their success in the imperial examination 7 Among the women scholars, Tung Ch'ing-t'ien 1251, Sung Liang-chen 1281, Ssu-t'u Pin-erh 1271, Liao Hsi-ch'un 1281, Tzu Yao-m'ai 1291, Chiang Ch'iu-hui 1301 and Mi Lan-fen 131J all have knowledge of mathemaUes 8

The mathematical problems put forward by the women scholars include rule of false posiUon (ying-n suan-Ja 1321), calculaUon of the circumference (yan-chou suan-Ja 1331), the cyclic square (yan-Jang 1341), problems in arith-metical progression (ch'a-Jen Ja 1351), problems about 'pheasants and rabbits caged together' (chih-t'u t'ung-lung (S8IJ, multiplication by means of the gelosia (p'u-ti-chin 1371), calculation with counting rods (ch'ou-suan 1381), the division table (kuei-ch'u Ja [at] ), the value of n, as weil as the Indeterminate Analysis (Han Hsin tien-ping 1401 and Erh-shih-pa-hsiu nao K'un-yang 1411). Principles of science discussed by the women scholars concern the weight of matter and the velocity of sound.

Ching-hua-yan, Peking, Jen-min c:h'u-pan-she, 1957 ed., (hereafter CHY), 165. 7 CHY 565, 586--590, 712-713. 8 CHY 565.

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III. Analysis ol the Mathematical Problems and Principles ol Science raised in the Novel:

1. Rule ol False Position (ying-n suan-la)

A problern of this nature was put forward by Liao Hsi-ch'un and answered by Ssu-t'u Pin-erh ' The question reads:

A tray of fruit is distributed to a number of persons. lf everyone gets 7 [pieces of] fruit, there is 1 left over; if everyone gets 8, there are 16 short. How many persons and [pieces of] fruit are there?

The solutionwas given orally, and can be expressed as follows:

Number of persons = 1 + 16 8-7

17 = -- = 17 1

Number of [pieces of] fruit = 17 x 1 + 1 = 120 This confirms with the following equations in modern algebra:

Let a = number of persons; 7a + 1 =b ....... . 8a- 16 = b ...... . (2) - (1) : Ba - 7a = 16 + 1

16 + 1 :.a = 17 8-1

b = number of fruit (1) (2)

b = 7a + 1 = 11 x 1 + 1 = 120 The origin of ying-n suan-ta can be traced to Chiu-chang suan-shu [uJ,

(Nine Chapters on Mathematical Art) (50-100). In chapter 1 of this book under the heading 'ying-pu-tsu' 1431 (excess and deficit), 20 such examples and their solutions are given 10 Some other Chinese books on mathemaUes lik.e Sun-tzu suan-ching 1481 (Master Sun's arithmetical manual) (400), and Suan-fa t'ung tsung 1471 (Collections of works on arithmetical art) (1592) also contain such questions and solutions 11

2. Han Hsin tien-ping and Erh-shih-pa-hsiu nao K~un-.Yang

In Chapter _16, Sung Liang-chen referred to two arithmetical quizes. i. e., 'Han Hsin tien-ping' and 'Erh-shih-pa-hsiu nao K'un-yang', but their co11tents arenot mentioned 11

' CHY 565. 11 Chiu-dlang suan-shu l"l (in Suan-dling shih shu [UJ, Peking, Chung-hua shu-

dl, 1963 reprint, 205---219.) 11 Sun-tzu suan-dling l"l eh. 2, Problem 28, (in Suan-dling shih shu 307.) Ch'en Ta-wei l"l, Suan-/a t'ung-tsung (SOJ, 1858 ed., eh. 8, 1-2. tJ CHY 565.

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'Han Hsin tien-ping' is a specific indeterminate problern which is better known as 'Master Sun's problern' (SU:Il-tzu wen-t'i 1511), because it was first raised in Sun-tzu's book Sun-tzu suan-ching 13 The question reads:

"There are things of unknown whidl when divided by 7 leave 2, by 5 leave 3, by 7 Ieave 2. What is the number?"

The solutiongiven by Sun Tzu is: The nurober = 70x 2 + 21 x3 + 15 x2- 105x2 = 23 14

Master Sun's problern assumed various names sudlas Ch'in-Wang an-tien-ping (52) (The prince of Chin's secret rnethod of counting soldiers), chien-kuan-shu (ss) (Method of Chien and Kuan), Kuei-ku suan l5'l (Counting method of Kuei-ku-tzu 1551), etc. 15 Sorne verses were created to help memorize the solution, too. In Suan-fa t'ung-tsung, where Sun-tzu's problern is called wu-pu-chih-tsung 1511 (things of unknown number), and its solution was given inverse as 18 :

., Among 3 persans walking together, rarely one is 70 years of age, "5 plum trees bear 21 brandles, 7 talented men gathering at 15th of the first rnoon, "Minus 105 and the answer is known.

The verse appea.rs ra.ther unintelligible, but the three dividers in the question, as weil as the key numbers in the solution, are all contained in it.

lt can thus be seen that this problern or quiz was very popular in China, and it was therefore natural that the women scholars in Ching-hua-yan would refer to it in their discussion of arithmetical problerns.

The 'Erh-shih-pa-hsiu nao K'un Yang' could have been an arithmetic quiz of the same nature, but no mention of it elsewhere can be traced. It could also have been a term fabricated by Li Ju-chen for his story.

3. Calculation ol the Circumference of a circle (yan-chou suan-fa):

This problern is seen in Chapter 79 17 Mi Lan-fen calculated the circum-ference of a round table of diarneter 3 ch'ih (Chinese foot) 2 ts'un (Chinese inch}, and gave the answer as 10.048 ch'ih. She did the calculation by using

13 Sun-tzu suan-dting eh. 3, Problem 26. (in Suan-chlng shih shu, 318.) u Ang Tian-se in his article chinese Interest in Indeterminate Analysis and

lndeterminate Equations" gives a detailed analysis of this problem. See Majalah Pantal, University of Malaya, Kuala Lumpur 1971/72, 106-108. 15 See Li Yen [HJ and Tu Shih-jan (11) Chung-kuo ku-tai shu-hseh dtien-shih 1581,

Peking, Chung-hua shu-eh 1963, vol. 1, i23-124. te Suan-fa t'ung-tsung eh. 5, 29. 17 CHY 586-587.

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the formula (circumference = diameter x x), and took the value of n tobe 3.14. But she went on to say that the accurate value of 1t should be 3.14159265.

4. The Gelosia (P'u-ti-chin)

In working out multiplication (3.2 x 3.14), Mi Lan-fen introduced a method called p'u-ti-chin 1601. In terms of modern symbols, the p'u-ti-chin she drew is as follows 18 :

3 .l.

I 3

0 32 X 314 = 10048

0 4

4 8 P'u ti chin in its earliest form in China appears in Chiu-chang suan-fa pi-Jei

ta-ch'an 1611 (Comparative studies on nine chapters on arithmetical art) (1450) by Wu Ching l621 of the Ming Dynasty. In this book there is an introduction to hsieh-suan 162aJ, (writing calculation) and examples are given as follows in terms of modern symbols 11 :

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7 425 X 456,789 = 194,135,325

An example of p'u-ti-chin is also given in Suan-fa t'ung-tsung (1592) where it is called yin-ch' eng-t'u [831, In term of mo~em mathematical symbols it can be represented as follows 20 :

18 CHY 586--587. 11 See Li Yen and Tu Shih-jan, Chung-kuo ku-tai shu-hseh dtien-shih, vol. 2, 226. 10 Suan-Ja t'ung-tsung c:h. 12, ~.

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5. The Value ol it

Ancient Chinese mathematicians paid special attention to the value of it. The oldest Chinese book on arithmetic and astronomy in existence, Chou-pl suan-ching l681 (The Arithmetical classis of the gnomon and the circular paths of heaven) (-100) states the value of it to be 3 21 Since the Han Dynasty many attempts were made to obtain a more accurate valueza.

Among those who made contril;>utions towards this goal was Liu Hui (t 263) l741 of Wei Dynasty. Inmakingexplanarynotes in the Chiu-changsuan-shu, Liu Hui pointed out that one side of an equilateral hexagon inscribed in a circle is the samein lenght as the radius of the circle, hence the circum-

21 Tseng-pu wan-pao ch'an-shu rsJ, 1823 ed., c:h. 8, 3--5. n See Chou-pi suan-ching trJ, c:h. 1. (in Suan-ching shih shu, 35.) 23 Other Chinese mathematicians who made a study of the value of n include: Liu

Hsin 1181 (47 to + 23), Chang Heng l"J (78---139), Wang Fan [701 (219-257), and P'i Yen-tsung [71~ (?-445): see the L-li-c:hih [7!) of Sui-shu l711, Po na pen, ed., c:h. 16, 3b.

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ference must be Ionger than tbe perimeter of the bexagon. In other words, tbe value of 1t must be greater tban 3. Consequently, by calculating the peri-meter of a cyclic polygon of 96 sides, wbich is very close to tbe circum-

ference, he worked out an approximate value for "'of 3.14 ~ (deficit 625

value). On the other band, be drew polygons tbat touch the circle extemally, and, through tbe same process, obtained another approximate value for 1t

169 of 3.14 -- (excess value). He stated that tbe true value of "' must lie

625 somewbere between these two figures u.

Tsu Cb'ung-chih 1751 (429-500) made a more accurate calculation of the value of "' obtaining an ying-shu 1711 (excess value) of 3.1415927, and a n-shu l771 (deficit vale) of 3.1415926 26 Subsequently, Chinese matbematiciaM adopted the intermediate 3.14159265 as the accurate value of 1t 21

Even thougb tbe book Shu-li ching-yn 1871 (1723), wbich was published before tbe time of Li Ju-chen, bad given a more accurate value for 1t of 3.141592653 plus 27, and his contemporary the matbematician Chu HunglSIIJ, bad introduced a value of 1t being accurate up to 39 digits after the decimal point 21 Li Ju-chen considered it sufficiently accurate to take 1t to be 3.14159265.

14 See Chiu-chang suan-shu, dl. 1. (in Suan-ching shih shu, 103-105.) u See Sui-shu dl. 16, 4a; Ch'ou-jen-chuan [78) (Biographies of mathematicians and

astronomers), edited in 1799--1898, Shanghai, Shang-wu yin-shu-kuan reprint, eh. 1, 104.

11 See Yu Cheng-hsieh (71) Kuei-ssu ts'un-kao (80) edited in 1833, Shanghai, Sh.mg-wu yin-shu-kuan reprint, dl. 2, 180--181, quote: "Tsu Ch'ung-ehih, assistant ma-gistrate (ts'ung-shih l811) of Nan-hs-dlou [81J, made a more accurate calculation for the value of :t, giving an excess value of the circumference tobe 3.1415927 and an deficit value tobe 3.1415926, when the diameter is 1. The accurate value lies between these two values, i. e. 3.14159265." See also Li Yen, Chung-kuo suan-hseh shih l831, Shanghai, Shang-wu yin-shu-kuan, 1955, 27, whidl says, "Both in the L-li-dlih l8'1 of Sui-shu [851, eh. 16, and Chin-shu 1881, eh. 16, calculations are based on Tsu Ch'ung-ehih's value, :t = 3.14159265".

17 Shu-li ching-yn l881 (Essentials of mathematical theories), Shanghai, Shan-wu yin-shu-kuan ed., eh. 3, 695--696.

18 Dates of Chu Hung's birth and death cannot be traced. According to Ch'ou-jen chuan (dl. 7, 738), Chu Hung obtained eh-jen l"l in 1789, and tetired in 1830. So he should be over 20 by 1798 and still alive in 1830.

11 See Ch'ou-jen chuan dl. 7, 738.

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6. Cyclic Square (yan-Jang):

After working out the circumference of the round table, Mi Lan-fen went on the say that if the round table were cut into a square one, the length of each side would be 2.26 ch'ih (foot), which is quite accurate, to be proved with geometric equations,

L = Y2r! (where r = radius of the round table = 1.6 ch'ih) L = Jl2 X 1.2! = 1.414 X 1.6 = 2.262 Nevertheless, Mi Lan-fen did not specify the method of calculation, saying

only that she had calculated it by means of yan-lang [sn) (cyclic square) 80,

Ancient China showed great interest in the cyclic square, because it believed at that time that heaven was round and earth square 81 A k.now-ledge of the relations between the properties of round and squarewas there-fore necessary for the study of astronomy and the calendar 82 Furthermore, since it was difficult to calculate the area or circumference of a round object, attempts were made to obtain their values through the square inscribed in the circle. Chou-pi suan-ching contains two diagrams of a yan-/ang 121 (cyclic square), and a /ang-yan 131 (a circle inscribed in a square). Accord-ing to the explanary notes by Chao Shuang 1041 38, they were used to ascertain the circumference of the circle 8'. He stated, u the diameter of the circle is 1, the circumference is 3 i if a side of the cyclic square is 1, the circumference is 4" 35

Evidently the method applied by Chinese mathematicians subsequently in calculating the circumference of a circle by means of polygons inscribed in it was initiated from this idea, and Li Ju-chen's object of raising this simple geometric problem. in his novel is no other than to bring out this popular term -- yan-lang.

31 CHY 587. 31 See Chou-pi suan-ching dl. t. (in Suan-ching shih shu, 22). sz In ancient China, a considerable pa.rt of th.e studies in mathemaUes related to

astronomy an~ calendar, in whidl the emperors bad the greatest interest. Chou-pi suan-ching, for instance, begins with the question as how to measure the heaven the earth, and the answer is to start studying the relationship between the circle and the square. See Chou-p.J suan-ching dl. t. (in Suan-ching shih shu, 13).

aa The dates of Chao Shuang's birth and death are unknown. According to .the Suan-ching shih shu t'i-yao (15) (Essential points in the Suan-dling shih shu) dl. 1 5; and what Pao Kan-dlih 1"1 has to say in the Chou-pl suan-ching yin-11'1J (Phonology a~d explanations for Chou-pi suan-ching), in Suan-ching shih-shu, Taipei, Shang-wu ym-shu-kuan, Kuo-dli ed., vol. 130, 61, Chao Shuang was born in 222 or later and flo!ished during the Wei and Chin Dynasties and he annotated the Chou-pi suan-chmg.

34 Chou-pi suan-chlng m. 1 (in Suan-ching shih shu, Chung-hua shu-chu ed., 42--43).

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7. A Problem in Arithmetical Progression (ch'a-len-la):

Another arithmetical problern raised in Chapter 79 reads 38 :

~~ A set of 9 gold cups is rnade frorn gold weighing 126 Jiang (taels). Wbat is the weight of eadl cup? 11

This problern was also solved by Mi Lan-fen by giving the following rules, 119 plus 1 rnakes 10; 10 tirnes 9 is 90; "90 divided by 2 gives 45; ~~ 126 divided by 45 gives 2.8, whidl is the weight of the srnallest cup."

She then rnultiplied 2.8 by 2, 3, 4, 5, 6, 7, 8 and 9 to obtain the respective weights of the other cups:

2.8liang x 2 = 5.6 Jiang, 2.8liang x 3 = 8.4 Jiang 2.8 liang x 4 = 11.2 liang, 2.8Jiang x 5 = 14 Jiang 2.8liang x 6 = 16.8liang, 2.8liang x 7 = 19.6liang 2.8Jiang x 8 = 22.4 Jiang, 2.8liang x 9 = 25.2 liang Both the question and answer are badly worded and rather incornprehen-

sible. But from the answer given, it can be seen that this is a problern in arithrneticalprogression in the form:

S = a + 2a + 3a ........ + 9a n

or: S = -- (a + 9a) 2

where S = the total weight; a = the weight of the smallest cup; and n = 9

9 S = --(a+9a)

2 9

2 90a ---

2

This proves that Mi Lan-fen's solution is correct.

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45a

126 = 45a

:.a 126

45

28

Problems in arithmetical progressionarealso a common subject in ancient Chinese books on rnathematics. Sirnilar problerns and solutions can be found in Chiu-chang suan-shu81, Sun-Tzu suan-chingas, Chang Ch'iu-chien suan-ching l081 (Chang Ch'iu-dlien's arithmatiC:al mannual) (466--485) 30, Suan-fa t'ung-tsung'0, etc.

II CHY 587. 1~ See Chiu-dtang suan-shu dl. 1. (in Suan-chlng shih shu 131-141). 18 See Sun-tzu suan-ching eh. 2. (in Suan-ching shih shu 305--306). 11 See Chang Ch'iu-chien suan-ching dl. 2. (in Suan-ching shih shu 35~7). 40 See Suan-/a t'ung-tsung dl. 5, 10--14.

226

8. The Division Table (kuei-ch'u-la):

When Mi Lan-fen gave the oral solution of the above problem, and divided 126 by 45, she used a terminology in Chinese arithmetic, namely, ssu-kuei wu-ch'u rae] ('to be divided by 45') 41 By kuei-ch'u,f1001, or kuei-ch'u ko-cheh (tot) or chiu-kuei-ko 11021, it means a division table inverse, composed to facilitate memorization, just as chiu-chiu ko-cheh (tos) is used for multiplication. Kuei-ch'u ko-cheh appears in its earliest form in Ch' eng-ch'u t'ung-pien suan-pao z (to5) (A treasury of facile methods for multiplication and division) (1274) by Yang Hui 11011. Later it was improved upon by Chu Shih-chieh 11071 in bis Suan-hseh ch'i-meng 43 [tot) (Introduction to arithmetic) (1299), which has been generally adopted ever since, and is contained in general books on mathematics such as Suan-Ja t'ung-tsung 44, and even in the popular refer-ence work Wan-pao ch'an-shu 45.It is still in use in abacus computations.

9. The Counting Rods (suan-ch'ou):

Having calculated the weight of the smallest gold cup to be 2.8 liang, Mi Lan-fe~ introduced suan-ch'ou 11101 (counting rods) for the purpose of multi-plication in calculating the weights of the other cups 41 She used a 2 times and 8 times table and arranged the first on top of the second as follows:

q 7 b ~ + 3 l. 2.

~8 2 4- D ~ .:t 4- (:o ln this position, the figures in the semi-circles vertically arranged re-

present the products of 28 times 1, 2 through 9 respectively. The figurein the lower semi-circle of the top table is added to the figure in the upper semi-circle of the bottom table.

Thus: The 1st column to the right represents 28 x 1 = 28 The 2nd column to the right represents 28 x 2 = 56

tt CHY 587 . . u Yang Hui, Ch'eng-ch'u t'ung-pien suan-pao (104) (Facile methods of multiplica-

hon and division) eh. 2. 43 Chu Shih-ehieh, Suan-hseh ch'i-meng [tOS) (Introduction to arithmetic) eh. 1. 44 s f 45

uan- a t ung-tsung eh. 1, 8a-9a.

41 Wan-po ch'an-shu eh. 8, 2b-9b. CHY 587-589.

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227

The 3rd column to the right represents 28 x 3 = (6 + 2) 4 = 84

The 9th column to the right represents 28 x 9 = (8 + 7) 2 = 252 lt can be seen that this is a method of calculation evolved from the gelosia.

Dr. Joseph Needham believes that suan-ch'ou isaform of Napier's bones introduced into China in the 17th centuryn. Li Yen1111l also mentions in his Chung-kuo suan-hseh hsiao-shih 11121 (A Short History of Chinese Arith-metic) that suan-ch'ou was introduced into China by the Jesuit Jacques Rho (1593-1638). Jacques Rho wrote a book called Ch'ou-suan 11131 (Calculation with counting rods), which was collated by Johann Adam Schall von Bell (1591-1666), andlater collected intotheHsi-yanghsin-la Ji-shu 11141 (Western gregorian calendar) (1643) 48

Counting rods when first made in China were vertical with numbers written horizontally from left to right 41 :

47 See Joseph Needham, Science and Civillsatlon in China vol. 3, Cambridge University Press 1959, 72. .

" See U Yen, Chung-kuo suan-hseh hsiao-shih (111) Sbanghai, Shang-wu ym-shu-kuan, 1939, Wan-yu wen-k'u ed., 97-98.

" The original now kept in Peking Palace Museum, see U Yen and Tu Shih-jan, Chung-kuo ku-tal shu-hseh dlien-shih 216.

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228

Mei Wen-ting ruJ (1633-1721)

10. Problems about Pheasants and Rabbits caged tagether (chih-t'u t'ung-Iung):

The last arithmetical problern rnentioned in Ching-hua-yan appears in Chapter 93 51 and belongs to the chih-t'u t'ung-lung type of questions in Chinese books on arithmetic. It was raised by Pien Pao-yn and answered by Mi Lan-fen.

This type of problern can be seen in Sun-tzu suan-ching 54, Suan,..fa t'ung-tsung55, Yang Hui suan-la [tttJ (Arithmetical art by Yang Hui) 58 (1275), etc. Question 31 in Sun-tzu suan-ching is given below as an exarnple:

.. There are pheasants and rabbits of unk.nown numbers in a cage. 35 heads can be seen on the top, and 94 feet at the bottorn. How many pheasants and rabbits are there? ..

The solution given by Sun-tzu is: Nurnber of rabbits = 94 + 2-35 = 12 Number of pheasants = 35- 12 = 23

The question raised in Ching-hua-yan is rnore cornplicated, but belongs to the sarne category. It reads:

Two types of lantems are hung on the first floor. In one type, there are 3 big balls on the top of the lantern with srnall ballsbanging below. This type of lantern thus consists of 9 big and srnall balls. In the other type, there are 3 big balls on the top and 18 srnall ballsbanging below, thus consisting of 21 big and srnall balls. Another two different of lanterns are hung on the ground floor. In one type, there is one big ball with 2 srnall balls banging below. In the other type, there is one ball with 4 srnall balls banging below. All together there are 396 big balls and 1,440 srnall balls on the first floor, and 360 big balls and 1,200 srnall balls on the ground floor. How rnany lantems are there on the first floor and the ground floor?

Mi lan-fen said that she was using the forrnula of the 'pheasants and rabbits caged together' type of question to solve this problern. Hersolution is:

Lanterns on the ground floor: Nurnber of lantems with 4 srnall balls = 1,200 + 2 - 360 = 240 Nurnber of lantems with 2 srnall balls = 360 - 240 = 120

Lanterns on the lirst floor: Number of lantems with 18 srnall balls = (1,440 + 2- 396) + 6 =54 Number of lanterns with 6 srnall balls = (396 - 54 x 3) + 3 = 78

51 CHY 712-713. " See Sun-tzu suan-ching dl. 3, problem 31 (in Suan-chlng shlh shu 320). 15 See Suan-la t'ung-tsung dl. 7, 26. . " See Yang Hui suan-la [1"1 (Yang Hui's mathematical art), in Hs-ku cha1-dti

suan-la (1111 {Supplement to ancient arithmetic wonders), Yl-chia-t'ang ts'ung-shu ed., vol. 2, 1.

(129) tl-~ (130) .... (!

230

The answer proves correct as follows:

Lanterns on the ground floor:

Let a = number of lanterns with 4 small balls b = number of lanterns with 2 small balls Total number of small balls = 4a + 2 (360- a) = 1,200

:.a = (1,200 + 2) - 360 = 240 b = 360 - a = 360 - 240 = 120

Lanterns on the first lloor:

Let a = nurober of lanterns with 18 small balls b = number of lanterns with 6 small balls

Total number of small balls = 18a + 6 (396 + 3- a) = 1,440 18a - 6a = 1,440 - 396 x 2

. .a = (1 ,440 + 2 - 396) + 6 = 54 b = 396 + 3 - a = 396 + 3 - 54 = 78

11. Specific Weight of Matter

In Chapter 79 Mi Lan-fen calculated the weight of a piece of hung ma-nao 11321 (red agate) of cubic 3 ts'un (Chinese inch) as 59 liang (taels) and 4 ch'ien (mace); and the weight of a piece of pai ma-nao 11331 (white agate) of the same volume to be 62 liang and 2 ch'ien 57 She also gave the respective weights of some other material as follows:

Silver Copper ..... Pai t'ung ss (134) Huang t'ung 5t (137) White agate . . . Red agate ....

9 liang per cubic ts'un 1liang 5 ch'ien per cubic ts'un 61iang 9 ch'ien 8 fen per cubic ts'un 61iang 8 ch'ien per cubic ts'un 2 Jiang 3 ch'ien per cubic ts'un 21iang 2 ch'ien per cubic ts'un

Some ancient Chinese books on mathemaUes contain records of 'the weights of certain metals and other matters with a cubic ts'un as the unit volume. For example:

In Sun-tzu suan-chingeo:

57 CHY 589.

Gold .. Silver Copper

1 chin (kati) . 141iang

1liang 5 ch'ien

~ 'Pai-t'ung is an alloy of copper and arsenic. - According to Sung Ying-hsi.ng 113'1, T'ien-kung k'ai-wu [138] eh. 3, 1; English translation by B-tu Zen Sun and Sh1ou-chuan Sun, T'ien-kung k'ai-wu, Chinese Technology in the Seventeenth Century, The Pennsylvania State University Press, 1966. P. 242.

51 'Huang-t'ung is an alloy of copper (60'/o) and tin (30/o)' - According to B. C. Read and C. P. Pah., Minerals and Stones in the Pen-ts'ao kang-mu, Peking, 1936, 4.

80 Sun-tzu suan-ching dl. 1. (in Suan-dting shih shu 282).

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231

In Suan-la t'ung-tsung'1:

In Shu-li ching-yn tl:

Gold . Silver Copper

Gold .. Silver Copper . Pal t'ung Bronze . White agate . Red agate ..

16liang 14Jiang 1liang 5 ch'ien

16liang 8 ch'ien 9Iiang 1liang 5 ch'ien 6liang 9 ~'Jen 8 fen 6liang 8 ch'ien 2Iiang 3 ch'ien 2liang 2 ch'ien

It is apparent that the weights of the different kinds of material per cubic ts'un given in Ching-hua-yan are reproduced from the Shu-li ching-yn.

In the three books mentioned above, the weight per cubic ts'un of copper is identical, the weight per cubic ts'un of gold is quite close, but the weight per cubic ts'un of silver in the first two books differs materially from that in the third book, possibly because of differences in quality.

To compare the densities of matters given in the Shu-li ching-yn with what we know today, we may find that the records in this book are quite accurate:

If we take 1 cubic ts'un = 32.765 c.c. and 11iang = 37.3 gm.

For gold 16 liang 8 ch'ien per cubic ts'un = 16.8 liang per cubic ts'un

37.3gm = 16.8 x 32.765 c.c. ~ 19.12 gm per c.c.

(The actual density is 19.3 gm per c.c.)

For silver 9Jiang per cubic ts'un

37.3 gm. 9 x 32.765 c.c. = 10.242 gm. per c.c.

(The actual density is 10.5 gm. per c.c.)

For copper 1.51iang per cubic ts'un

= 7.5 x 37.3 gm. 5 = 8.53 gm. per c.c. 32.765 c.c.

(The actual density is 8.96 gm per c.c.)

Other substances in the Iist are alloys or other compounds, and their actual densities are hard to determine due to qualitative variations in their basic

11 Suan-fa t'ung-tsung eh. 1, 5. 0 Shu-Ji dllng-yn dl. 10, 1195-1196.

232

elements. However; to judge by the respective densities of their elements, the values given in Shu-Ji ching-yn arealso fairly significant u.

12. Velocity ol Sound: In Chapter 79 Mi Lan-fen showed a knowledge of the velocity of sound

through the medium of lightning and thunder". She said: u is anestablished rule that the sound of thunder travels 128 chang 5 ch'ih, 7 ts'un in one second (i. e., 1,285.7 ch'ih per second).

In Europe, lssac Newton (1642-1727) gave a theoretical value 6f 979 feet per second, and an experimental value of 1,142 feet per second for the velocity of sound. Pierre Gassend (1592-1655) gave 1,473 Paris feet per second, while Gassini, Picard, Romer and Huygene gave 1,172 Paris feet per second.

Marin Mersenne (1588-1648), a member of the Minim order with whom Matteo Ricci had corresponded on scientific matters, determined the velocity of sound in air by the difference in time between the flash and the report of fire-arms at known distances and arrived at 1,380 feet per second, whidl is some 20 per cent !arger than the modern value.

At the time of Li Ju-dlen, knowledge of natural science had been intro-duced into China, along with mathematics, for more than 200 years. A number of European books on science had been translated into Chinese. Li Ju-dlen might have learned the velocity of sound from some of these translated sources. But it is more likely that he copied it from Shu-li ching-yn. In Part 11, Vol. 3, Chapter 3 of this book an example of 'direct propor-tion' is given as follows U:

using a time-piece to determine the velocity of sound of a cannon blast, it records 7 seconds from the time one sees the smoke till one hears the report, when the cannon is 5 li (1 Ii = 1,800 ch'ih) away. How many 1i is the cannon away from us if the time difference is 14 seconds?

The velocity of sound derived from this example is thus, 5 X 1,800

7 ch'ih

per second = 1,285.7 ch'ih per second, whidl is exactly the value given by Li Ju-dlen. According to Li Shan-lan (lUJ the ch'ih referred to in the Shu-li ching-yan must be the Mandlu official foot. Wu Ch'eng-lo l111l estimated this to be equivalent to 30.9 cm ... Hence the velocity of sound given here

, aa Densities of Arsenic, Tin and Silica, respective elements of 'pai-t"ung' 'huang-t ung' and agate are:

Arsenic . . . . . . . . . . . 5.72 gm/c.c. Tin ............. 7.3 gmlc.c.

" Silica (Si01) 2.33 gm/c.c. CHY589.

: See Shu-li ching-yn eh. 1, 297. See Wu Ch'eng-lo (1DJ, Chung-kuo tu-Ilang-heng shih (1411, Shanghai 1937, 295.

(138) *~ (139) ~*~ (140) ~~ 233

is about 397.3 metres per second, some 20 per cent higher than the modern value. The Shu-11 ching-yan must have incorporated the result transmitted by Marin Mersenne to the J esuits in China.

VI. Level o.J knowledge oi Mathematics and Science in China during the Time oi Li Ju-chen:

Sinee 1581 when ltalian Jesuit, Matteo Ricci (1552-1610), came to Peking and jointly translated Chi-ho yan-pen 87 11421 (Elementary geometry, 1607) with Hs Kuang-ch'i 11431 (1562-1633), as well as r'ung-wen suan-chih es (U4J (Mathematies translated from foreign language) with Li Chih-tsao l145J (1565-1630), until1723, knowledge of mathemaUes and natural seience bad been eontinuously introdueed into China from Europe by J esuits for more than 140 years. During this period, Chinese scholars such as Hs Kuang-ch'i, Li Chih-tsao, Huang Tsung-hsi 11481 (1.610-1695) and Mei Wen-ting [147J (1633-1721), ete. bad imbided such new knowledge extensively and pro-dueed a number of translated works. Emperors such as Wan-li 11481 and Ching-chen 11491 of Ming Dynasty, and K'ang-hsi 11501 of the Ch'ing Dynasty bad also given great eneouragement to such study and aeeorded the Jesuits with special privileges. This was especially so with K'ang-hsi who was hirnself deeply interested in the study of mathematics. He went one step further by marshalling eontemporary mathematicians to edit Shu-li ching-yn whidl is the most eomprehensive book ever produced dealing with ancient Chinese and western mathematies.

When K'ang-hsi's son, Yung-Cheng 11511, took over the throne in 1723, he imposed a close-door poliey for politieal reasons - banning trade with foreigners, forbidding Chinese to leave or return to China, and banishing the Jesuits in Chinato Maeaou. From this date until the end of Opium War (1842) the influx of western knowledge was entirely blocked.

Nevertheless this does not mean that the development of mathematieal knowledge was retarded in China. On the eontrary it witnessed a signifieant advance in another direction. Under the new scholastic atmosphere that emphasized textural research, Chinese scholars systematically rearranged ancient works on mathematics, adding explanations and annotations where necessary, and at the same time; western knowledge that was introdueed to China earlier was now thoroughly digested. The study of mathemaUes readled its zenith and certain schools of students even regarded mathemaUes as a compulsory subjeet for study. A nurober of ~minent mathematicians

87 Translated from Clavius, Euclidis Eiementarum Libri, XV, 1574 ed. 88 Translated from Clavius, Epitome. Arithmeticae Praticae, 1585 ed.

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were born, of whom quite a few belonged to the same scbool as Li Ju-cben. Among them we may mention Li Ju-cben's teacber Ling T'ing-k'an 811 (15ZJ (1757-1809), a recognized mathematician and astronomer, who helped Juan Yuan (153) in compiling the Ch'ou-jen chuan 115'1 (Biographies of mathemaU-cians and astronomers). Ling in tun studied under Tai Chen 70 11551 (1723-1777), the distinguished author of many works including the Ts'e-suanf156J (Counting rods calculation) and the Kou-ku ko-yan chi 11571 (Notes on sphe-rical triangles). He also collated the Hsi-yang hsin-la li-shu 11581 (Western Gregorian Calendar), the Ssu-k'u ch'an-shu 115111 and a number of ancient Chinese books on mathematics, e. g. the Chou-pi, Chiu-chang, Sun-tzu, Hai-tao, Wu-ts'ao, Hsia-hou Yang, Wu-ching, etc. Tai Chen's teacber was Chiang Yung71 I160J (1681-1762), who wrote the Shu-hseh 11611, the T'uein-pu la-chieh !1821 (Explanation of the method of calculating the movement of heaven-ly bodies and other works.) Ch'eng Yao-t'ien 72 11831 (1725-1814) another of Chiang Yung's pupils, wrote the Chou-pi ch-shu t'u-chu 11"1 (Illustration to the properties of the square in Chou-pi) and the Chou-pi yung-ch shu-yen 11651 (Elaboration on the application of the square in Chou-pi). Chiao Hsn 73 11661 (1763-1820) was one of Tai Chen's pupils. His work include the Ch'eng-/ang shih-li 11671 (Illustration on squares), the Chia-chien ch'eng-ch'u chieh 11681 (Explanations of addition, subtraction, multiplication and division), the T'ien-yan i-shih 116111 (Explanations of algebraical equations), the Shih-hul170l (On arc), the Shih lun 11711 (On wheel), the Shih-t'ol1721 (On ovael), and the K'ai-fang t'ung-shih 11731 (General explanations of square roots). K'ung Kuang-sen 74 11741 (1752-1786) was another pupil of Tai Chen. He wrote the Shao-kuang chen-fu-shu 11751 (Positive and negative values of square roots), and collated the Ts' e-yan hai-ching {176) (A book on mathematics dealing with spherical triangles and algebraical equations by the 13thcenturymathe-matician Li Chih 11771). Hs Kuei-lin 75 (1781 (1778-1821) was Li Ju-cben's

111 See Ch'ou-jen dtuan dl. 49,647, and ECCP, 402,514-515. 70 ECCP 695--699. . 71 ECCP 695. 72 ECCP 695. 73 ECCP 144-145. " ECCP 434. 75 ECCP 472.

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brother-in-law. His works included the Li-t'ien yuan-1 tao-ch'iao 11711 (lnstrue-tion in algebrical equations) and the Suan-yu U&OJ (Arithmetic enlighten-ment).

V. Conclusion

Knowledge of mathemaUes in China during the time of Li Ju-chen bad already made eonsiderable progress due to the earlier introduetion of westernmathemaUes by the Jesuits as weil as the extensive research into ancient Chinese mathematics. Many books and translated works bad been produeed, while the study of mathemaUes bad beeome quite popular with scholars. Amongst Li Ju-chen's teacher and friends were a good number of anonymaus mathematicians. Natural scienee was relatively negleeted, but there was also in existenee some knowledge in this field which was intro-dueed from the West.

The mathematieal problems and principles of seienee mentioned by Li Ju-chen in the Ching-hua-yan arenot very advaneed eompared to the know-ledge of bis friends and eontemporaries just meritioned. However such referenees give a very truthful narrative of the general interest of scholars in the early 19th eentury in China. They adopted a scientifie attitude in their aeademic pursuits, not only in depth, but also in width. Many scholars, therefore, invariably touch on mathemaUes in -their study. Li Ju-chen's Ching-hua-yan is a refleetion of general attitude of Chinese schalarship of bis time.

* I wish to thank Professor Ho Peng Yoke of Griffith University, Brisbane, for his

kind suggestions, and to Professor Colin Mackerras of Griffith University, Professor Wolfgang Franke of University Hamburg, Mr. Ang Tian-se of University of Malaya, for reading through the draft of this paper.

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236