(Received 13 January, 1978) China Has a Long-Standing Algebraic Tradition. the Earliest Extant Chinese Mathematical Text Is

THE JADE MIRROR OF THE FOUR UNKNOWNS - SOME REFLECTIONS*

J. Hoe

(received 13 January, 1978)

China has a long-standing algebraic tradition. The earliest extant Chinese mathematical text isZhoubt the suanjrng ~ The arithmetical classic of the gnomon and the circular paths of heaven, thought to be of the Han period (206BC to 220AD), but almost certainly containing material dating back some thousand years. Although primarily a text on astronomy, it opens with a discussion of the theorem of Pythagoras, stated in essentially algebraic form. The second oldest extant mathematical text, also of the Han period,Jiiizhang the su&nshti - Nine chapters on the mathematical art, contains a chapter on the solution of systems of linear equations in up to five unknowns, using elementary column operations to reduce the matrix of coefficients to triangular form. Systems of indeterminate equation are also discussed, and the text contains the earliest known reference to negative numbers. The Chinese algebraic tradition reached its height in the latter part of the Song dynasty (960 to 1279AD) and the early part of the Yuan dynasty (1279 to 1368AD). The work of Zhu Shiji6 is the culmination of this tradition. Only two of Zhu Shijie's works are extant:

1. Suarucue qvmeng 'Jx„ - Introduction to mathematics, 1299AD;

2. Styuan yujian rS? Aj i ^ - The jade mirror of the four unknowns, 1303AD,

* Invited address delivered at the Twelfth New Zealand Mathematics Colloquium, held at Wellington 9-12th May, 1977.

Math. Chronicle 7(1978) 125-156.

125 Sarton ([1] Vol • III, p. 703) says of theS'Lyuan yu^j'ian "that it is "the most important Chinese book of its kind, and one of the outstanding mathematical books of mediaeval times". However, as late as 1968, Marco Adamo [2] affirms that Chinese mathematics is a collection of ideas copied from the Greeks and Indians. He states that the reasoning is carried out in the language of discourse, and that the meaning of Chinese mathematics is as incomprehensible as its ideas, methodology and applications are incomplete, and that only in arithmetic are there some rare signs of originality. Which of these opposing views should one accept? In order to be able to judge between these opinions, one needs to be able to look at concrete examples of the work done by Chinese mathematicians. Unfortunately, until recently, few translations of Chinese mathematical texts have been made, so that it has not been easy for mathematicians, who do not read Chinese, to do more than note the judgements made by others, while wondering at the wide divergence in the views expressed. The situation is now beginning to change. For instance, studies of two Song mathematical texts have recently been made: one by Libbrecht [3] on theShhshU jiZzhdng 1247 by Qin Jius'hao > and the other by Lam Lay Yong [4] on the Y&ng Em suctnfft of 1274/5 by Y£ng Hui . In this address, I wish however, to confine myself to some reflections on the work published in 1303 by Zhu Shijie, entitledThe jade mirror of the four unknowns [5], in the hope of giving you some idea of how Sarton and Adamo can have come to hold such opposite views. First, how true is Sarton’s viewThe that jade mirror of the four unknowns is the most important Chinese book of its kind and one of the outstanding mathematical books of mediaeval times. In order to answer this, we need to know what other books on mathematics existed in China and something of their content. We do indeed have some idea of what books existed ([6] pp 18-53), for the Chinese have kept fairly complete historical records, and the twenty-four official histories, compiled over a period of some 2000 years, contain, in addition to the usual material one would expect to find in official annals, extensive bibliographies, listing what were considered to be important works. For example, the bibliography of theHbn Shu - History of the Han dynasty3 covering the period from 206BC to 9AD, lists 21 titles on astronomy and 18 on the calendar, of which 2 are specifically on mathematics ([7] pp 1763-1766). The bibliography of theSui ShU fi| - History of the Sui dynasty, which lasted from 581 to 618AD, lists 97 titles on astronomy and ICO on the calendar of which 27 deal with mathematics ([8] pp 1017-1026). For the Tang dynasty (618 to 907AD), there are two official histories. In Jitithe Tdng Shu - Old history of the Tang dynasty, 19 works on mathematics are listed in the bibliography ([9] pp 2036-1039), whereas theXZn Tdng ShU - New history of the Tang dynasty lists 35 works of mathematics among the 75 calendrical works listed in the calendrical section ([10] pp 1543-1548; [ll] p 35). In 656AD, ten mathematical works were edited into a single collection for the use of aspiring officials preparing for the state examinations, but already, one of the important works, praised in Historythe of the Sui dynasty had been lost. This was a work on astronomical calculation known as Zhut Shu by a mathematician Zu ChongzhI who lived from 430 to 501AD, and who today is known largely for having calculated the value of it as lying between 3.1415926 and 3.1415927. He has been honoured today by having a crater on the moon named after him. The number of titles listed in these official bibliographies gives us some idea of the extent of mathematical activity up to the end of the Tang dynasty, and also of the importance accorded this type of activity. Unfortunately, we have almost no idea of what was contained in these books, apart from what survives today in the collection known as theSuanjZng shishu - The ten mathematical classics. These include the two Han texts already cited,Zhoubi the suanjing and the Jiuzhang suanshu. Even so, not all the texts in this collection

127 are complete. After the Tang dynasty, mathematical activity in China went into decline. But in 1084 AD, a printed edition of the ten mathematical classics appeared - again for the use of candidates for the state examinations. These were later copied into an encyclopaedia of all knowledge compiled in some 22,000 volumes during the fifteenth century, and known as theYdng L& dhdi&n fKjfc. Z L & - The Yong Le Encyclopaedia. 36 chapters in this encyclopaedia (Chapters 16 329 to 16 365) were devoted to mathematics. Unfortunately, only about a hundred chapters of the encyclopaedia survive today, of which only two of the chapters on mathematics (Chapters 16 343 and 16 344). These are kept in the Cambridge University Library ([6] p32). A number of bibliographies exist for the Song period. From these, Li Yan ([1l] pp 87-90) lists some 70 titles on mathematics. There also exists a bibliography entitled theSut Chu Tang Catalogue , compiled by Ydu M&o (1127-1194 AD). 95 titles are listed in the mathematical section ([6] p 40). Today, only eight works survive from the post- Tang period, or more specifically from the late Song - early Yuan period, by four mathematicians in all. They are:

1. Qin Jiushao Shushu jiuzhcng Mathematical treatise in nine sections, 1247

2. Li Ye Ceyuan haijing "SA if] & Sea mirror of circle measurements3 1248

Ylgu yandurn * -£ if New steps in computation, 1259

3. Y£ng Hul Xifingjie jiuzhang su&nfa Detailed analysis of the mathematical rules in the 'Nine chapters 1261

Riyong suanfa 0 ^ ^ Computing methods for daily use, 1262 Y6ng E va , su.hr.fa %% ^ 'Ju Yang Hui rs computing methods, 1274/5

4. Zhu Shijie Suanccue qzmeng 'jjf Introduction to mathematics, 1299

N ^ *s - Siyuan yujian \tT] Jl The jade mirror of the four unknowns, 1303

We see then, that Sarton was judgingThe jade mirror of the four unknowns in relation to the very small number of mathematical texts that survive today, and that we have no way of telling what was in the books that have been lost. We find ourselves in the position of trying to determine and assess the nature of Chinese mathematical activity from studying the handful of school text-books that survive. It is in this context that we must judge Adamo's view that Chinese mathematics shows little originality. One does not normally expect mathematical originality from a text-book.The ten mathematical classics were, as already mentioned, compiled for the use of students preparing for the civil service examinations, while the preface to the 1303 edition of The jade mirror of the four unknowns specifically praises Zhu Shijie for his teaching ability, and makes no claim for great originality. A text-book should, of course, be clear. If it is true then, that mathematical argument in Chinese was carried out in the language of discourse and that the meaning of Chinese mathematics is incomprehensible, as Adamo suggests, then these books will have failed in their primary aim. The idea that the texts are in the language of discourse, and that mathematical reasoning as we know it is not therefore possible, arises, I think, from a misunderstanding regarding the nature of the Chinese written language. This misunderstanding is still rather widespread today, and it has been argued that it is impossible for China today to assimilate the concepts of modern science because the language forces her to try to do so through the seventeenth century language of discourse ([l2] p 437). It has even been seriously suggested that the Chinese cannot hope to rival Europeans in science,

129 engineering or scholarship until they abandon their ideographs ([13] p 26). For example, I have heard it contended that it is impossible to express a term such as "electron microscope" in Chinese without saying each time the equivalent of "a system of lenses and mirrors for revealing the shape of infinitesimal things by the use of particles of electricity". The flaw in the argument lies in the fact that this long-winded English phrase is expressed in Chinese in just five monosyllabic characters, diccnzt xiarriweijing ^ 3p % \ , in which

jing i f o means a system of mirrors and

wei tnL infinitesimally-sized things,

xian 3 . to reveal,

dian electricity,

Z 1 particle, so that the phrasedianzi xiartweijing really does mean "a system of lenses and mirrors for revealing the shape of infinitesimal things by the use of particles of electricity", but it takes no longer to pronounce than the six syllables of "electron microscope" in English. It is possibly true that the ideographs take longer to write than does "electron microscope" in English, but I am sure also, that anyone able to read Chinese will confirm that the ideographic form of the phrase conveys the meaning with a clarity and directness which a phonetic script cannot hope to rival. From this, you can get a preliminary idea of how the monosyllabic ideographic script makes it possible to express ideas in Chinese in a very concise form. This is further helped by the fact that Chinese words are uninflected, i.e. they do not change in form according to their grammatical function. For example, nouns do not vary according to person, tense or mood. These characteristics of the language are exploited in the ancient Chinese mathematical texts. To illustrate, let us take an actual exampleThe jadefrom mirror of the four unknowns. I have already mentioned that it is a text book. It is a text-book designed to teach the student, how, starting from given hypotheses, to set up either: a polynomial equation in one unknown, or: a system of polynomial equations in up to four unknowns, and then by a process of elimination to reduce it to a single polynomial equation in one unknown. It begins with four illustrative problems - illustrative in the sense that some brief hints are given for their solution. These four problems are then followed by 284 exercises for the student to practise on. Let us take the second of these illustrative problems. By going through it in some detail, we will be able to see how the inherent symbolism of the Chinese language is made use of in mathematical writing. The text of the problem is as follows:

/>- A % ii Jh A t A. ^ * V jln you gu mu jian xian jiao jiao yu gu cheng gou deng

a 4 V zi % % X'0 fe A 3¾ /¾ zhi yun gou mu jia xian jido he yu gou cheng xian tong

i'O XL fL •fa * • V wfen gu n he

The problem, which is stated in 30 monosyllables, is about a right- angled triangle. The Chinese thought of a right-angled triangle as standing like this

^L(gu)

(gou)

131 with a vertical side - corresponding to the vertical part of a gnomon - represented in Chinese by the monosyllabic character (gu), and a horizontal side - corresponding to the shadow cast by the gnomon - represented in Chinese by the monosyllabic characterifj (gCu) . The hypotenuse of the right-angled triangle is denoted in Chinese by the monosyllabic character (xi£n). The arithmetic operations used in this problem are denoted by the characters:

^ j ian minus

jia plus

cheng multiply.

The character ji§.o means either "difference", in general, or, according to context, "the difference in length of the two sides con taining the right angle". Similarly, jfcz h£ means either "sum" generally, or according to context, "the sum of the lengths of the two sides containing the right angle". The character ^ means in general "raised to some power", but here the context requires that it be interpreted as specifically "raised to the power of two". The phrase ...... y u...... deng means "is equal to" and is synonymous with the phrase ...... Is} yu...... tong, meaning "is the same as". jin you means "if we have that" and introduces the first hypothesis, while H % zhi yun, meaning literally "only say", i.e. "if in addition we have only that", introduces the second hypothesis, j*) wen means "ask" fij, ji he means "how much?" If now, knowing the meaning of each character of the text, we translate the problem in the same way as we translated the phrase for "electron microscope", we would get something like this;

Suppose we have that A" & len6th °f the vertical side of a right-angled ■'"T*' triangle, raised to the power of two minus the difference between the length of the hypotenuse and the difference between the lengths of the two sides containing the right-angle ii] is equal to rt* the length of the vertical side of the right-angled triangle ^ multiplied by ^ the length of the horizontal side of the right-angled triangle, * and suppose we also have that the length of the horizontal side of the right-angled " 'Y triangle raised to the power of two

3« Plu s the sum of the length of the hypotenuse and the difference si ft 4» between the lengths of the two sides containing the right-angle ^ is the same as j the length of the horizontal side of the right-angled ^ triangle multiplied by 5¾ the length of its hypotenuse, jiQ find the length of the vertical side of the right-angled ML triangle

The original 30 syllables in Chinese have been translated into over 200 syllables in English, and it is clear that, like the "electron microscope" example, the translation gives a completely false idea of the spirit of the Chinese original, but, what is worse, it also

133 obscures the meaning in a mass of verbiage. If, instead of translating

instead At by the symbol a ij by b '61 by o

by - /> by + by X

by - or a - b % i * by + or a + b % by the index 2

b y = h .. ...» ... :¾ b y = K b y "if" 5 a by "and if" • V M .. by "find", then by formally substituting these symbols for the Chinese characters, we get as our translation:

If a 2 - [c - (a - 20) = a x b

^ M \ lA 3¾ fe. -¾ - ¾ ^ ^ ^ and if b 2 + {c + {a - b)) = £ x e .,

c? find a, f» ] .'£L However, such a translation is also a betrayal, for not only does it make the symbolism of the Chinese appear much more advanced than it really is, but it also makes the meaning clearer than it really is, by removing the ambiguities which in the original Chinese have to be resolved from the context. A translation which gives a much better idea of what the Chinese really sounds like, without obscuring the meaning, can be achieved by translating the passage into a telegrammatic style, similar to that used in modern programming languages. This can be done by translating 4 1 by the symbol ALT by BASE by HYP

by minus by plus by times

by DIFF by SUM by squared "V

Jq .... f b y equals ISJ .... 1¾) b y same as

by if ,*? a by and if 1¾.... /L'fcj by find

This then gives the following translation:

If ALT-squared minus HYP-DIFF-DIFF equals ALT times BASE

and if BASE-squared plus HYP-DIFF-SUM same as BASE times HYP, ® ^ ^ f 3¼ ^ A. 1¾ find ALT.

1*1 ^

135 This semi-symbolic translation is not only much more faithful to the original Chinese in both meaning and spirit, it also has the added advantage that it enables us to reconstruct the kind of steps in reasoning which the Chinese of the period must have taken to solve their algebraic problems, even though they had no symbolic notation like our modern one, but relied solely on the symbolism inherent in the Chinese language itself. To see that this is so, let us have a look at the problem in more detail. We have in effect three equations:

a2 - (

b 2 + (c + (a - b)) = be

a2 + b2 = a2 from which we wish to solve for a. In the hints for the solution, Zhu Shijie suggests taking a as one unknown x , and b + a as the other unknown y. He then says, in effect:

"Obtain the equations:

x 3 + (2x + x 2)y + (-2 - x)y2 = 0

and x 3 + 2xy + (2 - x)y = 0,

and eliminate y to obtain:

-8 - 2x + x 2 = 0, whence x = 4

Of course, the equations were not written like this. How then, were they written? All the computations were done on a counting board, on which the digits from one to nine were represented by counting rods placed either as: i il II! ill! II T T ¥ ¥ or as: ^ so that 23, for example could be represented as = 1 1 1 , to avoid confusion with The decimal place-value system has existed in China since at least the 15th century BC, and a number such as 48576, for example, was represented as iiityniuj- On the counting board, a zero is indicated by a blank space. This was also done in the texts, at first, but later, from the 13th century AD on, the circular symbol for zero was used. At first, red rods were used to represent positive numbers, and black rods for negative numbers, but later, a negative number was indicated by placing a rod diagonally across the last digit. A polynomial was represented on the counting board by placing rods representing what we would call the coefficients of the polynomial in different rows. Thus the polynomial which we would write as:

-3888 + 729a;2 - 81a:3 - 9 ^ + a:5

is represented by placing the coefficients as follows:

-3888 & 0 0 729 or in rod-numerals T “un -81 m \ -9 1st 1 li where the symbol (tai) indicates which is the constant term. Rows below ^ are used for the coefficients of the positive powers of x, while rows above are used for the coefficients of the negative powers. The counting board notation did not distinguish between the polynomial A(x) and the corresponding equation A (~) = 0. Which is meant, has to be ascertained from the context. It is clear that what we have just described is the "detached coefficients" notation which we use today in elementary algebra. The Chinese not only developed the rules for adding, subtracting and multiplying such polynomials on the counting board, but also developed what is essentially Horner's method

137 for obtaining a numerical solution to a polynomial equation, except that the computations, instead of being done with pencil and paper, were carried out on the counting-board. This method of notation was subsequently extended to deal with polynomial equations in two unknowns. This was done by placing the rods in a two-dimensional array:

Columns to the right of jk. were used for negative powers of y, while rows above £ were used for negative powers of x. We have in effect a two-dimensional coordinate system for the coefficients of x y^ :

y <------

v X It is easy to develop the rules for adding, subtracting and multiplying such polynomials on the counting board. But, in addition, the Chinese mathematicians developed from this notation, a simple way of eliminating one of the unknowns from two such polynomial equations. For, if we number the columns from the right, we see that: the 1st column represents a polynomial ^qG*0 in x only, the 2nd column represents a polynomial the 3rd column represents a polynomial A 2 (x)*y2 , and so on. Suppose then we have two equations in which y appears only to the first degree, i.e.

4 0(aO + A 1 (x) y = 0 and S Q(a:) + (x) y = 0.

Then, eliminating y gives:

4 0(aO B l Or) - £ 0Or) A l (x) = 0 (a)

On the counting board, each polynomial is represented by a two-column array:

A 1(x) A q (x ) and

so that the expression (a) is computed by taking the product of the polynomials represented by the inner columns, and subtracting the product of the polynomials represented by the outer columns. This is precisely the algorithm used by Zhu Shijie. Moreover, suppose now we consider two three-column arrays, i.e. polynomial equations of the form:

V * ) + y + yz = °> ( i) and B 0(aO + (*) y + ^2(x) y2 = 0. (ii)

139 Then, we can eliminate the terms in y2 to obtain an equation of the form:

C M + Cj (a?) 2/ = 0 . (iii)

We can then apply (iii) to (i) to obtain another equation of the form:

D q (x ) + D l(x) y = 0. (iv)

The "inner-outer column" algorithm can then be applied to (iii) and (iv) to eliminate y. Alternatively, instead of eliminating the terms in y 2 in (i) and (ii), we could have begun by eliminating the terms ^ 0(aO and (a:) without y, and then, after dividing out by y, we would still obtain an equation of the form (iii), and so proceed as before. From the point of view of the counting board, what this means is that we have two three-column arrays:

A 2 {x ) A l (a:) i40(ar) and B2 0c) Bl (x) Bq (¢)

(i) (ii)

and that we use these to eliminate one of the outer columns of (ii) to obtain a two-column array:

C l (x) C Q (a:)

(iii)

We then apply (ii) to (i) in order to eliminate one of the outer columns of (i), and so obtain a second two-column array: Civ)

The final elimination is then effected by applying the "inner-outer column" algorithm to arrays (iii) and (iv). This process of elimination is known in Chinese as the huyin tongfen ft method, roughly translatable as the "hidden reduction" method. It was through counting- board techniques such as these, and with the help of the inherent symbolism of the Chinese language, that the solutions were obtained.

Let us go back to the first hypothesis of the problem cited. In semi-symbolic translation, this was: ALT-squared minus HYP-DIFF-DIFF equals ALT times BASE. The problem is to represent this on the counting board. Zhu Shijie, in his hint for the-solution, suggests taking ALT as one unknown, and BASE-HYP-SUM as the second unknown. In the counting board notation, this leads to the initial configurations shown in diagrams A1 and A2. In order to set up the equation given in the first hypothesis on the counting board, we need to obtain representations for:

(i) ALT-squared (ii) HYP-DIFF-DIFF (iii) ALT and (iv) BASE

141 The configuration for (iii) is shown in diagram Al.

A1 (iii) ALT £ 1 & A2 BASE-HYP-SUM 1

—

i B1 (i) ALT-squared £ 1 0

; Cl BASE-HYP-DIFF * 0 . 0 ' ■ 1

Dl-Twice (iv) = 1 & 0 Twice BASE 0 -1 1 1

1 E2 (ii) HYP-DIFF-DIFF -1

-1 £ F2 ALT-squared minus 1 HYP-DIFF-DIFF 1 1 G1 ALT times £ twice BASE 1 0 0 |

! o 1 I -1 i i

1 ! -2 H2 Twice ALT-squared minus 1 | 2 HYP-DIFF-DIFF

! I 2 !

i II First -2 0 hypothesis !-i 2 0 !

2 0 ,

! -2 0 i ( -1 2 0 i

2 0

K1 IF "2 0 & 2 0 * K2 ALSO -1 2 0 “1 2 0 2 0 0 1 1 0 k L2 ALSO minus IF C 0 0 I ! "2 0 0 2 0 * M2 ALSO minus IF divided 0 by 2 “1

143 - * N2 ALSO minus IF

I 2 0 j 0 I I ! -1

01 IF after ’ 2| * elimination of - i | * ■ i first column 2 i

PI IF -2 2 0 P2 ALSO -1 4 0

2 -1

Q1 Inner column I Q2 Outer column product 8 product

** 2

; ! 1

i R2 Final equation i "8

The configuration for (i) is obtained by moving the 1 in the ALT array one row down, as shown in diagram Bl. Zhu Shijie's hint suggests that instead of taking the theorem of Pythagoras in the usual form:

ALT-squared plus BASE-squared equals HYP-squared, we take it in the form:

BASE-HYP-DIFF equals ALT-squared divided by BASE-HYP-SUM. The student would be familiar with this form of the theorem from his previous study of the Zho'ubi suanjing. Dividing ALT-squared by BASE-HYP-SUM is achieved by moving the ALT-squared array one column to the left, as in diagram Cl. This therefore gives the array for BASF.-HYP-DIFF. By subtracting this from BASE-HYP-SUM, or equivalently by adding BASE-HYP-SUM to the negative of BASE-HYP-DIFF, we obtain the array for twice BASE, as shown in diagram Dl, i.e. for twice (iv). To obtain the array of (ii), we note that HYP-DIFF-DIFF equals BASE-HYP-SUM minus ALT. This leads to the array shown in diagram E2. Having thus obtained counting board representations for (i), (ii), (iii) and twice (iv), we are now ready to apply the first hypothesis. The left hand side of the equality, ;

ALT-squared minus HYP-DIFF-DIFF, requires that we add the array for ALT-squared to the negative of that for HYP-DIFF-DIFF, giving the array shown in diagram F2. Twice the right hand side of the equality

ALT times twice BASE, is accomplished by lowering all the rods of Dl by one row, as shown in diagram Gl. Hence, to apply the first hypothesis, we first multiply the array F2 by 2 to give the array of diagram H2, and then add it to the negative of array Gl, giving the array shown in diagram II. Since this is the representation of an equation, we can move the : one column to the right, this being equivalent to multiplying the equation through by y, and so obtain the array shown in diagram Jl. Let us call this the IF array, as Zhu Shijie does, since it is the counting board representation of the first hypothesis. By similar arguments, we can easily obtain a counting board representation for the equation implied in the second hypothesis, which we can, following Zhu Shijie, label as the ALSO hypothesis. The result is shown in diagram K2. We now have two arrays to which we can apply the "hidden reduction" method described earlier. It is clear

145 that we can eliminate the first column of the ALSO array by subtracting the IF array from it, as in diagram L2. The equation represented by this array remains the same if we divide through by 2, as in diagram M2. By comparing this with the IF array in Kl, we see that we can use it to eliminate the first column of.the IF array by first moving the rods of diagram M2 one row down and one column to the right, this being equivalent to multiplying the equation through by x y , (see diagram N2), and then adding the result to Kl. The result is shown in diagram 01. The equations represented by 01 and N2 are equivalent to those shown in diagrams PI and P2 respectively. The product of the inner columns is shown in diagram Ql, and the product of the outer columns is shown in diagram Q2. Application of the "inner-outer column" algorithm leads to the equation shown in R2, viz:

-8 -2x + x 2 = 0.

From the above description, we can see the reasoning whereby the Chinese, by relying on the inherent symbolism of the Chinese language and the techniques of the counting board, were able to arrive at equations of this sort from given hypotheses. Clearly, a generalisation of the counting board notation for polynomials to 3 unknowns is not practicable without a three-dimensional counting board. Nevertheless, the Chinese did try a partial generalisa tion. This was done by using the columns in the two-dimensional array to the left of to represent a third unknown. Zhu Shijie then tried to extend the method to 4 unknowns by using the rows in the two- dimensional array above the £ to represent the fourth unknown. Of course, neither of these extensions to three and four unknowns led to anything worthwhile, but they do at least point to the need felt by the mediaeval Chinese mathematicians to generalise, a characteristic of mathematical thought. I hope that the fairly detailed discussion of the problem in two dimensions will have shown that Adamo's contention that Chinese mathematics is written in the language of discourse, and that its meaning is incomprehensible, and its ideas and methods incomplete, does not do justice to what was achieved, at least not to the achievements as reflected in The jade mirror of the four unknowns. What about his view on the incompleteness of the applications? How useful was the material taught in this book? The four illustrative problems are followed by 284 exercises, grouped into three books, into a total of 24 chapters. Just over a hundred of the exercises are problems on the right-angled triangle and on rectangles, similar to the one which we have just discussed, and consequently of no great practical or theoretical interest from the point of view of providing us with information about triangles and rectangles. They are designed solely for training the student in setting up equations on the counting boards, starting from hypotheses stated in the normal Chinese written language, and then in applying the elimination procedures. To each of these exercises, Zhu Shijie gives a hint, in which he indicates what to take as the unknown, and then gives the final equation that has to be solved, together with one root of the equation. From the numerical answers, there is no doubt that the problems were constructed to give nice, simple answers - just as in our modern text-books. In some cases, he names the method to be used for solving the equation, but no details of these methods are spelt out, which again indicates that the mechanical solution of the equation is not the main point of the exercise, but rather, the setting up of the equation. We all know that this is an aspect of mathematics that cannot really be taught, but that it is a process learned only by repeated and often painful experience. That Chinese mathematicians were aware of this, is indicated in the preface, where we read that:

147 The gentleman who would understand mathematics,

In my opinion,

Should try to construct detailed solutions.

Only then will his knowledge be complete, /«

And not partial.

The remaining 180-odd problems o£ the text appear at first sight to be problems involving applications. There are, for instance, a large selection of problems on areas of fields and ponds, volumes of granaries, lengths of rolls of cloth, etc., seemingly involving problems of distribution and of tax-collection. There are problems on the construc tion of city walls, excavation of moats, measurement from afar of the height of city walls, forts, observation platforms, and so on. Other problems reatuire the number of pieces of fruit stacked in variously shaped heaps, the amount of grain in sheaves, the number of arrows in differently shaped bundles. There are problems concerning the organisa tion of men for corvee duty, for instance, the number of men to be employed per day in various construction projects, the amount of wages to be paid, the amount of grain needed to feed them, and so on. However, when one looks at the problems in closer detail, it soon becomes apparent that they are in fact, of little practical value. The field, ponds and granaries turn out to be of the most unlikely geometrical shapes. The piles of fruit, sheaves of grain and bundles of arrows are stacked with quite remarkable mathematical precision. In fact, the problems, like those in our elementary text-books today, were designed to make the students use his knowledge of geometry and of algebraic techniques such as series summation, finite differences, solution of systems of linear equations, etc., in order to practise his mathematical reasoning, and so are merely vehicles for training the student in mathematical thinking. But they are dressed up in seemingly practical vocabulary to motivate the student, or possibly to deceive the bureaucracy into keeping mathematics in the curriculum! Although the problems are of little practical interest, they are useful from a historical point of view, for, because Zhu Shijie... in each exercise gives the equation to be solved, the problems provide us with useful material for detective work, in trying to assess the kind of mathematical knowledge that the thirteenth century student was assumed to possess, and in trying to discover how, with the knowledge available to them, they arrived at their results. Working through the problems to obtain the equations given by Zhu Shijie shows, for instance, that the student was expected to know the standard formulae for the area of triangles, rectangles and other polygons, including Heron's formula for the area of the triangle a = Vs(s-a) (s-b) (s-c), formulae for the volumes of frusta of cones and of pyramids, and that incorrect formulae were used for the volume and for the surface area of a sphere. One of the interesting formulae used by the Chinese since the first century AD is that for the area of a segment of a circle:

A = ½ s(s + o') o in which s is the length of the sagitta and o the length of the chord. Interestingly enough, the Chinese terms for the sagitta (shi) and chord ^ (xi£n) have exactly the same meaning, as does the Chinese term for arc (gQ). Zhu Shijie makes use of this formula in a number of his problems, but in one problem, he uses instead the approximation

A = h s(s + o) + - -g1 3- c2 .

How this would have been obtained is not clear. I have mentioned a number of times my belief that the text is designed primarily for training the student in transforming hypotheses expressed in ordinary language into equations on the counting board. What we might call in our modern jargon - the mathematisation of hypotheses. This implies that the Chinese were more interested in

149 logical reasoning than they have often been given credit for. This concern for reasoning is not confined to The jade mirror of the four unknowns. Allied to this is a concern for clear definitions. For instance, in the MojZng ^ , a classical work ascribed to the philosopher Mozi -3" (c.479 to 381 BC), amongst a list of definitions, we also find definitions for mathematical terms such as ([14], p 62 ff [6], pp 93-94):

parallel ^ I tliat which is the same hei8ht

straight line , that with three points

equi-distant /*]& that which when compared has same exhaustion

centre '“f that which is equi-distant

circle IS - +1*1-M l that which has centre equi-distant

These definitions may not be very precise from a modern point of view but they do indicate that the Chinese mathematicians felt the need for definitions, as indeed did the Chinese philosophers. The first dictionary compiled in China dates from the first century BC. Once the first extant Chinese mathematical text, the Zhoubi suanjing became a set text for the examinations, it became the object of many commenta tors. One of these, Zhao Junqlng (3rd century AD), not only gives definitions of some of the important terms used, but also gives what must, I think, be called a proof of the theorem of Pythagoras. Translated into semi-symbolic language, the text says ([15], p 18): According to the diagram, multiply BASE by ALT to get two RED results.

Double to get four RED results.

Take BASE-ALT-DIFF squared to get YELLOW result

Add one DIFF result.

But same as HYP result.

In other words:

ab = 2 red areas 2cib = 4 red areas {a - 2?)2 = 1 yellow area But 2ab + (a - b)2 same as a2 .

In later editions of the book, the diagram was drawn for a 3-4-5 triangle. This has led some writers to conclude that the Chinese were familiar only with the 3-4-5 special case. But, as can be seen from the text, the proof makes no reference to any particular lengths. Indeed, a later commentator, Zhfen Lu£n (6th century AD), thought it necessary to add in explanation "for example, if we take BASE as 3, etc.,...." ([15], p 18). Some of the formulae given by Zhu Shijie in his hints are relatively complex, particularly in the section on the measurement of heights and distances. For example, one of the problems is about measuring the height of an observation post on a city wall by the use of two surveyors' posts. The geometrical figure involved is as follows:

151 E

In the solution, the length ZD is given by the formula:

(AAf - PjPj’ - A'A") ?1?2 ZD = ------P B - P A 2 1 a result which is easily shown using similar triangles. The question that arises is, were answers like this given so that the student could check up on the formula to apply, or were they given, like those in our text-books today, to provide him with a formula against which he could check his own reasoning? How we answer this question depends on whether or not we accept the view that Zhu Shijie’s text is a text-book on mathematical thinking. An aspect we have not yet touched on is, how true is Adamo’s view that Chinese mathematics is copied from the Greeks and the Indians. There is no doubt that there were periods in Chinese history during which China and India were in close contact, particularly during the period of Buddhist influence. It is also undeniable that many Indian mathematical works were translated into Chinese ([6], pp 146; [1l], pp 52-58). For example, the History of the Sui dynasty [8] lists a number of Brahmin works on astronomy and mathematics, but, unfortunately, these have all been lost. The nev) history of the Tang dynasty [10] lists four Indian astronomers, two of whom became President of the Chinese Astronomical Board. However, in the books which are extant, up till now, no single problem or rule of solution can be positively traced as being of Indian origin. On the other hand, there are problems in the Chinese extant texts, which appear at a later date, in Indian texts. For example, the proof of the theorem of Pythagoras in the form given by Zhao Junqing in the 3rd century, appears in a work by Baskhara in 1150. The segment formula cited earlier, which appears in the Han text of the 1st century, the Nine chapters on the mathematical art, appears in a work by Mahavira in the 9th century. One of the indeter minate problems of the Sunzt sudnjing -1ne arithmetical classic of Sunzi 3rd century AD, appears in Brahmagupta in the 7th century. Two geometry problems by Liu Hui $ jbj £ , the 3rd century commentator of the Nine chapters on the mathematical art3 are the same as in Mahavira, 9th century. The decimal place-value system of notation, in use in China since at least the 15th century BC, appears in India at the earliest in the late 6th century, one of the periods of close Chinese-Indian contact. Thus, there are reasons for believing that the interchange was not all one way. Later there was also Islamic influence on China through Baghdad and through the Mongol conquest of China from 1279-1368. Unfortunately, little has been studied on Chinese intellec tual contacts with India, Persia and other Islamic countries. Zhu Shijie is the last of the great Chinese mathematicians of the past whose works are extant. After him, mathematics in China went into decline, and in the 16th century, the Jesuits had to be called in to help in the reform of the calendar and other mathematical activities. The reasons for the rapid decline in mathematical activity have been sought in the economic and social conditions of mediaeval China, and in the prevailing philosophical attitudes of the period. However, the decline can also be partly ascribed to the kind of mathematics studied and the methods employed. We have already seen, for example, that the counting board notation led to a dead end, because it could not be extended beyond two dimensions. Moreover, the counting board procedures not only became increasingly complex, but, as in modern abacus computa tions, the intermediate steps are not retained, with the result that it is difficult to check o n e ’s calculations, short of re-working the whole

153 problem. We saw also that the problems to which the techniques were applied were of little practical use. The result of all this is that once the method is lost, it was not only difficult to recover, but there was little incentive to do so. So that it is not surprising that, after the Mongol conquest, Chinese mathematics did not revive, although the tradition remained alive in Japan and Korea until the 17t.h century. It is possible too, that the semi-symbolic nature of the Chinese language itself acted as a hindrance to progress. For though it proved to be an advantage at first, giving the Chinese mathematicians a lead over their contemporaries, they were not led to try to develop a true symbolism as was done in the West, where the alphabetic, inflected languages really did make reasoning in the language of discourse difficult, and hence encouraged the search for a truly symbolic notation.

To end this talk, I should perhaps briefly mention why a text-book on mathematics should be called a "jade mirror". The word "mirror" in Chinese conjures up the idea of something in which we see ourselves reflected and therefore, something in which we can see both our blemishes and our good points. It came then to symbolise something from which we can learn lessons,particularly lessons from the past. The significance of "jade" is explained in the preface of the work, where, in an oblique reference to the motion of the rods used in computation, we read:

It is called jade, because it resembles Zhu Shijie's powerful art. In motion, its sound is clear, far-reaching and prolonged. At rest, its inner qualities are discernible from every side. And there is nothing hidden or obscure. REFERENCES

1. G. Sarton, Introduction to the history of science3 Williams §■ Wilkins, Baltimore, Vol. I (1927); Vol. II (2 parts)(1931); Vol. Ill (2 parts)(1947-8).

2. M. Adamo, La matematica nell'antica Cina3 Osiris, 15(1968), 175-195.

3. U, Libbrecht, Chinese mathematics in the thirteenth century: The shu-shu chiu-chang3 MIT Press, 1973.

4. Lam Lay Yong, A critical study of The Yang Bui suanfa3 University of Malaya Press, Singapore (in press).

5. J. Hoe, Les syst&mes d'Equations polyndmes dans le Siyuan yujian3 M€moires de l'Institut des Hautes Etudes Chinoises, Vol. VI, Paris, 1977.

6. J. Needham, Science & civilisation in China (Vol. Ill, Mathematics § the sciences of the heavens § the earth), Cambridge University Press, 1970.

7. Hhn Shu > A # 3 History of the Han dynasty3 Zhonghua Shuju, Peking, 1975, 12 Vols.

8. Sui Shu j History of the Sui dynasty3 Zhonghua Shuju, Peking 1973, 16 Vols.

9. JiU Tdng Shu IQ $ 3Old history of the Tang dynasty3 Zhonghua Shuju, Peking 1975, 16 Vols.

10. XZn Tang Shu 3 New history of the Tang dynasty3 Zhonghua Shuju, Peking 1975, 20 Vols.

11. Li Yan h-ffi , Zhdngguo suanxuS shr History of Chinese mathematics3 Commercial Press, Shanghai, 1955 (First published 1937).

155 12. F. Bodmer, The loom of languages Allen § Unwin, 1943,

13. E.H. Sturtevant, An introduction to linguistic science, Yale University Press, 1947.

14. Gao Heng % , Mdjing jihoqudn The Canon of Mo3 edited and with commentaries s Taiping Shuju, Hong Kong, 1966.

15. Qian Baozong & ^ t SuhnjZng shishu, S fll +■ $ . The ten mathematical classics3 2 Vols. Zhonghua Shuju, Peking, 1963.

Victoria University of Wellington