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".