# The Jade Mirror of the four unknowns

*The Jade Mirror of the four unknowns*was written in 1303. Below we give some of Jock Hoe's work on this important book.

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Colloquium Address in 1977

Reviews of Jock Hoe's 1977 thesis.

Review of Jock Hoe's 1980 thesis Conference Address.

**1. Extract from Jock Hoe's address to the Twelfth New Zealand Mathematical Colloquium in 1977.**

**1.1. Jock Hoe, The Jade Mirror of the four unknowns - some reflections.**

*Mathematical Chronicle*

**7**(1978), 125-156.

Invited address to the Twelfth New Zealand Mathematical Colloquium, held in Wellington, New Zealand, 9-12 May 1977.

China has a long-standing algebraic tradition. The earliest extant Chinese mathematical text is the

*Zhoubi suànjing - The arithmetical classic of the gnomon and the circular paths of heaven*, thought to be of the Han period (206 BC to 220 AD), 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, the

*Jiuzhang suànshu*-

*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 l279 AD) and the early part of the Yuan dynasty (1279 to l368 AD). The work of Zhu Shijié is the culmination of this tradition. Only two of Zhu Shijie's works are extant:

*Suanxue qimeng*[Introduction to mathematics], 1299 AD, and

*Siyuan yujian*[The jade mirror of the four unknowns], 1303 AD

Sarton (

*Introduction to the history of science*, Vol. III (1947-48), page 703) says of the

*Siyuan yujian*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 [

*La matematica nell'antica Cina*(1968)] 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 [

*Chinese mathematics in the thirteenth century: The shu-shu chiu-chang*] on the

*Shushu jiuzhang*of 1247 by Qín Jiusháo, and the other by Lam Lay Yong [

*A critical study of The Yang Hui suanfa*] on the

*Yáng Hui suànfa*of 1274/5 by Yáng Hui. In this address, I wish however, to confine myself to some reflections on the work published in 1303 by Zhu Shijie, entitled

*The jade mirror of the four unknowns*[

*Les systèmes d'équations polynômes dans le Siyuan Yujian*], 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 view that

*The 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 ([J Needham,

*Science and civilisation in China*Vol III] pages 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 the

*Hàn Shii*-

*History of the Han dynasty*, covering the period from 206 BC to 9 AD, lists 21 titles on astronomy and 18 on the calendar, of which 2 are specifically on mathematics. The bibliography of the

*Suí Shu*-

*History of the Sui dynasty*, which lasted from 581 to 6l8 AD, lists 97 titles on astronomy and 100 on the calendar of which 27 deal with mathematics. For the Tang dynasty (618 to 907 AD), there are two official histories. In the

*Jiù Táng Shu - Old history of the Tang dynasty*, 19 works on mathematics are listed in the bibliography, whereas the

*Xin Táng Shu - New history of the Tang dynasty*lists 35 works of mathematics among the 75 calendrical works listed in the calendrical section. In 656 AD, 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 the

*History of the Sui dynasty*had been lost. This was a work on astronomical calculation known as

*Zhui Shù*, by a mathematician Zu Chongzhi who lived from 430 to 50l AD, and who today is known largely for having calculated the value of π 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 the

*Suànjing shishu*-

*The ten mathematical classics*, These include the two Han texts already cited, the

*Zhoubi suanjing*and the

*Jiuzhang suanshu*. Even so, not all the texts in this collection 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 the

*Yong Lè dàdian*-

*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). A number of bibliographies exist for the Song period. From these, Li Van (

*History of Chinese mathematics*, pages 87-90) lists some 70 titles on mathematics. There also exists a bibliography entitled the

*Suí Chu Táng Catalogue*, compiled by Yóu Mào (1127-1194 AD). 95 titles are listed in the mathematical section. 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:

- Qín Jiusháo, Mathematical treatise in nine sections (1247);

- Li Ye, Sea mirror of circle measurements (1248), New steps in computation (1259);

- Yáng Hui, Detailed analysis of the mathematical rules in the 'Nine Chapters' (1261), Computing methods for daily use (1262), Yand Hui's computing methods (1274/5);

- Zhú Shijié, Introduction to mathematics (1299), The jade mirror of the four unknowns (1303).

*The 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 textbooks 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 textbook. 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 ([F Bodmer,

*The loom of language*] page 437). It has even been seriously suggested that the Chinese cannot hope to rival Europeans in science, engineering or scholarship until they abandon their ideographs ([E H Sturtevant,

*An introduction to linguistic science*] page 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

*diànzi xianweijing*, in which

*jing*- means a system of mirrors and lenses

*wei*- infinitesimally-sized things

*xian -*to reveal,

*diàn -*electricity,

*zi*- particle,

so that the phrase

*dianzi xianweijing*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 example from

*The jade mirror of the four unknowns*. I have already mentioned that it is a textbook. It is a textbook 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.

**2. Reviews of the part of Jock Hoe's thesis published in 1977.**

**2.1. C Lee, Review: Les systèmes d'équations polynômes dans le Siyuan Yujian (1303), by John Hoe.**

*A bibliographical study of Western publications (1800-1985) on traditional Chinese Science*(Thesis, University of London).

*Les systèmes d'équations polynômes dans le Siyuan Yujian*(1303) by John Hoe [also known as Jock Hoe] (1977) begins with a quick glance at Western studies on traditional Chinese mathematics, followed by a short clarification of the Chinese title and an introduction to fundamental traits in Chinese mathematics. The main section of the book is taken up by the translation and delineation of the four important initial problems given in this textbook by Chu Shih-chieh. Through closely scrutinising and patiently elucidating these key problems, Hoe furnishes a detailed and finely crafted account of the methods by which Chu formulated and solved polynomial equations in 1, 2, 3, or 4 unknowns (which involve notations that are very difficult to comprehend, especially to those not acquainted with Chinese mathematics). Hoe then prepares a listing of the subject matter and topics covered in the many mathematical problems and exercises Chu provided for the student (e.g. problems involving areas and volumes, simultaneous linear equations, series). A distinctive characteristic of Hoe's translation and explication of the problems is his adoption of what François Hominal called "un langage 'semisymbolique'." The monograph concentrates on mathematical reasoning and problems. Little material and discussion is offered on Chu's life and work, on the role of the Ssu xüan yü chien in the Chinese mathematical tradition, or on social and cultural contexts.

**2.2. H Libbrecht, Review: A problem in the siyuán yùjiàn: The Jade Mirror of the Four Unknowns, by John Hoe.**

*Chinese Science*

**4**(1980), 65-68.

These are the first valuable investigations of the mathematical work of Chu Shih-chieh in European languages; the two articles previously available were much lower in quality. John Hoe, who teaches in the Mathematics Department of the Victoria University of Wellington, New Zealand, has written a profound study of the 'Ssu-yuan yü-chien'. He defended it as a doctoral dissertation at the University of Paris VII.

In the first chapter, the author elucidates the bases of Chinese algebra: the numeral notation, the operations on the counting-board, the 't'ien-yuan' method and the Pythagorean theorem. None of this is peculiar to Chu Shih-chieh, but it is necessary to explain this system to those who are not acquainted with the character of Chinese mathematics.

Chapter 2 is the most important part of the work. Indeed, the non-sinologist historian of mathematics, who until now had at his disposal only the explanation of Van Hée, could hardly understand Chu's algorithm. In this chapter, the author explains the four introductory problems in which Chu elucidates the method for solving non-linear simultaneous equations with one, two, three or four unknowns. The explanation is at first sight long-winded, but the detail is indispensable to full comprehension. The special notation used in Chu Shih-chieh's work is evidence of his great mathematical skill but it is also largely responsible for the stagnation of Chinese mathematics. It limited its own possibilities, and could never develop into a general mathematical notation.

Chapter 3 is devoted to the mathematical content of the work: numerical equations of higher degree, areas and volumes, series and interpolation, simultaneous linear equations, and a kind of proto-trigonometry - the usual topics of Sung and Yuan mathematical handbooks.

From the mathematical point of view, this is an extremely good study of Chu Shih-chieh's work. The explanation of the Chinese text is Cartesian in its clarity and the material is made accessible to non-sinologists.

The general conception of 'Les systèmes' calls for comment. The account of previous research (pp. 6-31) is concerned broadly with knowledge in the West of Chinese mathematics. Still Hoe has not related the work of Chu Shih-chieh to Sung mathematics. Even the connection of the Ssu-yuan yu chien with Chu's other work, the 'Suan-hsueh ch'i-meng', is not discussed. Nor does Hoe place the book he is studying against the cultural and historical background of the turn of the fourteenth century. What level had been attained in methods for solving non-linear simultaneous equations in other parts of the world at the same time, and later? The value of Chinese knowledge cannot be elucidated fully without comparing medieval European, Islamic, and Indian algorithms.

I was surprised to find neither bibliography nor index, so that references in the footnotes are not easily retrieved. The introductory sections of 'Ssu-yuan yu chien' are only partly translated in the text, and omitted from the appendices. Hoe does not mention the reprint of the Basic Sinological Series edition of the Ssu-yuan yu chien hsi-ts'ao (6 vols., Taipei: Commercial Press, 1968). In "L'algèbre" Hoe devotes nearly two hundred pages to the Chinese text. It is not a new edition, but the familiar 'hsi-ts'ao' recension transcribed in simplified characters. The point of this laborious "improvement" of the text is not explained.

These remarks do not detract from the admirably clear analysis in John Hoe's work. Chu Shih-chieh's work can now be incorporated alongside that of Ch'in Chiu-shao and Yang Hui in histories of mathematics; this is a substantial accomplishment. We can only hope that Hoe will publish a more general overview of Chu's endeavours, to prepare the way as Lam has done for a general and comparative study of Sung-Yuan mathematics. Of the four great mathematicians of the time, only Li Yeh has yet to be seriously studied in the West.

**2.3. B Poizat, Review: Les systèmes d'équations polynômes dans le Siyuan yujian (1303) par Chu Shih-chieh, by J Hoe.**

*Mathematical Reviews*MR0497474

**(58 #15816)**.

The author edits the first pages (the first four problems!) of a Chinese treatise published in 1303 AD, on the elimination in systems of polynomial equations of one to four unknowns. For each of these problems we will find the Chinese text, a transcription, a translation into a semi-symbolic language, which tries to reproduce the conciseness of the original text, and brings it closer to modern algebraic notations, then to a long series of mathematical comments. These comments occupy the greatest part of this book, which is a true introduction to Chinese mathematics of this period; the author explains in detail how the operations indicated in the solution of the problems were carried out using sticks placed on a counting board: this explanation is essential for the contemporary uninitiated reader. In addition, this work goes beyond the framework of Chinese studies, because in the opinion of the reviewer, the author's thesis that the symbolic character of Chinese writing provided mathematicians with a sufficient tool for their needs, and did not constrain them from forging a specific rating system (as was the case in the West), which ultimately limited the development of their science, deserves to be considered by all who want to study the symbolism of mathematical language.

**2.4. F Hominal, Review: Les systèmes d'équations polynômes dans le Siyuan Yujian (1303), by John Hoe.**

*T'oung Pao, Second Series*

**65**(1/3) (1979), 119-122.

This is the first serious study in a Western language of 'Siyuan Yujian', a treatise on algebra by Zhu Shi-jie published at the beginning of the 14th century. The work had been studied and annotated by Luo Shi-lin in the years 1823-1835, then it had been the subject, at the beginning of our century, of very questionable translations into French (van Hée) and into English (Chen Zai-xin and Konantz). Fortunately Jock Hoe has taken over the translation and commentary of this famous work in the history of Chinese algebra.

An introduction (pp. 1-40) positions the work presented in relation to the studies already published; the author underlines the small place occupied by Chinese mathematics in most general histories of mathematics. The first chapter (pp. 41-90) justifies the proposed translation of the work with the above title and specifies most of the mathematical knowledge assumed by Zhu Shi-jie in his work: the number system, operations on the counting board, expression of polynomials in one unknown, various versions of the "Pythagoras theorem".

The second chapter (pp. 91-247) contains the substance of the work. Indeed, the 'Siyuan Yujian' or 'Jade Mirror of the four unknowns' consists, on the one hand, of four problems for which a detailed solution is given, and, on the other hand, of two hundred and eighty-four problems where the methods are applied. This second chapter is a translation, commented with precision, of the four initial problems. Jock Hoe succeeds in a remarkable way in making us understand how the Chinese algebraists managed to express the problems in question in the form of equations by arranging the rods on a counting board, polynomial equations with one, two, three or four unknowns, then to eliminate the unknowns, before solving the system.

The third chapter (pp. 248-331), finally, brings together in a systematic way the mathematical knowledge used in the problems: in particular, areas and volumes of various plane and solid figures, methods of solving equations, calculation of finite numerical series of various types ... We learn that the cubic interpolation formula was already known to Zhu Shi-jie, whereas it is usually attributed to Guo Shoujing (1231-1316).

The four introductory problems are translated into semi-symbolic language, then transcribed into our modern algebraic symbolism. In his translation, Jock Hoe seeks to account for the capacity of the ancient Chinese language to lend itself to a "rhetorical algebra" whose symbols can be manipulated according to certain rules and form assemblages. For example, the characters [various Chinese characters], correspond respectively to the base, the height and the hypotenuse of a right-angled triangle (shown on its base); they have a more precise meaning than modern algebraic symbols (x, y, z) whose meaning, purely conventional, can change from one problem to another, but, unlike the corresponding French terms, they can enter into composition in assemblies such as 'gou cheng' (lit. "base-multiply-height", "base multiplied by height"). To maintain the conciseness of the original text and its aptitude for algebraic manipulations, Jock Hoe has created some twenty acronyms evoking certain methods of expression in programming languages. Thus, ALT designates the height of the right-angled triangle, MULT the operation of multiplication and BASE the height of the right-angled triangle, so that the above assembly is translated as BASE MULT. ALT. One can doubtless make many objections to such a translation and the effort which the reading of the Chinese text itself would require for a layman may not be so great as Jock Hoe thinks.

The author excels at describing the procedures employed by Chinese algebraists; he shows the impressive results they have achieved with their help, but he also highlights the handicap they presented at some stage in the development. He succeeds in rendering the Chinese text in an original way and he gives an overview of the mathematical knowledge which presupposes the setting in an equation of the problems contained in the 'Siyuan Yujian'.

However, the author provides little information on Zhu Shi-jie, on the sociological context of Chinese mathematics at the time, on the relationships that Zhu may have had with other mathematicians, etc. The author tries to show that the originality of the work is not in the presence so often mentioned of the arithmetic triangle which is its frontispiece, but one can regret that he did not reproduce this figure. It is also a pity that the prefaces to the work, so interesting from the point of view of the link between philosophical conceptions and mathematical methods, are not fully translated. Finally, Jock Hoe is obviously very sensitive to the assertions according to which Chinese mathematics is essentially turned towards the practical, even of an empirical nature; he seeks to show the various theoretical interests pursued by the Chinese, but his demonstration is not always convincing. The conclusion of his development on the Chinese number system is strange logic (pp. 56-57) and seems unaware that an Indian work describing the calculation methods used in India, and in particular the use of a point having the function of our zero, was introduced in China and translated into Chinese in the 13th century.

These criticisms should not hide the difficulties of the work: difficulty in understanding a text which, deprived of the master's oral commentary, is not always clear, difficulty of translation, difficulty of presentation to Western readers, difficulty of interpretation. The publisher specifies in his preface, that the appendices of Jock Hoe's thesis consisting of the reproduction of the Chinese text, its translation into semi-symbolic language and its translation into modern algebraic language "will eventually be published in a second part if this first volume meets with sufficient interest from the public." We hope that this volume will meet the interest it deserves and that these appendices will be published soon.

**2.5. F Hominal, Review: Les systèmes d'équations polynômes dans le Siyuan Yujian (1303), by J Hoe.**

*Revue d'histoire des sciences*

**31**(4) (1978), 374-377.

Thirty years ago, Sarton regretted in his

*Introduction to the history of science*the absence of commented translations of Chinese mathematical works. As for general studies, there was hardly any in a Western language at the time other than Mikami's work,

*The development of mathematics in China and Japan*, published in 1913 in Leipzig. Despite many flaws, it was reprinted in New York in 1961, shortly before the publication of two important works:

*The History of Mathematics in the Middle Ages*by Prof. A P Youschkevitch which describes medieval mathematics in different cultural areas (China, India, lands of Islam and the West) and volume 3 of J Needham's monumental work,

*Science and Civilization in China*, a volume devoted to Ancient Chinese mathematics, astronomy and earth sciences. But these works do not meet Sarton's wish: while they provide many elements for evaluating the Chinese mathematical tradition, they only contain translations of brief passages of the works described.

Several commented translations have been published in recent years in Western languages. Kurt Vogel translated and commented on the main Chinese mathematical work of Antiquity,

*Jiu-zhang suan-shu*, while three works of 13th century algebra, the height of ancient Chinese mathematics, caught the attention of mathematical sinologists: the

*Shu-shu jiu-zhang*of Qin Jiu-shao, the

*Arithmetic of Yang Hui*and the

*Siyuan Yujian*of Zhu Shi-jie, translated and commented by M Hoe.

Today we present the publication of an abridged version of J Hoe's thesis. This thesis contained a long commentary followed by three appendices giving successively the Chinese text, a translation in "semi-symbolic" language which we will describe later and a translation in modern algebraic language.

Zhu Shi-jie's work was annotated by Luo Shi-lin in the last century and translated into French and English in the first third of that century. Although one can hardly trust these translations, the work of Luo Shi-lin is a sure guide (for those who read Chinese) in the exploration of this difficult work which is the

*Siyuan Yujian*.

*The commentary*- The commentary is divided into four parts. An introduction (pp. 1-10) situates the work presented in relation to the studies already published; the author underlines the small place occupied by Chinese mathematics in most general histories of mathematics.

The first chapter (pp. 41-90) justifies the proposed translation of the work of the title and specifies most of the mathematical knowledge assumed by Zhu Shi-jie in his work: the number system, counting board operations, expression of polynomials in one unknown, various expressions of the "Pythagorean theorem."

The second chapter (pp. 91-247) contains the substance of the work. Indeed, the

*Siyuan Yujian*(or

*Jade Mirror of the Four Unknowns*) is made up, on the one hand, of four problems for which a detailed solution is given, and, on the other hand, of 284 problems to which the methods are applied. This second chapter is a translation commented with precision of the four initial problems; J Hoe succeeds in a remarkable way in making us understand how the Chinese algebraists of the Middle Ages managed to express the problems in question in the form of equations by putting sticks on a counting board, polynomial equations with one, two, three or four unknowns, then eliminate the unknowns before solving the system.

The third chapter (pp. 248-331), finally, collects in a systematic way the mathematical knowledge used in the problems: in particular areas and volumes of various plane and solid figures, methods of solving equations, calculation of finite numerical series of various types ...

We learn that the cubic interpolation formula was already known to Zhu Shi-jie, whereas it is usually attributed to a later astronomer and mathematician, Guo Shou-jing.

*Translation into "semi-symbolic" language.*- The four introductory problems are translated into a "semi-symbolic" language, then transcribed into our modern algebraic symbolism. In the translation, J Hoe seeks to account for the capacity of the ancient Chinese language to lend itself to a "rhetorical algebra" whose signs can be manipulated according to certain rules and form assemblies. For example, the characters gou, gu, xian correspond respectively to the base, to the height and to the hypotenuse of a right-angled triangle (shown on its base); they have a more precise meaning than modern algebraic symbols $(x, y, z)$ whose meaning, purely conventional, can change from one problem to another, but, unlike the corresponding French terms, they can enter into composition in assemblies such as gou cheng gu (lit. "base-multiply-height", "base multiplied by height"). To keep the conciseness of the original text and its aptitude for algebraic manipulations, J Hoe has created some twenty "words" written in capital letters evoking certain expression methods in programming languages. Thus ALT designates the height of the right-angled triangle, MULT the operation of the multiplication and BASE the height of the right-angled triangle, so that the above assembly is translated BASE MULT ALT. One can, no doubt, make many objections to such a translation. Is it preferable to insist on the readability of the text by means of an effort which is proportionate to the effort required to read the Chinese text itself.

J Hoe excels at describing the procedures employed by Chinese algebraists in the Middle Ages; he shows the impressive results they have achieved with their help, but he also highlights the handicap they presented at some stage of development. He succeeded in rendering the Chinese text in an original way and he gives an overview of the mathematical knowledge that presupposes the equation of the problems contained in the

*Siyuan Yujian*.

However, the author provides little information on Zhu Shi-jie, on the sociological context of Chinese mathematics at the time, on the relationships that Zhu may have had with other mathematicians, etc. The author tries to show that the originality of the translated work is not in the frontispiece reproduction of the arithmetic triangle so often mentioned, but we regret, with R Taton, that this reproduction is not given at all. It is also a pity that the prefaces to the work, so interesting from the point of view of the link between philosophical conceptions and mathematical methods, are not fully translated. Finally, J Hoe is obviously very sensitive to the assertions according to which Chinese mathematics is essentially turned towards the practical, even of an empirical nature; he seeks to show the various theoretical interests pursued by the Chinese, but his demonstration is not everywhere convincing. The conclusion of his development on the Chinese number system is an odd logic (pp. 56-57) and he seems completely unaware that an Indian work describing the calculation methods used in India, in particular the use of a point where we now put a zero, was introduced in China and translated into Chinese in the 13th century.

These critiques will not be able to hide from anyone the difficulties of J Hoe's work, difficulties in understanding a text which, deprived of the master's oral commentary, is not always clear, difficulty of translation, difficulty of presentation to Western readers, difficulty of interpretation. The publisher specifies, in his preface, that the appendices of J Hoe's thesis "will eventually be translated into a second part if this first volume meets sufficient interest from the public." Hopefully this volume will meet the interest that it deserves and that these appendices will be published shortly.

**3. Review of Jock Hoe's address to the First Australian Conference on the History of Mathematics in 1980.**

**3.1. W Ruppert, Review: Zhu Shijie and his Jade mirror of the four unknowns, by J Hoe.**

*Mathematical reviews*MR0653164

**(84h:01009)**.

Zhu Shijie's

*Jade mirror of the four unknowns*(published 1303) is one of the most important classics in the Chinese mathematical tradition. The present article offers a brief introduction to the content and style of this work (and indeed of ancient Chinese mathematics as well), intended for non-specialists in the field. The author chooses one of the 280-odd problems treated in the Jade mirror to give the reader an impression of the conciseness of the classical Chinese language (a conciseness which the author claims to amount to a "semisymbolic" expression of mathematical ideas) and the elegance of the Chinese counting-board methods in dealing with systems of algebraic equations; other aspects are touched on briefly. The problem presented would read in modern language and notation as follows:

Let $d$ be the diameter of the circle inscribed in a right triangle; $a, b$ the lengths of the two legs and $c$ the length of the hypotenuse of the triangle. Given that $dab = 24$ and $a + c = 9$ find $b$ (to the given data one has to add the elementary equalities $d = a + b - c$ and $(a + c)(a - c) = b^{2}$, which were well known to the Chinese). The main point in the problem is the reduction of the given system of equations to a single polynomial equation (which turns out to be of degree five; the Chinese solved such equations numerically by using a variant of Horner's scheme). This reduction was accomplished on the counting-board by an elimination procedure which the author describes in great detail; he also provides examples for later generalisations of the method.

Last Updated January 2021