When West knows East, and East knows West, 'Tis then the twain shall meet.

The above may be an optimistic response to Kipling's famous words but this book attempts to further the understanding of traditional Chinese culture, both in the West and in the East, by unveiling the secrets of the three Chinese cosmic boards. These methods of astronomical-cum-divinatory calculations had exerted a profound influence on traditional China over a very long period. They show their presence in some hitherto obscure passages in the literature and in various aspects of Chinese culture. To understand them it is necessary to take note of the cultural differences between past and present as well as between East and West. As pointed out by Lloyd (2000), science developed very differently in Babylonia, China and Greece, both in the nature of the investigations undertaken, and in terms of the social and intellectual institutions within which the investigators worked.

The traditional Chinese reached out to the stars in several ways, although obviously not physically. In mythology and in the novels, human beings turn holy immortals and travel among the stars; in Daoism the devotee reaches out to space by imagination during meditation or in liturgy; while in the three cosmic boards the operator contacted the stars by sheer calculation. Stars to the traditional Chinese were not entirely materialistic in the modern sense, but were composed of qi and occupied by spirits. The operator of the cosmic boards was thus able to attempt to derive data or messages from the stars and make interpretations therefrom. This process of calculations and making interpretations constituted the most advanced form of traditional Chinese mathematics.

Meaning of the term mathematics in traditional China

In the study of the history of science in China it is important to remember that Chinese terms do not necessarily convey exactly the same meaning to a modern scholar as they did in traditional China. A case in point is the word shuxue which is now a universally accepted Chinese equivalent for the modern English term mathematics. This is no surprise when one is reminded of the fact that in mediaeval Europe the same word did not mean exactly the same as it does to us. The Roman writer Boethius (c.480-c.525) defined mathematics as astronomy, arithmetic, geometry and music, which later became the quadrivium in mediaeval European universities. The Chinese term shuxue was first adopted as the technical term for mathematics only in the middle of the nineteenth century by Li Shanlan ^HiiS (1811— 1882) and was not officially adopted until the beginning of the twentieth century. In traditional China the term had a much broader meaning. Shuxue used to refer to what we now call mathematics, natural philosophy, numerology, divination, astronomy, astrology, fengshui jSUJc (geomancy) and music. Indeed, mathematics was only regarded as the elementary segment of shuxue, as clearly stated in the Preface of Qin Jiushao's (1202

1261) Shushu jiuzhang and in what Liu Hui §!jM wrote earlier in the year ad 260 in his Jiuzhang suanshu A^H®.1 The opening paragraph of Qin Jiushao's book contains the sentence 'If we aim at the great, we can be in touch with the spiritual powers and thus live conformably with our destinies; if we aim at the small, we can settle the affairs of this life, and by classification deal with the myriad phenomena.'2 Qin Jiushao goes on to classify mathematics into 'esoteric mathematics' (neisuan \H%) and 'exoteric mathematics' (waisuan What Qin refers to as esoteric mathematics is more often referred to as shushu ifrj§SC, a general term that included not only mathematics but also astronomy, astrology, music and divination, and exoteric mathematics is what modern scholars understand by the word 'mathematics'.

At first sight it seems that modern historians of mathematics can simply reverse the order of priority of the traditional Chinese and take only the exoteric component in Chinese mathematics. However, esoteric mathematics also included problems of calculating the unknown, such as the method in Qin Jiushao's Shushu jiuzhang to find out the number of years from the Great Epoch in Yixing's —fr Dayanli 'jz.^uM calendar system and the remainder theorem that first appeared in the Sunzi suanjing i^-pH&S. Modern historians of Chinese science have already taken this particular mathematical aspect of Chinese esoteric mathematics into account. Sinologists have also made substantial contributions to the study of Chinese divination and the Ytjing JIM (Book of Changes). What Joseph Needham (1956) writes under sections 13, 14 and 16 in Science and Civilisation in China, volume II, suggests that knowledge of shushu is a prerequisite for the understanding of his magnum corpus. In his Author's Notes, he also points out that 'the first necessity is to apprehend the deeply organic and non-mechanical quality of Chinese naturalism'.

INTRODUCTION Chinese divination

Interest of the traditional Chinese people in the prediction of human events is well known.3 Evidence of divination can be found in the earliest Chinese written records in the form of the oracle bones, with which the oracle officials made predictions for the Yin IS kings. In the Zhou dynasty (1030P-221 bc), the court also enlisted the service of diviners using milfoil, probably according to some system interpreted from the Yijing HM (Book of Changes), besides retaining the service of oracle officials. Even the heads of state in the period of the Warring States (480-221 bc) employed their own oracle bones and Yijing system experts. From the Zuozhuan (Master Zuo Qiuming's ¿EixBf] Enlargement of the Spring and Autumn Annals) one can note the vagueness, or flexibility if one may like to call it, of the interpretations of the Yijing system as well as the divergent messages from different groups of diviners. The systems themselves were seldom in doubt, but making correct interpretations was the issue. The system of the Yijing in fact became one of the bases of Chinese thought, while the oracle bones fell out of use. Many different schools of interpretation on the Yijing system were developed since the Han period (202 bc-ad 220), and captured the attention of the literati.

The oracle bones and the Yijing Hexagrams were not the only methods used in divination. A system of astrology was developed during the time of the Spring-and-Autumn period (722-480 bc) that was supposed to enable the emperor and the feudal lords to read their fortunes and destinies from the signs given by the heavenly bodies. Eclipses, conjunctions of planets, comets, meteors, clouds and vapours were dutifully observed by the ruling house in the belief of there being a mutual influence between men and heaven. Earthquakes, natural disasters and unseasonable rain and snow were also believed to come under the same influence.4 A form of astrology that was relevant only to the ruling class was thus developed. The common people also had their own forms of divination, although they were generally far less sophisticated than those employed by the ruling class. Modern archaeology provides evidence that the method of selecting auspicious days (zeri iP 0 ) was already practised during the time of the Warring States period. None of the methods mentioned above used for divination involved elaborate calculations, although the manipulation of milfoil in the Yijing system did require some amount of counting. Ultimate knowledge in traditional mathematics must have been that of the 'numbers of the heavens' and how to calculate them. This must have been referring to the traditional Chinese shushu par excellence. Such knowledge was employed in the three cosmic boards, which were not as commonly known as other divinatory systems.

The literati, for example, held much respect for the Yijing system and knew it far better than the common people. However, Chinese novels refer to the secret arts of the cosmic boards that were supposed to yield much more precise interpretations than the Yijing system. In the Hongloumeng ft(Dreams of the Red Mansions), Jia Rong HI? consults Mao Banxian for a prognosis of his mother's illness. Mao uses the Yijing system but finds the message too vague to make an accurate prediction. With the consent of Jia Rong, he follows this with Liuren 7\3r prognostication, which tells that the patient will remain sick for some time but will eventually recover. Although the Liuren system was probably already in use during the Warring States period, it has escaped the attention of the academic world until quite recently. There were also other systems. Ge Hong's (283?—343?) Baopuzi neipian ftiibil^ira refers to the mystic arts of Dunjia and Qimen niPI. A passage in the Nan Qi shu l^^lf (Official History of Southern Qi Dynasty) has something to do with the Taiyi jiugong suan method. These methods, being arcane as well as rather sophisticated, have hitherto received little attention from scholars, both traditional and modern.

There are several reasons why these sophisticated methods did not receive due attention from traditional scholars. The first was the general attitude towards studies beyond the requirements for the civil service examinations, in which the Confucian classics predominated. It is interesting to observe that because the Yijing was included among the Confucian classics, it received much respect from traditional scholars. The commentator of the Liuren daquan (Great Compendium on Liuren) in the Siku quanshu zongmu tiyao § £IJI must have considered that the

Yijing already contained all sufficient knowledge when he underrated the book under review by saying that Liuren was a mere derivative of the Yijing. Yet Gao E's ili!§ continuation of the Hongloumeng Slftl^ indicates the superiority of the Liuren method over that of the Yijing, and Li Ruzhen's Jinghuayuan iiTtlfc (Mirror of the Flowers) hints that

Liuren was an important item that should not be found missing from a scholar's repertoire of knowledge.5 A second reason was the confidential nature of these methods, as they were looked upon as having deep implications with military affairs. They were often regarded as part of the classified corpus of knowledge, taught and practised only within the walls of the Astronomical Bureau or the imperial court. Both scholarly neglect and bureaucratic secrecy have adversely affected the availability of books and written records on the subject to modern scholars. A third reason was the overwhelming amount of technical terms on the subject that would discourage the uninitiated from trying to understand these sophisticated systems.

Neo-Confucian natural philosophy

Neo-Confucian scholars before Zhu Xi vfcJi (1130-1200) had made a theoretical study of the natural world and merged natural philosophy with ethics in the light of the traditional Chinese concept of harmony between human beings and nature - tian ren he yi . Their ethics were based on the teaching of Confucius, while their natural philosophy was rooted in the Yijing. In Northern Song China there was also growing interest in two charts or diagrams mentioned in the Yijing, namely the 'River Chart' hetu and the 'River Luo Writings' Luoshu There developed the

'Symbolic Numerology' xiangshu school that was then devoted to the study of different relations among the hexagrams of the Yijing and these two diagrams.6 It was within this intellectual environment that some neo-Confucian thinkers became involved in the study of natural philosophy and ethics. Among the writings of the neo-Confucian scholars are found some interesting ideas that were also important in the Western scientific tradition. Zhu Xi, for example, touched on the idea of infinite greatness and infinite smallness, saying that what is infinitely big has nothing 'outside', and what is infinitely small has nothing 'inside': in other words, the infinitely small is indivisible. Zhu Xi explained the idea of inertia by taking the example of a cart. It would require an applied force to set it in motion, but once in motion the starting force would no longer be needed. He expressed his observation of relative motion by taking two wheels rotating about the same axis at different speeds. The one rotating more slowly would seem to an observer on the faster wheel to be rotating in the opposite direction. He also noted the existence of centrifugal force, observing that water in a revolving vessel would not spill even when the vessel was upside down.7

The three cosmic boards

There is a conspicuous lack of mathematical calculations in neo-Confucian studies of nature and natural phenomena. Even Shao Yong (10111077), who was reputed to be an expert on a form of the Liuren system known as meihuashu i§7t:!fi( (plum blossom numerology), did not use mathematics to explain science. It was not generally known that hidden within the walls of the Song Imperial Palace were methods other than the system of the Yijing that were used in the Astronomical Bureau. As the methods used in the Astronomical Bureau were often secret in nature, they were known only to a very few and were probably unknown to most neo-Confucian scholars except by name only. In eleventh-century Song China the Astronomical Bureau examinations syllabus included the method of the three cosmic boards (sanshi Hit), comprising the method of Taiyi JzZL ), the method of concealing the Yang Wood (Dunjia and the method of employing the six sexagenary cyclical members with the Yang Water stem (Liuren /\3r). These methods have hitherto received little attention from Sinologists and historians of Chinese science alike. They are not dealt with in Joseph Needham's Science and Civilisation in China. It was only in the second decade before the turn of the millennium that Yan Dunjie JRSft'ffi published an article on the cosmic boards in the Kaogu xuebao, which was preceded by a translation into Japanese by Hashimoto

Keizo and Sakade Yoshinobu.8 Due to their semi-secret nature the methods of the three cosmic boards have never been known by more than a very few people. Writings on these methods were obscure in style, comprehensible only within the profession and, at the same time, less accessible than other writings and more liable to become lost.

We are indebted to the patronage of the Song Emperor Renzong 1-1 (reigned 1023-1063) for the accessibility of the more important writings on the three cosmic boards. He ordered a team of officials in the Astronomical Bureau led by Yang Weide flltS"©- to compile monographs on the cosmic boards, namely the Jingyou Taiyi fuyingjing MftjfcZjfysMM. (Canon of Auspicious Responses from the Taiyi (Board) compiled during the Jingyou Reign-Period), the Jingyou Dunjia fuyingjing Jfti&jii¥ fflMM. (Canon of Responsive Dunjia Techniques compiled during the Jingyou Reign-Period), the Jingyou Liuren sendingjing JSIiiTvitt/EM (Canon of Spiritual Readings from the Liuren Boards compiled during the Jingyou Reign-Period), and others. This team of officials in the Astronomical Bureau also had to write chapters on the three cosmic boards for the military compendium Wujing zongyao sSiMiiSi? (Collection of the Most Important Military Techniques), again compiled by Zeng Gongliang ta^iffi in 1040 under imperial order. The writings of Yang that still remain are now the most valuable sources on the three cosmic boards. It would seem that the best approach for a formal investigation into the three cosmic boards is a full translation and annotations of Yang's writings. However, unfortunately some of Yang's writings do not survive now in their complete original form. For example, both the Jingyou Taiyi fuyingjing and the Jingyou Dunjia fuyingjing do not exist in their original complete text and we can no longer learn the procedures for the Liuren method from his Jingyou Liuren sendingjing. Thus, a preliminary general study of the three cosmic boards must precede any attempt to translate these works.9

Being the earliest text containing the Taiyi system as well as one universally accepted by the Chinese literati, a passage from the Nan Qi shu (Official History of the Southern Qi Dynasty) is selected to introduce and describe that cosmic board in Chapter 3. Certainly, the text itself does not tell us explicitly about the system, yet from the data it contains it is quite possible to trace the system employed by its sixth-century author and in the process demonstrate that it was essentially similar to that used during the time of Yang. The most important application of the Qimen Dunjia system was in the art of war and to the layman both its name and its system were shrouded in mystery. In Chapter 4 an arcane military manual, namely the Huangdi yinfujing JHfrPi^M, serves the purpose of introducing the system. It is the text of choice over a similar but more lengthy text, the Yanbo diaosouge aHiik, because of its brevity. In Chapter 4, three passages from Shen Gua's tfcfS (1031-1095) Mengxi bitan are used to introduce the Liuren cosmic board system and bring out an important point in the progress of scientific thinking. The translations and annotations of these passages are followed by a description of the manipulation of the Liuren cosmic board.

It is interesting to note that Yang Weide was a student of the astronomer Han Xianfu (938?—1013?) who constructed the earliest Song armillary sphere known as the Zhidaoyi Mil HI in the year 985. He was a senior official in the Astronomical Bureau during the time of the emperor Renzong and, as we have already noted, he was mainly responsible for the official texts on the three cosmic board systems. Eventually, he rose to the rank of directorship of the Astronomical Bureau.10 He was however later dismissed from his post in the Astronomical Bureau, and in the following decade Shen Gua was appointed to a similar post, and Shen Gua had a very bad opinion of Yang.11 Without passing judgement we need only to note Yang's contributions to the writings on the three cosmic board systems against just a few jottings on the Liuren system, irrespective of the great reputation enjoyed by the name Shen Gua among modern historians of science. Although Shen Gua does not say much about the manipulation of the Liuren cosmic board in his Mengxi bitan, he does touch upon some fundamental ideas on which the system is based and offers his criticisms and suggestions for improving the system. No reform in the system resulted from Shen Gua's comments, but his spirit of scepticism and courage to challenge established thinking is remarkable within the context of history of science.

Scientific activities involved knowledge of the natural world and the ability to change or harness it. Such knowledge and ability, for example, were accorded to Zhuge Liang fillS^; (ad 181-234) in the fourteenth-century novel Sanguo yanyi Hffl'^il (Romance of the Three Kingdoms). Other novels mention similar mystical arts and magical powers. Although they were the products of the writers' fertile imaginations they probably originated from knowledge that the authors had at least heard of. Some can be traced to knowledge that can be subjected to rational enquiries, but others not. Occasionally, one even finds passages referring to these arts in the literature. These passages seemingly have just been passed over by traditional Chinese scholars, judging from the lack of commentaries and annotations on them, where much could have been written. Those that could not be subjected to rational enquiries belong to the realm of mysticism and magic, and many of these were of Daoist origin. Understanding the three cosmic boards is the key which unlocks many obscure passages in Chinese literature that refer to them, and can enable the reader to appreciate what Qin Jiushao meant by his statement that 'we can be in touch with the spiritual powers and thus live conformably with our destinies', quoted above. Furthermore, a knowledge of the three cosmic boards can also provide scholars engaged in annotating Chinese classical literature with new tools for their work.12

The three cosmic boards will also be of special interest to historians of science. Predictions of meteorological events were regularly performed at the Astronomical Bureau and they were accomplished by calculations involving the use of astronomical data. Shortly before the scientific revolution in Europe, a great attraction of science was its ability to make it possible to calculate and predict the occurrence of natural phenomena. Tycho Brahé (1546-1601) observed the occurrence of a solar eclipse on 21 August 1560 at the precalculated time and became motivated to study this branch of science that permits such wonderful possibilities of prediction. Mathematical calculations became an indispensable tool to physical science. It is not widely known that traditional Chinese astronomers five centuries before Tycho had already attempted to calculate and predict the occurrence of natural phenomena, although their efforts did not ultimately lead towards a scientific revolution in traditional China. Perhaps this can be attributed to the Chinese preference for not discriminating between natural observations that were measurable and human actions that were not.

In the study of the history of science, due recognition to the role of magic has to be given in the early development of what we now call 'science'. The Cambridge anthropologist J.G. Frazer stated two laws on magic. For the ancient magician there was an unwritten law of similarity, of like producing like, and there was also one of contiguity or contagion, where things that have been in contact, but are no longer so, continue to act upon one another. Indeed these two laws, together with the law of chance, can account for much of the divinatory arts in traditional China. They also to a certain degree explain the three cosmic boards, after taking into consideration a new dimension that involves the employment of astronomical data and mathematical calculations. At least one modern writer has classified one of the three cosmic boards as a 'mystic art'.13 Indeed, to the traditional Chinese the three cosmic boards were regarded as arcane, and consequently were closely guarded secrets that were certainly not meant to be given public hearing. However, it is not the intention here to treat the cosmic board systems as mystic.

What science was to the traditional Chinese

Even today the word science does not have exactly the same meaning in different European languages. Its nearest German equivalent is Wissenschaft, which includes all systematic study, such as what we call science, but also what we call history, philosophy and philology. We should not be surprised to hear from a historian in Russia that he is doing scientific research. In China the word kexue derived probably from the Japanese kagaku, only came to be used for the word science in the middle of the nineteenth century. In this book we shall take science to mean two things, the general concept of the 'knowledge of the natural world' and the idea of 'controlling or harnessing nature and the environment'. The same thought must have existed in the minds of different peoples in the past, long before the terms science and kexue were established. Moreover, the concept of science has f varied in the course of time as well as in different civilizations. In traditional China, science came within the realm of the esoteric mathematics division of shuxue generally known as shushu

To a modern scientist, such as Needham, shushu may be considered a kind of pseudo-science. However, in traditional China it was what people meant by science. Even after the arrival of the Jesuits in the seventeenth century, Chinese thinkers equated Western science with the neo-Confucian li, qi and shu. Fang Kongzhao ^fiLi® (1591-1655), who became acquainted with Western science through his Jesuit friends, classified li into zhili MS (ultimate li), wuli #53 (material li) and zaili (presiding li), under wuli, a term used to denote Western science at that time.14 A school of teaching originated by Fang Kongzhao was developed by his son Fang Yizhi ^Wla (1611-1671) and his grandson Fang Zhongtong (1634-1688).

Another member of the Fang school, You Yi (1614?—1684), spoke about li as being both within shu (li zai shu zhong SftiSci3) and within qi (li zai qi zhong SSil^1)- His teaching of gewu qiongli f&#JlSi3 (distinguishing things and exhausting the li) influenced Japanese scientific thought in the seventeenth century. Wang Xichan 3EHMI (1628-1681) also referred to qiongli and also talked about li generating shu (li sheng shu 34t$0 and seeking li from shu (yin shu qiu li H®>!<;3).15

A basic difference between East and West

The three cosmic board systems use terms referring concurrently both to heavenly bodies and to spirits. In other words, the same term refers to both matter with form and matter without form. This is so when we take Zhu Xi's definition of shen if as 'qi in expansion' and gui M, as 'qi in contraction'. The Liuren system introduces yet another dimension of matter without form by referring to something that reminds us of what we now call the sixth sense and telepathy. Hence, what the traditional Chinese person viewed as science embodied the non-materialistic world as well as the tangible. It is interesting to note that the three cosmic board systems attempted to study both the material and the non-material worlds by setting up mathematical relations between the two. In the West, the best known among the ideas that have been considered as a new approach of modern science is perhaps Hegel's (1770-1831) 'Philosophy of Identity', so called because it proclaimed the identity not only of subject and object but also of contradictions, such as existence and non-existence. According to Hegel, both the spirit world and the material world resulted from thought on the part of a creative mind, which he supposed to be similar in kind to the human mind. Hence the human mind was competent to think the thoughts of the Creator and rediscover them by its own inner activities.16 Hegel did not receive enthusiastic support for this idea in Europe, where modern science was built on the foundation of the rationalism of the ancient Greek philosophers. Few people outside Japan know about a similar but perhaps even more subtle idea expressed some 30 years earlier than Hegel by Miura Baien HittëPIS (1723-1789). Buddhist philosophy and East Asian culture provided a more fertile ground for the germination of Baien's idea concerning the void than the West provided Hegel in his time. Nishida Kitaro HfflM^êP (1870-1945) put forward a proposition that the separation of logic from mysticism was in itself a factor that limited science in the West. The Chinese three cosmic board systems can be looked upon as a case in point, taken from East Asian history, that exemplifies Nishida's proposition.

Ancient Greek philosophers made a distinction between terrestrial and celestial motion. Isaac Newton (1642-1727) with his law of gravitation showed that a general law could be applied to both. Today, while generalization is considered highly desirable, the exponential growth of scientific knowledge has resulted in the compartmentalization of knowledge that has led to the segregation of scientists even within the same discipline. A general theory that is applicable across disciplines will indeed be a great breakthrough in science. The traditional East Asian view was far more universal than Newton's, for it extended beyond the material world. Perhaps it was one of the merits of traditional East Asian science, it may suffer by being in excess, while the Western scientific tradition is being restricted by its own self-imposed limitations. By venturing beyond such limitations, a new world of science may yet be opened to us. Today some people advocate 'fuzzy logic' and the 'chaos theory' as a departure from the rigid Greek rationalism. Perhaps this is a step forward in the direction of the East Asian tradition.17

"While I make no effort to advoçate the three cosmic boards, I would like to tell a story to anticipate some questions that are likely to arise in the minds of readers concerning the Chinese belief in these systems. Li Shimin who reigned as Emperor Taizong ^tk in Tang China between the years 626 and 649, had already asked the celebrated military commander and tactician Li Jing (571 ?—649?) about the same subject. The Tang Taizong Li Weigong wendui Jif^TK^^^Fnlit (Conversation between Emperor Tang Taizong and Li [Jing], the Duke Wei [guo] gong) informs us that Li Shimin once asked his favourite general whether the use of Yin and Yang and shushu could be discontinued in the art of war (Yin Yang shushu fei zhi bu ke bu Li Jing replied, 'No (bu ke ^ nj)'

and continued, saying, 'The art of war is an art of deception (bing zhe gui dao ye IbSÈ'È). Under the guise of Yin and Yang and shushu, the greedy and the simple-minded can be (easily) deployed. Hence they should not be abolished.' Li Jing went on to give examples of how astrological signs and divination could be taken advantage of by clever interpretations.18 Psychological warfare, misleading the enemy with false information, classification of secret material, and many other military strategies in modern times, seem to differ little from those in traditional China - there are only differences in terminology, hardware and dimension.

INTRODUCTION Textual matters

The purpose of this book is to explain the three cosmic boards. However, I do not intend to get the reader too heavily involved with past arguments that are repetitive or that go round in circles, without throwing any new light on the subject. At the same time, I try to avoid treading too much on ground that has already been well covered. Nevertheless, basic ideas that are essential to the understanding of the systems, such as Yin and Yang, wuxing, the jiugong magic square and the system of the Yijing, are briefly summarized in Chapter 2. Chapters 3, 4 and 5 form the main part of the book. Appendix I and Appendix II are brief accounts of traditional Chinese astrology and the Ziping method of fate-calculation. The romanization of Chinese characters follows the pinyin system, except for proper names where respect is given to known personal or local preferences for people and places outside mainland China.19 A conversion table lists pinyin romanization together with the Wade-Giles system and the modified Wade-Giles system employed by Needham in the Science and Civilisation in China series. The same respect for known personal preference also applies to non-Chinese names. 'References to historical Chinese geographical names' and 'A brief note on Chinese romanization' serve as useful reference points in the prelim material, while appendices illustrating 'Traditional Chinese astrology', 'The Ziping method of fate-calculation' and a 'Table of Chinese dynasties', as well as notes, bibliographies of Chinese and Western works and an index, are provided to complete the book.

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