Jing Zhai gu jin tu, (敬齋古今黈, Commentary of Jing Zhai on Things Old and New).
Third roll, p.33286: “When I came to Dongping, I came by a mathematical classic (算 經) that brilliantly explained the procedure of equal areas (如積之術). Using nineteen characters and the same amount of lines from top to bottom, it reads:287 “Xian, míng, xiāo, hàn, lěi, céng, gāo, shàng, tiān, rén, de, xià, dī, jiǎn, luò, shì, quán, àn, guǐ.” This probably means that the human being is taken as Tai Ji (太極) and heaven and earth respectively as Sources (Yuan, 元) that one can let ascend or descend. Although this explanation is superficial its principle facilitates the insight considerably. The diagrams (圖式) relating to the method of equal areas of all schools that I have inspected, all place the Tian Yuan at the top. When multiplying, one lets it ascend. When dividing, one lets it descend. Only Yan Cai (彥材)288 from Taiyuan places the Tian Yuan at the lowest position. All printed editions nowadays, which copy the models of other books and all place the Tian Yuan at the bottom, are simply a continuation of the method Yan Cai. Within mathematical studies (數學) Yan Cai’s position resembles that of an intellectual who is invading foreign territory. The method he adopts is directly opposed to the ancient methods because their importance derives from the fact that Tian is originally based at the top. If you move it, unfortunately it cannot ascend further. You therefore have to place it at the bottom so that, when it moves, it can slowly ascend. This resembles the divinatory diagrams [in the Book] of Mutations (易卦)289. Qian
286A complete transcription into simplified characters of the book with punctuation can be download from http://zh.wikisource.org/wiki/: “敬齋古今黈: 予至東平。得一算經。大概多明如積之術。以十九字志其上
287 What follows are the designations of the 19 positions on the table in which the arrangement of the characters probably represents a metaphor borrowed from Chinese Buddhism: 仙、明、霄、漢、壘、層、高、上、天、
人、地、下、低、減、落、逝、泉、暗、鬼. The enumeration of character could also be translated as a sentence: “The immortals shine light upon the highest regions of the Kingdom of Heaven. By overlapping steps they climb higher and higher into the heavens. While human beings on earth, who are climbing down towards lower regions, gradually fall down towards the percolating sources in the darkness of the demons”. The different steps could therefore also symbolize the degrees in the development of a Bodhisattva, according to which the Reign of the Enlightened is physically separated from the Underworld inhabited by spirits. [Andrea Breard, 1999], p.162.
288 Unfortunately, we could not find any information about this person.
289The Book of Mutation, or Book of Changes, or Yì Jīng, is one of the oldest of the Chinese classic texts. The book contains a divination system; it is still widely used for this purpose. Traditionally, the Yi Jing and its hexagrams were thought to pre-date recorded history, and based on traditional Chinese accounts, its origins trace back to the 3rd to the 2nd millennium BC. Historians suggest that the earliest layer of the text may date from the end of the 2nd millennium BC, but place doubts on the mythological aspects in the traditional accounts. Some consider the Yi Jing as the oldest extant book of divination, dating from 1,000 BC and before. The oldest
164 (乾) is located at the bottom, Kun (坤) at the top290. The reciprocal exchange of the two energies (氣) is the Tai (太). This is the reason why you take: “When multiplying, one lets it ascend. When dividing, one lets it descend.” If one is looking for an earthly unknown (地元), however, the process is the opposite”291.
manuscript that has been found, albeit incomplete, dates back to the Warring States Period (475-221 BC).
[Wilhelm Richard, Baynes Cary, 1967].
290 The trigrams Tian and Kun respectively symbolize the sky and the earth in the book of changes. The order of characters can be interpreted as a metaphor on the situation of man between sky and earth. [Breard Andrea, 2000], p. 261.
291 A translation into German of the same text can be found in [Andrea Breard, 1999], p.162.
165 益古演段自序
Preface of Development of Pieces [of Areas] of the Improvement of the Ancient [collection].
術數雖古六藝之末，而施之人事，則最為切務。故古之博雅君子，馬鄭之流，未有不 研精於此者也。其撰者成書者，無慮百家，然皆以「九章」為祖。而劉徽、李淳風又 加注釋，而此道益明。
Although being last on the list of six arts (六藝), mathematics (術數) is the most crucial in daily practices. Therefore ancient knowledgeable intellectuals like Ma and Zheng (馬, 鄭) all master the mathematics. When it comes to a mathematics book, regardless the mathematician's school, The Nine Chapter (九章) is commonly traced back to as the root.
Meanwhile, Liu Hui (劉徽) and Li Chun-Feng (李淳風)'s notes and comments on The Nine Chapter (九章) make the mathematics even more perspicuous.
今之為算者，未必有劉、李之工，而褊心跼見，不肎292曉然示人，惟務隱互錯糅，故
為溟涬黯黮，惟恐學者得窺其彷彿也。不然，則又以淺近觕俗，無足觀者，致使軒轅 隸首之術，三五錯綜之妙，盡墮於市井沾沾之見，及夫荒邨下里，蚩蚩之民，殊可憫 悼。
On the other hand, contemporary mathematicians (算者), who do not necessarily study as comprehensively as Liu Hui or Li Chun-Feng, are narrow-minded and short-sighted. Instead of making it clear, they prefer rendering it as implicit and intricate as possible in order to make the mathematics appear opaque and obscure. They prevent even a glimpse of its simulation being caught by others. Otherwise, some of them opt to deal with merely the basic and well-known part that does not worth looking into. Consequently, the methods (術) of the ancients Xuan Yuan (軒轅) and Li Shou (隸首)293 along with the sophisticated art of numbers (三五錯綜之妙) become something with which everyone in the town can be self-satisfied. It is such a pity that they actually know just as much as ignorant villagers.
近世有某者，以方圓移補成編，號「益古集」，真可與劉李相頡頏。余猶恨其悶匿而 不盡發，遂再為移補條段細繙圖式，使粗知十百者，便得入室啗其文，顧不快哉？
[For instance], a book entitled Collection Improving the Ancient [Knowledge] (益古集) was compiled recently with reshaped (移補) [solutions to geometric problems of] rectangles and circles. It is indeed an equivalent of Liu Hui and Li Chun-Feng. However, I detest its reserved style, and hence added detailed diagrams (細繙圖式294) of how to reshape the
292肎, I read 肯, “willing”
293 The Yellow Emperor's father was Shao Dian (Shaodian) 少典, his actual name was Gongsun Xuanyuan 公孫 軒轅 (Xuan-yuan might also be a place name where his clan dwelled). The Yellow Emperor was the first of a line of cultural heroes that are venerated for their inventions. Xuan Yuan is said to have invented - also with the help of his ministers - wells, mortars, bow and arrow, cattle breeding, carts and ships, clothing, divination, mathematics, astronomy and calendar, musical notes, medicine and writing. Li Shou, 隸首, is the name of the minister who created mathematics.
294I do not know if one has to read tushi, as « diagrams » as tu and shi, “diagrams and configuration”.
166 Section of Areas. Isn't it a great joy that the book will thus be easily accessible to anyone with basic knowledge now295?
客有訂於曰296 ：「子所述果能盡軒隸之秘乎？」余應之，曰：「吾所述，雖不敢追配
作者，誠令後生輩優而柔之，則安知軒隸之秘不是乎始？」客退，因書以為自序。
A guest asked after proofreading, “do you really think the secret (秘) of Xuan Yuan and Li Shou is fully expressed in your words?” I answered him, “I dare not challenge or match the author with my presentation. Nevertheless, if we leave the students and future generations uncertain and unclear of it, how could we not wonder that it is in this way the methods of Xuan Yuan and Li Shou started to become a secret?” When the guest left, I write thereupon this preface.
時大元己未夏六月二十有四日欒城李冶自序
Yuan Dynasty - Year of Ji-Wei (1259 AD) - Summer - 6th Month - 24th Day. Completed by Li Ye.
295See Introduction, Part I. [Gu Meisheng, 1999].
296客有訂於曰, I am not sure to understand.
167