Author's preface: the Heavenly Gate and the Metal Tiger; a prophetic song that shows disorder

餘以天門金虎，呼吸精泉，按《星經》云：昴者，西方白虎之宿；太白者，金之精也。太白入昴，金虎相薄，法有兵亂。周宣王時有人采薪於郊，聞歌曰：「金虎入門，呼長精，吸玄泉。」時人莫能知其義。老君曰：「太白入昴，兵其亂。」徐氏名嶽，東萊人。蓋以漢室版蕩，又譎詭見於天，將訪名山，自求多福也。

I, Xu Yue, was born in a time when the "Heavenly Gate and Metal Tiger" stood in the sky and the prophetic song of "breathing the essence-spring" circulated in the world—Zhen Luan notes: the Star Classic says that Mao is one of the seven lodges of the White Tiger in the west, and Taibai is the planet Venus, condensed from the essence of Metal qi (氣). When Taibai invades the Mao lodge, the two layers of Metal qi, "Metal" and "Tiger," press upon each other; by the method of watching for omens this governs armed disorder. In the time of King Xuan of Zhou, someone gathering firewood in the outskirts heard a song that sang: "The Metal Tiger enters the gate, calling the long essence, drinking the dark spring." At the time not a single person could explain the meaning of these lines. The prognostic saying of the Old Lord says: "When Taibai enters Mao, armed disorder is about to occur." The author of this book is surnamed Xu and named Yue, a man of Donglai commandery. He had probably seen the Han house rocking and in peril, and seen strange and deceptive images appear in the heavens one after another, and so intended to travel the famous mountains in search of a road of more fortune and less harm for himself.

羽檄星馳，郊多走馬，按：漢徵天下兵，必露檄插羽也。老君曰：「天下有道，卻走馬以糞；天下無道，戎馬生於郊也。」遂負帙遊山，蹠跡志道，蹠跡者，兩足共躡一足跡也。漢文（帝）〔時〕河上公蹠跡為士。傋歷丘嶽，林壑必過。乃於太山，見劉會稽博識多聞，於數術。餘因受業，頗染所由。

Urgent military letters with feathers inserted flew and were transmitted like shooting stars; the outskirts were everywhere full of war-horses on the run—Zhen Luan notes: when the Han summoned the troops and horses of the realm, they always left the dispatch exposed and inserted a feather, to show that the matter was of the utmost urgency; this is called a "feathered dispatch." The Laozi says: "When the realm has the Way, war-horses are returned to the fields to haul dung and plough; when the realm is without the Way, even a mare in foal must drop her foal on the outskirts of the battlefield." Thereupon Xu Yue shouldered his book-box and travelled the famous mountains, treading the footprints of those before him and resolving to seek the Way—Zhen Luan notes: so-called "treading the traces" means both feet stepping in the same string of footprints, describing following former worthies step for step; Hegong of the River in the time of Emperor Wen of Han became a lofty gentleman by thus "treading the traces" (in the base text "di" should be "shi," time). He traversed hills and high peaks; wherever there were mountains, forests, gullies and ravines he was certain to go. At last on Mount Tai he met Liu of Kuaiji, a man of broad learning, especially versed in the arts of numbering. I then bowed to him as teacher and received the work, and so picked up something of the origins and developments of the arts of numbering.

餘時問曰：「數有窮乎？」會稽曰：「吾曾遊天目山中。」會稽，官號，漢中人也。按：《歷志》稱靈帝光和中，穀城守門候太山劉洪造《乾象歷》，又制月行遲疾陰陽（歷）〔歷〕，自洪（治）〔始〕也。方於《太初》、《四分》，轉精密矣。洪後為會稽太守。劉洪付《乾象》於東萊徐嶽，又授吳中書令闞澤，澤甚重焉，為註解。今案《地記》，天目山在吳興之界。

I asked him at the time: "Has number an end?" Liu of Kuaiji answered: "I once travelled in the Tianmu mountains…" Zhen Luan notes: "Kuaiji" is an official title; this man was originally of Hanzhong. According to the Treatise on the Calendar, in the Guanghe years of Emperor Ling of Han, Liu Hong of Mount Tai, who held the post of gate-warden of Gucheng, produced the Qianxiang calendar and also established the calendar for calculating the moon's slow and fast motion and the changes of yin and yang; all this began with Liu Hong (in the base text "zhi" should be "shi," began). Compared with the Taichu calendar and the Quarter-Remainder calendar, its calculations of motion were more precise. Liu Hong later took office as Grand Warden of Kuaiji, and is therefore called "Liu of Kuaiji." Liu Hong transmitted the Qianxiang calendar to Xu Yue of Donglai, and also to Kan Ze, Palace Secretary of the state of Wu; Kan Ze valued the book extremely and wrote a commentary on it. Consulting the Geographical Notes now, Mount Tianmu lies within the bounds of Wuxing commandery.

見有隱者，世莫知其名，號曰天目先生。餘亦以此意問之。先生曰：世人言三不能比兩，乃云捐悶與四維。《藝經》云，捐悶者周公作也。先〔布〕本位，以十二時相從。其文曰：「（周）〔同〕有文章，虎不如龍。豕者何為，來入兔宮。王孫出卜，乃造黃鍾。犬就馬廄，非類相從。羊奔蛇穴，牛入雞籠。」

Liu of Kuaiji went on: in the Tianmu mountains I met a recluse; no one in the world knew his name, and he was only called "the Master of Tianmu." I too put to him the question whether number has an end. The Master said: what the world calls "three cannot be compared with two" refers to the two number-games "Juanmen" (捐悶) and "Four Corners" (四維). Zhen Luan notes: the Classic of Arts says that Juanmen was devised by the Duke of Zhou; the method of play is first to lay out the home positions, then to pair them in the order of the twelve double-hours (the twelve Earthly Branches) (the base text has dropped a "bu," to lay out). Its formula says: "Together they have patterned writing; the tiger is not the dragon's equal. What is the pig about, coming into the rabbit's palace? The princeling goes out to divine, and so makes the Yellow Bell. The dog goes to the horse's stable; unlike kinds follow one another. The sheep runs to the snake's hole; the ox enters the chicken's coop." This is a riddle composed by pairing the twelve animals of the zodiac two by two, shifted out of place: tiger with dragon, pig with rabbit, monkey (the "princeling" is the monkey) with rat (the Yellow Bell belongs to Zi), dog with horse, sheep with snake, ox with chicken—the twelve signs making exactly six pairs (in the base text "zhou" should be "tong," together).

徐援稱，捐悶乃是奇兩之術。發首即奇一，後乃奇兩者，即為疑更調曰：「大豬東行遁虎坑，兔子欲宿入馬廄，羊來入村狗所屯，大牛何知乘龍上，蛇往西方人猴鄉，雞鳥不止夜〔鼠藏〕。」

Xu Yuan held that Juanmen is in fact an art of "odd-and-pair" (pairing singles and doubles): the opening line puts out only one sign, which is "odd one"; thereafter they are paired two by two, becoming "odd-and-pair." He had doubts about the old formula and changed it to another tune, saying: "The great pig goes east and flees the tiger's pit; the rabbit, wishing to lodge, enters the horse's stable; the sheep comes into the village the dog holds; what does the great ox know, riding upon the dragon; the snake goes west into the monkey's country; the chicken-bird does not stop, and the night-rat hides." The intent is still pig with tiger, rabbit with horse, sheep with dog, ox with dragon, snake with monkey, chicken with rat, pairing the twelve signs two by two, shifted apart (in the base text the "ren" of "ren hou xiang" should be "ru," to enter; the last line has dropped the two characters "shu cang").

其言三不能比兩者，孔子所造也。布十干於其方，戊巳在西南維。其文曰：「火為木生甲呼丁，夫婦義重巳隨壬，貴遺則統領幸參南丙妻則須守乙後火戊子天癸就庚。」四維，東萊子所造也。布十二時，四維之一。其文曰：「天行星紀，石隨龍淵，風吹羊圈，天門地連，兔居蛇穴，馬到猴邊，雞飛豬鄉，鼠入虎廛。」摰亦有四維之戲，與此異焉。

As for what is called "three cannot be compared with two," tradition ascribes it to Confucius. Its layout is to place each of the ten Heavenly Stems in its own quarter, and to seat the two stems Wu and Ji at the southwestern corner (the southwestern angle). Its formula says: "Fire is born of Wood, Jia calls Ding; the duty of husband and wife is heavy, Si follows Ren; the noble bequeaths and then governs, fortunately taking part in southern Bing; the wife must keep Yi, fire after; Wu-Zi, Heavenly Gui, goes to Geng." This formula is rather corrupt and dropped in the base text, and it is already hard to make every line run; the general sense is still that the ten Heavenly Stems answer and pair with one another according to the quarters. The kind called "Four Corners" was devised by the Master of Donglai; the layout is to spread the twelve double-hours (the twelve Earthly Branches) so that one of them falls on one of the four corners—northeast, southeast, southwest, northwest. Its formula says: "Heaven travels, the star-marker; the stone follows the dragon's abyss; wind blows the sheepfold; the Heavenly Gate and the earth join; the rabbit dwells in the snake's hole; the horse reaches the monkey's side; the chicken flies to the pig's country; the rat enters the tiger's market." This too is a riddle of the twelve Earthly Branches paired out of place. There is also a "Four Corners" game transmitted by the Zhi family, whose method of play differs from this one of the Master of Donglai.

數不識三，妄談知十。三者，上、中、下也。十數昴一數也。於筅之意非止十等之名，將關大衍之旨事一也。猶川人（事）〔士〕，迷其指歸，乃恨司方之手爽。司方者，指南車也。

The Master of Tianmu went on: people of the world cannot even tell "three" clearly, yet rashly claim they understand "ten." Zhen Luan notes: the "three" here are the three methods of advancing place—the upper number, the middle number and the lower number; so-called "ten numbers," though the names are ten, the principle in fact returns to one consistent thread (the character "mao" in the base text is corrupt). The intent of calculation does not stop at these ten number-names; it also involves the purport of the "number of the Great Expansion" in the Book of Changes, and the two principles are originally one affair. It is like that man of the river-country who himself lost his direction and on the contrary blamed the south-pointing carriage for pointing wrong ("shou shuang" means the hand-method went astray; in the base text "shi" should be "shi," gentleman). Zhen Luan notes: so-called "director of the quarter" is the south-pointing carriage.

《狐疑論》稱：「黃帝將見大隗於具茨之山，至襄城之野，川谷之（山）〔形〕率多斜曲。川人曰：『積數之常，乃固以之，非指南車之為爽。』乃指謂〔曰〕：『擢司方所指者乃為我等之西也，然則指南豈其謬也？』乃行數里，川人又曰：『司方所指我等之東也。』眾共論之，為疑笑於時。容成子怪而問之，川人以其將白對。

The Discourse on Fox-like Doubt records: "The Yellow Emperor was about to go to Mount Juci to meet Dawei, and came to the outskirts of Xiangcheng, where the lie of the river-valleys was for the most part crooked and bent (in the base text "shan" should be "xing," form). A river-man in the train said: 'The constant number accumulated on the carriage was set thus from the first; it is not that the south-pointing carriage points wrong.' He then pointed at the carriage and said: 'See, what the south-pointing carriage points to is precisely our west; so how can the south-pointing carriage be wrong?' After walking a few more li the river-man changed his tune and said: 'What the south-pointing carriage points to has become our east.' Everyone discussed the matter together; for a time all were puzzled, and it was passed about as a joke. Master Rongcheng found it strange and came to inquire; the river-man then reported the whole course to him just as it was.

容成曰：『在此望之，具茨之山於汝住所復在何方？』川人又曰：『在我之東。』容成曰：『汝向言在西，今更在東，何言不常也！此非山川之移，〔蓋〕川曲之斜，人心之惑耳。』川人乃請於斜曲之以定東西南北之術。

Master Rongcheng asked him: 'Looking from here, on which side of your dwelling is Mount Juci?' The river-man answered: 'On my east.' Master Rongcheng said: 'You said before it was on the west, and now you say it is on the east; how can you on the contrary blame the south-pointing carriage for pointing without settling! It is not that the mountains and rivers themselves have shifted place; it is that the road, crooked and askew, has turned people about, and in the end it is only that the human mind itself is confused.' The river-man then asked him: in such a crooked and askew place, by what method can one fix east, west, south and north accurately.

容成曰：『當豎一木為表，以索繫之表，引索繞表畫地為規。日初出影長則出圓規之外，向中影漸短，入規之中。候西北隅影初人規之處則記之。乃過中，影漸長出規之外。候東北隅影初出規之處又記之。取二記之所，即正東西也。折半以指表，則正南北也。』川人志之，以為知方之術。」

Master Rongcheng said: 'One should set up a wooden pole as a "gnomon" (a rod for measuring shadows), fasten a cord to the gnomon, and, leading the cord around the gnomon, draw a circle on the ground; this circle is called the "compass." When the sun first comes out the shadow is longest, and the shadow-tip falls outside the circle; near noon the shadow gradually shortens, and the shadow-tip draws in within the circle. Keep watch on the northwest side, and when the shadow-tip has just stepped inside the circle, make a mark on the ground ("ren" should be "ru," to enter). After noon the shadow gradually lengthens again, and the shadow-tip stretches outside the circle; keep watch on the northeast side, and when the shadow-tip has just stepped out of the circle, make another mark. Join these two marks in a line, and that is the direction of due east and due west; take the midpoint of this joining-line and connect it with the gnomon-pole, and what it points to is due south and due north.' The river-man recorded this method and took it as a gate for recognising direction."

未識剎那之促，安知麻姑之桑田。按：《楞伽經》云：「稱量長短者，積剎那數以成日夜。」剎那量者，壯夫一彈日指過頃遙六十四剎那。二百四〔十〕剎那名一（恆）〔怛〕剎那，三十（恆）〔怛〕剎那名一婆羅，三十婆羅名一摩羅多，三十摩羅多（子）為一日一夜。其一日一夜有六百四十八萬剎那。

The Master of Tianmu said further: people of the world do not even recognise so extremely short a unit of time as a "ksana" (剎那); how then can they understand the long ages of which Magu spoke, when the grey sea became mulberry fields? Zhen Luan notes: the Lankavatara Sutra says: "To measure the length of time is to pile up ksana upon ksana until they make a day and a night." The measure of a ksana is fixed thus: in the time it takes a man in his prime to snap his fingers once, sixty-four ksana have already passed; two hundred and forty ksana make one "tatksana," thirty tatksana make one "vara," thirty vara make one "muhurta," thirty muhurta make one day and night (in the base text "heng" should be "da"; after "two hundred and four" a "ten" has dropped; after "muhurta" a superfluous "zi" has been added). Multiplying layer upon layer in this way, one day and night comes in all to six million four hundred and eighty thousand ksana.

《神仙傳》稱：「麻姑謂王方平曰：『自接（待）〔侍〕以來，見東海為桑田。向到蓬萊，水乃淺於往者略半也。豈復將為陵乎？』方平乃曰：『東海行復揚塵耳。』」

Zhen Luan further cites the Biographies of Divine Immortals: "Magu said to Wang Fangping: 'Since I have waited upon the ranks of the immortals, I have already seen with my own eyes the Eastern Sea become mulberry fields. Just now I went to Penglai, and the seawater there is again shallower than before by nearly half; can it be about to become dry land (a mound) again?' Wang Fangping then said: 'Before long the Eastern Sea will again raise dust.'" This is precisely to describe the length of the years, forming two poles with the brevity of a ksana (in the base text "dai" should be "shi," to wait upon).

不辨積微之為量，詎曉百億與大千。按《楞伽經》云：「積微成一阿耨，七阿耨為一銅上塵，七銅上塵為一水上塵，七水上塵為〔一〕兔毫上塵，七兔毫上塵為一羊毛上塵，七羊毛上塵為一牛毛上塵，七牛毛上塵為一向中由塵，七向中由塵成一蟣，七蟣成一蝨，七蝨成一麥橫，七麥橫成一指節，二十四指節為一肘，四肘為一弓，去（肘）〔村〕五百弓為阿蘭若。」

The Master of Tianmu said further: people of the world cannot even tell how motes of dust pile up into measures; how then can they comprehend such huge numbers as a "hundred kotis" and a "great thousand"? Zhen Luan notes: the Lankavatara Sutra says: "Gather the extremely minute to make one 'anu' (阿耨, an extremely fine mote); seven anu make one 'dust on copper,' seven dusts on copper make one 'dust on water,' seven dusts on water make one 'dust on a rabbit-hair,' seven dusts on a rabbit-hair make one 'dust on a sheep-hair,' seven dusts on a sheep-hair make one 'dust on a cow-hair,' seven dusts on a cow-hair make one 'dust in a sunbeam' (floating dust drifting in sunlight); seven dusts in a sunbeam make one 'nit,' seven nits make one 'louse,' seven lice make one 'barley-width' (the width of a grain of barley), seven barley-widths make one 'finger-joint'; twenty-four finger-joints make one 'forearm,' four forearms make one 'bow'; a place five hundred bows from the village is called an 'aranya' (a quiet place where a monk practises in stillness)." (In the sentence "dust on water makes dust on a rabbit-hair" the base text has dropped a "one"; the "forearm" in "five hundred bows from the forearm" should be "village.")

據若摩竭國人，一拘盧舍為五里，八拘盧舍為一由旬，一由旬計之為四十里也。及以算校之，正得一十七里。何者？計二尺為一肘，四肘為一弓，弓長八尺也。計五百弓，有四千尺也。八拘盧舍則有三萬二千尺。除之，得五千三百三十三步。以裡法三旦步除之，得一十七里，餘二百三十三步。

Zhen Luan goes on: by the calculation of the people of Magadha, one "krosa" equals five li, eight krosa make one "yojana" (由旬), and thus one yojana ought to come to forty li. But checking by actual calculation with the scale above, one yojana comes to only seventeen li. Why? By the former method two chi make one forearm, four forearms make one bow, so one bow is eight chi long; five hundred bows (that is, one aranya, one krosa) come to four thousand chi; eight krosa then have thirty-two thousand chi. Dividing by six chi to a pace, one obtains five thousand three hundred and thirty-three paces (with two chi remaining); dividing again by the li-measure of "three hundred paces to a li," one obtains seventeen li, remainder two hundred and thirty-three paces—so it can be seen that the two accounts differ widely (in the base text "san dan bu" should be "san bai bu," three hundred paces).

《華嚴經》云：「四天下共一日月，為一世界。有千世界有一小鐵圍山遶之，名曰小千世界。有一千小世界有中鐵圍山遶之，名曰中千世界。有〔一千〕中千世界有大鐵圍山遶之，名曰大千世界。此（三千）大千世界之中，有百億須彌山。」

Zhen Luan further cites the Avatamsaka Sutra: "The four great continents share one sun and moon, and together are called one 'world.' Outside a thousand worlds a small iron-enclosing mountain rings them, called a 'small-thousand world'; outside a thousand small-thousand worlds a middle iron-enclosing mountain rings them, called a 'middle-thousand world'; outside a thousand middle-thousand worlds a great iron-enclosing mountain rings them, called a 'great-thousand world.' Within this one great-thousand world there are in all a hundred kotis of Mount Sumerus." (Before "middle-thousand world" the base text has dropped the two characters "yi qian"; before "great-thousand world" the two characters "san qian" have been superfluously added.)

乃今校之，世有十億日月，十億須彌山。何者？置小千世界之中有一千日月，以一千乘之，得一百萬，即中千世界中日月數也。置中千（乘）〔世〕界日月之數以一千乘之，得即大千世界日月之數也。

Zhen Luan checks it by calculation and says: reckoning down layer upon layer by thousands in this way, one great-thousand world ought to have only ten kotis of suns and moons and ten kotis of Mount Sumerus, not a hundred kotis. Why? First set out the number of suns and moons in a small-thousand world—a thousand worlds are a thousand suns and moons; multiply that by a thousand and one obtains a million, which is the number of suns and moons in one middle-thousand world; then take the million suns and moons of the middle-thousand world and multiply by a thousand, and the ten kotis obtained are the number of suns and moons in the great-thousand world (in the base text "cheng" should be "shi," world).

又云：「四天下者，須彌山南曰閻浮提，山北曰鬱丹越，山東曰〔浮〕提，山西曰俱瞿耶尼山。其日月一日一夜照四天下，山南日中，山北夜半，山東日中，山西夜半。」及以成事驗之，則有疑矣。何者？按閻浮提人在須彌山南，及至二月、八月春、秋分畫夜停，以漏刻度之，則晝夜各五十刻也。然則日初出時，東向視日之當我之東，即漏刻，及其日（浸）〔沒〕當我之西〔則〕五十刻。其一日一迄之中，遶三天下而來，所以至曉亦得五十刻也。胡以十萬為億，有百倍日月，四天下等事，有所未詳也。

The Avatamsaka Sutra says further: "So-called four great continents means: south of Mount Sumeru is called Jambudvipa (the southern continent), north of the mountain Uttarakuru (the northern continent), east of the mountain Purvavideha (the eastern continent), west of the mountain Godaniya (the western continent). This one sun and moon in the course of a day and night shine throughout the four great continents: when it is noon south of Mount Sumeru, it is midnight north of the mountain; when it is noon east of the mountain, it is midnight west of the mountain." (In the base text "east of the mountain is called ti" a "fu" has dropped.) Zhen Luan tests it against measured fact and finds it doubtful. Why? The people of Jambudvipa dwell south of Mount Sumeru; on the two days of the spring and autumn equinoxes in the second and eighth months, day and night are equal, and measured by the water-clock day and night each obtain fifty marks. Then when the sun first comes out, looking east, the sun is precisely in my east, and from this one starts the clock; when the sun sinks in the west and the sun is precisely in my west, exactly fifty marks have been walked. In the course of this one day and night the sun must still circle the remaining three continents and come back, so from sunset to dawn is also exactly fifty marks—reckoned thus, the day and night of the four continents cannot be halved two and two as the sutra says. As for the people of the Western Regions counting a hundred thousand as one koti (and thus speaking of a hundred kotis of Mount Sumerus, a hundred times more suns and moons than ten kotis), and this sort of statement about the allotment of day and night under the four heavens, there are still places in the principle that have not been made clear (in the base text "jin" should be "mo," to sink; after "precisely my west" a "ze" has dropped).

黃帝為法，數有十等。及其用也，乃有三焉。十等者，億、兆、京、垓、秭、壤、溝、澗、正、載。三等者，謂上、中、下也。其下數者，十十變之，若言十萬曰億，十億曰兆，十兆日京也。中數者，萬萬變之，若言萬萬曰億，萬萬億曰兆，萬萬北曰京也。上數者，數窮則變，若言萬萬曰億，億億曰兆，兆兆曰京也。

The Master of Tianmu said: the Yellow Emperor created the method of numbering; the names of the large numbers are ten grades in all; yet when it comes to actual use, the manner of advancing place has three kinds. So-called "ten grades" are the ten number-names yi, zhao, jing, gai, zi, rang, gou, jian, zheng and zai. So-called "three ranks" are the three methods of advancing place: the upper number, the middle number and the lower number. The "lower number" advances one grade every time ten is filled, as when a hundred thousand is called "yi," ten yi are called "zhao," ten zhao are called "jing." The "middle number" advances one grade only when a myriad myriads (that is, one yi) are filled, as when a myriad myriads are called "yi," a myriad-myriad yi are called "zhao," a myriad-myriad zhao are called "jing" (in the base text "bei" should be "zhao"). The "upper number" changes only when the present grade is exhausted, that is, it advances place by self-multiplication, as when a myriad myriads are called "yi," yi times yi is called "zhao," zhao times zhao is called "jing."

按：《詩》云：「胡取禾三百億兮。」毛注曰「萬萬曰億」，此即中數也。鄭注云「十萬曰億」，此即下數也。徐援《受記》云：「億億曰兆，兆兆曰京也」，此即上數也。鄭（注）〔蓋〕以數為多，故合而言之。

Zhen Luan notes: the poem "Cutting Sandalwood" in the Airs of Wei in the Book of Songs says: "By what right do you take three hundred yi of grain?" Mao Heng in his tradition says "a myriad myriads make an yi," which is precisely the "middle number"; Zheng Xuan in his notes says "a hundred thousand make an yi," which is precisely the "lower number"; Xu Yuan in his Record of Receiving says "yi times yi make a zhao, zhao times zhao make a jing," which is precisely the "upper number." Thus the same character "yi" does not mean the same size in each house. Zheng Xuan probably found the numbers too large to be convenient to name, and so merged and simplified them in speaking (in the base text "zhu" should be "gai," probably).

從億至載，終於大衍。按《易》（經）「大衍之數五十，其用四十有九」。又云，「天一、地二、天三、地四、天五、地六、天七、地八、天九、地十」。天數五，地數五。天數二十有五，地數三十。凡天地之數，五十有五也。

The Master of Tianmu said: counting the ten grades of large numbers from "yi" all the way to "zai," they still in the end return to the "number of the Great Expansion" in the Book of Changes. Zhen Luan notes: the Upper Appended Remarks of the Book of Changes say "the number of the Great Expansion is fifty, of which what is truly set in motion is forty-nine"; they also say "Heaven one, Earth two, Heaven three, Earth four, Heaven five, Earth six, Heaven seven, Earth eight, Heaven nine, Earth ten"—that is, the odd numbers one, three, five, seven, nine belong to Heaven, the even numbers two, four, six, eight, ten belong to Earth. The heavenly numbers are five, the earthly numbers also five; the heavenly numbers added come to twenty-five, the earthly numbers added come to thirty; the numbers of Heaven and Earth together come in all to fifty-five (in the base text a superfluous "jing" has been added after "yi").

下數淺短，計事則不盡。上數宏廓，世不可用。故其傅業，推以中數耳。餘時問曰：「先生之言上數者數窮則變，既云終於大衍，大衍有限，此何得〔無〕窮？」先生笑曰：「蓋未之思耳。數之為用，言重則變，以小兼大，又加循環。循環之理，豈有窮乎。」小兼大者，備加董氏《三等術數》。加更載為煩，故〔略〕焉。

The "lower number," advancing by tens, can express numbers that are too shallow and short; when a great matter is met it cannot be calculated to the end. The "upper number," advancing by self-multiplication, yields numbers that are too vast, and the world has no actual use for them. Therefore in transmitting this study what is current is still the "middle number" ("fu" should be "chuan," to transmit). I then pressed the question: "The Master says the upper number changes only when it is exhausted, and also that the ten grades of number in the end return to the Great Expansion; yet the number of the Great Expansion is only fifty-five, which is limited—how can one speak of the inexhaustible?" The Master smiled and said: "That is because you have not yet thought it through. The use of number is that when numbers pile upon one another to a certain point they advance one place, using the small unit to govern the large unit, and adding to that a cycling that is pressed on and on. This principle of cycling—where would it have an end?" (In the base text "ci he de qiong" has dropped a "wu," without.) Zhen Luan notes: on "using the small to embrace the large," the Arts of Number in Three Ranks of the Dong family treats it very completely; to enter it item by item here would be tedious, so it is omitted (the base text has dropped a "lue").

餘又問曰：「為算之體皆以積為名為復，更有他法乎？」先生曰：隸首注術乃有多種。及餘遺忘，記憶數事而已。其一積（等）〔算〕，其一太乙，其一兩儀，其一三才，其一五行，其一八卦，其一九宮，其一運（算）〔籌〕，其一了知，其一成數，其一把頭，其一龜算，其一珠算，其一計（算）〔數〕。

I asked further: "The forms of methods of calculation all take 'piling' as their name, relying on repeated addition to advance place; besides this, is there another path?" The Master said: tradition holds that the methods of calculation transmitted and annotated by Lishou were originally of many kinds, only that I have forgotten the greater part; what I can remember are no more than fourteen (in the base text "ji deng" should be "ji suan," piled counting; "yun suan" should be "yun chou," moving rods; "ji suan" should be "ji shu," counting in the mind). The distinctions among these fourteen lie mainly in the form of the calculating instrument and the number of calculating places. One, "piled counting," is the rod-calculation current at the time, using bamboo calculating rods laid out lengthwise and crosswise, piled layer upon layer. Two, "Taiyi calculation": nine tracks are cut on a board, and one calculating bead comes and goes up and down among the nine tracks, hence "coming and going on nine tracks." Three "Two Modes," four "Three Powers," five "Five Phases," six "Eight Trigrams" calculations, respectively according to the numbers of the Two Modes (Heaven and Earth), the Three Powers (Heaven, Earth and Man), the Five Phases and the eight trigrams, cut two, three, five or eight places to seat the beads. Seven, "Nine Palaces calculation," lays out numbers according to the Nine Palaces (九宮) positions "nine on the head, one underfoot, three on the left, seven on the right, two and four as the shoulders, six and eight as the feet, five occupying the centre"; the positions are fixed and do not move, so the text below says it "keeps to one and does not shift." Eight, "moving-rod calculation," turns the calculating rods between the fingers and palm, using the small to govern the large. Nine "Knowing at a Glance," ten "Completed Numbers," eleven "Holding the Head," twelve "Turtle Calculation," either use the palm and bodily positions as calculating places, or take their image from the form of a vessel, so that once the beads stop one can read the number at a glance, sparing the tedium of adding step by step. Fourteen, "counting in the mind," sets the instruments aside and calculates directly in the mind. The most important of them is thirteen, "bead calculation," which is the earliest surviving documentary record of the Chinese abacus: a wooden board is cut into three compartments; the upper and lower compartments are used to park beads that can travel, the middle compartment to mark the calculating places; each place seats five beads, one in the upper compartment of a different colour from the four in the lower, the upper one counting as five, each of the lower four as one; pushing the beads up and down one can add and subtract—precisely the embryo of the later abacus form "one beam dividing, one above and four below."

此等諸法隨須更位。唯有九宮守一不移，位依行色，並應無窮。從積〔算〕以來至珠算，從一至於百、千巳上，位更不變改。位依行色者，位依五行之色。北方水色黑，數一。東方木色青，數三。南方火色赤，數二。西方金色白，數四。中央土色黃，數五。

In operation these methods of calculation all change calculating place as the size of the number changes. Only "Nine Palaces calculation" holds fast to the home position and does not move; it distinguishes rank of place by the colour of the beads, and can therefore meet numbers infinitely large. From "piled counting" down to "bead calculation," whether one is calculating units, tens, or hundreds, thousands and above, the form of the calculating place itself is not further altered; once the places are many one must add further calculating places (the base text has dropped a "suan"; "si" should be "yi," already). Zhen Luan notes: so-called "places according to the colours of the phases" means settling calculating place by the colours paired with the Five Phases—the north belongs to Water, its colour black, its number one; the east belongs to Wood, its colour green, its number three; the south belongs to Fire, its colour red, its number two; the west belongs to Metal, its colour white, its number four; the centre belongs to Earth, its colour yellow, its number five.

言位依行色〔者〕，各一位第一用玄珠，十位第二用赤珠，百位第三用青珠，千位第四用白珠，萬位第五用黃珠〔十萬位以赤線繫黃株，百萬位以青蜒繫黃珠〕，千萬位以白綖繫黃珠，萬萬位曰億，以黃綖繫黃珠。自餘諸位唯兼之，故曰並應無窮也。

Zhen Luan goes on to explain the concrete use of "places according to the colours of the phases": the units place is the first place, using a black bead (a dark bead); the tens place is the second, using a red bead; the hundreds place is the third, using a green bead; the thousands place is the fourth, using a white bead; the myriads place is the fifth, using a yellow bead—exactly matching this set of Five Phases colours, Water black one, Fire red two, Wood green three, Metal white four, Earth yellow five. Further up, the hundred-thousand place uses a red silk thread tied to a yellow bead, the million place a green thread tied to a yellow bead, the ten-million place a white thread tied to a yellow bead, the myriad-myriad place is called "yi" and uses a yellow thread tied to a yellow bead (in the base text "zhu" should be "zhu," bead; "yan" should be "yan," that is, a silk thread). The still higher places, too, are no more than pairing bead-colour and thread-colour in this way and using them together, so it is said that Nine Palaces calculation "can meet inexhaustible numbers."