February 28, 2005

History topic: Overview of Chinese Mathematics

By Dr. Frank J. Collazo


Several factors led to the development of mathematics in China being, for a long period, independent of developments in other civilisations. The geographical nature of the country meant that there were natural boundaries (mountains and seas) which isolated it. On the other hand, when the country was conquered by foreign invaders, they were assimilated into the Chinese culture rather than changing the culture to their own. As a consequence there was a continuous cultural development in China from around 1000 BC and it is fascinating to trace mathematical development within that culture. There are periods of rapid advance, periods when a certain level was maintained, and periods of decline.

The first thing to understand about ancient Chinese mathematics is the way in which it differs from Greek mathematics. Unlike Greek mathematics there is no axiomatic development of mathematics. The Chinese concept of mathematical proof is radically different from that of the Greeks, yet one must not in any sense think less of it because of this. Rather one must marvel at the Chinese approach to mathematics and the results to which it led.

Chinese mathematics was, like their language, very concise. It was very much problem based, motivated by problems of the calendar, trade, land measurement, architecture, government records and taxes. By the fourth century BC counting boards were used for calculating, which effectively meant that a decimal place valued number system was in use. It is worth noting that counting boards are uniquely Chinese, and do not appear to have been used by any other civilisation.

Our knowledge of Chinese mathematics before 100 BC is very sketchy although in 1984 the Suan shu shu (A Book on Arithmetic) dating from around 180 BC was discovered. It is a book written on bamboo strips and was found near Jiangling in Hubei province. The next important books of which we have records are a sixteen chapter work Suanshu (Computational prescriptions) written by Du Zhong and a twenty-six chapter work Xu Shang suanshu (Computational prescriptions of Xu Shang) written by Xu Shang. Neither of these texts has survived and little is known of their content. The oldest complete surviving text is the Zhoubi suanjing (Zhou Shadow Gauge Manual) which was compiled between 100 BC and 100 AD (see the article on The Ten Classics). It is an astronomy text, showing how to measure the positions of the heavenly bodies using shadow gauges which are also called gnomons, but it contains important sections on mathematics. It gives a clear statement on the nature of Chinese mathematics in this period (see for example [2]:-

The method of calculation is very simple to explain but has wide application. This is because a person gains knowledge by analogy, that is, after understanding a particular line of argument they can infer various kinds of similar reasoning ... Whoever can draw inferences about other cases from one instance can generalise ... really knows how to calculate... . To be able to deduce and then generalise.. is the mark of an intelligent person.

The Zhoubi suanjing contains a statement of the Gougu rule (the Chinese version of Pythagoras's theorem) and applies it to surveying, astronomy, and other topics. Although it is widely accepted that the work also contains a proof of Pythagoras's theorem, Cullen in [39] disputes this, claiming that the belief is based on a flawed translation given by Needham in [13].

In fact much Chinese mathematics from this period was produced because of the need to make calculations for constructing the calendar and predicting positions of the heavenly bodies. The Chinese word 'chouren' refers to both mathematicians and astronomers showing the close link between the two areas. One early 'choren' was Luoxia Hong (about 130 BC - about 70 BC) who produced a calendar which was based on a cycle of 19 years.

The most famous Chinese mathematics book of all time is the Jiuzhang suanshu or, as it is more commonly called, the Nine Chapters on the Mathematical Art. The book certainly contains contributions to mathematics which had been made over quite a long period, but there is little in the original text to distinguish the precise period of each. This important work, which came to dominate mathematical development and style for 1500 years, is discussed in the article Nine Chapters on the Mathematical Art. Many later developments came through commentaries on this text, one of the first being by Xu Yue (about 160 - about 227) although this one has been lost.

A significant mathematical advance was made by Liu Hui (about 220 - about 280) who wrote his commentary on the Jiuzhang suanshu or Nine Chapters on the Mathematical Art in about 263. Dong and Yao write [24]:-

Liu Hui, a great mathematician in the Wei Jin Dynasty, ushered in an era of mathematical theorisation in ancient China, and made great contributions to the domain of mathematics. From the "Jiu Zhang Suan Shu Zhu" and the "Hai Dao Suan Jing" it can be seen that Liu Hui made skilful use of thinking in images as well as in logical and dialectical ways. He solved many mathematical problems, pushing his mathematical reasoning further along the dialectical way.

Liu Hui gave a more mathematical approach than earlier Chinese texts, providing principles on which his calculations are based. He found approximations to using regular polygons with 3 cross2n sides inscribed in a circle. His best approximation of was 3.14159 which he achieved from a regular polygon of 3072 sides. It is clear that he understood iterative processes and the notion of a limit. Liu also wrote Haidao suanjing or Sea Island Mathematical Manual (see the article on The Ten Classics) which was originally an appendix to his commentary on Chapter 9 of the Nine Chapters on the Mathematical Art. In it Liu uses Pythagoras's theorem to calculate heights of objects and distances to objects which cannot be measured directly. This was to become one of the themes of Chinese mathematics.

About fifty years after Liu's remarkable contributions, a major advance was made in astronomy when Yu Xi discovered the precession of the equinoxes. In mathematics it was some time before mathematics progressed beyond the depth achieved by Liu Hui. For example Sun Zi (about 400 - about 460) wrote his mathematical manual the Sunzi suanjing which on the whole provides little new. However, it does contains a problem solved using the Chinese remainder theorem, being the earliest known occurrence of this type of problem.

This text by Sun Zi was the first of a number of texts over the following two hundred years which made a number of important contributions. Xiahou Yang (about 400 - about 470) was the supposed author of the Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual) which contains representations of numbers in the decimal notation using positive and negative powers of ten. Zhang Qiujian (about 430 - about 490) wrote his mathematical text Zhang Qiujian suanjing (Zhang Qiujian's Mathematical Manual) some time between 468 and 486. Its 92 problems illustrate the formula for summing an arithmetic progression. Perhaps it is most famous for presenting the 'Hundred fowls problem' which is an indeterminate problem with three non-trivial solutions.

One of the most significant advances was by Zu Chongzhi (429-501) and his son Zu Geng (about 450 - about 520). Zu Chongzhi was an astronomer who made accurate observations which he used to produce a new calendar, the Tam-ing Calendar (Calendar of Great Brightness), which was based on a cycle of 391 years. He wrote the Zhui shu (Method of Interpolation) in which he proved that 3.1415926 < p < 3.1415927. He recommended using 355/113 as a good approximation and 22/7 in less accurate work. With his son Zu Geng he computed the formula for the volume of a sphere using Cavalieri's principle (see [25]). The beginnings of Chinese algebra is seen in the work of Wang Xiaotong (about 580 - about 640). He wrote the Jigu suanjing (Continuation of Ancient Mathematics), a text with only 20 problems which later became one of the Ten Classics. He solved cubic equations by extending an algorithm for finding cube roots. His work is seen as a first step towards the "tian yuan" or "coefficient array method" or "method of the celestial unknown" of Li Zhi for computing with polynomials.

Interpolation was an important tool in astronomy and Liu Zhuo (544-610) was an astronomer who introduced quadratic interpolation with a second order difference method. Certainly Chinese astronomy was not totally independent of developments taking place in the subject in India and similarly mathematics was influenced to some extent by Indian mathematical works, some of which were translated into Chinese. Historians argue today about the extent of the influence on the Chinese development of Indian, Arabic and Islamic mathematics. It is fair to say that their influence was less than it might have been, for the Chinese seemed to have little desire to embrace other approaches to mathematics. Early trigonometry was described in some of the Indian texts which were translated and there was also development of trigonometry in China. For example Yi Xing (683-727) produced a tangent table.

From the sixth century mathematics was taught as part of the course for the civil service examinations. Li Chunfeng (602 - 670) was appointed as the editor-in-chief for a collection of mathematical treatises to be used for such a course, many of which we have mentioned above. The collection is now called The Ten Classics, a name given to them in 1084.

The period from the tenth to the twelfth centuries is one where few advances were made and no mathematical texts from this period survive. However Jia Xian (about 1010 - about 1070) made good contributions which are only known through the texts of Yang Hui since his own writings are lost. He improved methods for finding square and cube roots, and extended the method to the numerical solution of polynomial equations computing powers of sums using binomial coefficients constructed with Pascal's triangle. Although Shen Kua (1031 - 1095) made relatively few contributions to mathematics, he did produce remarkable work in many areas and is regarded by many as the first scientist. He wrote the Meng ch'i pi t'an (Brush talks from Dream Brook) which contains many accurate scientific observations.

The next major mathematical advance was by Qin Jiushao (1202 - 1261) who wrote his famous mathematical treatise Shushu Jiuzhang (Mathematical Treatise in Nine Sections) which appeared in 1247. He was the first of the great thirteenth century Chinese mathematicians. This was a period of major progress during which mathematics reached new heights. The treatise contains remarkable work on the Chinese remainder theorem, gives an equation whose coefficients are variables and, among other results, Heron's formula for the area of a triangle. Equations up to degree ten are solved using the Ruffini-Horner method.

Li Zhi (also called Li Yeh) (1192-1279) was the next of the great thirteenth century Chinese mathematicians. His most famous work is the Ce yuan hai jing (Sea mirror of circle measurements). written in 1248. It contains the "tian yuan" or "coefficient array method" or "method of the celestial unknown" which was a method to work with polynomial equations. He also wrote Yi gu yan duan (New steps in computation) in 1259 which is a more elementary work containing geometric problems solved by algebra. The next major figure from this golden age of Chinese mathematics was Yang Hui (about 1238 - about 1298). He wrote the Xiangjie jiuzhang suanfa (Detailed analysis of the mathematical rules in the Nine Chapters and their reclassifications) in 1261, and his other works were collected into the Yang Hui suanfa (Yang Hui's methods of computation) which appeared in 1275. He described multiplication, division, root-extraction, quadratic and simultaneous equations, series, computations of areas of a rectangle, a trapezium, a circle, and other figures. He also gave a wonderful account of magic squares and magic circles.

Guo Shoujing (1231-1316), although not usually included among the major mathematicians of the thirteen century, nevertheless made important contributions. He produced the Shou shi li (Works and Days Calendar), worked on spherical trigonometry, and solved equations using the Ruffini-Horner numerical method. He also developed a cubic interpolation formula tabulating differences of the accumulated difference as in Newton's forward difference interpolation method.

The last of the mathematicians from this golden age was Zhu Shijie (about 1260 - about 1320) who wrote the Suanxue qimeng (Introduction to mathematical studies) published in 1299, and the Siyuan yujian (True reflections of the four unknowns) published in 1303. He used an extension of the "coefficient array method" or "method of the celestial unknown" to handle polynomials with up to four unknowns. He also gave many results on sums of series. This represents a high point in ancient Chinese mathematics.

The decline in Chinese mathematics from the fourteenth century was not by any means dramatic. The Nine Chapters on the Mathematical Art continued to be the model for mathematical learning and new works based in it continued to appear. For example Ding Ju published the Ding ju suan fa (Ding Ju's arithmetical methods) in 1355, He Pingzi published the Xiangming suan fa (Explanations of arithmetic) in 1373, Liu Shilong published the Jiu zhang tong ming suanfa (Methods of calculation in the 'Nine Chapters') in 1424, and Wu Jing published the Jiu zhang suan fa bi lei da quan (Complete description of the 'Nine Chapters') in 1450. Wu Jing was an administrator in the province of Zhejing and his arithmetical encyclopaedia contained all the 246 problems of the Nine Chapters. Again Cheng Dawei (1533 - 1606) published the Suanfa tong zong (General source of computational methods) in 1592 which is written in the style of the Nine Chapters on the Mathematical Art but provides an even larger collection of 595 problems.

The books we have just listed show mathematical activity, but they did not take forward the methods of polynomial algebra. On the contrary, the deep works of the 13th century ceased to be even understood much less developed further. Xu Guangqi (1562 - 1633) certainly recognised exactly this and offered possible explanations including scholars neglecting practical computational tools and an identification of mathematics with mystical numerology under the Ming dynasty. Other factors must be that the books describing the advanced methods were, in the Chinese tradition, very terse, and without teachers to pass on an understanding it became increasingly difficult for scholars to learn directly from the texts. Xu Guangqi was the first native of China to publish translations of European books in Chinese. Collaborating with Matteo Ricci he translated Western books on mathematics, hydraulics, and geography. Certainly this does not mark the end of the Chinese mathematics tradition, but from the time of Matteo Ricci and other Western missionaries China was greatly influenced by other mathematical traditions.

It is impossible in an article of this length to mention many of the numerous contributions from this period on. Let us mention one important family, however, namely the Mei family. The most famous member of this family was Mei Wending (1633-1721) and his comment on the golden section is typical of the sensible attitude he took towards Western mathematics (see for example [9]):-

After having understood how to make use of the golden section, I began to believe that the different geometrical methods could be understood and that neither the missionaries attitude of considering this simple technique as a divine gift, nor the Chinese attitude of rejecting it as heresy is correct.

Mei chose not to take a government post as most mathematicians did, but rather decided to devote himself to mathematics and its teaching. He travelled widely throughout China and gained great fame leading to many people becoming his pupils. Two of his brothers, Mei Wenmi and Mei Wennai, worked on astronomy and mathematics. Mei Wending was assisted later in his life by his son Mei Yiyan. Mei Juecheng (1681-1763), who was Mei Wending's grandson, was asked in 1705 by Emperor Kangxi to be editor-in-chief of the major mathematical encyclopaedia Shuli jingyun (Collected basic principles of mathematics) (1723). Mei Juecheng also edited his grandfather Mei Wending's work producing the Meishi congshu jiyao (Collected works of the Mei family) in 1761.

Certain people from the eighteenth century onwards did an excellent job in recording the Chinese tradition so that much of it is still accessible to us today. For example Dai Zhen (1724 - 1777) became an editor for the Siku quanshu (Complete library of the four branches of literature) which was a project set up by Emperor Qianlong in 1773. He edited a new edition of the Nine Chapters on the Mathematical Art after copying the complete text as part of this project. Ruan Yuan (1764 - 1849) produced his famous work the Chouren zhuan or Biographies of astronomers and mathematicians containing biographies of 275 Chinese and 41 Western "mathematicians". Many biographical details of Chinese mathematicians recorded in this Archive are known through this work. Li Rui (1768 - 1817) assisted Ruan Yuan. He was a highly productive mathematician who died when at the height of his abilities. His most important work is Lishi suan xue yi shu (Collected mathematical works of Li Rui).

It is to the credit of Chinese mathematicians that they did not let their mathematical tradition be replaced by the western tradition. For example Li Shanlan (1811-1882) is important as a translator of Western science texts but he is most famous for his own mathematical contributions. He produced his own versions of logarithms, infinite series, and combinatorics which did not follow the style of western mathematics but his research naturally developed out of the foundations of Chinese mathematics. There were many other efforts to promote Chinese mathematics, and in particular a mathematics journal, the Suanxue bao, was set up in 1899. The editors wrote:-

Western methods should not be adulated and Chinese methods despised.

Western mathematicians began lecturing in China during the early years of the twentieth century. For example Knopp taught there between 1910 and 1917, and Turnbull between 1911 and 1915. Chinese students began to study mathematics abroad and in 1917 Minfu Tah Hu obtained a doctorate from Harvard. China was represented for the first time at the International Congress of Mathematicians in Zürich in 1932. The Chinese mathematical Society was formed in 1935.

Article by: J J O'Connor and E F Robertson

December 2003

MacTutor History of Mathematics



History topic: Chinese numerals

In 1899 a major discovery was made at the archaeological site at the village of Xiao dun in the An-yang district of Henan province. Thousands of bones and tortoise shells were discovered there which had been inscribed with ancient Chinese characters. The site had been the capital of the kings of the Late Shang dynasty (this Late Shang is also called the Yin) from the 14th century BC. The last twelve of the Shang kings ruled here until about 1045 BC and the bones and tortoise shells discovered there had been used as part of religious ceremonies. Questions were inscribed on one side of a tortoise shell, the other side of the shell was then subjected to the heat of a fire, and the cracks which appeared were interpreted as the answers to the questions coming from ancient ancestors.

The importance of these finds, as far as learning about the ancient Chinese number system, was that many of the inscriptions contained numerical information about men lost in battle, prisoners taken in battle, the number of sacrifices made, the number of animals killed on hunts, the number of days or months, etc. The number system which was used to express this numerical information was based on the decimal system and was both additive and multiplicative in nature. Here is a selection of the symbols that were used.

By having multiplicative properties we mean that 200 is represented by the symbol for 2 and the symbol for 100, 300 is represented by the symbol for 3 and the symbol for 100, 400 is represented by the symbol for 4 and the symbol for 100, etc. Similarly 2000 is represented by the symbol for 2 and the symbol for 1000, 3000 is represented by the symbol for 3 and the symbol for 1000, 4000 is represented by the symbol for 4 and the symbol for 1000, etc. There was also a symbol for 10000 which we have not included in the illustration above but it took the form of a scorpion. However larger numbers have not been found, the largest number discovered on the Shang bones and tortoise shells being 30000.

The additive nature of the system was that symbols were juxtaposed to indicate addition, so that 4359 was represented by the symbol for 4000 followed by the symbol for 300, followed by the symbol of 50 followed by the symbol for 9. Here is the way 4359 would appear:  

Now this system is not a positional system so it had no need for a zero. For example the number 5080 is represented by:  

Because we have not illustrated many numbers above here is one further example of a Chinese oracular number. Here is 8873:  

There are a number of fascinating questions which we can consider about this number system. Although the representation of the numbers 1, 2, 3, 4 needs little explanation, the question as to why particular symbols are used for the other digits is far less obvious. Two main theories have been put forward.

The first theory suggests that the symbols are phonetic. By this we mean that since the number nine looks like a fish hook, then perhaps the sound of the word for 'nine' in ancient Chinese was close to the sound of the word for 'fish hook'. Again the symbol for 1000 is a 'man' so perhaps the word for 'thousand' in ancient Chinese was close to the sound of the word for 'man'. To take an example from English, the number 10 is pronounced 'ten'. This sounds like 'hen' so a symbol for a hen might be appropriate, perhaps modified so that the reader knew that the symbol represented 'ten' rather than 'hen'.

A second theory about the symbols comes from the fact that numbers, and in fact all writing in this Late Shang period, were only used as part of religious ceremonies. We have explained above how the inscriptions were used by soothsayers, who were the priests of the time, in their ceremonies. This theory suggests that the number symbols are of religious significance. Of course it is possible that some of the symbols are explained by the first of these theories, while others are explained by the second. Again symbols such as the scorpion may simply have been used since swarms of scorpions meant "a large number' to people at that time. Perhaps the symbol for 100 represents a toe (it does look like one), and one might explain this if people at the time counted up to ten on their fingers, then 100 for each toe, and then 1000 for the 'man' having counted 'all' parts of the body.

The symbols we have illustrated evolved somewhat over time but were surprisingly stable in form. However a second form of Chinese numerals began to be used from the 4th century BC when counting boards came into use. A counting board consisted of a checker board with rows and columns. Numbers were represented by little rods made from bamboo or ivory. A number was formed in a row with the units placed in the right most column, the tens in the next column to the left, the hundreds in the next column to the left etc. The most significant property of representing numbers this way on the counting board was that it was a natural place valued system. One in the right most column represented 1, while one in the adjacent column to the left represented 10 etc.

Now the numbers from 1 to 9 had to be formed from the rods and a fairly natural way was found.

Here are two possible representations:

The biggest problem with this notation was that it could lead to possible confusion. What was ||| ? It could be 3, or 21, or 12, or even 111. Rods moving slightly along the row, or not being placed centrally in the squares, would lead to the incorrect number being represented. The Chinese adopted a clever way to avoid this problem. They used both forms of the numbers given in the above illustration. In the units column they used the form in the lower row, while in the tens column they used the form in the upper row, continuing alternately. For example 1234 is represented on the counting board by:   and 45698 by:  

There was still no need for a zero on the counting board for a square was simply left blank. The alternating forms of the numbers again helped to show that there was indeed a space. For example 60390 would be represented as:  

Ancient arithmetic texts described how to perform arithmetic operations on the counting board. For example Sun Zi, in the first chapter of the Sunzi suanjing (Sun Zi's Mathematical Manual), gives instructions on using counting rods to multiply, divide, and compute square roots.

Xiahou Yang's Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual) written in the 5th century AD notes that to multiply a number by 10, 100, 1000, or 10000 all that needs to be done is that the rods on the counting board are moved to the left by 1, 2, 3, or 4 squares. Similarly to divide by 10, 100, 1000, or 10000 the rods are moved to the right by 1, 2, 3, or 4 squares. What is significant here is that Xiahou Yang seems to understand not only positive powers of 10 but also decimal fractions as negative powers of 10. This illustrates the significance of using counting board numerals.

Now the Chinese counting board numbers were not just used on a counting board, although this is clearly their origin. They were used in written texts, particularly mathematical texts, and the power of the place valued notation led to the Chinese making significant advances. In particular the "tian yuan" or "coefficient array method" or "method of the celestial unknown" developed out of the counting board representation of numbers. This was a notation for an equation and Li Zhi gives the earliest source of the method, although it must have been invented before his time.

In about the fourteenth century AD the abacus came into use in China. Certainly this, like the counting board, seems to have been a Chinese invention. In many ways it was similar to the counting board, except instead of using rods to represent numbers, they were represented by beads sliding on a wire. Arithmetical rules for the abacus were analogous to those of the counting board (even square roots and cube roots of numbers could be calculated) but it appears that the abacus was used almost exclusively by merchants who only used the operations of addition and subtraction.

Here is an illustration of an abacus showing the number 46802.

For numbers up to 4 slide the required number of beads in the lower part up to the middle bar. For example on the right most wire two is represented. For five or above, slide one bead above the middle bar down (representing 5), and 1, 2, 3 or 4 beads up to the middle bar for the numbers 6, 7, 8, or 9 respectively. For example on the wire three from the right hand side the number 8 is represented (5 for the bead above, three beads below).

One might reasonably ask why each wire contains enough beads to represent 15. This was to make the intermediate working easier so that in fact numbers bigger than 9 could be stored on a single wire during a calculation, although by the end such "carries" would have to be taken over to the wire to the left.

Article by: J J O'Connor and E F Robertson

January 2004

MacTutor History of Mathematics



History topic: Chinese problems

We give here a collection of Chinese problems which are extracted from various articles in our archive on Chinese mathematics or Chinese mathematicians. Many of the problems have answers given in the corresponding article, and some have a description of the method. Each problem has a reference to the article in which it occurs.

Problem 1: See Nine Chapters

A good runner can go 100 paces while a poor runner covers 60 paces. The poor runner has covered a distance of 100 paces before the good runner sets off in pursuit. How many paces does it take the good runner before he catches up the poor runner.

Problem 2: See Cheng Dawei

Boy shepherd B with his one sheep behind him asked shepherd A "Are there 100 sheep in your flock?". Shepherd A replies "Yet add the same flock, the same flock again, half, one quarter flock and your sheep. There are then 100 sheep altogether." How many sheep is in shepherd A's flock?

Problem 3: See Yang Hui

Now 1 cubic cun of jade weighs 7 liang, and 1 cubic cun of rock weighs 6 liang. Now there is a cube of side 3 cun consisting of a mixture of jade and rock which weighs 11 jin. Tell: what are the weights of jade and rock in the cube. [Note 1 jin = 16 liang]

Problem 4: See Sun Zi

Suppose that, after going through a town gate, you see 9 dykes, with 9 trees on each dyke, 9 branches on each tree, 9 nests on each branch, and 9 birds in each nest, where each bird has 9 fledglings and each fledgling has 9 feathers with 9 different colours in each feather. How many are there of each?

Problem 5: See Nine Chapters

Certain items are purchased jointly. If each person pays 8 coins, the surplus is 3 coins, and if each person gives 7 coins, the deficiency is 4 coins. Find the number of people and the total cost of the items.

Problem 6: See Nine Chapters

There are two piles, one containing 9 gold coins and the other 11 silver coins. The two piles of coins weigh the same. One coin is taken from each pile and put into the other. It is now found that the pile of mainly gold coins weighs 13 units less than the pile of mainly silver coins. Find the weight of a silver coin and of a gold coin.

Problem 7: See Nine Chapters

There is a square town of unknown dimensions. There is a gate in the middle of each side. Twenty paces outside the North Gate is a tree. If one leaves the town by the South Gate, walks 14 paces due south, then walks due west for 1775 paces, the tree will just come into view. What are the dimensions of the town.

Problem 8: See Sun Zi

Suppose we have an unknown number of objects. When counted in threes, 2 are left over, when counted in fives, 3 are left over, and when counted in sevens, 2 are left over. How many objects are there?

Problem 9: See Nine Chapters

A cistern is filled through five canals. Open the first canal and the cistern fills in 1/3 day; with the second, it fills in 1 day; with the third, in 21/2 days; with the fourth, in 3 days, and with the fifth in 5 days. If all the canals are opened, how long will it take to fill the cistern?

Problem 10: See Li Zhi

Given a circular walled city of unknown diameter with four gates, one at each of the four cardinal points. Two persons A and B start from the west gate. B walks a distance of 256 pu eastwards. Then A walks a distance of 480 pu south before he can see B. Find the diameter of the town.

Problem 11: See Li Zhi

Given a circular walled city of unknown diameter with four gates, one at each of the four cardinal points. Person A leaves the west gate and walks south for 480 pu. B leaves the east gate and walks straight ahead a distance of 16 pu, when he just sees A. Find the diameter of the town.

Problem 12: See Li Zhi

Given a circular walled city of unknown diameter with four gates, one at each of the four cardinal points. 135 pu directly out of the south gate is a tree. If one walks 15 pu out of the north gate and then turns east for a distance of 208 pu, the tree comes into sight. Find the diameter of the town.

Problem 13: See Qin Jiushao

Given a circular walled city of unknown diameter with four gates, one at each of the four cardinal points. A tree lies three li north of the northern gate. If one turns and walks eastwards for nine li immediately on leaving the southern gate, the tree just comes into view. Find the circumference and the diameter of the city wall.

Problem 14: See Li Zhi

A square farm has a circular pond in the centre. The land area is 13 mou and 71/2 tenths of a mou. The pond is 20 pu from the edge. Find the length of the side of the farm and the diameter of the pond.

Problem 15: See Cheng Dawei

Now a pile of rice is against the wall with a base circumference 60 chi and an altitude of 12 chi. What is the volume? Another pile is at an inner corner, with a base circumference of 30 chi and an altitude of 12 chi. What is the volume? Another pile is at an outer corner, with base circumference of 90 chi and an altitude of 12 chi. What is the volume?

Problem 16: See Cheng Dawei

A small river cuts right across a circular field whose area is unknown. Given the diameter of the field and the breadth of the river find the area of the non-flooded part of the field.

Problem 17: See Cheng Dawei

In the right-angled triangle with sides of length a, b and c with a > b > c, we know that a + b = 81 ken and a + c = 72 ken. Find a, b, and c.

Problem 18: See Zhu Shijie

A right-angled triangle has area 30 bu. The sum of the base and height of the triangle is 17 bu. What is the sum of the base and hypotenuse?

Problem 19: See Wang Xiaotong

Let a right angled triangle have sides a, b, c where c is the hypotenuse. If a times b is seven hundred and six and one fiftieth, and if c is thirty six and nine tenths more than a. What are the values of the three sides.

Problem 20: See Zhang Qiujian

A circular road around a hill is 325 li long. Three persons A, B, and C run along the road. A runs 150 li per day, B runs 120 li per day, and C runs 90 li per day. If they start at the same time from the same place, after how many days will they meet again.

Problem 21: See Zhang Qiujian

There are three persons, A, B, and C each with a number of coins. A says "If I take 2/3 of B's coins and 1/3 of C's coins then I hold 100". B says If I take 2/3 of A's coins and 1/2 of C's coins then I hold 100 coins". C says "If I take 2/3 of A's coins and 2/3 of B's coins, then I hold 100 coins". Tell me how many coins do A, B, and C hold?

Problem 22: See Zhang Qiujian

Cockerels costs 5 qian each, hens 3 qian each and three chickens cost 1 qian. If 100 fowls are bought for 100 qian, how many cockerels, hens and chickens are there?

Problem 23: See Yang Hui

100 coins buy Wenzhou oranges, green oranges, and golden oranges, 100 in total. If a Wenzhou orange costs 7 coins, a green orange 3 coins, and 3 golden oranges cost 1 coin, how many oranges of the three kinds will be bought?

Problem 24: See Yang Hui

A number of pheasants and rabbits are placed together in the same cage. Thirty-five heads and ninety-four feet are counted. Find the number of pheasants and rabbits.

Problem 25: See Zhu Shijie

Given the relations 2yz = z2 + xz and 2x + 4y + 4z = x(y2 - z + x) between the sides of a right angled triangle x, y, z where z is the hypotenuse, find d = 2x + 2y.

Problem 26: See Zhu Shijie

If the cube law is applied to the rate of recruiting soldiers and it is found that on the first day 3 cubed are recruited, 4 cubed on the second day, and on each succeeding day the cube of a number one greater than the previous day are recruited, how many soldiers in total will have been recruited after 15 days? How many after n days?

Problem 27: See Zhu Shijie

Let d be the diameter of the circle inscribed in a right triangle (you should use the relation d = x + y - z where x, y, z are as defined below). Let x, y be the lengths of the two legs and z the length of the hypotenuse of the triangle. Given that dxy = 24 and x + z = 9 find y.


Article by: J J O'Connor and E F Robertson

December 2003

MacTutor History of Mathematics