The Nine Chapters on the Mathematical Art Rod Calculus

9 Chapters on the Mathematical Art

The Jiuzhang suanshu or 9 Chapters on the Mathematical Fine art is a practical handbook of mathematics consisting of 246 problems intended to provide methods to be used to solve everyday issues of engineering, surveying, trade, and taxation. It has played a fundamental function in the evolution of mathematics in Cathay, not unlike to the role of Euclid'due south Elements in the mathematics which developed from the foundations gear up by the ancient Greeks. At that place is i major difference which we must examine right at the start of this article and this is the concept of proof.

It is well known what that Euclid, for example, gives rigorous proofs of his results. Failure to encounter similar rigorous proofs in Chinese works such as the Nine Chapters on the Mathematical Fine art led to historians assertive that the Chinese gave formulas without justification. This however is simply an case of historians well versed in mathematics which is essentially derived from Greek mathematics, thinking that Chinese mathematics was inferior since it was different. Recent work has begun to correct this false impression and understand that there are different understandings of "proof". For instance in [ 8 ] Chemla shows that Chinese mathematicians certainly understood how to requite convincing arguments that their methodology for solving particular problems was correct.

Let u.s. at present give a short description of each of the nine chapters of the book.

Chapter i: Land Surveying.
This consists of 38 problems on state surveying. It looks beginning at area problems, and then looks at rules for the addition, subtraction, multiplication and division of fractions. The Euclidean algorithm method for finding the greatest common divisor of two numbers is given. It then proceeds to further surface area problems which do non use the cloth on fractions which appears somewhat misplaced. The types of shapes for which the area is calculated include triangles, rectangles, circles, trapeziums. In Problem 32 an authentic approximation is given for π. This is discussed in detail in Liu Hui'due south biography.

Affiliate ii: Millet and Rice.
This chapter contains 46 issues concerning the exchange of appurtenances, especially the commutation rates amid twenty dissimilar types of grains, beans, and seeds. The mathematics involves a study of proportion and percentages and introduces the rule of three for solving proportion problems. Many of the problems seem simple an alibi to give the reader practise at handling difficult calculations with fractions.

Chapter 3: Distribution by Proportion.
Here in that location are xx problems which again involve proportion, many involving dissimilar sums given to or owed by officials of various dissimilar ranks. Direct proportion, changed proportion and compound proportion are all studied. In particular arithmetic and geometric progressions are used in some of the problems.

Affiliate iv: Short Width.
This affiliate contains 24 problems and takes its name from the first eleven problems which ask what the length of a field volition be if the width is increased but the area kept constant. These first eleven problems involve unit fractions are all of the following blazon, where $n = ii, 3, iv, ..., 12$ :

Suppose a field has width $ane+ \large\frac{i}{two}\normalsize + \large\frac{1}{three}\normalsize + ... + \large\frac{1}{northward}\normalsize$ . What must its length be if its area is one?

Problems 12 to 18 involve the extraction of foursquare roots, and the remaining problems involve the extraction of cube roots. Notions of limits and infinitesimals announced in this chapter. Liu Hui whose commentary of 263 Advertising has become part of the text attempts to find the volume of a sphere, gives an guess formula which he shows to be incorrect, then charmingly writes:-

Let us leave the problem to whoever can tell the truth.

Chapter 5: Ceremonious Applied science.
Here at that place are 28 problems on the structure of canals, ditches, dykes, etc. Volumes of solids such as prisms, pyramids, tetrahedrons, wedges, cylinders and truncated cones are calculated. Liu Hui, in his commentary, discusses a "method of burnout" he has invented to find the correct formula for the book of a pyramid.

Affiliate vi: Fair Distribution of Goods.
This affiliate contains 28 problems involving ratio and proportion. The problems are varied and business organization bug near travelling, tax, sharing etc. Problem 12 is a pursuit problem:-

A good runner tin can get 100 paces while a poor runner covers 60 paces. The poor runner has covered a distance of 100 paces earlier the proficient runner sets off in pursuit. How many paces does it take the adept runner before he catches up the poor runner.
[Answer: 250 paces]

Trouble 26 has become a classic blazon still used today:-

A cistern is filled through five canals. Open the commencement canal and the cistern fills in $\large\frac{i}{3}\normalsize$ day; with the second, it fills in 1 twenty-four hour period; with the third, in $2\large\frac{1}{2}\normalsize$ days; with the quaternary, in 3 days, and with the fifth in 5 days. If all the canals are opened, how long volition it take to fill up the cistern?
[Answer: $\large\frac{fifteen}{74}\normalsize$ of a day]

Affiliate vii: Excess and Arrears.
The 20 bug give a rule of double false position. Essentially linear equations are solved past making two guesses at the solution, so computing the correct answer from the two errors. For example to solve

$ax + b = c$

we try $x = i$ , and instead of $c$ we go $c + d$ . So nosotros try $ten = j$ , and instead of $c$ nosotros obtain $c + e$ . Then the correct solution is

$10 = (jd - ie)/(d - e)$ .

The first problem essentially contains the "guesses" in its formulation:-

Sure items are purchased jointly. If each person pays 8 coins, the surplus is three coins, and if each person gives 7 coins, the deficiency is iv coins. Detect the number of people and the total cost of the items.
[Respond: There are 7 people and the full cost of the items is 53 coins.]

Trouble 18, although not formulated every bit a "guessing trouble" is solved in that way:-

There are two piles, 1 containing 9 gold coins and the other 11 silver coins. The two piles of coins weigh the same. One money 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 argent coins. Detect the weight of a argent coin and of a gold coin.

Affiliate 8: Calculation by Square Tables.
Hither xviii problems which reduce to solving systems of simultaneous linear equations are given. However the method given is basically that of solving the system using the augmented matrix of coefficients. The issues involve upwardly to six equations in half dozen unknowns and the merely difference with the modern method is that the coefficients are placed in columns rather than rows. The matrix is then reduced to triangular form, using elementary column operations every bit is washed today in the method of Gaussian elimination, and the answer interpreted for the original problem. Negative numbers are used in the matrix and the affiliate includes rules to compute with them.

Affiliate ix: Right angled triangles.
In this last chapter at that place are 24 problems which are all based on right angled triangles. The offset thirteen problems are solved using an application of Pythagoras'southward theorem, which the Chinese knew every bit the Gougu dominion. Two problems study what are now called Pythagorean triples, while the remainder use the theory of similar triangles. Here is an instance of ane using similar triangles; information technology is Problem 20:-

There is a foursquare town of unknown dimensions. There is a gate in the centre of each side. 20 paces outside the North Gate is a tree. If ane leaves the town by the Due south Gate, walks 14 paces due south, and then walks due westward for 1775 paces, the tree will just come into view. What are the dimensions of the town.

In the diagram the North Gate is $Due north$ , the South Gate is $S$ , and the tree is $A$ . Walking southward from $Due south$ fourteen paces reaches $B$ , turn w and walk 1775 paces to $C$ . From $C$ the tree at $A$ is only visible so the line $CA$ passes through the corner $D$ of the square.

Now triangles $AND$ and $ABC$ are similar so

$AN/ND = AB/BC$

giving

$20/(x/two) = (xx + x + fourteen)/1775$ .

And so $x^{two} + ten(xx + 14) = 2 (xx\times 1775)$ , or

$x^{2} + 34x = 71000$ .
[Answer: The side of the boondocks is 250 paces]

Quadratic equations are considered for the first fourth dimension in Chapter nine, are solved by an analogue of division using ideas from geometry, in fact from the Chinese square-root algorithm, rather than from algebra.

Having looked at the content of the work, let us think next about its date. Liu Hui wrote a commentary on the Ix Chapters on the Mathematical Fine art in 263 AD. He believed that the text which he was commentating on was originally written around 1000 BC but incorporated much fabric from subsequently eras. He wrote in the Preface:-

In the past, the tyrant Qin burnt written documents, which led to the destruction of classical noesis. Afterwards, Zhang Cang, Marquis of Peiping and Geng Shouchang, Vice-President of the Ministry building of Agriculture, both became famous through their talent for calculation. Because of the ancient texts had deteriorated, Zhang Cang and his squad produced a new version removing the poor parts and filling in the missing parts. Thus, they revised some parts with the result that these were different from the former parts ...

Let united states of america give some dates for the events Liu Hui describes. The Qin dynasty preceded the Han dynasty and it was the Qin ruler Shih Huang Ti who tried to reform education past destroying all earlier learning. He ordered all books to be burnt in 213 BC and Zhang Cang, who Liu Hui refers to, did his reconstruction around 170 BC. Most historians, all the same, would non believe that the original text of the 9 Chapters on the Mathematical Art was nearly as quondam every bit Liu Hui believed. In fact most historians think that the text originated effectually 200 BC later on the burning of the books by Shih Huang Ti. Others give dates betwixt 100 BC and 50 Advertisement.

What methods are used to effort to appointment the material? Perhaps the most of import is to examine the units of length, volume and weight which appear in the diverse bug. Standard decimal units of length were established in Prc around 200 BC and later further subdivisions occurred. That the bones units are used, but not the afterward subdivisions, leads to a date of shortly later on 200 BC. In Liu Hui's commentary subdivisions introduced around 250 AD are used, which is in line with this commentary beingness written in 263 Advertizement.

Of grade, the dating using units of length is not conclusive. Consider the fact that U.k. inverse to a decimal currency in 1970. If you choice upwards a book with mathematics issues given in decimal currency then we could argue as above and say that the volume was written later 1970. Notwithstanding new editions of popular textbooks were brought out when the currency changed, and so many older books appeared in decimal editions. The 9 Chapters on the Mathematical Art was certainly an important text, and then may have had its units of length brought up to date equally it evolved.

Is there other evidence for dating parts of the Ix Chapters on the Mathematical Art other than units of measurement? Yes, there are. Bug contain references to taxes, methods of distributing goods, towns, and parks which all point to slightly different dates for different parts of the text only 206 BC to 50 Advertising covering these different dates.

In addition to Liu Hui'south commentary of 263, there was some other important later commentary, namely that of Li Chunfeng whose commentary was written around 640 when he headed a team asked to comment The Ten Classics. Li Chunfeng corrected and clarified some of Liu Hui'south comments, expanding on much of what had been pretty concisely written.

The 9 Chapters on the Mathematical Art [ 4 ] :-

... has dominated the history of Chinese mathematics. It served equally a textbook not only in Cathay simply too in neighbouring countries and regions until western science was introduced from the Far E around 1600 Ad.

Now although European science does not appear to have reached China in sixteenth century, it has been pointed out that a number of mathematical formulas and rules which were widely used in Europe during that century are essentially identical to formulas written down in the Nine Chapters on the Mathematical Art. This leads to an interesting question which historians have every bit however no convincing respond, namely were the European formulas taken directly from those of China.

Additional Resources (evidence)

Written by J J O'Connor and E F Robertson
Final Update December 2003

purselconfled.blogspot.com

Source: https://mathshistory.st-andrews.ac.uk/HistTopics/Nine_chapters/

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