The ancient Chinese counting board

Transcription

1 Chapter 5 The ancient Chinese counting board 5.1 The counting rods The numeration system using counting rods is a decimal positional system. A number is represented by a collection of counting rods. This representation is simplified by first placing all the rods in the rightmost position in a (horizontal) row of squares, and replacing every group of ten rods by a single one in its immediate left neighbour. By applying this rule to every group of ten rods, beginning with the rightmost position one eventually obtains the decimal representation of the number, consisting of a horizontal array of groups of counting rods, each not more than nine, and possibly empty. To avoid confusion, the rods in adjacent squares are arranged in perpendicular directions. If a square contains more than five rods, then five of these rods are replaced by one placed in a perpendicular direction vertical horizontal For example, six thousand thirty seven is represented by with an empty space in the hundreds position.

2 28 The ancient Chinese counting board 5.2 Addition and subtraction It is clear that addition and subtraction can be easily carried out simply by adding and removing rods. For subtraction, one often has to replace a rod in a square by ten rods in its immediate right neighbour. 5.3 Multiplication Problem I.1 of Sunzi Suanjing gives an example of how multiplication is carried out with counting rods. 1 It must be emphasized that although a succession of tables is shown here, the actual calculation is done in the same three lines, moving through a number of stages. Nine nines are eighty one. Find the amount when this is multiplied by itself. Answer: Method: Set up the two positions: [upper and lower]. The upper 8 calls the lower 8: eight eights are 64, so put down 6400 in the middle position. The upper 8 calls the lower 1: one eight is 8, so put down 80 in the middle position. Shift the lower numeral one place [to the right] and put away the 80 in the upper position. The upper 1 calls the lower 8: one eight is 8, so put down 80 in the middle position. The upper 1calls the lower 1: one one is 1, so put down 1 in the middle position. Remove the numerals in the upper and lower positions leaving 6561 in the middle position. (1) The multiplication of large number depends on the multiplication of single digit numbers. The standard Chinese begins with nine nines is 81 down to one one is 1. (2) Multplication is performed on a three-row counting board. multiplicand product multiplier The unit of the multiplier is initially placed directly under the leftmost digit of the multiplicand. 1 Translation by Lam and Ang, p.196.

3 5.3 Multiplication 29 multiplicand product multiplier multiplicand product multiplier multiplicand product multiplier multiplicand product multiplier multiplicand product multiplier

4 30 The ancient Chinese counting board multiplicand product multiplier 5.4 Division Problem I.2: If 6561 is divided among 9 persons, find how much each gets. Answer: 729. Method: First set 6561 in the middle position to be the dividend (shi). Below it, set 9 persons to be the divisor (fa). Put down 700 in the upper position. The upper 7 calls the lower 9: seven times nines are 63, so remove 6300 from the numeral in the middle position. Shift the numeral in the lower position one place [to the right] and put down 20 in the upper position. The upper 9 calls the lower 9: nne nines are 81, so remove 81 from thenumeral in the middle position. There is now no numeral in the middle position. Put away the numeral in the lower position. The result in the upper position is what each person gets. quotient dividend divisor (1) The divisor is initially placed in the bottom line that it can be subtracted by the number above it in the dividend. (2) The quotient is called shang in Chinese, meaning guess and see to figure out. In each case, it amounts to finding the largest number which multiplied to the divisor does not exceed the dividend. This can be done by repeatedly subtracting the divisor from the dividend. The Chinese term for division is chu, removal.

5 5.4 Division 31 quotient dividend divisor quotient dividend divisor quotient dividend divisor quotient dividend divisor quotient dividend divisor

6 32 The ancient Chinese counting board quotient dividend divisor quotient dividend divisor quotient dividend divisor Lam considered the replacement of counting rods by the abacus (in the 16th century) a cause of decline of mathematics in China. The use of the abacus to shorten the time of calculation necessitated the rote learning of numerous mathematical methods. The rigorous step-by-step reasoning so essential to the development of mathematics is discarded, ad inevitably, mathematics declined.

7 Chapter 6 Fractions in Fangtian 6.1 Jiuzhang Suanshu I.1-4 I.1 There is a field 15 bù wide and 16 bù long. Question: How much is (the area of) the field? Answer: 1 mǔ. I.2 There also is a field 12 bù wide and 14 bù long. Question: How much is (the area of) the field? Answer: 168 [square] bù. Rectangular field rule: Multiply the width and the length to obtain the (number of) square bù. L1 Divide this by 240, the standard of mǔ. This is (the area in) number of mǔ. One hundred mǔis one qǐng. L1: This product is the area. Every product of a width and a length is called mì. I.3 There is a field 1 lǐ wide and 1 lǐ long. How much is (the area of) the field? Answer: 3 qǐng 75 mǔ. I.4 There also is a field 12 bù wide and 14 bù long. How much is (the area of) the field? Answer: 168 [square] bù. Rule of field in lǐ Multiply the numbers of lǐ in width and length to obtain (the number of) square lǐ; multiply this by 375. This is (the area in) number of mǔ. L2 L2: Note that in this method the numbers of lǐ in width and length multiply to give the number of square lǐ. There are 3 qǐng 75 mǔ in a square lǐ, so multiplying by this number, one obtains the number in mǔ.

8 34 Fractions in Fangtian 6.2 Reduction of fractions I.5 There are 12 parts of 18. How much are these reduced to? Answer: 2 parts of 3. I.6 There also are 49 parts of 91. How much are these reduced to? Answer: 7 parts of Rule of reduction of fractions L3 If (both quantities are) halvable, half them. If not (both) halvable, place the denominator and numerator side by side. Keep on subtracting the lesser from the more, to find (two) equal numbers. Now, reduce (the denominator and the numerator) by the (value of the) equal numbers. L4 L3: As to reduction of fractions, note that the quantities of things may not always be whole, and it is necessary to speak of fractions. Fractions are difficult to use if they are too complicated. Suppose there are 2 parts of 4. In a more complicated expression, these are the same as 4 parts of 8. However, in reduced for, this is 1 part of 2. The expressions might be different, but as numbers, they serve the same end. In comparing the divisor with the dividend, one often finds unevenness. Therefore, the practitioner must first learn to deal with fractions. L4: To reduce by the děng shù is to divide by it. Those (numbers) involved in the subtractions being all multiples of the děng shù, we reduce (the given quantities) by it. 6.3 Addition of fractions I.7 There are one parts of three, two parts of five. How much do we get by adding them? Answer: eleven parts of fifteen. I.8 There are also two parts of three, four parts of seven, and five parts of nine. How much do we get by adding them? Answer: get one and fifty parts of sixty three. I.9 There are also two parts of two, two parts of three, three parts of four, and four parts of five. How much do we get by adding them? Answer: get two and forty three parts of sixty. Rule of addition of fractions Let the denominators cross multiply (hùchéng) (each) numerator; combine the results as dividend. Multiply the denominators together for the divisor. L5 Compare

9 6.4 Subtraction of fractions 35 the dividend to the divisor as unit. If there is any remainder, make it the numerator of a fraction with the divisor as denominator. L6 When all denominators are the same, directly add up (the numerators). L5: Let the denominators cross multiply (hùchéng) (each) numerator, (A fraction) in its reduced form has a coarse subdivision (of its denominator), whereas with a complicated expression, such a subdivision is fine. Fine or coarse, they represent the same fraction. Given a number of fractions, intricate and mixed (in that their denominators are different), one cannot combine them without finely subdividing (their denominators). Multiply (both denominator and numerator of each fraction by the same number) to sàn the fractions, so as to communicate them. With a common denominator, (the numerators) can be combined together. Letting the denominators cross multiply (hùchéng) (each) numerator is called homogenize (qì). Multiplying the denominators together is called uniformize (tóng). To uniformize (Tóng) is to multiply the denominator and the numerator of each fraction by the same number, so as to make the fractions uniformize (tóng), i. e. sharing a common denominator. To homogenize (Qí) is to align the numerator and the denominator; it is essential not to alter the value (of the fraction). Methods are classified into types; things are divided into groups. Numbers of the same kind are not far; whereas those of different kinds are not near. For those with the same denominator, even their numerators are far in that they occupy different places (on a calculating board), these can be added together. (On the other hand), for those with different denominators, even if their numerators are so near as to be in the same column, they are still at odd with one another. The methods of homogenize (qì) and uniformize (tóng) are very important indeed! The (given) fractions may be in disaccord; but (by such methods) we can bring them into harmony. This is like equipping with an ivory cone for untying knots, always managing (the fractions) orderly. Multiplying to sàn fractions, dividing to reduce them, homogenize (qì) and uniformize (tóng) to make them tōng. Is this not the guiding principle of calculation? As to the method, we may let lù be the product of all but one denominators, and multiply this lù to the numerator to get homogenize (qì). L6: Now to find the dividend, first homogenize (qì) the numerators an make the denominators common. Compare (this dividend) with the divisor as unit. Reduce any remainder (with the divisor) by the děng shù. That is it. This is to say the common divisor as denominator, and the remainder from the dividend as numerator. All (additions of fractions) follow these examples. 6.4 Subtraction of fractions I.10 There are 8 parts of 9. Subtract from it 1 part of 5. How much is the remainder? Answer: 31 parts of 45.

10 36 Fractions in Fangtian I.11 There also are 3 parts of 4. Subtract from it 1 p of 3. How much is the remainder? Answer: 5 parts of 12. Rule for subtraction of fractions Let the denominators cross multiply (hùchéng) the numerators. Subtract the smaller (product) from the greater, L7 (and take) the remainder as dividend. Multiply the denominators to form the divisor. (Compare) the dividend to the divisor as unit. I.12 There are 5 parts of 8, and 16 parts of 25. Which is more, by how much? Answer: 16 parts of 25 is more, by 3 parts of 200. I.13 There also are 8 parts of 9, and 6 parts of 7. Which is more, by how much? Answer: 8 parts of 9 is more, by 2 parts of 63. I.14 There are 8 parts of 21, and 17 parts of 50. Which is more, by how much? Answer: 8 parts of 21 is more, by 13 parts of Rule for comparison of fractions Let the denominators cross multiply (hùchéng) the numerators. Subtract the smaller (product) from the greater, L7 (and take) the remainder as dividend. Multiply the denominators to form the divisor. (Compare) the dividend to the divisor as unit. That is the difference L7: (To let) the denominators cross multiply (hùchéng) the denominator is to homogenize (qì) the numerators. (We may) subtract the smaller from the greater since the numerators are homogeneous (qì). To multiply the denominators together as the divisor is to make a common denominator. Since the denominator and the numerator are aligned, (we may) compare (the difference) to the denominator as unit. QED. 6.5 I.15 There are one part of three, two parts of three, and three parts of four. How much should be subtracted from the more to benefit the fewer to make them the same level? Answer: subtract two from three parts of four, one from two parts of three, combine them to benefit one part of three; then they are all level at seven parts of twelve. I.16 There are one part of two, two parts of three, and three parts of four. How much should be subtracted from the more to benefit the fewer, to make them

11 the same level? Answer: subtract one from two parts of three, four from three parts of four, combine them to benefit one part of two; then they are all level at twenty three parts of thirty six Rule for averaging of fractions Let the denominators cross multiply (hùchéng) each numerator, L8. Combine (the products) to be the péngshí. Mutliply the denominators together to form the divisor. L9 Multiply each summand (of the péngshí) by the number of columns to form the column dividends. Also multiply the divisor by the number of columns. Subtract the péngshí from the column dividends. Reduce the difference and take these for (the amounts) to be subtracted. Combine the amounts subtracted to benefit the smallest one. Form a fraction with the péngshí as numerator and the division as denominator. (Now), each fraction achieves the average. L8: This is to align the numerators. L9: (Note on) multiplying the denominators together for the divisor: after aligning their numerators, now make their denominators common. L10: Here, we should have divided the péngshí by the number of columns. But then, we would run into compound fractions. So, instead, we multiply the divisor by the number of columns. 6.6 I.17 There are seven persons dividing eight qián, one part of three qián. How much does each one get? Answer: each gets one qián, four parts of twenty one qián. I.18 There also are three persons, one part of three persons, dividing six qián, one part of three qián, and three parts of four qián. How much does each one get? Answer: each gets two qián, one part of eight qián. Rule for division of fractions Method The number of persons as divisor, the number of qián as dividend, compare the dividend with the divisor as unit. Convert any mixed fractions into improper fractions. L11 For compound fractions, make he denominators (of the denominator and the numerator) common. L12 L11: To let the denominators cross multiply (hùchéng) the numerators is to align the numerators; to multiply the denominators is to make the denominators common. To uniformize (tóng) with the denominators is to multiply the whole number by the denominator and to bring (the product) into the numerator. Multiplying the whole

12 38 Fractions in Fangtian number (by the denominator) is to sàn it into a jīfèn. Then the jīfèn and the fraction part are tōng, and can be added up together. Every number used to multiply to both the denominator and the numerator of a fraction is called 1ù. The (various) 1ù are so chosen to make the fractions tōng. If there are mixed fractions, convert them into improper fractions. If the denominator and the numerator of) the fraction contain common multiples, reduce them. Divide the divisor and the dividend by the děny shù. This děny shù is the lù to multiply (to the denominator and the numerator) to make the fraction (assume its current form). Therefore, to sàn the fractions (in the denominator and the numerator), we let both denominators multiply the divisor and the dividend. L12: Also, (when one of the numerator and the denominator is a whole number), multiply the dividend by the denominator of the divisor, or the visor by the denominator of the dividend, (as appropriately). This (last statement in the Method) is to say, if the divisor and the dividend are both mixed fractions, then we multiply each denominator to the whole number and bring it into the numerator, and then let he denominators cross multiply (hùchéng) (the numerators) in the top and the bottom (of the compound fraction). 6.7 I.19 There is a field 7 parts of 9 bù wide, 9 parts of 11 bù long. How much is (the area of) the field? Answer: 7 parts of 11 bù There also is a field 4 parts of 5 bù wide, 5 parts of 9 bù long. How much is (the area of) the field? Answer: 4 parts of 9 bù. Rule for multiplication of fractions Multiply the denominators together for the divisor; multiply the numerators together for the dividend. Compare the dividend to the divisor as unit. L13 L13: Whenever (the dividend) does not measure up to the divisor, (one obtains a proper fraction with) denominator. and numerator (only). If there is (second) fraction which, when multiplied to it, makes the dividend exceed the divisor, then there results in a whole number as well. Also, as the numerators are multiplied together, the denominators must be effected in division, i.e., comparing the dividend to the divisor as unit. Since multiplying the numerators together requires at the same time that the denominators be effected into division, we multiply the denominators together and perform division subsequently. In speaking about (areas of rectangular) fields, (one encounters two analogous quantities) width and length, and it is diffi-

13 cult to illustrate (with such the connection between this and the previous method). Consider, therefore, the problem: 20 horses are worth 12 jin of gold. Now, 20 horses are sold and the profit is shared among 35 persons. How much does each one get? Answer: 12 parts of 35 jin. This follows from the division of fractions method, with 12 jin of gold as dividend and 35 persons as divisor. Let us now reformulate the problem: 5 horses are worth 3 jin of gold. Now 4 horses are sold and the profit is shared among 7 persons. How much does each one get? Answer: 12 parts of 35 jin of gold. This follows by aligning the numbers of (jin) gold and persons, and then the rule for division of fractions, as in the previous problem. Now, to multiply the numerators for the dividend is like aligning the (number of jin) gold, and multiplying the denominators for the divisor is like aligning the (number of) persons. The common denominator 20 has no relevance in the calculation. One simply wants to align the (numbers of jin of gold and persons). Also, for 5 horses worth 3 jin of gold, these number are all whole. In terms of fractions, a horse is worth 3 part of 5 jin of gold. (Since) 7 persons sell 4 horses, (one may regard) each person selling 4 parts of 7 horses. (The answer is now obtained from) the numerator (the dividend in the division above) and turning upside down (the divisor, the fraction obtained by considering) persons (as above), by multiplying together. The expressions are different, but all three methods lead to the same answer. 6.8 I.21 There is a field 3 bù and 1 part of 3 bù wide, 5 bù and parts of 5 bù long. How much is (the area of) the field? Answer: 18 bù. I.22 There also is a field 7 bù and 3 parts of 4 bù wide, 15 b and 5 parts of 9 bù long. How much is (the area of) the field? Answer: 120 bù and 5 parts of 9 bu. I.23 There also is a field 18 bù and 5 parts of 7 bù wide, 2 bù and 6 parts of 11 bù long. How much is (the area of) the field? Answer: 1 mu 200 bù and 7 parts of 11 bu.

14 40 Fractions in Fangtian Rule for most general field Multiply each denominator t the whole number, and add to the numerator. L14 Multiply these (sums) for the dividend, and multiply the denominators for the divisor. L15 Compare the dividend to the divisor as unit. L16 L14: Multiply each denominator to the whole number and ad to the numerator. This is to tōng the whole number of bù and bring into the numerator. Then both the denominator and the numerator are whole numbers. L15: (This is) like the multiplication of fractions (above). L16: In this Method, both width and length are mixed f actions. Each (of these) should be converted into an improper fraction. As the denominators have be brought into (their own numerators), they need to be brought out again (after the multiplication). We therefore multiply the denominators for the divisor and carry out the division subsequently.

15 Chapter 7 Ancient Chinese extraction of square root In ancient China, arithmetic operations were performed using calculating rods arranged on a counting board. The extraction of a square root is carried out in a 4-row arrangement, each column for one digit: (i) The second row is labelled Dividend. One begins by entering in this row the number whose square root is to be extracted. (ii) The fourth row is labelled Unit. Beginning with a counting rod in the column of the rightmost digit, one repeatedly shifts this 2 places to the left before it reaches beyond the leftmost digit of the dividend. (iii) The first row is labelled Quotient, in this row shall appear the square root in question. (iv) The third row, labelled Divisor, is auxiliary in the computation. We illustrate this with Problem I.16 of Jiuzhang Suanshu on the extraction of the square root of 3,972,150,625. Quotient Dividend Divisor 1 Unit The computation begins with finding the largest number whose square does not exceed the number with unit indicated in the bottom row. In this case, the largest square not exceeding 39 is 6 2 =36. Here one enters 6 in the first row, 1 and subtract 36 from the second row. 1 The square root in question being a 5-digit number, the leftmost digit 6 should be entered in the 5th column from the right. There is, however, definite advantage in aligning it with the unit rod in the bottom row.

16 42 Ancient Chinese extraction of square root 6 Quotient Dividend Divisor 1 Unit Multiply the quotient by 2, enter it in the third row (as divisor), shifting one place to the right. Shift also the unit rod in the bottom row two places to the right. 6 Quotient Dividend 1 2 Divisor 1 Unit In the column of the unit rod, put the largest number which, when multiplied to the number with the same units digit in the third row, gives a product not exceeding the number in the second row. In the present case, this number is 3. Subtract the product from the second row. 6 3 Quotient Dividend Divisor 1 Unit Double the number in the first row, enter it in the third row, 2 shifting one place to the right. Shift the unit rod in the bottom row two places to the right. 6 3 Quotient Dividend Divisor 1 Unit Repeat the same procedure till the unit rod appears at the end. The number appearing in the first row would be the square root. In the present example, the next number in the first row is 0. Thus, one simply shifts the divisor one place to the right, and the unit rod two places to the right Quotient Dividend Divisor 1 Unit Quotient Dividend Divisor 1 Unit 2 This has the same effect as doubling the unit digits of the divisor in the third row.

17 Quotient Dividend Divisor 1 Unit The next number should be 5. Since = , this leaves the second row blank, and the calculation terminates, giving for the square root Quotient Dividend Divisor 1 Unit If the dividend appearing in the last step is a nonzero number r, the square root in question is not exact. In this case, the ancient Chinese adopted one of the following options. r (i) Round off the square root with the fraction or r, q being the quotient 2q+1 2q in the first row. In other words, q2 + r q + r 2q +1 or q + r 2q. (ii) Continue the calculation beyond the decimal point by treating a unit as 10 subunits (fen), 100 sub-subunits (li), 1000 sub-sub-subunits (hou), etc. Exercise Find the square roots of the following numbers (i) 55225, (ii) 25281, (iii) 71824, (iv) 564, A diagrammatic explanation of the procedure of extraction of square roots In Jiuzhang Suanshu, all square roots are exact. In explaining the extraction of square root, the text says dang yi mian ming zhi. Earlier commentators interpreted this as rounding off the square root with the r fractions or r. LI Jimin, however, explains that mian refers to the square root 2q+1 2q itself, so that what the text means is that Q = + r. Q

18 44 Ancient Chinese extraction of square root Appendix: A paper-and-pencil algorithm for square roots To find the square root of an integer a, divide the digits of a into blocks of two digits beginning with the right hand side. We represent this as a 1 a 2 a n where each a k, k =2,...,nis a 2-digit number, and a 0 has either 1 or 2 digits. (1) Set b 1 := a 1, and let q 1 be the largest integer such that r 1 := a 1 q Set Q 1 := q 1. (2) Suppose b n, r n and Q n have been defined. Form Find the largest integer q n+1 such that Set Q n+1 := 10Q n + q n+1. b n+1 := 100r n + a n+1. r n+1 := b n+1 (20Q n + q n+1 )q n

19 Chapter 8 Areas of plane figures 8.1 Problems I Here are some area problems from the first chapter of The Nine Chapters on the Mathematical Art. 1 In these problems, bu is a unit of length. It literally means step. mu is a unit of area. One mu consists of 240 square bu. I.19 Now given a field, 4 bu in breadth and bu in length. I.20 Again, given another field, 7 9 Question: What is the area? 9 bu in breadth and Answer: 11 bu in length. I.21 Again, given another field, 4 bu in breadth and bu in length. I.22 Now given a field, 3 1 bu in breadth and bu in length. I.23 Given another field, 7 3 bu in breadth and bu in length. I.24 Given another field, 18 5 bu in breadth and in length. I.25 Now given a triangular field with base 12 bu and altitude 21 bu. 1 Answers: I.19: [square] bu. I.20: 7 11 [square] bu. I.21: 4 9 [square] bu. I.22: 18 [square] bu. I.23: [square] bu. I.24: 1 mu [square] bu. I.25: 126 [square] bu. I.26: [square] bu. I.27: 9 mu 144 [square] bu. I.28: 23 mu 70 [square] bu. I.29: 1 mu 135 [square] bu. I.30: 46 mu [square] bu. I.31: 75 [square] bu. I.32: 11 mu [square] bu. I.33: 120 [square] bu. I.34: 5 mu [square] bu. I.35: 1 mu [square] bu. I.36: 2 mu [square] bu. I.37: 2 mu 55 [square] bu. I.38: 4 mu [square] bu.

20 46 Areas of plane figures I.26 Given another triangular field with base 5 1 bu and altitude 8 2 bu. 2 3 I.27 Now given a right-angled trapezoidal field with bases 30 bu and 42 bu respectively and altitude 64 bu. I.28 Given another right-angled trapezoidal field with bases 72 bu and 100 bu respectively and width 65 bu. I.29 Now given a trapezoidal field with bases 20 bu and 5 bu respectively and altitude 30 bu. I.30 Given another trapezoidal field with bases 117 bu and 50 bu respectively and altitude 135 bu. 8.2 The rules The rule for multiplying fractions Multiply the denominators as divisor; multiply the numerators as dividend. Divide. LIU Hui s commentary: When the dividend is smaller than the divisor, then [one] gets a [proper] fraction. When the numerator is multiplied by the dividend, the product may be larger than the denominator (divisor), thus yielding an integer. If the numerator is multiplied, the product should be divided by the denominator. Since the product of [all the] numerators is taken as dividend, it should be divided by the product of [all the] denominators, i.e. take the continued product of the denominators as divisor. The mere length and width of a rectangular field leave no room for further explanation. Let someone ask: 20 horses are valued at 12 jin of gold. Now 20 horses are sold and the proceeds are shared by 35 persons. How much does each one get? The answer is 12 jin. By the Rule for Division, 12 jin is the 35 dividend and 35 persons the divisor. Let someone ask again: 5 horses are valued at 3 jin of gold. Now 4 horses are sold and [the proceeds] shared by 7 persons. How much does each one get? The answer is 12 jin each. By the Homogenization and 35 Uniformization Rule one can get the same answer as by the Rule for Division. Here, the product of [all] the numerators is taken as dividend means homogenizing the gold; the product of [all the] denominators...asdivisor means homogenizing the persons and uniformizing the horses. Though the common denominator 20 [tong] is not in question, it is merely for homogenizing when in the case of 5 horses valued at 3 jin of gold, both the numerator and denominator are integers. Alternatively,

21 8.2 The rules 47 using fractions, 1 horse is worth 3 jin of gold. 7 persons selling 4 horses means 5 each sells 4 horse. The numerical relationship between horses, gold and persons 7 may be considered in different ways, yet the results are identical all three ways The general rule for rectangular fields Multiply each denominator by its integral part; then add the corresponding numerators. Multiply the sums as dividend; multiply the denominators as divisor. Divide. LIU Hui s commentary: (1) Multiply each denominator by its integral part; then add the corresponding numerators. In this manner numerator and denominator are in the dividend. (2) The rule is the same as that for the multiplication of fractions. (3) Here both the breadth and length have fractional parts, so first reduce each to a common denominator. The product of numerators is increased by multiplying by the product of the denominators, so the product of the denominators is regarded as the divisor, i.e., continued division The rule for triangular fields Multiply half the base by the altitude. LIU Hui s commentary: Halving the base means filling the vacancy with the surplus, transforms [the triangle] into a rectangle. Alternatively, multiply half the altitude by the base. Multiplication of half the base by the altitude is for getting its mean. Divide the area by the number of [square] bu per mu giving the number of mu The rule for right-angled trapezoidal fields Multiply half the sum of the bases by the altitude, or half the altitude by the sum [of the bases]; divide by the number of [square] bu per mu. LIU Hui s commentary: Halving the sum means filling the vacancy with the surplus The rule for trapezoidal fields Multiply half the sum of the upper and lower bases by the altitude, divide by the number of [square] bu per mu. LIU Hui s commentary: A trapezoid can be divided into two right-angled trapezoids, so that the rules for both are alike. Alternatively, multiply the sum of the two base by half the altitude.