〔一〕今有穿地积一万尺。问为坚、壤各几何？
荅曰：为坚七千五百尺。
为壤一万二千五百尺。术曰：穿地四，为壤五，为坚三，为墟四。以穿地求壤，五之；求坚，三之，皆四而一。以壤求穿，四之；求坚，三之，皆五而一。以坚求穿，四之；求壤，五之，皆三而一。城、垣、隄、沟、、渠，皆同术。
术曰：并上下广而半之，以高若深乘之，又以袤乘之，即积尺。
现在有一个挖掘的土地体积共计一万尺。问其中坚硬部分和松软部分各是多少？
答案是：坚硬部分为7500尺，松软部分为12500尺。
解题方法：挖掘土地的体积中，4份为松软土壤，5份为坚硬土壤，3份为虚空，4份为废墟。要求得松软土壤的体积，用总体积乘以5再除以4；要求得坚硬土壤的体积，用总体积乘以3再除以4。反之，如果已知松软土壤或坚硬土壤的体积，也可以用相应的比例关系求出总体积。城墙、院墙、堤坝、沟渠等的计算方法都与此相同。
具体算法是：将上下宽度相加后除以2，乘以高度或深度，再乘以长度，就得到了体积的尺数。
Now, there is a dug-out land with a total volume of 10,000 cubic feet. What are the volumes of the solid and loose parts respectively? The answer is: The solid part is 7,500 cubic feet, and the loose part is 12,500 cubic feet. Methodology: In the volume of excavated land, 4 parts are loose soil, 5 parts are solid earth, 3 parts are voids, and 4 parts are ruins. To find the volume of loose soil, multiply the total volume by 5 and divide by 4; to find the volume of solid earth, multiply the total volume by 3 and divide by 4. Conversely, if the volume of loose soil or solid earth is known, the total volume can also be calculated using the corresponding ratios. The calculation methods for city walls, perimeter walls, embankments, ditches, and canals are all the same. The specific algorithm is: Add the upper and lower widths and divide by 2, multiply by the height or depth, and then multiply by the length to obtain the volume in cubic feet.
〔二〕今有城下广四丈，上广二丈，高五丈，袤一百二十六丈五尺。问积几何？
荅曰：一百八十九万七千五百尺。
现在有一个城墙，底部宽四丈，顶部宽二丈，高五丈，长一百二十六丈五尺。问这个城墙的体积是多少？
答案是：一百八十九万七千五百尺。
Now, there is a city wall that is four zhang wide at the bottom, two zhang wide at the top, five zhang high, and one hundred and twenty-six zhang and five chi long. What is the volume of this city wall? The answer is: 1,897,500 cubic chi.
〔三〕今有垣下广三尺，上广二尺，高一丈二尺，袤二十二丈五尺八寸。问积几何？
荅曰：六千七百七十四尺。
现在有一段墙，底部宽3尺，顶部宽2尺，高1丈2尺，长22丈5尺8寸。问这段墙的体积是多少？
答案是：6774立方尺。
解题方法：这种问题涉及到梯形体积的计算。由于墙的上下宽度不同，需要将墙视为一个梯形体进行体积计算。算法是先求出墙的平均宽度，然后乘以高度和长度得到体积。
Now, there is a section of wall that is 3 feet wide at the base, 2 feet wide at the top, 12 feet high, and 225 feet and 8 inches long. What is the volume of this wall? The answer is: 6,774 cubic feet. Solution method: This problem involves calculating the volume of a trapezoidal prism. Since the wall tapers at the top and bottom, it needs to be treated as a trapezoidal body for volume calculation. The algorithm involves finding the average width of the wall, then multiplying by the height and length to obtain the volume.
〔四〕今有隄下广二丈，上广八尺，高四尺，袤一十二丈七尺。问积几何？冬程人功四百四十四尺。问用徒几何？
荅曰：七千一百一十二尺。
荅曰：一十六人、一百一十一分人之二。
术曰：以积尺为实，程功尺数为法，实如法而一，即用徒人数。
现在有一道堤坝，底部宽两丈，顶部宽八尺，高四尺，长十二丈七尺。问这个堤坝的体积是多少？冬季工程每人每天工作量为444尺。问需要多少工人？
答案是：7112尺。
答案是：16人又111分之2人。
计算方法如下：以体积的尺数作为被除数，以每人每天的工作量尺数作为除数，将被除数除以除数，即可得到需要的工人人数。
Now, there is a dike that is two zhang wide at the bottom, eight chi wide at the top, four chi high, and twelve zhang and seven chi long. What is the volume of this dike? The answer is: 7,112 cubic chi. In winter, the daily work output per person is 444 cubic chi. How many workers are needed? The answer is: 16 people and 2/111 of a person. The method is as follows: Take the volume in cubic chi as the dividend, and the daily work output per person in cubic chi as the divisor; divide the dividend by the divisor to get the number of workers needed.
〔五〕今有沟上广一丈五尺，下广一丈，深五尺，袤七丈。问积几何？
春程人功七百六十六尺，并出土功五分之一，定功六百一十二尺、五分尺之四。问用徒几何？
荅曰：四千三百七十五尺。荅曰：七人、三千六十四分人之四百二十七。
术曰：置本人功，去其五分之一，余为法。以沟积尺为实。实如法而一，得用徒人数。
现在有一条沟，顶部宽1丈5尺，底部宽1丈，深5尺，长7丈。问这条沟的体积是多少？
春季每人每天的工作量是766立方尺，包括出土工作量的五分之一，确定工作量为612立方尺又五分之四尺。问需要多少工人？
答案是：4375立方尺。答案是：7人又3064分之427。
解题方法：将每人的工作量减去其五分之一，余下的作为标准。以沟的体积作为被除数。将被除数除以标准，得到需要的工人人数。
Now, there is a ditch that is 15 feet wide at the top, 10 feet wide at the bottom, 5 feet deep, and 70 feet long. What is the volume of this ditch? The answer is: 4,375 cubic feet. The spring workload per person per day is 766 cubic feet, including one-fifth of the earthwork, with a set workload of 612 cubic feet and four-fifths of a foot. How many workers are needed? The answer is: 7 people plus 427/3064. Method: Subtract one-fifth of each person's workload, and the remainder is used as the standard. Take the volume of the ditch as the dividend. Divide the dividend by the standard to get the number of workers needed.
〔六〕今有堑上广一丈六尺三寸，下广一丈，深六尺三寸，袤一十三丈二尺一寸。问积几何？
夏程人功八百七十一尺。并出土功五分之一，沙砾水石之功作太半，定功二百三十二尺、一十五分尺之四。问用徒几何？
荅曰：一万九百四十三尺八寸。荅曰：四十七人、三千四百八十四分人之四百九。术曰：置本人功，去其出土功五分之一，又去沙砾水石之功太半，余为法。以堑积尺为实。实如法而一，即用徒人数。
现在有一条沟渠，顶部宽1丈6尺3寸，底部宽1丈，深6尺3寸，长13丈2尺1寸。问这条沟渠的体积是多少？
夏季每人每天的工作量是871尺。包括出土工作量的五分之一，沙砾水石的工作量占大半，确定工作量为232尺又十五分之四尺。问需要多少工人？
答案是：19438立方尺又8分。答案是：47人又3484分之409。
解题方法：将每人的工作量减去其出土工作量的五分之一，再减去沙砾水石工作量的大半，余下的作为标准。以沟渠的体积作为被除数。将被除数除以标准，得到需要的工人人数。
Now, there is a trench that is 16 feet and 3 inches wide at the top, 10 feet wide at the bottom, 6 feet and 3 inches deep, and 132 feet and 1 inch long. What is the volume of this trench? The answer is: 19,438 cubic feet and 8/10 of a foot. The summer workload per person per day is 871 cubic feet. Including one-fifth of the earthwork, with the majority of the work involving sand, gravel, water, and stone, the set workload is 232 cubic feet and four-fifteenths of a foot. How many workers are needed? The answer is: 47 people plus 409/3484. Method: Subtract one-fifth of the earthwork from each person's workload, and subtract the majority of the work involving sand, gravel, water, and stone. The remainder is used as the standard. Take the volume of the trench as the dividend. Divide the dividend by the standard to get the number of workers needed.
〔七〕今有穿渠上广一丈八尺，下广三尺六寸，深一丈八尺，袤五万一千八百二十四尺。问积几何？
秋程人功三百尺，问用徒几何？
一千人先到，问当受袤几何？
荅曰：一千七万四千五百八十五尺六寸。 荅曰：三万三千五百八十二人功。內少一十四尺四寸。荅曰：一百五十四丈三尺二寸、八十一分寸之八。
术曰：以一人功尺数，乘先到人数为实。并渠上下广而半之，以深乘之为法。实如法得袤尺。
现在有一条渠道，顶部宽1丈8尺，底部宽3尺6寸，深1丈8尺，长51824尺。问这条渠道的体积是多少？
秋季每人每天的工作量是300尺，问需要多少工人？
如果有1000人先到达，问他们应负责多长的长度？
答案是：174585立方尺又6分。答案是：33582人的工作。其中少了14尺4寸。答案是：154丈3尺2寸又81分之8寸。
解题方法：用每人工作量的尺数乘以先到的人数作为被除数。将渠道的上下宽度相加后除以2，乘以深度作为除数。将被除数除以除数得到长度的尺数。
Now, there is a canal that is 18 feet wide at the top, 3 feet and 6 inches wide at the bottom, 18 feet deep, and 51,824 feet long. What is the volume of this canal? The answer is: 174,585 cubic feet and 6/10 of a foot. In the autumn, the daily workload per person is 300 linear feet. How many workers are needed? The answer is: The work of 33,582 people. This is 14 feet and 4 inches short. If 1,000 people arrive first, how long of a stretch should they be responsible for? The answer is: 154 rods, 3 feet, and 2 inches, plus 8/81 of an inch. Method: Multiply the number of linear feet each person is expected to work by the number of people who arrived first to get the dividend. Add the top and bottom widths of the canal and divide by two, multiply by the depth to get the divisor. Divide the dividend by the divisor to get the length in linear feet.
〔八〕今有方堡壔方一丈六尺，高一丈五尺。问积几何？
荅曰：三千八百四十尺。
术曰：方自乘，以高乘之，即积尺。
现在有一个方形的烽火台，底面边长为一丈六尺，高一丈五尺。问这个烽火台的体积是多少？
答案是：3840尺。
计算方法如下：将底面边长自乘，然后乘以高，即可得到体积的尺数。
Now, there is a square beacon tower with a base side length of one zhang and six chi, and a height of one zhang and five chi. What is the volume of this beacon tower? The answer is: 3,840 cubic chi. The method is as follows: Square the base side length and then multiply by the height to get the volume in cubic chi.
〔九〕今有圆堡壔，周四丈八尺，高一丈一尺。问积几何？
荅曰：二千一百一十二尺。
术曰：周自相乘，以高乘之，十二而一。
现在有一个圆形的土堡垒，周长是4丈8尺，高度是1丈1尺。问这个土堡垒的体积是多少？
答案是：2112立方尺。
解题方法：将周长自乘一次，然后乘以高度，最后除以12得到体积。
Now, there is a circular earthen fortress with a circumference of 48 feet and a height of 11 feet. What is the volume of this earthen fortress? The answer is: 2,112 cubic feet. Methodology: Square the circumference, multiply by the height, and divide by twelve to obtain the volume.
〔十〕今有方亭，下方五丈，上方四丈，高五丈。问积几何？
荅曰：一十万一千六百六十六尺、太半尺。
术曰：上下方相乘，又各自乘，并之，以高乘之，三而一。
现在有一个方亭，底面宽5丈，顶面宽4丈，高5丈。问这个方亭的体积是多少？
答案是：101666立方尺又三分之二尺。
解题方法：将底面和顶面的边长相乘，然后分别自乘，相加后乘以高度，最后除以3得到体积。
Now, there is a square pavilion with a base width of 50 feet, a top width of 40 feet, and a height of 50 feet. What is the volume of this square pavilion? The answer is: 101,666 cubic feet and two-thirds of a foot. Methodology: Multiply the lengths of the base and top by each other, then multiply each by itself, add them together, multiply by the height, and divide by three to obtain the volume.
〔一一〕今有圆亭，下周三丈，上周二丈，高一丈。问积几何？
荅曰：五百二十七尺、九分尺之七。
术曰：上、下周相乘，又各自乘，并之，以高乘之，三十六而一。
现在有一个圆形的凉亭，底部直径为三丈，顶部直径为二丈，高一丈。问这个凉亭的体积是多少？
答案是：五百二十七尺又九分之七尺。
计算方法如下：将顶部和底部的周长相乘，再各自平方，然后相加，乘以高，最后除以三十六。
Now, there is a round pavilion with a base diameter of three zhang and a top diameter of two zhang, and a height of one zhang. What is the volume of this pavilion? The answer is: 527 cubic chi and 7/9 of a cubic chi. The method is as follows: Multiply the perimeters of the top and bottom, square each result, add them together, multiply by the height, and then divide by thirty-six.
〔一二〕今有方锥下方二丈七尺，高二丈九尺。问积几何？
荅曰：七千四十七尺。术曰：下方自乘，以高乘之，三而一。
现在有一个方锥，底面宽2丈7尺，高2丈9尺。问这个方锥的体积是多少？
答案是：7470立方尺。
解题方法：将底面边长自乘一次，然后乘以高度，最后除以3得到体积。
Now, there is a square pyramid with a base width of 27 feet and a height of 29 feet. What is the volume of this square pyramid? The answer is: 7,470 cubic feet. Methodology: Square the base width, multiply by the height, and divide by three to obtain the volume.
〔一三〕今有圆锥下周三丈五尺，高五丈一尺。问积几何？
荅曰：一千七百三十五尺、一十二分尺之五。术曰：下周自乘，以高乘之，三十六而一。
现在有一个圆锥，底面周长3丈5尺，高5丈1尺。问这个圆锥的体积是多少？
答案是：1735立方尺又12分之5尺。
解题方法：将底面周长自乘一次，然后乘以高度，最后除以36得到体积。
Now, there is a circular cone with a base circumference of 35 feet and a height of 51 feet. What is the volume of this circular cone? The answer is: 1,735 cubic feet and five-twelfths of a foot. Methodology: Square the base circumference, multiply by the height, and divide by thirty-six to obtain the volume.
〔一四〕今有堑堵下广二丈，袤一十八丈六尺，高二丈五尺。问积几何？
荅曰：四万六千五百尺。
术曰：广袤相乘，以高乘之，二而一。
现在有一个沟渠，底部宽两丈，长十八丈六尺，高二丈五尺。问这个沟渠的体积是多少？
答案是：46500尺。
计算方法如下：将宽度和长度相乘，然后乘以高，最后除以二。
Now, there is a ditch that is two zhang wide at the bottom, eighteen zhang and six chi long, and two zhang and five chi high. What is the volume of this ditch? The answer is: 46,500 cubic chi. The method is as follows: Multiply the width by the length, then multiply by the height, and finally divide by two.
〔一五〕今有阳马，广五尺，袤七尺，高八尺。问积几何？
荅曰：九十三尺、少半尺。
术曰：广袤相乘，以高乘之，三而一。
现在有一个阳马（直角四棱锥），底面宽5尺，长7尺，高8尺。问这个阳马的体积是多少？
答案是：93立方尺又三分之一尺。
解题方法：将底面的宽和长相乘，然后乘以高度，最后除以3得到体积。
Now, there is a "yangma" (right-angled pyramid) with a base width of 5 feet, a length of 7 feet, and a height of 8 feet. What is the volume of this "yangma"? The answer is: 93 cubic feet and one-third of a foot. Methodology: Multiply the base width by the length, then multiply by the height, and divide by three to obtain the volume.
〔一六〕今有鱉臑下广五尺，无袤，上袤四尺，无广，高七尺。问积几何？
荅曰：二十三尺、少半尺。术曰：广袤相乘，以高乘之，六而一。
现在有一个三角形的物体，底部宽五尺，没有长度，上部的长度为四尺，没有宽度，高七尺。问这个物体的体积是多少？
答案是：23尺又1/2尺。
计算方法如下：将宽度和长度相乘，然后乘以高，最后除以六。
Now, there is a triangular object that is five chi wide at the bottom without length, four chi long at the top without width, and seven chi high. What is the volume of this object? The answer is: 23 cubic chi and a half cubic chi. The method is as follows: Multiply the width by the length, then multiply by the height, and finally divide by six.
〔一七〕今有羡除，下广六尺，上广一丈，深三尺，末广八尺，无深，袤七尺。问积几何？
荅曰：八十四尺。术曰：并三广，以深乘之，又以袤乘之，六而一。
现在有一个梯形柱体（羡除），底面宽6尺，顶面宽1丈，深3尺，末（顶）面宽8尺，没有深度，长7尺。问这个梯形柱体的体积是多少？
答案是：84立方尺。
解题方法：将三个宽度相加，然后乘以深度，再乘以长度，最后除以6得到体积。
Now, there is a trapezoidal prism (xianchu) with a base width of 6 feet, a top width of 10 feet, a depth of 3 feet, and a top (end) width of 8 feet without depth, and a length of 7 feet. What is the volume of this trapezoidal prism? The answer is: 84 cubic feet. Methodology: Add the three widths together, multiply by the depth, then multiply by the length, and divide by six to obtain the volume.
〔一八〕今有芻甍，下广三丈，袤四丈，上袤二丈，无广，高一丈。问积几何？
荅曰：五千尺。
术曰：倍下袤，上袤从之，以广乘之，又以高乘之，六而一。
芻童、曲池、盘池、冥谷，皆同术。术曰：倍上袤，下袤从之，亦倍下袤，上袤从之，各以其广乘之，并，以高若深乘之，皆六而一。其曲池者，并上中、外周而半之，以为上袤；亦并下中、外周而半之，以为下袤。
现在有一个刍甍（古代建筑中的一种屋顶结构），底部长四丈，顶部长二丈，高一丈。问这个刍甍的体积是多少？
答案是：5000尺。
计算方法如下：将底部长度翻倍，再加上顶部长度，然后乘以宽度，再乘以高度，最后除以六。
刍童、曲池、盘池、冥谷等物体都使用相同的计算方法。计算方法是：将顶部长度翻倍，再加上底部长度，同时也将底部长度翻倍，再加上顶部长度，各自乘以它们的宽度，然后相加，接着乘以高度或深度，最后都除以六。对于曲池来说，将上中和外周相加后除以二，作为上部的长度；同样地，将下中和外周相加后除以二，作为下部的长度。
Now, there is a "chuma" roof that is four zhang wide at the bottom, two zhang long at the top, and one zhang high. What is the volume of this "chuma"? The answer is: 5,000 cubic chi. The method is as follows: Double the length at the bottom, add to it the length at the top, then multiply by the width, and also multiply by the height, finally dividing by six. For "chutou," curved ponds, round pools, and deep valleys, all use the same method. The method is: Double the length at the top, add to it the length at the bottom, also double the length at the bottom, add to it the length at the top, each multiplied by their respective widths, then added together, and multiplied by the height or depth, finally divided by six. For a curved pond, add the inner and outer circumferences at the top and divide by two to get the upper length; do the same for the bottom to get the lower length.
〔一九〕今有芻童，下广二丈，袤三丈，上广三丈，袤四丈，高三丈。问积几何？
荅曰：二万六千五百尺。
现在有一个刍童（草堆），底面宽2丈，长3丈，顶面宽3丈，长4丈，高3丈。问这个刍童的体积是多少？
答案是：26500立方尺。
Now, there is a "chutong" (grass mound) with a base width of 20 feet, a length of 30 feet, a top width of 30 feet, a length of 40 feet, and a height of 30 feet. What is the volume of this "chutong"? The answer is: 26,500 cubic feet. Methodology: Add the base and top widths together, then multiply by the length, and multiply by the height to obtain the volume.
〔二０〕今有曲池，上中周二丈，外周四丈，广一丈，下中周一丈四尺，外周二丈四尺，广五尺，深一丈。问积几何？
荅曰：一千八百八十三尺三寸、少半寸。
现在有一个曲池，上中周长2丈，外周4丈，宽1丈，下中周长1丈4尺，外周2丈4尺，宽5尺，深1丈。问这个曲池的体积是多少？
答案是：1883立方尺又三分之二立方尺。
Now, there is a curved pond with an upper middle circumference of 20 feet, an outer circumference of 40 feet, a width of 10 feet, a lower middle circumference of 14 feet, an outer circumference of 24 feet, a width of 5 feet, and a depth of 10 feet. What is the volume of this curved pond? The answer is: 1,883 cubic feet and two-thirds of a cubic foot. Methodology: Add the upper middle and lower middle circumferences together, then multiply by the width, and multiply by the depth to obtain the volume.
〔二一〕今有盘池，上广六丈，袤八丈，下广四丈，袤六丈，深二丈。问积几何？
负土往来七十步，其二十步上下棚除。棚除二当平道五，踟躕之间十加一，载输之间三十步，定一返一百四十步。土笼\积一尺六寸，秋程人功行五十九里半。问人到、积尺、用徒各几何？
荅曰：七万六百六十六尺、太半尺。荅曰：人到二百四尺。用徒三百四十六人、一百五十三分人之六十二。
术曰：以一笼\积尺乘程行步数为实。往来上下，棚除二当平道五。置定往来步数，十加一，及载输之间三十步以为法。除之，所得即一人所到尺。以所到约积尺，即用徒人数。
现在有一个盘池，顶部宽六丈，长八丈，底部宽四丈，长六丈，深二丈。问这个盘池的体积是多少？
每二步棚除相当于平地五步，犹豫之间每十步多加一步，运输之间走三十步，确定一次往返为一百四 十步。一笼子积土一尺六寸，秋季每人每天的工程进度为五十九里半。问每个人的进度、积累的尺数和使用的工人数分别是多少？
答案是：76666尺又2/3尺。背负土来往走70步，其中20步用于上下棚除（古代筑城时用的脚手架）。答案是：每个人进度为二百四尺。
使用工人数为346人又153分之62人。
计算方法如下：以一笼子积土的尺寸乘以每日行走的步数作为被除数。考虑往返上下，每二步棚除相当于平地五步的情况。将确定的往返步数、每十步加一步，以及运输之间的三十步作为除数。除以后得到的就是每个人的进度尺数。用总积累尺数除以每个人的进度尺数，得到的就是需要的工人数。
Now, there is a round pond with a top width of six zhang, a length of eight zhang, a bottom width of four zhang, and a length of six zhang, at a depth of two zhang. What is the volume of this round pond? The answer is: 76,666 cubic chi and 2/3 of a cubic chi. To carry soil back and forth takes 70 steps, with 20 steps for getting on and off the scaffolding (an ancient construction method used in fortification). For every two steps on the scaffolding, it equals five steps on flat ground. During hesitation, add one step for every ten steps, and during carrying, take thirty steps. One round trip is set to be 140 steps. One basket of accumulated soil measures one chi and six cun (a traditional Chinese unit of length), and in autumn, the daily work progress per person is fifty-nine li and a half. How much progress does each person make, what is the total accumulation in chi, and how many workers are used? The answer is: Each person makes a progress of two hundred and four chi. The number of workers used is 346 people plus 62/153 of a person. The method is as follows: Multiply the size of one basket of accumulated soil by the number of steps walked daily to get the dividend. Considering the back-and-forth walking, where every two steps on the scaffolding equals five steps on flat ground. Take the determined number of steps for a round trip, add one step for every ten steps, and include the thirty steps for carrying as the divisor. Dividing will give the progress in chi for each person. Divide the total accumulated chi by the progress per person to get the number of workers needed.
〔二二〕今有冥谷上广二丈，袤七丈，下广八尺，袤四丈，深六丈五尺。问积几何？
载土往来二百步，载输之间一里，程行五十八里，六人共车，车载三十四尺七寸。问人到积尺及用徒各几何？
荅曰：五万二千尺。荅曰：
人到二百一尺、五十分尺之十三。
用徒二百五十八人、一万六十三分人之三千七百四十六。术曰：以一车积尺乘程行步数为实。置今往来步数，加载输之间一里，以车六人乘之，为法。除之，所得即一人所到尺。以所到约积尺，即用徒人数。
现在有一个冥谷，顶部宽两丈，长七丈，底部宽八尺，长四丈，深六丈五尺。问这个冥谷的体积是多少？
运土来回走200步，运输之间距离一里，全程行走58里，六人共乘一辆车，每车载土34尺7寸。问每个人运送的积土尺寸和所需的工人数分别是多少？
答案是：52000尺。答案是：
每个人运送210尺又13/50尺。
需要工人258人又1063分之3746人。
计算方法如下：以一车所载积土的尺寸乘以全程行走的步数作为被除数。将来回走的步数加上运输之间的一里，再乘以六人为一辆车的人数，作为除数。除以后得到的就是每个人运送的积土尺寸。用总积土尺寸除以每个人运送的尺寸，得到的就是所需的工人数。
Now, there is a deep valley that is two zhang wide at the top, seven zhang long at the top, eight chi wide at the bottom, four zhang long at the bottom, and six zhang and five chi deep. What is the volume of this deep valley? The answer is: 52,000 cubic chi. Carrying soil back and forth takes 200 steps, with a distance of one li between transportations, covering a total distance of 58 li. Six people share one cart, and each cart carries 34 chi and 7 cun of soil. How much soil does each person transport and how many workers are needed? The answer is: Each person transports 210 chi and 13/50 of a chi. The number of workers needed is 258 people plus 3746/1063 of a person. The method is as follows: Multiply the amount of soil carried by one cart by the total number of steps walked to get the dividend. Add the number of steps walked back and forth to the one li between transportations, then multiply by the number of people per cart (six), to get the divisor. Dividing will give the amount of soil each person transports. Divide the total amount of soil by the amount each person transports to get the number of workers needed.
〔二三〕今有委粟平地，下周一十二丈，高二丈。问积及为粟几何？
荅曰：
积八千尺。为粟二千九百六十二斛、二十七分斛之二十六。
现在有一堆谷物平铺在地面上，底面直径12丈，高2丈。问这堆谷物的体积和有多少斛？
答案是：
体积8000立方尺。谷物共有2962斛又27分之26。
Now, there is a pile of grain spread out on the ground, with a base diameter of 120 feet and a height of 20 feet. What is the volume of this pile of grain and how many hu (an ancient Chinese unit of volume) does it make? The answer is: The volume is 8000 cubic feet. The grain amounts to 2962 hu and 26/27 of a hu.
〔二四〕今有委菽依垣，下周三丈，高七尺。问积及为菽各几何？
荅曰：
积三百五十尺。
为菽一百四十四斛、二百四十三分斛之八。
现在有一堆豆子靠着墙，底部是三角形，边长三丈，高七尺。问体积和豆子数量各是多少？
答案是：
体积为350尺。
豆子数量为144斛又243分之8斛。
Now, there is a pile of beans leaning against a wall, forming a triangle with a base length of three zhang and a height of seven chi. What are the volume and the quantity of beans? The answer is: The volume is 350 cubic chi. The quantity of beans is 144 hu plus 8/243 of a hu.
〔二五〕今有委米依垣內角，下周八尺，高五尺。问积及为米几何？
荅曰：
积三十五尺、九分尺之五。
为米二十一斛，七百二十九分斛之六百九十一。委粟术曰：下周自乘，以高乘之，三十六而一。其依垣者，十八而一。其依垣內角者，九而一。
程粟一斛，积二尺七寸。其米一斛，积一尺六寸、五分寸之一。其菽、荅、麻、麦一斛，皆二尺四寸、十分寸之三。
现在有米堆积在墙角，底面为一个半圆形，直径8尺，高5尺。问这堆米的体积和有多少斛？
答案是：
体积35立方尺又9分之5立方尺。
米共有21斛又729分之691斛。
计算堆积粮食的体积的一般方法是：将底面的周长自乘，再乘以高，然后除以36。如果是靠墙边堆积的，就要除以18。对于堆积在墙角的情况，要除以9。
规定一斛粟的体积是2尺7寸（即2.7尺）。一斛米的体积是1尺6寸又5分之1寸（即1.6+0.2/5=1.64尺）。一斛豆类、小豆、麻或麦的体积都是2尺4寸又10分之3寸（即2.4+0.3/10=2.43尺）。
Now, there is rice piled up in the corner of a wall, with the base being a semicircle with a diameter of 8 feet and a height of 5 feet. What is the volume of this pile of rice and how many bushels are there? The answer is: The volume is 35 cubic feet and 9/5 cubic feet. There are 21 bushels of rice and 729/691 bushels. The general method for calculating the volume of grain piles is to square the base perimeter, multiply by the height, and then divide by 36. If it's piled up against the wall, you divide by 18. For piles in the corner of a wall, you divide by 9. It is stipulated that one bushel of millet has a volume of 2 feet 7 inches (i.e., 2.7 feet). One bushel of rice has a volume of 1 foot 6 inches and 5/1 inch (i.e., 1.6 + 0.2/5 = 1.64 feet). One bushel of legumes, small beans, hemp, or wheat all have a volume of 2 feet 4 inches and 3/10 inches (i.e., 2.4 + 0.3/10 = 2.43 feet).
〔二六〕今有穿地，袤一丈六尺，深一丈，上广六尺，为垣积五百七十六尺。问穿地下广几何？
荅曰：三尺、五分尺之三。
术曰：置垣积尺，四之为实。以深、袤相乘，又三之，为法。所得倍之，减上广，余即下广。
现在有一个挖地的问题，长一丈六尺，深一丈，上宽六尺，为墙的体积是五百七十六尺。问挖地的下宽是多少？
答案是：三尺又五分之三。
计算方法如下：将墙的体积尺寸乘以四作为被除数。将深度和长度相乘，再乘以三，作为除数。所得的结果乘以二，减去上宽，余数就是下宽。
Now, there is a problem with digging the ground, the length is one zhang and six chi, the depth is one zhang, the top width is six chi, and the volume of the wall is 576 chi. What is the width at the bottom of the excavation? The answer is: three chi and three-fifths of a chi. The method is as follows: Take the volume of the wall, multiply it by four to get the dividend. Multiply the depth and length together, then triple it to get the divisor. Double the result obtained, subtract the top width, and the remainder is the bottom width.
〔二七〕今有仓广三丈，袤四丈五尺，容粟一万斛。问高几何？
荅曰：二丈。
术曰：置粟一万斛积尺为实。广袤相乘为法。实如法而一，得高尺。
现在有一个仓库，宽三丈，长四丈五尺，能容纳粟米一万斛。问仓库的高度是多少？
答案是：二丈。
计算方法如下：将粟米一万斛的体积尺寸作为被除数。将宽度和长度相乘作为除数。用被除数除以除数得到高度尺寸。
Now, there is a granary that is three zhang wide and four zhang and five chi long, which can hold ten thousand hu of grain. What is the height of the granary? The answer is: two zhang. The method is as follows: Take the volume of ten thousand hu of grain as the dividend. Multiply the width and length together to get the divisor. Divide the dividend by the divisor to obtain the height in chi.
〔二八〕今有圆困，高一丈三尺三寸、少半寸，容米二千斛。问周几何？
荅曰：五丈四尺。
术曰：置米积尺，以十二乘之，令高而一，所得，开方除之，即周。
现在有一个圆形的囷（古代一种储粮设施），高为一丈三尺三寸少半寸，可以容纳两千斛（古代容量单位）的米。问它的周长是多少？
答：五丈四尺。
方法：将米的体积换算成尺，乘以十二，然后除以高度，得到的结果开方后就是周长。
Now there is a round granary, with a height of one zhang and three chi, three fen and less than half fen, which can hold two thousand hu of rice. What is its circumference? Answer: Five zhang and four chi. Method: Convert the volume of rice into chi, multiply by twelve, divide by the height, take the square root of the result, which gives the circumference.