"Sun Tzŭ's Computational Classic: Volume II" 《孫子算經卷中》 §23

《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§23. Volume of a trapezoidal prism (3)

This section gives a worked example of computing the volume of a trapezoidal prism.

The relevant unit conversions for length are

\begin{aligned} 1 \unit{mile~(里)} &= 300 \unit{paces~(步)} \\ 1 \unit{pace~(步)} &= 6 \unit{rules~(尺)} \\ 1 \unit{rod~(丈)} &= 10 \unit{rules~(尺)} \\ 1 \unit{rule~(尺)} &= 10 \unit{inches~(寸)}. \end{aligned}

Translation

Chinese source text: Version A, Version B, Version C, Version D.
Unless noted otherwise, I follow the text from Version D, 《知不足齋叢書》本.

Source text	Target text	Notes
今有穿渠、長二十九里一百四步、上廣一丈二尺六寸、下廣八尺、深一丈八尺。	Suppose there be [a] boring of [a] canal, of length twenty-nine miles [and] one hundred [and] four paces, upper breadth one rod, two rules, [and] six inches, lower breadth eight rules, [and] depth one rod [and] eight rules.
秋程人功三百尺。問須功幾何。	[The] autumn quota [for a] person's output [is] three hundred rules. [We] ask, how many [people's] output [be] needed?	三百尺： three hundred rules In modern language these would be cubic rules. Version A has 湏 for 須.
答曰、三萬二千六百四十五人、不盡六十九尺六寸。	Answer saith: three myriad two thousand six hundred [and] forty-five people, remainder sixty-nine rules [and] six inches.	Version B has 功 "output" instead of 人 "people". The remainder 69 \unit{rules} + 6 \unit{inches} is actually a volume, 69.6 \unit{rules}^3. Classical Chinese mathematics did not have a proper concept of dimensional analysis.
術曰、置里數、以三百步乘之、內零步、六之、得五萬二千八百二十四尺。	Method saith: put [down the] number of miles, [and] multiply it by three hundred paces; admitting [the] residual paces, [and] sextupling it, resulteth in five myriad two thousand eight hundred [and] twenty-four rules.	內： admitting Used for 納. Here we have a unit conversion of the canal length, \begin{aligned} L &= 29 \unit{miles} + 104 \unit{paces} \\ &= \left( 29 \unit{miles} \cdot \frac{300 \unit{paces}}{\unit{mile}} + 104 \unit{paces} \right) \cdot \frac{6 \unit{rules}}{\unit{pace}} \\[\tallspace] &= 52824 \unit{rules}. \end{aligned}
并上下廣、得二丈六寸、半之、以深乘之、得一百八十五尺四寸。	Combining [the] upper [and] lower breadths, resulteth in two rods [and] six inches; halving it, [and] multiplying it by [the] depth, resulteth in one hundred [and] eighty-five rules [and] four inches.
以長乘、得九百七十九萬三千五百六十九尺六寸。	Multiplying by [the] length, resulteth in nine hundred [and] seventy-nine myriad three thousand five hundred [and] sixty-nine rules [and] six inches.
以人功三百尺除之、即得。	Dividing it by [a] person's output three hundred rules, [we] are done.	In modern notation, the effort required to excavate a trapezoidal prism of upper breadth A = 12.6 \unit{rules}, lower breadth B = 8 \unit{rules}, depth H = 18 \unit{rules}, and length L = 52824 \unit{rules}, at rate R = 300 \unit{rules}^3 / \unit{person}, is \begin{aligned} W &= \frac{A + B}{2} \cdot H L \div R \\[\tallspace] &= \frac{12.6 \unit{rules} + 8 \unit{rules}}{2} \times 18 \unit{rules} \times 52824 \unit{rules} \div \frac{300 \unit{rules}^3}{\unit{person}} \\[\tallspace] &= 32645 \unit{people} + 69.6 \unit{rules}^3 \div \frac{300 \unit{rules}^3}{\unit{person}}. \end{aligned}

Source text

Target text

Notes

今有穿渠、長二十九里一百四步、上廣一丈二尺六寸、下廣八尺、深一丈八尺。

Suppose there be [a] boring of [a] canal, of length twenty-nine miles [and] one hundred [and] four paces, upper breadth one rod, two rules, [and] six inches, lower breadth eight rules, [and] depth one rod [and] eight rules.

秋程人功三百尺。問須功幾何。

[The] autumn quota [for a] person's output [is] three hundred rules. [We] ask, how many [people's] output [be] needed?

三百尺： three hundred rules
In modern language these would be cubic rules.
Version A has 湏 for 須.

答曰、三萬二千六百四十五人、不盡六十九尺六寸。

Answer saith: three myriad two thousand six hundred [and] forty-five people, remainder sixty-nine rules [and] six inches.

Version B has 功 "output" instead of 人 "people".
The remainder 69 \unit{rules} + 6 \unit{inches} is actually a volume, 69.6 \unit{rules}^3. Classical Chinese mathematics did not have a proper concept of dimensional analysis.

術曰、置里數、以三百步乘之、內零步、六之、得五萬二千八百二十四尺。

Method saith: put [down the] number of miles, [and] multiply it by three hundred paces; admitting [the] residual paces, [and] sextupling it, resulteth in five myriad two thousand eight hundred [and] twenty-four rules.

內： admitting
Used for 納.
Here we have a unit conversion of the canal length,
\begin{aligned} L &= 29 \unit{miles} + 104 \unit{paces} \\ &= \left( 29 \unit{miles} \cdot \frac{300 \unit{paces}}{\unit{mile}} + 104 \unit{paces} \right) \cdot \frac{6 \unit{rules}}{\unit{pace}} \\[\tallspace] &= 52824 \unit{rules}. \end{aligned}

并上下廣、得二丈六寸、半之、以深乘之、得一百八十五尺四寸。

Combining [the] upper [and] lower breadths, resulteth in two rods [and] six inches; halving it, [and] multiplying it by [the] depth, resulteth in one hundred [and] eighty-five rules [and] four inches.

以長乘、得九百七十九萬三千五百六十九尺六寸。

Multiplying by [the] length, resulteth in nine hundred [and] seventy-nine myriad three thousand five hundred [and] sixty-nine rules [and] six inches.

以人功三百尺除之、即得。

Dividing it by [a] person's output three hundred rules, [we] are done.

In modern notation, the effort required to excavate a trapezoidal prism of upper breadth A = 12.6 \unit{rules}, lower breadth B = 8 \unit{rules}, depth H = 18 \unit{rules}, and length L = 52824 \unit{rules}, at rate R = 300 \unit{rules}^3 / \unit{person}, is
\begin{aligned} W &= \frac{A + B}{2} \cdot H L \div R \\[\tallspace] &= \frac{12.6 \unit{rules} + 8 \unit{rules}}{2} \times 18 \unit{rules} \times 52824 \unit{rules} \div \frac{300 \unit{rules}^3}{\unit{person}} \\[\tallspace] &= 32645 \unit{people} + 69.6 \unit{rules}^3 \div \frac{300 \unit{rules}^3}{\unit{person}}. \end{aligned}

《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II" §23. Volume of a trapezoidal prism (3)

Translation

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《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§23. Volume of a trapezoidal prism (3)