"Sun Tzŭ's Computational Classic: Volume II" 《孫子算經卷中》 §12

《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§12. Volume of a cylinder (2)

This section gives a worked example of computing the volume of a cylinder.

The relevant unit conversions for length are

\begin{aligned} 1 \unit{rod~(丈)} &= 10 \unit{rules~(尺)} \\ 1 \unit{rule~(尺)} &= 10 \unit{inches~(寸)} \\ 1 \unit{inch~(寸)} &= 10 \unit{tenths~(分)}. \end{aligned}

The conversion between capacity units and volume units is given by

1 \unit{barrel~(斛)} = 1.62 \unit{rules~(尺)}^3.

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
今有圓窖、周五丈四尺、深一丈八尺。問受粟幾何。	Suppose there be [a] circular cellar, of circumference five rods [and] four rules, [and] depth one rod [and] eight rules. [We] ask, how much grain beareth [it]?
答曰、二千七百斛。	Answer saith: two thousand seven hundred barrels.
術曰、先置周五丈四尺、自相乘、得二千九百一十六尺。	Method saith: first put [down the] circumference five rods [and] four rules, [which], multiplied with itself, resulteth in two thousand nine hundred [and] sixteen rules.	Version A, Version B, and Version D are erroneously missing 自 in 自相乘.
以深一丈八尺乘之、得五萬二千四百八十八尺。	Multiplying it by [the] depth one rod [and] eight rules, resulteth in five myriad two thousand four hundred [and] eighty-eight rules.
以一十二除之、得四千三百七十四尺。	Dividing it by twelve, resulteth in four thousand three hundred [and] seventy-four rules.
以斛法一尺六寸二分除之、即得。	Dividing it by [the] barrel divisor, one rule, six inches, [and] two tenths, [we] are done.	斛法： [the] barrel divisor This is the conversion between the capacity unit "barrel", huk (斛), and the volume unit "cubic rule", chʻek (尺); see §10 Extended commentary. In modern notation, the volume (or capacity) of a cylinder of circumference C = 54 \unit{rules} and depth H = 18 \unit{rules} is \begin{aligned} V &\approx \frac{C^2 H}{12} \\[\tallspace] &= \frac{(54 \unit{rules})^2 \cdot (18 \unit{rules})}{12} \div \frac{1.62 \unit{rules}^3}{\unit{barrel}} \\[\tallspace] &= 2700 \unit{barrels}. \end{aligned} Here \pi \approx 3 (so that 4\pi \approx 12), see Vol. I §5.

Source text

Target text

Notes

今有圓窖、周五丈四尺、深一丈八尺。問受粟幾何。

Suppose there be [a] circular cellar, of circumference five rods [and] four rules, [and] depth one rod [and] eight rules. [We] ask, how much grain beareth [it]?

答曰、二千七百斛。

Answer saith: two thousand seven hundred barrels.

術曰、先置周五丈四尺、自相乘、得二千九百一十六尺。

Method saith: first put [down the] circumference five rods [and] four rules, [which], multiplied with itself, resulteth in two thousand nine hundred [and] sixteen rules.

Version A, Version B, and Version D are erroneously missing 自 in 自相乘.

以深一丈八尺乘之、得五萬二千四百八十八尺。

Multiplying it by [the] depth one rod [and] eight rules, resulteth in five myriad two thousand four hundred [and] eighty-eight rules.

以一十二除之、得四千三百七十四尺。

Dividing it by twelve, resulteth in four thousand three hundred [and] seventy-four rules.

以斛法一尺六寸二分除之、即得。

Dividing it by [the] barrel divisor, one rule, six inches, [and] two tenths, [we] are done.

斛法： [the] barrel divisor
This is the conversion between the capacity unit "barrel", huk (斛), and the volume unit "cubic rule", chʻek (尺); see §10 Extended commentary.
In modern notation, the volume (or capacity) of a cylinder of circumference C = 54 \unit{rules} and depth H = 18 \unit{rules} is
\begin{aligned} V &\approx \frac{C^2 H}{12} \\[\tallspace] &= \frac{(54 \unit{rules})^2 \cdot (18 \unit{rules})}{12} \div \frac{1.62 \unit{rules}^3}{\unit{barrel}} \\[\tallspace] &= 2700 \unit{barrels}. \end{aligned}
Here \pi \approx 3 (so that 4\pi \approx 12), see Vol. I §5.

《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II" §12. Volume of a cylinder (2)

Translation

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《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§12. Volume of a cylinder (2)