"Sun Tzŭ's Computational Classic: Volume III" 《孫子算經卷下》 §3

《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§3. Volume of a cone

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

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
今有平地聚粟、下周三丈六尺、高四尺五寸。問粟幾何。	Suppose there be gathered grain [on] level ground, of lower circumference three rods [and] six rules, [and] height four rules [and] five inches. [We] ask, how much [be the] grain?
答曰、一百斛。	Answer saith: one hundred barrels.
術曰、置周三丈六尺、自相乘、得一千二百九十六尺。	Method saith: put [down the] circumference three rods [and] six rules, [which], multiplied with itself, resulteth in one thousand two hundred [and] ninety-six rules.
以高四尺五寸乘之、得五千八百三十二尺。	Multiplying it by [the] height four rules [and] five inches, resulteth in five thousand eight hundred [and] thirty-two rules.
以三十六除之、得一百六十二尺。	Dividing it by thirty-six, resulteth in one hundred [and] sixty-two 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 Vol. II §10 Extended commentary. In modern notation, the volume (or capacity) of a cone of circumference C = 36 \unit{rules} and depth H = 4.5 \unit{rules} is \begin{aligned} V &\approx \frac{C^2 H}{36} \\[\tallspace] &= \frac{(36 \unit{rules})^2 \cdot (4.5 \unit{rules})}{36} \div \frac{1.62 \unit{rules}^3}{\unit{barrel}} \\[\tallspace] &= 100 \unit{barrels}. \end{aligned} Here \pi \approx 3 (so that 12\pi \approx 36), see Vol. I §5.

Source text

Target text

Notes

今有平地聚粟、下周三丈六尺、高四尺五寸。問粟幾何。

Suppose there be gathered grain [on] level ground, of lower circumference three rods [and] six rules, [and] height four rules [and] five inches. [We] ask, how much [be the] grain?

答曰、一百斛。

Answer saith: one hundred barrels.

術曰、置周三丈六尺、自相乘、得一千二百九十六尺。

Method saith: put [down the] circumference three rods [and] six rules, [which], multiplied with itself, resulteth in one thousand two hundred [and] ninety-six rules.

以高四尺五寸乘之、得五千八百三十二尺。

Multiplying it by [the] height four rules [and] five inches, resulteth in five thousand eight hundred [and] thirty-two rules.

以三十六除之、得一百六十二尺。

Dividing it by thirty-six, resulteth in one hundred [and] sixty-two 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 Vol. II §10 Extended commentary.
In modern notation, the volume (or capacity) of a cone of circumference C = 36 \unit{rules} and depth H = 4.5 \unit{rules} is
\begin{aligned} V &\approx \frac{C^2 H}{36} \\[\tallspace] &= \frac{(36 \unit{rules})^2 \cdot (4.5 \unit{rules})}{36} \div \frac{1.62 \unit{rules}^3}{\unit{barrel}} \\[\tallspace] &= 100 \unit{barrels}. \end{aligned}
Here \pi \approx 3 (so that 12\pi \approx 36), see Vol. I §5.

《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III" §3. Volume of a cone

Translation

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《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§3. Volume of a cone