《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§10. Volume of a cylinder (1)

This section gives a worked example of computing the volume of a cylinder, and contains the first mention of the conversion between the capacity unit "barrel", huk (斛), and the volume unit "[cubic] rule", chʻek (尺):

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

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}

and, for capacity,

1 \unit{barrel~(斛)} = 100 \unit{quarts~(升)}.

See Vol. I §1 (Units of length) and Vol. I §3 (Units of capacity).

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、下周二百八十六尺、深三丈六尺。問受粟幾何。 Suppose there be [a] circular cellar, of lower circumference two hundred [and] eighty-six rules, [and] depth three rods [and] six rules. [We] ask, how much grain beareth [it]?
  • 窖： cellar

    窖、居效切、 Cantonese: kaau3, Mandarin: jiào

答曰、一十五萬一千四百七十四斛七升二十七分升之一十一。 Answer saith: fifteen myriad one thousand four hundred [and] seventy-four barrels, seven quarts, [and] eleven twenty-sevenths of [a] quart.
術曰、置周二百八十六尺、自相乘、得八萬一千七百九十六尺。 Method saith: put [down the] circumference two hundred [and] eighty-six rules, [which], multiplied with itself, resulteth in eight myriad one thousand seven hundred [and] ninety-six rules.
以深三丈六尺乘之、得二百九十四萬四千六百五十六。 Multiplying it by [the] depth three rods [and] six rules, resulteth in two hundred [and] ninety-four myriad four thousand six hundred [and] fifty-six.
以一十二除之、得二十四萬五千三百八十八尺。 Dividing it by twelve, resulteth in twenty-four myriad five thousand three hundred [and] eighty-eight 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.
  • 斛法一尺六寸二分： [the] barrel divisor, one rule, six inches, [and] two tenths

    This is the conversion between the capacity unit "barrel", huk (斛), and the volume unit "cubic rule", chʻek (尺). Rules, inches, and tenths are units of length, so at face value 一尺六寸二分 represents the length 1.62 rules. However, it is intended to represent the volume 1.62 cubic rules; see Extended commentary.

  • In modern notation, the volume (or capacity) of a cylinder of circumference C = 286 \unit{rules} and depth H = 36 \unit{rules} is
    \begin{aligned} V &\approx \frac{C^2 H}{12} \\[\tallspace] &= \frac{(286 \unit{rules})^2 \cdot (36 \unit{rules})}{12} \div \frac{1.62 \unit{rules}^3}{\unit{barrel}} \\[\tallspace] &= 151474 \tfrac{2}{27} \unit{barrels} \\ &= 151474 \unit{barrels} + 7 \tfrac{11}{27} \unit{quarts}. \end{aligned}
    Here \pi \approx 3 (so that 4\pi \approx 12), see Vol. I §5.

Extended commentary

The 斛法 "barrel divisor" gives the conversion between the capacity unit "barrel", huk (斛), and the volume unit "cubic rule", chʻek (尺).

If we parse the barrel divisor 一尺六寸二分 literally, it is the length

1 \unit{rule~(尺)} + 6 \unit{inches~(寸)} + 2 \unit{tenths~(分)} = 1.62 \unit{rules~(尺)}.

Now Chinese mathematics did not have a fully developed notion of dimensional analysis, so square units and cubic units were simply referred to as units. In the case of the barrel divisor, 一尺六寸二分 is actually intended to represent the volume

\begin{aligned} 1.62 \unit{rules}^3 &= (1 \unit{rule} + 6 \unit{inches} + 2 \unit{tenths}) \cdot 1 \unit{rule}^2 \\ &= 1 \unit{rule}^3 + 600 \unit{inches}^3 + 20000 \unit{tenths}^3 \\ &\ne 1 \unit{rule}^3 + 6 \unit{inches}^3 + 2 \unit{tenths}^3. \end{aligned}

Evidently Chinese mathematicians knew that a cubic inch wasn't equal to a tenth of a cubic rule, but they had no means of expressing this in words.

In modern (dimensionally consistent) notation,

1 \unit{barrel} = 1.62 \unit{rules}^3,

so that the barrel divisor is

1 = \frac{1.62 \unit{rules}^3}{\unit{barrel}}.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume II §10". <https://yawnoc.github.io/sun-tzu/ii/10> Accessed yyyy-mm-dd.