"Sun Tzŭ's Computational Classic: Volume II" 《孫子算經卷中》 §14

《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§14. Area of a square of given circumradius

This section gives a worked example of computing the area of a square given its circumradius.

The relevant unit conversions for area and length are

\begin{aligned} 1 \unit{yardland~(頃)} &= 100 \unit{acres~(畝)} \\ 1 \unit{acre~(畝)} &= 240 \unit{paces~(步)}^2. \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
今有方田、桑生中央、從角至桑、一百四十七步。問為田幾何。	Suppose there be [a] square field, [a] mulberry-tree growing [in the] centre, [and] from [a] corner unto [the] mulberry-tree, one hundred [and] forty-seven paces. [We] ask, how much field be [this]?
答曰、一頃八十三畝奇一百八十步。	Answer saith: one yardland [and] eighty-three acres remainder one hundred [and] eighty paces.	頃： yardland Kʻing (頃), equal to 100 mu (畝) or Chinese acres. Difficult to render, and I have chosen "yardland" because it sounds more natural, even though the imperial oxgang (which is only half an imperial yardland or virgate) is closer in size.
術曰、置角至桑一百四十七步、倍之、得二百九十四步。	Method saith: put [down the] corner unto [the] mulberry tree, [even] one hundred [and] forty-seven paces; doubling it, resulteth in two hundred [and] ninety-four paces.
以五乘之、得一千四百七十步。以七除之、得二百一十步、自相乘、得四萬四千一百步。	Multiplying it by five, resulteth in one thousand four hundred [and] seventy paces. Dividing it by seven, resulteth in two hundred [and] ten paces, [which], multiplied with itself, resulteth in four myriad [and] four thousand one hundred paces.
以二百四十步除之、即得。	Dividing it by two hundred [and] forty paces, [we] are done.	In modern notation, the area of a square with circumradius (or semi-diagonal) s = 147 \unit{paces} is \begin{aligned} A &\approx \left( 2s \cdot \frac{5}{7} \right)^2 \\[\tallspace] &= \left( 2 \times 147 \unit{paces} \times \frac{5}{7} \right)^2 \div \frac{240 \unit{paces}^2}{\unit{acre}} \\[\tallspace] &= 183 \unit{acres} + 180 \unit{paces}^2 \\ &= 1 \unit{yardland} + 83 \unit{acres} + 180 \unit{paces}^2. \end{aligned} Here \sqrt{2} \approx 7/5, see Vol. I §5.

Source text

Target text

Notes

今有方田、桑生中央、從角至桑、一百四十七步。問為田幾何。

Suppose there be [a] square field, [a] mulberry-tree growing [in the] centre, [and] from [a] corner unto [the] mulberry-tree, one hundred [and] forty-seven paces. [We] ask, how much field be [this]?

答曰、一頃八十三畝奇一百八十步。

Answer saith: one yardland [and] eighty-three acres remainder one hundred [and] eighty paces.

頃： yardland
Kʻing (頃), equal to 100 mu (畝) or Chinese acres. Difficult to render, and I have chosen "yardland" because it sounds more natural, even though the imperial oxgang (which is only half an imperial yardland or virgate) is closer in size.

術曰、置角至桑一百四十七步、倍之、得二百九十四步。

Method saith: put [down the] corner unto [the] mulberry tree, [even] one hundred [and] forty-seven paces; doubling it, resulteth in two hundred [and] ninety-four paces.

以五乘之、得一千四百七十步。以七除之、得二百一十步、自相乘、得四萬四千一百步。

Multiplying it by five, resulteth in one thousand four hundred [and] seventy paces. Dividing it by seven, resulteth in two hundred [and] ten paces, [which], multiplied with itself, resulteth in four myriad [and] four thousand one hundred paces.

以二百四十步除之、即得。

Dividing it by two hundred [and] forty paces, [we] are done.

In modern notation, the area of a square with circumradius (or semi-diagonal) s = 147 \unit{paces} is
\begin{aligned} A &\approx \left( 2s \cdot \frac{5}{7} \right)^2 \\[\tallspace] &= \left( 2 \times 147 \unit{paces} \times \frac{5}{7} \right)^2 \div \frac{240 \unit{paces}^2}{\unit{acre}} \\[\tallspace] &= 183 \unit{acres} + 180 \unit{paces}^2 \\ &= 1 \unit{yardland} + 83 \unit{acres} + 180 \unit{paces}^2. \end{aligned}
Here \sqrt{2} \approx 7/5, see Vol. I §5.

《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II" §14. Area of a square of given circumradius

Translation

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《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§14. Area of a square of given circumradius