"Sun Tzŭ's Computational Classic: Volume III" 《孫子算經卷下》 §22

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
今有地、長一千步、廣五百步、尺有鶉、寸有鷃。問鶉鷃各幾何。	Suppose there be land, of length one thousand paces, [and] breadth five hundred paces, [each] rule having [a] quail, [and each] inch having [a] birdie. [We] ask, how many each [be the] quails [and the] birdies?	鷃： birdie; or quail 鶉 and 鷃 are both types of quail; 鷃 has been rendered "birdie" to avoid duplication.
答曰、鶉一千八百萬、鷃一億八千萬。	Answer saith: [the] quails one thousand eight hundred myriad, [and the] birdies one square-myriad [and] eight thousand myriad.	The number of birdies (鷃) is incorrect in all four versions of the source text, and I have faithfully reproduced this error here. The correct number is eighteen square-myriad (十八億). Indeed, whilst a rule (尺) is 10 times an inch (寸), 尺有鶉 and 寸有鷃 specify area densities, so the number of birdies should be a 10^2 = 100 times the number of quails. This is why dimensional analysis is important.
術曰、置長一千步、以廣五百步乘之、得五十萬步。	Method saith: put [down the] length one thousand paces; multiplying it by [the] breadth five hundred paces, resulteth in fifty myriad paces.	In modern notation, the area of the land is \begin{aligned} A &= 1000 \unit{paces} \times 500 \unit{paces} \\ &= 50 \times 10^4 \unit{paces}^2. \end{aligned}
以三十六乘之、得一千八百萬尺、即得鶉數。	Multiplying it by thirty-six, resulteth in one thousand eight hundred myriad rules, which is resulting in [the] number of quails.	Here, the text correctly performs a unit conversion between square paces and square rules: \begin{aligned} A &= 50 \times 10^4 \unit{paces}^2 \times \roundbr{\frac{6 \unit{rules}}{\unit{pace}}}^2 \\[\tallspace] &= 50 \times 10^4 \times 36 \unit{rules}^2 \\ &= 1800 \times 10^4 \unit{rules}^2. \end{aligned}
上十之、即得鷃數。	Decupling it upward, doth result in [the] number of birdies.	Here, the text incorrectly uses a value of 10 for the ratio between birdie density and quail density. The correct ratio is 100, because \frac{1 / {\unit{inch}}^2}{1 / {\unit{rule}}^2} = \roundbr{\frac{\unit{rule}}{\unit{inch}}}^2 = 10^2 = 100. Version C erroneously has 赉 for 鶉.

Source text

Target text

Notes

今有地、長一千步、廣五百步、尺有鶉、寸有鷃。問鶉鷃各幾何。

Suppose there be land, of length one thousand paces, [and] breadth five hundred paces, [each] rule having [a] quail, [and each] inch having [a] birdie. [We] ask, how many each [be the] quails [and the] birdies?

鷃： birdie; or quail
鶉 and 鷃 are both types of quail; 鷃 has been rendered "birdie" to avoid duplication.

答曰、鶉一千八百萬、鷃一億八千萬。

Answer saith: [the] quails one thousand eight hundred myriad, [and the] birdies one square-myriad [and] eight thousand myriad.

The number of birdies (鷃) is incorrect in all four versions of the source text, and I have faithfully reproduced this error here.
The correct number is eighteen square-myriad (十八億). Indeed, whilst a rule (尺) is 10 times an inch (寸), 尺有鶉 and 寸有鷃 specify area densities, so the number of birdies should be a 10^2 = 100 times the number of quails. This is why dimensional analysis is important.

術曰、置長一千步、以廣五百步乘之、得五十萬步。

Method saith: put [down the] length one thousand paces; multiplying it by [the] breadth five hundred paces, resulteth in fifty myriad paces.

In modern notation, the area of the land is
\begin{aligned} A &= 1000 \unit{paces} \times 500 \unit{paces} \\ &= 50 \times 10^4 \unit{paces}^2. \end{aligned}

以三十六乘之、得一千八百萬尺、即得鶉數。

Multiplying it by thirty-six, resulteth in one thousand eight hundred myriad rules, which is resulting in [the] number of quails.

Here, the text correctly performs a unit conversion between square paces and square rules:
\begin{aligned} A &= 50 \times 10^4 \unit{paces}^2 \times \roundbr{\frac{6 \unit{rules}}{\unit{pace}}}^2 \\[\tallspace] &= 50 \times 10^4 \times 36 \unit{rules}^2 \\ &= 1800 \times 10^4 \unit{rules}^2. \end{aligned}

上十之、即得鷃數。

Decupling it upward, doth result in [the] number of birdies.

Here, the text incorrectly uses a value of 10 for the ratio between birdie density and quail density. The correct ratio is 100, because
\frac{1 / {\unit{inch}}^2}{1 / {\unit{rule}}^2} = \roundbr{\frac{\unit{rule}}{\unit{inch}}}^2 = 10^2 = 100.
Version C erroneously has 赉 for 鶉.

《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§22. Unit conversion (3)

Translation

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《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III" §22. Unit conversion (3)

Translation

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《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§22. Unit conversion (3)