"Sun Tzŭ's Computational Classic: Volume II" 《孫子算經卷中》 §27

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
今有女子善織、日自倍、五日織通五尺。問日織幾何。	Suppose there be [a] lady good at weaving, [each] day self-doubling [her output], [and in] five days weaving through five rules. [We] ask, how much weaveth [she each] day?	Version C erroneously has 扣 before 問. In modern notation, the problem is to determine the lengths L, 2L, 2^2 L, 2^3 L, 2^4 L such that L + 2L + 2^2 L + 2^3 L + 2^4 L = 5 \unit{rules} = 50 \unit{inches}.
答曰、初日織一寸三十一分寸之一十九、次日織三寸三十一分寸之七、次日織六寸三十一分寸之一十四、次日織一尺二寸三十一分寸之二十八、次日織二尺五寸三十一分寸之二十五。	Answer saith: [the] initial day [she] weaveth one inch [and] nineteen thirty-firsts of [an] inch; [the] next day [she] weaveth three inches [and] seven thirty-firsts of [an] inch; [the] next day [she] weaveth six inches [and] fourteen thirty-firsts of [an] inch; [the] next day [she] weaveth one rule, two inches, [and] twenty-eight thirty-firsts of [an] inch; [the] next day [she] weaveth two rules, five inches, [and] twenty-five thirty-firsts of [an] inch.	Version A is missing 寸 in 三十一分寸之二十八.
術曰、各置列衰、副并、得三十一為法。	Method saith: put each [into a] row of waning, [which], combined subsidiarily, resulteth in thirty-one as [the] divisor.	列衰： row of waning The weights (1, 2, 2^2, 2^3, 2^4), which sum to 31. See the chapter 〈衰分〉 'Waned sharing', of 《九章算術》 "Nine Chapters [on] Computational Methods". So called because the weights are usually listed in descending order (which is opposite to what we have here).
以五尺乘未并者、各自為實。	Multiplying those not yet combined by five rules, each [on its] own be [a] dividend.
實如法而一、即得。	[Taking the] dividends as [per the] divisor [being] one, [we] are done.	In modern notation, \begin{aligned} (L, 2L, 2^2 L, 2^3 L, 2^4 L) &= \frac{(1, 2, 2^2, 2^3, 2^4) \cdot 50 \unit{inches}}{31} \\[\tallspace] &= \left( 1 \tfrac{19}{31}, 3 \tfrac{7}{31}, 6 \tfrac{14}{31}, 12 \tfrac{28}{31}, 25 \tfrac{25}{31} \right) \unit{inches}. \end{aligned}

Source text

Target text

Notes

今有女子善織、日自倍、五日織通五尺。問日織幾何。

Suppose there be [a] lady good at weaving, [each] day self-doubling [her output], [and in] five days weaving through five rules. [We] ask, how much weaveth [she each] day?

Version C erroneously has 扣 before 問.
In modern notation, the problem is to determine the lengths L, 2L, 2^2 L, 2^3 L, 2^4 L such that
L + 2L + 2^2 L + 2^3 L + 2^4 L = 5 \unit{rules} = 50 \unit{inches}.

答曰、初日織一寸三十一分寸之一十九、次日織三寸三十一分寸之七、次日織六寸三十一分寸之一十四、次日織一尺二寸三十一分寸之二十八、次日織二尺五寸三十一分寸之二十五。

Answer saith: [the] initial day [she] weaveth one inch [and] nineteen thirty-firsts of [an] inch; [the] next day [she] weaveth three inches [and] seven thirty-firsts of [an] inch; [the] next day [she] weaveth six inches [and] fourteen thirty-firsts of [an] inch; [the] next day [she] weaveth one rule, two inches, [and] twenty-eight thirty-firsts of [an] inch; [the] next day [she] weaveth two rules, five inches, [and] twenty-five thirty-firsts of [an] inch.

Version A is missing 寸 in 三十一分寸之二十八.

術曰、各置列衰、副并、得三十一為法。

Method saith: put each [into a] row of waning, [which], combined subsidiarily, resulteth in thirty-one as [the] divisor.

列衰： row of waning
The weights (1, 2, 2^2, 2^3, 2^4), which sum to 31. See the chapter 〈衰分〉 'Waned sharing', of 《九章算術》 "Nine Chapters [on] Computational Methods". So called because the weights are usually listed in descending order (which is opposite to what we have here).

以五尺乘未并者、各自為實。

Multiplying those not yet combined by five rules, each [on its] own be [a] dividend.

實如法而一、即得。

[Taking the] dividends as [per the] divisor [being] one, [we] are done.

In modern notation,
\begin{aligned} (L, 2L, 2^2 L, 2^3 L, 2^4 L) &= \frac{(1, 2, 2^2, 2^3, 2^4) \cdot 50 \unit{inches}}{31} \\[\tallspace] &= \left( 1 \tfrac{19}{31}, 3 \tfrac{7}{31}, 6 \tfrac{14}{31}, 12 \tfrac{28}{31}, 25 \tfrac{25}{31} \right) \unit{inches}. \end{aligned}

《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§27. Weighted sharing (2)

Translation

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《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II" §27. Weighted sharing (2)

Translation

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《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§27. Weighted sharing (2)