"Sun Tzŭ's Computational Classic: Volume II" 《孫子算經卷中》 §28

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
今有人盜庫絹、不知所失幾何。但聞草中分絹、人得六匹、盈六匹、人得七匹、不足七匹。	Suppose there be people [having] stolen warehoused strong-silk, [and we] know not how much that lost [be]. But [we] hear [a] sharing of [the] strong-silk amidst [the] grass: [each] person getting six lengths, [be there] excess six lengths; [each] person getting seven lengths, [be there] not sufficient seven lengths.
問人絹各幾何。	[We] ask, how much each [be the] people [and the] strong-silk?	Version A and Version D have 得幾何. I have taken 各幾何 per Version B and Version C. In modern notation, suppose the length of strong-silk is l, stolen by p people, with surplus Y_\mathrm{s} = 6 \unit{lengths} if each person takes X_\mathrm{s} = 6 \unit{lengths}, and deficit Y_\mathrm{d} = 7 \unit{lengths} if each person takes X_\mathrm{d} = 7 \unit{lengths}. Then \begin{aligned} l - p X_\mathrm{s} &= +Y_\mathrm{s} \\ l - p X_\mathrm{d} &= -Y_\mathrm{d}. \end{aligned}
答曰、賊一十三人、絹八十四匹。	Answer saith: [the] thieves [be] thirteen people, [and the] strong-silk eighty-four lengths.
術曰、先置人得六匹於右上、盈六匹於右下、後置人得七匹於左上、不足七匹於左下。	Method saith: first put [each] person getting six lengths upon [the] upper right, [and the] excess six lengths upon [the] lower right; [and] afterward put [each] person getting seven lengths upon [the] upper left, [and the] not sufficient seven lengths upon [the] lower left.
維乘之、所得、并之、為絹。	[In] linkage multiply them; [of] those which result, combining them, maketh [the] strong-silk.	Having formed the matrix \begin{pmatrix} X_\mathrm{d} & X_\mathrm{s} \\ Y_\mathrm{d} & Y_\mathrm{s} \end{pmatrix}, we are told to form the products X_\mathrm{s} Y_\mathrm{d} and X_\mathrm{d} Y_\mathrm{s}, and add them together to obtain l. This does not make dimensional sense; area cannot equal length. The modern equivalent is to apply Cramer's rule to the system above, which yields l = \frac{ \begin{vmatrix} +Y_\mathrm{s} & -X_\mathrm{s} \\ -Y_\mathrm{d} & -X_\mathrm{d} \end{vmatrix} }{ \begin{vmatrix} 1 & -X_\mathrm{s} \\ 1 & -X_\mathrm{d} \end{vmatrix} } = \frac{ X_\mathrm{s} Y_\mathrm{d} + X_\mathrm{d} Y_\mathrm{s} }{ X_\mathrm{d} - X_\mathrm{s} } = \frac{6 \times 7 + 7 \times 6}{7 - 6} \frac{\unit{lengths}^2}{\unit{lengths}}. We see that Sun Tzŭ's prescription only works because the denominator is X_\mathrm{d} - X_\mathrm{s} = 7 \unit{lengths} - 6 \unit{lengths} = 1 \unit{length}, so the division by X_\mathrm{d} - X_\mathrm{s} can be omitted if all computations are done in units of \unit{length~(匹)}.
并下盈不足、為人。	Combining [the] excess [and the] not sufficient below, maketh [the] people.	We are told that the sum Y_\mathrm{s} + Y_\mathrm{d} forms the number of people. Again this does not make dimensional sense, and it only works because X_\mathrm{d} - X_\mathrm{s} = 7 \unit{lengths} - 6 \unit{lengths} = 1 \unit{length}. Indeed Cramer's rule gives p = \frac{ \begin{vmatrix} 1 & +Y_\mathrm{s} \\ 1 & -Y_\mathrm{d} \end{vmatrix} }{ \begin{vmatrix} 1 & -X_\mathrm{s} \\ 1 & -X_\mathrm{d} \end{vmatrix} } = \frac{ Y_\mathrm{s} + Y_\mathrm{d} }{ X_\mathrm{d} - X_\mathrm{s} } = \frac{6 + 7}{7 - 6} \frac{\unit{lengths}}{\unit{lengths}} for the number of people.

Source text

Target text

Notes

今有人盜庫絹、不知所失幾何。但聞草中分絹、人得六匹、盈六匹、人得七匹、不足七匹。

Suppose there be people [having] stolen warehoused strong-silk, [and we] know not how much that lost [be]. But [we] hear [a] sharing of [the] strong-silk amidst [the] grass: [each] person getting six lengths, [be there] excess six lengths; [each] person getting seven lengths, [be there] not sufficient seven lengths.

問人絹各幾何。

[We] ask, how much each [be the] people [and the] strong-silk?

Version A and Version D have 得幾何. I have taken 各幾何 per Version B and Version C.
In modern notation, suppose the length of strong-silk is l, stolen by p people, with surplus Y_\mathrm{s} = 6 \unit{lengths} if each person takes X_\mathrm{s} = 6 \unit{lengths}, and deficit Y_\mathrm{d} = 7 \unit{lengths} if each person takes X_\mathrm{d} = 7 \unit{lengths}. Then
\begin{aligned} l - p X_\mathrm{s} &= +Y_\mathrm{s} \\ l - p X_\mathrm{d} &= -Y_\mathrm{d}. \end{aligned}

答曰、賊一十三人、絹八十四匹。

Answer saith: [the] thieves [be] thirteen people, [and the] strong-silk eighty-four lengths.

術曰、先置人得六匹於右上、盈六匹於右下、後置人得七匹於左上、不足七匹於左下。

Method saith: first put [each] person getting six lengths upon [the] upper right, [and the] excess six lengths upon [the] lower right; [and] afterward put [each] person getting seven lengths upon [the] upper left, [and the] not sufficient seven lengths upon [the] lower left.

維乘之、所得、并之、為絹。

[In] linkage multiply them; [of] those which result, combining them, maketh [the] strong-silk.

Having formed the matrix
\begin{pmatrix} X_\mathrm{d} & X_\mathrm{s} \\ Y_\mathrm{d} & Y_\mathrm{s} \end{pmatrix},
we are told to form the products X_\mathrm{s} Y_\mathrm{d} and X_\mathrm{d} Y_\mathrm{s}, and add them together to obtain l. This does not make dimensional sense; area cannot equal length. The modern equivalent is to apply Cramer's rule to the system above, which yields
l = \frac{ \begin{vmatrix} +Y_\mathrm{s} & -X_\mathrm{s} \\ -Y_\mathrm{d} & -X_\mathrm{d} \end{vmatrix} }{ \begin{vmatrix} 1 & -X_\mathrm{s} \\ 1 & -X_\mathrm{d} \end{vmatrix} } = \frac{ X_\mathrm{s} Y_\mathrm{d} + X_\mathrm{d} Y_\mathrm{s} }{ X_\mathrm{d} - X_\mathrm{s} } = \frac{6 \times 7 + 7 \times 6}{7 - 6} \frac{\unit{lengths}^2}{\unit{lengths}}.
We see that Sun Tzŭ's prescription only works because the denominator is X_\mathrm{d} - X_\mathrm{s} = 7 \unit{lengths} - 6 \unit{lengths} = 1 \unit{length}, so the division by X_\mathrm{d} - X_\mathrm{s} can be omitted if all computations are done in units of \unit{length~(匹)}.

并下盈不足、為人。

Combining [the] excess [and the] not sufficient below, maketh [the] people.

We are told that the sum Y_\mathrm{s} + Y_\mathrm{d} forms the number of people. Again this does not make dimensional sense, and it only works because X_\mathrm{d} - X_\mathrm{s} = 7 \unit{lengths} - 6 \unit{lengths} = 1 \unit{length}. Indeed Cramer's rule gives
p = \frac{ \begin{vmatrix} 1 & +Y_\mathrm{s} \\ 1 & -Y_\mathrm{d} \end{vmatrix} }{ \begin{vmatrix} 1 & -X_\mathrm{s} \\ 1 & -X_\mathrm{d} \end{vmatrix} } = \frac{ Y_\mathrm{s} + Y_\mathrm{d} }{ X_\mathrm{d} - X_\mathrm{s} } = \frac{6 + 7}{7 - 6} \frac{\unit{lengths}}{\unit{lengths}}
for the number of people.

END of Volume II

《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§28. Two-point method of false position (1)

Translation

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《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II" §28. Two-point method of false position (1)

Translation

Cite this page

《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§28. Two-point method of false position (1)