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

This section gives a worked example of the two-point method of false position for a system of linear equations in two variables.

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 ll, stolen by pp people, with surplus Ys=6 ⁣lengthsY_\mathrm{s} = 6 \unit{lengths} if each person takes Xs=6 ⁣lengthsX_\mathrm{s} = 6 \unit{lengths}, and deficit Yd=7 ⁣lengthsY_\mathrm{d} = 7 \unit{lengths} if each person takes Xd=7 ⁣lengthsX_\mathrm{d} = 7 \unit{lengths}. Then
    lpXs=+YslpXd=Yd.\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
    (XdXsYdYs),\begin{pmatrix} X_\mathrm{d} & X_\mathrm{s} \\ Y_\mathrm{d} & Y_\mathrm{s} \end{pmatrix},
    we are told to form the products XsYdX_\mathrm{s} Y_\mathrm{d} and XdYsX_\mathrm{d} Y_\mathrm{s}, and add them together to obtain ll. 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=+YsXsYdXd1Xs1Xd=XsYd+XdYsXdXs=6×7+7×676 ⁣lengths2 ⁣lengths.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 XdXs=7 ⁣lengths6 ⁣lengths=1 ⁣lengthX_\mathrm{d} - X_\mathrm{s} = 7 \unit{lengths} - 6 \unit{lengths} = 1 \unit{length}, so the division by XdXsX_\mathrm{d} - X_\mathrm{s} can be omitted if all computations are done in units of  ⁣length (匹)\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 Ys+YdY_\mathrm{s} + Y_\mathrm{d} forms the number of people. Again this does not make dimensional sense, and it only works because XdXs=7 ⁣lengths6 ⁣lengths=1 ⁣lengthX_\mathrm{d} - X_\mathrm{s} = 7 \unit{lengths} - 6 \unit{lengths} = 1 \unit{length}. Indeed Cramer's rule gives
    p=1+Ys1Yd1Xs1Xd=Ys+YdXdXs=6+776 ⁣lengths ⁣lengthsp = \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

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume II §28". <https://yawnoc.github.io/sun-tzu/ii/28> Accessed 2025-07-17.