《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§18. A system of linear equations (2)

This section gives the solution to a specific system of linear equations.

The relevant unit conversions for length are

\begin{aligned} 1 \unit{rod~(丈)} &= 10 \unit{rules~(尺)} \\ 1 \unit{rule~(尺)} &= 10 \unit{inches~(寸)}. \end{aligned}

See Vol. I §1 (Units of length).

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
今有木不知長短。引繩度之、餘繩四尺五寸、屈繩量之、不足一尺。問木長幾何。 Suppose there be wood [we] know not [the] length of. Drawing [a] cord [and] measuring it, [is there] excess cord four rules [and] five inches; [but] bending [the] cord [and] gauging it, [is it] one rule not sufficient. [We] ask, how much [be the] length of [the] wood?
  • 度： measuring

    度、入聲、 Cantonese: tok9, Mandarin: duò (Government-regulated 統讀: duó)

  • 餘： [is there] excess; or [there] remaineth
  • 屈： bending

    Specifically bending in half.

  • 量： gauging

    量、平聲、 Cantonese: lœng4, Mandarin: liáng

  • Version B is missing 木長 before 幾何.
  • In modern notation, we have the system
    \begin{aligned} c &= w + 4.5 \unit{rules} \\ c/2 &= w - 1 \unit{rule} \end{aligned}
    in c the length of the cord and w the length of the wood.
答曰、六尺五寸。 Answer saith: six rules [and] five inches.
術曰、置餘繩四尺五寸、加不足一尺、共五尺五寸。倍之、得一丈一尺。減餘四尺五寸、即得。 Method saith: put [down the] excess cord four rules [and] five inches, [and] add [the] one rule not sufficient, altogether five rules [and] five inches. Doubling it, resulteth in one rod [and] one rule. Subtracting [the] excess four rules [and] five inches, [we] are done.
  • In modern notation, this result is arrived at by first doubling both equations and taking the difference, giving
    c = 2 \times (4.5 \unit{rules} + 1 \unit{rule}) = 11 \unit{rules},
    and then substituting into the first equation to get
    w = c - 4.5 \unit{rules} = 6.5 \unit{rules}.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume III §18". <https://yawnoc.github.io/sun-tzu/iii/18> Accessed yyyy-mm-dd.