《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§29. Two-point method of false position (3)

This section gives a worked example of the two-point method of false position for determining the input to an affine function that produces a given output.

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
今有百鹿入城。家取一鹿、不盡、又三家共一鹿、適盡。問城中家幾何。 Suppose there be [an] hundred deer entering [a] city. [Each] family taking one deer, [there be a] remainder; [each] three families together again [taking] one deer, just exhausteth [them]. [We] ask, how many [be the] families admist [the] city?
  • 不盡： [there be a] remainder; lit. exhausteth not
  • Version A erroneously has 城下中 for 城中.
  • In modern notation, we seek x the number of families such that
    f(x) = x + \frac{x}{3} = 100.
答曰、七十五家。 Answer saith: seventy-five families.
術曰、以盈不足取之。 Method saith: take it by [the method of] the excess [and] the not sufficient.
假令七十二家、鹿盈四。令之九十家、鹿不足二十。 Supposing that [there be] seventy-two families, [the] deer excess [be] four. Supposing them ninety families, [be the] deer twenty not sufficient.
  • Version A erroneously has 鹿盡四 for 鹿盈四.
  • Version B and Version C erroneously have 鹿不盡四 for 鹿盈四.
  • 令之： supposing them: lit. making them
  • In modern notation, for surplus input X_\mathrm{s} = 72 and deficit input X_\mathrm{d} = 90, we have output surplus and deficit
    \begin{aligned} Y_\mathrm{s} &= 100 - f(X_\mathrm{s}) = 100 - \roundbr{72 + \frac{72}{3}} = +4 \\[\tallspace] -Y_\mathrm{d} &= 100 - f(X_\mathrm{d}) = 100 - \roundbr{90 + \frac{90}{3}} = -20. \end{aligned}
    Since 0 = 100 - f(x), and f is affine, we have
    \frac{x - X_\mathrm{s}}{Y_\mathrm{s}} = \frac{X_\mathrm{d} - x}{Y_\mathrm{d}}.
    Therefore
    x = \frac{ X_\mathrm{d} Y_\mathrm{s} + X_\mathrm{s} Y_\mathrm{d} }{ Y_\mathrm{s} + Y_\mathrm{d} }.
置七十二於右上、盈四於右下。置九十於左上、不足二十於左下。 Put seventy-two upon [the] upper right, [and the] excess four upon [the] lower right. Put ninety upon [the] upper left, [and the] twenty not sufficient upon [the] lower left.
  • In modern notation, the matrix
    \begin{pmatrix} X_\mathrm{d} & X_\mathrm{s} \\ -Y_\mathrm{d} & +Y_\mathrm{s} \end{pmatrix} = \begin{pmatrix} 90 & 72 \\ -20 & +4 \end{pmatrix}
    has determinant X_\mathrm{d} Y_\mathrm{s} + X_\mathrm{s} Y_\mathrm{d} which is the numerator of the expression for x.
維乘之、所得、并為實。并盈不足為法。 [In] linkage multiply them; [of] those which result, combine [them] as [the] dividend. Combine [the] excess [and the] not sufficient as [the] divisor.
除之、即得。 Dividing them, [we] are done.
  • In modern notation,
    x = \frac{ X_\mathrm{d} Y_\mathrm{s} + X_\mathrm{s} Y_\mathrm{d} }{ Y_\mathrm{s} + Y_\mathrm{d} } = \frac{ 90 \times 4 + 72 \times 20 }{ 4 + 20 } = 75.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume III §29". <https://yawnoc.github.io/sun-tzu/iii/29> Accessed yyyy-mm-dd.