《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§15. Two-point method of false position (2)

This section gives a worked example of the two-point method of false position for a system of linear equations in two variables. See Vol. II §28 for a more detailed discussion of the method.

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
今有三人共車、二車空、二人共車、九人步。問人與車各幾何。 Suppose there be three people together [a] chariot, [and] two chariots empty; [but] two people together [a] chariot, [and] nine people afoot. [We] ask, how many each [be the] people and [the] chariots?
  • 步： afoot; lit. step
  • In modern notation, suppose there are p people and c chariots. Then
    \begin{aligned} p &= 3(c - 2) \\ p &= 2c + 9. \end{aligned}
答曰、一十五車、三十九人。 Answer saith: fifteen chariots, [and] thirty-nine people.
術曰、置二車、以三乘之、得六、加步者九人、得車一十五。 Method saith: put [down the] two chariots; multiplying it by three, resulteth in six; adding those afoot, [even the] nine people, resulteth in chariots fifteen.
  • In modern notation,
    c = 3 \times 2 + 9 = 15.
  • Version A, Version B, and Version D all erroneously have 置二人 for 置二車; the latter is correct because the 2 arises not from the two people per chariot, but from the two chariots empty. This is more easily seen if we write the computation dimensionally:
    \frac{ 3 \unit{people} / {\unit{chariot}} \times (2 \unit{chariots}) + 9 \unit{people} }{ (3 - 2) \unit{people} / {\unit{chariot}} } = 15 \unit{chariots}.
欲知人者、以二乘車、加九人、即得。 Wishing to know [the] people: multiplying [the] chariots by two, [and] adding [the] nine people, [we] are done.
  • In modern notation,
    p = 2 \times 15 + 9 = 39.
  • Version D erroneously has 加九十 for 加九人.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume III §15". <https://yawnoc.github.io/sun-tzu/iii/15> Accessed yyyy-mm-dd.