《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 pp people and cc chariots. Then
    p=3(c2)p=2c+9.\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=3×2+9=15.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 22 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:
    3 ⁣people/ ⁣chariot×(2 ⁣chariots)+9 ⁣people(32) ⁣people/ ⁣chariot=15 ⁣chariots.\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=2×15+9=39.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 2025-07-10.