《孫子算經卷上》 "Sun Tzŭ's Computational Classic: Volume I"
§5. \pi \approx 3, \sqrt{2} \approx 7/5

This section gives the approximate values of 3 for \pi and 7/5 for \sqrt{2}.

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
周三徑一、方五邪七。
見邪求方、五之、七而一。
見方求邪、七之、五而一。		 [The] circumference three: [the] diameter one. [The] square's [edge] five: [the] diagonal seven.
Seeing [the] diagonal [and] seeking [the] square's [edge], quintuple it, [with] seven [being] one.
Seeing [the] square's [edge and] seeking [the] diagonal, septuple it, [with] five [being] one.
  • 邪： diagonal; lit. incline

    Here, 邪 is interchangeable with 斜.

  • 五之： quintuple it

    Multiply by five.

  • 七而一： seven [being] one

    Divide by seven.

Extended commentary

Here, 3 and 7/5 are used as approximations of \pi and \sqrt{2}, which arise in the geometry of the circle and square respectively.

For a circle of circumference C and diameter d we have

\frac{C}{d} = \pi \approx \frac{3}{1}.
Diagram of a circle of circumference C and diameter d.

For a square of side length L and diagonal D we have

\frac{D}{L} = \sqrt{2} \approx \frac{7}{5},

so

\begin{aligned} L & \approx \frac{5D}{7} \\[\tallspace] D & \approx \frac{7L}{5}. \end{aligned}
Diagram of a square of side length L and diagonal D.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §5". <https://yawnoc.github.io/sun-tzu/i/5> Accessed yyyy-mm-dd.