《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.

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}.

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

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

\begin{aligned} L & \approx \frac{5D}{7} \\[\tallspace] D & \approx \frac{7L}{5}. \end{aligned}

Conway (2023). "Sun Tzŭ's Computational Classic: Volume I §5". <https://yawnoc.github.io/sun-tzu/i/5> Accessed 2025-04-30.