Zhao Youqin's π algorithm



Zhao Youqin's π algorithm was an algorithm devised by  Yuan dynasty Chinese astronomer and mathematician Zhao Youqin (赵友钦, ? – 1330) to calculate the value of π in his book Ge Xiang Xin Shu (革象新书).

Algorithm
Zhao Youqin started with an inscribed square in a circle with radius r.

If $$\ell$$ denotes the length of a side of the square, draw a perpendicular line d from the center of the circle to side l. Let e denotes r &minus; d. Then from the diagram:


 * $$d=\sqrt{r^2-\left(\frac{\ell}{2}\right)^2}$$


 * $$e=r-d=r-\sqrt{r^2-\left(\frac{\ell}{2}\right)^2}.$$

Extend the perpendicular line d to dissect the circle into an octagon; $$\ell_2$$ denotes the length of one side of octagon.


 * $$\ell_2=\sqrt{\left(\frac{\ell}{2}\right)^2+e^2}$$


 * $$\ell_2=\frac{1}{2}\sqrt{ \ell^2 +4\left(r-\frac{1}{2} \sqrt{4r^2-\ell^2}\right)^2}$$

Let $$l_3$$ denotes the length of a side of hexadecagon


 * $$\ell_3=\frac{1}{2}\sqrt{ \ell_2^2 +4\left(r-\frac{1}{2}\sqrt{4r^2-\ell_2^2}\right)^2   }$$

similarly


 * $$\ell_{n+1}=\frac{1}{2}\sqrt{ \ell_n^2 +4\left(r-\frac{1}{2}\sqrt{4r^2-\ell_n^2}\right)^2} $$

Proceeding in this way, he at last calculated the side of a 16384-gon, multiplying it by 16384 to obtain 3141.592 for a circle with diameter = 1000 units, or


 * $$\pi =3.141592. \,$$

He multiplied this number by 113 and obtained 355. From this he deduced that of the traditional values of π, that is 3, 3.14, $22⁄7$ and $355⁄113$, the last is the most exact.