Zhao Youqin's π algorithm

Zhao Youqin's π algorithm

A page from Zhao Youqin's book Ge Xiang Xin Shu vol 5

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 − 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.

See also

• Liu Hui's π algorithm