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 by Yuan dynasty astronomer-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.^[1]

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.^[2]

References

↑ Yoshio Mikami, Development of Mathematics in China and Japan, Chapter 20, The Studies about the Value of $π$ etc., pp 135–138
↑ Yoshio Mikami, p136

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Zhao Youqin's π algorithm

Algorithm

See also

References