Zhao Youqin's π algorithm explained

Zhao Youqin's algorithm was an algorithm devised by Yuan dynasty Chinese astronomer and mathematician Zhao Youqin (Chinese: 赵友钦, ? – 1330) to calculate the value of in his book Ge Xiang Xin Shu (Chinese: 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( (\ell)/(2)} \right) ^{2}} e = r - d = r - \sqrt{r ^{2- \left( (\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( (\ell)/(2)} \right) ^{2+e} ^{2}} \ell _{2= (1)/(2)} \sqrt{ \ell ^{2} +4 \left(r - (1)/(2) \sqrt{ 4 r ^{2-\ell} ^{2}\right)} ^{2}}

Let

l _{3} denotes the length of a side of hexadecagon\ell _{3= (1)/(2)} \sqrt{ \ell ^{2} _{2} +4 \left(r - (1)/(2) \sqrt{ 4 r ^{2 }\right)} _{2} ^{2} }

similarly

\ell _{n+1} = (1)/(2) \sqrt{ \ell ^{2} _{n} +4 \left(r - (1)/(2) \sqrt{ 4 r ^{2 }\right)} _{n} ^{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, that is 3, 3.14, and, the last is the most exact.^[2]

See also

• Liu Hui's algorithm

References

1. [Yoshio Mikami]
2. Yoshio Mikami, p136