# 割圆术 (赵友钦)

（重定向自赵友钦割圆术

${\displaystyle d={\sqrt {r^{2}-({\frac {\ell }{2}})^{2}}}}$

${\displaystyle e=r-d=r-{\sqrt {r^{2}-({\frac {\ell }{2}})^{2}}}}$

d 的延长线与圆周相交点将圆周等分为正八边形。

${\displaystyle \ell _{2}={\sqrt {(\ell /2)^{2}+e^{2}}}}$

${\displaystyle \ell _{2}={\frac {1}{2}}*{\sqrt {\ell ^{2}+4*(r-{\frac {1}{2}}*{\sqrt {4*r^{2}-\ell ^{2}}})^{2}}}}$

${\displaystyle \ell _{3}}$ 为分割圆成正16边形之边长，赵友钦正确地推断${\displaystyle \ell _{3}}$${\displaystyle \ell _{2}}$的迭代关系：

${\displaystyle \ell _{3}={\frac {1}{2}}*{\sqrt {\ell ^{2}+4*(r-{\frac {1}{2}}*{\sqrt {4*r^{2}-(\ell _{2})^{2}}})^{2}}}}$

${\displaystyle \ell _{n+1}={\frac {1}{2}}*{\sqrt {\ell ^{2}+4*(r-{\frac {1}{2}}*{\sqrt {4*r^{2}-(\ell _{n})^{2}}})^{2}}}}$

${\displaystyle \ell _{1}={\sqrt {(}}2)}$

${\displaystyle \ell _{2}={\sqrt {2-{\sqrt {(}}2)}}}$

${\displaystyle \ell _{3}={\sqrt {2-{\sqrt {2+{\sqrt {(}}2)}}}}}$

${\displaystyle \ell _{4}={\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {(}}2)}}}}}}}$

${\displaystyle \ell _{5}={\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {(}}2)}}}}}}}}}$

……

## 圆周率

${\displaystyle \pi ={\frac {3141.592}{1000}}}$


4 3.121445
8 3.136548
16 3.140331
32 3.141277
64 3.141513
128 3.141572
256 3.141587
512 3.141591
1024 3.141592
2048 3.141592
16384 3.141592+

## 密率

${\displaystyle \pi \approx {\frac {355}{113}}}$

${\displaystyle \pi \approx {\frac {3141.592}{1000}}}$

${\displaystyle 113*\pi \approx {\frac {3141.592}{1000}}*113=355}$

${\displaystyle \pi \approx {\frac {355}{113}}}$

## 参考文献

1. ^ 李俨 《中国数学史》 第六章《宋元数学》 144-145页 商务印书馆 1998 ISBN 978-7-100-01474-3
2. ^ Yoshio Mikami The Development of Mathematics in China and Japan p135-136