割圆术 (赵友钦)

赵友钦割圆术是元代数学家赵友钦在所著的《革象新书》卷五《乾象周髀》篇研究的割圆术。与刘徽从内接正六角形开始不同，赵氏割圆术从分割内接正方形开始^[1]。

赵友钦割圆术

赵友钦《革象新书》卷五《乾象周髀》篇割圆术书影

如图，圆的半径为r; 内接正方形的边长为 ${\displaystyle \ell }$,由圆心到正方形一边倒垂直距离为 d

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

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

d 的延长线与圆周相交点将圆周等分为正八边形。

令正八边形的边长为${\displaystyle \ell _{2}}$

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

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

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

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

推而广之：

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

令 r=1;

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

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

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

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

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

……