割圓術 (趙友欽)維基百科,自由的 encyclopedia 趙友欽割圓術是元代數學家趙友欽在所著的《革象新書》卷五《乾象周髀》篇研究的割圓術。與劉徽從內接正六角形開始不同,趙氏割圓術從分割內接正方形開始[1]。 趙友欽割圓術 趙友欽《革象新書》卷五《乾象周髀》篇割圓術書影 如圖,圓的半徑為r; 內接正方形的邊長為 ℓ {\displaystyle \ell } ,由圓心到正方形一邊倒垂直距離為 d d = r 2 − ( ℓ 2 ) 2 {\displaystyle d={\sqrt {r^{2}-({\frac {\ell }{2}})^{2}}}} e = r − d = r − r 2 − ( ℓ 2 ) 2 {\displaystyle e=r-d=r-{\sqrt {r^{2}-({\frac {\ell }{2}})^{2}}}} d 的延長線與圓周相交點將圓周等分為正八邊形。 令正八邊形的邊長為 ℓ 2 {\displaystyle \ell _{2}} ℓ 2 = ( ℓ / 2 ) 2 + e 2 {\displaystyle \ell _{2}={\sqrt {(\ell /2)^{2}+e^{2}}}} ℓ 2 = 1 2 ∗ ℓ 2 + 4 ∗ ( r − 1 2 ∗ 4 ∗ r 2 − ℓ 2 ) 2 {\displaystyle \ell _{2}={\frac {1}{2}}*{\sqrt {\ell ^{2}+4*(r-{\frac {1}{2}}*{\sqrt {4*r^{2}-\ell ^{2}}})^{2}}}} 設 ℓ 3 {\displaystyle \ell _{3}} 為分割圓成正16邊形之邊長,趙友欽正確地推斷 ℓ 3 {\displaystyle \ell _{3}} 與 ℓ 2 {\displaystyle \ell _{2}} 的迭代關係: ℓ 3 = 1 2 ∗ ( ℓ 2 ) 2 + 4 ∗ ( r − 1 2 ∗ 4 ∗ r 2 − ( ℓ 2 ) 2 ) 2 {\displaystyle \ell _{3}={\frac {1}{2}}*{\sqrt {(\ell _{2})^{2}+4*(r-{\frac {1}{2}}*{\sqrt {4*r^{2}-(\ell _{2})^{2}}})^{2}}}} 推而廣之: ℓ n + 1 = 1 2 ∗ ( ℓ n ) 2 + 4 ∗ ( r − 1 2 ∗ 4 ∗ r 2 − ( ℓ n ) 2 ) 2 {\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; ℓ 1 = ( 2 ) {\displaystyle \ell _{1}={\sqrt {(}}2)} ℓ 2 = 2 − ( 2 ) {\displaystyle \ell _{2}={\sqrt {2-{\sqrt {(}}2)}}} ℓ 3 = 2 − 2 + ( 2 ) {\displaystyle \ell _{3}={\sqrt {2-{\sqrt {2+{\sqrt {(}}2)}}}}} ℓ 4 = 2 − 2 + 2 + ( 2 ) {\displaystyle \ell _{4}={\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {(}}2)}}}}}}} ℓ 5 = 2 − 2 + 2 + 2 + ( 2 ) {\displaystyle \ell _{5}={\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {(}}2)}}}}}}}}} ……
趙友欽割圓術是元代數學家趙友欽在所著的《革象新書》卷五《乾象周髀》篇研究的割圓術。與劉徽從內接正六角形開始不同,趙氏割圓術從分割內接正方形開始[1]。 趙友欽割圓術 趙友欽《革象新書》卷五《乾象周髀》篇割圓術書影 如圖,圓的半徑為r; 內接正方形的邊長為 ℓ {\displaystyle \ell } ,由圓心到正方形一邊倒垂直距離為 d d = r 2 − ( ℓ 2 ) 2 {\displaystyle d={\sqrt {r^{2}-({\frac {\ell }{2}})^{2}}}} e = r − d = r − r 2 − ( ℓ 2 ) 2 {\displaystyle e=r-d=r-{\sqrt {r^{2}-({\frac {\ell }{2}})^{2}}}} d 的延長線與圓周相交點將圓周等分為正八邊形。 令正八邊形的邊長為 ℓ 2 {\displaystyle \ell _{2}} ℓ 2 = ( ℓ / 2 ) 2 + e 2 {\displaystyle \ell _{2}={\sqrt {(\ell /2)^{2}+e^{2}}}} ℓ 2 = 1 2 ∗ ℓ 2 + 4 ∗ ( r − 1 2 ∗ 4 ∗ r 2 − ℓ 2 ) 2 {\displaystyle \ell _{2}={\frac {1}{2}}*{\sqrt {\ell ^{2}+4*(r-{\frac {1}{2}}*{\sqrt {4*r^{2}-\ell ^{2}}})^{2}}}} 設 ℓ 3 {\displaystyle \ell _{3}} 為分割圓成正16邊形之邊長,趙友欽正確地推斷 ℓ 3 {\displaystyle \ell _{3}} 與 ℓ 2 {\displaystyle \ell _{2}} 的迭代關係: ℓ 3 = 1 2 ∗ ( ℓ 2 ) 2 + 4 ∗ ( r − 1 2 ∗ 4 ∗ r 2 − ( ℓ 2 ) 2 ) 2 {\displaystyle \ell _{3}={\frac {1}{2}}*{\sqrt {(\ell _{2})^{2}+4*(r-{\frac {1}{2}}*{\sqrt {4*r^{2}-(\ell _{2})^{2}}})^{2}}}} 推而廣之: ℓ n + 1 = 1 2 ∗ ( ℓ n ) 2 + 4 ∗ ( r − 1 2 ∗ 4 ∗ r 2 − ( ℓ n ) 2 ) 2 {\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; ℓ 1 = ( 2 ) {\displaystyle \ell _{1}={\sqrt {(}}2)} ℓ 2 = 2 − ( 2 ) {\displaystyle \ell _{2}={\sqrt {2-{\sqrt {(}}2)}}} ℓ 3 = 2 − 2 + ( 2 ) {\displaystyle \ell _{3}={\sqrt {2-{\sqrt {2+{\sqrt {(}}2)}}}}} ℓ 4 = 2 − 2 + 2 + ( 2 ) {\displaystyle \ell _{4}={\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {(}}2)}}}}}}} ℓ 5 = 2 − 2 + 2 + 2 + ( 2 ) {\displaystyle \ell _{5}={\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {(}}2)}}}}}}}}} ……