割圆术 (赵友钦)
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。
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)}}}}}}}}}
  
……