いずれの公式も  arctan(1/1)  の逐次展開で求める事ができる。
以下に実例を示す。
 

[Euler の公式]

arctan(1/1)
           ↓
arctan(1/2)+arctan(1/3)

[Dahse の公式]

arctan(1/2)+arctan(1/3)
           ↓
arctan(1/2)+arctan(1/5)+arctan(1/8)

[Vega の公式]

arctan(1/2)+arctan(1/3)
           ↓
2arctan(1/2)-arctan(1/7)

[Clausen の公式]

arctan(1/2)+arctan(1/3)
           ↓
2arctan(1/3)+arctan(1/7)

[Gauss の公式]

arctan(1/2)+arctan(1/3)
           ↓
2arctan(1/3)+arctan(1/7)
           ↓
2arctan(1/5)+arctan(1/7)+2arctan(1/8)
           ↓
2arctan(1/5)+3arctan(1/7)-2arctan(1/57)
           ↓
5arctan(1/7)+2arctan(1/18)-2arctan(1/57)
           ↓
5arctan(1/12)+5arctan(1/17)+2arctan(1/18)-2arctan(1/57)
           ↓
10arctan(1/17)+2arctan(1/18)+5arctan(1/41)-2arctan(1/57)
           ↓
12arctan(1/18)+5arctan(1/41)-2arctan(1/57)+10arctan(1/307)
           ↓
12arctan(1/18)+5arctan(1/41)+8arctan(1/57)-10arctan(1/70)
           ↓
12arctan(1/18)+8arctan(1/57)-5arctan(1/70)+5arctan(1/99)
           ↓
12arctan(1/18)+8arctan(1/57)-5arctan(1/239)

[Machin の公式]

12arctan(1/18)+8arctan(1/57)-5arctan(1/239)
           ↓
12arctan(1/18)+8arctan(1/70)-5arctan(1/239)+8arctan(1/307)
           ↓
8arctan(1/17)+4arctan(1/18)+8arctan(1/70)-5arctan(1/239)
           ↓
4arctan(1/5)-4arctan(1/7)+8arctan(1/17)+8arctan(1/70)-5arctan(1/239)
           ↓
4arctan(1/5)-4arctan(1/12)+4arctan(1/17)+8arctan(1/70)-5arctan(1/239)
           ↓
4arctan(1/5)-4arctan(1/41)+8arctan(1/70)-5arctan(1/239)
           ↓
4arctan(1/5)+4arctan(1/70)-4arctan(1/99)-5arctan(1/239)
           ↓
4arctan(1/5)-arctan(1/239)

[Rutherford の公式]

4arctan(1/5)-arctan(1/239)
           ↓
4arctan(1/5)-arctan(1/70)+arctan(1/99)

[Vega の公式]

4arctan(1/5)-arctan(1/239)
           ↓
4arctan(1/5)-arctan(1/408)-arctan(1/577)
           ↓
4arctan(1/5)-2arctan(1/408)+arctan(1/1393)

 
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