アークタンジェント公式
π
4= arctan 1
2+ arctan 1
3Euler
π
4= 4 arctan 1
5− arctan 1
239Machin
π
4= 2 arctan 1
3+ arctan 1
7Hutton
π
4= arctan 1
2+ arctan 1
5+ arctan 1
8Strassnitzky
π
4= 2 arctan 1
2− arctan 1
7Hermann
π
4= arctan 1
2+ arctan 1
4+ arctan 1
13Størmer
π
4= 2 arctan 1
3+ arctan 1
12+ arctan 1
17Størmer
π
4= 2 arctan 1
3+ arctan 1
8+ arctan 1
57Størmer
π
4= 2 arctan 1
3+ arctan 1
9+ arctan 1
32Størmer
π
4= 2 arctan 1
2− arctan 1
5+ arctan 1
18Størmer
π
4= 2 arctan 1
2− arctan 1
9− arctan 1
32Størmer
π
4= 2 arctan 1
2− arctan 1
6+ arctan 1
43Størmer
π
4= 2 arctan 1
2− arctan 1
8− arctan 1
57Størmer
π
4= arctan 1
2+ arctan 1
5+ arctan 1
13+ arctan 1
21Matsumoto
π
4= arctan 1
2+ arctan 1
4+ arctan 1
23+ arctan 1
30Matsumoto
π
4= arctan 1
3+ arctan 1
5+ arctan 1
7+ arctan 1
8Takano
π
4= arctan 1
3+ arctan 1
4+ arctan 1
7+ arctan 1
13Takano
π
4= arctan 1
2+ arctan 1
5+ arctan 1
9+ arctan 1
73Takano
π
4= arctan 1
2+ arctan 1
4+ arctan 1
15+ arctan 1
98Takano
π
2= 2 arctan 1
2+ arctan 4
7+ arctan 1
8Newton
π
4= arctan 1
7+ arctan 1
8+ arctan 4
7
11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020 2121
11
22
33
44
55
66
77
88
99
1010
1111
00
Euler
another solution of Nelsen (1993) page 39
100100 200200 300300 400400 500500 600600 700700 800800 900900 10001000 11001100 12001200 13001300
100100
200200
300300
400400
500500
600600
700700
800800
00
Machin
another solution of Heo (2015) and Nelsen (1990)
11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919
11
22
33
44
55
66
77
88
99
1010
1111
1212
1313
1414
1515
00
Hutton
Reference: Nelsen (2015) page 75
11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020 2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232 3333 3434 3535 3636 3737 3838 3939 4040 4141 4242 4343 4444 4545
11
22
33
44
55
66
77
88
99
1010
1111
1212
1313
1414
1515
1616
1717
1818
1919
2020
2121
2222
2323
2424
2525
00
Strassnitzky
Reference: Nelsen (2015) page 75
11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616
11
22
33
44
55
66
77
88
99
1010
1111
1212
1313
1414
1515
1616
1717
1818
00
Strassnitzky
I improved Nelsen (2015) page 75.
Hermann
I slightly improved Unal (2014a).
11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020 2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232 3333
11
22
33
44
55
66
77
88
99
1010
1111
1212
1313
1414
1515
1616
1717
1818
1919
2020
2121
2222
2323
2424
2525
00
Stormer
11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020 2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232 3333 3434 3535
11
22
33
44
55
66
77
88
99
1010
1111
1212
1313
1414
1515
1616
1717
1818
1919
2020
2121
2222
00
Stormer
1010 2020 3030 4040 5050 6060 7070 8080 9090 100100 110110 120120 130130 140140 150150
1010
2020
3030
4040
5050
6060
00
Stormer
1010 2020 3030 4040 5050 6060 7070 8080 9090 100100 110110 120120 130130
1010
2020
3030
4040
5050
6060
7070
00
Stormer
11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020 2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232 3333 3434 3535 3636 3737 3838 3939 4040 4141 4242 4343 4444 4545 4646 4747 4848 4949 5050 5151 5252 5353 5454 5555 5656 5757 5858 5959 6060 6161 6262 6363 6464
112233445566778899
101011111212131314141515161617171818191920202121222223232424252526262727282829293030313132323333343435353636
00
Stormer
55 1010 1515 2020 2525 3030 3535 4040 4545 5050 5555 6060 6565 7070 7575 8080
55
1010
1515
2020
2525
3030
3535
4040
4545
5050
5555
6060
00
Stormer
55 1010 1515 2020 2525 3030 3535 4040 4545 5050 5555 6060 6565 7070
55
1010
1515
2020
2525
3030
3535
4040
4545
5050
5555
6060
00
Stormer
1010 2020 3030 4040 5050 6060 7070 8080 9090 100100
1010
2020
3030
4040
5050
6060
7070
8080
9090
00
Stormer
11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020 2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232 3333 3434 3535 3636 3737 3838 3939 4040 4141 4242
11
22
33
44
55
66
77
88
99
1010
1111
1212
1313
1414
1515
1616
1717
1818
1919
2020
2121
2222
2323
2424
2525
2626
2727
2828
2929
3030
00
Matsumoto
11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020 2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232 3333 3434 3535 3636 3737 3838 3939 4040 4141 4242 4343 4444 4545 4646 4747 4848 4949 5050 5151 5252 5353 5454 5555 5656 5757 5858
11
22
33
44
55
66
77
88
99
1010
1111
1212
1313
1414
1515
1616
1717
1818
1919
2020
2121
2222
2323
2424
2525
2626
2727
2828
2929
3030
3131
3232
3333
3434
3535
3636
3737
00
Matsumoto
11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020 2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232 3333 3434 3535 3636 3737 3838 3939 4040 4141 4242 4343 4444 4545 4646 4747 4848
11
22
33
44
55
66
77
88
99
1010
1111
1212
1313
1414
1515
1616
1717
1818
1919
2020
2121
2222
2323
2424
2525
2626
2727
2828
2929
3030
3131
00
Takano
55 1010 1515 2020 2525 3030 3535 4040 4545 5050 5555 6060 6565 7070 7575 8080
55
1010
1515
2020
2525
3030
3535
4040
4545
00
Takano
1010 2020 3030 4040 5050 6060 7070 8080 9090 100100 110110 120120 130130 140140 150150 160160
1010
2020
3030
4040
5050
6060
7070
8080
9090
00
Takano
55 1010 1515 2020 2525 3030 3535 4040 4545 5050 5555 6060 6565 7070 7575 8080 8585 9090 9595 100100 105105 110110 115115 120120 125125 130130 135135 140140 145145
55
1010
1515
2020
2525
3030
3535
4040
4545
5050
5555
6060
6565
7070
7575
8080
8585
9090
9595
100100
105105
110110
115115
120120
125125
00
Takano
11 22 33 44 55 66 77 88 99 1010 1111 1212 1313 1414 1515 1616 1717 1818 1919 2020 2121 2222 2323 2424 2525 2626 2727 2828 2929 3030 3131 3232 3333 3434 3535 3636 3737 3838 3939 4040 4141 4242 4343 4444 4545 4646 4747 4848 4949 5050 5151
11
22
33
44
55
66
77
88
99
1010
1111
1212
1313
1414
1515
1616
1717
1818
1919
2020
2121
2222
2323
2424
2525
2626
2727
2828
2929
3030
3131
00
Newton
55 1010 1515 2020 2525 3030 3535 4040 4545 5050 5555 6060 6565 7070 7575 8080
55
1010
1515
2020
2525
3030
3535
4040
4545
00
I improved Unal (2014b).
参考文献Heo, n. g. (2015). Proof Without Words : Machin’s Formula. Mathematics And
Computer Education, 49:91.Nelsen, R. (1993). Proofs Without Words: Exercises in Visual Thinking. Classroom
resource materials. Mathematical Association of America.Nelsen, R. (2015). Proofs Without Words III: Further Exercises in Visual Thinking.
Classroom Resource Materials. Mathematical Association of America.Nelsen, R. B. (1990). A Geometric Proof of Machin’s Formula. Mathematics Magazine,
63(5):336–337.Unal, H. (2014a). Proof Without Words: A Formula of Hermann. The College
Mathematics Journal, 45(5):387–387.Unal, H. (2014b). Proof Without Words: An Arctangent Identity. The College
Mathematics Journal, 45(5):357–357.