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.