Euler's Estimate of Pi

choose formula for π/4

-1

tan

1

2

+

-1

tan

1

3

2

-1

tan

1

3

+

-1

tan

1

7

3

-1

tan

1

7

+ 2

-1

tan

2

11

5

-1

tan

1

7

+ 2

-1

tan

3

79

decimals

27

round off

number of terms of part 1

8

number of terms of part 2

8

show Euler's calculation

term	1	2400000000000000000000000000
––––	2	0160000000000000000000000000
––––	3	0012800000000000000000000000
––––	4	0001097142857142857142857142
––––	5	0000097523809523809523809523
––––	6	0000008865800865800865800865
––––	7	0000000818381618381618381618
––––	8	0000000076382284382284382284
Part I.		2574004427231435231435231432
term	1	0560000000000000000000000000
––––	2	0007466666666666666666666666
––––	3	0000119466666666666666666666
––––	4	0000002048000000000000000000
––––	5	0000000036408888888888888888
––––	6	0000000000661979797979797979
––––	7	0000000000012221165501165501
––––	8	0000000000000228128422688422
Part II.		0567588218416651314125874122
I. + II.		3141592645648086545561105554
π		3141592653589793238462643383

In [1] Euler derived the formula

arctan(t)=

t

1+

2

t

∞

∑

i=0

(2i)!!

(2i+1)!!

i

2

t

1+

2

t

. He claimed that his formula was better for calculation than the Leibniz–Gregory formula

arctan(t)=

∞

∑

i=0

2i-1

t

2i-1

, since for

t=

1

3

,

1

7

,

3

79

, the factor

2

t

1+

2

t

in the series has values

1

10

,

2

100

,

144

100000

, which are simpler to calculate with. He illustrated this with the formula

π=8arctan

1

3

+4arctan

1

7

. He calculated eight terms of the sum for each of the arc tangents on the right to 27 decimal places each and concluded that

π≈3.14159265

. On the next page he calculated terms 9–16 of the first part and terms 9–10 of the second part and concluded that

π≈3.14159265358979315

. To eighteen places,

π≈3.14159265358979324

. To 30 places,

π≈3.14159265358979323846264338328

.