 About 314 results  books.google.com 1830 Machin's Formula 17 5 1.90938 18 570 1.87698 19 1 1.94899 20 11
1.95716 21 1 1.98938 Total 1500 1.51244 See also EULER'S MACHINLIKE
FORMULA, GAUSS'S MACHINLIKE FORMULA, GREGORY NUMBER,
HERMANN'S ... 

 books.google.com The method by which Euler derived his evaluations of E"=1 l/n2k is outlined in
Exercise 7. This is to be found in Euler's Introducio in Analysin Infinitorum of 1748
. The Machinlike formula (11.1.14) it = 20 arctan (JjJ + 8 arctan [jrj coupled with ... 

 books.google.com It would actually require a separate monograph to do justice to the wealth of
Euler's discoveries on TT. ... TT is irrational in [79], His proof begins with the
statement of an infinite continued fraction for tanu, which he expressed like this:
tanu= l 1 (13.45) 1 : v i 3:v ... His arctan formula became extremely popular: TT =
16 arctan   4 arctan — (13.46) Machin calculated each of the two arctan
expressions using ... 

 books.google.com Tewodros Amdeberhan, Luis A. Medina, Victor H. Moll  2010  Preview We use (5 + i)4 239 + i to prove Machin's formula; it is enough to make the
arguments equal. ... It is like the preceding ones, but it involves one more term. In
2002 Kanada and his ... Finally, Euler proved in 1736 that its value is π2/6. To do
it he ... 

 books.google.com Jonathan M. Borwein  1997  Preview 5 coupled with a formula which allows small x to be used, like M This particular
formula is due to Machin 16 and was ... Thus  = 4 arctan(  )  arctan( ) (Machin, 1
706) 4 5 2<39  = arctan(  ) + arctan(  ) (Euler, 1738) 4 2 .5 — = 2arctan() ... 

 books.google.com + + Until recently, the most popular algorithms used to generate π have been
variations of Machin's 1709 formula: π 4 = 4tan −1 ... For example, Euler's
equation only converges at one bit per term. ... from Burnside's Lemma, they are
nongermane transformations (like time of day or colour of the circle in Jaynes'
straw and ... 

 books.google.com Euler used the identity ^7r = tan1J + tan1J (10), which gives = ( 2 " 3721 + 5T25
~ ' ' ') + (3 ~ O5 + 5T35 ~ ' ' ' ) " ' ( 1 1 ^ Machin had previously employed the
formula ^ = 4tan'tan»TJlr (12), which, like (10), is proved in most elementary ... 

 books.google.com gave it in a letter to John Machin, dated July 26, 1712. ... Taylor gave a singular
solution of a differential equation and the method of finding that solution by
differentiation of the differential equation. ... He wrote also a work on linear
perspective, a treatise which, like his other writings, suffers for want of fulness
and clearness ... perfected by Leonhard Euler in his Introductio in analysin
infinitorum, 1748, Vol. 

 books.google.com Petr Beckmann holds up this mirror, giving the background of the times when pi made progress  and also when it did not, because science was being stifled by militarism or religious fanaticism. 

 books.google.com Euler used the identity Jt = tan1} + tan1!, (6) which gives ^ = (2 ~ O + 572s ~ ) +
(3 " + O5" )—<7> Machin had previously employed the formula i» = 4tan4tan'I
T, (8) which, like (6), is proved in most elementary books on Trigonometry. 

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