Isaac Newton
(1642  1727)
Newton's Formula
(1) (2)
A similar sequence :
(3)
Slices of his life
Isaac Newton was born on Christmas day
1642. From a humble familly, he quickly became very inventive in
his childhood. Entering the University of Cambridge in 1660, he
met a profesor and friend who will become inportant during his
formation, Isaac Barrow.
Newton quickly mastered the mathematics of his time. But the Great
Plague (16651666) force him to go back to his natal village of
Woolsthorpe, which give him some time for his mathematics and physics
research.
And what research ! Law of motion (universal attraction), optical law
are brilliantly tackled and Newton discover differential calculus (as
simple as that) thanks to his method of fluxon (x+x^{.}o or
x+dx in modern notation, well that is said that it was invented by
his greatest rival Leibniz !)
Greatest rival because he and Newton would violently argue on the
authorship on the invention of differential analysis that lasted till
the end of their lives. The notations x^{.} or x¨
of Newton are still being used in physics...
The end of Newton the 20 march 1727 produced a scientifical
mourning that has never been equaled in England. As a matter of fact he
is buried at Westminster abbey.
Around
What interest us here, is of course and those two formula that followes from a clever analysis
but which is un fact a particular case of the derivative of arcsin
:
Proof
Let us proof the first formula in a "modern"
and quick way so to leave space for the ingenuity of the
original method by Newton for the second formula!
If we let y=Arcsin(x), that is x=sin(y) for 1<x<1,
we easily get :
dx=cos(y)dy=(1sin^{2}(y))^{1/2}dy hence
:
By a recurrence that I qualify as "nearly"
straightforward, we change it:
Since it's a power serie with convergence
radius of 1 (also straightforward from Alembert's criteria) we
can integrate bewteen 0 and x on an open interval ]1,1[
and we the get :
And here all fantasies are allowed !
We can replace x by anything, as long as 1<x<1...
For x=1/2, we have :
which is the first formula !
Or we can choose
For the second formula, let us go back to the world of geometry :
Newton took a very original reasoning that we'll inscribe here:
Consider the circle of diameter AB=1 and OC=1/4 (hence =/3) :
Newton then looked to calculate the area of the region in bold ACD
(Denoted a(ACD), a complete personal notation)
1. First, =/3 hence 6a(AOD)=R^{2}=/4 since R=1/2
Wherer a(AOD)=/24 but ,
2. Secondly, Newton considered that a(ACD) is equal to the area
swiped by the segment [MN] between the point A and the
segment [CD],which can easily be seen !!
But by pathagorus AN^{2}+NB^{2}=AB^{2}
but AN^{2}=AM^{2}+MN^{2} and NB^{2}=MB^{2}+MN^{2}
Hence AM^{2}+MN^{2}+MB^{2}+MN^{2}=AB^{2}
and so we then get 2MN^{2}=AB^{2}AM^{2}MB^{2}=1x^{2}(1x)^{2}=2x(1x)
We can then conclude that :
Since we are dealing with a power serie, we are perfectly allowed to
integrate term by term. So we find :
Great, no ?
By taking into account the equation of the quarter circle (1x^{2})^{1/2}
for x varing between 0 and 1, we similarly get :
Trials
Those sequence have a nearly linear
convergence that can be noticed just by looking at them... So the
trials are by default not that interesting!
n=... 
Sequence 1 
Sequence 2 
Sequence 3 
5 
3,1415767 (4) 
3,1415950 (5) 
3,170 (1) 
10 
3,141592646 (7) 
3,1415926541 (8) 
3,15256 (1) 
50 
33 correct
decimals

34 correct
decimals

3,1426 (2) 
100 
63 correct
decimals

65 correct
decimals

3,141965 (3) 
The two first are what we expected with respecctive convergence of
roughly 0,63n and 0,65n (the logarithm sums are
neglible !). On the other hand, sequence 3 is very
disapointing. Well, let's see why....
Stirling's Equivalence gives us:
Which confirms the extreme slowness of sequence 3 (logarithmic
convergence) and the relative speed of sequence 1 and 2
(linear convergence!). Furthermore, Log(4)=0,60... which also
comply with the observations !
So be warn about series in factorial, a few can have a logarithm
convergence if the denominator is more poweful in the equivalent term
of the serie..... Luckely this is rare....
Note that the Delta2 of Aitken is
practically useless in this kind of sequence (we only gain one decimal
maximum for n100). This is not worth it... So, no
chapter "Acceleration of the convergence"!
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