Nicolas of Cues
(1401 - 1464)

Slices of life

Nicolas de Cues was born in 1401 at Kues, Nikolaus Krebs is a German theologian known by the name gallicized of Nicolas of Cues. His main action was to support the popes and the principle of the infallibility against councils. While touching mathematics, he also left an important theological and philosophical work.
He dies in 1464, leaving behind him an approximation method of Pi in the lineage of the sequences from archimedian origin.
As Leibniz and Descartes, his theological and scientific thought are often connected. he considers as well that the search for the solution of the quadrature of the circle is comparable in search of the truth!

About

As I have yet wrote in the history of the geometrical period of Pi. This formula poses me problem. it is indeed often given as the "official" version of Archimedes' formula (Le fascinant nombre Pi ), and nevertheless I found it on an exercice of mathematics entitled "Cues's method ". And in Le Petit Archimède, this formula is considered as the one of Gregory... Then, I renew my appeal, if somebody knows the real paternity of this sequence, I wish he clarifies it to me!
Let us note however that this sequence uses the perimeter of a polygone having a fixed value. We try then to estimate the radius of inscribed and circumscribed circles what is still more close to the Descartes's method of isoperimeters. But after all, I decided to choose De Cues to make known a little this remarkable person!

Proof

Although this algorithm much looks like the arithmetico - geometrical mean (see Salamin), convergence is (regrettably!) in no way comparable and more close to a linear convergence than to a quadratic convergence!
As for Archimede,a polygone in 2ⁿ sides is considered with perimeter equal to 1 and we note a_n and b_n the respective radius of the inscribed and circumscribed circles.

So we have :





From which, because the perimeter of the polygone is 1, one deducts from it : 2ⁿA₁A₂=1 and

according to previously, one has so :


Let us calculate now :




Let us show that these two series are adjacent :


Because tan > sin.
As well, so b_n+1-b_n is of the sign of because cos < 1 for n > 0.

From which (b_n) is decreasing...
Furthermore, tan(x) ~ x and sin(x) ~ x in 0 so it is immediate that :



and we also have :

so b_n>a_n then a_n and b_n are adjacent.

Finally, so we have :

Let us estimate the convergence speed now :


so :

That gives us a linear convergence (if we apply the log). Let us verify all this by some attempts...

Trials

Nevertheless it is not necessary to be discouraged of this lineat convergence because sequence is all the same easy to calculate and to accelerate...

	an=	bn=
n=5	3,1517 (1)	3,136 (1)
n=10	3,14160 (3)	3,141587 (4)
n=20	3,141592653599 (10)	3,14159265358606 (11)
n=50	28 decimals exact	29 decimals exact

A convergence in about 3n / 5 is obtained here

Acceleration of the convergence

Aitken and the Delta2 is here particularly effective. Let us see it!

	Delta2(an)=	Delta2(bn)=
n=5	3,1422 (2)	3,14163 (3)
n=10	3,14159265418 (8)	3,14159265362 (9)
n=20	20 décimales justes	22 décimales justes
n=50	56 décimales exactes	58 décimales exactes

It is rather incredible, but the Delta2 doubles the speed of convergence! A convergence of the order 1.2n is obtained indeed

Then let us continue and let us iterate the same process (Delta2 applied twice !)

	Delta22(an)=	Delta22(bn)=
n=5	non available (div by zero)	3,1415953 (5)
n=10	12 decimals exact	14 decimals exact
n=20	32 decimals exact	32 decimals exact
n=50	87 decimals exact	87 decimals exact

Factor of 50 % is gained again! One has then a convergence 1.75n that is very honorable for a series of this visible simplicity!
We will so be able to remember this series among archimedian's as one of the fastest.