René Descartes
(1596 - 1650)

Slices of life

René Descartes was born at Hague in Touraine exactly on March 31, 1596. He was certainly the one of the most famous French philosopher! But he was not only that...
After a licence of right in 1616, he chooses the military profession in Holland then in Duke de Bavaria service until 1620. Back in in France in 1625, he draws up there his philosophical works - famous, but it is not the object of this site! - and publish scientific works about optics, astronomy, biology and especially geometry. In 1631, appears Geometry in that it defines the Cartesian coordinates of a point. Let us note by the way that it is for Descartes that we have to the habit to represent quantities known by the first letters of the alphabet a, b, c, d and the unknowns by x, y, z.... He dies in 1650.

About

After his death, the method of isoperimeters will be found in his papers. It consists in making the opposite of Archimede's method it should to tell to determine the radius of a circle of which the perimeter is fixed in advance. It is a wholly geometrical construction...

Proof

It would be better to say construction, because it is not a real mathematical proof...
A suite of regular polygons is considered P₀ ,P₁, P₂ ... P_n of respectively 2² , 2³,..., 2ⁿ⁺² sides, having - it is important - the same perimeter L

The following figure, with A_n B_n =C_n and OH_n =r_n .
Let us look for a relation of recurrence between r_n+1 and r_n knowing that r_n tends towards the radius of the circle.
One knows that 2ⁿ⁺²C_n =L because L is the perimeter of the polygone P_n of side C_n. This relation being valid for every nN,we have also 2²C₀=L.
For P₀ , we got a square OH₀ =C₀/2=r₀, so r₀=C₀/2=L/(2.2²)=L/8
Let E the middle of the small arc A_nB_n . the segment which joins the circles, the middles A_n+1 and B_n+1 of [EA_n] and [EB_n] is the side of P_n+1. All the history is going to be geometrical, then let us concentrate!

We have A_n+1 B_n+1=C_n+1=L / 2ⁿ⁺³ : Indeed, by the theorem of this dear Thales,

In the right-angled triangle OEA_n+1 (Because OA_nE is isosceles), we have A_n+1H_n+1²=EH_n+1 *H_n+1O
But let us show it if it is not evident!
We have on the first hand EO²=(EH_n+1+H_n+1O)²=EH_n+1²+2EH_n+1*H_n+1O+H_n+1O²
So .
On the other hand, A_n+1H_n+1²=A_n+1E² - EH_n+1² by Pythagore and A_n+1H_n+1²=OA_n+1² - OH_n+1², so we have :

Now always by pythagore EO² =A_n+1E² + OA_n+1²
So A_n+1H_n+1²=EH_n+1*H_n+1O (openly saddened for heaviness of notations!!)

Now

and EH_n+1=H_n+1H_n (obvious by Thales!)
=OH_n+1-OH_n =r_n+1-r_n and again, H_n+1O=r_n+1
So :

Well, Here we are, our relation of recurrence!
It is moreover a polynomial in r_n+1, that is obviously positive.
The only positive root of the polynomial is extracted and we obtain:

When n increases, the polygone P_n tends to become merged with the circle of perimeter L=8r₀=2r_n (2*radius...)
so :

Interesting, no?
And not so bad in term of efficiency!

Trials

Let us look at it closer...
Expression is reminiscent of the area of a rectangle of r_n+1 and r_n+1-r_n sides. The geometrical series of the areas of these rectangles would be of reason 1/4. A priori, the relation between r_n+1 and r_n should also have to behave as a geometrical series, and convergence should be linear (-Log (r_n) =a*n+b)... Let us verify by taking L=8, and so r₀=1 (the choice of L does not influence the result because the relation between r_n+1 and r_n is homogeneous in L) there:

n=5	3,1422 (2)
n=10	3,1415932 (5)
n=50	28 decimals exact
n=100	60 decimals exact

Totaly, a good small convergence in 3n / 5, here is that is very honorable!

Acceleration of the convergence

What there is well with Aitken's Delta2, it is that there is always an acceleration, so small is it. But then when it is gigantic, How euphoric! Let us look at attempts:

	Without Aitken	With Aitken	With Aitken iterated
n=5	3,1422 (2)	3,14159508 (5)	3,1415926559 (8)
n=10	3,1415932 (5)	3,141592653592 (10)	16 decimals exact
n=20	3,1415926535903 (10)	23 decimals exact	35 decimals exact
n=50	28 decimals exact	59 decimals exact	90 decimals exact

It is incredible! Aitken multiplies by more 2 the performance of the series which reaches a convergence of 1.2n !
I feel that it is the best result obtained with Aitken for series converging toward Pi.
And look at the results with Aitken iterated (Delta2 is applied twice) because of the limit preciseness of my computer (100 decimals) and the sensibility of Delta2, it is even possible to obtain a better result.
We reache with Aitken iterated an upper preciseness in 1.6n and that goes better!