## Wednesday, 16 February 2011

### Euler's Gem....Descartes' Pearl?

One of the great lost papers of mathematics was the Progymnasta de solidorum elementis [Exercises in the elements of solids] of Descartes.  It is in this paper that he did, or did not, depending on who's argument most impresses you, first give the famous theorem by Euler on the relation between the faces, edges and vertices of polyhedra.

In 1649 Descartes went to Sweden to serve as the tutor of Princess Christina.  After his death he was buried in Sweden, but his possessions were sent back to France, but the box containing his manuscripts fell into the river.  Many of them were rescued, and this particular one was recopied by Leibniz.  Afterward, the original seems to have been lost, and Leibniz copy was undiscovered until 1860.  The manuscript is unquestionably the first known study of polyhedra.  It certainly had something close to Euler's famous V+F=E+2, at least to the modern eye.  It was just as certainly not known to Euler or any other mathematician of the period.

So did Descartes discover Euler's Gem?

Here is what Ed Sandifer Writes about the paper:

So, what did Descartes do? He studied something closely related to Euler’s formula for the sum
of the plane angles of a polyhedron. In Descartes’ time, people had a concept of a solid angle called the deficiency. The deficiency of a solid angle is the amount by which the sum of the plane angles at the solid angle fall short of four right angles. In the case, for example, of a solid right angle, formed by three right angles, the deficiency will be one right angle. For a cube, which contains eight solid right angles, the total deficiency is eight right angles. Descartes’ main result is that this always happens:

Theorem: The sum of the deficiencies of the solid angles of a polyhedron is always eight right angles. It is an almost trivial step from this to Euler’s theorem, that the sum of the plane angles is four times the number of solid angles, less eight right angles, that is 4V – 8 right angles.
Descartes’ other interesting result is more subtly related, but still remotely equivalent to V – E +
F = 2. Descartes writes:  "Dato aggregato ex omnibus angulis planis et numero facierum,
numerum angulorum planorum invenire: Ducatur numerus facierum per 4, et productum addatur aggregato ex omnibus angulis planis, et totius media pars erit numeris angulorum planorum."
.. Given the sum of all the plane angles and the number of faces, to find the number of plane angles: The number of faces is multiplied by 4, and to the product is added the sum of all the plane angles, and the half part of this total will be the number of plane angles.

It is easy, but not obvious, to transform this rule into Euler’s V – E + F = 2,

In his "Proofs and Refutations", Imre Lakatos believes that the small step from Descartes to Euler was not so small a step in the period of Descartes.  Descartes failed to seize upon the concept of dimensionality that Euler grasped, a connection between the zero dimensional points, the one dimensional edges and the two dimensional faces.
Here is how Lakatos stated it, as quoted in Descartes Mathematical Thought by Chikara Sasaki