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


Dave said...

Pat, if you want to read more about exactly what Descartes wrote, you should check out Federico's nice book "Descartes on Polyhedra: a study of the de solidorum elementis" (

Pat's Blog said...

Thanks, Your recommendation goes a long way with me... maybe it will become my 2nd favorite book about the study of Polyhedra... You wrote my first one..

Steven Colyer said...

Or you can read chapter 9 of Dave's book, Euler's Gem, which I just read in order to respond here, but the Master beat me to it.

That's still a great chapter, because it tells of the long journey of Descartes' notes (though water and time), and how he tried to cure himself of pneumonia (wine and tobacco), except his cure didn't work, and he died.

What a way to go though.

I'm going with it being Euler's discovery. Edges are edges, not plane angles. Maybe if Descartes hadn't taken the smokey cure, he's have survived and discovered it, and Topology would be 100 years farther along than it is. And maybe not.

Pat's Blog said...

I completely agree, Dave's book is the most enjoyable and informative I have ever found on the topic. My only question is "When's the next one coming out?... Dave?"

Steven Colyer said...

Everyone should own this book. Everyone. What's also cool for bookworms like me is the "recommended" reading part at the end. I haven't gone to the books he recommends yet, ... I'm still reading his book!

Very assessable, very "common-speech" if you will. It very much shows how WELL so many mathematicians, including Dave Richeson, have a flair with the words, expostulating how BEAUTIFUL what I like to call "The Infinite Field (of study)" really is. Barry Mazur of Harvard is another I wish to call attention to. There are many more such people.

And Math is beautiful because from the simplest things, such as the pre-school teachable Euler's Formula, so many other things emerge.

And what is life if not "emergence"? Be it the Theory of Evolution, new politicians emerging from the old, young people wising up with age, it is all emergence, IMO.

Of course there are other fields of study besides Mathematics, but if you reckon back in time, it always comes back to Math.

Social Anthropology for example, uses Linear Algebra, and that's just the tip of the iceberg.