I am teaching a class on elementary differential geometry and I would like to know, for myself and for my students, something more about the history of Euler Theorem and Euler equation: the curvature of a normal section is determined by the principal curvatures and the angle that the tangent to the curve makes with a principal direction via Euler equation.
This gives a rather rigid, albeit infinitesimal, description of the geometry of a surface at a point. I have been wondering how Euler might have thought that something like this should be true. Does it come from linear algebra and results about eigenvalues and eigenvectors of symmetric matrices?
I did have a look at Euler's original paper (English translation). It was interesting to see how mathematics research was done and discussed at the time but it doesn't give any insight on Euler's geometric intuition.
