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?

  • $\begingroup$ The wikipage links to an outside page that contains both the original paper by Euler (1760) in French, and a translation into English by Jerry Lodder of NMSU. $\endgroup$ – Benjamin Dickman Jan 7 '15 at 16:20
  • $\begingroup$ Thank you both. Thanks Mark for your suggestion. This is the first question I post and I am still feeling my way through the sites. I will post a question where you suggest. Thanks Benjamin. I did have a look at the paper before posting this question. While it was interesting to see how maths was discussed at the time, it does not give any insight into Euler's geometric intuition. $\endgroup$ – Giuseppe Jan 7 '15 at 19:26
  • $\begingroup$ See page 9 of math.nmsu.edu/hist_projects/curvature.pdf I suspect the general discussion here about Euler, Bernouilli's etc. the study of variational calculus and all that helps give you a sense of why Euler looked at it. If I read correctly, Euler consider a graph $z=f(x,y)$ then reduced the problem to curvature in the intersected plane where much was already known by 1760. $\endgroup$ – James S. Cook Jan 7 '15 at 19:54
  • $\begingroup$ Thanks James. A very interesting reading. Unfortunately it still does not resolve the crux of my question. For instance, did Euler suspect, before doing the computation, that the principal directions had to be perpendicular and if so why? $\endgroup$ – Giuseppe Jan 8 '15 at 10:37
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    $\begingroup$ This question is also posted on MO mathoverflow.net/questions/193355/… and ME matheducators.stackexchange.com/questions/7181/… I'd like to post an answer but I am not sure where. I think the question is most suitable for HSM but ME already has some helpful comments, and MO even has an answer (not accepted). Can we please consolidate it here? $\endgroup$ – Conifold Jan 9 '15 at 23:57

Euler was most certainly not motivated by matrices and eigenvalues, the chain of causation goes the other way. Sylvester only introduced the term "matrix" to denote an array of numbers in 1850, and did not do much with them. Only Cayley discovered in 1857 that each matrix satisfies equation of its own order, which led to "characteristic values". However, already in 1829 Cauchy "proved that symmetric matrices have real eigenvalues". How did he accomplish such a feat? What he actually proved was that any quadratic form can be reduced to a weighted sum of squares by a linear change of variables. Cauchy even used Lagrange multipliers to find extremal values of quadratic forms subject to normalization constraint, and obtained equation for the "characteristic values" identical to Cayley's for the corresponding matrix. More on history of matrices and eigenvalues in Victor Katz's article in Learn from the Masters.

Now back to Euler. Cauchy considered quadratic forms in any number of variables, but the case of two and three was known long before that because they correspond to conics and quadrics. Euler's Curvature of Surfaces paper appeared in 1760, but 12 years earlier in an appendix to the second volume of the famous Introductio in Analysin Infinitorum Euler showed how to reduce equations of quadrics to the simplest form, and gave explicit but cumbersome formulas for the transformations that accomplish that.

So the quadratic expression in sines and cosines in the denominator of the "osculatory radius" (i.e. the curvature) would have been familiar to him from even simpler case of conics in polar coordinates. He knew that ellipses and hyperbolas have perpendicular principal axes, that their equations look simplest with those as coordinate axes, and how to change the coordinates. This also explains the word "principal", now attached to directions and curvatures. In the paper Euler finds the extrema the modern way, by taking the derivative and setting it equal to zero, and then transforms coordinates by two reflections, rather than a single rotation used today.

We should also add that curvature of curves was first introduced in 1659 by Huygens via osculating circles to construct evolutes for his ideal pendulum, and was published in his famous Horologium Oscillatorum of 1673 that Euler was certainly familiar with. Jerry Lodder's Curvature in the Calculus Curriculum gives a step by step breakdown of both Huygens's and Euler's curvature computations along with insightful commentary.

What Euler suspected before starting the computation calls for speculation of course, so I will entertain some. If we take a generic point and zoom in on its vicinity it will approximately look like a paraboloid, elliptic or hyperbolic, standing on its vertex. In modern notation we get its equation by keeping terms up to order two in the Taylor expansion of $z=f(x,y)$. Cross-sections parallel to the surface will then be approximately ellipses or hyperbolas, and mentally fitting osculating circles perpendicular to the surface at the vertex suggests, with elliptic one at least, that they have the largest and the smallest radii along their principal axes. That Euler had this in mind is supported by the fact that after considering curves obtained by intersecting a surface with an arbitrary plane in general coordinates, he switches to normal planes and chooses $x,y$ in the tangent plane to the surface. Of course, this intuition was made fully explicit only much later in Dupin indicatrix, but considering Euler's facility with infinitesimals and quadrics it is not too far fetched that he had some idea of what to expect from geometry.


You can read about the origin of Euler's 1760 study on curvature in Mathematical Masterpieces: Further Chronicles by the Explorers (page 187 and following). The idea of curvature goes back several decades, Alexes Clairaut studied it for one-dimensional curves in his 1731 book Recherces sur les courbes à double courbure. Euler was the first to apply this concept to higher-dimensional objects. The formula $k=k_1\cos^2\alpha+k_2\sin^2\alpha$ relating the curvature $k$ of a normal section to the principal curvatures $k_1,k_2$ and the inclination $\alpha$ was implicit in Euler's 1760 paper, but first written down in 1813 by Charles Dupin.

The book I mentioned above gives Euler's step-by-step derivation of his formula in English translation. I show the first paragraph, to appreciate the lucid and systematic way he approached the problem:

In order to know the curvature of curved lines, the determination of the radius of the osculating circle offers the proper method, which for each point on the curve provides us with a circle whose curvature is the same. But when one asks for the curvature of a surface, the question is rather equivocal, and not at all subject to a definitive answer, as in the previous case. It is only spherical surfaces for which one can measure the curvature, considering that the curvature of a sphere is the same as that of its great circles, and its radius can be considered as the proper measure of curvature. But for other surfaces, one would not even know how to compare the curvature of the surface with that of a sphere, as one can always compare the curvature of a curved line with that of a circle. The reason for this is evident, since through each point of a surface, there are infinitely many different curves. One only need consider the surface of a cylinder, where along directions parallel to the axis, there is no curvature, while cross sections perpendicular to the axis, which are circles, have the same curvature, and all other sections taken obliquely to the axis yield particular values of the curvature. Similarly for all other surfaces, where it can even happen that in one direction the curvature might be convex and in another concave, as for surfaces which resemble a saddle.

  • $\begingroup$ Thanks Carlo. A very interesting book to look at. Unfortunately the discussion about Euler Theorem still does not resolve the crux of my question. For instance, did Euler suspect, before doing the computation, that the principal directions had to be perpendicular and if so why? $\endgroup$ – Giuseppe Jan 8 '15 at 10:40
  • $\begingroup$ For example, my understanding of Gauss Remarkable Theorem, as it is narrated in Spivak, is that Gauss knew what is now called Gauss-Bonnet Theorem for geodesic triangles, that is the integral of the Gauss curvature over the interior of a geodesic triangle gives the sum of the angles. Since all the quantities involved in this computation except for, possibly, the Gauss curvature were intrinsic, he realised that that also must be intrinsic. Motivated by this, he started his long computation and proved Gauss Remarkable Theorem. $\endgroup$ – Giuseppe Jan 8 '15 at 10:40

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