Gauss proved that the so-called Gaussian curvature is an intrinsic invariant of the surface, even though it is defined extrinsically as the product of the principal curvatures of a Euclidean embedding. But apart from the issue of its intrinsic nature, did anyone consider the invariant given by the product of the principal curvatures before Gauss?


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The answer is not really. Before Euler's Recherches sur la Courbure des Surfaces (1760, published 1767, the publication and partial translation are available on Euler Archive) only curvature of curves was considered, see How was curvature originally defined and calculated? Euler introduced principal directions and curvatures by cutting a surface with perpendicular sections, and explicitly stated that both numbers are required to characterize curvature of a surface at a point in contrast to curves. The product of principal curvatures does appear in some formulas, but Euler places no special significance on it. Euler returned to surfaces in De Solidis Quorum Superficiem in Planum Explicare Licet (1771), where he introduced parametric representation for the first time, and studied developable surfaces.

According to one source, which I do not trust too much, Gauss named the product of principal curvatures "total curvature" after Euler, Struik's Lectures on Classical Differential Geometry only confirms that the term was used in this way. It was later renamed into modern "Gaussian curvature", and total curvature was appropriated for the latter's integral over the entire surface. If so this is not in Euler's 1760 paper, one would have to check the 1771 paper which is only available in Latin, and it seems that it lacked applications. Monge studied developable surfaces starting in 1770s, his work is summarized in Application de L'analyse à la Géométrie written in 1795-1801. He wrote a differential equation for developables, which is equivalent to Gaussian curvature being 0, but it would be a stretch to say that he "considered" it. Monge represented surfaces by implicit equations, so principal curvatures are not apparent in his differential equation, and one does not need to multiply them to express that one of them is 0.

Another combination of principal curvatures however, their average now called mean curvature, had better luck. In 1760, the same year as Euler, Lagrange also published a paper about surfaces that minimize area among those with the same boundary. Like Monge he used implicit equations and his differential equation for minimal surfaces was messy, he could not produce any non-trivial solutions (trivial solutions are flat surfaces). But according to Struik some time in early 1770s Monge showed Euler's papers to his student Meusnier (later French revolutionary general who died at the siege of Mayence). Meusnier wrote the now well known paper in 1776 (published in Mém. prés. par div. Etrangers. Acad. Sci. Paris, 10 (1785) pp. 477–510), where he proved the theorem about the curvature of non-perpendicular sections, now named after him, and simplified Lagrange's differential equation to show that it amounted to the mean curvature being 0 (it is unclear if he used the name). This allowed him to produce the first non-trivial examples of minimal surfaces, catenoid and helicoid.

The mean curvature then came up in the work of Sophie Germain that won her a kilogram of gold prize in 1816. It was a generalization of Euler-Bernoulli theory of elastic beams to thin plates, the first work to correctly derive elastic force in them, see Lodder's Curvature in the Calculus Curriculum. In 1821 she reformulated it by introducing the "referent sphere" to represent the curvature of surfaces in the same way that the osculating circle represents it for curves, but the curvature it represents is the mean curvature. Germain's correspondence with Gauss in 1820s shows that his treatment of surface geometry was novel even to her. Using parametric representation of surfaces to discover their intrinsic geometry was Gauss's major breakthrough in 1825-27 papers. Gaussian curvature is an important invariant of intrinsic geometry, but there does not seem to have been much use for it before that.

  • $\begingroup$ You write that "The product of principal curvatures does appear in some formulas, but Euler places no special significance on it." Do you have more specific references? There are places where the product of the radii of curvature appears, but that seems to be just because the sum of the curvatures is brought to common denominator. $\endgroup$ Commented Jan 4, 2016 at 9:15
  • $\begingroup$ @katz This is what I meant, it appears in the 1760 paper only as a byproduct of calculations. Once mean curvature was isolated it could be plausible to look for "complementary" second invariant for which the product is a natural candidate (so is sum of squares), but other than a mention in an old popular book with no references I did not find it before Gauss. It might be useful to look at Gauss's papers to see if he indeed defers to Euler when introducing "total curvature". $\endgroup$
    – Conifold
    Commented Jan 4, 2016 at 21:05

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