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 fourth order were investigated by Kummer, and Fresnel's wave-surface, studied by Hamilton, is a particular case of Kummer's quartic surface, with sixteen canonical points and sixteen singular tangent planes.[56]

The infinitesimal calculus was first applied to the determination of the measure of curvature of surfaces by Lagrange, Euler, and Meunier (1754–1793) of Paris. Then followed the researches of Monge and Dupin, but they were eclipsed by the work of Gauss, who disposed of this difficult subject in a way that opened new vistas to geometricians. His treatment is embodied in the Disquisitiones generales circa superficies curvas (1827) and Untersuchungen über gegenstände der höheren Geodäsie of 1843 and 1846. He defined the measure of curvature at a point to be the reciprocal of the product of the two principal radii of curvature at that point. From this flows the theorem of Johann August Grunert (1797–1872; professor in Greifswald), that the arithmetical mean of the radii of curvature of all normal sections through a point is the radius of a sphere which has the same measure of curvature as has the surface at that point. Gauss's deduction of the formula of curvature was simplified through the use of determinants by Heinrich Richard Baltzer (1818–1887) of Giessen.[69] Gauss obtained an interesting theorem that if one surface be developed (abgewickelt) upon another, the measure of curvature remains unaltered at each point. The question whether two surfaces having the same curvature in corresponding points can be unwound, one upon the other, was answered by F. Minding in the affirmative only when the curvature is constant. The case of variable curvature is difficult, and was studied by Minding, J. Liouville (1806–1882) of the Polytechnic School in Paris, Ossian Bonnet of Paris (died 1892). Gauss's measure of curvature, expressed as a function of curvilinear co-ordinates, gave an impetus to the study of