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 differential-invariants,
or differential-parameters, which have been investigated by Jacobi, C. Neumann, Sir James Cockle,
Halphen, and elaborated into a general theory by Beltrami,
S. Lie, and others. Beltrami showed also the connection between the measure of curvature and the geometric axioms.
Various researches have been brought under the head of "analysis situs." The subject was first investigated by
Leibniz, and was later treated by Gauss, whose theory of
knots (Verschlingungen) has been employed recently by J. B. Listing, O. Simony, F. Dingeldey, and others in their "topologic
studies." Tait was led to the study of knots by Sir William Thomson's theory of vortex atoms. In the hands
of Riemann the analysis situs had for its object the determination
of what remains unchanged under transformations brought about by a combination of infinitesimal distortions. In continuation of his work, Walter Dyck of Munich wrote on
the analysis situs of three-dimensional spaces.
Of geometrical text-books not yet mentioned, reference should be made to Alfred Clebsch's Vorlesungen über Geometrie,
edited by Ferdinand Lindemann, now of Munich; Frost's
Solid Geometry; Durège's Ebene Curven dritter Ordnung.