Synthetic geometry has been studied with much success by Luigi Cremona, professor in the University of Rome. In
his Introduzione ad una teoria geometrica delle curve piane he developed by a uniform method many new results and proved synthetically all important results reached before that time by analysis. His writings have been translated into German by M. Curtze, professor at the gymnasium in Thorn.
The theory of the transformation of curves and of the correspondence of points on curves was extended by him to three dimensions. Ruled surfaces, surfaces of the second order,
space-curves of the third order, and the general theory of surfaces have received much attention at his hands.
Karl Culmann, professor at the Polytechnicum in Zürich, published an epoch-making work on Die graphische Statik, Zürich, 1864, which has rendered graphical statics a great rival of analytical statics. Before Culmann, B. E. Cousinery
had turned his attention to the graphical calculus, but he made use of perspective, and not of modern geometry.62 Culmann is the first to undertake to present the graphical calculus as a symmetrical whole, holding the same relation to the new geometry that analytical mechanics does to higher analysis. He makes use of the polar theory of reciprocal figures as expressing the relation between the force and the funicular polygons. He deduces this relation without leaving the plane of the two figures. But if the polygons be regarded as projections of lines in space, these lines may be treated as reciprocal
elements of a "Nullsystem." This was done by Clerk Maxwell in 1864, and elaborated further by Cremona.63 The
graphical calculus has been applied by O. Mohr of Dresden
to the elastic line for continuous spans. Henry T. Eddy, of
the Rose Polytechnic Institute, gives graphical solutions of problems on the maximum stresses in bridges under concentrated loads, with aid of what he calls "reaction polygons."
A standard work, La Statique graphique, 1874, was issued by Maurice Levy of Paris.
Descriptive geometry (reduced to a science by Monge in
France, and elaborated further by his successors, Hachette,
Dupin, Olivier, J. de la Gournerie) was soon studied also in
other countries. The French directed their attention mainly to the theory of surfaces and their curvature; the Germans and Swiss, through Schreiber, Pohlke, Schlessinger, and particularly
Fiedler, interwove projective and descriptive geometry.
Bellavitis in Italy worked along the same line. The
theory of shades and shadows was first investigated by the French writers just quoted, and in Germany treated most exhaustively by Burmester.62