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 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 ''0. 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]

During the present century very remarkable generalisations have been made, which reach to the very root of two of the oldest branches of mathematics,—elementary algebra and geometry. In algebra the laws of operation have been extended; in geometry the axioms have been searched to the bottom, and the conclusion has been reached that the space defined by Euclid's axioms is not the only possible non-contradictory space. Euclid proved (I. 27) that "if a straight line falling on two other straight lines make the alternate angles equal to one another, the two straight lines shall be parallel to one another." Being unable to prove that in every other case the two lines are not parallel, he assumed this to be true in what is generally called the 12th "axiom," by some