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 To Chasles we owe the introduction into projective geometry of non-projective properties of figures by means of the infinitely distant imaginary sphero-circle.[61] Remarkable is his complete solution, in 1846, by synthetic geometry, of the difficult question of the attraction of an ellipsoid on an external point. This was accomplished analytically by Poisson in 1835. The labours of Chasles and Steiner raised synthetic geometry to an honoured and respected position by the side of analysis.

Karl Georg Christian von Staudt (1798–1867) was born in Rothenburg on the Tauber, and, at his death, was professor in Erlangen. His great works are the Geometrie der Lage, Nürnberg, 1847, and his Beiträge zur Geometrie der Lage, 1856–1860. The author cut loose from algebraic formulæ and from metrical relations, particularly the anharmonic ratio of Steiner and Chasles, and then created a geometry of position, which is a complete science in itself, independent of all measurements. He shows that projective properties of figures have no dependence whatever on measurements, and can be established without any mention of them. In his theory of what he calls "Würfe," he even gives a geometrical definition of a number in its relation to geometry as determining the position of a point. The Beiträge contains the first complete and general theory of imaginary points, lines, and planes in projective geometry. Representation of an imaginary point is sought in the combination of an involution with a determinate direction, both on the real line through the point. While purely projective, von Staudt's method is intimately related to the problem of representing by actual points and lines the imaginaries of analytical geometry. This was systematically undertaken by C. F. Maximilien Marie, who worked, however, on entirely different lines. An independent attempt has been made recently (1893) by F. H. Loud of Colorado