The full value of these principles of Chasles was not brought out until the appearance, in 1879, of the Kalkül der Abzählenden Geometrie by Hermann Schubert of Hamburg. This
work contains a masterly discussion of the problem of enumerative geometry, viz. to determine how many geometric figures of given definition satisfy a sufficient number of conditions. Schubert extended his enumerative geometry to -dimensional space.55
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
- 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
College. Von Staudt's geometry of position was for a long
time disregarded, mainly, no doubt, because his book is extremely condensed. An impulse to the study of this subject was given by Culmann, who rests his graphical statics upon
the work of von Staudt. An interpreter of von Staudt was at last found in Theodor Reye of Strassburg, who wrote a
Geometrie der Lage in 1868.