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 previous researches, and published in 1861 his ''Vorlesungen über die Analytische Geometrie des Raumes, insbesondere über Flächen 2. Ordnung.'' More elementary works soon followed. While in Heidelberg he elaborated a principle, his "Uebertragungsprincip." According to this, there corresponds to every point in a plane a pair of points in a line, and the projective geometry of the plane can be carried back to the geometry of points in a line.

The researches of Plücker and Hesse were continued in England by Cayley, Salmon, and Sylvester. It may be premised here that among the early writers on analytical geometry in England was James Booth (1806–1878), whose chief results are embodied in his Treatise on Some New Geometrical Methods; and James MacCullagh (1809–1846), who was professor of natural philosophy at Dublin, and made some valuable discoveries on the theory of quadrics. The influence of these men on the progress of geometry was insignificant, for the interchange of scientific results between different nations was not so complete at that time as might have been desired. In further illustration of this, we mention that Chasles in France elaborated subjects which had previously been disposed of by Steiner in Germany, and Steiner published researches which had been given by Cayley, Sylvester, and Salmon nearly five years earlier. Cayley and Salmon in 1849 determined the straight lines in a cubic surface, and studied its principal properties, while Sylvester in 1851 discovered the pentahedron of such a surface. Cayley extended Plücker's equations to curves of higher singularities. Cayley's own investigations, and those of M. Nöther of Erlangen, G. H. Halphen (1844–1889) of the Polytechnic School in Paris, De La Gournérie of Paris, A. Brill of Tübingen, lead to the conclusion that each higher singularity of a curve is equivalent to a certain number of simple singularities,—the node, the ordinary cusp, the double tangent,