and the inflection. Sylvester studied the "twisted Cartesian," a curve of the fourth order. Salmon helped powerfully towards the spreading of a knowledge of the new algebraic and geometric methods by the publication of an excellent series of text-books (Conic Sections, Modern Higher Algebra, Higher Plane Curves, Geometry of Three Dimensions), which have been placed within easy reach of German readers by a free translation, with additions, made by Wilhelm Fiedler of the Polytechnicum in Zürich. The next great worker in the field of analytic geometry was Clebsch.
Rudolf Friedrich Alfred Clebsch (1833–1872) was born at Königsberg in Prussia, studied at the university of that place under Hesse, Richelot, F. Neumann. From 1858 to 1863 he held the chair of theoretical mechanics at the Polytechnicum in Carlsruhe. The study of Salmon's works led him into algebra and geometry. In 1863 he accepted a position at the University of Giesen, where he worked in conjunction with Paul Gordan (now of Erlangen). In 1868 Clebsch went to Göttingen, and remained there until his death. He worked successively at the following subjects: Mathematical physics, the calculus of variations and partial differential equations of the first order, the general theory of curves and surfaces, Abelian functions and their use in geometry, the theory of invariants, and "Flächenabbildung."[68] He proved theorems on the pentahedron enunciated by Sylvester and Steiner; he made systematic use of "deficiency" (Geschlecht) as a fundamental principle in the classification of algebraic curves. The notion of deficiency was known before him to Abel and Riemann. At the beginning of his career, Clebsch had shown how elliptic functions could be advantageously applied to Malfatti's problem. The idea involved therein, viz. the use of higher transcendentals in the study of geometry, led him to his greatest discoveries. Not only did he apply Abelian
