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 functions to geometry, but conversely, he drew geometry into the service of Abelian functions.

Clebsch made liberal use of determinants. His study of curves and surfaces began with the determination of the points of contact of lines which meet a surface in four consecutive points. Salmon had proved that these points lie on the intersection of the surface with a derived surface of the degree $\scriptstyle{11n-24}$, but his solution was given in inconvenient form. Clebsch's investigation thereon is a most beautiful piece of analysis.

The representation of one surface upon another (Flächenabbildung), so that they have a (1,1) correspondence, was thoroughly studied for the first time by Clebsch. The representation of a sphere on a plane is an old problem which drew the attention of Ptolemæus, Gerard Mercator, Lambert, Gauss, Lagrange. Its importance in the construction of maps is obvious. Gauss was the first to represent a surface upon another with a view of more easily arriving at its properties. Plücker, Chasles, Cayley, thus represented on a plane the geometry of quadric surfaces; Clebsch and Cremona, that of cubic surfaces. Other surfaces have been studied in the same way by recent writers, particularly M. Nöther of Erlangen, Armenante, Felix Klein, Korndörfer, Caporali, H. G. Zeuthen of Copenhagen. A fundamental question which has as yet received only a partial answer is this: What surfaces can be represented by a (1,1) correspondence upon a given surface? This and the analogous question for curves was studied by Clebsch. Higher correspondences between surfaces have been investigated by Cayley and Nöther. The theory of surfaces has been studied also by Joseph Alfred Serret (1819–1885), professor at the Sorbonne in Paris, Jean Gaston Darboux of Paris, John Casey of Dublin (died 1891), W, R. W. Roberts of Dublin, H. Schröter (1829–1892) of Breslau. Surfaces of the