VI.
All the works we have enumerated, others to which we shall return later, find their origin and, in some sort, their first motive in the conceptions of modern geometry; but the moment has come to indicate rapidly another source of great advances for geometric studies. Legendre's theory of elliptic functions, too much neglected by the French geometers, is developed and extended by Abel and Jacobi. With these great geometers, soon followed by Riemann and Weierstrass, the theory of Abelian functions which, later, algebra would try to follow solely with its own resources, brought to the geometry of curves and surfaces a contribution whose importance will continue to grow.
Already, Jacobi had employed the analysis of elliptic functions in the demonstration of Poncelet's celebrated theorems on inscribed and circumscribed polygons, inaugurating thus a chapter since enriched by a multitude of elegant results; he had obtained also, by methods pertaining to geometry, the integration of Abelian equations.
But it was Clebsch who first showed in a long series of works all the importance of the notion of deficiency (Geschlecht, genre) of a curve, due to Abel and Riemann, in developing a crowd of results and elegant solutions that the employment of Abelian integrals would seem, so simple was it, to connect with their veritable point of departure.
The study of points of inflection of curves of the third order, that of double tangents of curves of the fourth order and, in general, the theory of osculation on which the ancients and the moderns had so often practised, were connected with the beautiful problem of the division of elliptic functions and Abelian functions.
In one of his memoirs, Clebsch had studied the curves which are rational or of deficiency zero; this led him, toward the end of his too short life, to envisage what may be called also rational surfaces, those which can be simply represented by a plane. This was a vast field for research, opened already for the elementary cases by Chasles, and in which Clebsch was followed by Cremona and many other savants. It was on this occasion that Cremona, generalizing his researches on plane geometry, made known not indeed the totality of birational transformations of space, but certain of the most interesting among these transformations.
The extension of the notion of deficiency to algebraic surfaces is already commenced; already also works of high value have shown that the theory of integrals, simple or mutiple, of algebraic differentials will find, in the study of surfaces as in that of curves, an ample field of important applications; but it is not proper for the reporter on geometry to dilate on this subject.
