Page:Popular Science Monthly Volume 66.djvu/434

430 geometry, of projective geometry and of infinitesimal geometry. Many other orthogonal systems have been made known. Among these it is proper to signalize the cyclic systems of Ribaucour, for which one of the three families admits circles as orthogonal trajectories, and the more general systems for which these orthogonal trajectories are simply plane curves.

The systematic employment of imaginaries, which we must be careful not to exclude from geometry, has permitted the connection of all these determinations with the study of the finite deformation of a particular surface.

Among the methods which have permitted the establishment of all these results it is proper to note the systematic employment of linear partial differential equations of the second order and of systems formed of such equations. The most recent researches show that this employment is destined to renovate most of the theories.

Infinitesimal geometry could not neglect the study of the two fundamental problems set it by the calculus of variations.

The problem of the shortest path on a surface was the object of masterly studies by Jacobi and by Ossian Bonnet. The study of geodesic lines has been followed up; we have learned to determine them for new surfaces. The theory of ensembles has come to permit the following of these lines in their course on a given surface.

The solution of a problem relative to the representation of two surfaces one on the other has greatly increased the interest of discoveries of Jacobi and of Liouville relative to a particular class of surfaces of which the geodesic lines could be determined. The results concerning this particular case led to the examination of a new question: to investigate all the problems of the calculus of variations of which the solution is given by curves satisfying a given differential equation.

Finally, the methods of Jacobi have been extended to space of three dimensions and applied to the solution of a question which presented the greatest difficulties: the study of properties of minimum appertaining to the minimal surface passing through a given contour.

Among the inventors who have contributed to the development of infinitesimal geometry, Sophus Lie distinguishes himself by many capital discoveries which place him in the first rank.

He was not one of those who show from infancy the most characteristic aptitudes, and at the moment of quitting the University of Christiania in 1865, he still hesitated between philology and mathematics.

It was the works of Pluecker which gave him for the first time full consciousness of his veritable calling.