488 MATHEMATICS 



complexes of the first and second order and calling attention to 

 their application to mechanics and optics. 



The most important advance over Pliieker has been made by 

 Klein, who takes as coordinates six-line complexes in involution. 

 Klein also observed that line geometry may be regarded as a point 

 geometry on a quadric in five-way space. Other laborers in this 

 field are Clebsch, Reye, Segre, Sturm, and Konigs. 



Differential Geometry 



During the first quarter of the century this important branch of 

 geometry was cultivated chiefly by the French. Monge and his 

 school study with great success the generation of surfaces in vari- 

 ous ways, the properties of envelopes, evolutes, lines of curvature, 

 asymptotic lines, skew curves, orthogonal systems, and especially the 

 relation between the surface theory and partial differential equations. 



The appearance of Gauss's Disquisitiones generates circa super- 

 ficies curvas, in 1828, marks a new epoch. Its wealth of new ideas 

 has furnished material for countless memoirs, and given geometry 

 a new direction. We find here the parametric representation of a 

 surface, the introduction of curvilinear coordinates, the notion of 

 spherical image, the Gaussian measure of curvature, and a study of 

 geodesies. But by far the most important contributions that Gauss 

 makes in this work is the consideration of a surface as a flexible, 

 inextensible film or membrane, and the importance given quadratic 

 differential forms. 



We consider now some of the lines along which differential geometry 

 has advanced. The most important is perhaps the theory of differen- 

 tial quadratic forms with their associate invariants and parameters. 

 We mention here Lame", Beltrami, Menardi, Codazzi, Christoffel, 

 and Weingarten. 



An especially beautiful application of this theory is the immense 

 subject of applicability and deformation of surfaces, in which Mind- 

 ing, Bauer, Beltrami, Weingarten, and Voss have made important 

 contributions. 



Intimately related with the theory of applicability of two surfaces 

 is the theory of surfaces of constant curvature which play so import- 

 ant a part in non-Euclidean geometry. We mention here the work 

 of Minding, Beltrami, Dini, Backlund, and Lie. 



The theory of rectilinear congruences has also been the subject 

 of important researches from the standpoint of differential geometry. 

 First studied by Monge as a system of normals to a surface and then 

 in connection with optics by Mains, Dupin, and Hamilton, the gen- 

 eral theory has since been developed by Kummer, Ribaucour, 

 Guichard, Darboux, Voss, and Weingarten. An important applica- 

 tion of this theorv is the infinitesimal deformation of a surface. 



