MATHEMATICS IN THE NINETEENTH CENTURY 489 



Minimum surfaces have been studied by Monge, Bonnet, and 

 Enneper. The subject owes its present extensive development prin- 

 cipally to Weierstrass, Riemann, Schwarz, arid Lie. In it we find 

 harmoniously united the theory of surfaces, the theory of functions, 

 the calculus of variations, the theory of groups, and mathematical 

 physics. 



Another extensive division of differential geometry is the theory of 

 orthogonal systems, of such importance in physics. We note espe- 

 cially the investigations of Dupin, Jacobi, Darboux, Combescure, 

 and Bianchi. 



Other Branches of Geometry 



Under this head we group a number of subjects too important 

 to pass over in silence, yet which cannot be considered at length for 

 lack of time. 



In the first place is the immense subject of algebraic curves and 

 surfaces. To develop adequately all the important and elegant 

 properties of curves and surfaces of the second order alone would 

 require a bulky volume. In this line of ideas would follow curves 

 arid surfaces of higher order and class. Their theory is far less 

 complete, but this lack it amply makes good by offering an almost 

 bewildering variety of configurations to classify and explore. No 

 single geometer has contributed more to this subject than Cayley. 



A theory of great importance is the geometry on a curve or sur- 

 face inaugurated by Clebsch in 1863. 



Expressing the coordinates of a plane cubic by means of elliptic 

 functions and employing their addition theorems, he deduced with 

 hardly any calculation Steiner's theorem relating to the inscribed 

 polygons and various theorems concerning conies touching the curve. 

 Encouraged by such successes, Clebsch proposed to make use of 

 Riemann's theory of Abelian functions in the study of algebraic 

 curves of any order. The most important result was a new classifica- 

 tion of such curves. Instead of the linear transformation, Clebsch 

 in harmony with Riemann's ideas employs the birational transforma- 

 tion as a principle of classification. From this standpoint we ask 

 what are the properties of algebraic curves which remain invariant 

 for such transformation. 



Brill and Nother follow Clebsch. Their method is, however, alge- 

 braical, and rests on their celebrated Residual theorem which in 

 their hands takes the place of Abel's theorem. We mention further 

 the investigation of Castelnuovo, Weber, Krause, and Segre. An 

 important division of this subject is the theory of correspondences. 

 First studied by Chasles for curves of deficiency in 1864, Cayley, 

 and, immediately after, Brill extended the theory to the case of any 

 p. The most important advance made in later years has been made 



