490 MATHEMATICS 



by Hurwitz, who considers the totality of possible correspondences 

 on an algebraic curve, making use of the corresponding integrals of 

 the first species. 



Alongside the geometry on a curve is the vastly more difficult and 

 complicated geometry on a surface, or more generally, on any algebraic 

 spread in n-way space. Starting from a remark of Clebsch (1868), 

 Nother made the first great step in his famous memoir of 1868 

 74. Further progress has been due to the French and Italian mathe- 

 maticians. Picard, Poincare, and Humbert make use of transcend- 

 ental methods, in which figure prominently double integrals which 

 remain finite on the surface and single integrals of total differentials. 

 On the other hand, Enriques and Castelnuovo have attacked the 

 subject from a more algebraic-geometric standpoint by means of 

 linear systems of algebraic curves on the surface. 



The first invariants of a surface were discovered by Clebsch and 

 Nother; still others have been found by Castelnuovo and Enriques 

 in connection with irregular surfaces. 



Leaving this subject, let us consider briefly the geometry of n 

 dimensions. A characteristic of nineteenth-century mathematics 

 is the generality of its methods and results. When such has been 

 impossible with the elements in hand, fresh ones have been invented; 

 witness the introduction of imaginary numbers in algebra and the 

 function theory, the ideals of Kummer in the theory of numbers, 

 the line and plane at infinity in projective geometry. The benefit 

 that analysis derived from geometry was too great not to tempt 

 mathematicians to free the latter from the narrow limits of three 

 dimensions, and so give it the generality that the former has long- 

 enjoyed. The first pioneer in this abstract field was Grassmann (1844) ; 

 we must, however, consider Cayley as the real founder of n-dimen- 

 sional geometry (1869). Notable contributions have been made by 

 the Italian school, Veronese, Segre, etc. 



Non-Euclidean Geometry 



Each century takes over as a heritage from its predecessor a 

 number of problems whose solution previous generations of mathe- 

 maticians have arduously but vainly sought. It is a signal achieve- 

 ment of the nineteenth century to have triumphed over some of the 

 most celebrated of these problems. 



The most ancient of them is the Quadrature of the Circle, which 

 already appears in our oldest mathematical document, the Papyrus 

 Rhind, B.C. 2000. Its impossibility was finally shown by Lindemann 

 (1882). 



Another famous problem relates to the solution of the quintic, 

 which had engaged the attention of mathematicians since the middle 

 of the sixteenth century. The impossibility of expressing its roots by 



