34 KANSAS UNIVERSITY QUARTERLY. 



The other, the analytical school, has as its representatives: Pliicker, 

 Hesse, Aronhold, Gordan, Cayley, and many others. 



Meanwhile the brilliant results of modern synthetic and analytic 

 geometry have had a great influence upon pure analysis. The 

 modern theory of functions was created, and by the investigations 

 of Jacobi, Abel, Cauchy, Riemann, Hermite, and Weierstrass, it has 

 reached a dominant position in almost all branches of mathematics. 

 It became more and more a prevailing opinion that in fact the syn- 

 thetic and analytic methods in geometry are identical and it is now 

 generally acknowledged that the two methods differ only in their 

 formal representation. Fiedler in 1874 defined the homogeneous 

 co-ordinates as anharmonic ratios which lead at once from 

 synthetic to analytic, and from analytic to synthetic geom- 

 etry. The greatest step in overcoming the difficulties between 

 synthetic and analytic methods was however taken by Klein and 

 Lie about 1871. Klein in his "Erlanger Programm " clearly 

 outlined the standpoint from which the problems of modern math- 

 ematics have to be considered. The fundamental idea of higher 

 geometry is to find all the "groups" and to investigate their 

 properties, i. e., to find geometrical truth. As to Lie, it is well 

 known that the achievements of this great mathematician concern- 

 ing the theory of groups, since about 1874, influenced and still 

 influence many of the most important fields of mathematics. 



The old conception of invariants is al)andoned and its place 

 has been taken by the conception of groups. 



In this paper it has been attempted to make a little contribution 

 to geometr}^ by applying the theory of groups to the well-known 

 subject of perspective. In works on groups, which hitherto has 

 been published, the treatment is almost exclusively analytical and 

 it may be pointed out that our paper is the first bearing on groups 

 in which the synthetical method is used. 



Lie divided all projective transformations into five types. Before 

 the investigations of Lie were known, onl}' two of these types have 

 usually been treated, the general projective transformation (col- 

 lineation), and the perspective collineation. Our perspective 

 collineations make up two of those five types: perspective, and 

 elation. As has been said already in the preface, the other three 

 types are being investigated in the same way by Prof. Newson. 



For references we give the following list of books: 



Sophus Lie, \'orIesungen iiber continuierliche Gruppen, Leipzig, 

 Teubner, 1894. 



