230 KANSAS ACADEMY OF SCIENCE. 



also well known that the latter are subgroups of the general project- 

 ive group. From the constructive study of collineation we have thus 

 arrived at the conception of the continuous projective groups of trans- 

 formation. The method which we have followed makes it again pos- 

 sible to follow the train of reasoning in the discussion of groups by 

 illustrative constructions. It may, of course, be extended to space. 



V. CONCLUSION. 



A well-arranged parallelism of descriptive, synthetic and analytic 

 methods in organic connection, a method chiefly cultivated by Fied- 

 ler, seems to be most valuable for a rapid introduction into the fields 

 of higher geometry. The introduction of critical discussion concern- 

 ing the foundations of geometry into elementary treatises has a 

 tendency to confuse the student. The establishment of the funda- 

 mental principles of projective geometry independent of metrical 

 relations or of the eleventh axiom of Euclid may follow an introduc- 

 tion as outlined in this paper, v. Standt's construction, Fiedler's 

 projective coordinates, Caley's and Klein's absolute geometry, or non- 

 Euclidian geometry, must form indispensable parts of such an ad- 

 vanced study. In the method followed by us, and which is partly, 

 also, that of Poncelet, Steiner, and Charlps, the projective properties 

 of the circle are easily established and transferred to conies by per- 

 spective. It is, however, necessary to show that all curves of the 

 second order defined as products of projective ranges and pencils, or 

 analytically by equations of the second degree, are conies. There is 

 no difficulty in doing this. 



Descriptive analytic methods are also of invaluable service for the 

 ■study of congruences and complexes of rays and for higher geometry 

 in general. In this respect I may mention the treatment of linear 

 complexes, the congruence of bisecants of a twisted cubic, of the 

 ■"Null system," by descriptive methods, and their elegant representa- 

 tion by certain partial differential equations.* 



There is one branch of mathematics which is rarely mentioned in 

 connection with projective geometry, namely, kinematics. In the 

 hands of Penucellier, Kempe, Sylvester, Hart, and, in recent times, 

 especially by Professor Koenigs, of Paris, kinematics has rendered 

 valuable services to modern geometry. Starting from the beautiful 

 theorem j that every plane and twisted algebraic curve and every alge- 

 braic surface may be described by a linkage, Koenigs invented a plani- 

 graph, and quite recently, also, a link-motion perspectivograph, real- 

 izing collineation. A short treatment of these interesting linkages 

 would form a valuable addition to any text- book on projective geometry. 



* See my paper " On the Congruences of Rays (3,1) and (1,3)," Annals of Mathematics; 

 also, S, Lie: Geometrie der Beruhrungs-transformationen, vol. I, p. 326. 



t Koenigs: Lecons de Cinematique, Paris, 1897, pp, 271, 297, 305. 



