PRESENT PROBLEMS OF GEOMETRY 579 



is far more complicated, and awaits solution even in the case of 

 polyhedral surfaces. 



Beltrami's visualization of Lobachevsky 's geometry by pictur- 

 ing the straight lines of the Lobachevsky plane as geodesies on 

 a surface of constant negative curvature is well known. However, 

 since the known surfaces of this kind, like the pseudosphere, have 

 singular lines, this method really depicts only part of the plane. In 

 fact Hilbert (Transactions of the American Mathematical Society 

 for 1900), by very refined considerations, has shown that an analytic 

 surface of constant negative curvature which is everywhere regular 

 does not exist, so that the entire Lobachevsky plane cannot be 

 depicted by any analytic surface. 1 There remains undecided the 

 possibility of a complete representation by means of a non-analytic 

 surface. The partial differential equation of the surfaces of negative 

 constant curvature is of the hyperbolic type and hence does admit 

 non-analytic solutions. 2 (This is not true for surfaces of positive 

 curvature, since the equation is then of elliptic type.) The discussion 

 of non-analytic curves and surfaces will perhaps be one of the really 

 new features of future geometry, but it is not yet possible to indicate 

 the precise direction of such a development. 3 



Other theories belonging essentially to geometry im Grossen 

 are the questions of analysis situs, or topology, to which reference has 

 been made on several occasions, and the properties of the very 

 general convex surfaces introduced by Minkowski in connection 

 with his Geometrie der Zahlen. 



Systems of Curves Differential Equations 



Although projective geometry has for its domain the investigation 

 of all properties unaltered by collineation, attention has been con- 

 fined almost exclusively to the algebraic configuration, so that pro- 

 jective is often confused with algebraic geometry. To the more 

 general projective geometry belong, for example, the ideas of oscu- 

 lating conic of an arbitrary curve and the asymptotic lines of an 

 arbitrary surface, and Mehmke's theorem which asserts that when 

 two surfaces touch each other, the ratio of their Gaussian curvatures 

 at the point of contact is an (absolute) projective invariant. The 

 field for investigation in this direction is of course very extensive, 

 but we may mention as a problem of special importance the deriva- 



1 The entire projective plane, on the other hand, can be so depicted on a sur- 

 face devised by W. Boy (Inaugural Dissertation, Gottingen, 1901). 



2 According to Bernstein (Math. Annalen, vol. LIX, 1904, p. 72), the proof given 

 by Liitkemeyer (Inaugural Dissertation, Gottingen, 1902) is not valid, though 

 the conclusion is correct. 



3 Lebesgue (Comptes Rendus, 1900, and These, 1902) has examined the theory 

 of surfaces applicable on a plane without assuming the existence of derivatives 

 for the denning functions, and thereby obtains an example of a non-ruled develop- 

 able. 



