ON POSSIBLE SYSTEMS OF GEOMETRY. 
211 
to the fact that all known surfaces of constant negative 
curvature have singularities. The elliptic geometry has its 
interpretation upon a sphere, a surface of constant positive 
Gaussian curvature, free from singularities. Evidently, if a 
surface of constant negative curvature, free from singulari¬ 
ties, existed, it was important to discover it. In the paper 
mentioned, Hilbert proposed this problem to himself and, 
solved it in the negative. He proved that there exists no sur¬ 
face of constant negative Gaussian curvature free from singu¬ 
larities. 
(b) I wish to call attention in this connection to a matter 
that appears to cause confusion in regard to a paper of 
Dehn’s * which appeared some three years ago. 
It had up to that time been taken as demontsrated that the 
statement 11 the sum of the three angles of a plane triangle is 
greater , equal to , or less than , two right angles ” was the full 
equivalent of the statement that “through a point no , one or 
two parallels could be drawn to a given line.” 
Dehn establishes the existence of a geometry in which the 
last statement for two parallels is not necessarily true. Much 
of what has been written about this geometry of Dehn’s 
would, in my judgment, tend to the belief that there was an 
error in logic in the conclusions up to the time of Dehn’s 
discovery. This is not the case. The conclusions are both 
sound; the premises are different. The work up to Dehn’s 
time postulated continuity of space; this he states definitely 
on page 432 of his article. Hilbert pointed out, in his 
“ Grundlagen der Geometrie,” the possibility of a geometry 
in which continuity was not postulated, and to this geometry 
he gave the name “ Non-Archmedian Geometry.” It is this 
geometry which Dehn investigated under Hilbert’s direction. 
It is important that the relation of the postulate of continuity to 
this question be kept in mind. 
(c) In regard to non-Euclidian properties of curves and of 
surfaces, let me call attention to a paper on the “ Classifica- 
x 'Dehn: Die Legendres’che Satze Uber die Winkelsumme im Dreieck. 
Math. Annalen, Bd. 53, p. 405. 
