ON THE FOUNDATIONS OF GEOMETRY, AND ON 
POSSIBLE SYSTEMS OF GEOMETRY. 
BY 
Henry F. Stecker. 
[Read before the Society January 17, 1903.] * 
INTRODUCTION. 
The word geometry is used here in its strict sense as relating 
to the three-dimensional space of experience, and as revealed 
to us by logical reasoning based, as fai; as possible, upon such 
facts in regard to space as are in our possession. 
Our knowledge of space is limited sharply in two ways: 
(a) It is inexact ,f consisting only of the crude facts of ex¬ 
perience ; 
( b) It is restricted to a limited portion of space. 
The geometry of space, which is mathematically exact, 
cannot be builded on such a foundation. It is necessary to 
“ idealize ” these inexact facts of experience; make them mathe- 
ematically exact. In short, assume the existence of mathe¬ 
matical points, lines, etc.; the continuity of space, the in¬ 
variance of the properties of figures when displaced, and 
many other things. The essential thing to remember is that 
they are assumptions, and that without assumptions there is 
no possibility of constructing a geometry. 
It is to be noticed that we say “ a geometry ” and not “ the 
geometry,” because when premises contain an assumption, 
*In the absence of Dr. Stecker the paper was read by Mr. Radelfinger. 
f This is necessarily very brief, for details in this connection see: Klein, 
Nicht-Euklidsche Geometrie, Bd. I, p. 354-358. Also, Zur Ersten Ver- 
theilung des Lobatschewsky—Preises, Math. Annalen. Bd. 50, p. 583. 
30—Bull. Phil. Soc., Wuah., Vol, 14. (205) 
