ON THE FOUNDATION OF GEOMETRY. 207 
of Hilbert’s system with those offered in its place leads to 
the conclusion that thus far it has withstood all attacks and 
remains not only apparently sound in logic, but the simplest of 
such systems as have thus far been constructed . 
There has also appeared recently * a paper by F. R. Moul¬ 
ton under the title, “A simple Non-Desarguessian Geometry,” 
which is important since the construction of certain logical 
systems in which all except one of a certain set of premises 
holds, formed an important part of the method used by Hil¬ 
bert in his paper. 
This activity directed toward and growing out of Hilbert’s 
classic paper turned his attention again toward that field of 
work, and led him to lecture on the foundations of geometry 
at Gottingen during the past year, and culminated in his 
second great paper f which appeared October 13 of this year 
under the title “ Uber die Grundlagen der Geometrie.” 
In order to understand the paper it is necessary to recall 
that Lie j had established a system of axioms, based upon 
the notion of a group, which through the medium of his 
theory of transformation groups, were sufficient to furnish a 
foundation for geometry. This work is classic and indispen¬ 
sable in establishing the theory of Non-Euelidian Geometry. 
But it had a possible defect, which was all the more serious 
because of the impossibility of avoiding the use of these re¬ 
sults of Lie’s. 
Lie assumed that (a) the functions defining his groups were 
differentiable; ( b ) that the group of motions is generated by in¬ 
finitesimal transformations. So that the question was not 
settled whether or not these follow from the notion of a 
group and the other necessary assumptions of geometry. 
The problem which Hilbert proposed to himself was to 
remedy this serious defect, and is stated by him as follows: 
“ I have sought to establish a system of axioms for plane geom¬ 
etry which should likewise rest upon the conception of a group, 
* Trans, of the Am. Math. Soc., vol. 3, p. 192. 
f Math. Annalen, Bd. 56, pp. 281-422. 
t See Theorie der Transformations gruppen, Bd. 3, chap. 5. 
