640 
Proceedings of the Royal Society 
On Pangeometry. 
I know of no question possessing more interest for a thinker, and 
none of more importance for a mathematician, than the well-worn 
one of the origin of the axioms of geometry. 
Passing over the discussions of mental philosophers, which, so far 
as I am acquainted with them, are of little mathematical or physical 
interest, we find two great modern contributions to this interesting- 
subject ; one by the mathematicians headed by Gauss, Lobatschewsky, 
Bolyai, and Riemann : the other by the physiologists represented by 
Helmholtz. 
The mathematical investigators may be taken as representing the 
subjective side of the subject, the physiologists as representing the 
objective ; although, in point of fact, Helmholtz, the personal 
representative of the latter, is a happy union of both classes of 
philosopher. 
Any purely abstract science starts with certain data called defini- 
tions and axioms ;* and of these materials reason builds the fabric 
of the science. 
I do not intend to take up the question of the origin of axioms 
directly. On the contrary, I shall lay down axioms, and the only 
argument against me, so far, will be to prove the inconsistency of my 
conclusions with my premises, or with one another. 
The absence of such inconsistency is what I mean by conceiv- 
ability. I do not deny that other meanings may be attached to this 
word, and that the question of the conceivability of axioms might be 
profitably discussed from other points of view. We might discuss 
it as a purely personal question, each man to be judge and jury, or it 
might be granted, as I, for the most part in what follows, take it to 
be, that any axioms that can be made the foundation of a consistent 
reasoned system are given d priori. I suspect that this would be 
* In Euclid’s Geometry the functions of definition and axiom are not always 
clearly separated ; at all events, some of liis definitions serve purposes for which 
others are unfit, and this must be kept in view in what follows. With postu- 
lates I have at present nothing to do, as I am concerned solely with geometrical 
theorems. The mixture of problems with theorems is a peculiarity of Euclid’s 
method for which there is no absolute necessity, and which is certainly incon- 
venient in an elementary text-book. Geometrical constructions are in a sense 
the applications of geometrical theory, and ought to be kept by themselves. 
The Society for the Improvement of Geometrical Knowledge have acted wisely, 
I think, in following this arrangement in their syllabus. 
