of Edinburgh, Session 1879 - 80 . 639 
he wrote. Accordingly, although the subject is treated very ably in 
his paper, it is treated from only one point of view ; and, indeed, one 
side of it is left out of sight altogether. The relation of the whole 
theory to the question of the origin and mutual independence of 
the axioms of geometry has been made much clearer of late, and I 
believed that some account of the more modern views might be of 
interest. 
I am particularly desirous of bringing pangeometrical speculations 
under the notice of those engaged in the teaching of geometry. In 
discussing with schoolmasters the difficult problem of the reform of 
geometrical teaching, I have met with much enlightened and some 
unenlightened criticism. The former kind of criticism has convinced 
me that many teachers of mathematics will be glad to have this subject 
made more accessible ; and I believe that a knowledge of what great 
mathematicians have thought on the subject would destroy criticism 
of the latter kind altogether. 
It will not be supposed that I advocate the introduction of 
pangeometry as a school subject ; it is for the teacher that I advocate 
such a study. It is a great mistake to suppose that it is sufficient 
for the teacher of an elementary subject to be just ahead of his pupils. 
hTo one can be a good elementary teacher who cannot handle his 
subject with the grasp of a master. Geometrical insight and wealth 
of geometrical ideas, either natural or acquired, are essential to a 
good teacher of geometry ; and I know of no better way of cultiva- 
ting them than by studying pangeometry. 
The following sketch is addressed to those already familiar with 
Euclid’s geometry. I have made no attempt to give a detailed ac- 
count of modern researches, or to build up a systematic treatise. 
I have simply tried to give in a synthetic way a general idea 
of what is known in a certain department of a now very 
widely developed subject. In so doing I have used the mate- 
rials and methods of Euclid as much as I consistently could, 
at some sacrifice of elegance, no doubt, but with obvious practical 
advantage. 
I have not attempted to give any bibliographical details, for the 
simple reason that any one who wants them will find nearly all that 
can be desired in two papers by Mr Halsted in the first volume of 
the “ American Journal of Mathematics.” 
4 G 
VOL. X. 
