20 
Journal of the Mitchell Society. 
\_May 
cian. Most interesting- comparisons between the different 
types of geometry flow from a study of certain surfaces. 
Since the sum of the three angles of a spherical triangle is 
greater than two right angles, it is evident that the charac- 
teristic geometry of the sphere is Riemannean ; it has been 
known, since Eobatchevsky and Bolyai, that the characteris- 
tic geometry of the orisphere is Euclidian; since Beltrami, that 
of the Euclidian pseudo-sphere is Lobatchevskian.* Such 
generalizations as Barbarin’s Theorem, for example, link to- 
gether the various types of geometry in a most succinct and 
illuminative fashion, exhibiting with great clarity their fun- 
damental distinctions and similarities. Text books in non- 
Euclidian geometry are now being written ; Professor Hal- 
sted entitles a popular article Ihe Non-Euclidian Geometry 
Inevitable. The first step toward the popularization of non- 
Euclid ian geometry is the clear enunciation, at the proper 
place in our ordinary text-books of geometry, of the principle 
on which the Euclidian geometry rests : that from the stand- 
point of pure logic the parallel-postulate is a mere choice be- 
tween alternatives. “In all the books put into the hands of 
students,” as M. Barbarin has said, “the hypothetical and 
wholly factitious character of the Euclidian postulate (should) 
be put well into relief.”! 
The second great gain from the discovery of the non-Euclid- 
ian geometry is the possibility of the formulation of the prin- 
ciples of the general geometry. It is most instructive and 
stimulating to the mathematical student to see the theories of 
Euclidian geometry emerge as special cases of the more gen- 
eral and comprehensive theories of Pan-Geometry. The 
*If we consider the tubes or surfaces equidistant from a straight line, 
and make that distance infinite, we have theorispheres; the pseudo- spheres 
are surfaces of revolution which have for meridians a tractrix or line of 
equal tangents. A pseudo-sphere finds its approximate counterpart in na- 
ture in a morning-glory whose stem is infinitely prolonged; for a figure, 
cf. Elements of Trigonometry, by Phillips and Strong, p. 126. 
t On the Utility of Studying Non-Euclidian Geometry , by P. Barbarin; Le 
Mathematiche, May, 1901. 
