22 Journal of the Mitchell Society. [ May 
Lastly, the discovery of the non-Euclidian geometry virtu- 
ally fixes upon the Euclidian geometry its practical and em- 
pirical character. “In connecting a geometry with experi- 
ence,” to cite the view of the most confirmed of non-Euclid- 
ians, “there is involved a process which we find in the theo- 
retical handling of any empirical data, and which therefore 
should be familiarly intelligible to any scientist. The results 
of any observations hold good, are valid, always only within 
definite limits of exactitude and under particular conditions. 
When we set up the axioms, we put in place of these results 
statements of absolute precision and generality. In this ideal- 
ization of the empirical data our addition is at first only re- 
stricted in its arbitrariness in so much as it must seem to ap- 
proximate, must apparently fit, the supposed facts of experi- 
ence, and, on the other hand, must introduce no logical con- 
tradiction. Thus to-day the ordinary triply-extended space 
of our experience may be purely Bolyaian, or purely Eu- 
clidian, or purely Cliffordian, or purely Riemannian.”* To 
put it extravagantly, the non-Euclidian geometer, like a crou- 
pier, cries out to his audience: “Here are three assumptions 
in regard to the angle sum of a triangle; from not one of the 
three do any logical contradictions follow; which one will you 
take ? Messieurs, faites vos jeux /” The result is, not that the 
mathematical world singles out one to the exclusion of the 
others — but studies all three, their inter-actions, inter-relations, 
and mutual dependencies. And yet if the “man in the street” 
impatiently cries out: “I am not interested in what may be 
the possible nature of space in the vicinity of Mars, or even the 
possible character of geometrical figures on the planet Jupiter, 
or in the tortuous reasonings of a mathematical Alice in Won- 
derland. Tell me, what is the character of the space I occu- 
py, the nature of the physical world in which I live and move 
and have my being?” And the answer of mathematicians 
throughout the world, with certain distinguished exceptions, 
*The Appreciation of non-Euclidian Geometry, by G. B. Halsted; Science , 
March 22, 1901, pp. 462-465. 
