Impossibility of the Conception. 
247 
much is the effect of the former inconceivability diminished. 
So too the occasionally observed negative parallax of stars may 
perhaps be allowed as evidence in favor of the negative curva¬ 
ture of that space which our ordinary experience reports to be 
space of zero curvature. But perhaps this is balanced by Zoll- 
ner’s work upon the darkness of the sky which he considered to 
give evidence in favor of a probable positive curvature. So in 
the face of these conflicting witnesses the question returns 
"What is curved space?” We are referred to the analogy of 
curved surfaces, and also to the impossibility of forming a con¬ 
ception of infinite space. We are reminded that a spherical 
surface, although unbounded, is finite, and that it is reason¬ 
able to infer an analogous attribute of curved space. But the 
curved space still declines acquaintance. Even if we seek to 
cultivate such acquaintance along the converging lines of the 
eleventh axiom, which after all are never fated to meet, the 
two lines with the intersecting line persist in maintaining 
themselves in a plane which, however, evidently cannot be the 
plane of our experimental knowledge, but some sort of a curved 
surface; and our space becomes still more inscrutable, as a mul¬ 
titude of curved surfaces stretch away beyond our mental 
vision. 
Assume the possibility of such space, and the discussion of 
the relations between contained magnitudes is undoubtedly cor¬ 
rect. If we close our minds against all questions of actual fact, 
the way is clear. But the old question remains, unless we are 
ready to accept an ancient form of assent to theological dogma, 
" I believe because it is impossible. ” Kant made all space 
a transcendental form of intuition, independent of experience, 
and considered the axioms of Euclid to be therefore necessarily 
true. But Gauss in opposition declared, " If number is merely 
a product of our mind, space has a reality beyond our mind, of 
which we can not fully foreordain the laws a priori. ” Loba¬ 
chevsky gave a most emphatic assent to the views of Gauss, 
basing his rejection of the parallel axiom on the assertion that 
its truth could be determined only by experience. In his ad¬ 
dress at the time of entering upon his work as rector of the 
University of Kasan, about one year after his presentation of 
