Statement of Principles. 
248 
a + V a 2 _x 2 _ y a ,_ s 2 on which the sum of the angles of a 
x 
triangle is less than two right angles. This last space is the 
space of Lobachevsky, while the space of positive curvature has 
been investigated by Rieman. Prof. Halsted inclines to the be¬ 
lief that the space of our experience is negatively curved, al¬ 
though so slightly that its curvature has thus far defied deter¬ 
mination. And it would indeed seem that the attempt to measure 
this supposed curvature would not be unlike an investigation of 
the curvature of the surface of the earth by means of a careful 
measurement of the convexity of the surface of the water in 
an ordinary tub caused by the force of gravity. This belief 
about the familiar space about us is not, however, essential to 
the Non-Euclidian geometry, which only assumes the possibil¬ 
ity of the existence of such spaces somewhere, and proceeds 
to develop corresponding systems of logical reasoning. 
It only remains to add that, while curved spaces have only 
three dimensions, they are supposed to be contained in four di¬ 
mensional space, as surfaces varying in respect to curvature 
are contained in space of three dimensions; and we now have 
the essentials of the entire system, which may justly be termed 
transcendental, as being beyond experience, even though our 
familiar Euclidian space be one of the contained spaces. 
Entering now upon the central thought of this paper, it 
must not in candor be forgotten that a very small number of 
persons are to be found who claim that they possess a concep¬ 
tion of four dimensional space, or more correctly, of four di¬ 
mensional matter; since, curiously enough, for some reason they 
seem to need the material element to support their mental steps 
in the so greatly widened fields. But by far the larger part, even 
of those whose are inclined to believe such space to be possible, 
perhaps even probable, would assent in respect to their personal 
experience to the words of one of America’s greatest astronomers, 
when he said that while he could not say what space of four 
dimensions might mean to him in another state of existence, it 
had no meaning in this. In fact it will hardly be denied that, 
to our minds with their present limited conditions of heredity 
and environment, transcendental space, whether it be trans- 
