242 
Chandler—Transcendental Space. 
without doubt earnestly engaged in his unappreciated efforts, ex¬ 
tending over more than a quarter century, to convince the 
world of the truth of sundry unique statements like his pet dis¬ 
covery that the area of a circle is just three-fourths of the cir¬ 
cumscribed square. 
Yet the situation has certainly been remarkable,-— an ex¬ 
tended system of accepted truths resting upon an axiom which 
could not but be considered as “off color,” and it is no wonder 
that there came a sense of relief at the assertion of the Non- 
Euclidian reformers that the parallel axiom is by no means a 
blemish on Euclid’s work, but on the contrary an additional 
token of the perfect logic characterizing his thought. They as¬ 
sert as the cause of his failure to demonstrate the truth of this 
so-called axiom that it is not necessarily true, that it rests not 
upon pure reason, but upon experience, that Euclid attempted 
and claimed nothing more than “ perfect deduction from as¬ 
sumed hypotheses, ” and that “ in favor of the external reality or 
truth of these assumptions he said no word. ” 
They then proceed to develop a system of geometry in which 
the two lines of the eleventh axiom need not meet, or, trans¬ 
ferring the thought from this conception to one less cumbrous, 
but involving the same revolutionary change, a system in which 
the sum of the angles of a triangle need not be two right 
angles. Prof. Halsted considers it among the possibilities that 
instruments for the measurement of angles may sometime be 
devised sufficiently delicate to allow an experimental demon¬ 
stration that the sum of these angles differs from its long ac¬ 
cepted value by at least an appreciable fraction of a second. 
A new space is thus presented for our consideration, differing 
form the space of previous thought in that it has an attribute 
of curvature, whatever that may mean. The space heretofore 
recognized, Euclidian space, is said to be space of zero curva¬ 
ture, like the plane of accepted properties. But there may be 
it is said, space of positive curvature, analogous to the surface 
of a sphere, on which the sum of the angles of a triangle is 
greater than two right angles; and there may be, also, space of 
negative curvature, analogous to the so-called pseudo-spheri¬ 
cal surface formed by the revolution of the curve y= a log 
