244 
Chandler—Transcendental Space. 
cendental on account of its curvature or its many dimensions, 
is incomprehensible. We may follow the steps of the mathemati¬ 
cal reasoning by which its properties are demonstrated; but 
when we have done, we have a less definite idea of the archi¬ 
tecture of that space than a following of the dream of Cole¬ 
ridge along the windings of “Alph, the Sacred River” gives us 
of the “stately pleasure dome ” decreed in Xanadu by Kubla 
Khan. 
Still, evidently this does not justify a mental consignment 
of the asserted systems to the realm of fantasy. The fact that 
mathematicians of acknowledged pre-eminence treat them with 
respect forbids any such summary disposal. Moreover, it is 
well for each one to recall the experience of his early years 
along a mental route not greatly varying from that which had 
been traveled by his ancestors within the period of recorded 
mathematical history, and to remember how the unreal negative 
quantity became a real conception; how those quantities, the 
general estimate of which as mathematical fantasies is still em¬ 
balmed in their title as “ imaginary” quantities, came to as¬ 
sume a sturdy reasonable existence; how narrow truths so 
broadened that the evidently true became untrue, as in the case 
of the commutative law of multiplication, when the conception 
of that operation was so extended as to include something more 
than successive additions; and from these and many similar ex¬ 
periences learn to expect further changes. 
But with all this caution we can hardly avoid the question 
concerning such extension of our powers as will make trans¬ 
cendental space really an element of our thought; such as will 
make it a legitimate realm of mathematical research. Or the 
■consideration may be presented in another form asking whether 
the mathematicians who with such patient ingenuity and su¬ 
preme faith in logical truth have developed their conceptions of 
the possibility of absolute geometry, may not possibly have 
come to value the instruments of their profession so highly on 
account of their usefulness, that they have mistaken mathe¬ 
matical symbols for actual existences, and so have woven merely 
“the baseless fabric of a dream?" May it be that the words 
with which Bolyai, at the age of twenty, announced his dis- 
