240 
Chandler — Transcendental Space. 
greatly increase its power and widely vary its methods 
and details, they surely can affect in no respect the principles 
long considered to be firmly established. 
That the last half of the nineteenth century should bring into 
notice mathematical views quite revolutionary in character is 
certainly a most unexpected fact, and one which has failed to 
receive the large general attention usually bestowed upon new 
views, partly on account of the greater interest along other than 
mathematical lines, and in part also because of an incredulity 
as to any possible place for such views as it has been thought, 
could be worth the attention of such students only as are 
specially devoted to the work produced by the genus “ crank. ” 
So it has been largely true that theories of multi-dimensional 
space have been treated as merely ingenious products of a 
vagrant imagination, worthy perhaps, if ably presented, of an 
honorable position among fairy stories but, to most minds, of 
nothing more than that. In fact only imperfectly is it gener¬ 
ally realized that these theories present a field for serious con¬ 
sideration. 
In a similar spirit have the various presentations of the geom¬ 
etry of “ absolute space ” been received, although in a somewhat 
less marked degree, p'robably because the last-named subject 
lends itself less readily to a graphic, or even romantic, treat¬ 
ment, such as is employed in books like “Flatland, ” “A Plane 
World,” and other attempts to make higher space seem possible. 
Still during the past quarter century the efforts of Dr. Balt- 
zer of Giessen, and Prof. Halsted of the University of Texas, 
have brought into notice the work, now nearly seventy-five years 
old, of the Bussian, Lobachevsky, and the Hungarian, Bolyai; 
and have secured for them a respectful and even highly appre¬ 
ciative recognition, in many cases indeed, as it seems to some 
who strive to consider the subject candidly, even unduly eulog¬ 
istic. A hearty assent to the words of Prof. Loud of Colorado, 
when he writes to Prof. Halsted ‘‘You have made it impossible 
for American teachers of any spirit to shut their eyes to the 
4 hypothesis anguli acutV ” may perhaps not be inconsistent 
with a pause for careful thought when we hear Prof. Clifford of 
Cambridge say “What Vesalius was to Galen, what Copernicus 
