2 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
perience alone, and can not be even proved by experience. Its 
adequacy as a subjective form for experience has not yet been dis- 
proved, but mig-ht in future be disproved. It can never be proved. 
The realities which with the aid of our subjective space form 
we understand under motion and position, may, with the coming- 
of more accurate experience, refuse to lit in that form. Our math- 
ematical reason may decide that they would be fitted better by a 
non-Euclidean space form. But we are, and will be, helpless to 
g-et such a space form from any experience whatever. 
Space is presupposed in all human notions of motion or position. 
We may drop out such specifications from our space form as ren- 
der it specifically Euclidean, but we can not replace them by non- 
Euclidean. Euclidean space is a creation of that part of mind 
which has worked and works yet unconsciously. 
It is not the shape of the straig-ht lines which makes the ang-le- 
sum of a rectilineal triangle a straig-ht angle. With straig-ht lines 
of precisely such shape but in a non-Euclidean space, this sum 
. may be greater or less. In non-Euclidean spaces, if one edg-e of a 
flat ruler is a straig-ht line the other edg-e is a curve, if the ruler be 
everywhere equally broad. In any sense in which it can be prop- 
erly said that we live in space, it is probable that we really live in 
such a space. What becomes of the dog-ma that fundamental 
axioms are derived from experience alone? 
George Bruce Halsted. 
INTRODUCTION. 
Every one knows, that in Geometry the theory of parallels has 
remained, even to the present day, incomplete. 
The futility of the efforts which have been made since Euclid’s 
time during- the lapse of two thousand years to perfect it awoke in 
me the suspicion that the ideas employed mig-ht not contain the 
truth soug-ht to be demonstrated, and whose verification, as with 
other natural laws, could be helped only by experiments, as for 
example astronomic observations. 
When finally I had convinced myself of the correctness of my 
supposition, and believed myself to have completely solved the dif- 
ficult question, I wrote a paper on it in the year 1826.* 
* Exposition succincte des principes de la G4om6trie, avec une demonstra- 
tion rigoureuse du th^oreme des paralleles, read February 12, 1826, in the 
stance of the physico-mathematic Faculty of the University of Kazan, but not 
printed 
