Doctrine of Parallels. 89 
Ant. [X.—Attempt to Demonstrate the Assumed Point. in the 
Doctrine of Parallels ; by A. C. Twrsine, Prof. of Math., Nat. 
Philos. and Civil Engineering, in Middlebury College. 
A concise and rigorous demonstration of what is called the 
postulatum of Evctip,—that is to say, that two straight lines 
which make, with a third line, interior angles together less than 
two right angles, will meet if indefinitely produced,—is still ac- 
counted among geometrical writers a desideratum. The author 
of the able Treatise on Geometry put forth in England by the 
“ Society for the Diffusion of Useful Knowledge,” not only an- 
nounces the difficulty in the text itself, but declares, in a scho- 
lium, that it is agreed by geometers that some assumption is in- 
dispensable. The numerous although abortive attempts, how- 
ever, to resolve the difficulty, extending from the earliest periods 
of the science quite down to our own times, and still im process 
of continuance, evince that a hope at least is still entertained by 
the lovers of exact reasoning of wiping away the reproach, as 
they esteem it, of their favorite branch of knowledge. 
It is here worth an inquiry why the one difficulty in the doc- 
trine of parallels has monopolized attention and anxiety to the 
exclusion (to say nothing of a plain assumption in the 21st of the 
first book of Euciim, and the same in the corresponding 9th of the 
first book of Lecenpre, which however I expect at some subse- 
quent opportunity to exhibit, as reducible to demonstration in a 
distinct proposition) of two equally palpable infirmities that sub- 
sist in the definitions themselves; one in the ordinary definitions 
of the straight line, and the other in the definition of a plane 
surface. Proof, as it would seem, may justly be demanded that 
there can be lines of such a property that two cannot coincide 
in two points without coinciding throughout—or, as the axiom 
shapes it, cannot “enclose a space.” Neither can the adopted 
“shortest distance between two points” relieve Lecenpre’s Sys- 
tem; for, although most evidently there is a “ shortest distance” 
in amount, yet what geometer, unless a very late one, has shown 
that there is but one specific path in space to which that least 
amount can be attributed. Again, in what estimation shall the 
exact reasoner hold the ordinary definition of a plane? It would 
indeed be admissible, supposing a straight line properly defined, 
Seconp Sxnres, Vol. I, No. 1.—Jan. 1846. 12 
. 
