222 Prof. Stevelly on t/ie Doctrine of Parallel Lines 



wDx at right angles to AD, and at E the line tu^Ex' perpendi- 



Fig. 1. 



Fig. 2. 



cular to BE. Then if d and c?' can be conceived not to be in 

 the line wx, they must stand either equally above it, as in fig. 1, 

 or equally below it, as in fig. 2 (as may be shown in a round- 

 about manner by a mere application of the 4th of Euclid^s first 

 book, or), as may be (more) readily seen to be evident by con- 

 ceiving wl^Kad to be folded over on the line AD ; when wl^ 

 must coincide with a:'D, ak. with a' A, and ad with cid^ ; therefore 

 d with d' and the point where ad meets wx, with the point where 

 a'd! meets it. The same is true of the perpendicular line lu'Ex', 

 and the points e' and e. And since one line Cu stands perpen- 

 dicular to AB at C, which at the same time bisects ab, AB, and 

 a'b', we may conceive the left-hand side of either figure to be so 

 folded over Cu as to coincide in all similar lines and correspond- 

 ing points with the right-hand side of that figure ; and therefore 

 fig. 1 or fig. 2 must represent the circumstances, if the lines wx, 

 wx', dd\ and e^ can be supposed not to be coincident through- 

 out : also on that supposition, wx, if produced indefinitely, could 

 never meet dnJe-y nor could dmd' ever meet idEx', Hence, 

 if we now join D with m' by a straight line, no part of that join- 

 ing line can in fig. 1 fall below Da?, nor in fig. 2 above it (other- 

 wise two straight lines would enclose a space). Nor can any 

 part of the same Dm' fall in fig. 1 above eW, nor in fig. 2 below 



