238 On the Theory of Angular Geometry. [No. 123. 



exterior angle of the polygon. Hence the exterior angles together 

 amount to 4 right angles. 



The above eight Propositions comprise all the properties of intersec- 

 ting lines which are independent of the consideration of length and size. 

 They shew how possible it is to translate the spirit of the principle of 

 homogeneity from analytical into geometrical inquiries ; for our results 

 being altogether free from the comparison of triangles or the length of 

 lines, the interweaving of those subjects in our processes raises a suspi- 

 cion, that we are not proceeding so simply as we might do, but are 

 embarrassed with matters really foreign to the direct truth. We might 

 extend the same course to parallel lines. 



Definition. — Straight lines that never intersect each other, are called 

 parallel lines. 



Proposition IX. (Fig. 10 .) 



If a straight line meet two others, so as to make the exterior angle 

 equal to the interior and opposite on the same side, these two others 

 shall be parallel. 



LetCBE meet AB and DE making ABC = DEB, then DE 

 must be parallel to A B. For the angular space DC = DA-f AC 

 and DC is DEC, and AC is ABC, 



.-. DEC = DA + ABC, butDED = ABC. 



.*. D A is zero, or D E and A B contain no angle, therefore they 

 never meet, for if they met, they must contain an angle ; hence they are 

 parallel. 



Cor. — This proposition proves the possible existence of parallels. 



Proposition X. (Fig. 10.) 



If a straight cuts a pair of parallels, it makes an exterior angle equal 

 to an interior and opposite one on the same side. 



For as before D A C = D A + A C, but since D E and A B never 

 meet, they contain no angle, i. e. D A is zero; hence D A C = A C or 

 the angle DEB = ABC. 



Cor. — It would be a waste of space to deduce from this, the other 

 usual properties of parallels. 



Proposition XI. (Fig. 12. J 

 Straight lines parallel to the same are parallel to each other. 



