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



a similar reason, the space on that side of A E B towards C is equal 

 to two right angles. These two angular spaces being constantly equal, 

 take away from both the common angular space A E C, therefore the 

 remainders are equal, viz. the angles A E D and C E B. 



Scholium. 



It may be useful to devote an angular space by two letters, one from 

 each side, if the angle be less than two right angles, or by three if 

 the angle be two right angles or more, to prevent the confusion of 

 direct and reverse angles. Thus in Fig. 4, C B would stand for the 

 angular space corresponding to C E B ; A C B for the two right angles 

 between E A and E B : and D A C B for the reverse angular space be- 

 tween E D and E B. The demonstration of III may then be made short- 

 er, and perhaps clearer, thus : C E D being a straight line, D A C = 2 

 right angles; also because AEB is straight, ACB = 2 right angles, 

 Hence D A C = A C B ; take away the common part A C, then AD=CB. 

 that is the angle A E D = C E B. 



Proposition IV. (Fig. 5. J 



If the angle contained by two straight lines is equal to two right an- 

 gles, those straight lines form but one continued line. 



For if A B, A C including an angle equal to two right angles, are 

 not in the same straight line, let A D be the continuation of A B : 

 then D C B is two right angles, but C B is the same by hypothesis, 

 .*. D C B = C B an absurd result ; therefore A C and A B form but one 

 line. 



Proposition V. (Fig. 6. J 



If any number of straight lines tend towards the same parts, the 

 angle made by the extremes is equal to the sum of the angles made by 

 the successive pairs of lines. 



Let A, B, C and D be straight lines, tending towards the same parts, 

 then the angle A H D is equal to the sum of the angles A E B, B F C, 

 C G D formed by the successive pairs of lines. For the angular space 

 A B C D is equal to the sum of the three angular spaces A B, B C and 

 C D. But A B C D is the angle A H D and the constituent angular 

 spaces A B, B C, CD are respectively identical with the angles AEB, 

 BFC, CGD. Hence A HD =AEB+BFC + CGD. 



