1842.] On the Theory of Angular Geometry. 235 



asBK. ThenGFE = CBK ^ CBA ^1 (its = ) ABD^KBD 

 ^EFH. 



But GFE = E F H by definition, hence the supposition, that F E does 

 not coincide with B A involves absurd consequences, .'. F E does coincide 

 with B A, and G F E with C B A, G F E is therefore equal to C B A. 



Cor. 1. — A right angle is therefore a constant in angular magnitude. 



Cor. 2. — The space on one side of a straight line, considered as an 

 angle at some given point in the line, is two right angles. 



Scholium. (Fig. S.J 



Consider a line O A fixed, and another line O R, having a point in 

 common with O A, but being itself in a state of rotation round O. 

 When in the position O R, it will have generated an acute angle 

 A O B : as it proceeds, it will coincide with the perpendicular O B, 

 and will have described a right angle. In the position O R 2 , the angle 

 generated is an obtuse one. The generating line then coincides in its 

 progress with O C, the continuation of O A. It will in such position 

 have described two right angles (A O B and BO C). Supposing the 

 rotation to continue, O R will fall below O C, as O R3, having described 

 the whole of the coloured angular space, which is greater than two 

 right angles. Such angle is called a reverse angle. During the pro- 

 gress of the line, the reverse angle continues to increase, equals 

 3 right angles, exceeds that amount, and at length equals 4 right angles, 

 when O R has completed an entire circuit. By conceiving the line to 

 move on, still revolving, and with the aid of a contrivance like the 

 spiral twisted palm-leaf fans, used by the Natives, the beginner may 

 obtain the idea of angles greater than four right angles, and generally 

 of (2 n ii + A) which some find it difficult to understand in their later 

 trigonometrical studies, and perfect acquaintance with which is so indis- 

 pensable to the comprehension of periodic functions. 



Proposition III. (Fig. 4. J 



The vertically opposite angles made by two intersecting straight 

 lines, are equal to one another. 



The vertically opposite angles A E D and C E B made by the inter- 

 sectors A B and C D at E are equal. For C E D being a straight line, 

 the angular space on the side of it towards A is two right angles. For 



