﻿Elements of Geometry. 23 



definition must be rejected ; it must be regarded merely as 

 a generalized form of Prop. I. 20. 



8. Angle is the measure of the crude conception of 

 direction. The following are the definitions of equality and 

 addition for the angle between two intersecting straight 

 lines : — The angle between two straight lines A, B is equal 

 to that between C, D if it is possible to bring A into con- 

 tinguity with C and B with D. The angle between A, B is 

 the sum of the angles between C, D and E, F, if, when 

 A is brought into contiguity with C and D with E, I) lying 

 between C and F and in the same plane with them, F can 

 be brought into contiguity with B. These definitions are 

 satisfactory only if the straight lines are in rigid bodies; 

 or, in other words, there are surfaces which satisfy the 

 conditions for the measurement of length and also those 

 for the measurement of angle. 



But even if the surfaces are those of rigid bodies, the 

 definitions are not wholly satisfactory and the laws of 

 measurement not entirely true. We must distinguish 

 angles according as the two straight lines which they 

 relate are or are not prolonged on both sides of the 

 common point : the latter class may be termed " corners/' 

 the former " crossings." Angles between edges are always 

 corners ; those between scratches may be either corners or 

 crossings. If we try to include both corners and crossings 

 in the same class as a single magnitude, the law of equality 

 is not true ; for two corners which are both, according 

 to the definition, equal to a crossing may not be equal 

 to each other ; as we say now, one angle may be the 

 supplement of the other. But if we treat corners and 

 crossings as separate magnitudes this difficulty disappears ; 

 the law of equality is true for either taken apart from the 

 other. Actually we take corners only as magnitudes ; 

 crossings we measure by the corners with which they can 

 be made contiguous. Each crossing then has four angles 

 (i. e. corners) associated with it. It is an important experi- 

 mental fact that the " opposite" angles are equal; it is 

 best taken as a primary law, instead of being proved from 

 other axioms as in Prop. I. 15. It is a law of measurement, 

 because if it were not known, we should need four and not 

 two angles to measure a crossing ; it is thus inherent in our 

 system of measurement. 



But though the law of equality is now true, the first law of 

 addition is false ; it is false for both corners and crossings. 

 The whole which is the sum of the parts may be equal to 



