﻿24 Dr. Norman Campbell on the 



one of the parts : e. </., if both of two parts, being corners, 

 are what we now call 120°. Some kind of spiral space can 

 be imagined in which the law would be true ; but actually 

 it is very important that it is false. For, apparently in- 

 separable from its falsity is the fact that the angle between 

 two portions of the same straight line can be measured and 

 given a finite value in terms of a unit which is the angle 

 between two intersecting lines. This fact is described by 

 the assertion that there are right angles and that a per- 

 pendicular can be drawn to any straight line from any 

 point in it, a right angle being defined as in Def. 1. 10. 

 (Axiom 1. 11 follows from this definition, regarded as an 

 existence theorem, and our axiom Prop. I. 15.) Since the 

 existence of right angles is vital to geometry, we cannot 

 avoid the falsity of the first law of equality by some 

 alteration of the definition. We can only recognize that 

 the law is true in some conditions, and be careful to apply 

 it in deduction only when it is true. It is true when all the 

 lines making the added angles lie on the same side of (or 

 contiguous with) a single straight line passing through 

 their common point ; this condition can be expressed, 

 though with some complexity, in terms of the fundamental 

 notion of between. Thus, in proving Prop. I. 16 we need 

 to know that OF and CD both lie on the same side of AC 

 This law, and perhaps others of the same nature, are laws of 

 measurement, defining the conditions in which angle can be 

 measured uniquely. They require explicit mention. 



The ambiguity which the falsity of the first law of 

 addition introduces into numerical measurement is removed 

 by certain conventions. These need not be considered here 

 for we are not assigning numerical values. 



If the length of curved lines were measurable funda- 

 mentally, angle might be measured as a pure derived 

 magnitude, e. g. by the ratio of the arc to the radius of 

 a circle in virtue of the numerical law, established experi- 

 mentally, that the arc is proportional to the radius. But 

 since curved lines cannot be so measured, we must take 

 angle to be fundamental. We cannot use right-angled 

 triangles with straight sides to measure angle as derived, 

 because we need fundamental measurement to determine 

 what angles are right. Of course we might define for 

 this purpose a right angle as an angle between some two 

 lines arbitrarily chosen as standard ; but such measurement 

 would be intolerably artificial and nothing whatever could 

 be deduced from such a definition. 



