﻿26 Dr. Norman Campbell on the 



axiom of parallels that the matter requires rather more con- 

 sideration. Two questions are involved. First, would it 

 be possible to measure the angle between non-intersecting 

 lines without assuming some proposition logically deducible 

 from the axiom? It would be if, and. only if, some 

 property, common to all lines between which the angle 

 is the same, can be found which is determinable by direct 

 experiment not involving parallel lines. There may be 

 such a property, but I have not been able to think of it. 

 Second, if the axiom were not actually true — but we may 

 stop there. In a pure experimental science, there is no 

 sense in asking what would happen if the world were other 

 than it actually is. Theory is necessary to give such a 

 question a meaning, by suggesting what might remain 

 unaltered during the change. For our present purpose 

 the axiom is as necessary as any other of those we are 

 considering. 



10. Area is distinguished from all other fundamental 

 magnitudes because the definitions of equality and addition 

 are inseparable. They may be expressed thus. The areas 

 of two bounded plane surfaces are equal if (but not only if) 

 their boundaries can be brought into complete contiguity 

 with each other or with the same third boundary. (A 

 bounded surfaca is a part of a surface which includes all 

 portions which can be traversed without crossing the 

 boundary line.) The area of A is the sum of the areas 

 of B and C, if when parts of the boundaries of B and C 

 are brought into contiguity with each other, the remaining 

 parts of the boundary can be brought into contiguity with 

 the boundary of A. In virtue of the fact that parts of the 

 boundaries of two surfaces can be brought into contiguity 

 in many different ways, there may be many different 

 bounded surfaces, of which the boundaries cannot be made 

 contiguous, which are the sum of the same bounded sur- 

 faces. If the measurement of area is to be satisfactory, 

 these surfaces must also be deemed to have equal area, and 

 the definition of equality must be extended correspondingly. 

 With this extension the laws of equality and addition are 

 true, and the measurement is satisfactory. 



In order that all bounded plane surfaces should have 

 areas, some rule must be found for choosing the shape of 

 the members of the standard series and for grouping them 

 in such a way that some sum of them is equal to any area. 

 We use for this purpose rules based on the axiom of 

 parallels, and that axiom is therefore again a law of the 



