﻿22 Dr. Norman Campbell on tlie 



branch of geometry with which we (and, according to our 

 view, Euclid) are concerned may be fitly described as the 

 study of the surfaces of rigid bodies. It is thus dis- 

 tinguished from a wholly different branch of geometry, 

 with which we are not here concerned, ihat is not confined 

 to rigid bodies ; this is the geometry of position. 



It is important to notice that not all pairs of straight 

 lines can be brought into contiguity, and that the law of 

 equality cannot therefore be tested universally. It might 

 have turned out that there was some material difference 

 between those which can and those which cannot be brought 

 into contiguity with a given line ; and that if we assumed 

 that the law of equality is universally true, we should be led 

 to inconsistencies. It is an experimental fact that no such 

 inconsistencies do arise when we extend our definition of 

 equality so that lengths are equal when they are equal to 

 to the same length, although they cannot be brought into 

 contiguity with each other. This is, of course, one of the 

 most important laws that make measurement possible. A 

 similar remark applies to all the geometric magnitudes and 

 need not be repeated. 



7. The length of lines that are not straight can be 

 measured approximately as fundamental magnitudes by 

 means of flexible but inextensible strings. But the laws 

 of such measurement are not strictly true, because (as we 

 say now) no string is infinitely thin and the surface never 

 coincides with the neutral axis. Another possible way, 

 perhaps more accurate but of limited application, would 

 be to roll curved edges on some standard edge, which 

 need not be straight. But in truth there is no perfectly 

 satisfactory way of measuring fundamentally the length of 

 curved lines. All the measurements which we make on them 

 are derived from measurement of straight lines ; they involve 

 numerical laws between fundamentally measured magnitudes. 

 One of these laws is that the perimeters of the circumscribed 

 and inscribed regular polygons tend to a common limit as 

 the number of sides is increased. That law is therefore a 

 law of measurement if curved lines are to be measured. 



The question whether curved lines can be measured 

 fundamentally is important, because, if they could be, it 

 would be possible to define a straight line as the shortest 

 distance between two points. (The definition would have 

 to be put in some other form, since distance, a conception 

 belonging to the geometry of position, implies the mea- 

 surement of length.) But since they cannot be, that 



