﻿Elements of Geometry. 27 



measurement of area. The rule might possibly be dis- 

 pensed with, it we were prepared to spend unlimited time 

 in selecting by trial and error shapes for the members of 

 the standard series which fulfil the necessary conditions ; 

 but actually we could never measure area except by making 

 use of similar figures, the production and properties of 

 which depend wholly on the axiom of parallels. Further, 

 it is the use of that axiom which enables us nowadays to 

 calculate area from the linear dimensions of a surface 

 without resorting at all to fundamental measurement. 

 But of course all the numerical laws on which that cal- 

 culation depends have to be established by means of 

 fundamental measurement. It is only by defining area 

 as we have done, and assuming the axiom of parallels, 

 that we can prove by deduction that the area of a rect- 

 angle is proportional to the product of its sides, or equal if 

 the units are suitably chosen. 



The areas of surfaces that are not plane cannot be mea- 

 sured fundamentally, even to the extent that the length of 

 curved lines can be. For there are no inextensible surfaces 

 which can be brought into contiguity with surfaces of any 

 curvature. Measurement of curved area is always derived 

 and estimated by the limit of the circumscribed polyhedra 

 as the number of their sides is increased. But the whole 

 matter is obscure, because it is much more difficult to 

 establish experimentally that there is a limit or to say what 

 the limit is ; for there is here no inscribed polyhedron 

 tending to the same limit. There is singularly little experi- 

 mental evidence for the assertion that the area of a sphere 

 is 47rr 2 , and there is great difficulty in saying exactly what 

 we mean by such an assertion ; curved area is almost always 

 a hypothetical idea and not an experimental magnitude 

 at all. 



11. Volume is a property of complete surfaces. Since 

 complete surfaces can never be brought into complete con- 

 tiguity, volume cannot be measured fundamentally by any 

 process at all similar to those applicable to the magnitudes 

 we have considered so far. Volume is measured (1) as a 

 fundamental magnitude by means of incompressible fluids, 

 or (2) as a derived magnitude by means of the lengths and 

 angles characteristic of the surface. The second method 

 depends upon numerical laws established by means of the 

 first. In certain cases these laws can be related closely 

 to other geometric laws by means of the following propo- 

 sitions : — (1) Two complete surfaces with equal dimensions, 



