﻿28 Dr. Norman Campbell on the 



i. e. with equal lengths and equal angles between them, have 

 equal volumes. (2) If two complete surfaces have each 

 one part plane, and the boundary of the plane part of one 

 can be brought into complete contiguity with the plane part 

 of the other, then the complete surface which has dimensions 

 equal to that of the complete surface so formed has a volume 

 equal to the sum of the volumes of the original surfaces. 

 These propositions could be used as definitions of equality 

 and addition in a system of measurement, which would be 

 independent of the measurement of length and angle (and 

 therefore not derived), because it involves only equality, and 

 not addition, of length and angle. But it is of limited scope 

 and, in particular, would not permit the measurement of 

 the volumes of curved surfaces. Since we do undoubtedly 

 attribute a meaning to the volume of such surfaces, in a 

 way that we do not to their area, measurement by incom- 

 pressible fluids, which is not geometric, cannot be wholly 

 avoided. But the propositions, which are those on which 

 Euclid bases his treatment of volume, are actually used in 

 modern practice, and are therefore regarded permissibly as 

 laws of measurement. 



12. In deducing Euclid's propositions from the laws 

 of measurement of these magnitudes, subsidiary laws are 

 required, corresponding roughly to his postulates, expressed 

 and implied. First, we need "existence theorems" corre- 

 sponding to each of the definitions; for example, the 

 definition of a plane surface justifies the conclusion that 

 a straight edge can be placed contiguously to any two 

 portions of such a part of a surface. Second, we need the 

 assumption that we can make an object having a magnitude 

 equal to that of any object presented to our notice. All 

 these propositions are laws of measurement : the first group, 

 because ail definitions in experimental science are nothing 

 but existence theorems ; the second, because it is implied in 

 the fact that we can make a standard series by which we can 

 measure any magnitude. 



Euclid's three expressed postulates are all untrue. I 

 cannot " draw a straight line " from this room to the next 

 when the door is closed. Moreover his constructional 

 propositions, closely connected with the postulates, are 

 unsatisfactory because they are all directed to the drawing 

 of scratches, rather than to the making of edges. The 

 hypothetical experiments by means of which the deductions 

 are effected are carried out much more easily with edges 



