﻿J 8 Dr. Norman Campbell on the 



amendment in the thesis might be made, and it might 

 be said that Euclid's assumption is that the laws are true 

 which would make measurement possible if there were no 

 incommensurable lengths — for these laws, though necessary 

 to measurement, may not be sufficient. But the difficulty 

 vanishes entirely, if it is remembered what is meant by 

 " equality " in experimental measurement. When it is 

 said that A is equal to B, it is meant that there is no 

 possible means of deciding which of the two is the greater. 

 If then I say that the diagonal of a square is \/2 times 

 the side, I mean that, if I measure the diagonal in terms 

 of the side as unit, there is no means of deciding whether 

 the value obtained, when multiplied by itself according to 

 the multiplication table, will be greater or less than 2. 

 That statement is not in the least inconsistent with my 

 assigning to particular diagonals values of which the square 

 is not 2 ; it is only inconsistent if a law can be found 

 by which I can tell in particular cases whether the square 

 will be greater or less than 2. My assertion is that* there 

 is no such law ; and that assertion is true. In its appli- 

 cation to all magnitudes except number, equality must be 

 interpreted in this, slightly statistical, sense. 



3. There is then no preliminary objection to the view 

 that Euclid's propositions are deductions from the laws in 

 virtue of which the geometrical magnitudes are measurable. 

 We now proceed to ask what those laws are. 



Geometrical conceptions are derived ultimately from 

 our immediate sensations of muscular movement, just as 

 dynamical conceptions are derived from our sensations of 

 muscular exertion and thermal conception from our sense 

 of hot and cold. We have an instinctive and indescribable 

 appreciation of differences in direction of various movements ; 

 we appreciate that one direction may be between two others ; 

 and if other sensations (e. g. those of hot and cold or rough 

 and smooth) vary with movement along a certain direction, 

 we appreciate that of the varying sensations some are between 

 others. The notions of direction and of the two kinds of 

 betweenness are the foundations of geometry. It is a vitally 

 important fact that there is an intimate relation connecting 

 betweenness determined by one kind of muscular motion (e. g. 

 that of the hand) and that determined by another {e.g. that 

 of the eye). The relation is much too complex for any 

 account of it to be attempted here ; but it is only because it 

 exists that "space" explored visually or by our different limbs 

 is always the same. 



