MEASUREMENT 127 



of numerals ; it is also characteristic of the properties 

 represented by numerals in the manner we are 

 considering now. This is our feature which makes the 

 " measurement " significant. Thus, in our example, bodies 

 have a natural order of density which is independent 

 of actual measurement. We might define the words 

 " denser " or " less dense " as applied to liquids (and the 

 definition could easily be extended to solids) by saying 

 that the liquid A is denser than B, and B less dense than 

 A, if a substance can be found which will float in A but 

 not in B. And, if we made the attempt, we should find 

 that by use of this definition we could place all liquids 

 in a definite order, such that each member of the series 

 was denser than the preceding and less dense than the 

 following member. We might then assign to the first 

 liquid the density i, to the second 2, and so on ; and we 

 should then have assigned numerals in a way which would 

 be physically significant and indicate definite physical 

 facts. The fact that A was represented by 2 and B by 7 

 would mean that there was some solid body which would 

 float in B, but not in A. We should have achieved some- 

 thing that might fairly be called measurement. 



Here again it is important to notice that the possibility 

 of such measurement depends upon definite laws ; we 

 could not have predicted beforehand that such an arrange- 

 ment of liquids was possible unless we knew these laws. 

 One law involved is this : If B is denser than A, and C 

 denser than B, then C is denser than A. That sounds 

 like a truism ; but it is not. According to our definition 

 it implies that the following statement is always true : 

 If a body X floats in B and sinks in A, then if another body 

 Y sinks in B it will also sink in A. That is a statement 

 of facts ; nothing but experiment could prove that it is 

 true ; it is a law. And if it were not true, we could not 

 arrange liquids naturally in a definite order. For the 

 test with X would prove that B was denser than A, 



