MEASUREMENT 133 



the " arbitrary " measurement of density, depending 

 simply on the arrangements of the substances in their 

 order (p. 127), would serve equally well. The true 

 answer to our question is seen by remembering the con- 

 clusion, at which we arrived in Chapter III, that \lie terms 

 between which laws express relationships are themselves 

 based on laws and represent collections of other terms 

 related by laws. When we measure a property, either 

 by the fundamental process or by the derived process, 

 the numeral which we assign to represent it is assigned as 

 the result of experimental laws ; the assignment implies 

 laws. And therefore, in accordance with our principle, 

 we should expect to find that other laws could be dis- 

 covered relating the numerals so assigned to each other 

 or to something else ; while if we assigned numerals 

 arbitrarily without reference to laws and implying no 

 laws, then we should not find other laws involving these 

 numerals. This expectation is abundantly fulfilled, and 

 nowhere is there a clearer example of the fact that the 

 terms involved in laws themselves imply laws. When we 

 can measure a property truly, as we can volume (by the 

 fundamental process) or density (by the derived process) 

 then we are always able to find laws in which these pro- 

 perties are involved ; we find, e.g., the law that volume is 

 proportional to weight or that density determines, in a 

 certain precise fashion, the sinking or floating of bodies. 

 But when we cannot measure it truly, then we do not find 

 a law. An example is provided by the property " hard- 

 ness " (p. 128) ; the difficulties met with in arranging 

 bodies in order of hardness have been overcome ; but 

 we still do not know of any way of measuring, by the 

 derived process, the property hardness ; we know of no 

 numerical law which leads to a numeral which always 

 follows the order of hardness. And so, as we expect, we 

 do not know any accurate and general laws relating hard- 

 ness to other properties. It is because true measurement 



