128 WHAT IS SCIENCE? 



while the test with Y (floating in A, but sinking in B) 

 would prove that A was denser than B. Are we then to 

 put A before or after B in the order of density ? We 

 should not know. The order would be indeterminate 

 and, whether we assigned a higher or a lower numeral to 

 A than to B, the assignment would represent no definite 

 physical fact : it would be arbitrary. 



In order to show that the difficulty might occur, and 

 that it is an experimental law that it does not occur, an 

 instance in which a similar difficulty has actually occurred 

 may be quoted. An attempt has been made to define 

 the " hardness " of a body by saying that A is harder 

 than B if A will scratch B. Thus diamond will scratch 

 glass, glass iron, iron lead, lead chalk, and chalk butter ; 

 so that the definition leads to the order of hardness : 

 diamond, glass, iron, lead, chalk, butter. But if there is 

 to be a definite order, it must be true in all cases that if 

 A is harder than B and B than C, then A is harder than 

 C ; in other words, if A will scratch B and B C, then A 

 will scratch C. But it is found experimentally that there 

 are exceptions to this rule, when we try to include all 

 bodies within it and not only such simple examples as 

 have been quoted. Accordingly the definition does not 

 lead to a definite order of hardness and does not 

 permit the measurement of hardness. 



There are other laws of the same kind that have to be 

 true if the order is to be definite and the measurement 

 significant ; but they will not be given in detail. One 

 of them the reader may discover for himself, if he will 

 consider the propert}^ colour. Colour is not a property 

 measurable in the way we are considering, and for this 

 reason. If we take all reds (say) of a given shade, we 

 can arrange them definitely in an order of lightness and 

 darkness ; but no colour other than red will fall in this 

 order. On the other hand, we might possibly take all 

 shades and arrange them in order of redness pure red, 



