NUMERICAL LAWS AND MATHEMATICS 137 



represented when we were drawing up the table. And if 

 any other numerals are presented to our notice, it is 

 possible and legitimate to ask whether these numerals, 

 whatever they may represent, are in fact related as are 

 the numerals in the multiplication table. In particular, 

 when we are seeking a numerical relation between the 

 columns of Table I, we may inquire, and it is natural 

 for us to inquire, whether by means of the multiplication 

 we can find a rule which will enable us to arrive at the 

 numeral in the second column starting from that in the 

 first. 



That explains why it is so natural to us to try division 

 when we are seeking a relation between numbers. But 

 it does not answer the second part of the question ; for 

 in the numerical law that we are considering, the relation 

 between the things represented by the numerals is not 

 that which we have just noticed between things counted. 

 When we say that, by dividing the volume by 7, we can 

 arrive at the weight, we do not mean that the weight is 

 the volume of each of the things at which we arrive by 

 dividing the substance into 7 portions, each having the 

 same volume. For a weight can never be a volume, any 

 more than a soldier can be a number ; it can only be repre- 

 sented by the same numeral as a volume, as a soldier 

 can be represented by a numeral which also represents a 

 number. 



The distinction is rather subtle, but if the reader is to 

 understand what follows, he must grasp it. The relation 

 which we have found between weight and volume is a 

 pure numerical relation ; it is suggested by the relation 

 between actual things, namely collections which we 

 count ; but it is not that relation. The difference may be 

 expressed again by means of the distinction between 

 numbers and numerals. The relation between actual 

 things counted is a relation between the numbers which 

 are physical properties of those things ; the relation 



