144 WHAT IS SCIENCE? 



numerical law, the rule involved in it will be changed by 

 changing the unit in which we measure the properties 

 represented by the two columns ; but the change will 

 only consist in the substitution of one constant numeral 

 for another. If we chance to choose the units in some 

 particular way, that constant numeral may turn out to 

 be i and so will disappear from sight. But it will always 

 be there. There must be associated with every numerical 

 law, involving a rule for arriving at the numerals in one 

 column from those in the other, some constant numeral 

 which is applicable to all members of the column alike. 

 And this constant may always, as in the case of density, 

 be the measure of some property to which derived 

 measurement is applicable. Every numerical law there- 

 fore this is the conclusion to be enforced may give 

 rise to a system of derived measurement ; and as a matter 

 of fact all important numerical laws do actually so give 

 rise. 



CALCULATION 



But though the establishment of system of derived 

 measurement is one' use of numerical laws, they have 

 also another use, which is even more important. They 

 permit calculation. This is an extremely important 

 conception which deserves our close attention. 



Calculation is the process of combining two or more 

 numerical laws in such a way as to produce a third numeri- 

 cal law. The simplest form of it may be illustrated by the 

 following example. We know the following two laws which, 

 in rather different forms, have been quoted before : 

 (i) the weight of a given volume of any substance is 

 proportional to its density ; (2) the density of a gas is pro- 

 portional to the pressure upon it. From these two laws 

 we can deduce the third law : the weight of a given 

 volume of any gas is proportional to the pressure upon it. 



