NUMERICAL LAWS AND MATHEMATICS 145 



That conclusion seems to follow directly without any need 

 for further experiments. Accordingly we appear to have 

 arrived at a fresh numerical law without adducing any 

 fresh experimental evidence. But is that possible ? 

 All our previous inquiry leads us to believe that laws, 

 whether numerical or other, can only be proved by experi- 

 mental inquiry and that the proof of a new law without 

 new experimental evidence is impossible. How are we 

 to reconcile the two conclusions ? When we have 

 answered that question we shall understand what is the 

 importance of calculation for science. 



Let us first note that it is possible, without violating 

 the conclusions already reached, to deduce something 

 from a numerical law by a process of mere thought with- 

 out new experiment. For instance, from the law that 

 the density of iron is 7, I can deduce that a portion of it 

 which has a volume i will have a weight 7. But this 

 deduction is merely stating in new terms what was 

 asserted by the original law ; when I said that the density 

 of iron was 7, 1 meant (among other things) that a volume 

 i had a weight 7 ; if I had not meant that I should never 

 have asserted the law. The " deduction " is nothing 

 but a translation of the law (or of part of it), into different 

 language, and is of no greater scientific importance than 

 a translation from (say) English into" French. One kind 

 of translation, like the other, may have useful results, 

 but it is not the kind of useful result that is obtained 

 from calculation. Pure deduction never achieves any- 

 thing but this kind of translation ; it never leads to 

 anything new. But the calculation taken as an example 

 does lead to something new. Neither when I asserted 

 the first law, nor when I asserted the second did I mean 

 what is asserted by the third ; I might have asserted the 

 first without knowing the second and the second without 

 knowing the first (for I might have known what the density 

 of a gas was under different conditions without knowing 



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