24 Presidential Address 



The divisions on a foot rule, divided as 

 closely as you please, represent commen- 

 surable fractions, but they represent none 

 of the length. No matter how numerous 

 they are, all the length lies between them; 

 the divisions are mere partitions and have 

 consumed none of it ; nor do they connect 

 up with each other, they are essentially 

 discontinuous. The interspaces are in- 

 finitely more extensive than the barriers 

 which partition them off from one another ; 

 they are like a row of compartments 

 with infinitely thin walls. All the incom- 

 mensurables lie in the interspaces; the 

 compartments are full of them, and they 

 are thus infinitely more numerous than 

 the numerically expressible magnitudes. 

 Take any point of the scale at random, 

 that point will certainly lie in an inter- 

 space: it will not lie on a division, for the 

 chances are infinity to I against it. 



Accordingly incommensurable quanti- 

 ties are the rule in physics. Decimals do 

 not in practice terminate or circulate, 

 in other words vulgar fractions do not 

 accidentally occur in any measurements, 



