12 PRESIDENTS ADDRESS. 
whose co-ordinates are commensurable were alone regarded as real, the 
in-circle of a square and the diagonal of the square would not intersect, 
since the co-ordinates of the points of intersection are incommensur- 
able.’) 
I want to explain how commensurable fractions do not connect up 
numbers, nor remove their discontinuity in the least. The divisions on 
a foot rule, divided as closely as you please, represent commensurable 
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 infinitely 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 incommensurables lie in the interspaces; the com- 
partments 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 interspace: it will 
not lie on a division, for the chances are infinity to 1 against it. 
Accordingly incommensurable quantities ave the rule in physics. 
Decimals do not in practice terminate or circulate, in other words vulgar 
fractions do not accidentally occur in any measurements, for this would 
mean infinite accuracy. We proceed to as many places of decimals as 
correspond to the order of accuracy aimed at. 
Whenever, then, a commensurable number is really associated with 
any natural phenomenon, there is necessarily a noteworthy circum- 
stance involved in the fact, and it means something quite definite and 
ullimately ascertainable. Every discontinuity that can be detected and 
counted is an addition to knowledge. It not only means the discovery 
of natural units instead of being dependent on artificial ones, but it 
throws light also on the nature of phenomena themselves. 
For instance :— 
The ratio between the velocity of light and the inverted square root 
of the product of the electric and magnetic constants was discovered by 
Clerk Maxwell to be 1; and a new volume of physics was by that dis- 
covery opened. 
Dalton found that chemical combination occurred between quantities 
of different substances specified by certain whole or fractional numbers; 
and the atomic theory of matter sprang into substantial though at first 
infantile existence. 
The hypothesis of Prout, which in some modified form seems likely 
to be substantiated, is that all atomic weights are commensurable 
numbers; in which case there must be a natural fundamental unit 
underlying, and in definite groups composing, the atoms of every form 
of matter. 
