Sbptembee 19, 1913] 



SCIENCE 



385 



as so many "souls." And atoms of matter 

 can be counted too. 



Certainly however there is an illusion of 

 continuity. We recognize it in the case of 

 water. It appears to be a continuous 

 medium, and yet it is certainly molecular. 

 It is made continuous again, in a sense, by 

 the ether postulated in its pores; for the 

 ether is essentially continuous. Though 

 Osborne Eeynolds, it is true, invented a 

 discontinuous or granular ether, on the 

 analogy of the seashore. The sands of the 

 sea, the hairs of the head, the descendants 

 of a patriarch, are typical instances of 

 numerable, or rather of innumerable, 

 things. The difficulty of enumerating 

 them is not that there is nothing to count, 

 but merely that the things to be counted 

 are very numerous. So are the atoms in a 

 drop of water — they outnumber the drops 

 in an Atlantic Ocean — and, during the 

 briefest time of stating their number, fifty 

 millions or so may have evaporated; but 

 they are as easy to count as the grains of 

 sand on a shore. 



The process of counting is evidently a 

 process applicable to discontinuities, i. e., 

 to things with natural units ; you can count 

 apples and coins, and days and years, and 

 people and atoms. To apply number to a 

 continuum you must first cut it up into 

 artificial units; and you are always left 

 with incommensurable fractions. Thus 

 only is it that you can deal numerically 

 with such continuous phenomena as the 

 warmth of a room, the speed of a bird, the 

 pull of a rope or the strength of a current. 



But how, it may be asked, does discon- 

 tinuity apply to number? The natural 

 numbers, 1, 2, 3, etc., are discontinuous 

 enough, but there are fractions to fill up 

 the interstices; how do we know that they 

 are not really connected by these fractions, 

 and so made continuous again? 



(By number I always mean commensur- 



able number; incommensurables are not 

 numbers : they are just what can not be ex- 

 pressed in numbers. The square root of 2 

 is not a number, though it can be readily 

 indicated by a length. Incommensurables 

 are usual in physics and are frequent in 

 geometry; the conceptions of geometry are 

 essentially continuous. It is clear, as Poin- 

 care says, that ' ' if the points whose coordi- 

 nates are commensurable were alone re- 

 garded as real, the in-circle of a square and 

 the diagonal of the square would not inter- 

 sect, since the coordinates of the points of 

 intersection are incommensurable.") 



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 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 dis- 

 continuous. The interspaces are infinitely 

 more extensive than the barriers which par- 

 tition them off from one another; they are 

 like a row of compartments with infinitely 

 thin walls. All the incommensurables lie 

 in the interspaces; the compartments are 

 full of them, and they are thus infinitely 

 more numerous than the numerically ex- 

 pressible magnitudes. Take any point of 

 the scale at random, that point will cer- 

 tainly lie in an interspace: it will not lie 

 on a division, for the chances are infinity 

 to 1 against it. 



Accordingly incommensurable quantities 

 are the rule in physics. Decimals do not 

 in practise 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 



