PRESIDENTS ADDRESS, 11 
and numerical properties, and discontinuity once more makes its 
appearance. Then we invent the ether 'and are impressed with con- 
tinuity again. But this is not likely to be the end; and what the ulti- 
mate end will be, or whether there is an ultimate end, is a question 
difficult to answer. 
The modern tendency is to emphasise the discontinuous or atomic 
character of everything. Matter has long been atomic, in the same 
sense as Anthropology is atomic; the unit of matter is the atom, as the 
unit of humanity is the individual. Whether men or women or chil- 
dren—they can be counted as so many ‘ souls.’ And atoms of matter 
can be counted too. 
Certainly however there is an illusion of continuity. We recognise 
it in the cose 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 Reynolds, it is true, invented a discontinuous or 
granular Hther, on the analogy of the sea shore. 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 courted 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 discon- 
tinuities, 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 discontinuity 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 commensurable number; incommensur- 
ables are not numbers: they are just what cannot be expressed 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 essen- 
tially continuous. It is clear, as Poincaré says, that ‘if the points 
