30 
NATURE 
[SEPTEMBER II, 1913 
prompted in part by general philosophical views, in 
the direction that the theoretical constructions of 
physical science are largely factitious, that instead of 
presenting a valid image of the relations of things on 
which further progress can be based, they are still 
little better than a mirage... . 
“The best method of abating this scepticism is to 
become acquainted with the real scope and modes of 
application of conceptions which, in the popular 
language of superficial exposition—and even in the 
unguarded and playful paradox of their authors, 
intended only for the instructed eye—often look 
bizarre enough.” 
One thing is very notable, that it is closer and 
more exact knowledge that has led to the kind of 
scientific scepticism now referred to; and that the 
simple laws on which we used to be working were thus 
simple and discoverable because the full complexity of 
existence was tempered to our ken by the roughness 
of our means of observation. 
Kepler’s laws are not accurately true, and if he 
had had before him all the data now available he 
could hardly have discovered them. A planet does 
not really move in an ellipse but in a kind of hypo- 
cycloid, and not accurately in that either. 
. So it is also with Boyle’s law, and the other simple 
laws in physical chemistry. Even Van der Waals’ 
generalisation of Boyle’s law is only a further 
approximation. 
In most parts of physics simplicity has sooner or 
later to give place to complexity; though certainly 
I urge that the simple laws were true, and are still 
true, as far as they go, their inaccuracy being only 
detected by further real discovery. The reason they 
are departed from becomes known to us; the law is 
not really disobeyed, but is modified through the action 
of a known additional cause. Hence it is all in the 
direction of progress. ‘ 
It is only fair to quote Poincaré again, now that I 
am able in the main to agree with him :— 
“Take, for instance, the laws of reflection. Fres- 
nel established them by a simple and attractive theory 
which experiment seemed to confirm. Subsequently, 
more accurate researches have shown that this veri- 
fication was but approximate; traces of elliptic 
polarisation were detected everywhere. But it is 
owing to the first approximation that the cause of 
these anomalies was found, in the existence of a 
transition layer; and all the essentials of Fresnel’s 
theory have remained. We cannot help reflecting 
that all these relations would never have been noted 
if there had been doubt in the first place as to the 
complexity of the objects they connect. Long ago it 
was said: If Tycho had had instruments ten times 
as precise, we would never have had a Kepler, or a 
Newton, or astronomy. It is a misfortune for a 
science to be born too late, when the means of 
observation have become too perfect. That is what 
is happening at this moment with respect to physical 
chemistry ; the founders are hampered in their general 
grasp by third and fourth decimal places; happily 
they are men of robust faith. As we get to know 
the properties of matter better we see that continuity 
reigns. . . . It would be difficult to justify [the belief 
in continuity] by apodeictic reasoning, but without 
[it] all science would be impossible.” 
Here he touches on my own theme, Continuity; for 
if we had to summarise the main trend of physical 
controversy at present, I feel inclined to urge that it 
largely turns on the question as to which way ultimate 
victory lies in the fight between continuity and dis- 
continuity. - 
On the surface of nature at first we see discon- 
tinuity; objects detached and countable. Then we 
realise the air and other media, and so emphasise con- 
NO. 2289, VOL. 92] 
tinuity and flowing quantities. Then we detect atoms 
and numerical properties, and discontinuity once more 
makes its appearance. Then we invent the zther and 
are impressed with continuity again. But this is not 
likely to be the end; and what the ultimate end will 
be, or whether there is an ultimate end, is a question 
difficult to answer. 
The modern tendency is to emphasise the dis- 
continuous or atomic character of everything. Matter 
has long been atomic, in the same sense as anthropo- 
logy is atomic; the unit of matter is the atom, as the 
unit of humanity is the individual." Whether men or 
women or children—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 case of water. It appears to 
be a continuous medium, and yet it is certainly mole- 
cular. It is made continuous again, in a sense, by 
the zther postulated in its pores; for the ther is 
essentially continuous. Though Osborne Reynolds, it 
is true, invented a discontinuous or granular zther, 
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 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 incom- 
mensurable 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, &c., 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 con- 
tinuous again? 
(By number I always mean commensurable number ; 
incommensurables 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 concep- 
tions of geometry are essentially continuous. It is 
clear, as Poincaré says, that ‘‘if the points whose 
coordinates are commensurable were alone regarded 
as real, the in-circle of a square and the diagonal of 
the square would not intersect, 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 discon- 
tinuity in the least. The divisions on a foot rule, 
divided as closely as you please, represent commensur- 
able 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 mare 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 inter- 
1 In his recent Canadian address, Lord Haldane emphasised the fact that 
though humanity is individually discontinuous it possesses a social and 
national continuity. 
