ee ape 
SEPTEMBER II, 1913] 
spaces; the compartments are full of them, and they 
are thus infinitely more numerous than the numeric- 
ally 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 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, for this would 
mean infinite accuracy. We proceed to as many 
places of decimals as correspond to the order of accu- 
racy aimed at. 
Whenever, then, a commensurable number is really 
associated with any natural phenomenon, there is 
necessarily a noteworthy circumstance involved in the 
fact, and it means something quite definite and ulti- 
mately 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 
discovery 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 under- 
lying, and in definite groups composing, the atoms 
of every form of matter. 
The small number of degrees of freedom of a mole- 
cule, and the subdivision of its total energy into equal 
parts corresponding thereto, is a theme not indeed 
without difficulty but full of importance. It is re- 
sponsible for the suggestion that energy too may be 
atomic ! ; 
Mendelejefi’s series again, or the detection of a 
natural grouping of atomic weights in families of 
seven, is another example of the significance of 
number. 
Electricity was found by Faraday to be numerically 
connected with quantity of matter; and the atom of 
electricity began its hesitating but now brilliant career. 
Electricity itself—i.e., electric charge—strangely 
enough has proved itself to be atomic. There is a 
natural unit of electric charge, as suspected by Fara- 
day and Maxwell and named by Johnstone Stoney. 
Some of the electron’s visible effects were studied by 
Crookes in a vacuum; and its weighing and measur- 
ing by J. J. Thomson were announced to the British 
Association meeting at Dover in 1899—a fitting pre- 
lude to the twentieth century. 
An electron is the natural unit of negative elec- 
tricity, and it may not be long before the natural unit 
of positive electricity is found too. But concerning 
the nature of the positive unit there is at present some 
division into opposite camps. One school prefers to 
regard the unit of positive electricity as a homo- 
geneous sphere, the size of an atom, in which elec- 
trons revolve in simple harmonic orbits and constitute 
nearly the whole effective mass. Another school, 
while appreciative of the simplicity and ingenuity and 
beauty of the details of this conception, and the skill 
with which it has been worked out, yet thinks the 
evidence more in favour of a minute central positive 
NO. 2289, VOL. 92] 
NATURE 
37 
nucleus, or nucleus-group, of practically atomic mass; 
with electrons, larger—i.e. less concentrated—and 
therefore less massive than itself, revolving round it 
in astronomical orbits. While from yet another point 
of view it is insisted that positive and negative elec- 
trons can only differ skew-symmetrically, one being 
like the image of the other in a mirror, and that the 
mode in which they are grouped to form an atom 
remains for future discovery. But no one doubts that 
electricity is ultimately atomic. 
Even magnetism has been suspected of being 
atomic, and its hypothetical unit has been named in 
advance the magneton; but I confess that here I have 
not been shaken out of the conservative view. 
We may express all this as an invasion of number 
into unsuspected regions. 
Biology may be said to be becoming atomic. It 
has long had natural units in the shape of cells and 
nuclei, and some discontinuity represented by body- 
boundaries and cell-walls; but now, in its laws of 
heredity as studied by Mendel, number and discon- 
tinuity are strikingly apparent among the reproductive 
cells, and the varieties of offspring admit of numeri- 
cal specification and prediction to a surprising extent; 
while modification by continuous variation, which 
seemed to be of the essence of Darwinism, gives place 
to, or at least is accompanied by, mutation, with 
finite and considerable and in appearance discon- 
tinuous change. 
So far from Nature not making jumps, it becomes 
doubtful if she does anything else. Her hitherto 
placid course, more closely examined, is beginning to 
look like a kind of steeplechase. 
Yet undoubtedly continuity is the backbone of 
evolution, as taught by all biologists—no artificial 
boundaries or demarcations between species—a con- 
tinuous chain of heredity from far below the amceba 
up to mag. Actual continuity of undying germ- 
plasm, running through all generations, is taught 
likewise; though a strange discontinuity between this 
persistent element and its successive accessory body- 
plasms—a discontinuity which would convert indi- 
vidual organisms into mere temporary accretions or 
excretions, with no power of influencing or conveying 
experience to their generating cells—is advocated by 
one school. 
Discontinuity does not fail to exercise fascination 
even in pure mathematics. Curves are invented which 
have no tangent or differential coefficient, curves 
which consist of a succession of dots or of twists; 
and the theory of commensurable numbers seems to 
| be exerting a dominance over philosophic mathe- 
matical thought as well as over physical problems. 
And not only ‘these fairly accepted results are 
prominent, but some more difficult and unexpected 
theses in the same direction are being propounded, 
and the atomic character of energy is advocated. We 
had hoped to be honoured by the presence of Prof. 
Planck, whose theory of the quantum, or indivisible 
unit or atom of energy, excites the greatest interest, 
and by some is thought to hold the field. 
Then again radiation is showing signs of becoming 
atomic or discontinuous. The corpuscular theory of 
radiation is by no means so dead as in my youth 
we thought it was. Some radiation is certainly 
corpuscular, and even the ethereal kind shows indica- 
tions, which may be misleading, that it is spotty, or 
locally concentrated into points, as if the wave-front 
consisted of detached specks or patches; or as J. J. 
Thomson says, ‘the wave-front must be more 
analogous to bright specks on a dark ground than 
to a uniformly illuminated surface,” thus suggesting 
that the ather may be fibrous in structure, and that 
a wave runs along lines of electric force; as the 
