FOUR-DIMENSIONAL TUBES 
XI. 2] 
I5I 
The four-dimensional world is usually regarded as a con 
tinuum of points. For a line to be continuous there must be 
no gaps or “ holes in it; the ideal line of the geometer is not 
a series of discrete black pencil marks on a sheet of paper, but 
is a one-dimensional continuum. The same idea may be ex 
tended to space of two or of three dimensions, and may be 
generalized so as to apply to any aggregate of elements numeric 
ally determined in such a way as to leave no holes. Such a 
set of elements of any kind forms a continuum. Now it is com 
monly assumed that the four-dimensional space-time world of 
Minkowski is a continuum, so that we may pass from one element 
to another—from one event to another—by traversing a path 
involving “ successive ” events. But it is just here that the 
quantum theory introduces a difficulty. According to Poincare * 
the hypothesis of quanta is the only hypothesis leading to the 
law of Planck, and it involves the following proposition. 
" A physical system is only susceptible of a finite number of 
distinct states ; it jumps from one of these states to another 
without passing through a continuous series of intermediate 
states.” 
It may be added that Poincare has shown that no small, or 
even finite, departures from the formula of Planck will enable 
us to escape from such discontinuities. The necessity for dis 
continuity is bound up with the fact that the total radiation at 
a given temperature is finite and not infinite. 
2. Continuity and Discontinuity 
The ultimate analysis of the notions of continuity and dis 
continuity will no doubt provide material for age-long discussion 
from the metaphysical side. We cannot, however, altogether 
escape the difficulties inherent in the quantum theory by shifting 
them on to the shoulders of the metaphysician. To a certain 
extent they have already burdened the physicist in connection 
with the atomic constitution of matter. Thus Larmor f writes : 
" The difficulty of imagining a definite uniform limit for divisi 
bility of matter will always be a philosophical obstacle to an 
atomic theory, so long as atoms are regarded as discrete particles 
moving in an empty space. But as soon as we take the next 
step in physical development, that of ceasing to regard space as 
mere empty geometrical continuity, the atomic constitution of 
matter (each ultimate atom consisting of parts which are incap 
* Poincaré, Dernières pensées (Flammarion, Paris, 1913) ; Jeans, 
Report on Radiation and the Quantum Theory, pp. 35-6 (1914). 
t Larmor, Aether and Matter, § 46, 1900.