xviii. 2] INTERPRETATION OF THE QUANTUM 247 
in Chapter XI. “ The theories of electricity and thermodynamics 
which are actually current show a tendency to substitute the 
discontinuous for the continuous ; the hypothesis of quanta 
seems to explain phenomena better than hypotheses based on 
continuity. Does this mean that we should introduce this 
discontinuity into geometry itself ? The question is perhaps 
somewhat premature, but we can hardly avoid putting it ” 
(Emile Borel*). 
It may be that our conception of continuous motion is an 
illusion produced by our senses, the apparatus available for 
perception being too coarse to appreciate the minute jumps 
which actually constitute observed phenomena. The question 
has been stated by Poincar6 as follows : 
“ Is discontinuity destined to reign over the physical universe, 
and will its triumph be final ? Or will it finally be recognized 
that this discontinuity is only apparent, and a disguise for a 
series of continuous processes ? The first observer of a collision 
thought he was witnessing a discontinuous process, but we know 
to-day that what he saw was the result of changes which, although 
very rapid, were continuous.” 
The final answer to any series of questions is inevitably— 
“because the world is so constructed” The final representation 
of the universe to which the quantum theory seems to point is 
a four-dimensional space-time world of events, in which 
unexpected (and perhaps in some respects undesirable) flaws or 
cracks are to be found. As the late Lord Rayleigh once remarked 
with reference to this very theory : “ It is not the question what 
we may like or dislike, but what the facts demand.” 
It appears then that we may be compelled to look upon the 
four-dimensional world as essentially discontinuous in char 
acter. In three dimensions we have emphasized the suggestion 
that the quantum theory points to the existence of discrete tubes 
of magnetic induction. Applying the same ideas to the four 
dimensional tubes of force of Whittaker, we are led to the view 
that the world of events is not a continuum, but is built up of 
individual tubes of force or “ calamoids.” In his Kelvin lecture 
Jeans has put this idea in picturesque form by saying that the 
quantum theory represents a quality of the four-dimensional 
continuum, which is somehow analogous to the scaliness of a 
crocodile skin. 
In considering such revolutionary ideas as are implied in 
the quantum theory there are two opposite dangers to be guarded 
against, as Planck has pointed out in the Preface of the second 
edition of his Heat Radiation. “ Any one who would make his 
attitude concerning the hypothesis of quanta depend on whether 
* Emile Borel, Space and Time, 1926. 
i* ’ I