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G. P. Thomson 

ELECTRONS BEHAVING AS (1) Particles, in passing through a gas; 
(2) Waves, in passing through a thin metal film (p. 198) 

New Background of Science 



M.A., D.Sc., Sc.D. 





All ri^its reserved no part of this book may be 
reproduced in any form without permission in 
writing from the publisher) except by a reviewer 
who wishes to quote brief passages in connection 
with a review written for inclusion in magazine 
or newspaper. 

Set up and electrotype*. 
Published May, IQ33- 


Preface page vii 

Chapter I The Approach to the External World 1 

II The Methods of Science 45 

III The Framework of the External World 

Space and Timfe 70 

IV Mechanism 111 

V The Texture of the External World 

Matter and Radiation 146 

VI Wave-Mechanics 193 

VII Indeterminacy 230 

VIII Events 261 

Index 297 


After undergoing a succession of kaleidoscopic changes, 
theoretical physics appears to have attained a state of 
comparative quiescence, in which there is fairly general 
agreement about essentials. In the following pages I 
have tried to depict the present situation in broad outline 
and in the simplest possible terms. I have drawn my 
picture against a roughly sketched background of rudi- 
mentary philosophy the philosophy of a scientist, not 
of a metaphysician because I believe, in common with 
most scientific workers, that without a background of this 
kind we can neither see our new knowledge as a consistent 
whole, nor appreciate its significance to the full. State- 
ments made without reference to such a background 
as, for instance, that " an electron consists of waves of 
probability " or that " the principle of indeterminacy 
shews that nature is not deterministic " can convey at 
best only a minute fraction of the truth. 

I have tried to exhibit the new knowledge in such a way 
that every reader can form his own judgment as to its 
philosophical implications. There is room for much 
legitimate difference of opinion as to what precisely these 
are; yet few, I think, will be found to doubt that some 
reorientation of scientific thought is called for. I have 
not suppressed my own view that the final direction of 
change will probably be away from the materialism and 
strict determinism which characterised nineteenth-cen- 



tury physics, towards something which will accord better 
with our everyday experience. This part of my work 
may be regarded as an amplification and clarification 
of parts of my earlier small book, The Mysterious Universe. 
I have hoped that the present book may serve a serious 
scientific purpose, and prove of interest and value both to 
students of physics and to other more general readers. 
Unhappily I found it impossible to attain the necessary 
precision of thought and statement without occasionally 
using a few mathematical symbols and formulae; at the 
same time I have tried to arrange that the general purport 
of these shall be made clear to the non-mathematical 
reader, who will, I hope, find most of the book intelligible. 


January 19th, 1933 



Twentieth-Century Physics 

A century which has run less than a third of its course has 
already witnessed two great upheavals in physical science. 
These are associated with the words Relativity and 
Quanta, and have forced the physicist of to-day to view 
nature against a background of ideas which is very differ- 
ent from that of his nineteenth-century predecessor. 

The latter thought of nature as an assemblage of ob- 
jects located in space and continually changing with the 
passage of time. It was something entirely detached 
from, and external to, himself; something which he could 
study and explore from a distance as the astronomer 
studies the surface of the sun through his telescope, or 
the explorer the desert from his aeroplane. He thought 
of the apparatus of his laboratory as the astronomer thinks 
of his telescope, or the explorer of his field-glass; it 
shewed him things which were there whether he looked 
at them or not, which had been there before the first man 
appeared on earth, and would still be there after the last 
man had been frozen to extinction. Finally he accepted 
a "common-sense" view of nature, believing that there 
was no great difference between appearance and reality; 
the possibility that things were not as they seemed might 
provide an admirable subject for a debating society of 
philosophers, but was of as little practical concern to the 
scientist as. to the farm-labourer. 



Although he may not have realised it, this complex 
of beliefs constituted a philosophical creed in itself. No 
attempt was made to justify it by abstract argument; so 
long as it worked satisfactorily none seemed to be needed, 
the success of the science based upon it providing a suffi- 
cient justification. If ever it ceased to work, there would 
be time enough to probe its foundations and perhaps look 
for a new philosophy. 

That time has now come. The old philosophy ceased 
to work at the end of the nineteenth century, and the 
twentieth-century physicist is hammering out a new phi- 
losophy for himself. Its essence is that he no longer sees 
nature as something- entirely distinct from himself. 
Sometimes it is what he himself creates, or selects or 
abstracts; sometimes it is what he destroys. 

In certain of its aspects, which are revealed by the new 
theory of quanta, nature is something which is destroyed 
by observation. It is no longer a desert which we explore 
from the detached position of an aeroplane; we can only 
explore it by tramping over it, and we raise clouds of 
dust at every step. Trying to observe the inner workings 
of an atom is like plucking the wings off a butterfly to see 
how it flies, or like taking poison to discover the conse- 
quences. . Each observation destroys the bit of the uni- 
verse observed, and so supplies knowledge only of a 
universe which has already become past history. 

In certain other aspects, especially its spatio-temporal 
aspects as revealed by the theory of relativity, nature is 
like a rainbow. The ancient Hebrew the analogue 
of the nineteenth-century physicist saw the rainbow 
as an objective structure set in the heavens for all men 
to behold, the token of a covenant between God and 


man, and as objective as the signature to a cheque. We 
now know that the objective rainbow is an illusion. 
Raindrops break sunlight up into rays of many colours, 
and the coloured rays which enter any man's eyes form 
the rainbow he sees; but as the rays which enter one 
man's eyes can never enter those of a second man, no two 
men can ever see the same rainbow. Each man's rain- 
bow is a selection of his own eyes, a subjective selection 
from an objective reality which is not a rainbow at all. 
And it is the same with the nature which each man sees. 

Again, just as a man's rainbow follows him about as he 
moves round the country-side, so nature follows us about. 
At whatever speed we move, we find nature adjusting 
itself to bur motion, so that this motion makes no differ- 
ence to its laws. 

Yet the analogy fails in one respect. A rainbow will 
disclose our own motion to us by the speed with which it 
moves against a background of distant forests and hills, 
but physical science can find no such background for 
nature. The whole of nature appears to follow us about. 

Imperfect though these analogies are, they will shew 
that the physicist of to-day must needs have some 
acquaintance with ideas which used to be considered the 
exclusive preserve of metaphysics. 

one of the foremost workers in modern theoretical 
physics, Professor Heisenberg of Leipzig, has described 
the present situation in the following words:* 

"With the advent of Einstein's relativity theory it was neces- 
sary for the first time to recognize that the physical world 
differed from the ideal world conceived in terms of everyday 

* The Physical Principles of the Quantum Theory (Univ. of Chicago Press, 
1930), p. 62. 


experience. . . . The experimental material resulting from 
modern refinements in experimental technique necessitated 
the revision of old ideas and the acquirement of new ones, 
but as the mind is always slow to adjust itself to an extended 
range of experience and concepts, the relativity theory seemed 
at first repellantly abstract. None the less, the simplicity of 
its solution for a vexatious problem has gained it universal 
acceptance. As is dear from what has been said, the resolu- 
tion of the paradoxes of atomic physics can be accomplished 
only by further renunciation of old and cherished ideas. . . . 
"To mold our thoughts and language to agree with the 
observed facts of atomic physics is a very difficult task, as it 
was in the case of the relativity theory. In the case of the latter, 
it proved advantageous to return to the older philosophical 
discussions of the problems of space and time. In the same 
way it is now profitable to review the fundamental discussions, 
so important for epistemology, of the difficulty of separating 
the subjective and objective aspects of the world. Many of 
the abstractions that are characteristic of modern theoretical 
physics are to be found discussed in the philosophy of past 
centuries. At that time these abstractions could be dis- 
regarded as mere mental exercises by those scientists whose 
only concern was with reality, but to-day we are compelled by 
the refinements of experimental art to consider them seriously". 

This is not meant in any way to suggest that an ob- 
jective nature does not exist^ but merely that it is at 
present beyond our purview. We can only see nature 
blurred by the clouds of dust we ourselves make; we can 
still only see the rainbow, but a sun of some sort must 
exist to produce the light by which we see it. 

Writing in 1899, * F. H. Bradley proposed to define the 
nature of metaphysics as 

"the bare physical world, that region which forms the object 

of purely physical science, and appears to fall outside of all 

* Appearance and Reality p. 261. 


mind. Abstract everything psychical, and then the remainder 
of existence will be Nature". 

A few lines farther on, he brings us to the crux of 
the present situation in physical science when he writes: 

"We sometimes forget that this world [of nature], in the 
mental history of each of us, once had no existence. There 
was a time when the separation of the outer world, as a thing 
real apart from our feeling, had not even been begun. The 
physical world, whether it exists independently or not, is, for 
each of us, an abstraction from the entire reality". 

A nineteenth-century physicist, reading this, would 
have identified the "time when the separation of the 
outer world had not even been begun" with a few days 
in his extreme infancy, and would little suspect that he, a 
scientist of mature years, had not yet effected the sepa- 
ration completely. It was left for twentieth-century 
physics under the lead of Einstein, Bohr and Heisenberg 
to discover how large a subjective tinge entered into the 
nineteenth-century description of nature; recognising 
this, it tries to discard our human spectacles and study 
the objective reality that lies beyond. only in this way 
has it proved possible to give a consistent description of 
nature. Thus the history of physical science in the 
twentieth century is one of a progressive ; emancipation 
from the purely human angle of vision. 

The physicist who can discard his human spectacles, 
and can see clearly in the strange new light which then 
assails his eyes, finds himself living in an unfamiliar 
world, which even his immediate predecessors would 
probably fail to recognise. 

We must now try to explain how this change of thought 
has come about, examine its implications, and describe, 


in so far as this is possible, the new world of twentieth- 
century physics. 

The World of Sense-impressions 

We may properly approach this world by imagining the 
entry into life of a child endowed with consciousness, 
with a mind capable of experiencing sensations and 
desires, and with a capacity for thought. 

At first it has no consciousness except of its own exist- 
ence; no knowledge of an outer world of nature, as 
something distinct from and clearly separated from itself, 
its thoughts and its sensations; no past experiences to 
form a background to its thoughts or with which to com- 
pare its present sensations. Gradually the passage of 
time provides past experiences, which memory fixes in its 
mind to form the needed background. It begins to view 
its sensations against this background, and discovers that 
they continually change. They fall into the two cate- 
gories of pleasurable sensations, which it desires to in- 
crease, repeat or perpetuate, and painful sensations, 
which it desires to diminish or avoid. Soon it makes the 
melancholy discovery that it cannot by its own volition 
make all its sensations pleasurable; it finds that it has 
needs, such as for food and warmth; when these are not 
adequately satisfied, its sensations are less pleasurable 
than when they were. These needs introduce it to the 
hard facts of life, for it finds they can only be satisfied 
from outside itself. Definite acts, such as sucking sugar 
or running a pin into its hand, produce still more acute 
sensations of pleasure or pain; the materials for these 
sensations, the sugar or the pin-point, also come to it 
from outside. 


From such experiences the child infers the existence of 
an environment which is not part of itself in brief, of 
an external world. It has every inducement to try to 
understand the workings of this external world, in which 
it believes all physical pains and pleasures to originate. 
It soon learns, when burnt, to dread the fire; once bitten, 
it is twice shy. Through such experiences, it finds law 
and order in the external world, and discovers the prin- 
ciple which it will describe in later years as the "uni- 
formity of nature" like causes produce like effects. 
Finally, in its efforts to understand the external world, 
it begins tentatively to endow this world with certain 
qualities, properties and occupants. The inference that 
an external world exists obviously stands on a higher 
level of probability than the conjecture that any special 
qualities, properties or occupants are associated with it. 

For the child has definite knowledge only of the sensa- 
tions in its own mind. If these originated solely in its 
own mind, it could choose to make them all pleasurable; 
since it cannot do this, it is on fairly safe ground in sup- 
posing that something external must exist to produce and 
control these sensations. on the other hand, the nature 
of this something can never be more than guessed. The 
child will never be able to test the absolute truth of its 
conjectures; the most stringent test available is that of 
their consistency with one another and with the phe- 
nomena which they attribute to the external world. 
Such a test may disprove, but can never prove. 

Throughout its whole life, the child will assume that 
an external world exists, and will make conjectures with 
a view to understanding its workings. When it does this 
in a logical and systematic manner, we call it a scientist. 


The child's sensations reach its mind through five chan- 
nels, which we call the five senses sight, hearing, 
smell, taste and touch. These all function in similar 
ways. Something external produces an impression on 
some part of the body the retina, the ear-drum, the 
nostrils, the palate or the skin and this impression is 
transmitted along a complicated nervous system to the 
brain. Up to this stage the impression has been conveyed 
by atomic changes, but it now crosses what we may de- 
scribe as the "mind-body" bridge, and when it appears 
on the other side, it is as a mental sensation, accompanied 
by such mental attributes as pleasure or pain, satisfac- 
tion or irritation, ecstasy or despair. 

The nerves may be compared to a number of tele- 
phone wires transmitting electric currents into a prison- 
cell, which suitable instruments subsequently metamor- 
phose into messages of sound, television, etc. The child 
is a prisoner inside the cell, and is doomed to remain a 
prisoner all its life. It can have no knowledge of the 
outer world except through the messages received over 
the wires. These may give truthful reports of the events 
occurring outside the prison cell, but its occupant will 
only be able to interpret them in terms of the contents of 
its cell, which consist of thoughts and sensations. A 
mind which is directly acquainted only with thoughts and 
sensations may be as little able to form a true picture of 
an outer world as a blind man is able to understand the 
beauty of a sunset or a deaf man to grasp the meaning of 
a symphony. Even a superior being coming direct from 
the outer world might still be unable to explain its 
nature to the prisoner, for the simple reason that they 
would have no common language in which to converse. 


Nevertheless, from the fragmentary messages which his 
senses send to him over his nerves, the prisoner may 
attempt to form a consistent picture of the external world 
for himself, in terms of the concepts with which his mind 
is familiar. Science merely attempts to build up such a 
picture in a systematic, organised way. 

The External World 

The first messages which the child receives from its senses 
teach it to regard the external world as a collection of 
objects, each possessing a certain degree of persistence or 
continuity in time. It soon finds that these fall into dis- 
tinct categories. First come other human beings, simi- 
lar to itself except for differences in age, size and other 
characteristics. There are also animals, birds, fishes and 
insects, then plants and trees, and finally objects which 
consist of inanimate matter. 

The child's mind is not only occupied by its sensations 
but also by its volitions, which are desires to increase or 
diminish particular sensations according as it finds them 
pleasurable or the reverse. Having discovered that its 
sensations come to it from the disposition of the objects 
of the external world, it would like to alter this dis- 
position, so as to avoid pain and increase pleasure. It 
finds, or thinks it finds, that it is possessed of a will-power, 
through which it may at least try to effect the changes it 

It soon discovers an essential difference between ani- 
mate and inanimate objects. After a little experience, 
it can catch a rolling marble without difficulty, because 
this has no will-power to set in opposition to its own, but 
as soon as it tries to catch a crawling fly or a crawling 


wasp, it becomes conscious of an opposing will-power; 
the fly tries to avoid capture, the wasp resents capture. 
Finally it finds that other children have a will-power of 
the same kind as its own. As it believes its will-power to 
emanate from its mind, it infers that the external world is 
controlled in part by minds other than its own, but simi- 
lar to its own; it concludes that it is not the only mind in 
the universe. 

When it establishes contact with these other minds, it 
learns that they experience sensations and desires similar 
to its own; not only are they endowed with similar senses 
but also, most important of all, they perceive objects 
similar to those which it perceives. 

Not only are these objects similar in kind; often they 
are obviously identical. If I count that there are six 
chairs in my room, the normal event will be for my com- 
panion also to count six chairs. Repeated experiences 
such as these suggest that the chairs he sees are identical 
with those which I see. The knowledge that a chair can 
be perceived by a mind is extended to the knowledge that 
the same chair can be perceived by two minds, and we 
conclude that the chairs have what we may call an 
"objective" existence an existence outside our indi- 
vidual minds. Something outside both of us, which we 
loosely describe as a chair, can produce in both of us the 
sense-impression we describe as seeing a chair. At this 
stage we naturally begin to inquire "What is this 
object which we call a chair?" We turn to the physicist 
for an answer, because he has devoted his life to investi- 
gating such problems. 



He tells us in the first place that all sense-impressions 
which come to us from the external world originate in 
what he calls "matter". This cannot of itself make a 
direct impression on our senses; such impressions are 
only made by physical "events" occurring in matter. 
Strictly speaking, we do not see the sun; we see events 
taking place in the sun. The sun only affects our senses 
because a continuous re-arrangement of electrons in the 
solar atoms results in the emission of light. In the same 
way, we do not see a chair, but the event of daylight or 
electric light falling on a chair. If we stumble against the 
chair in the dark, we do not feel the chair, but the event 
of a transfer of energy and momentum between the chair 
and our bodies. 

Both chronologically and causally the act of percep- 
tion starts at the end of the chain remote from the per- 
cipient in the sun, the electric light, or the chair. We 
must not, for instance, compare the act of vision, as 
Descartes did, to a poking about in space, as a blind man 
pokes about with a stick; the object is the starting-point, 
not the terminus, of an act of perception. 

A mental impression may be produced either by the 
activities of the mind itself, as when I dream, or by 
external events which originate in matter and subse- 
quently operate on my mind through my senses. When 
many of us experience the same, or very similar, mental 
impressions, we usually attribute them to external events. 
When only one person receives an impression, although 
others were equally in a position to do so if it had origi- 
nated in external events, we may safely attribute it to 


the activities of the percipient's own mind, stimulated 
possibly by events in his body, as with the nightmares of 
the man who has dined too well, or the waking illusions 
of the man who has drunk too well. 

Thus matter may be defined as that which is capable 
of originating objective sensations sensations which 
can be perceived by anyone who is suitably conditioned 
to receive them as, for instance, by sending rays of 
light into our eyes. The chairs in my room are material 
because my companion and I can both see them if we look 
in the proper direction with our eyes open. But if he 
claims to see red snakes or pink rats which I cannot see 
when I look where he directs me to look, I shall conclude 
that his sensations are peculiar to himself; the supposed 
snakes and rats are creations solely of his imagination, 
and do not consist of matter. For practical purposes, the 
test of the photographic plate is usually taken to be final. 
A hundred people may say they see an Indian climbing 
up a rope into heaven, but if a suitably exposed photo- 
graphic plate shews no image of the Indian and his rope, 
we refuse to classify these as material. 

In our less reflecting moments we are apt to claim 
a very intimate acquaintance with matter. Reflection 
shews through how many intervening stages our knowl- 
edge of it must come matter, events, effect on our 
senses, travel along our nerves, passage over the mind- 
body bridge before it reaches our minds. For this 
reason the matter in which events originate may often be 
very different from the matter we think we see or hear or 
feel all magic, conjuring and unconscious self-decep- 
tion rest on the possibility of this distinction. We may see 
or photograph a rainbow, but the light by which we do 


this does not originate in the rainbow we think we see; it 
originates in the sun, whose rays are reflected into our 
eyes or our camera by the drops of rain which make 
the rainbow. We could photograph a ghost if this con- 
sisted of moonlight reflected from a white curtain, but 
the light which affected our photographic film would not 
come from a disembodied spirit, but from the sun. 

Primary and Secondary Qualities of Matter 

Even when my companion and myself both see an un- 
mistakably objective chair, the sensations which this pro- 
duces in him will never be quite identical with those it 
produces in me. This may be due in part to our looking at 
the chair from different positions, but even if we look at it 
|ja succession from the same position, there will still be 
differences. My perception of the chair owes something 
to the chair, but something also to myself. 

The philosophers, who took this question in hand be- 
fore there was much exact scientific knowledge to guide 
them, proceeded by discussing all objects and material 
substances in terms of certain characteristic qualities or 
properties with which they were supposed to be endowed. 
A chair, for instance, was supposed to be possessed of 
hardness, brownness, squareness, and so on; sugar of 
hardness, sweetness and whiteness. They divided these 
Dualities into two categories which they labelled as 
primary and secondary, or sometimes with a different 
shade of meaning as substantive and adjective. In 
brief, the secondary or adjective qualities were "sense- 
qualities", which made, or could make, a direct appeal 
to our senses. Such qualities might vary with the con- 
ditions of perception, or with the state of the senses of the 


percipient; sugar might look white on one occasion, but 
yellowish or greyish on another, when it was viewed 
in a different light, or by a sick man. These secondary 
or adjective qualities were supposed to result from certain 
primary or substantive qualities, which were not directly 
perceived in themselves, but persisted independently of 
the perceiver, and so also of his idiosyncrasies. These 
existed in their own right, even when the object was not 
perceived at all; they were the residue after all the sec- 
ondary qualities had been stripped away, the bedrock 
underlying the ever-shifting sands of appearance. These 
primary qualities could only exist attached to some sub- 
stratum or foundation of real substance. 

There is no obvious a priori justification for dividing 
qualities into two sharply defined categories in this way, 
and neither does science know of any. And as no clear- 
cut division can be found in practise, there has been no 
general agreement as to which qualities were primary 
and which were secondary. 

Descartes, for instance, maintained that the only 
primary qualities were extension in space and motion 
"Give me extension and motion and I will construct the 
Universe". Locke, relying on Newton's teaching that 
an unchangeable mass was associated with every object, 
added mass to the list. Others have maintained that ex- 
tension in space is the only primary quality, and that all 
the observed qualities of objects emanate from this. In 
a later chapter we shall see how the theory of relativity 
has shewn that neither mass nor motion nor extension in 
space can qualify as true primary qualities. They de- 
pend one and all on the special circumstances of the per- 
cipient, so that the mass, motion and size of a body are as 


much secondary qualities as the brownness of a chair or 
the whiteness of sugar. Thus the theory of relativity 
makes it clear that if primary qualities exist, we must 
commence the search for them afresh. 

Long before the days of relativity, Bishop Berkeley 
(1685-1753) and his school of thought held that there 
were no primary qualities at all, or, more precisely, that 
there was no real distinction between primary and 
secondary qualities. They maintained that an object 
was nothing more than the sum of the impressions it 
made in our minds, so that it had no existence at all ex- 
cept in so far as it was perceived by a mind or existed in a 
mind; nothing had more substance than the things we 
see in a dream. This led to a philosophy of idealism or 
mentalism, to use a more modern term according to 
which all matter, as ordinarily understood, is an illusion; 
nothing exists in reality except mind. 


Let us now start the search for primary qualities anew, 
rnaVi'ngr use o f our scientific knowledge of the physical 
structure and properties of matter. 

Many Greek philosophers, from Democritus onward, 
had imagined matter to consist, in the last resort, of hard 
indivisible pellets, each of which possessed in itself all the 
characteristic properties of the substance. These pellets 
were at first called atoms (d-r^im?, incapable of being 
divided), but are now known as molecules. Gold, for 
instance, was supposed to be hard and yellow because it 
consisted of hard yellow atoms; it appeared yellow, not 
because our eyes saw it yellow, but because it was yellow 
in itself. Yellowness was a primary quality of gold. . 


The atomic hypothesis remained little more than a 
philosophic speculation until the eighteenth century, 
when John Dalton shewed how it illuminated and ex- 
plained Lavoisier's work on the foundations of chemistry. 
It gained still further in vigour in the second half of the 
nineteenth century, when Maxwell and others shewed 
how it gave a simple and natural explanation of many of 
the known properties of gases. It has now become an 
essential ingredient of physical science. 

It is known that an object may be either a homogene- 
ous mass of a single substance, such as water, or a com- 
bination or mixture of different substances, as for instance 
a cup of tea. Here the cup may consist of a single sub- 
stance known to the public as china, and to science as 
kaolinite, while the tea inside it is a mixture of water and 
tea, with perhaps sugar and milk. It is found that every 
ample substance, such as water or china, is formed of 
exactly similar molecules, each of which possesses the 
same chemical properties as the substance as a whole. 
Even a small amount of the substance consists of a vast 
number of molecules; a china tea-cup will consist of 
about a hundred thousand million million million mole- 
cules of kaolinite, and can contain an even greater 
number of molecules of water. 

Each molecule is built up of still simpler units, to which 
the name "atom** has now been transferred. Chem- 
istry, which has methods for resolving all known sub- 
stances into their constituent atoms, finds that all mole- 
cules are combinations of only 90 kinds of atoms, al- 
though reasons of an abstract kind suggest that two others 
will probably be found in time, and possibly even a few 


The atoms themselves are in turn built up of still 
simpler units. There are believed to be only two kinds 
of these, known as protons and electrons. Both are 
charged with electricity, the charge on each electron 
being the same in amount as that on each proton but of 
the opposite sign; it is conventionally agreed to describe 
the charge on the proton as positive, and that on the 
electron as negative. The protons stay permanently at 
the centre of the atom, where, in combination with a 
certain number of electrons, they form the compact 
structure we describe as the "nucleus" of the atom. 
Outside this are more electrons, most of which are kept 
near to the nucleus by the attraction of opposite kinds of 
electricity for one another, although the outermost are 
gripped so loosely that they may easily become detached 
from the atom to which they belong. 

Wherever an atom contains more protons than elec- 
trons, its total charge is positive, and it attracts further 
negative electricity to itself from outside, in the form of 
electrons, until the excess charge is neutralised. only 
when this has occurred is the atom in its normal per- 
manent state. Thus the normal atom must always con- 
tain just as many electrons as protons. The simplest 
atom of all, that of hydrogen, contains only one proton 
and one electron; the next simplest, that of helium, con- 
tains four electrons and four protons; the atom of oxygen 
contains sixteen of each, and so on. 

These electrified protons and electrons form the basic 
units of which all material objects are built. The 
' physical properties of a particular substance are deter- 
mined by the way in which these units, or their combi- 
nations the atoms or molecules are arranged. 


If, for instance, these are spaced widely apart, it is easy 
to crush them closer together, and we say that the sub- 
stance is soft or yielding. If they are already so close to- 
gether that a great deal of pressure is needed to get them 
still closer, we say the substance is hard. Thus diamond 
is hard, but carbon and lamp-black, which consist of sim- 
ilar atoms in more open spacing, are relatively soft. 

Again, the 18 protons and 18 electrons which form a 
molecule of water are so arranged that they do not ob- 
struct the passage of light; hence water is colourless and 
transparent. on the other hand, the 258 electrons and 
258 protons which form a molecule of kaolinite are 
arranged in such a way that very little light can pass 
through. As a consequence light which falls on a 
kaolinite surface is merely turned back not regularly, 
like light reflected from a mirror, but irregularly and in 
all directions, like the water splashed from a wall on 
which a fire-hose is playing. If we look in the right direc- 
tion, it is as certain that some of this light will enter our 
eyes, as it is that we shall get wet if we stand near the 
wall. White light, such as sunlight, is a mixture of lights 
of all colours, so that when sunlight falls on kaolinite, a 
mixture of lights of all colours is reflected back into our 
eyes, and we say that the kaolinite looks white. on the 
other hand, when kaolinite is illuminated by blue light, 
it can only reflect blue light because there is no light 
of other colours for it to reflect and so looks blue. We 
see that the whiteness of china in sunlight is a property of 
the illumination rather than of the substance itself. The 
same is true of other substances, such as paper and linen, 
which look white in sunlight; all these merely assume the 
colour of the light by which they are illuminated. 


Other substances have distinctive colours of their own. 
For instance, the redness of a rose is not a mere quality 
of the illumination by which we see it. Its petals absorb 
light of all colours except red, but any red light which 
falls on them is splashed back and may enter our eyes. 
When we see the rose in ordinary sunlight, nothing enters 
our eyes but red light, and we say that the rose looks red. 
on the other hand^ if it is illuminated by blue light, there 
is no red light to be turned back into our eyes, so that it 
looks colourless or black. In the same way a man who is 
colour-blind to red will see and describe a red rose as 
colourless or black, in all lights. For the rose can send no 
light into his eyes except red light, and this can make no 
impression on his mind. Thus the redness of a rose de- 
pends on three factors a redness in the rose itself, 
a redness in the light by which it is illuminated, and 
a capacity for seeing redness on the part of the per- 

This may seem to suggest that colour is a secondary 
quality of objects, because it depends on the senses of the 
percipient. Science is, however, possessed of a colour- 
scale which is entirely independent of the imperfections 
of human perceptions. We shall see later how light con- 
sists of waves of different lengths. In normal eyes the 
longest waves produce the colour-impressions we describe 
as various shades of red and orange, the shortest produce 
shades of indigo, violet and blue, while those of inter- 
mediate lengths produce shades of yellow and green; in 
abnormal eyes they may of course produce other im- 
pressions. Thus, although our sense-estimation of colours 
may be partly subjective, we can measure the exact 
lengths of the waves which constitute light, and so obtain 


a perfectly definite, perfectly precise and perfectly ob- 
jective scale of colour. 

A succession of waves in which crests and troughs 
occur at perfectly regular intervals is described as a uni- 
form train of waves, and the distance between any two 
successive crests, or any two successive troughs, is known 
as its "wave-length". The scientist describes light as 
being of a perfectly pure colour, or "monochromatic 35 , 
when it consists entirely of waves of one uniform wave- 
length. In general he will not say that light is red, jxcept 
as a brief and convenient way of making a rough state- 
ment; in his more scientific moments he will speak of light 
of wave-length say .00006562 cm., and in so doing will 
specify a precise shade of colour in a way which is per- 
fectly objective, and is limited only in precision by the 
number of decimals he uses. The lights by which we 
ordinarily see things sunlight, electric light, candle- 
light are all mixtures of waves of different lengths. 
They may be specified as made up of various pure colours 
of light, each specified by its wave-length, combined in 
stated proportions. The instrument known as the spec- 
troscope actually effects the analysis for us, dividing up 
any kind of light into its constituent pure colours. The 
simplest spectroscope of all is a drop of water; a more 
powerful spectroscope is formed by a multitude of drops 
of water, such as the dew on the grass or the shower, 
which break up sunlight into the many-coloured light of 
the rainbow. In this we see all the pure colours arranged 
in the order of their wave-lengths red, orange, yellow, 
green, blue, violet, indigo. 

These two examples shew that if we still wish to divide 
the qualities of an object into primary and secondary, the 


existence and mode of arrangement of its protons and 
electrons must be held responsible for, and indeed consti- 
tute, the primary qualities of an object; such qualities as 
colour result from these in conjunction with die special 
circumstances under which the object is perceived. Yet, 
underlying every red we perceive, there is a true ob- 
jective red associated with either the object we perceive 
or its illuminant. 

Mechanism of Sense-perception 

Before we can study the properties which objects possess 
in their own right, we must learn how to allow for the 
special circumstances of the perceiver and the act of 
perception. This makes it important to understand how 
external objects act on our five senses. 

The senses of smell and taste are affected by direct 
contact. When I say I smell ammonia, I mean that mole- 
cules of ammonia are entering my nose and being ab- 
sorbed by its membranes, thereby affecting certain nerves 
which transmit a message to my brain. This message 
produces the sensation I describe as smelling ammonia. 
The process of tasting is similar; to taste sugar I must 
place particles of sugar in contact with my palate; the 
absorption of these particles sends a message to my brain 
which produces the sensation I describe as a sweet taste. 
Touch also operates through direct contact; I do not feel 
an object until there is actual contact between part of it 
and my skin. 

on the other hand, we hear distant objects without 
their coming into contact with our sense-organs. When I 
hear a bell, it is not through bits of the bell striking my 
car-drums; it is through waves of sound, which the bell 


initiates, striking against my ear-drums. The vibrations 
of the bell set up vibrations in the surrounding air, and 
these set my ear-drums also into vibration. This produces 
the sensation which I describe as hearing the sound of a 
bell, although actually it is feeling the effect of waves of 
condensation and rarefaction of the air inside my ears. 
All sounds are heard by a similar process. 

Thus three of our senses smell, touch and taste 
perceive an object by direct contact, while a fourth, hear- 
ing, perceives an object by means of the waves it excites in 
a medium of communication, which is usually the air. 
Howdoes the fifth sense of sight operate? The obvious but 
superficial answer is that it operates through light falling 
upon a part of our bodies, the retina, which is sensitive to 
light; but this merely raises the further question: What 
is light? The story of efforts to answer this question forms 
a very long chapter in the history of science. 

The Nature of Light 

The outstanding and most superficially obvious property 
of light is its tendency to travel in straight lines we all 
are familiar with the straight outline of the beam of a 
searchlight, and the straight shafts of light which the sun 
shoots through a hole in the clouds, and we all shield our 
eyes from a strong light by interposing an opaque object. 
This led the early scientists to suppose that light consisted 
of a shower of small particles, emitted from a luminous 
object like shot from a gun. Newton adopted this view 
and elaborated it in his Corpuscular Theory of Light; he sup- 
posed that we see the sun because it is continually throw- 
ing off little bits of itself, some of which enter our eyes 
just as we smell ammonia through its continually throw- 


ing off little bits of ammonia, some of which enter our 

Yet it proved exceedingly hard to fit all the facts of 
observation into such a theory. It is found that a big 
object casts a shadow, inside which everything is pro- 
tected from the light, just as though rays of light were like 
gun-shot. on the other hand, a very small object affords 
no such protection; the rays of light bend round it and 
re-unite behind, so that there is no region of perfect 
shadow to which the light does not penetrate at all. Now 
this property of bending round an obstacle is one which 
we associated with waves, rather than with projectiles. 
When a gun is fired, an intervening obstacle may save us 
from being hit by the shot, but it will not save us from 
hearing the noise of the gun. This is because sound 
travels in the form of waves, and waves can bend round 
an obstacle. 

This similarity between light and sound led scientists to 
suppose that light, like sound, must consist of waves. 
Just as we hear a bell because it sends out waves of sound, 
so, it was thought, we see the sun and a candle-flame be- 
cause they send out waves of light. This concept formed 
the basis of the Undulatory Theory of Light, which regarded 
light as consisting of waves. Newton, who had consist- 
ently maintained that light was of the nature of particles, 
opposed this theory, but after Fresnel had disposed of his 
objections, the theory was developed in great detail, and 
was found to explain all the facts which the corpuscular 
theory had failed to explain, as well as many other known 
properties of light. Throughout the greater part of the 
eighteenth and nineteenth centuries, no single fact was 
known to be in opposition to the Undulatory Theory, and 


it was regarded as providing a final and complete expla- 
nation of the nature of light. 

It has since become clear that the explanation was 
neither final nor complete. We now know that there was 
a great amount of truth in the old corpuscular theory of 
light, and the corpuscular and undulatory concepts of 
light must be regarded as complementary rather than 
antithetical. Viewed in one aspect, light has all the 
appearance of waves; but viewed in another aspect, it has 
the appearance of particles somewhat as a comb may 
look like either a row of points or a solid bar, when viewed 
from different directions. 

We shall see below (pp. 162, 189) that there is a single 
self-consistent mathematical description of light which 
accounts for all its known qualities, both wave-like and 
particle-like. But for the moment we can only describe 
the nature of light by analogies. 

A partial although only partial, and in many ways 
misleading analogy is provided by an ordinary swell 
at sea. In a sense this consists of waves, but in another 
sense of particles the molecules of the sea. The 
analogy is misleading because sea-waves admit of an 
objective description which shews that, quite apart from 
our observation of them, they consist of waves and of 
particles at the same time. This is not so with light. It 
can be viewed so as to look like either particles or waves, 
but never like both. In so far as we make it assume the 
properties of particles, we make it shed those of waves, and 
vice-versa. And when we discard our human spectacles 
entirely, we find that light is neither waves nor particles. 

In another respect, however, the analogy is a good one. 
We may regard the water of the sea either from a sta- 


tistical, or from an individual, aspect. Statistically it 
consists of waves, but individually of molecules. In the 
same way, when light is viewed statistically, it exhibits 
many of the properties of waves; when viewed individ- 
ually, of particles. A very intense light may be treated 
as consisting of waves, but we find it necessary to think 
of a minute amount of light as consisting of separate 
particles. Because nineteenth-century science did not 
concern itself with such minute amounts, it found the 
undulatory theory satisfactory; it could treat light as a 
continuous stream. But the minute amounts which are 
so important to twentieth-century science may more 
properly be compared to shot fired from a gun, almost 
exactly as the old corpuscular theory supposed. We shall 
discuss all this more fully in a later chapter (p. 215). 

If, then, we regard light as consisting of particles, we 
may say we see the sun because it is firing shot at us. We 
have seen how the material structure of the sun consists 
of atoms, which are in turn built up of protons and 
electrons. It is, however, neither atoms nor protons nor 
electrons which the sun shoots off; there is a further con- 
stituent to all matter, which we call energy, without 
knowing in the least what it is. It may exist either asso- 
ciated with matter, or as "free" energy not attached to 
matter. Energy may pass from one piece of matter to 
another, but it may also break loose from matter entirely, 
and travel through space as free energy, when we describe 

it as radiation. 


If we regard light as consisting of particles, we must 
regard the particles as consisting of energy. These parti- 
cles of freely travelling energy, or bullets of radiation, 


are known as "photons 95 . Each photon has associated 
with it a definite mathematical quantity of the nature of 
a length, and when this quantity has the same value for 
every member of a swarm of photons, the swarm as a 
whole is found to shew many of the properties which 
would be shewn by waves having this as the distance from 
crest to crest of successive waves. For this reason this 
quantity is usually described as the "wave-length" of the 
photons. We shall see what it means if we think of ordi- 
nary radio waves, which are of course merely a special 
kind of radiation, characterised by having a specially long 
wave-length. An average transmitting aerial sends out 
about 10 32 photons* every second, each having a "wave- 
length" of, say, 500 metres. only a minute fraction of 
this torrent of photons falls on a distant receiving aerial, 
and yet even this fraction consists of so many individual 
photons that it may be treated as a continuous stream; 
this stream behaves like a succession of waves of wave- 
length 500 metres. 

Like all other forms of energy, photons possess the 
property of inertia or mass. For this reason they exert 
pressure on anything they strike, here again behaving like 
shot from a gun. A regiment of men could be mown 
down by a sufficiently strong light just as surely as by the 
stream of shot from a machine gun. The sun discharges 
about 250 million tons of energy every minute. on die 
corpuscuia*view this consists of tiny massive bullets 
travelling at 1 86,000 miles a second. Some of these enter 
our eyes, and, impinging on our retinas, transfer their 

*10 3 * means the number 10000..., in which 32 zeros follow the 
initial digit 1. Also 10-'* means unity divided by 10. Thus 
10~ 0-000001.. 


energy to our optic nerve, and give us the sensation we 
describe as seeing the sun. The filament in an electric 
light bulb discharges somewhat similar photons, although 
in this case only at the rate of a fraction of an ounce per 
million years. Some of these, entering our eyes directly, 
strike our retinas, and we see the filament; others falling 
on our tables and chairs are turned back from these to 
pass on to our retinas, and we say we see our tables and 
chairs by electric light. Thus seeing is similar to smelling, 
except that the distance is traversed by photons, which 
are bullets of energy, instead of by molecules, which are 
bullets of matter. 

Yet the mechanism of sight is far more intricate, and 
gives us far more detailed knowledge, than that of smell. 
The molecules which affect our sense of smell travel over 
zig-zag paths as they are buffeted about by other mole- 
cules, and so reach our noses from all directions; we 
cannot usually say that a smell comes from a certain 
direction, but merely that the air is pervaded by a smell, 
or, at best, that the air which reaches us from a certain 
vague direction is so pervaded. Photons differ from 
molecules in that they do not interact with one another; 
nothing but matter can stop a photon, or deflect it from 
its course. Thus photons travel through empty space in 
straight lines, and we know the direction from which 
light reaches us with the utmost precision. Just as the 
lens of a camera arranges that all photons which come 
from the same direction shall be thrown on to the same 
point of a photographic plate, and so produce a picture 
of the world outside the camera, so the crystalline lens 
of the eye arranges that all photons which arrive in the 
same direction, and so come from the same object, shall 


fall on the same spot of the retina. In this way the light 
falling on our retina constitutes a sort of picture of all the 
objects which are affecting our vision at any instant, and 
we see these objects arranged in the right order relative to 
one another. 

When we smell several objects at the same time, we are 
conscious of little more than an unassorted medley of 
smells; we speak of the smell of the East, or the smell of 
a ship, without being able to enumerate the separate con- 
stituent smells. It is much the same with our palates; we 
taste the dish rather than its separate ingredients, which 
are known only to the cook. Our ears do somewhat 
better for us. When we hear a number of sounds simul- 
taneously, our ears analyse the resultant sound into its 
constituent tones of different pitch; it is in this way that 
we recognise individual voices and separate musical 
instruments, that the ordinary ear can concentrate on the 
voice of a companion to the exclusion of much louder 
sounds, and that the trained musical ear can analyse a 
chord into its constituent notes. But our eyes form 
enormously less effective analysing instruments than any 
of these. They can only inform us as to the direction 
from which light arrives, and have no capacity at all for 
analysing a beam of mixed light into its constituent 
colours. . 

Just as there are sounds too deep or too acute for us to 
hear, so there are photons which we cannot see. Some 
are of too short a wave-length to be seen; none of our 
sense-organs apprehends theSe directly, although they 
may make painfiil burns on our skin. Others are of too 
long a wave-length to be seen; many of these represent 
heat rather than light, and their impact on our skin tells 


us of the warmth of the sun or the heat of a fire. We 
see, then, that our sense of touch can perceive photons 
as well as material objects. 

The Outer World 

Thus all our five senses act in essentially the same way; 
something ponderable from the outer world something 
of which we can say that its weight is so-and-so comes 
into contact with our sense-organs. We feel, taste and 
smell sugar by the direct contact of our skin and mem- 
branes with small particles of sugar. We hear a bell when 
particles of air, set into rhythmic motion by the bell, 
strike upon our ear-drums. We see the sun by certain of 
the photons which it emits striking our retina. We feel its 
heat by certain other of its photons impinging on our skin. 

In general, then, we may say that we experience the 
outer world through small samples of it coming into con- 
tact with our sense-organs. The outer world consists of 
matter and energy; samples of this outer world consist 
of molecules and photons. 

Yet not all samples of the outer world affect our sense- 
organs. Our ear-drums are affected by ten octaves, at 
most, out of the endless range of sounds which occur in 
nature; by far the greater number of air-vibrations make 
no effect on them. Our eyes are even more selective; 
speaking in terms of the Undulatory Theory of Light, 
these are sensitive to only about one octave out of the 
almost infinite number which occur in nature. 

It is often maintained that, as we cannot experience the 
whole of nature, we can never hope to understand it. 
Animals exist whose senses are very different from our 
own; bats and cats are said to hear and see different 


things from ourselves, while dogs obviously smell different 
things. The world must seem very different to them. In 
the same way, if the sensitiveness of our organs were 
shifted to different ranges, or if we were endowed with 
other senses in place of these we now possess, or if our 
present meagre channels of communication with the outer 
world were opened wider, this world would seem very 
different to us. We can at best, so the argument runs, 
view the world through coloured spectacles which shut 
off all light except of those colours to which our senses are 
attuned beings which could experience the full light of 
day would give a very different account. 

Laboratory Data 

Science has of course provided us with methods of ex- 
tending our senses both in respect of quality and quantity. 
We can only see one octave of light, but it is easy to 
imagine light-vibrations some thirty octaves deeper than 
any our eyes can see. While philosophy is reflecting how 
different the world would appear to beings with eyes 
which could see these vibrations, science sets to work to 
devise such eyes they are our ordinary wireless sets. 
We also have means for studying vibrations far above 
any our eyes can see. Actually a range of vibrations 
extending over about 63 octaves can be detected and has 
been explored 63 times the range of the unaided eye. 
And even this limit is not one of the resources of science, 
but of what nature provides for us to see. In the same 
way, the spectroscope makes good the deficiency of our 
eyes for analysing a beam of light into its constituent 
colours, and further enables us to measure the wave- 
length of each colour of light to a high degree of accuracy . 


Science has extended the range and amplified the 
powers of our other senses in similar ways, in quality as 
well as in quantity. We cannot touch the sun to feel how 
hot it is, but our thermocouples estimate its temperature 
for us with great accuracy. We cannot taste or smell the 
sun, but our spectroscopes do both for us or at any rate 
give us a better acquaintance with the substance of the 
sun than any amount of smelling or tasting could do. We 
are entirely wanting in an electric sense, but our galva- 
nometers and electroscopes make good the deficiency. 

Nevertheless, no one would claim to be able to imagine 
all the kinds of senses that we might possibly possess, or 
maintain that science has provided us with substitutes for 
them all. We can imagine beings who could neither see, 
hear, smell, taste nor touch,' and yet were endowed with 
other senses, of kinds not only unknown to us but totally 
unimagined by us. Would their world be at all like ours? 

The reply is that the instruments of research provided 
by physical science disclose a fairly self-contained region 
of phenomena. We may properly suppose that a reality, 
which we may describe as the physical universe, underlies 
it. Whether this is the whole of reality is a matter for 
debate. Some biologists, for instance, believe that this 
domain includes the whole domain of biology; others 
prefer to think that a connecting passage-way leads from 
this domain of physics to a whole new domain of life. 
Again, those of a purely materialistic outlook maintain 
that the domain of physics comprises the whole of reality, 
while those who believe in the reality of a world of the 
spirit the poet, the artist, the mystic are at one 
in believing that there are other domains than that of 


In their more irrational moments, these latter may feel 
inclined to maintain that these other domains are entirely 
distinct from that of physics; that there are no connecting 
ways between them, so that the universe is not one but 
many. Such a contention can hardly survive serious 
reflection. The artist may often claim that his creations 
are on a higher plane" than the purely physical, but he 
can hardly claim they are totally disconnected from it; 
no one knows better than he how much his imaginings 
depend on the state of his physical health and the con- 
dition of his physical tools and instruments. No poet will 
write quite the same "Ode to Joy" when he has a cold 
in his head as when he has not. And the preacher who 
has just told us how hard it is for the rich man to enter the 
Kingdom of God must not, at any rate in the same sermon, 
tell us that worldly riches are on a different plane from, 
and entirely unconnected with, the Kingdom of God. 

For the moment let us merely remark that physical 
science is competent to discuss these questions. If pas- 
sage-ways connect the domain of physics with the domains 
of life or of spirit, physics ought in time to discover these 
passage-ways, for they start from her own territory. 
When physicists are urged to investigate the claims of 
psychical science to produce ectoplasm, to speak with a 
"direct voice", to agitate tables and produce other ma- 
terial phenomena by non-material means, they are in 
effect being invited to decide as to the reality or not of 
alleged channels of communication of precisely this kind. 

The Study of Nature 

It will be convenient to conclude the present chapter by 
reviewing, very briefly, the history of man's efforts to 


understand the workings of the external world. We may 
distinguish three broad epochs, the nature of which may 
be suggested by the words animistic, mechanical and 

The animistic period was characterised by the error of 
supposing that the course of nature was governed by the 
whims and passions of living beings more or less like man 
himself. Before our infant can distinguish between ani- 
mate and inanimate objects, he is destined to pass through 
a stage of confusing the two. He will fail to catch the 
rolling marble just as he failed to catch the crawling fly, 
and will assign the same reason like the fly the marble 
was anxious not to be caught. He will trap his finger in 
the door, and attribute his sorrows to the naughtiness of 
the door. Because personality is the concept of which he 
has most immediate and direct experience, he begins by 
personifying everything. 

As the history of the individual is merely the history 
of the race writ small, our race did much the same in its 
infancy as its individuals still do in theirs. Sometimes 
they endowed the inanimate objects of nature with wills 
of their own, sometimes they supposed them governed 
by the caprices of gods, goddesses, wood-nymphs and 
demons. A storm at sea was not the result of a depression 
moving eastward from the Atlantic, but of Poseidon and 
Boreas playing schoolboy jokes on their fellow-Immortals, 
or possibly even interfering in human affairs. As Mene- 
laos is dragging the naturally reluctant Paris to slaughter 
by his helmet, the chin-strap gives way not because its 
tensile strength was unequal to the pull of the indignant 
husband, but because (or so Homer tells us) Aphrodite 
herself loosens the strap as a return for favours previously 


conferred on her in the shape of a golden apple. This 
anthropomorphic fallacy permeated man's whole view of 
nature, as it still does in primitive races until scientific 
knowledge supersedes it. Such views of nature were un- 
reflecting and almost instinctive, arising in part from 
man's projecting his own personality on to nature, with a 
resultant confusion between man and nature, and in part 
from a mere fixation of infantile ideas. 

Then in Ionian Greece, six centimes before Christ, the 
human intelligence began consciously to apply itself to 
the study of nature. It felt very little desire to increase its 
factual knowledge of nature, so that Greek science con- 
sisted in the main of mere vague questionings and specu- 
lations as to why things came to be as they were rather 
than otherwise. 

It was not until the time of Galileo that science turned 
from cosmology to mechanics, and from speculation to 
experiment. The simplest way of affecting inanimate 
matter was to push it or pull it by means of muscular 
effort. So long as men could only experiment with 
objects which were comparable in size with their own 
bodies, they found inanimate nature behaving as though 
its constituent pieces exerted pushes and pulls on one 
another, like those we exert on them by the action of our 
muscles. In this way the science of mechanics came into 
being. Pieces of matter were supposed to exert "forces" 
on one another, and these forces were the causes of the 
motions of the bodies in question, or rather, the changes 
in their motions. And it was found that the behaviour of 
every object was determined, entirely and completely, by 
the pushes and pulls to which it was subjected; there was 
no longer any room for the intervention either of gods or 


of demons. A chin-strap broke as soon as the pull on it 
exceeded its tensile strength; no number of golden apples 
given to Aphrodite could have made it break sooner or 
later. The wind became high and the sea rough as soon 
as the barometric gradient exceeded a certain intensity 
and so on. Bodies moved just as they were pushed or 
pulled by other bodies; nothing else mattered. 

Science, having established these laws for objects of 
tangible size, went on to imagine that they governed the 
whole of nature. Thus when Newton (1687) had ex- 
plained the motions of comets by mechanical concepts, he 
expressed a wish that the whole of nature might in time 
be explained on similar lines. Three years later, Huygens 
described the principle which was to guide physical 
science wrongly as we now know for the next two 
centuries, in the words:* 

" In true philosophy, the causes of all natural phenomena 
are conceived in mechanical terms. We must do this, in my 
opinion, or else give up all hope of ever understanding any- 
thing in physics ". 

Closely connected with this view of the workings of 
nature was the principle described as "The uniformity of 
nature". This asserted that, when the same experiment 
was performed any number of times on exactly similar 
objects in exactly similar circumstances, the result was 
necessarily always the same. The simple explanation was 
of course that the bodies under observation were sub- 
jected to the same pushes and pulls on the various 
occasions, and so behaved in the same way. Science ad- 
mitted no exceptions to this uniformity; alleged violations 

* Traiti de la Lumiere (Leyden, 1690), Chapter 1. 


of it were adjudged to be miracles, frauds or self-decep- 
tions according to circumstances and the mentality of 
the judge. And just because observation and everyday 
experience seemed to establish this principle so firmly, 
scientists were wholly convinced that their simple me- 
chanical explanation of it was the true one bodies 
moved just as they were pushed or pulled by other bodies. 

Causality ', Determinism and Free-Will 

This view of nature was soon seen to have far-reaching 
implications, and to raise far-reaching questions. When 
one object pushed or pulled another, the conditions pre- 
vailing at the moment determined the intensity of the 
push or pull. Science had now discovered that the inten- 
sity of this push or pull, and this alone, determines the 
ensuing motion of the affected object, which in turn 
determines the conditions that will prevail at the next 
moment, and so on. Thus conditions at one moment de- 
termine those at the next, these determine the conditions 
at the succeeding moment, and so on ad infnitum. The 
universe appears as a mere machine, wound up to go like 
a machine and destined to run down like a machine. Its 
whole future is inherent in its state at any moment, just 
as this state must have been inherent in its state at its 

This of course supposes that there is no intervention 
from outside, as, for instance, by the will-power of living 
things. Is it, however, conceivable that will-power should 
intervene? If the closed system of nature provides no 
opening for the activities of gods and goddesses, is it likely 
to leave a loophole for the similar activities of animals and 
men? The physiologists tell us that the brain is part of, 


and continuous with, the body; apart from reflexes, the 
atoms of our brains direct the motions of our bodies. on 
the mechanical view of nature, these atoms must move 
precisely as they are pushed and pulled about like the 
atoms of a motor car. The butcher kills a lamb, and im- 
mediately the brain which had just been directing the 
creature's leaps and bounds, becomes mere mechanical 
matter sheep's brains, to be thrown on the scales and 
sold by the pound. The metamorphosis has been accom- 
plished without the loss or gain of a single atom. Why, 
then, should the quality of the pushes and pulls on the 
atoms of the brain change so abruptly just at the moment 
when the mind leaves the body? 

The obvious suggestion is that these atoms experience 
pushes and pulls from mental as well as from material 
sources in brief that our volitions can affect the atoms 
of our brains, and through them the atoms of our bodies. 
The plain man accepts this without even pausing to con- 
sider any alternative. He is quite sure that his mind is, 
within limits, free to guide his body, so that he can keep 
his appointments, and put a X where he pleases on his 
ballot-paper. He beats his dog for not coming when he 
whistles for it, and sends the forger to gaol because his 
fingers have written someone else's name at the foot of a 
cheque. All this provides him with a self-consistent 
scheme which is agreeable not only to his intuitions, but 
also to his moral sense. 

The challenge to this scheme did not come, in the first 
place, from science but from philosophy; it originated 
with Descartes (1596-1650). His philosophy regarded 
mind and matter as entirely independent "substances", 
each existing in its own right apart from the other, and of 


such essentially different natures that they could not 
possibly interact the one, for instance, existed in space, 
the other out of space. He accordingly thought of mind 
and matter as moving, so to speak, on parallel yet entirely 
different sets of rails, completely without interaction, and 
yet synchronised after the manner of a cinematograph 
film and its "talkie" accompaniment, so that the appro- 
priate mental thoughts, moods and emotions always 
come at the right moment to suit the prevailing arrange- 
ment of atoms and the associated events. 

A child seeing a speaking film for the first time might 
well think the words were the natural outcome of the 
events occurring before its eyes; it would be hard to 
believe that words which seemed to fit the events so natu- 
rally had been planned to match long in advance. So it 
is with our thoughts and the atoms of our world; they not 
only seem to match, but to. emanate one from the other. 
Descartes, however, insisted, as in a different way did 
Leibnitz at a later date, that at the first morning of 
creation, a supremely benevolent God had miraculously 
arranged for a perfect and continuous synchronisation 
between bodily and mental events. Faith cannot really 
move mountains, because one is imponderable and the 
other so very ponderable, but the good God lets us think 
it can. 

Descartes accordingly compared the body to "an 
earthly machine" actuated by a sort of reflex mechanism: 

"You may have seen in the grottoes and fountains which are 
in our royal gardens, that the force with which the water moves 
when issuing from its source is of itself enough to set various 
machines in motion, and to make various instruments play or 
utter words, according to the different arrangements of the 


tubes which convey the water. We may compare the nerves 
of the machine which I am describing with the tubes of the ma- 
chines of these fountains, the muscles and tendons with the 
other engines and springs which move the machines, and the 
animal spirits, the source of which is the heart and of which 
the cavities of the brain are the reservoirs, with the water which 
sets them in motion. Moreover, breathing and similar acts, 
which are natural and usual to the machine, and depend on 
the flow of the spirits, are like the movements of a water-clock 
or mill, which the ordinary flow of water keeps continually in 
motion. External objects, which by their presence act on the 
sense-organs of the machine and so determine it to move in 
different ways according to the disposition of the parts of the 
brain, are like strangers who enter one of the grottoes and are 
themselves the unwitting cause of the movements they witness. 
For on entering they tread on certain tiles or plates which are 
so arranged that if they approach a bathing Diana they cause 
her to hide in the rose bushes, and if they try to follow her they 
cause a Neptune to come towards them threatening them with 
his trident. Or if they pass in another direction they make a 
sea-monster spring forward and spout water in their faces, or 
things of a like kind according to the caprice of the engineers 
who constructed them". 

He further, and quite inconsistently, compared the mind 
to an engineer in a control tower, who, by manipulating 
taps, could change the course of the water from one 
pipe to another with a minimum of effort, and went 
dangerously near to repudiating his own philosophy when 
he conjectured that the whole mechanism of our bodies 
was worked by "animal spirits 9 ' which were, he said, like 
"a very subtle air 5 *, the subtlety being so marked that they 
were on the verge of ceasing to be material. Later and 
more scientific writers have remarked that even in a 
completely mechanical world, large amounts of energy 
can have their courses changed by the expenditure of a 


minimum of energy, as, for instance, in moving railway 
points or turning over an electric switch. 

These and many other elaborate fabrications arose out 
of a natural desire to avoid the implications of a me- 
chanical view of nature. So long as his science appeared 
to tell him that his intuitive beliefs were erroneous, and 
the actions he based on them irrational, even the scientist 
himself could not but feel that his studies were divorced 
from reality, or at best were concerned only with a small 
corner of reality which had but little connection with his 
everyday life. Bradley, writing in 1899, summed up the 
current feeling in the words:* 

"Nature to the common man is not the Nature of the 
physicist; and the physicist himself, outside his science, still 
habitually views the world as what he must believe it cannot 

And, even as late as 1926, we find Whitehead writing: f 

"The Western people exhibit on a colossal scale a pecu- 
liarity which is popularly supposed to be more especially 
characteristic of the Chinese. Surprise is often expressed that 
a Chinaman can be of two religions, a Confucian for some 
occasions and a Buddhist for other occasions. Whether this is 
true of China I do not know; nor do I know whether, if true, 
these two attitudes are really inconsistent. But there can be 
no doubt that an analogous fact is true of the West, and that the 
two attitudes involved are inconsistent. A scientific realism, 
based on mechanism, is conjoined with an unwavering belief 
in the world of men and of the higher animals as being com- 
posed of self-determining organisms. This radical inconsist- 
ency at the basis of modern thought accounts for much that is 
half-hearted and wavering in our civilisation. It would be 

* Appearance and Reality, p. 262. 

t Science and the Modern World, p. 94. 


going too far to say that it distracts thought. It enfeebles it, 
by reason of the inconsistency lurking in the background. 
After all, the men of the Middle Ages were in pursuit of an 
excellency of which we have nearly forgotten the existence. 
They set before themselves the ideal of the attainment of a 
harmony of the understanding. We are content with super- 
ficial orderings from diverse arbitrary starting-points. For 
instance, the enterprises produced by the individualistic energy 
of the European peoples presupposes physical actions directed 
to final causes. But the science which is employed in their 
development is based on a philosophy which asserts that 
physical causation is supreme, and which disjoins the physical 
cause from the final end. It is not popular to dwell in the 
absolute contradiction here involved. It is the fact, however 
you gloze it over with phrases". 

Clearly a science which involved such implications 
and entailed such consequences was in need of a new 
background, such as should reconcile the nature of the 
laboratory and the text-books with the nature of every- 
day experience. Happily it has acquired such a new 
background within recent years. 

The New Physics 

Throughout the mechanical age of science, scientists had 
proceeded on the same general lines as the child and the 
unreflective savage. Out of the impressions registered 
through their senses, they had built an inferential world 
of objects which they believed to be real, and affected by 
events of much the same kind as occurred in their every- 
day experience. They described this as the "common- 
sense" view of science; and defined science as "organised 
common-sense". Any scientific theory which could not 
be explained in terms of the familiar concepts of everyday 
life was said to be contrary to common-sense, and could 


hope for but a cold and unsympathetic reception, either 
from laymen or scientists. Then new refinements of 
experimental technique brought new observational 
knowledge, which shewed that the workings of nature 
could not be explained in terms of the familiar concepts of 
everyday life. New and unfamiliar concepts were found to 
be necessary; the age of common-sense science had passed. 

Had science continued to pursue its old methods, it 
might have tried to draw concrete pictures of these new 
concepts. Some scientists indeed tried, by introducing a 
multitude of small changes here and there, to modify the 
old view of nature so that it could meet the new demands 
upon it. But they were trying to confine new wine in old 
bottles; their efforts met with no success, and the main 
stream of scientific thought followed a very different 
course. For it was just about this time that science, 
mainly under the guidance of Poincare, Einstein and 
Heisenberg, came to recognise that its primary, and 
possibly its only proper, objects of study were the sensa- 
tions that the objects of the external universe produced 
in our minds; before we could study objective nature, we 
must study the relation between nature and ourselves. 
The new policy was not adopted of set purpose or choice, 
but rather by a process of exhaustion. Those who did not 
adopt it were simply left behind, and the torch of knowl- 
edge was carried onward by those who did. 

This new line of advance has led us to a science which 
is no longer in flat contradiction with our intuitions and 
the experiences of everyday life; the physicist need no 
longer feel that his laboratory door divides his life into 
two watertight compartments as scientist and as 
human being. In particular, mechanism, with its im- 


plications, has dropped out of the scheme of science. The 
mechanical universe in which objects push one another 
about like players in a football scrimmage has proved to 
be as illusory as the earlier animistic universe in which 
gods and goddesses pushed objects about to gratify their 
own caprices and whims. We are beginning to see that 
man had freed himself from the anthropomorphic error of 
imagining that the workings of nature could be compared 
to those of his own whims and caprices, only to fall head- 
long into the second anthropomorphic error of imagining 
that they could be compared to the workings of his own 
muscles and sinews. Nature no more models her be- 
haviour on the muscles and sinews of our bodies than on 
the desires and caprices of our minds. 

Whether determinism has also been banished from 
nature is still a question for debate. We shall see later 
that the answer is probably something more subtle than a 
mere "Yes" or "No"; possibly we could make either 
answer true by suitable definitions of determinism and 
nature. But that those particular causes which seemed 
until recently to compel determinism have gone this is 
hardly open to question. 

We shall see the fundamental contrast between the old 
science and the new very clearly if we compare the begin- 
ning of Newton's Principia, in which the mechanistic view 
of nature was first put in perfect logical form, with the 
beginning of Dirac's Quantum Mechanics, which represents 
the most complete exposition of the new theory of Quanta 
at present in existence. 

Newton wrote in 1687: 

"Every body perseveres in its state of rest, or of uniform 
motion in a right line, unless it is compelled to change that 


state by forces impressed thereon. The alteration of motion 
is ever proportional to the motive force impressed . . ."; 

and Dirac in 1930: 

"Whetf an observation is made on any atomic system that 
has been prepared in a given way and is thus in a given state, 
the result will not in general be determinate, i.e. if the experi- 
ment is repeated several times under identical conditions 
several different results may be obtained. If the experiment 
is repeated a large number of times it will be found that each 
particular result will be obtained a definite fraction of the total 
number of times, so that one can say there is a definite proba- 
bility of its being obtained any time the experiment is per- 



We have already noticed the inadequacy of the definition 
which describes science as organised common-sense. We 
ought perhaps rather to define it as organised knowledge. 
Such a definition makes it clear that the first stage in the 
development of any science must necessarily be the accu- 
mulation of facts. The facts may be either particular or 
universal. Some sciences, such as botany and pathology, 
still find it important to record exceptional and unusual 
occurrences which at first sight appear to form exceptions 
to the general scheme of nature. In the more exact and 
more highly developed sciences, such as physics and 
astronomy, there are none such to record; here nature 
appears to be governed by immutable laws. The aim of 
science is to discover and interpret these laws. 

Scientific Synthesis 

When a sufficient number of facts have been collected in 
any particular branch of science, the next stage is to try 
and cover them all by a general principle, which may or 
may not admit of an explanation in terms of familiar 
concepts. To be ultimately satisfactory, such a general 
principle or explanation must not only cover all the facts 
already known, but also all the facts which remain to be 
found out. It is accordingly first put forward in the form 
of a hypothesis. A scientist says in effect "Observation 
shews that the following facts are true; I find that a cer- 



tain hypothesis as to their origin is consistent with them 
all". He and his colleagues may now set to work to 
obtain more accurate or extensive data bearing on the 
original facts, or entirely new facts may be discovered. 
The hypothesis may be tested by examining whether the 
extended and new facts can be covered, as the old were, 
by the proposed general principle or explanation. When 
two separate and conflicting hypotheses are in the field, it 
is sometimes possible to devise an experimentum cruets to 
decide between them. Suppose it can be shewn that if 
hypothesis A is true, a phenomenon X will occur, and that 
if hypothesis B is true, the phenomenon X will not occur. 
Then we can decide between the two hypotheses by per- 
forming an experiment, or taking an observation, to find 
whether the phenomenon X occurs or does not. 

Interrogating Nature 

Such an experiment, like every other, amounts in. effect 
to asking a question of nature. This question can never 
be "Is hypothesis A true?" but "Is hypothesis A ten- 
able?" ^Nature may answer our question by shewing us 
a phenomenon which is inconsistent with our hypothesis 
or by shewing us a phenomenon which is not inconsistent 
with our hypothesis. She can never shew us a phenome- 
non which proves it; one phenomenon is enough to dis- 
prove a hypothesis, but a million million do not suffice to 
prove it. For this reason, the scientist can never claim 
to know anything for certain, except direct facts of 
observation. Beyond this, he can only proceed by build- 
ing up hypotheses^ each of which covers more phe- 
nomena than its predecessor, but each of which may have 
to give place to another hypothesis in due course. Strictly 


speaking, the time for replacing a hypothesis by a claim to 
certainty never arrives. 

We have just considered the simplest possible instance 
of the process of interrogating nature. It is not always 
possible to frame a question which permits only of the 
answers "Yes" or "No 35 . More difficult problems arise 
when the experimenter is deceived by hypotheses of his 
own imagining, and tries to obtain an answer to a non- 
sensical question. If his experiment can be carried 
through at all, it must give an answer of some kind, but 
the answer, when it comes, may well seem as nonsensical 
to the questioner as we can imagine the original question 
did to nature. 

For instance, let us imagine a race of men equipped 
with perfect scientific instruments, but with very little 
scientific intelligence or knowledge. They see a rainbow 
in the sky, and wish to discover how far away it is. Treat- 
ing it as though it were a piece of stage scenery, they 
instruct a party of surveyors to discover the distance of 
their cardboard rainbow. Observations taken with per- 
fect instruments give a precise and unequivocal answer 
the distance is minus 93,000,000 miles. Those in authority 
might decapitate their surveyors for incompetence, or 
harangue against the untrustworthiness of the observa- 
tional method "It is absurd to suppose that a distance 
can ever be negative, and anyhow 93,000,000 miles is 
preposterous, since the foot of the rainbow obviously lies 
between us and the mountain over there 5 '. But let them 
instead change the form of their question to nature, and 
express it in the form "How far in front of us is the source 
of the light we see in the rainbow?" and the answer, minus 
93,000,000 miles, becomes full of significance. The pre- 


liminary minus now tells them that the source of light does 
not lie in front of them at all, but behind them, and as its 
distance is 93,000,000 miles they can at once identify it 
with the sun. It is frequently more difficult to frame a 
sensible question than to obtain an answer to a nonsen- 
sical one. And if the question was not rightly framed in 
the first place, it may be inconceivably difficult to inter- 
pret the answer aright. 

To avoid the dullness and indefiniteness of a general 
discussion, let us step across forthwith to two particular 
instances, both of which will figure largely in the dis- 
cussion that is to follow. 

Astronomy and Relativity 

The Greeks and Egyptians had collected a great array of 
facts concerning the apparent motions of the sun, moon 
and planets across the sky. About A.D. 150 Ptolemy of 
Alexandria attempted to cover them all by a single hy- 
pothesis. Contrary to the earlier views of Aristarchus of 
Samos and the Pythagoreans, he imagined the earth to 
form a fixed centre to the whole system, while sun, moon 
and planets revolved round it, the sun and moon revolv- 
ing in circles, but the planets in a complicated system of 
cycles and epicycles. No new facts were brought forward 
to test this hypothesis, but in A.D. 1543 Copernicus 
brought forward an alternative hypothesis which ap- 
peared to explain the same facts in a simpler way; he 
supposed the sun, instead of the earth, to be the centre of 
the solar system, and earth, moon and planets to describe 
circles round it, the motions of the planets still being 
complicated to some extent by epicycles. 
Two hypotheses were now in the field, and Copernicus 


devised an experimentum crucis to decide between them. 
If Ptolemy's hypothesis were correct, Venus could never 
appear as less than a half-circle of light. on the other 
hand, if Venus circled round the sun, its appearance, 
as seen from the earth, ought to shew phases like that of 
the moon, varying from a full circle down to a crescent 
as thin as that of the new moon. In 1609, the newly 
invented telescope provided the means of asking nature to 
decide between the two hypotheses. As soon as Galileo 
saw Venus appearing as a thin crescent of light, he knew 
that Ptolemy's hypothesis was untenable. 

This did not of course establish the truth of the 
Copernican hypothesis. Indeed, new and more precise 
facts began to accumulate which threw doubt upon it 
In particular, Kepler studied the motion of Mars in some 
detail, and found that this was inconsistent with the 
Copernican hypothesis. This led him to propound the 
new hypothesis that the planets did not move round the 
sun in cycles and epicycles, but in ellipses having the sun 
as common focus. For a time this hypothesis fitted all the 
facts known to astronomy. 

Half a century later, Newton tried to combine these 
and other facts under the cover of a still wider hypothesis. 
He imagined every object in the universe to attract every 
other object with a force, the force of gravitation, which 
varied inversely as the square of the distance between the 
two objects, and supposed that the planets moved merely 
as these forces compelled them to move. He shewed that 
this hypothesis explained the elliptical orbits of the 
planets, and an immense range of other facts and phe- 
nomena as well the motion of the moon round the 
earth, the fall of an apple to the ground, the parabolic 


trajectory of the cricket ball in flight, and even the ebb 
and flow of the tides. Finally it was found to account for 
the motion of comets. These fearsome and mysterious 
apparitions, which had hitherto been dreaded as portents 
of evil 'or symbols of divine displeasure, were now shewn 
to be mere chunks of inert matter, driven to describe 
paths round the sun by exactly the same forces as pre- 
scribed the orderly motions of the planets. 

New data continued to accumulate, all of which fitted 
into Newton's theory, until in the middle of the nineteenth 
century the astronomer Leverrier found a discrepancy in 
the motion of the planet Mercury. Newton's hypothesis 
required a planet continually to repeat its path round the 
sun; it ought to describe the same ellipse again and again 
like a small boy's engine running round and round the 
same track. Leverrier found that Mercury did not do 
this, but described an ellipse which itself turned round in 
space once every three million years or thereabouts. It 
was as though the track on which the toy locomotive ran 
was itself laid on a turntable, which slowly rotated in 
space while the locomotive ran rapidly round the track. 

In time Einstein propounded yet another new hypoth- 
esis, the theory of relativity, which not only explained 
all the phenomena which Newton's theory of gravitation 
had previously explained, but also gave an accurate 
account of the motion of Mercury, and explained a great 
number of other scientific facts as well. It was possible to 
devise experiments and observations to provide crucial 
tests between the new theory of Einstein and the older 
theory of Newton, and in every case nature ruled out the 
latter and decided in favour of the new theory. Other 
crucial experiments were designed to compare the new 


theory with the physical theories then prevailing, such as 
that light was propagated as waves in an all-pervading 
ether, and that electric and magnetic forces were trans- 
mitted as pressures and tensions through such an ether. 
Again, nature decided in every instance in favour of the 
theory of relativity. To-day, Einstein's theory provides 
an explanation of an enormous range of natural phe- 
nomena, and no single fact of nature is known to be in- 
consistent with it. 

The general aim of science is to progress towards, and 
ultimately achieve, such theories. We can never say that 
any theory is final or corresponds to absolute truth, be- 
cause at any moment new facts may be discovered and 
compel us to abandon it. Although this seems unlikely, 
facts as yet undiscovered may in time compel us to 
abandon the theory of relativity. But even if this occurs, 
the time spent in constructing it will not have been 
wasted; it will have provided us with a stepping-stone to 
a still wider theory, which will fit still more of the phe- 
nomena of nature. The layman sees Science, as it seems 
to him, for ever changing her mind, hesitating, turning 
back on her tracks, and repudiating her earlier opinions. 
The scientist sees her ever progressing through a suc- 
cession of theories, each of which covers more phenomena 
than the predecessor it displaced, towards the goal of a 
single theory which shall embrace all the phenomena of 
nature. If such a theory is ever attained, it will give us a 
hypothetical scheme of the external world which will be 
capable of reproducing all the phenomena of the external 
world and no others. 


Atomic Physics and Quantum Theory 

Before turning to discuss the significance and value of 
such a scheme, let us take a second example of scientific 
progress, drawn this time from physical science. 

When a mass of hydrogen gas is raised to incandescence 
whether in the atmosphere of a hot star, or in an 
electric discharge in a terrestrial laboratory the pho- 
tons it emits prove to be of many different kinds, which 
can be specified by many different and distinct wave- 
lengths. A spectroscope will sort out the photons accord- 
ing to their wave-length, much as a potato-sieve sorts out 
potatoes according to their size, but with incomparably 
greater accuracy; the wave-length of the hydrogen pho- 
tons can be measured to an accuracy of about one part 
in a hundred thousand. 

There are reasons for thinking that each individual 
photon comes from a single hydrogen atom, which is 
believed to consist of one proton and one electron. For a 
long time it was difficult to see how so simple a structure 
could emit photons at all. The electron and proton were 
believed to be mere electrified particles which attracted 
one another according to the inverse square law. In this 
case, current theories of electric action shewed that the 
electron would describe an ellipse round the far more 
massive proton just as a planet does round the sun 
and would emit a continuous stream of radiation in so 
doing. There was therefore the primary objection that 
the emission of radiation would be gradual and not by 
complete photons. There was also the further objection 
that a gradual emission of energy would cause a gradual 
shrinkage in the size of the atom so that, contrary to 


observation, there could be no definiteness either in the 
size of the atom or in the quality of the photons it 

In 1913, Dr. Bohr of Copenhagen put forward a hy- 
pothesis which seemed for a time to dispose of all these 
difficulties. He supposed that hydrogen atoms could exist 
in a great number of different but quite distinct states, 
different amounts of energy being associated with each. 
There could be no gradual transitions between these 
states, but the atom might occasionally jump discon- 
tinuously from one to another, giving out energy in the 
form of a complete photon as it did so. 

Some years later, Franck and Hertz of Gottingen 
obtained direct experimental evidence that such distinct 
states really existed. They found that when electrons 
collided with atoms, the latter might either take up cer- 
tain large amounts of energy from the electrons, or none 
at all; they never took up a small amount of energy, so 
that a continuous dribble of energy was a fortiori impos- 
sible. The encounters were like a series of commercial 
transactions; money changed hands at each, but always 
by complete coins, so that each individual always had a 
certain number of complete coins in his pocket; fractions 
did not come into the question at all. And the amounts 
of energy associated with the different states were found 
to be precisely those required by Bohr's hypothesis. 

Although this hypothesis was never quite consistent 
logically, it seemed to fit all the facts as known at the 
time. Then more refined measures of the wave-lengths of 
photons were obtained, and it was found that these did 
not completely agree with the predictions of the hypoth- 
esis. The hypothesis predicted the right results for the 


hydrogen atom under ordinary conditions, but the wrong 
results when the atom was put between the poles of a 
powerful magnet. It also gave wrong results for the 
normal helium atom, which is the simplest atom of all 
after hydrogen. 

Recently a new hypothesis, forming what is known as 
the "new quantum theory", has removed at a single 
stroke both the logical difficulty and the whole of the 
observational discrepancies. The new theory is purely 
mathematical in form, dealing only with measurable 
quantities and the relations between them, but it admits 
of several physical interpretations. The best known of 
these, generally described as "wave-mechanics", supposes 
that electrons and protons are not mere particles of hard 
matter, as had previously been imagined, but that 
much in the same way as photons they possess many 
of the properties of waves. 

Unlike Bohr's older hypothesis, this new hypothesis 
assigns to the atom properties which are in no way in- 
consistent with the inverse-square attraction of its elec- 
trons and protons; rather they are additive to it. The 
great merit of the new hypothesis is, however, that its 
predictions agree exactly with observation in every case 
in which comparison has been found possible. To con- 
sider the case of hydrogen light alone, it is probably an 
under-statement to say that twenty kinds of photons can 
have their wave-lengths measured to one part in a hun- 
dred thousand, and that in every case the measures agree, 
to within one part in a hundred thousand, with the values 
predicted by the new quantum theory. Now if this were 
a perfectly random hypothesis, having no relation at all 
to truth, only a piece of astounding good luck could 


enable it to predict even a single wave-length to an 
accuracy of one part in a hundred thousand; indeed there 
would be odds of something like a hundred thousand (10 5 ) 
to one against. The odds against the same luck holding 
good for a run of twenty wave-lengths would be some- 
thing like 10 100 to one against. Such at least would be the 
case if the wave-lengths were not inter-connected. Actu- 
ally there is a certain inter-connection, since both the 
wave-lengths demanded by theory and those observed 
in practice fall into regular series. This circumstance 
obviously calls for a large reduction in the odds just men- 
tioned, yet, even so, they remain enormously large 
unthinkable millions to one against the agreement being 
a mere chance coincidence. 

It should be added that the new quantum theory goes 
far beyond the explanation of the hydrogen-spectrum, or 
indeed of spectra of any kind; it explains a great number 
of phenomena in many departments of physics which had 
previously defied explanation, while not a single fact of 
observation is known to be inconsistent with it. Again, 
we see science moving towards a hypothesis which will 
cover all known facts with complete accuracy if in- 
deed it has not already attained such a goal. 

The Search Jar Reality 

Suppose, however, that two or more hypotheses prove 
equally well able to explain the whole range of phe- 
nomena. This is not a mere flight of fancy; in some re- 
stricted branches of science the electromagnetic field 
equations, for instance such a situation exists to-day* 
Is the scientist to rest content with two distinct and 
possibly inconsistent hypotheses, or shall he try to dis- 


cover which of the two comes nearer to the realities of the 
external world? 

The answer must of course depend to a large extent on 
what we regard as the ultimate aim of science. What is it 
that urges one set of scientists to spend arduous lives in 
discovering new facts to destroy old hypotheses, while 
another spends even more arduous lives in framing new 
hypotheses, destined to be destroyed in their turn by yet 
newer facts of observation? Up to now, the raison d'etre 
of science has been irrelevant to our discussion. 

Part of the value of science is of course utilitarian; it 
enriches our lives, and shews us how to live more com- 
fortably and more happily in brief, it lessens our pains 
and increases our pleasures. This is the obvious extension 
of the rudimentary science by which the one-day-old 
child tries to adjust itself to the hard facts of life. 

Part of the value of science is intellectual. It would be 
a dull mind that could see the rich variety of natural 
phenomena without wondering how they are inter- 
related. Quite apart from all questions of practical 
utility, the modern mind feels strongly urged to synthesise 
the phenomena it observes, to try to combine happenings 
in the external world under general laws. This impels 
Karl Pearson to describe the function of science as "the 
classification of facts, the recognition of their sequence 
and relative significance". In the same spirit, Einstein 
writes: "The object of all. science is to coordinate our 
experiences and bring them into a logical system". This 
view of the aims of science may take very extreme forms, 
as for instance when Dirac says that "the only object of 
theoretical physics is to calculate results that can be com- 
pared with experiment" in other words to gratify 


intellectual curiosity, since otherwise it would be simpler 
to gain the required knowledge from the experiments 

These views regard science as being concerned solely 
with the phenomena of nature; the underlying reality 
from which the phenomena originate does not come into 
the question at all. And indeed many specifically main- 
tain that the phenomena and their laws constitute the 
whole province of science science, in brief, is concerned 
with what happens, not with what is. They hold that 
when science has included all phenomena in one single 
all-embracing hypothesis, she has run her course and 
nothing more remains for her to do. If two or more such 
hypotheses are in the field, well and good; either of them 
satisfies all requirements, and it is impossible to escape 
from the prison-house of the senses to discover which of 
the hypotheses agrees most closely with the external 
world. If we had a single picture which represented all 
the phenomena quite perfectly, we should have no means 
of investigating whether it represented reality or not. 

Such considerations as this prove quite convincingly 
that we can have no certain knowledge of reality. They 
do not, however, touch the question of knowledge of 

For instance, we ask the question, "Can we know that 
the new quantum theory gives the true origin of the 
hydrogen spectrum?" The argument quoted above gives 
the answer, "No; we can know nothing of the external 
world", and a very satisfying answer it is to one who 
does not wish to go any further. Science amplifies the 
answer, saying "No; we can know nothing of the external 
world for certain. At best we can only deal in probabilities. 


Yet the predictions of the new quantum theory agree so 
well with the observed spectrum of hydrogen that the 
odds in favour of the scheme having some correspond- 
ence with reality are enormous. Indeed, we may say the 
scheme is almost certain to be quantitatively true; that is 
to say, true to reality in those features which it is impos- 
sible to alter, in any way whatsoever, without destroying 
the numerical agreement of the theory with observation". 
A probability which reaches so close a proximity to 
certainty is generally good enough for the ordinary affairs 
of life. It is far better than the kind of probability which 
lawyers describe as "good enough to hang a man on". 
Indeed, as Laplace remarked with reference to another 
scientific problem, it is far better than the probabilities 
in favour of the best attested events in history. We are 
accustomed to accept as an indisputable fact that Queen 
Anne is dead. The metaphysical argument that we can 
have no certain knowledge of anything beyond the con- 
fines of our prison-house, because we cannot go there to 
see, will of course prove that we cannot know this for 
certain. Indeed, it will take us as much further than this 
as we like; it can prove the impossibility of knowing that 
Queen Anne ever lived. But if we assume that she once 
lived, then no conceivable calculation can make the odds 
that she is dead anything like the odds of 10 100 to 1, which 
we had occasion to mention just now. We may then 
argue that there is a better justification for supposing that 
the scheme we just discussed for the origin of the hydrogen 
spectrum is true in its numerical essentials than for 
supposing that Queen Anne is dead. 

This particular argument only shews that we can ac- 
quire knowledge of numerical essentials i.e. of factors 


which cannot be altered without destroying numerical 
agreement with observation. Other arguments might 
conceivably be devised to shew that other factors in 
reality can be known, at least to a high degree of 

Yet the possibility of acquiring knowledge of ultimate 
reality is obviously restricted by considerations which 
have already been mentioned. We cannot claim to have 
knowledge unless we can explain it to other beings with 
minds like our own. And we cannot explain, and so can- 
not know, the ultimate nature of external things except 
in the a priori improbable event of these proving to be of 
the same nature as something with which our knowing 
minds are familiar. For otherwise there is no standard 
of comparison, no language in which to describe it, for 
language can only describe experiences we have in com- 
mon. Trying to explain reality, whether to ourselves or 
to one another, would be like trying to explain a wireless 
outfit to a savage. He would have no difficulty in under- 
standing the phenomena, the voices or music that issue 
from the set, for he is accustomed to voices and music. 
He may even understand the atmospherics, for he is 
accustomed to thunder. Our troubles begin when we 
try to explain grid-bias, tuned circuits and high-tension 
batteries to him. And, except in the a priori improbable 
event just mentioned, we must expect to encounter similar 
difficulties when we try to explain reality either to our- 
selves or to others. It is this kind of difficulty, rather than 
the bleak metaphysical argument that we can have no 
certain knowledge of what lies beyond the confines of 
our prison-house, that constitutes the true barrier to 


Pictures of Nature 

What we have so far described as a hypothesis might 
equally well be described as a picture, or a representation, 
or a model, of nature. It does not attempt to portray the 
reality of nature, but only what we see of nature the 
phenomena of nature. It may reproduce all the phe- 
nomena within our cognisance with perfect fidelity, and 
yet may differ from reality in its essence just as much as 
a photographic print differs from a living face colour, 
extension in a further dimension and all vital qualities 
may be lacking. The elements of this picture are neces- 
sarily concepts with which our minds are familiar, other- 
wise we could not have drawn the picture at all. on the 
other hand, the elements of reality need not be so, and if 
we cannot make our minds familiar with such elements as 
can exist in reality, we shall never understand reality. 
But it would seem that science might legitimately pro- 
gress along the road from phenomena to reality by thinking 
over unfamiliar concepts until they become familiar, the 
concepts being selected in the first instance on grounds of 
probability, as appearinglikely to figure in ultimate reality. 
For instance, when the intelligent child is first told that 
the world is round, it at once protests that, if it were, the 
people on the far side would fall off; at a later age the con- 
cept of a round earth presents no difficulties. In the same 
spirit, the Victorian physicist used to say that he could 
never understand a physical concept of which he could 
not make a working model; Lord Kelvin explained that 
this was why he could not get hold of the electromagnetic 
theory of light. Yet the average physicist of to-day has 
somehow contrived to acquire a tolerably clear under- 


standing of this theory, not because of any inborn intel- 
lectual superiority but because he has thought about these 
concepts for longer. For the same reason, he has formed 
a far better picture of four-dimensional space and of 
tensors of the second rank than his predecessor of a gener- 
ation ago. And there is no reason why the process should 
not continue indefinitely; after all, it is merely a contin- 
uation of the process by which the scientist has already 
differentiated himself from his medicine-man predecessor. 

Our minds are indisputably familiar with the concept 
of quantity, and from this we can pass by an easy transi- 
tion to the mathematical treatment of quantity. When 
we say that there are two constituents to a hydrogen 
atom, namely the electron and proton, the words hydro- 
gen atom, electron and proton merely represent some- 
thing in our phenomenal world. The concept two may, 
however, be common both to this world and the world of 
reality. Thus it is not foolish to suppose that if the above 
statement were translated from the language of phe- 
nomena into that of reality, the concept "two" would 
figure in the reality. 

Again, our minds are familiar with a succession of 
changes in time, since our sensations continually experi- 
ence such changes. Thus there is no a priori reason 
against our obtaining knowledge of measurable quantities 
in reality and of the way they change with the progress of 
time. (The possible a posteriori objection that neither 
time nor measurable quantities may ultimately prove to 
exist in reality need not concern us at present.) More- 
over, the mathematician can prove, by an argument 
which assumes nothing as to the nature of external reality, 
that all changes with time can be pictured in terms of 


wave-motion; this concept will, then, enable us to picture 
such changes. If a certain kind of wave-motion seems 
capable of describing something in reality to a very high 
degree of probability, we may proceed to discuss the 
further question "Waves of what?" 

Here, for the first time, we are confronted with diffi- 
culties, since the real essence of the "What" must neces- 
sarily remain unknown to us, unless it should prove to be 
of the same general nature as something already existent 
in our minds, such as a thought or mental concept, a wish 
or an emotion. 

To anticipate for a moment, we shall find later that the 
waves which are most important of all in physics can 
quite unexpectedly be interpreted as being of this type. 
They are waves of something which the scientist loosely 
describes as "probability", but may be more explicitly 
described as "uncertainties or imperfections of knowl- 
edge" a concept with which our minds are only too 
familiar. This may create a suspicion that our minds 
have merely forced a priori upon the waves one of the very 
few interpretations which a posteriori they would be able 
to comprehend. This may be so, and other and less 
easily intelligible interpretations may be possible, but in 
any case the "probability" interpretation fits the facts of 
observation. Given the waves, we know the probabilities, 
so that, in a sense, the waves really are waves of proba- 
bility. Some may wish to interpret this as shewing that 
these waves have no existence in reality at all, but merely 
in our imperfect knowledge of reality. 

This, however, brings IK right up to the question which 
has been lurking in the background all the time "What 
is reality?" I think it is possible that science and philoso 


phy would answer this question in slightly different ways* 
The metaphysician is, I think, more inclined to regard 
reality and phenomena as detached and distinct, like a 
man and his image in a mirror, or an aeroplane and its 
shadow on the ground: to use a number of grotesque 
expressions, an entity may have either an ontal or phe- 
nomenal existence, but nothing in between. on the other 
hand, the scientist is more inclined to regard reality and 
phenomena as the two ends of a continuous road, along 
which it is his job to travel. The metaphysician may dis- 
miss the statement that waves really are waves of proba- 
bility as ignorant nonsense, while the scientist applauds it 
as a step towards final truth. 

Other types of waves may not prove so easily intel- 
ligible as those we have just been discussing. Yet even 
here we may perhaps be able to discover certain proper- 
ties which we may then try to visualise in terms of familiar 
concepts until finally our progress is stopped by some- 
thing which we can neither picture, imagine nor describe. 
The Victorian physicist, for instance, used to picture 
light-waves as similar to the shakings of a jelly, or the 
waves of an earthquake, until he found that his picture 
did not agree with the facts of observation. 

Again, we are familiar with the concepts of space, 
extension in space, limited extension in space, and so by a 
process of abstraction can pass to the concept of a parti- 
cle. If our picture of nature proves to consist in part of 
particles, we again ask "Particles of what?" and may or 
may not be able to arrive at a partial answer. 

Finally, we are familiar with the concept of mechanism 
through the interaction of our volitions and the muscles 
of our bodies. 


It might conceivably have proved possible to picture 
the whole external world, completely and perfectly, in 
terms of familiar concepts such as waves, particles and 
mechanism; indeed nineteenth-century physics aimed 
consciously and deliberately at such a representation, not 
sufficiently realising how great the odds were against its 
being possible. 

Had the attempt succeeded, science was all ready 
to identify the representation with the reality. Indeed, 
most scientists did this without waiting to see whether the 
representation could be made to fit all the facts of obser- 
vation. It was usual to assert at this time that all dis- 
crepancies were sure to be cleared up in time, and those 
who taught science seldom allowed any other possibility 
to enter the mental field of vision of their pupils. Behind 
the scientists whole schools of philosophers, realists and 
materialists, were identifying reality with particles, waves 
and so forth out there in space. The few others who 
urged that neither the known facts nor any possible facts 
could compel or warrant any such identification were felt 
to be valiant defenders of a lost cause. Their voices 
passed almost unheeded, not because they could not 
prove their case, or because their opponents could prove 
a case against them, but because the probabilities at that 
time seemed overwhelmingly against them. 

Our present observational knowledge shews that no 
representation of this kind can fit the phenomena, so that 
the question of identification with reality does not arise* 
The external world has proved to be farther removed 
from the familiar concepts of everyday life than nine- 
teenth-century science had anticipated, and we are now 
finding that every effort to portray it brings us up im- 


mediately against concepts which we can neither picture, 
imagine, nor describe. We have already seen that radia- 
tion cannot be adequately portrayed either as waves or as 
particles, or in terms of anything that we can imagine, 
and we shall soon find that the same is true also of matter. 

Subjective Nature 

The very real difficulties of modern physical science 
originate, in large degree, in the facts just cited. Physical 
science set out to study a world of matter and radiation, 
and finds that it cannot describe or picture the nature of 
either, even to itself. Photons, electrons and protons have 
become about as meaningless to the physicist as #, y> z 
are to a child on its first day of learning algebra. The 
most we hope for at the moment is to discover ways of 
manipulating x,y, z without knowing what they are, with 
the result that the advance of knowledge is at present 
reduced to what Einstein has described as extracting one 
incomprehensible from another incomprehensible. 

Apart from this, science knows of only one way of 
proceeding so as to avoid a complete deadlock. Dividing 
the world up into (a) ourselves, (b) our experiments on 
the external world, and (c) the external world, it can 
leave off concerning itself with (c), and can concentrate 
on (b) 9 our knowledge of the world as disclosed by experi- 
ments which we ourselves perform. The metaphysical 
argument mentioned above (p. 57) will suggest one 
obvious advantage of this procedure; it is that our knowl- 
edge of (c) can never consist of more than probabilities, 
whereas that of (b) will consist of certainties. But there is 
an even more immediate gain. However little we may be 
able to know the ultimate reality of external nature, and 


however unintelligible the imagined reality may be, the 
results of the experiments we perform on nature must 
necessarily be both knowable and expressible in terms of 
familiar concepts, since if the concepts had not previously 
been familiar, the experiments themselves would have 
made them so. 

For instance, we experiment with light, and obtain 
results which are expressible in terms of the familiar con- 
cepts, waves and particles. The experiments do not tell 
us what the true nature of light is; they do not, for in- 
stance, tell us that it consists of waves, or of particles. 
They merely shew us light behaving in a way which 
reminds us sometimes of waves and sometimes of particles. 
We infer that the whole nature of light cannot be ex- 
pressed by either of the words particles or waves, and as 
we do not know of any common object which is some- 
times like waves and sometimes like particles, it may be 
that the true nature of light is for ever beyond our powers 
of imagining; quite certainly it is so now. Thus we can- 
not reason about light, only about the results of our 
experiments on light, 

It is much the same with electrons and protons. We 
experiment with these (cf. frontispiece), and find that 
their behaviour reminds us sometimes of waves and some- 
times of particles. As with light, one has yet imagined 
a consistent picture of what the electron and proton 
really are. At present the most we can do is to express 
quantitatively and in mathematical terms the prop- 
erties of electrons and photons which our experiments 

There is no compelling reason why this stage should be 
final, and it may possibly represent only a very transitory 


phase in the development of our knowledge. Our experi- 
ments on nature provide an obvious connecting bridge 
between ourselves and nature, and in exploring nature we 
naturally start from our own end of this bridge. Because 
the bridge involves ourselves as well as nature, it is hardly 
surprising that our present knowledge of nature should 
still possess a subjective tinge. For, after all, we only 
started on the right road a third of a century ago. 

When we look into the future we see two possibilities. 
It may be that nature goes on her way regardless of us, 
and that it is only our imperfect present knowledge which 
involves ourselves as well. We can still only explore na- 
ture by stamping it with our own footprints and raising 
clouds of dust, so that our present pictures of nature shew 
our human stamp over it all. In time we shall perhaps 
learn how to remove our own footprints from the picture 
and shall then see that nature has a real existence, as 
much outside ourselves and independent of ourselves as 
the Sahara. The essentials of the Sahara are its particles 
of sand; the clouds we raise are transitory accidents. In 
1899 most scientists would have unhesitatingly averred 
that nature was like this. Yet we shall see that up to the 
present science has hardly been able to find any solid 
ground behind the clouds. 

There is another kind of desert in which cloud forms 
are the essentials, and the medium in which they are 
expressed an accident such are an artist* s concept, or 
the traveller's recollections, of a desert. Nature may, too, 
be like this. 

Broadly speaking, these two conflicting alternatives 
represent objectivist and subjectivist views of nature, or 
again realist and idealist schemes of philosophy. 

Bradley wrote of the latter alternative:* 

"It may be objected that we have now been brought into 
collision with common sense. The whole of nature, for com- 
mon sense, is; and it is what it is, whether any finite being 
apprehends it or not. on our view, on the other hand, . . . 
the world of physical science is not something independent, but 
is a mere element in one total experience. And, apart from 
finite souls, this physical world, in the proper sense, does not 
exist. But, if so, we are led to ask, what becomes of natural 
science? Nature there is treated as a thing without soul and 
standing by its own strength. And we thus have been appar- 
endy forced into collision with something beyond criticism. 
But the collision is illusive, and exists only through mis- 

Since this was written, science has gradually discovered 
that its nature "standing by its own strength" was an 
assumption rather than an ascertained fact, and so is 
more ready to admit that the collision may be illusive. 

Yet the difficulties of the idealist position are almost too 
obvious to need description. A being who had no means 
of communicating with his fellow-men would have no 
means of knowing whether or not the nature he saw was a 
creation of his own mind; he might well credit it with no 
more real existence in its own right than the objects 
he saw in a dream. We, on the contrary, must somehow 
fit into our scheme of nature the fact that, broadly speak- 
ing, innumerable other minds all observe the same nature 
as we do. Realism explains this very simply and natu- 
rally by supposing that nature exists outside of, and in- 
dependently of, all our minds we all see the same moon 
because the moon is out there, outside ourselves, for us all 
to see. Idealism cannot avail itself of this simple explana- 

* Appearance and Reality, pp. 279, 283. 


tion; it has to suppose that our minds are in some way 
all members of one body, and so arc all attuned to per- 
ceive the same concepts. They must be interconnected in 
some way perhaps as the branches of a tree are inter- 
connected, through having a common root or perhaps 
again as the members of a shower of photons are inter- 
connected; in some aspects these appear as a crowd of 
distinct individuals, in others as a continuous progression 
of light. 

We leave the question here and proceed to discuss the 
findings of modern science, bearing in mind that they are 
a description, not of nature, but of human questionings 
of nature. 



We have already pictured the new-born child trying to 
correlate the events and objects which affect its senses, 
thereby taking its first steps towards becoming a scientist. 
Gradually it makes the discovery which we express by 
saying that the events can be arranged in time, while the 
objects in which they appear to originate can be arranged 
in space. Thus space and time form a sort of framework 
for the sense impressions which the child receives from 
the external world. The child does not of course concern 
itself with metaphysical questions as to the fundamental 
nature of space and time, and neither shall we here; only 
the simplest properties of space and time, as perceived by 
us, are relevant at the present stage of our discussion. 

Rudimentary Views of Space and Time 

The child finds that the events of its day come in simple 
sequence, like beads on a string. The string is what we 
call time, and the order of events relative to one another 
can be fully described by the words "earlier" and "later". 
Adjacent events need not be contiguous; just as there 
may be stretches of a string which are not occupied by 
beads, so the child may experience uneventful periods of 
time. Time passes through our minds like tape through a 
chronograph; any small fragment of it may or may not 
have events impressed on it. Somewhere in our physio- 



logical processes, a sort of clock ticks moments, and so 
gives us a sense of the passage of time. Through the 
tickings of this mental clock, our minds judge time-inter- 
vals to be long or short; we find that time passes through 
our minds in a way which is, approximately at least, the 
same for all of us, so that we are led to think of time as 
something outside ourselves, flowing past or through the 
consciousness of each of us as a river flows past the piers 
of a bridge. Science measures the flow of this supposed 
river of time more precisely by counting evenly spaced 
events the passage of the sun or stars across the me- 
ridian, the ticking of a clock, the vibrations of a quartz 
crystal, or the oscillations of a tuned electric system* 
Until the theory of relativity compelled us to reconsider 
our position, we intuitively regarded time as an ever-roll- 
ing stream, whose flow could be measured in such ways as 

Our intuitive conception of space is very different. 
Light enters our eyes from external objects, and our 
crystalline lenses arrange that all the photons (p. 27) 
which come from what we describe as "the same direc- 
tion" shall be projected on to the same point of the retina. 
Our first classification of objects is accordingly by the 
points of the retina they affect, and, as the retina is a two- 
dimensional surface, we get the impression of objects 
arranged in a two-dimensional array of directions 
angular space. 

Yet we know that objects cannot be fully located by 
their directions alone* As we move about, they change 
their directions; sometimes a number may lie in the same 
direction, as seen by our eyes, and so interfere with one 
another's visibility. Looking in one single direction, I 


may see, one behind the other, tobacco smoke, a dirty 
window, a butterfly, a tree, a hilltop, a cloud, the sun. 
I arrange them in this particular order because of the way 
in which they interfere with one another's visibility. The 
arrangement is like that of events in time a one- 
dimensional arrangement. The different directions of 
two-dimensional angular space each contain a one-di- 
mensional arrangement of objects, so that objects, as they 
appear to me, form a three-dimensional array, which I 
can arrange in a three-dimensional "space". 

Each of my two eyes makes such an arrangement in- 
dependently for my mind, but something further is 
needed if both eyes are to make the same arrangement 
as they must if the objects exist in their own right in the 
external world. We find that, just as consecutive events 
are not usually contiguous in time, so consecutive objects 
are not usually contiguous in space; the butterfly is not 
contiguous with my window, nor the cloud with die sun. 
Consecutive objects may be separated by "distance", 
just as consecutive events may be by time. Counting the 
ticks of a clock will give a measure of the time between 
events, and in the same way counting the number of end- 
to-end juxtapositions of a measuring-rod will give us a 
measure of the distance between objects. This particular 
way of measuring distance is, of course, independent of 
our sense of sight, and indeed of the properties and even 
of the existence of rays of light. Beings deprived of all 
senses but that of touch could still map out the arrange- 
ment of bodies in space, armed only with their sense of 
touch and a measuring-rod. Their arrangement might or 
might not agree with that of other beings who used only 
their eyes. It would agree if the straightness of rays of 


light was the same as that of the straight edge of a meas- 
uring-rod; otherwise not. The distinction is important 
because we shall see later that light does not always travel 
in such straight lines. Thus it is already clear that the 
arrangement of objects in space may have a subjective 
tinge about it; a blind man might make a different ar- 
rangement from one who could see and used no instru- 
ment except his seeing. 

In some such way as this our individual consciousnesses 
first apprehend time and space. And once again the 
history of the individual is that of the race writ small. 

Pre-relativity Views of Space and Time 

We have seen how man as an individual only gradually 
becomes, and as a race only gradually became, aware of 
the existence of an objective nature, external to and inde- 
pendent of himself. What Professor Cornford describes as 
the "discovery of Nature . . . one of the greatest achieve- 
ments of the human mind 95 * occurred in Ionian Greece 
six centuries before Christ. It is important although, 
for the scientist, difficult to realise that space and time 
were also human "discoveries" of about the same epoch. 
Jowett writes: f 

"Our idea of space, like our other ideas, has a history. The 
Homeric poems contain no word for it; even the later Greek 
philosophy has not the Kantian notion of space, % but only the 
definite 'place' or 'the infinite'. . . . When therefore we speak 
of the necessity of our ideas of space, we must remember that 
this is a necessity which has grown up with the growth of the 
human mind, and has been made by ourselves. . . . 

* Before and after Socrates, p. 15. 

t The Dialogues of Plato, vol. iv, Introduction to Theatetus, p. 162. 

t See p. 97 Wow. 


"Within or behind space there is another abstraction in 
many respects similar to it time, the form of the inward, as 
space is the form of the outward. As we cannot think of out- 
ward objects of sense or of outward sensations without space, 
so neither can we think of a succession of sensations without 
time. It is the vacancy of thoughts or sensations, as space 
is the void of outward objects. . . . Like space it has been 
realized gradually: in the Homeric poems, or even in the 
Hesiodic cosmogony, there is no more notion of time than of 

Plato (Timaeus) describes space as* 

"that which receives all bodies. It must be called ever self- 
same, for it never departs from its own quality. . . . Were 
it like anything that enters into it, when things of opposite or 
wholly different character came to it and were received in it, 
it would reproduce them amiss, as its own features would shine 
through. Therefore also that which is to receive all kinds in 
itself must be bare of all forms, just as in the manufacture of 
fragrant ointments the artist first contrives the same initial 
advantage; he makes the fluids which are to receive his per- 
fumes as scentless as he can. So, too, those who essay to model 
figures in some soft vehicle permit no figure whatsoever to be 
already visible there, but first level the surface and make it 
as smooth as they may. . . . Space never perishes but pro- 
vides an emplacement for all that is born; it is itself appre- 
hended without sensation, by a sort of bastard inference, and 
so is hard to believe in. 5 Tis with reference to it, in fact, that 
we dream with our eyes open when we say that all that is must 
be in some place and occupy some space, and that what is 
neither on earth nor yet in the heavens is nothing". 

This view prevailed throughout the period of Greek 
science and until the time of Descartes (1596-1650); 
nature was conceived as consisting of solid objects inter- 

* Plato, Timaeus, Taylor's translation, pp. 49-51. 


spaced with a characterless void, and the space of our 
intuition was regarded as a mere empty framework for 
the arrangement of these substantial objects. 

Descartes introduced a new conception of space. It 
was fundamental to his philosophy that all substances fell 
into the non-overlapping and non-interacting categories 
of mind and matter, the essence of mind being thought, 
which did not occupy, and was not arranged in, space, 
while the essence of matter was occupancy of space and 
extension in space. He further maintained that all space 
must be occupied by something, arguing that empty space 
would fulfil no function, and that it was contrary to the 
perfection of design shewn throughout the universe that 
anything should exist without a purpose. Thus although 
the spaces between the stars might appear empty they 
could not be so, and must be occupied by some sort of 
continuous substance having a real existence and char- 
acteristic properties of its own. Space ceased to be a mere 
empty framework, and became an objective reality exist- 
ing in its own right. This led Descartes to maintain that 
extension in space and motion through space were the 
true primary qualities of objects (p. 14). 

In accordance with these ideas, Descartes abandoned 
the corpuscular theory of light, and imagined light to be 
of the nature of a pressure transmitted through this all- 
pervading substance to our eyes. At a later date, sci- 
entists also rejected the corpuscular theory of light in 
favour of the undulatory theory, which imagined light 
to be of the nature of waves. The all-pervading sub- 
stance of Descartes could now perform the function of 
transmitting these waves. It was accorded a real exist- 
ence, and described as the "luminiferous ether* 5 . 


Location in Space 

We have already noticed how two individuals might 
make different arrangements of the objects in space, ac- 
cording to whether they relied on their sense of sight or 
their sense of touch. It now appeared that nature, too, 
had her own special way of arranging objects in space, 
and this made all individual arrangements unimportant. 
They became right or wrong according as they agreed, 
or did not agree, with nature's own arrangement. Ob- 
jects could not only be arranged in space; they could be 
located in space by their positions in the ether, just as 
objects in England can be located by their positions on 
English soil. 

I can say for instance that an object is 50 yards north 
of the twentieth milestone on the Great North Road. If 
I tie my handkerchief to an object at this spot, take a walk, 
and come back to find my handkerchief still attached to 
the same object, I can say I have come back to the spot 
from which I started. on the other hand, if I drop my 
handkerchief overboard at sea, row about, and come 
back to my handkerchief, I am not entitled to say I have 
come back to the same spot, since currents and winds are 
likely to have moved my handkerchief. I can only fix a 
position at sea by taking bearings, directly or indirectly, 
from the land. 

If space is occupied by an ether, we can locate a spot 
in space by the former method. We can, in imagination 
at least, tie a handkerchief to a particle of ether, and if 
we come back to the handkerchief we may say we have 
come back to the same point in space. We need not fear 
that currents and winds will have moved the hand- 


kerchief, for if light consists of waves travelling through 
an ether, its speed of travel shews that this ether must 
be far more rigid than steel. 

If there is no ether, we can only locate a spot in space 
by its bearings from fixed landmarks, but where are such 
landmarks to be found? Not in the planets, for these are 
moving round the sun at speeds which range from 3 to 
30 miles a second. Not in the sun and stars, which move 
past one another even more rapidly. Not in the great 
nebulae, the most distant objects known, for these are 
rushing away from us and from one another at still greater 
speeds of many thousands of miles a second. Nowhere in 
the whole of space can we find fixed landmarks from 
which to take our bearings, with the result that it is 
impossible to fix a position in space. Newton was fully 
alive to this difficulty, for he wrote: 

"It is possible that in the remote regions of the fixed stars 
or perhaps far beyond them, there may be some body abso- 
lutely at rest, but impossible to know, from the positions of 
bodies to one another in our regions, whether any of these do 
not keep the same position to that remote body. It follows 
that absolute rest cannot be determined from the position of 
bodies in our regions". 

He also saw how an all-pervading ether might provide 
a solution, for he continued: 

"I have no regard in this place to a medium, if any such 
there is, that freely pervades the interstices between the parts 
of bodies". 

And indeed the existence of such a medium would seem 
to provide the only solution of the problem; its particles 
would provide fixed standard positions, against which the 
positions of moving objects could be measured at any 


instant. If there is no such medium, we can only define 
rest in space in an arbitrary way. 

Location in Time 

The precise identification of instants of time presents 
problems of a similar kind. 

We soon learn to regard the time of our own individual 
experience as an ever-rolling stream, and it used to be 
tacitly assumed that the same stream rolled on in the 
same way throughout the universe, so that events could 
be "located" in time, just as objects could be located in 
the ether. If, for instance, on January 1st, 1901, an 
astronomer saw a sudden outburst on a star which he 
believed to be 100 light-years' distant, he would say this 
outburst had occurred on January 1st, 1 801 . He believed 
that the outburst could be "located 39 in thestream of time, 
and that there was a definite meaning in saying that it 
had occurred at the precise instant of time at which the 
nineteenth century opened on earth. 

Let us, however, consider what is implied in such a 
belief. It will be enough to consider a single illustration, 
taken from the every-day operations of practical astron- 
omy. Let us suppose that British astronomers at Green- 
wich wish to compare their astronomical observations 
with those made by American astronomers at Annapolis, 
something more than 3000 miles to the west, and, with a 
view to doing this, set about synchronising their clocks. 
The obvious plan is to send some kind of a signal between 
the two places. If any known kind of signal travelled 
with literally infinite speed, the operation would be 
simple enough the Annapolis astronomers would send 
out a signal when their clocks shewed noon, and if the 


Greenwich clocks shewed exact noon when this was 
received, the clocks would already be synchronous; if not, 
they could easily be adjusted to be so. The essential 
difficulties of the problem arise from the circumstance 
that no signal can travel with infinite speed, since it is a 
fundamental principle of physics that no signal can ever 
travel faster than light. Actually, astronomers use the 
fastest signals available, namely wireless signals, which 
travel at the speed of light. Yet if Annapolis sends out a 
wireless signal when their clocks shew noon, it will al- 
ready be somewhat after noon at Annapolis by the time 
the signal reaches Greenwich. In practise the Greenwich 
astronomers say that as a wireless signal travels at about 
186,000 miles a second, it takes approximately a fiftieth 
of a second to come from Annapolis. They therefore 
regard their clocks as adequately synchronised if they 
point to a fiftieth of a second after noon at the moment 
when the Annapolis signal reaches them. 

This is near enough for the practical needs of astron- 
omy, but it is not absolutely exact. To obtain perfect 
synchronism, it would be necessary to know the exact 
time which the signal took on its journey. 

Now let us suppose that wireless signals consBIRJI 
waves travelling through the ether, and imagine t^t the 
earth is also travelling through the ether, let us say in the 
direction from Greenwich to Annapolis. Then Green- 
wich would be advancing through the ether to meet the 
signal sent out from Annapolis, and so would meet it 
sooner than if the earth were standing at rest in the ether. 
But to know by how much sooner, and so discover the 
exact time of travel of the signal, it would be necessary to 
know the speed of the earth's motion through the ether. 


The Michelson-Morley Experiment 

The famous Michelson-Morley experiment tried to 
measure this speed in the most direct and most obvious 
way. If signals travelled through the ether at 186,000 
miles a second, and the earth travelled through the ether 
from east to west at 1000 miles a second, signals travelling 
from west to east would have their rate of travel over the 
earth's surface increased from 186,000 to 187,000 miles a 
second because the earth would be moving to meet the 
signal, but that of a return signal from east to west would 
be decreased from 186,000 to 185,000 miles a second. A 
signal which made the double journey would be ex- 
pedited on the outward journey, but retarded on the 
return journey. For each thousand miles of path, the 
outward journey takes jfar second, the homeward 
journey ^fa* second, so that we have as the total, per 
thousand miles of path: 

Outward time yfy sec. = 0.005347594 sec. 
Return time = sec. =* 0.005405406 sec. 

Total time = 0.010753000 sec. 

on the other hand, if the earth were at rest in the ether, 
the total time would be: 

Total time = ^ sec. = 0.010752690 sec. 

We see that the gain of time on the outward journey does 
not quite make up for the delay on the return journey; 
there is a net delay of about a three-millionth part of a 

Conversely, if the net delay could be measured, and 
proved to be a three-millionth part of a second, we should 


know that the speed of the earth's motion through the 
ether was 1000 miles a second. 

Actually there was of course no means of comparing 
the time of the double journey with the time it would 
have taken had the earth been made to stand still. It 
was, however, possible to compare the times of two double 
journeys, both performed simultaneously on the moving 
earth, the one on the east-west course we have already 
considered, and the other on a course of equal length at 
right angles to this, and this comparison is found to give 
the needed information equally well. 

To be precise, if we denote the speed of light by c 9 
and the speed of the earth's motion through the ether 
by u, the loss of time per unit length of path on the double 
journey in the direction of the earth's motion is found 
to be 

1 + 1 2, 

+ U C U C 

Simple algebra shews that this is equal to 

2r ',-1' 

Also simple geometry shews that the corresponding loss 
of time on the double journey in a direction perpendicu- 
lar to the earth's motion is 

The former quantity is easily seen to be very approxi- 
mately double the latter, and if observation gives the 
difference between them, it is easy to deduce the value of 


u. In the actual experiment, the path of the rays was of 
course far less than the 1000 miles which we have taken 
for purposes of illustration; it was only a few yards, so 
that a speed of even a thousand miles a second would only 
have produced a time-difference of about a million- 
millionth part of a second. It is such minute times as 
this that have changed our whole outlook on the universe. 

Michelson and Morley hoped to measure this small dif- 
ference of time with accuracy, although naturally not 
with ordinary clocks or stop-watches. They performed 
their experiment with light of great purity of colour, the 
waves of which oscillate many millions of millions of times 
a second at a perfectly uniform rate, and these oscilla- 
tions provided a very perfect clock. The experiment 
consisted in starting two beams of light simultaneously to 
run the out and home course in the two directions, and 
observing which got back to the starting-point first, and 
by how much it won. 

If the difference of times had proved large, it would 
have shewn that the earth was moving rapidly through 
the ether; if small, that it was moving slowly. The one 
result that was never contemplated was that the time- 
difference should prove to be nothing at all. For the 
earth's motion round the sun alone gave it a speed of 
19 miles a second, and the delicacy of the apparatus was 
such as to disclose a speed of only about one mile a 
second. Yet it was the unexpected that happened. When 
the experiments were performed, absolutely no time-dif- 
ference could be detected. They have been repeated time 
after time, at different times of day and of the year (so 
as to get the apparatus pointing to different positions in 
space, and to get the earth at different parts of its orbit 


round the sun), under different conditions of temperature, 
altitude and so forth, but nature has consistently given 
the answer that she knows of no motion of the earth 
through the ether. The times given by formulae (A) and 
(B) are always precisely equal, so that u = 0. 

At first such an answer seemed to be pure nonsense. 
The obvious inference (which, however, it took a very 
long time to reach) was that the question also had 
been nonsense in brief, the concept of light as waves 
travelling through an ether had provided the wrong 
background for the experiment. The success of the 
undulatory theory shews that light has many wave-like 
properties, but these experiments seemed to shew that 
its mode of travel through space is not one of them. 

The corpuscular theory had implied a different mode 
of travel. For if light travelled like waves through a sea 
of ether, its speed of travel would always be the same 
relative to the sea of ether. on the other hand, if it travelled 
like particles shot out from a gun, then its speed of travel 
would be always the same relative to the gun from which it 

Perhaps then the question to nature ought to have been 
put in the form "Does light travel like waves or like 
particles? " When the question is framed in this way, the 
Michelson-Morley experiments unambiguously support 
the latter alternative. 

Yet if light travelled like particles, the photons emitted 
by two bodies moving at different speeds would them- 
selves move at different speeds. Now astronomical ob- 
servation shews that the photons emitted by the two 
components of a binary star travel at precisely equal 
speeds,, so that, in this case at least, light does not travel 


like particles* Clearly our last way of framing the ques- 
tion still assumed something we had no right to assume. 
It assumed that light must necessarily travel through 
space either as waves or as particles. Observation now 
seems to suggest that it does not travel as either. 

How, then, does light travel through space? We shall 
see shortly how Einstein solved the puzzle by giving us a 
new conception, not of light but of space. 

First, however, we must go somewhat back in the his- 
tory of science. In 1873 Maxwell shewed that light was 
one special form of electric action, and the question of 
how light was propagated through space became only 
one aspect of a far wider problem. Both Maxwell and 
Faraday had tried to shew that all electric action was 
transmitted through space in the form of disturbances in 
the ether. Now it was obvious that, if the earth were 
travelling through the ether at 1000 miles a second, there 
would be what may be described as an "ether-wind" 
sweeping past and through all objects on the earth at a 
speed of 1000 miles a second. It seemed inconceivable 
that such a wind should not affect the transmission of 
electric action, yet experiments seemed to shew that it 
did not. A whole array of experiments on electric action 
in general gave information similar to that which the 
Michelson-Morley experiments had given about light. 
They not only failed to disclose the speed of the earth's 
motion through the ether, but seemed to indicate that 
no such motion existed. At any rate, the supposed 
ether-wind was found to have absolutely no effect on 
terrestrial phenomena. 


Newtonian Relativity 

Before discussing the significance of this, let us consider 
a simpler problem of the same nature, which had been 
discussed by Newton. It is well known that when a ship 
or train or other vehicle moves steadily forward at a 
uniform speed, objects inside it behave precisely as though 
it were at rest. If we play tennis on board ship, the player 
who is facing towards the bows of the ship gains no 
advantage from the ship's motion. Any advantage he 
gains in imparting speed to the ball is exactly neutralised 
by the extra effort needed to check its motion when it 
first impinges on his racquet. Actually it is a matter of 
common observation that the ball rebounds from our 
racquets exactly as though the ship were at rest. Newton 
expressed this fact of observation in the following words: 

"The motions of bodies included in a given space are the 
same among themselves, whether that space is at rest, or moves 
uniformly forwards in a right line without any circular motion. 

"A clear proof of which we have from the experiment of a 
ship; where all motions happen after the same manner, 
whether the ship is at rest, or is carried uniformly forwards in 
a right line". 

and shewed why this must be in the following words: 

"For the differences of the motions tending towards the 
same parts [i.e. in the same direction] and the sums of those 
that tend towards contrary parts, are, at first (by supposition), 
in both cases the same; and it is from those sums and differ- 
ences that the collisions and impulses do arise with which the 
bodies mutually impinge one upon another. Wherefore (by 
Law 2) the effects of those collisions will be equal in both 
cases; and therefore the mutual motions of the bodies among 
themselves in the one case will remain equal to the mutual 
motions of the bodies among themselves in the other"* 


The same situation occurred when the action was 
electrical instead of mechanical; the motion of the earth 
was found to have no effect on the observed phenomena. 
Towards the end of the nineteenth century, a great num- 
ber of physicists were engaged in investigating how this 
could be, and Professor Lorentz of Leyden announced a 

very remarkable conclusion in 1895. 


The Lorentz Transformation 

To make as vivid a picture as possible, let us imagine that 
a professor of physics discovered certain laws of electric 
action in a laboratory on earth, at some epoch when this 
happened to be standing still in the ether. Let us suppose 
that he formulated them in terms of measurements made 
in time and space. We may suppose he would follow the 
usual mathematical practice of specifying a point in space 
by its distances x, y, z from three perpendicular planes, 
and the passage of time by a quantity t which measures 
the interval which has elapsed since a specified zero hour. 
He can then express his law as a relation connecting 
certain quantities which admit of observation and 
measurement with x, y, z and t. If we want a concrete 
example to fix our thoughts, we may take the law of 
magnetic induction, which Maxwell expressed by the 
equation lda = dr_< 

c dt dz dy' 

Here c is the velocity of light; a is the magnetic induction 
in a certain direction, and T, are electric forces in two 
other directions perpendicular to the first. Thus the law 
connects changes in the measurable quantities a, T 9 
with changes in x,y, z and t. 


Now let us imagine that our physicist is subsequently 
shot out into space in a rocket which travels through space 
in the direction of x with a speed we may call u. He had 
discovered his laws in a laboratory through which no 
ether-wind blew, and so could hardly expect them to be 
true under his new conditions. Yet Lorentz was able to 
shew, from the known laws of electric action, that, not- 
withstanding the ether-wind, any laws of electric action 
which the physicist had discovered on earth would still 
be qualitatively true in the moving rocket. In a certain 
restricted sense they would also be quantitatively true. 
If he re-investigated these laws in the moving rocket, he 
would find that they could be expressed with perfect 
accuracy in precisely the same mathematical formulae as 
he had used on the earth at rest. The only point of dif- 
ference would be that x, j, z and t would not have quite 
the same significance as they had on earth, although, as 
we shall shortly see, the physicist would never be able to 
discover this. 

Let us reserve the symbols x,y, , t for the measurement 
of space and time on earth; when the corresponding 
quantities are measured in the moving rocket, let us 
denote them by #',y, ', t'. Then Lorentz shewed that 
the same laws will be obtained in the rocket as on earth, 
provided the co-ordinates *',/, *', /' of the moving rocket 
are related to the co-ordinates x,y, z, t of the stationary 
earth by the equations 


These equations express what is known as the "Lorentz 
transformation". Every term in them deserves careful 

The symbol c still denotes the velocity of light as 
measured on earth; we shall soon see that it is also the 
velocity of light as measured in the moving rocket. When 
we are discussing problems of ordinary mechanics and 
astronomy, we need not trouble about the velocity of 
light at all; light moves so much faster than everything 
else, that we may quite properly think of it as travelling at 
infinite speed. Thus for the discussion of such problems, 
we may put c equal to infinity, as the mathematicians say, 
which means that everything divided by c becomes equal 
to zero. When we do this, the equations of the Lorentz 
transformation assume the much simpler form 

*' = x ut, / = y, z' = , f = /. 

Thus, so far as mechanical experiments were concerned, 
our experimenter could use precisely the same co-ordi- 
nates in the rocket as he had previously used on earth, 
except for the difference x x 1 = ; ut> which arises natu- 
rally from the circumstance that the rocket is increasing its 
distance from the earth at a speed u. This merely means 
that positions must be measured relative to the new labo- 
ratory, the rocket, and not relative to the old laboratory 
left behind on earth, as would naturally be done in any 

Thus when the velocity of light is enormously greater 
than all the other velocities concerned, Lorentz's result 
becomes exactly identical with that which Newton had 
found more than two centuries earlier all phenomena 


happen after the same manner, whether the laboratory is 
at rest, or moves uniformly forward. 

Lorentz was, however, concerned primarily with elec- 
trical phenomena, which are known to be propagated 
with precisely the velocity of light, and so was not able to 
treat the velocity of light as infinite; this is why c appears 
in his formulae. 

I ^ 

It first occurs in the factor Af 1 - in the denomi- 

* c 2 

nator of #'. This means that if is measured in different 
units from those in which the original x is measured. 

Just as twelve inches make a foot, so Af 1 ~ of the lat- 


ter units make one of the former. This factor is some- 
times described as the Fitzgerald-Lorentz contraction, 
because, while scientists still thought in terms of an ether 
pervading all space, Fitzgerald (1893) and Lorentz 
(1895) had independently suggested that an object which 
moved through the ether with a velocity u might undergo 
a contraction of this amount in the direction of its motion 
the ether-wind might compress a body moving into it, 
just as the pressure of ordinary wind must compress a 
football kicked into it. 

However we explain it, such a contraction is found to 
account exactly for the result of the Michelson-Morley 
experiment; the shortening of the apparatus in the up- 
and-down direction of the ether stream exactly com- 
pensates for the slower average speed of light on this 

We shall see this at once if we turn back to the formu- 
lae (A) and (B) on p. 81. If the apparatus is shortened 
lengthwise by a factor K when it is moving through the 


ether, the formula (A) giving the loss of time on an up- 
and-down journey of unit length must be replaced by 


- 1 

and when K has the value \ 1 -, this becomes exactly 


identical with formula (B), which gives the loss of time 
on a crosswise journey. 

No similar factor appears in the values of y* and ', so 
that there is no contraction in these directions; an object 
is only shortened in the direction of its motion, and not 
in directions at right angles to this. This leads to the odd 
result that motion alters the shape of an object; a billiard 
ball may be truly spherical when at rest, but ellipsoidal 
when in play. If Fitzgerald and Lorentz had been right, 
Gilbert's "elliptical billiard balls" would have described 
a sober scientific fact, provided that "elliptical" was 
meant to describe an ellipsoid of revolution. An object 
which moved with the speed of light would have been 
flattened to nothing at all in the direction of its mo- 
tion; a sphere to a mere disc, a cube to a square, and 
so on. 

The same shortening factor reappears in the value for 
*', so that the experimenter in the moving rocket must 
measure his time also in units different from those he used 
in his laboratory on earth, if the motion of his laboratory 
is not to affect his description of the observed laws of 

nature. Again \ 1 - ^ of the latter units will make one 


of the former. 


The value of the time t f as measured in the rocket is 
not only complicated by the shortening factor in its 
denominator; the numerator also is complicated, de- 
pending not only on t, the time on earth, but also on x 9 
the distance travelled from earth. This means that at 
any single moment on earth, when t has a known definite 
value, there is no corresponding definite value for t r which 
is the same at all points of space. For the man in the 
rocket, time varies at different points of space, just as 
"local time", or sun-time, varies at different points of the 
earth's surface. For this reason, Lorentz described the 
value of t f as the "local time" of the experimenter in the 
rocket. The astronomer's local time is propagated round 
the globe at such a rate that it is always "local" noon 
directly under the sun; Lorentz's formula shews that the 
physical experimenter's local time is propagated through 

, c* 
space at a speed 

Here we come upon a speed which is enormous even 
compared with the speed of light. If a rocket is moving 
at a ten-thousandth part of the speed of light, which is 
roughly the speed of the earth in its orbit round the sun, 
the "local time" for the rocket is propagated through 
space at ten-thousand times the speed of light. We shall 
see later that this speed of propagation plays a very im- 
portant part in modern physics. 

Let us take two concrete illustrations to explain the 
physical meaning of the Lorentz transformation. The 
law of electromagnetic induction given on p. 86 is ex- 
pressed in terms of the co-ordinates x> y, z and t. Since we 
know the relation between these and *', /, *', *', it is 
merely a matter of algebra to express the law in terms of 

these latter coordinates. We find that it is expressed by 
the equation 1 da' 

where a\ T', % have slightly different meanings from the 
original a, T 9 %. Physically this means that the experi- 
menter in the moving rocket might re-investigate mag- 
netic induction, and re-discover Maxwell's law. If he 
did, he could express it in precisely the same mathe- 
matical form as he had previously used on earth, although 
all the symbols except c would mean something a little 
different from what they had previously meant on earth. 
As a second illustration, let us imagine that the ex- 
perimenter in the rocket re-investigates the speed of 
propagation of light. Let us suppose that he ignites some 
magnesium powder at a point out in space which we may 
call the origin (x Q,y = 0, z 0), and at an instant 
which we may take to be zero-hour (t = 0). The flash of 
light produced in this way will set out to travel in all 
directions of space equally, at the same speed c. After a 
time-interval t, it will have travelled a distance ct in every 
direction, so that if it has reached the point x, y, , whose 
distance from the origin is V x*+y 2 + z 2 , we must have 

From this we can easily deduce, by using the equations 
of the Lorentz transformation, that 

+ *' 2 - cf. 

Between them these two equations shew that whether 
the experimenter uses the modes of measurement appro- 
priate to the moving rocket or those appropriate to the 
earth at rest, light will still appear to travel at the same 


uniform speed c in all directions. In other words, no 
number of Michelson-Morley experiments could possibly 
disclose the speed u with which the rocket was moving 
through space. 

The Theory of Relativity 

In 1905, Einstein gave a new and quite revolutionary 
turn to the whole problem. Lorentz had based his inves- 
tigation on the concept of an ether filling all space, and 
consequently of an ether-wind blowing through every 
experiment. Consequently he had imagined that in some 
way the time t used by our observer at rest in space was 
real time, nature's own time, while the "local time" t f of 
the man flying through space in a rocket was merely a 
convenient fiction, introduced to allow for the ether-wind. 

Yet if a new generation of men were born in the rocket 
as it moved through space, they would soon forget about 
the true time t they had left behind them on earth and 
would know of no time except the "local time" t'\ this 
would be "the time" for them. In the same way, our 
human race knows only one time; we call it "the time", 
but actually it must be merely the "local time" of (nor 
rocket, the earth, as it moves through space. When we 
say that light from Sirius takes 8-65 years to reach us, 
we mean 8-65 years of the local time of our earth. When 
we say that an outburst on a certain distant star was 
synchronous with the beginning of the nineteenth cen- 
tury, we must be speaking in terms of the "local time" of 
our earth. It might seem obvious that we have no right 
to identify this with the "true time" of nature. 

At this state Einstein asked cc Why not? What reason 
have we for supposing that our time is inferior to any 


other? " If the laws of nature are to be the same through- 
out space, all the various rockets moving through space 
with different speeds must have different local times, but 
there is no evidence that any true time exists which is 
superior to them all. Indeed, all the evidence points in 
precisely the opposite direction. True time implies the 
existence of a body at rest in space. Not only have we no 
means of discovering when a body is at rest in space, but 
there is every reason to suppose the phrase is meaningless. 

on these grounds, Einstein maintained that all time is 
"local 35 ; there are as many local times as there are rockets, 
or planets, or stars, moving through space, and none of 
them is more fundamental than any other. 

This implies that it is just as impossible to locate an 
event in time in an objective way, as to locate an object 
in space in an objective way. Einstein accordingly pro- 
posed abandoning the concepts of objective, or absolute, 
time and space, and putting in their place the supposition, 
which all experimental evidence appeared to confirm, 
that "Nature is such that it is impossible to measure an 
absolute velocity by any means whatever". In brief, 
nature is concerned only with relative velocities; there 
is no fixed background of points in space against which 
motion can be measured in absolute terms, and conse- 
quently no absolute flow of time against which intervals 
of time can be measured. 

The theory of relativity starts from this hypothesis, and 
proceeds to develop its logical consequences by strict 
mathematical analysis. If the hypothesis is true to nature, 
these consequences will agree exactly with the facts of 
nature. Many of them can be directly tested by experi- 
ment, and in every such case, without a single exception, 


nature has confirmed the theory the consequences de- 
duced from it have proved to be true. If ever one of these 
proved not to be true, it would at once become possible 
to measure an absolute velocity in space, and the observa- 
tion in question would provide us with a framework of 
absolute space and absolute time. So far not a single 
physical experiment has done this, so that the picture 
which modern physical science draws of nature contains 
no reference to either absolute space or absolute time. 
We shall, however, see later that when astronomical 
science studies the universe as a whole, it may draw a 
slightly different picture. 

This does not of course mean that we must abandon the 
intuitive concepts of space and time which we derive from 
individual experience. These may mean nothing to na- 
ture, but they still mean a good deal to us. Whatever 
conclusions the mathematicians may reach, it is certain 
that our newspapers, our historians and story-tellers will 
still place their truths and fictions in a framework of 
space and time; they will continue to say this event 
happened at such an instant in the course of the ever- 
flowing stream of time, this other event at another instant 
lower down the stream and so on. 

Such a scheme is perfectly satisfactory for any single 
individual, or for any group of individuals whose ex- 
periences keep them fairly close together in space and 
time and, compared with the vast ranges of nature, all 
the inhabitants of the earth form such a group. The 
theory of relativity merely suggests that such a scheme 
is private to single individuals or to small colonies of 
individuals; it is a parochial method of measuring, and 
so not suited for nature as a whole. It can represent all 


the facts and phenomena of nature, but only by attach- 
ing a subjective taint to them all; it does not represent 
nature so much as what the inhabitants of one rocket, or 
of one planet, or better still an individual pair of human 
eyes, see of nature. Nothing in our experiences or experi- 
ments justifies us in extending either this or any other 
parochial scheme to the whole of nature, on the supposi- 
tion that it represents any sort of objective reality. 

We used to think of space as something real and objec- 
tive in the region "out there" from which messages came 
to our senses; it even seemed to acquire a sort of sub- 
stantiality from the ether which we imagined to occupy 
its every point. We thought of time as something equally 
real and objective, flowing past our senses in a way en- 
tirely beyond our control. Yet when we question nature 
through our experiments, we find she knows nothing of 
either a space or of a time which are common to all men. 
When we interpret these experiments in the new light of 
the theory of relativity, we find that space means nothing 
apart from our perception of objects, and time means 
nothing apart from our experience of events. Space be- 
gins to appear merely as a fiction created by our own minds, 
an illegitimate extension to nature of a subjective con- 
cept which helps us to understand and describe the 
arrangement of objects as seen by us, while time appears 
as a second fiction serving a similar purpose for the 
arrangement of events which happen to us. 

This is of course in striking contrast with the earlier 
views of Kant which had dominated metaphysics until 
the advent of the theory of relativity. These may be 
summarised as follows:* 

* Sidgwick, The Philosophy of Kant, p. 38. 


"(1) The notion of Space cannot be derived from external 
experience; because, in order that I may apprehend things as 
out of me and out of each other, I must have the notion of 
Space already in my mind; 

"(2) the notion of Space is a necessary, a priori one; for I 
cannot imagine Space annihilated, though I can very well 
think it emptied of objects ". 

In brief, for Kant, as also for Descartes and Newton, 
objects cannot exist without space; for Einstein, space 
cannot exist without objects. 

Objective Space-time 

We have seen our ordinary space and time becoming 
reduced to mere frameworks of human origin, against 
which we see and record our individual sense-experiences. 

If we are to study objective nature, we clearly need an 
objective framework, which shall be independent of the 
motion of our particular rocket through space. Such a 
framework was all the time lying latent in the Lorentz 
transformation, although the genius of Einstein and 
Minkowski were needed to point it out. It is nothing 
more nor less than a four-dimensional space, having 
#, y, z and t for its four co-ordinates in other words, the 
ordinary everyday space of any individual we please, 
extended by the addition of a fourth dimension, the 
ordinary time of the same individual. When the individ- 
ual space and individual time of any particular individual 
are welded together in this way, the individual is found 
to drop out altogether the constituents are subjective 
to a particular individual, but the product is objective. 

An analogy from the ordinary three-dimensional space 
of everyday life will shew how this can be. We can divide 


ordinary space up in as many ways as we like; for many 
purposes it is found convenient to divide it into horizontal 
(two dimensions) and vertical (one dimension). Such a 
division is, of course, "local" to particular spots on the 
earth's surface; one man's vertical is not every man's 
vertical, and the division at London will not be the sarne 
as at Paris. Yet if an inhabitant of London combines 
his two-dimensional horizontal with his one-dimensional 
vertical, he will obtain just the same space of three 
dimensions as an inhabitant of Paris would obtain by the 
same procedure. Horizontal and vertical were local con- 
cepts, relative to London or Paris, but there is nothing 
local about the resulting space. Sometimes other modes 
of division may be more convenient than the horizontal- 
vertical division. An architect at work on the leaning 
tower of Pisa would probably use the division "per- 
pendicular to the axis of the tower" and "along the axis 
of the tower 95 . This would differ substantially from the 
horizontal-vertical division of the other inhabitants of 
Pisa, but would agree with the horizontal-vertical division 
of the inhabitants of Naples. The Pisan architect has a 
perfect right to use this division whenever he finds it 
convenient; it is not specially .reserved for Neapolitans. 
In the same way, each of us may divide our new four- 
dimensional space up into individual spaces and times in 
as many ways as we please. An individual often finds it 
convenient so to divide it that he regards himself as at 
rest; he thinks of the world as passing by him, rather than 
of himself as journeying through the world. At other 
times he may find other divisions more convenient. For 
instance, a terrestrial mathematician studying the motion 
of Jupiter's satellites would almost certainly choose a 


division which reduced Jupiter to rest in space he 
would, so to speak, imagine himself living on Jupiter. 
Yet, however the division is made, when each man com- 
bines the space he has chosen with the corresponding 
time, the four-dimensional space he obtains will always 
be the same. The relation between one man's space and 
time and another man's space and time, or between the 
two spaces and times the same man may select for himself 
on different occasions, is of course given by the formulae 
of the Lorentz transformation. 

Minkowski has shewn that this relation can be ex- 
pressed in an even simpler form. If we write r for ict, 
where t stands as usual for the square root of 1, and 
c is the velocity of light, the equations of the Lorentz 
transformation can be written in the form 

x' = x cos 6 T sin 8; y 1 = y^ 
r r = x sin & + r cos 8; z' , 

where the angle 8 is defined to be such that tan 8 iu/c. 
Every mathematician will see that these formulae repre- 
sent a rotation of the axes of co-ordinates through an 
angle 8. To interpret them geometrically, we must think 
of a four-dimensional space in which x 9 j>, z and T figure 
as co-ordinates, just as x, y and z do in ordinary three- 
dimensional space; in fact, our new space is merely this 
ordinary space extended to a fourth dimension, having T 
or ict for fourth co-ordinate. We now see that by turning 
the axes round so that they point in some new direction 
in this four-dimensional space i.e. by rotating his indi- 
vidual directions of space and time in this four-dimen- 
sional space one man may change his own space and 
time into those appropriate to another man, who is 


travelling through space at a different speed just as, 
in ordinary space, by rotating his directions of horizontal 
and vertical through a certain angle, the Pisan may 
change his horizontal and vertical into those of the 
Neapolitan; his leaning tower has already rotated to 
shew him how. 

The work of Lorentz, Einstein and Minkowski shewed 
in effect that although beings who are travelling at differ- 
ent speeds relative to one another will naturally divide up 
this four-dimensional space in all these different ways, 
they will all find the same laws of nature. In other words, 
nature herself has no special way of dividing it up. She 
is concerned only with the undivided four-dimensional 
space, in which she treats all directions equally. Such a 
space is generally described as a continuum. Clearly it 
forms the canvas on which we must draw our pictures of 
nature, if they are to be true pictures, free from all sub- 
jective bias. Indeed, we shall be able to test their truth 
by examining whether they treat all directions equally; 
as is said to be the case with modern cubist pictures, they 
must not suffer by being hung upside down or askew. 
Every picture or hypothesis which fails to satisfy this test 
must tie discarded. 

For instance, Newton's law of gravitation that the 
force varies inversely as the square of the distance fails 
to satisfy it. This is hardly surprising, since the "distance 35 
between two objects has no precise meaning when we 
cannot synchronise time at the two objects. Coulomb's 
similar law of electric attraction also fails by itself, but 
magnetic forces step in to make good the deficiency, and 
electric and magnetic forces in conjunction are found to 
satisfy the test perfectly, as indeed we have already seen. 


Objective Nature 

Thus nature knows nothing of space and time separately, 
being concerned only with the four-dimensional con- 
tinuum in which space and time are welded inseparably 
together into the product we may designate as " space- 
time' 9 . Our human spectacles divide this into space and 
time, and introduce a spurious differentiation between 
them, just as an astigmatic pair of spectacles divides the 
field of vision of a normal man into horizontal and 
vertical, and introduces a spurious differentiation be- 
tween these directions. With astigmatic spectacles on, 
we incline our head and see the scene in front of us re- 
arrange itself. Yet we know that nothing has happened 
to the objects in the scene. These are objective; our view 
of them through our spectacles is subjective. 

When we take our human spectacles off, we see that 
an event no longer occurs at a point in space and at an 
instant of time, but rather exists at a point of the con- 
tinuum, this point identifying both the time and place 
of its occurrence; we discover that the primary ingre- 
dients of nature are not objects existing in space and 
time, but events in the continuum. An object which was 
formerly characterised by continuity of existence in time 
may now be treated as a continuous succession of events 
each event being the existence of the object at one 
instant of time, and one point of space. Thus an object 
is associated with a continuous succession of points, i.e. 
a line, in the continuum. This is commonly called the 
"world-line 55 of the object, its shape and position repre- 
senting the motion of the object throughout its whole 
existence. Objects which are acted on by no forces, and 


so for ever move uniformly through space in straight lines, 
have straight lines in the continuum for their world-lines. 
If their speeds are the same, their world-lines are parallel; 
if different, they are inclined. 

Two different events are of course represented at two 
different points, and the amount by which they are 
separated is known as their "interval". With our human 
spectacles on, we say that the interval between the depar- 
ture of the Flying Scotsman from King's Cross and its 
arrival at Edinburgh consists of 7 hours in time, and 
400 miles in space, but this is merely a private and sub- 
jective description indeed, the fireman might describe 
it differently as 7 hours of hard work tied to a single spot 
the footplate of the engine. When we take our human 
spectacles off, space and time fade away from view, and 
we see the departure of the train represented by a single 
point in the continuum, while another point represents 
its arrival. In the same way, with our human spectacles 
on, we say that the emission of a photon in Sirius and the 
reception of it by our eyes and instruments are separated 
by 51 million million miles, and by 8-65 years. When we 
take them off, we can only say that the two events are 
separated by so much interval in the continuum. 

Certain philosophers object to this mode of treating the 
question, on the grounds that it presupposes that space 
and time have no existence in their own right, but only 
as seen by a conscious mind. They protest that the 
separation of the continuum into its two ingredients is 
physical .and not psychological, so that, for instance, it 
does not require the mind, but only the body, of an 
observer just as the selection of a rainbow out of the 
rays of the sun does not require the mind, but only the 


physical eye, of the observer; even a camera lens is ade- 
quate. We can test this contention by putting a dead 
body, say Imperial Caesar dead and turned to clay, into 
the continuum. We now have the continuum and the 
world-line of Caesar's body, and nothing else, and it is 
hard to find any sense in which "space-time" has been 
separated into space and time* We may of course agree 
to take the direction of the world-line at each point of it 
as the direction of time, and the other three directions as 
space, so that if Caesar returned to his body, he would not 
think of himself as travelling through the world but of 
the world as processing past him. Yet if we do this, the 
separation has not been effected by Caesar's dead body, 
but by our live minds. We cannot argue that Caesar's 
mind would necessarily effect the separation in this way, 
if he returned to life. When I am climbing a mountain, 
I do not choose my space and time in this way; I think of 
myself as going up the mountain and not of the mountain 
as coming down to me, and we need not doubt that 
Caesar used to do the same. 

We must recollect that the space and time with which 
the theory of relativity deals admit of perfectly precise 
definition; they are the space and time which an observer, 
discovering or verifying or discussing laws of nature, 
chooses with his conscious mind as the framework against 
which to record his observations. The space and time of 
his choice may or may not coincide with the space and 
time of his conscious perception at the moment. The 
theory of relativity knows nothing of the latter, so that if 
we identify the two, it is at our own risk. 

The space and time of relativity are definite and pre- 
cise; often those of our conscious perception are not 


When we voyage through a rough sea, the solid structure 
of the ship suggests one space visually to our conscious- 
ness, the horizon suggests another, while the combination 
of gravity and the ever-varying accelerations of the ship 
suggests a rapid succession of quite different others, the 
continual conflict adding much to the woes of the un- 
seasoned traveller. To take a more placid example of the 
same thing, while an astronomer is taking observations, a 
driving clock keeps his telescope pointing in a fixed 
direction in space, but his body shares in the earth's 
rotation. He will almost certainly choose to record his 
observations with reference to the fixed direction in space 
of the telescope, but unless he allows the space of his 
conscious perception to alternate repeatedly between this 
and the terrestrial space in which his body is at rest, he 
will find the telescope running away from his seat 
When he jumps off a moving omnibus, he must change 
the space and time of his perception with extreme alacrity 
or else he will fall. Yet if he subsequently wishes to 
understand why he fell, he must choose either the omni- 
bus or the road as the framework for his calculations, 
and must definitely confine himself to the one or the 
other: he must in fact pass from the space-time of his per- 
ceptions to that of the theory of relativity. Because the 
two are so entirely different, the technique of avoiding a 
fall is the exact opposite of that of understanding it after 
it has occurred. 

Past 9 Present and Future 

Even when space and time are completely welded to- 
gether in the continuum, we can still distinguish two 
distinct kinds of interval. It is a dearly established law of 


physics that no material object can travel faster than a 
ray of light, so that the speed of light which we have 
already (p. 93) seen to be objective, the same for all 
travellers in space provides an absolute maximum 
speed. If two events are so located in the continuum that 
a body can be present at both, although not travelling at a 
speed greater than that of light, we say that the interval 
between them is "time-like". Thus the interval between 
any two events on the world-line of the same body as 
for instance the departure of the Flying Scotsman from 
King's Cross and its arrival at Edinburgh must always 
be a time-like interval. In the same way, all the events 
which affect the individual consciousness of any one of us 
are separated by time-like intervals. It is from this we 
get our intuitive conception of the flow of time. 

on the other hand we say that two events are con- 
nected by a "space-like" interval when a messenger 
would have to move faster than light to be present at both. 
A boundary line between the two kinds of intervals is 
formed by cases in which a messenger could be present at 
both events by travelling with exactly the speed of light. 
For mathematical reasons, which do not concern us here, 
the interval in such a case is described as a "zero- 
interval". When events are separated from us by a time- 
like interval e.g. the death of Queen Anne or the 
Coronation of King George we can only know of them 
by the exercise of memory, or by the use of records which, 
by their permanence, arrest the flow of time. When 
events are separated from us by a space-like interval, we 
cannot know of them at all; more time must elapse until 
the interval becomes first zero, and then time-like, so that 
we can know the events. But when the interval is exactly 


zero, we can have direct and immediate knowledge of the 
events the knowledge of seeing them with our own eyes. 

So long as a river of time was supposed to flow equably 
through all points of space, events could be divided per- 
fectly sharply into past, present and future. All the 
events of the world could be represented in a continuum 
constructed of three directions of objective space and one 
of objective time. A surface drawn through the three 
directions of space at any instant of time had the whole 
of the past on one side of it, and the whole of the future 
on the other. Itself, it contained the whole of the 

The theory of relativity has shewn us that such a di- 
vision is merely the private choice of a particular indi- 
vidual. The surface through the three directions of space 
of any individual still forms the "now" for that individual 
and divides his subjective time into his past, present and 
future. It is sometimes suggested that by changing his 
speed through space, any man can wave his "now" about 
in the space-time continuum, much as the man in charge 
of a searchlight can wave his beam of light about in 
ordinary space; he can re-divide the continuum into past, 
present and future, much as the searchlight operator can 
re-divide space into darkness, light and darkness. Indeed, 
he need himself do nothing. If he sleeps for eight hours, 
the rotation of the earth will have changed his speed 
through space by several hundreds of miles an hour, and 
will have rearranged his division of the continuum ac- 
cordingly. It may well have shifted ten years of time on a 
distant nebula from the past into the future, and so may 
seem to give its inhabitants ten years of their lives to re- 
live for good or for evil or would it be merely to re-live 


precisely as they had already lived them before? The 
paradox disappears if we remember that the time in- 
volved is merely that which an individual chooses for the 
recording of his observations of nature. It is not the time 
of his consciousness, still less that of the consciousness of 
the inhabitants of the nebula. We cannot wave anything 
about in the continuum which is more tangible than our 
own thoughts. 

Nevertheless, we see that time, as one dimension of the 
continuum, may be lacking in one of the properties with 
which our uncritical intuitions had endowed it, namely, 
its sharp division by a "now 59 into past and a future. 
We cannot prove either that such a division exists or does 
not; the theory of relativity merely suggests that when our 
intuitions suggested that it certainly did, they may have 
been misleading us. 

Again, it has been suggested that if this sharp line dis- 
appears, the concept of evolution in time may lose all 
meaning. We used to think of the universe evolving 
much as a pattern is woven in a loom. Space and time 
were the warp and the woof of its weaving. At any one 
instant, so much and no more has been irrevocably fixed; 
the rest still lay hidden in the loom, the womb of time, to 
be brought forth in due course. on the mechanistic view 
of nature, the loom had been set to work according to 
certain unalterable laws, so that the complete pattern 
was potentially existent from the outset and evolution 
became a mere synonym for the disclosing of predeter- 
mined changes, the tapping out of a pattern already 
designed. on a non-mechanistic view the loom was 
guided no one knew how, and might produce no one 
knew what. 


The theory of relativity in no way compels us to give 
up this simple picture of evolution, but it certainly casts 
doubt on the intuitive concepts on which it was based. 
And many have thought that, because of this, the whole 
comparison between the evolution of the world and the 
weaving of a pattern is faulty. Time, they say, no longer 
remains time or involves change; it is merely a geo- 
metrical direction of our own choice in the continuum. 
The pattern is not being continually woven piece after 
piece in a time which no longer exists, but is spread before 
us .complete in a continuum in which future events have 
just the same kind of existence as past events. We say that 
Australia exists although we. are not there to see we 
have perhaps never been there yet, but shall go there 
some day. In the same way, they argue, may we not say 
that the year 1942 exists? We have not been there yet, 
but perhaps we shall get there some day. Indeed an in- 
habitant of the nebula we just mentioned can wave his 
"now** through the continuum until its intersection with 
the world-line of our earth passes instantaneously from 
1932 to 1942, like the searchlight operator waving his 
beam of light over the clouds. The clouds are there 
whether the searchlight falls on them or not, and, so it is 
said, is the year 1942. 

Again we must recall that the space and time with 
which the theory of relativity deals are merely a time and 
space selected by our own minds for the discussion of 
natural laws. The theory of relativity does not assert that 
anything more tangible than our own thoughts can im- 
pinge on the year 1942, and this can hardly be said to 
endow the year 1942 with a real existence at the present 


The theory of relativity is built upon a perfectly 
definite and concise experimental basis; in the analogy we 
have already used, it is that the whole of physical nature 
follows us about like a rainbow, or like our own shadow. 
The result is that we cannot find evidence of our own 
motion by questioning physical nature, so that absolute 
space and absolute time do not enter into the nature we 
study in our physical laboratories. But this is not to say 
that they cannot exist in a wider external world than that 
of pure physics. Indeed, we shall see later that astro- 
nomical nature finds some evidence of absolute space and 
absolute time, and this has a bearing upon the questions 
we have just discussed. We shall return to this later 

(p- 142). 

From the time of Plato onwards, philosophic thought 
has repeatedly returned to the idea that temporal changes 
and the flux of events belong to the world of appearances 
only and do not form part of reality. The reality, it is 
thought, must be endowed with permanency, otherwise it 
would not be real, and we could have no knowledge of 
it behind the kaleidoscopic changes of nature there 
must be a permanent kaleidoscope, imparting a unity to 
the flux of events. 

For this kind of reason philosophers have insisted that 
reality must be timeless, and time merely, in Plato's 
phrase, "a moving image of eternity". Bradley,* for 
instance, writes: 

"Change, as we saw, must be relative to a permanent. 
Doubtless here was a contradiction which we found was not 
soluble. But, for all that, the fact remains that change de- 
mands some permanence within which succession happens. 

* Appearance and Reality, pp. 207, 209. 


I do not say that this demand is consistent, and, on the con- 
trary, I wish to emphasize the point that it is not so. It is 
inconsistent, and yet it is none the less essential. And I urge 
that therefore change desires to pass beyond simple change. 
It seeks to become a change which is somehow consistent with 
permanence. Thus, in asserting itself, time tries to commit 
suicide as itself, to transcend its own character and to be taken 
up in what is higher". 

And again, two pages later: 

"Time is not real as such, and it proclaims its unreality by 
its inconsistent attempt to be an adjective of the timeless. It is 
an appearance which belongs to a higher character in which 
its special quality is merged. Its own temporal nature does 
not there cease wholly to exist but is thoroughly transmuted. 
It is counterbalanced and, as such, lost within an all-inclusive 
harmony. ... It is there, but blended into a whole which we 
cannot realize 95 . 

We may notice how the absorption of space and time 
into a higher unity, the space-time continuum, which 
transcends both and is changeless, satisfies the require- 
ments of the philosophers, although only at the expense of 
relegating evolution to the realm of appearance. 



Action at a Distance 

Primitive man saw nature as a collection of objects which 
acted on one another, if at all, by direct contact; he was 
familiar with the pressure of wind and water on his body, 
the fall of raindrops on his skin, the thrust by an enemy, 
but action at a distance was somewhat of a rarity in his 
scheme of things. 

Early science hardly advanced on this view, picturing 
matter as consisting of hard objects, no two of which 
could occupy the same space because one invariably 
pushed the other out of the way by direct contact. The 
science of a later era, however, found many instances of 
action at a distance. A magnet attracts iron filings to 
itself from a distance, and is itself acted on by the yet 
more distant magnetic poles of the earth; two electrified 
bodies attract or repel one another across the intervening 
space according as they are charged with opposite or 
similar kinds of electricity; the sun attracts the planets, 
and the earth the falling apple. In none of these cases 
can anything tangible be found to transmit the attractions 
and repulsions. It is true that the space between the 
interacting objects will often be occupied by air, but thig 
does not transmit the action; electrified bodies and mag- 
nets attract rather more forcibly in a perfect vacuum than 
in air, while an apple falls more freely and rapidly when 
there is no air-resistance to break its fall. The sun attracts 



the planets across a space which is practically void of 

At a still later period, matter was found to be wholly 
electrical in its structure, consisting of particles which 
carried electrical charges, and of nothing else. These 
particles were so minute that an object occupied enor- 
mously more space than the aggregate of the amounts 
occupied by its separate particles. Roughly a ton of 
bricks occupies a cubic yard, while the millions of 
particles which form this ton of bricks occupy only about 
a cubic inch; all the rest is empty space. The particles of 
the brick hold one another at arm's length through the 
electric forces they exert on one another. If these forces 
could be abolished, we could pack all the particles of a ton 
of bricks within a cubic inch of space. In the interiors of 
the densest stars the particles are packed as closely as 
this; the electric repulsions are not actually abolished, 
but they count for nothing against the immense forces 
resulting from the pressure of the star itself. 

In ordinary everyday life, however, these electric forces 
maintain their supremacy against all others, and the 
pushes and pulls of common objects are as much the out- 
come of action at a distance as is the attraction of a mag- 
net for iron filings or of the pole for the compass-needle. 
When the wind blows on my face, the molecules of air 
come to within about a thousand-millionth part of an 
inch of my skin, but no nearer; at this distance the mole- 
cules of my skin repel them so violently that they turn 
back the way they came. The sensation of the impact of 
the wind on my face is the outcome of the reaction of the 
electric forces exerted by the molecules of my own skin 
just as I feel a reaction in my foot when I kick a football. 


It is the same throughout nature. When we look at 
it through a sufficiently powerful mental microscope, we 
find no instances of actual contact; nature appears to have 
only one mechanism, which is action at a distance 
action across intervening space* 

For a long time it was thought that the ether, which 
had been originally introduced to transmit waves of light, 
might transmit all these other actions as well, and so serve 
as the general mechanism underlying nature. Innumer- 
able experiments were tried in the hope of discovering 
any signs of the existence of an ether. They one and all 
failed, which of course only amounts to saying that all 
the phenomena of nature were found to conform to the 
principle of relativity. Space appeared to be entirely 
empty except in the isolated regions which were occupied 
by objects. 

How, then, was the action transmitted from the mag- 
net to the iron filings, from the earth to the falling apple, 
from the moon to the tides, from the cricket bat to the 
ball? If the ether was no longer available for this pur- 
pose, something else must be found to take its place. The 
story of the quest for this new something brings us to the 
very heart of modern science. 

The Curvature of Space 

We have an intuitive belief that space is flat or Euclidean 
parallel lines never meet, and so on. This is based on 
our everyday experience. Yet all that this actually tells 
us is that we can bring law and order into the arrange- 
ment of the objects with which our everyday life is con- 
cerned by imagining them arranged in a space of this 
kind. We might try other arrangements of objects, as for 


instance arrangement in a four-dimensional space, and 
should soon discover that the ordinary Euclidean three- 
dimensional space, which the layman describes as 
"space" without any adjectives, had some sort of pre- 
eminence, at any rate for the arrangement of such objects 
as we encounter in ordinary life. Yet we have no right 
to assume that the whole universe could be reduced to 
law and order by being arranged in such a space; if we do 
so, we merely repeat the old mistake of thinking that all 
nature is like the small fragments of it with which we are 
familiar the "common sense" view of nature. 

There is a certain peculiar sect whose members insist 
that the earth's surface is flat, so tSat parallel lines drawn 
on it can never meet. Their intuitive concept of the 
earth's surface is like ours of space; both are based upon 
an imperfect acquaintance with the whole. And, just 
because the concept is intuitive, no amount of abstract 
argument will persuade the man who holds it that it is 
faulty. The little bit of the earth in which his daily walk 
or daily labour lies is flat, and he absorbs this flatness 
into his mental make up, until he is unable to conceive 
any possibility except flatness, which he then wrongly 
extends to the whole earth. It becomes a matter of com- 
mon sense to him that the earth not only is, but must 
be, flat. 

Suppose, however, that a member of this sect took to 
travel. He might find it of interest to draw a map of the 
earth on which to record his journeys it would of 
course be a flat map, as for instance an ordinary Mercator 
projection, which can be found in any school-atlas. 
When he travelled by sea, he might copy down the ship's 
position day by day, and in this way record the course of 


the ship on his map. Now the shortest course between 
two points in the northern hemisphere always bends 
towards the north pole, so as to "take advantage of the 
shorter degrees of longitude", while in the southern hemi- 
sphere there is a corresponding deflection towards the 
south pole. For instance, the shortest course from South- 
ampton to New York goes farther north than either 
Southampton or New York; the shortest course from 
Gape Town to Cape Horn goes farther south than either. 
When such courses are mapped on a Mercator projection 
they look very curved; a shortest course only looks straight 
if it happens to lie either due north and south, or else 
dead along the equator. No doubt our traveller would 
at first be surprised to find that all the steamers he 
patronised seemed to follow very curved tracks; he might 
imagine that they were pulled out of their direct courses 
by forces emanating from the two poles of the earth. 

one day, however, he might shew the curved tracks on 
his map to a friend, and discuss their meaning with him. 
It would be a tremendous revelation if his friend took him 
to a spherical globe on which the countries of the earth 
were marked in their proper positions, stretched bits of 
string from point to point, and shewed him that when the 
string was pulled tight so as to give the shortest path from 
point to point, it invariably lay exactly over the course 
which had been followed by the steamers in his travels. 
He would then see that the ships had actually been fol- 
lowing the shortest courses on a curved earth. Their 
tracks had appeared curved on his map, not because 
forces were pulling them out of their courses, but because 
the framework of latitude and longitude, which actually 
is twisted by the curvature of the earth, had been arti- 


ficially untwisted in his Mercator projection. In brief, he 
had been trying to describe his journeys against a back- 
ground which was not true to nature. He would discover 
that, although a flat map was perfectly suited to the 
arrangement of places in the immediate neighbourhood 



Fig. 1 Kg. 2 

Notwithstanding its apparent curvature, the course from A to B shewn 

in fig. 1 is the shortest possible. If it is mapped out on a globe, as shewn 

in fig. 2, a tightly-stretched string will be found to cover it exactly. 

of his own home, it was not at all suited to the representa- 
tion of the whole of the earth's surface. 

Newton's theory of gravitation had explained the 
curved paths of planets, comets and cricket balls precisely 
as our flat-earth traveller had explained the curvature of 
his steamer tracks. The latter imagined that his steamers 
sailed in a flat sea and were drawn out of their straight 
courses by a pull emanating from the poles of the earth. 
Newton imagined that the planets swam in a flat space 
and were drawn out of their straight courses by a pull 
emanating from the sun; he imagined that the cricket 


ball was thrown in a flat space but that its course curved 
earthward because of a gravitational pull emanating from 
the earth. 

We have already noticed that this theory of gravitation 
does not conform to the requirements of the theory of 
relativity. Einstein replaced it by a new theory which 
does; it is an extension of the simple or "restricted" theory 
of relativity which we discussed in our previous chapter, 
and is generally known as the "generalised" theory 
of relativity. It does not picture the planet and the 
cricket ball as describing curved paths in a straight (or 
Euclidean) space, but shortest paths in a curved space. 
Actually what is curved is not primarily the space of our 
ordinary life, but the four-dimensional continuum, the 
objective blend of space and time which we considered in 
our last chapter. The theory supposes that gravitating 
bodies, such as the earth, curve this up in their neigh- 
bourhood, the word neighbourhood now implying 
proximity in time as well as in space. This curvature de- 
flects the planet and the cricket ball much as the mole- 
hill on the bowling-green deflects the bowl. Just as the 
friend of the flat-earth heretic was able to shew him a 
curved surface in this case a sphere in which all his 
complicated curves became shortest courses, so Einstein 
has shewn the scientific world a curved continuum in 
which the complicated tracks of planets, cricket balls, 
rays of light, and so forth, all reduce to shortest courses. 
As the continuum is curved, the space of our everyday 
life, which is a cross-section of it, must also be curved. 

We notice that action at a distance has fallen out of the 
picture. If we fix our attention on the three space-dimen- 
sions of the continuum, we may say that the earth keeps 


space in its neighbourhood continually curved, so that 
when the cricket ball is thrown through this space it de- 
scribes a curved path, much as though it were rolled along 
a hill-side. The proximate cause of the curvature of its 
path is the contiguous space, not the distant earth. True 
action at a distance action transmitted instantaneously 
across intervening space inevitably had to drop out, 
because of the impossibility of synchronising time at two 
distant points. 

The generalised theory is concerned with precisely the 
same space and time as the restricted theory, namely, the 
space and time which an investigator or scientist chooses 
with his conscious mind for the recording of his observa- 
tions of nature. He has a right, just as our traveller had, 
to record these on any kind of map he pleases. When our 
traveller tried a flat map he had to introduce a compli- 
cated system of forces to explain his facts of observation, 
and when Newton tried a flat space he had to do the 
same. Just as the traveller's friend shewed him a better 
kind of map, so Einstein has shewn us a better kind of 
map. Using this new kind of map, Einstein has been 
able to dispense with gravitational pulls, and at the same 
time draw a far simpler and far more accurate picture of 

Shortest Courses 

Although the concept of a curved map of nature first 
entered science with Einstein, the complementary con- 
cept, of material objects and rays of light following the 
shortest possible course, goes back to the earliest ages of 
science, and has a most respectable pedigree behind it. 
Revolutionary though Einstein's new theory seemed 
when it was first announced, it merely put science back 


on the road it had been travelling for two thousand years 
before Newton, This may not be of much interest to 
the practical scientist except as a matter of history, but it 
is of considerable interest to one who wishes to under- 
stand the philosophical implications of modern science. 

Three hundred years before Christ, Euclid had defined 
a straight line as the shortest distance between two points, 
and announced that light travelled in straight lines. 
Thus he knew that light took the shortest path from point 
to point at any rate under ordinary terrestrial con- 
ditions. He also knew that light could be deflected from 
its path by a mirror, and discovered the laws it obeyed 
when this happened it followed the same path as a 
perfectly hard ball bouncing off the mirror. 

Euclid saw these only as two totally disconnected facts. 
About a century later, Hero of Alexandria combined 
them in a very significant synthesis, shewing that even 
when light was reflected at a mirror, its path was still the 
shortest by which it was possible to travel to the mirror 
and back again. Mirror or no mirror, light followed the 
shortest path. 

Light can experience other accidents besides reflection. 
For instance, we can see the setting sun after it has passed 
well below the horizon in the geometrical sense. We say 
we see it by rays which are "refracted", or bent, by the 
earth's atmosphere. These rays cannot be following the 
shortest path, for a bent path can never be as short as a 
straight one. 

Actually, they are following the quickest path. When 
light passes through any material substance, such as glass 
or air, the particles of matter slow down its motion, and 
the denser the substance, the greater the slowing down. 


Thus when light has to travel through a number of dif- 
ferent substances, it can often save time by travelling 
through a less dense substance even though it has to travel 
an additional distance to get to and from this substance 
just as the steamer saves time by going farther north 
than either its starting-point or its port of arrival, and 
teVing advantage of the short degrees of longitude up 
north. As an example of this, the quickest path for a ray 
from the setting sun to us is one which avoids travelling 
overmuch through the dense air near the earth's surface, 
the rays bending round this rather than travelling directly 
through it. It is found to be a quite general law of nature 
known as Fermat's principle that when rays of light 
have to make their way through a retarding substance, 
they take the quickest path from point to point. 

Hero had stated his law in the form that light, whether 
reflected or not, takes the shortest course from point to 
point. He might have said with equal truth that it takes 
the quickest, and had he done so, his statement would 
have been true for refracted light also. Indeed it would 
have been true for light moving in all the ways known to 
science; all known light takes the quickest path from point 

to point. 


The undulatory theory, which interpreted light as waves 
in an undulating ether, provided a very simple explana- 
tion of this. When waves or ripples travel over a pond, 
points on the surface of the water are alternately de- 
pressed and elevated beyond the normal level of the 
undisturbed water. When two sets of waves are travers- 
ing the surface at the same time, one may tend to elevate 
a particular spot on the surface, while the other tends at 


the same moment to depress it. When the two effects 
neutralise one another in this way, the two sets of waves 
may jointly produce less disturbance than either would 
produce alone. This is the phenomenon known as 
"interference 33 ; its essence is that the disturbances pro- 
duced by sets of waves must be treated as algebraic 
quantities elevations (positive) and depressions (nega- 
tive) and not as mere arithmetical quantities, and of 
course the sum of two large algebraic quantities may be 
small, or even zero, if they are of opposite signs. 

It is easily shewn that wo waves which started from the 
same source at the same instant, and so have travelled 
for the same length of time, will be always in step with one 
another or, to use the technical expression, in the same 
"phase". This will also be the case if their times of 
travel differ by the times of one, two, three or any exact 
number of complete oscillations. Now if two waves which 
are in the same phase, or nearly in the same phase, meet 
at any point, their joint effect is greater than that of 
either singly we may, for instance, have two crests 
superposed, or two troughs superposed; in either case the 
disturbance is intensified. on the other hand, if the 
waves are in opposite phases, crest is superposed on 
trough, or trough on crest, and we have destructive 

When waves are sent out from any specified point, we 
can imagine them travelling in circular ripples until 
something occurs to disturb their regular forward motion; 
they may, for instance, encounter an obstacle. As soon as 
this occurs, it is convenient to think of each little disturb- 
ance as itself forming a centre for new waves; these new or 
"secondary" waves now spread out from every point of 


the primary waves, the whole complex system crossing 
and recrossing one another and either reinforcing or 
destroying each other by interference at every point. 

Even when no such obstacles exist, we are still free to 
imagine each wave breaking up into a multitude of new 
waves at each point of its journey, so that the whole of the 
disturbed region can be thought of as filled with waves 
crossing and recrossing one another. Mathematical 
analysis shews that these waves will reinforce one another 
all along the path of quickest journey, while they neutral- 
ise one another along all other paths. It is not strictly 
true that the waves travel only along the quickest path; 
they travel along all possible paths, but destroy one 
another on all paths except the quickest. 

This is not quite the whole story, since we have seen 
that waves whose times of travel differed by exactly one, 
two, three, or any integral number of complete oscilla- 
tions, would also reinforce one another when they met, 
and so ought to be visible, although neither had travelled 
by the quickest path. The undulatory theory had the 
great triumph of its life when this possibility was found to 
account exactly and completely for all known phenom- 
ena of diffraction and interference. These phenomena, 
which had dealt a fatal blow to the corpuscular theory, 
seemed to provide incontrovertible proof of the truth 
of the undulatory theory. 

Least Action 

It was also found that material objects could be brought 
under the same synthesis as rays of light. Aristotelian 
doctrine had asserted that substances tended to rise or 
sink, according as they were light or heavy; every object 


moved so as to find its own proper place in the ordained 
scheme of nature. Galileo and Newton made it clear that 
this was not a universal law, but a mere local effect 
resulting from the gravitational pull of the earth. In the 
absence of such forces all objects moved in straight lines 
with uniform speed; like light they took the shortest path 
from point to point, or, if their speed of motion was 
assigned, they took the quickest path from point to point 
again like light. 

When forces were in operation, their effect was, in 
Newton's words, to "draw bodies off from their rectilinear 
path", in which event it was clear that their path could 
neither be the shortest nor the quickest possible. 

The French mathematician-philosopher Maupertuis 
argued that even in such cases the path must exhibit some 
perfection worthy of the mind of God. When there were 
no forces in action, it was already known that this per- 
fection took the form of either the distance or time of the 
motion being a minimum; hence when forces were in 
action, something else must still be a minimum. Our 
modern minds find it a strange line of attack, but it suc- 
ceeded. Maupertuis discovered a quantity known as the 
"action", which proved always to have the minimum 
possible value. This quantity is associated with the mo- 
tion either of a single object or of a group of objects; just 
as each bit of travel on a railway involves a certain ex- 
penditure in railway fare, so each bit of motion involves 
a certain expenditure of "action". Our expenditure on 
railway fares is not usually exactly proportional to either 
the time or the distance we travel, and in the same way, 
when forces were in operation, the expenditure of action 
was not exactly proportional to either the time or the 


distance of the journey. Yet it was easily calculable, and 
Maupertuis shewed that objects invariably moved in such 
a way as to make the total expenditure of action a 

When no forces were in operation, the expenditure of 
action was exactly proportional to the time, and the new 
principle of least action absorbed the older principle of 
least time. Thus the new principle provided a synthesis 
to cover the motion both of objects and of light of 
matter and radiation, the two constituents of the physical 

Incidentally, this principle enables us to take a first step 
at least towards understanding the puzzle of the appar- 
ently dual natures of radiation and matter, both of which 
sometimes remind us of waves and sometimes of particles. 

We have already seen that any picture we may draw 
of nature must be built up of concepts already existing in 
our minds. The number of such concepts is very limited, 
but waves and particles happen to be two of the most 
familiar of them, with the result that we tend to think in 
terms of waves and particles. For a time it seemed as 
though radiation could be pictured quite perfectly as con- 
sisting of waves, and matter as consisting of particles. We 
now know that nature is not as simple as this; neither 
matter nor radiation can be pictured either as pure 
particles or as pure waves. A wider view shews us both 
radiation and matter as entities whose behaviour con- 
forms to the mathematical principle of least action. 
Thus to obtain a true picture of nature, we must try to 
picture both matter and radiation in terms of familiar 
things whose behaviour also conforms to this principle, 
but we again find our choice almost limited to particles 


and waves. Thus we picture radiation sometimes as 
particles and sometimes as waves, and also, as we shall 
see later, do the same with electrons and protons. 

Least Interval 

Nevertheless a reservation must be made the simple 
synthesis of "least action" did not provide a perfect ex- 
planation of nature. At first it was found possible to alter 
and extend it, so as to bring one new phenomenon after 
another under its scope, but ominous sign ! each ex- 
tension made it more intricate and, to all appearances, 
more artificial, until finally it broke loose from the facts al- 
together; nothing could make it fit. Even in its most intri- 
cate form, it still predicted that gravitating bodies should 
bend starlight only half as much as they are observed to do, 
and that the perihelion of Mercury should stand still in* 
stead of advancing round the sun at about 80 miles a year. 

Then the theory of relativity came to the rescue, first 
explaining why the principle failed, and then shewing 
how it could be put right The restricted theory, de- 
scribed in our previous chapter, shewed that the principle 
was bound to fail. For any true picture of nature, or 
principle to explain the workings of nature, must permit 
of representation in the undivided four-dimensional con- 
tinuum. The principle of least action, on the other hand, 
did not permit of representation in this framework until 
it had been divided up into space and time. 

There was found to be only one entity capable of repre- 
sentation in the new framework which could possibly re- 
place action. This was the "interval", the blend of space 
and time which lay between two events, so that the only 
principle with any self-consistent meaning is one 


of "least interval". It was the essence of the generalised 
theory of relativity that this interval must be measured in 
a curved continuum. The world-line of a particle or 
other moving body could be obtained like the steamer 
tracks in our earlier analogy by stretching a tight 
string from point to point. 

When the principle is amended in this way, it takes 
upon itself the role which at one time seemed to be filled 
by the principle of least action, and is found to govern 
and to predict the whole motion of the universe, in so far 
as this is determined by what we used to describe as the 
forces of gravitation. It seems possible, although by no 
means certain, that electrical forces admit of explanation 
in terms of the same principle, so that when an electron 
appears to be compelled to describe a curved orbit by 
electric forces, it also is finding the shortest possible path 
through a curved continuum. Einstein's recent "Unitary 
Field-theory 55 attempts to specify the exact kind of con- 
tinuum necessary for such an explanation, but its success 
is not yet established. If ever complete success is achieved 
in this direction, the principle will equally govern the 
motion of a ray of light and of a moving body, and will 
remain valid whatever physical agencies are in action, so 
that we shall be able to combine all the operations of 
nature in one synthesis; they will all have become shortest 
courses in a curved four-dimensional space. 

Generalised Relativity 

In this way, Hero's first simple synthesis of the two laws 
of Euclid has been gradually extended and modified until 
it has finally emerged as a general principle covering all 
the large-scale phenomena of nature, and possibly sub- 


atomic phenomena as well. We used to think of the 
principle as applying to particles of matter and to rays of 
light existing and travelling in a framework of space and 
time. But in the process of making a perfect fit between 
the principle and the observed facts of nature, we have 
had to discard space and time as objective realities, forces 
and mechanism have dropped out of the picture alto- 
gether, and we are left only with empty space and empty 
time, first welded together to form a four-dimensional 
continuum differing in quality from either space or time, 
and then curved and contorted. We can no longer think 
of the varied phenomena of nature as arising from a blind 
dance of atoms as they are pushed and pulled about by 
mechanical force; we must attribute them to efforts of we 
know not what to find the shortest path through the 
tangled maze of the space-time continuum. 

At this point it becomes natural to inquire whether 
Einstein's picture represents anything in ultimate reality, 
or merely provides a convenient way of describing phe- 
nomena. For we must remember that the most con- 
venient description is not always that which is closest to 
reality. Although ships' captains are aware that the 
world is round, they still find it most convenient to map 
out the tracks of their ships on flat Mercator projections, 
as though the earth was flat. In the same way, Einstein's 
straight paths in a curved space may conceivably be 
merely convenient pictures which represent the phe- 
nomena but not the reality behind. 

The Einstein Universe 

Mathematical analysis shews that there are more ways 
than one of curving the continuum so as to explain the 


paths of astronomical bodies and rays of light. These all 
give equally good pictures of the astronomical phenomena 
of the solar system, or any other small part of the universe, 
yet one at most can represent ultimate reality. Many are 
disqualified because they lead to obvious absurdities when 
the universe is considered as a whole. Einstein found one 
way which he considered free from such objections. 

This postulated two distinct kinds of curvature. The 
first was a curvature inherent in the continuum itself, 
which rolled the whole continuum up into a closed 
surface, much as the whole surface of the earth is rolled 
up into a closed globe. The second consisted of local 
irregularities which were superposed on to the main cur- 
vature, just as a curvature of hills, mounds and molehills 
may be superposed on to the main curvature of the earth's 
surface. These smaller irregularities were caused by, or 
at any rate associated with, the presence of matter, and 
were responsible for the observed curvatures of the paths 
of planets and rays of light. Over a small region of space, 
such as the solar system, the main curvature produced too 
small an effect to permit of observation. 

If the continuum were curved in this way, then space, 
being a cross-section of the continuum, was also curved. 
It was moreover curved in such a way that there was 
only a finite amount of it. This representation possessed 
certain definite advantages over all older views of space, 
which had always been confronted with the dilemma 
that, although it was impossible to imagine any limit to 
space, yet unlimited space was objectionable on purely 
scientific grounds. If matter extended through unlimited 
space, there would be an infinite amount of it exerting its 
attraction on planets, stars and galaxies, and this would 


cause them to move at speeds far greater than those 
actually observed at infinite speeds, in fact. The only 
escape would be by supposing that there was only a finite 
amount of matter, and as this could only occupy a finite 
amount of space, it left an infinite amount of space en- 
tirely devoid of matter. Such a concept could not be dis- 
proved as being in any way ridiculous or impossible, but 
it was certainly not convincing by its inherent reason- 
ableness. Kant had dismissed it on the grounds that an 
infinite empty space would contain nothing by whicK to 
locate the position of a finite material world. If the 
question "Where is the finite matter in infinite space?" 
admitted of no answer, then there could not, according to 
Kant, be finite matter in infinite space. 

However serious these difficulties were, Einstein re- 
moved them all by his concept of a closed finite space. 

He found that, when the average density of matter in 
space is assigned, there is one and only one radius at 
which space can stand in equilibrium without either ex- 
panding or contracting. He accordingly supposed this to 
be the actual radius of space; it could of course be calcu- 
lated as soon as the average density of matter in space 
was discovered observationally. 

Since space was supposed to retain the same radius 
through all time, the curvature of the space-time con- 
tinuum could not be geometrically like that of the surface 
of the earth; it was rather like that of a roll of paper, or 
better still that of a single sheet of paper pasted so as to 
form a cylinder of paper like a postal-tube (cf. fig. 3). 
In this model of the continuum, any cross-section the 
paper itself, not the circular area it encloses represents 
space at any instant, while the passage of time is repre- 


sented by lengthwise motion along the cylinder. Thus 
space became finite and constant in amount, while time 
remained infinite, extending from an eternity back in the 
past, through the present to an eternity in the future. 

The Expanding Universe 

Recent mathematical investigations have shewn that the 
continuum cannot be represented by so simple a model. 

Fig. 3. Fig. 4. Fig. 5. 

Diagrammatic representations of space-time to exhibit various 
theoretical possibilities. 

Einstein originally introduced his large-scale curvature 
into the universe to keep it in equilibrium. Friedmann, 
Lemaitre and others have shewn that a universe whose 
equilibrium was secured in this way would not stay in 
equilibrium. It would be unstable, in the sense that space 
would at once start expanding or contracting the 
general mathematical theory leaves both alternatives 


open and that the process would continue with ever- 
increasing speed. Thus we must not picture space-time 
by a cylindrical roll of paper, but rather by a cone or 
horn-shaped surface, such as the cardboard surface of a 
megaphone (cf. fig. 4). Time is still represented by the 
central axis. Space, the cross-section of the horn, is still 
finite, but for ever changes its dimensions as we move 
about in time* In other words, space cannot remain of 
constant size, as Einstein originally imagined, but must 
be for ever expanding or contracting. 

If Einstein's molehill curvature represents anything in 
ultimate reality, and is not a mere convenient means of 
picturing the paths of the planets, then the large-scale 
curvature, which follows almost as a logical corollary to 
it, ought also to represent something in ultimate reality. 
Clearly it is important to look for observational evidence 
of its existence. 

The most obvious property of this large-scale curvature 
is that it closes space up, so that if we tried to travel on 
for ever through space, we should merely come back to 
our starting-point, as Drake did when he circumnavigated 
the globe. It is of course no good our trying to obtain a 
proof of the curvature of space by an actual circum- 
navigation of space for one thing life is too short. A 
ray of light might have a better chance, for it travels at 
ten million miles a minute and is not limited to a lifetime 
of three-score years and ten. It was at one time thought 
that a sufficiently powerful telescope might enable us to 
look round space and see our own galaxy by light which, 
starting many millions of years ago, had travelled round 
the whole of space, and finally come back to its starting- 
point. Such an experience would of course constitute a 


very direct and convincing proof of the curvature of space, 
but we no longer believe it to be possible. 

It is not, however, necessary to travel round space to 
obtain a convincing proof of its curvature, any more than 
it is necessary to travel round the earth to prove that its 
surface is curved. If we draw a circle on a perfectly flat 
piece of paper, we know that its circumference is x times 
its diameter, where v denotes the number 3-14159.... 
This is true, no matter how large or how small our circle 
may be, provided always that we draw it on a perfectly 
flat surface. It is not, however, true of a circle drawn on 
a curved surface such as that of the earth. A circle of 
synajl diameter still has a circumference equal to 3-14159 
times its diameter, but the ratio becomes less as the circle 
is made larger. A circle of 1000 miles diameter does not 
have a circumference of 3141-59 miles, but only of about 
3110 miles. If a surveyor were to draw such a circle on 
the earth's surface, and then measure its circumference, 
he would obtain a ready proof of the curvature of the 
earth's surface. 

In theory it is possible to test the curvature of space 
in a similar way. If we construct a small sphere of any 
substance, its surface will be T times the square of its 
diameter, where v is still the same number 3- 141 59. Now 
if space were uncurved, the surface of any sphere would 
always be 3-14159 times the square of its diameter, no 
matter how great this diameter might be. But if space is 
curved, the ratio continually decreases as the sphere gets 
larger, just as on the earth's surface the ratio of circum- 
ference to diameter decreases as the circle gets larger. 

If then we could map out an immense sphere in space 
and measure up the total area of its surface, we should 



have an immediate means of testing whether space is 
really curved in the way that Einstein imagined. Yet 
even if the curvature exists, its scale is so large that its 

Fig. 6. 

Fig. 7. 

effects are inappreciable in the solar system, and we 
should have to make a sphere millions of millions in 
diameter before we could hope to detect it. And, apart 


from the practical difficulties of mapping out a circle of 
such dimensions, there are two theoretical difficulties 
first (p. 76) we have no means of locating points in space, 
and second (p. 72) we have no objective means either of 
drawing straight lines or of measuring their lengths. 

Although this line of thought will not enable us to test 
whether space is curved, it goes some way towards help- 

Fig, 8. 

ing us to imagine the kind of curvature postulated by the 
generalised theory of relativity space contracts as we 
get farther from home, so that the content of a sphere of 
assigned radius is always less than it would be if space 
were flat. We can construct a flat area in our imagina- 
tions by joining a number of triangles together at their 
vertices, as in fig. 6, but if we want to imagine a curved 
(say a spherical) surface, we must replace our triangles by 
areas shaped like the leather sectors which are stitched 
together to make a football, as in fig. 7. In much the 
same way we can imagine a flat space formed by joining 


a number of sugar-cones together at their vertices, but if 
we want to form a curved (spherical) space, we must 
replace our sugar-cones by spindle-shaped bodies, as in 
fig. 8. If the reader can imagine enough spindles tied 
together at A to fill the whole of the space surrounding .4, 
and then (this is where the difficulty comes) imagine them 
all bent about, equally and similarly, until all their other 
ends 5, B', B" meet in a point, he will have made for 
himself a sort of mental picture of spherical space. More 
likely, however, he will be unable to imagine this at all, 
because of the difficulty of conducting his imagination out 
of ordinary three-dimensional space; he will then have a 
proof of the impossibility, to which we have so often re- 
ferred, of either picturing or describing things except in 
terms of concepts made familiar by our everyday life. 

As the curvature of space cannot be directly tested in 
either of the geometrical ways we have just described, 
we must fall back on the more indirect way of examining 
whether the various mathematical consequences of this 
curvature are to be found by observation. If the curved 
continuum merely provides a convenient means of pictur- 
ing phenomena, there is no reason why all its mathe- 
matical consequences should be found in nature; mere 
representation must be expected to part company with 
reality somewhere. on the other hand, if this curved 
continuum has a real existence in nature, all the mathe- 
matical consequences of this existence ought to be con- 
firmed by ob ervation. And the principal of these is that 
space must be either expanding or contracting at a uni- 
form rate throughout its whole extent. 

Now the great nebulae out beyond the Milky Way 
provide just the means of testing this prediction of theory. 


They are the largest and most distant objects known to 
astronomy, and yet, in relation to the universe as a whole, 
they are mere straws floating in the stream of space, and 
ought to shew us in what way, if any, its currents are 
flowing. If the continuum is curved in the way we have 
described, these nebulae ought all to be receding from us, 
or else all rushing towards us, the speed of each nebula 
being exactly proportional to its distance from us. 

At this point observation takes up the tale. If recent 
astronomical observations can be taken at their face 
value, these nebulae are all receding from us, and .this at 
quite terrific speeds. Moreover, their speeds are almost 
precisely proportional to their distances, exactly as de- 
manded by theory. Nebulae whose light takes a million 
years to reach us are receding at (in round numbers) 100 
miles a second, nebulae at twice this distance at double 
this speed, and so on. Nebulae whose distance is esti- 
mated to be 135 times as great as this so that their 
light takes 1 35 million years to reach us have just been 
found to be receding from us at the colossal speed of 
15,000 miles a second, the greatest speed so far known to 

These speeds are so great that many astronomers have 
doubted whether they are real surely, they say, the ob- 
servations must permit of some other and less sensational 
interpretation. It may be so; we are still a long way 
from being able to pronounce a final judgment on these 

Sir Arthur Eddington* has recently tried to investigate, 
in a purely theoretical manner, the speeds with which the 
nebulae ought to move if the universe were expanding in 

* The Expanding Umvarse (1933), chap. iv. 


the way required by the theory of relativity. The speeds 
he calculates agree with those actually observed to within 
a factor of about 2, which is as good an agreement as 
could reasonably be expected. The whole investigation is 
extremely speculative and does not yet, I think, command 
the general assent of mathematicians. If ever these cal- 
culations can be put beyond criticism, they will provide 
a very strong confirmation of the whole theory of the 
expanding universe, as developed by Friedmann and 

on the other hand, there are very grave astronomical 
objections to accepting the observed speeds of recession as 
real. If they are real, the universe must be changing very 
rapidly; it is doubling its dimensions every 1,300 million 
years or so. If we assume that the speeds have always 
been as at present, and trace the motion back for 2,000 
mil linn years, we find the whole universe concentrated in 
a quite small region of space. Actually the theory of the 
expanding universe she\vs that the speeds would diminish 
as we go backwards in time, and that there is no definite 
limit to the time during which expansion can have been 
in progress, but it also suggests rather forcibly that this 
time can hardly be more than about a hundred thousand 
million years. Against this, the time needed for the 
universe to attain its present stage of evolution can be 
estimated in a great number of ways, and all agree in 
indicating a period of millions of millions of years. It is 
exceedingly difficult although perhaps not absolutely 
impossible to imagine that the universe can have been 
evolving for ten or a hundred times longer than space has 
been expanding. It is even more difficult although 
again perhaps not absolutely impossible to imagine 


that space can have been expanding for millions of mil- 
lions of years. The difficulty is so grave as to cast real 
discredit on the whole mathematical theory of the ex- 
panding universe. 

A recent short note by Einstein and de Sitter may be 
found to contain a means of escape from this very serious 
dilemma. We have seen how Einstein originally thrust 
his large-scale curvature on to the universe because he 
saw no other way of keeping it quiet, and restraining it 
from either exploding or collapsing. Now that space ap- 
pears to be exploding in spite of all Einstein's efforts to save 
it, the inherent large-scale curvature seems to play a less 
essential part in the scheme of things than it once did. 

Einstein and de Sitter have accordingly examined what 
reasons, if any, remain for supposing that space possesses 
this inherent curvature. They find none at all. It is no 
longer needed to keep space at rest, because space is not 
at rest, and neither the mathematical equations nor the 
observed recessions of the nebulae in any way require it. 
Thus we become free to suppose that space would be flat 
if it were perfectly empty of matter, and that it owes the 
whole of its curvature, both coarse and fine, to the objects 
which occupy it. While this does not carry us much 
further towards a positive understanding of the nebular 
motions, it brings a whole new class of possibilities into 
the field. It has, for instance, been suggested that the 
universe may be undergoing a succession of alternate ex- 
pansions and contractions (cf. fig. 5, p. 130); this would 
account for the observed recessions of the nebulae, and 
yet give us all the time we want for the evolution of the 
universe; there is no longer any conflict with the general 
evidence of observational astronomy. 


We must not regard any of the foregoing speculations 
or conclusions as in any way final or established. Indeed, 
science is only just entering upon its latest and most com- 
prehensive problem the study of the universe as a 
single entity and it would be folly to treat the first 
tentative results as final. Yet, although these can hardly 
be said to have led to definite conclusions so far, they 
nevertheless hold out hope that conclusions may not be 
very distant. And they illustrate once again that it is 
usually the totally unexpected that happens in science 
the unaided human mind can seldom penetrate far into 
the darkness which lies beyond the small circle of light 
formed by direct observational knowledge. 

Even the meagre results so far obtained seem to shew 
that nature is one and not many. The different sciences 
have each drawn their own pictures of small fragments of 
nature which form their special objects of study, and we 
now find that these fragmentary pictures piece together 
to form a consistent whole. An experiment performed by 
two physicists, Michelson and Morley, with a view to 
measuring a time interval of less than a million millionth 
part of a second, or a length of less than a thousandth part 
of an inch, led, through the theory of relativity, to a pic- 
ture of the whole vast universe, which depicts it as explod- 
ing like a burst shell, its most distant objects unanimously 
rushing away from us. We examine these objects through 
our largest telescopes and find that they are, to all appear- 
ances, rushing away in precisely the way predicted by 

theory. _ 

The Nature of Space 

The point which is of immediate interest to our present 
discussion is the following. Unless this apparent agree- 


ment between theory and observation is wholly illusory, 
it provides us with evidence of a contact between the 
theory of relativity and reality at the furthest point to 
which this theory has so far been pushed. It suggests very 
strongly although of course it does not prove that 
the curved continuum postulated by this theory has more 
reality that that of a mere convenient explanation of the 
apparently curved paths of planets and cricket balls, 
just as the curved surface of the earth has more reality 
than that of a mere convenient explanation of the appar- 
ently curved tracks of steamers. 

Even so, it only tells us of the metrical properties of 
space, and nothing as to its essential nature. Indeed, 
there would appear to be little advantage in discussing 
this latter problem. After 2,000 years of metaphysical 
discussion, the question stands much as Plato left it in the 
Timaeus (pp. 74, 144); the growth of scientific knowledge 
has done little more than negative the speculations of 
subsequent philosophers. Of all external entities, per- 
haps space is the one whose essential nature is least likely 
to be understood by the human mind, since it is hardly 
probable that what is completely external to the mind, 
and without effect on the mind, will admit of being 
pictured in terms of familiar concepts inside the mind. 

Although the new curved continuum is still a blend 
of space and time, these constituents no longer enter 
it in similar or even symmetrical ways. In our simple 
diagrammatic analogy, space was the cross-section and 
time the axis of a cone, and however much new knowl- 
edge may change the details, some such distinction seems 
likely to persist. Such a continuum does not satisfy the 
invariant condition, which was found to be essential iii 


the restricted theory, of giving a picture which does not 
suffer by being hung askew. It therefore contains in itself 
a unique mode of separation into space and time, which 
we may now designate as absolute space and absolute 
time. The restricted theory of relativity, which we dis- 
cussed in our previous chapter, shewed that any division 
into space and time was subjective in respect of such phe- 
nomena as we could observe and measure in our labo- 
ratories. The generalised theory which we are now 
discussing suggests that just as our individual conscious- 
nesses recognise a sharp and clear-cut distinction be- 
tween space and time, so also does nature on the grand 
scale. This distinction, which we first find in our own 
minds, vanishes for a time when we study objective 
nature on the small scale, but apparently reappears in the 
cosmos as a whole. 

Neither the mathematical theory we have just de- 
scribed, nor the interpretation of the astronomical obser- 
vations, are sufficiently certain to w ? arrant the drawing of 
any conclusions except as almost random conjectures, 
but a simple analogy may suggest the kind of conjecture 
that presents itself. 

Let us, very unpoetically, compare the human race to 
a race of worms living inside the earth, and capable of 
burrowing about in it but never reaching its surface. 
As their bodies would be subject to gravitational forces, 
their minds would be conscious, through their nervous 
systems, of a distinction between horizontal and vertical 
directions inside the earth. They would not be able to 
pick out an absolutely permanent and unaltering hori- 
zontal and vertical, since a worm who was moving with 
an acceleration would experience a different horizontal 


and vertical from his fellows who were not. In spite of 
this, the worms might still feel sure that the horizontal 
was somehow essentially different from the vertical. 
Suppose now that they took to science and built labora- 
tories, still inside the earth, in which to study electro- 
magnetism and optics. They would be unable to detect 
any distinction between horizontal and vertical in their 
laboratories, because the laws of electromagnetism and 
optics treat all three directions in space equally. If, then, 
they knew of no sciences but these, they would be unable 
to discover any scientific justification for their intuitive 
feeling that horizontal and vertical were really dissimilar. 
Finally, to come to the climax, one of them might burrow 
his way out to the surface of the earth and discover that 
their intuitive feeling was based on a real fact of nature. 
He would then see that nature contained something more 
than the sciences they had studied in their laboratories, 
and would realise that they had all the time been in 
contact with nature through this something more. 

In the same way, our minds are conscious of a radical 
distinction between space and time which does not 
appear to extend to physical phenomena; these seem so 
similar in the continuum and so dissimilar when appre- 
hended by our minds. Through our consciousnesses, we 
break up the space-time product into space and time, 
while electrons and protons and radiation cannot. If this 
distinction is ultimately found to be real, as our present 
vague and uncertain knowledge seems to suggest, we may 
be tempted to conjecture that our minds are in contact 
with reality through other than purely physical channels. 
Finally, we may notice that, if a more complete knowl- 
edge of the continuum as a whole is ultimately found to 


restore a meaning to absolute space and absolute time, 
the problems which were indicated at the end of Chapter 
m do not arise. 

In that chapter we pointed out how the concept of the 
space-time continuum neither space nor time being 
complete in themselves, and only acquiring objective 
reality when blended into a single whole was in ac- 
cordance with a view \vhich certain metaphysicians had 
taken of space. In the present chapter we have con- 
sidered the properties of this blend of space and time in 
more detail. We have seen that, according to the picture 
drawn by the generalised theory of relativity, space must 
be finite in amount, and must possess a texture, defined 
by the different curvatures at its various points, so that 
it is in some way differentiated from mere emptiness* 
Again, these qualities satisfy the requirements of the 
metaphysician. Writing twenty years before the general- 
ised theory of relativity appeared, Bradley described these 
in the following words:* 

"Empty space space without some quality (visual or 
muscular) which in itself is more than spatial is an unreal 
abstraction. It cannot be said to exist, for the reason that 
it cannot by itself have any meaning. When a man realizes 
what he has got in it, he finds that always he has a quality 
which is more than extension. But, if so, how this quality is 
to stand to the extension is an insoluble problem". 

And again, with reference to finite space: 

"For take space as large and as complete as you possibly 
can. Still, if it has not definite boundaries, it is not space; 
and to make it end in a cloud, or in nothing, is mere blindness 
and our mere failure to perceive. A space limited, and yet 

* Appearance and Reality, pp. 37, 38. 


without space that is outside, is a self-contradiction. But the 
outside, unfortunately, is compelled likewise to pass beyond 
itself; and the end cannot be reached. And it is not merely 
that we fail to perceive, or fail to understand, how this can be 
otherwise. We perceive and we understand that it cannot be 
otherwise, at least if space is to be space. We either do not 
know what space means; and, if so, certainly we cannot say 
that it is more than appearance. Or else, knowing what we 
mean by it, we see inherent in that meaning the puzzle we are 
describing. Space, to be space, must have space outside itself. 
It for ever disappears into a whole, which proves never to be 
more than one side of a relation to something beyond". 

This quotation raises a metaphysical dilemma which 
science alone cannot claim to solve. If the whole con- 
tinuum is finite, what can there be outside the continuum 
except more continuum? which proves that our 
original continuum was not the whole continuum. And 
how can space be expanding, since there is nothing for it 
to expand into except more space? which proves that 
what is expanding cannot be the whole of space, and so 
on. We shall return to this in a later chapter. 

Finally, those who hold that "out of Plato come all 
things that are still debated among men of thought" may 
be tempted to claim that Plato anticipated Einstein in 
evolving the whole of nature out of the metrical texture 
of space. They may even claim that he anticipated Fried- 
mann and Lemaltre in respect of the instability of the 
Einstein universe. For he wrote*: 

"Even before the birth of a heaven, there were these several 
ree being, space, becoming. Hence as the foster-mother 
of becomingf was liquefied and ignited and received the shapes 

* Timaeus, Taylor's translation, p. 52. 
T /.*. Space. 


of earth and air and underwent further affections consequent 
on this, she took on many motley guises. And since the forces 
with which she was filled were neither alike nor equipoised, 
there was no equipoise in any region of her; she was swayed 
and agitated with utter irregularity by these her contents, and 
agitated them in turn by her motion". 



We must now leave the vastness of astronomical space, 
to pass to the other extreme of the scale of size and explore 
the innermost recesses of the ultra-microscopic atom. 
While the phenomena of astronomy may shew us the 
nature of space and time, it is here, if anywhere, that we 
may hope to discover the true nature of matter and of 
material objects, the contents of space and time. 

The Structure of Matter 

We have seen how the atomic concept of matter gradually 
gained scientific recognition, and finally appeared to be 
securely established when Maxwell and others shewed 
that a gas could be pictured as consisting of hard bullet- 
like atoms or molecules flying about indiscriminately at 
speeds comparable with those of ordinary rifle bullets. 
The impact of these bullets produced the pressure of the 
gas; the energy of their motion was the heat-energy of the 
gas, so that heating up the gas resulted in its bullets 
travelling faster; the viscosity of a gas was caused by the 
drag of one bullet on another on the rare occasions on 
which actual collisions occurred, and so on. These con- 
cepts made it possible to explain a great number of 
the observed properties of gases, both qualitatively and 
quantitatively, with great exactness. Yet a residue ob- 
stinately defied explanation, and it is only recently that 



an explanation of these has been obtained, in terms of new 
and very different concepts to which we shall shortly pass. 

This picture of matter as consisting of hard indivisible 
atoms had to be modified when Sir J. J. Thomson and his 
followers began to break up the atom. They shewed that 
the atom was far from indivisible; small fragments of it 
could be knocked out by bombardment, or pulled out 
by sufficiently intense electric forces. These fragments 
proved to be all similar the electrons. They all had 
the same mass, and carried the same electric charge, 
which was conventionally described as being of negative 
sign. It was subsequently found that the remaining in- 
gredients of the atom were also similar electrically 
charged particles the protons. Their charges were 
opposite in kind to the charges on the electrons, and so 
were described as positive in sign. 

There were many reasons for supposing all the atomic 
constituents to be of minute size. For instance, it was 
found that radio-active substances shot off two kinds of 
projectiles, a less massive kind known as j8-particles, which 
proved to be rapidly moving electrons, and a more mas- 
sive kind known as a-particles, which proved to be 
identical with the central nucleus of the helium atom. 
This is known to consist of four protons and two electrons. 
When particles of either kind were shot at matter they 
penetrated it to a considerable depth, which suggested 
that they were of very small dimensions. A tennis ball 
weighs about the same as a rifle bullet, yet if we fire both 
at the same piece of wood, the bullet will penetrate a 
considerable distance, because its mass is concentrated in 
a very small, space, while the tennis ball will not penetrate 
at all; both a- and jS-particles were found to behave like 


rifle bullets rather than like tennis balls. Not only so, 
but when they were fired at a thin film of metal, the 
majority passed through without being substantially 
deflected from their courses, which seemed to shew that 
the electron and protons of the metal film were them- 
selves of minute size. Thus there appeared to be fairly 
conclusive evidence that the ultimate ingredients of 
matter were of the nature of small particles carrying 
highly concentrated charges of electricity. 

It has never been found possible to measure the sizes 
of these particles directly. It is often supposed that the 
diameter of the electron must be about 4 X 10~ 18 cms.; 
it cannot be less than this, for if the electrical charge of 
the electron were compressed into any smaller volume, 
the inertia resulting from this alone would necessarily be 
greater than that of the total observed mass of the elec- 
tron. Yet this raises a serious difficulty. According to 
the generally accepted theory, the nuclei which form the 
centres of the most massive atoms, such as gold or ura- 
nium, must contain a large number of electrons as well as 
protons. These nuclei are, however, themselves so small 
in size that they could not contain the requisite number 
of electrons of the size just mentioned inside them, even if 
the protons occupied no space at all. This shews that the 
concept of electrons and protons as small charged parti- 
cles is at best only a picture, and a picture which cannot 
be true to nature in all particulars. Nevertheless, a small 
spherical particle of radius 2 X lO" 13 centimetres and 
charged with 4-777 X 1Q- 10 units of electricity repro- 
duces many of the properties of the electron, and proba- 
bly no one has ever regarded it as providing a complete 
picture which was true in all particulars. 



As regards their material structure, all objects are built 
up of these two kinds of electrified particles, but they con- 
tain also the intangible constituent of energy, which may 
be set free from all association with matter, when it 
travels through space in the form of radiation. We have 
seen how, throughout the nineteenth century, radiation 
was pictured as waves in the ether. This picture not only 
failed to describe the propagation of radiation in ways 
which have already been described; it also failed to 
account for some of the most fundamental properties of 
the radiation itself. 

We know how a pendulum swinging in air continually 
loses energy to the molecules of air which impinge on it; 
unless it is kept in motion by clockwork it soon comes to 
rest, the energy of its motion being transformed into 
waves of the surrounding air which are subsequently 
dissipated into heat. In the same way a steamer soon 
comes to rest when its engines are stopped, the energy of 
its motion being used in setting up waves in the surround- 
ing sea. And, again in the same way, it can be shewn 
that if material bodies were surrounded by a sea of ether, 
their energy would be rapidly dissipated in setting up 
waves in the ether. Calculation shews that this process 
would continue until the material bodies, like the pen- 
dulum and the steamer, had no energy left at all; their 
whole energy would have passed into the ether, where it 
would take the form of radiation of very short wave- 
length. This is true of all kinds of energy, so that a hot 
body ought speedily to lose all its heat-energy to the ether, 
and fall to the absolute zero of temperature. 


Instead of this, experiment shews that a state of equi- 
librium is soon attained in which a body receives back 
from the surrounding space exactly as much radiation as 
it pours out into it. For instance, disregarding certain 
small internal stores of heat, the average temperature of 
the earth is such that it loses just as much energy by 
radiation into space as it receives back from space in the 
form of solar light and heat. If the earth were suddenly 
made hotter than it now is, it would cool down to its 
present temperature, but not to the absolute zero; if it 
were made cooler, it would warm up to its present tem- 
perature. To take a more precise case, if a heating 
system maintains all the walls of a closed room at exactly 
60 F., then every object in the room will stay per- 
manently at exactly 60 F. this is why we can say that 
a thermometer gives "the temperature of the room". 

In such a state of equilibrium, every object gives out 
just as much radiation as it receives. In the idealised case 
of an object which has no reflecting power at all, and so 
appears perfectly black (p. 19), the radiation is known 
technically as "black-body radiation", and is said to 
have the temperature of the object which emits it. 
Radiation of this kind can be analysed into its different 
constituents with great accuracy, and its quality is found 
to be as unlike as possible to what it would be if radiation 
consisted of waves in a substantial ether. 


In the last years of the nineteenth century, Planck tried 
to discover the reason for this divergence, and, just as the 
century was closing, he put forward the ideas out of which 
the vast structure of the quantum theory has since arisen. 


He shewed that it was possible to account exactly for the 
observed state of equilibrium between matter and radia- 
tion, by the assumption that radiation was atomic in its 
nature. He supposed it to occur only in complete mul- 
tiples of a unit which he called the "quantum". This 
unit was not the same in amount for all kinds of radiation, 
but depended on the wave-length of the radiation, and so 
also on its period of oscillation or its "frequency" the 
number of oscillations performed in a second. To be 
precise, radiation which oscillated v times a second was 
supposed to occur only in complete units of energy of 
amount hv, where h was a quantity, now universally 
known as Planck's constant, which is found to pervade the 
whole of atomic physics. Thus blue or violet light, being 
of high frequency, consisted of quanta of great energy, 
while red light, which is of low frequency, consisted of 
quanta of small energy. The greater the energy of the 
quanta, the greater their capacity of producing atomic 
change. This is why blue light causes pigments to fade 
and affects photographic plates, where red light is in- 

The recently-discovered X-radiation was known to be 
of enormously high frequency, so that on Planck's theory 
its quanta ought to possess exceptionally great energy. 
It was soon remarked that when this radiation was passed 
through a gas, a few of the molecules of the gas were 
shattered, but the vast majority remained entirely un- 
affected by the passage of the rays. Had the rays 
consisted of waves travelling through an all-pervading 
ether, it might reasonably have been expected that they 
would treat all the molecules they encountered in the 
same way, or at least in approximately the same way; 


actually, less than one molecule in a billion seemed 
to be singled out for destruction. It was further found 
that doubling the intensity of the radiation did not double 
the damage done to each molecule, but merely doubled 
the number of molecules that were damaged. This was 
subsequently found to be true of radiation of all kinds, 
including ordinary visible light. It was exactly what was 
to be expected if radiation consisted of small point-like 
atoms of radiation, like the old Newtonian corpuscles. 


In 1905 Einstein crystallised these concepts and hypoth- 
eses in his theory of light-quanta, according to which all 
radiation consisted of discrete bullet-like units, which he 
called "light-quanta" at the time, although we now call 
them "photons". When an atom was struck by a photon, 
it might be either disturbed or shattered, according to the 
amount of energy which the photon brought to the attack, 
and by observing the amount of damage done to the 
atom, it became possible to calculate the energy of the 
individual photons. This invariably proved to be exactly 
one quantum if the incident radiation which attacked 
the atoms was of frequency v, the change produced in 
each affected atom represented an expenditure of en- 
ergy to. 

one of the fundamental consequences of the theory of 
relativity is that every kind of energy has mass associated 
with it. Thus a photon must possess mass of its own, and 
it is just as accurate to speak of the mass of a photon as 
of the mass of an atom or of a motor car. As photons are 
always in motion, we may also speak of the momentum 
of a photon, much as we speak of the momentum of a 


motor car, although there is the essential difference that 
photons always move with the same speed, the speed of 
light, whereas motor cars move with variable and differ- 
ent speeds. 

Professor Compton of Chicago has recently found very 
direct evidence of the existence of this mass, and has been 
able to measure its exact amount. When a photon strikes 
an atom, its energy is not always completely absorbed by 
the atom; it may occasionally strike a particular electron 
in an atom and rebound from it like a perfectly hard 
bullet. In such cases the photon loses part, but not all, 
of its energy to the electron with which it has collided. 
Compton found that when this happens to a photon, its 
frequency changes in such a way that after the collision 
the energy is precisely h times the frequency of the radia- 
tion, as of course it also was before the collision. The cir- 
cumstances of the recoil made it possible to calculate the 
momentum, as well as the energy, of the photon, and this 
proved to be hv/c. This is exactly the amount of momen- 
tum which the theory of relativity predicts must be 
associated with energy hv moving with the speed of light. 

These various experiments suggest that radiation may 
be pictured as consisting of bullet-like units, which travel 
through space very much like shot fired from a gun and 
have nothing to do with any supposed ether. In this new 
picture, the constant speed of 186,000 miles a second at 
which radiation travels is no longer regarded as the speed 
of waves; we have instead to imagine that photons of 
radiation are endowed with inertia, like a bullet or an 
electron. This inertia keeps them moving in a straight 
line with a uniform speed, although nothing in this pic- 
ture explains why this speed should always be 186,000 


miles a second. This last fact shews that the particle 
picture by itself is incomplete. 

Quantitatively, the experiments shew that the mo- 
mentum of a photon is connected with its wave-length by 
the relation 

momentum X wave-length A, 

while its energy is connected with its period of oscillation 
by the relation 

energy X period of oscillation = h. 

Finally the wave-length and period of oscillation are con- 
nected by the relation, which survives from the undula- 
tory theory, 

wave-length = period of oscillation X , 

where c is the uniform speed of light. 

The evidence which these experiments provide for the 
real existence of photons is of the same general nature as 
that which other experiments provide for the existence of 
electrons. In each case experiment suggests an indivisi- 
ble entity having definite quantities associated with it 
e and m for the electron, and h and c for the photon 
and measurement of these quantities yields uniformly 
consistent values. No experiment yet performed has sug- 
gested that fractions of either entity can exist independ- 
ently; fractions of a photon are as unknown as fractions 
of an electron. 

The radiation with which we are usually concerned in 
atomic physics is produced by disturbances or upheavals 
of single atoms, and it is found to be a general law that 
every such disturbance produces one, and only one, com- 
plete photon. As mass is conserved through a change of 


this kind, the atom must lose mass exactly equal to the 
mass of the photon it emits. When the disturbance con- 
sists only of a rearrangement of the outermost electrons 
of an atom, the resulting change of mass is only a few 
millionths of the mass of a single electron, and the photon 
has the wave-length of visible light it is by the entry 
of such photons into our eyes that we see things. A re- 
arrangement of the inner electrons of the atom produces 
X-radiation, in which each photon has a mass of perhaps 
the 10,000th part of the mass of an electron. If the nu- 
cleus of the atom rearranges itself, we have the still more 
penetrating 7-radiation, in which each photon has a mass 
comparable with the whole mass of an electron. Finally, 
the hardest constituent of cosmic radiation, the most 
penetrating radiation known, has photons of mass about 
equal to that of a complete atom of helium, while the 
next most penetrating constituent has photons of mass 
about equal to that of a hydrogen atom. It is possible, 
then, that these photons may be produced by the total 
annihilation of atoms of helium and hydrogen, or, more 
probably, by the annihilation of electrons and protons to 
an equivalent extent in more complex atoms. 

The Kinetic Theory of Radiation 

Just as Maxwell was able to explain many of the proper- 
ties of a gas by picturing it as a medley of bullet-like 
molecules, so we can explain many of the properties of 
radiation by picturing it as a medley of bullet-like pho- 
tons. The pressure of a gas can be pictured as resulting 
from the impacts of its molecules, and in the same way 
the pressure of radiation can be pictured as resulting from 
the impacts of its photons. And again, just as the energy 


of a gas is the sum of the energies of its molecules, so the 
energy of the radiation in any space is the sum of the 
energies of the photons in that space. 

When we picture radiation as consisting of waves, the 
quantity kno\vn as the "intensity" of the radiation is pro- 
portional to the energy of the waves at any point, or, even 
more pictorially, to the "storminess" of a sea of ether. 
When we picture radiation as consisting of photons, we 
can no longer interpret the intensity in this way. We 
may, however, give it a statistical interpretation; we can 
define it as proportional to the chance of finding a photon 
at the point in question, just as the density of the gas at 
a point is a statistical concept, and is proportional to the 
chance of finding a molecule there. The temperature of 
radiation, like the temperature of a gas, is also a statistical 
concept. We cannot speak of the temperature of a single 
photon, any more than of that of a single molecule. We 
say that "black-body radiation", which experiences no 
change either of quality or quantity when it interacts with 
heated matter, has the same temperature as the matter, 
but the temperature belongs to the crowd of photons 
and not to the individuals separately. 

Such radiation may be pictured as a crowd of pho- 
tons moving equally and indiscriminately in all direc- 
tions, just as a gas in equilibrium may be pictured as a 
crowd of molecules moving equally and indiscriminately 
in all directions. The energies of the separate photons 
conform to the statistical law known as Planck's law, 
just as the energies of the molecules in a gas conform to 
Maxwell's law. Various other concepts, such as those of 
the two specific heats, of their ratio, and of adiabatic 
changes, mean much the same for the radiation as for the 


gas, and permit of the same pictorial representation, 
photons of course replacing molecules. 

We may picture the photons as retaining their indi- 
vidual identities through all changes except that of being 
completely absorbed into, or emitted out of, an atom or a 
molecule. They may change their energies, but then they 
adjust their frequencies to their energies so that each 
photon remains a complete unit. Suppose for instance 
that "black-body radiation" is darting about inside an 
enclosure, whose volume can be varied by a cylinder- 
piston arrangement, and is being continually reflected 
from its walls. When we compress a gas inside a cylinder 
the advancing piston does work against the pressure of the 
gas, and this work reappears as an increase in the energy 
of the separate molecules i.e. as heat. When the cyl- 
inder is filled with radiation, the advancing piston does 
work against the pressure of the radiation, and this re- 
appears as an increased energy of the photons. Let us 
imagine that we reduce the volume accessible to the 
radiation to one-eighth, equivalent to an all-round reduc- 
tion of linear dimensions to half. It can be shewn that 
each photon will double its energy, and so also will double 
its frequency and halve its wave-length. Thus wave- 
lengths and enclosure are uniformly reduced to half-scale, 
and the new radiation is at double the original tempera- 
ture. If we suppose the photons to have retained their 
identity, there are eight times as many per unit volume, 
so that the density of energy has increased sixteen-fold 
as the fourth power of the temperature. This is exactly 
what is observed, the fourth-power law being known as 
Stefan's law. And, again as with a gas, the pressure is 
proportional jointly to the density and temperature, so 


that this also varies as the fourth power of the temper- 
ature, which again agrees with observation. 

Just as we can picture "black-body radiation" as a 
random crowd of photons, so we can picture a beam of 
radiation as a regular shower of photons, all moving in 
parallel paths. This, of course, corresponds to a blast of 
gas in which all the molecules move in parallel paths, 
their ordinary heat-motion being either neutralised or 
neglected. on the other hand, a beam of light is in one 
respect more intricate than a blast of gas, since in addition 
to its motion through space it possesses the property we 
describe as polarisation. 

The nineteenth-century picture of radiation attributed 
polarisation to angular momentum of the ether; in our 
present picture of radiation, we must attribute it to 
angular momentum in the separate photons which form 
the radiation. In brief, not only must our radiation move 
through space like bullets, but each bullet must have a 
spin like that caused by rifling. Planck's constant h has 
the same physical dimensions as angular momentum, and 
we find that we must picture all photons as spinning with 
the same angular momentum A/2?r, which may be in 
either direction right-handed or left-handed. In a 
beam of circularly polarised radiation, we picture the 
photons as all spinning in the same direction. If the beam 
is elliptically polarised, more photons spin in one direc- 
tion than in the other; if plane-polarised, the numbers 
are equal. If it is not polarised at all, the proportion of 
the two kinds continually varies at random, but the laws 
of probability secure that the actual ratio shall never 
wander very far from unity. This spin has recently been 
detected and measured by Raman and Bhagavantam. 


Inadequacy of the Particle Picture of Radiation 

This picture of radiation as a crowd of bullet-like pho- 
tons has many advantages, but also suffers from many 
limitations, which shew that it does not present us with a 
complete picture of reality, but at best only of certain 
aspects of reality. We have already mentioned one con- 

Fig. 9. (This is purely diagrammatic and is not 
drawn to scale.) 

spicuous instance of its failure: nothing in the particle 
picture explains the most fundamental of all the proper- 
ties of radiation its uniform speed of travel. 

A second instance of its failure is provided by an 
experiment which can be performed in any laboratory. 
Let S be a source of light, emitting light of approximately 
pure colour, and let an opaque screen, punctured by two 
tiny pinholes, A y B> be set up in front of , so that A and 
B are at equal distances from S. If we picture the light 
which S emits as bullet-like photons, then two points P 9 
Q, on the laboratory wall will be under fire from S, and we 
shall expect to find the wall illuminated at the points 
P, Q and dark everywhere else. In a general way this 
describes what will usually happen; yet if we make our 
holes A, B near enough together, the description fails 


entirely. The most brightly lighted region of the wall will 
no longer be the points P and , but the single point R 
midway between them, although this is out of both of the 
lines of fire SAP and SBQ if light really consisted of 
bullets, we should expect R to be completely dark* Not 
only so, but for light of one particular colour, P and Q, 
which ought to be most brilliantly lighted of all if the 
light consisted of bullets, may be completely dark. 

We can obtain yet more surprising results by blocking 
and unblocking one of the pinholes, say B. We shall 
find that so long as B is blocked up, P is brightly illumi- 
nated, but the moment B is unblocked, P becomes dark 
letting more light in on P changes light into darkness. 

The old undulatory theory had provided a perfect ex- 
planation of all this; it is, indeed, a special instance of the 
general principle already explained on p. 121 . Let us for 
the moment think of our diagram as representing the 
surface of a rectangular piece of water, such as a swim- 
ming pool, while A, B is a wall built across the pool with 
small apertures at A and B. When a swimmer splashes 
about at S, he will cause ripples to spread over the pond. 
Some will pass through the apertures A 9 5, and set up 
new systems of ripples in the space beyond. These will 
spread out in circles from the points 4, 5, and as these 
two points are symmetrically placed with respect to S 9 
the two sets of ripples will be exactly similar. 

Now the point R is symmetrically placed with respect 
to A and 5. Thus the crest of a ripple from A will arrive 
at R simultaneously with the crest of a ripple from JJ, 
and their combined effect at R will be just twice what it 
would be if only A or B were open separately. on the 
other hand, P is not symmetrically placed with respect 


to A and 5, so that the two sets of ripples from A and B 
will not in any case reinforce one another quite so per- 
fectly at P as they did at R. In an extreme case, crests 
of ripples from A may arrive coincidently with troughs of 
ripples from F, so that the two will exactly neutralise one 
another, and the water at P will remain unagitated. If 
we block up the opening B, the water at P is agitated by 
the waves which reach it from A. If we now reopen B, 
and so let more waves in on to P, the water at P becomes 
quiescent, because we have added a second set of waves 
which exactly neutralises the first. 

These are precisely the results obtained in an actual 
experiment. They seem nonsensical to the last degree 
when we picture light as bullets, but perfectly natural and 
inevitable when we picture it as waves. 

Yet suppose we carry our experiment a stage further, 
and put a sensitised photographic plate against the wall 
PRQ of our laboratory. If the apertures A and B are both 
open, this will of course give us a permanent record of 
the light at R and of the darkness at P and Q. At R 
the grains of the plate will be changed by the incidence 
of the two sets of waves, one from A and one from B, 
which reinforce one another. on the other hand, the 
grains of the plate at P and Q, will undergo no change, 
because the two sets of waves neutralise one another. 

Yet the grains of the plate can only absorb radiation by 
complete photons, as is shewn by the fact that blue light 
makes more impression on it than red. Thus, to make 
our picture consistent, we must suppose that light travels 
through space in the form of waves, but breaks up into 
photons as soon as it encounters matter. We shall find 
later that there is an exactly complementary picture for 


electrons and protons. This shews us electrons and pro- 
tons behaving as particles while they travel freely through 
space, and as waves when they encounter matter. 

There is a complete mathematical theory which shews 
how in all such cases the particle- and wave-pictures 
are merely two aspects of the same reality, so that light 
can appear sometimes as particles and sometimes as 
waves, but never as both at the same time.* It also ex- 
plains how the same can be true of electrons and protons. 
It is hardly possible to give even the vaguest account of 
this highly intricate theory in non-mathematical terms, 
but the following considerations will perhaps provide 
something of a bridge between the particle- and wave- 
pictures of radiation, and shew how both may represent 
partial aspects of a unity which transcends both particles 
and waves. 

Free Vibrations 

Any system which is capable of vibration is set into 
vibration when it is disturbed from outside. If the dis- 
turbance finally fades away or is withdrawn, the system 
does not come to rest immediately, but continues to 
vibrate for a time. The vibrations which it now executes 
are known as the "free vibrations 35 of the system, and the 
periods of these vibrations are described as the "free 
periods" of the system. The simplest instance of this is 
provided by a tuning-fork. For all practical purposes, 
this has only one period of free vibration, which deter- 
mines the pitch of the note emitted by the fork. Suppose, 
for instance, this is middle C, which corresponds to 256 
vibrations a second. If the fork is disturbed in any way 

* See, for instance, Hefeenberg, The Physical Principles of the Quantum 
Theory, p. 177. 


whatever, as by the impact of a blow or the friction of a 
violin bow, it will be set into vibration, and after the 
disturbance is over it will be left vibrating at 256 vibra- 
tions a second. It will, so to speak, have forgotten 
what caused it to vibrate, and remember only its own 
period of free vibration hence its utility for musical 

A piano string provides a more complicated example 
of the same thing. When we sound middle G of the piano, 
the hammer strikes a string which has an infinite number 
of free vibrations, these being at the rates of 256, 512, 
768, 1024, ... vibrations a second. These frequencies are 
in the ratio 1:2:3:4: ..., and the corresponding musical 
notes are called the "harmonics" of middle C. They are 
the C above, the G and C above this, then the E, G, Bk 
C, D, E and so on in succession, their respective wave- 
lengths being a half, a third, a quarter, and so on, of the 
length of the string. Again, these tones are associated 
with the string itself, not with the striking of it. In what- 
ever way the hammer strikes the middle C string, these 
same harmonic notes are always sounded; only the pro- 
portion of their intensities depends on the way in which 
the hammer strikes the string. 

A still more complicated example is the air in a concert 
hall. This has innumerable free vibrations, their wave- 
lengths ranging from the whole length of the concert hall 
down to a minute fraction of an inch. When a pianist 
plays middle C in the concert hall, the hammer momen- 
tarily strikes three strings of the piano, and sets them into 
vibration, after which they are left performing free vibra- 
tions of all the wave-lengths and frequencies just men- 
tioned. These vibrations do not persist for ever, because 


their energy is gradually transferred to the surrounding 
air through the medium of the sound-board of the piano. 
We can describe what happens by saying that the piano 
string sends out waves of sound into the concert hall, but 
it is an equally fair description to say that the energy of 
the string is transferred to the free vibrations of the air in 
the hall; actually it will be transferred almost exclusively 
to those having the same frequencies of vibration as the 
string itself, namely middle G and its harmonics. Thus 
we have two pictures the wave picture and the free 
vibration picture each of which represents the facts 
equally well, although each represents only one special 
aspect of the facts. 

We have mentioned these cases merely as stepping- 
stones to a system of still greater complexity, namely the 
optical laboratory represented in fig. 9 (p. 159). Radia- 
tion can travel through this laboratory just as sound can 
travel through a concert hall, and again this radiation can 
be represented equally well either as waves or as free 
vibrations of definite wave-lengths and frequencies. We 
need not associate the vibrations with any special under- 
lying mechanism, since a theorem of pure mathematics, 
similar to that already mentioned on p. 62, shews that, 
quite apart from any special type of mechanism, or indeed 
of any mechanism at all, every kind of disturbance can be 
pictured as made up of free vibrations. 

If the source of light at S emits radiation of any single 
definite frequency, the energy it emits is transferred to 
various free vibrations of the same frequency in the 
laboratory. If we study these vibrations after the manner 
of p. 160, we find that, under the special conditions and 
with the special arrangement of apparatus already postu- 


lated, there will be a violent disturbance at R, but no 
disturbance at all at P or Q. A source of light at S can 
add to the energy of these vibrations by emitting radia- 
tion, and, by the same process reversed, a molecule at R 
can subtract from their energy by absorbing radiation. 
Molecules at P and Q, cannot, however, do this; we have 
seen that the vibrations produce no disturbance at P or Q, 
which shews that there is no coupling between the free 
vibrations of the laboratory set up by the source of light 
at S and the molecules at P and Q. 

If we now picture the energy of these free vibrations as 
that of photons, we can say that the source at S emits 
photons of a definite known frequency, and that mole- 
cules at R can absorb these photons, while molecules at 
P and Q cannot. Thus if we expose a photographic plate 
on the wall PRQ, the points P and ( will appear dark on 
the plate, while R will appear light. We can make our 
picture of the process more vivid by saying that S emits 
photons which fall on R but not on P and Q, We are now 
picturing the light as consisting of bullets of energy, but 
only in a limited sense. We may picture it as bullets 
when it leaves , and again when it arrives at R, and this 
picture will give us a true account of the phenomena 
actually observed. on the other hand, we must not 
picture it as bullets while it is passing through the aper- 
tures A and B\ if we make this mistake we shall expect 
to find P and ? light and R dark, exactly contrary to the 
facts of observation. If we want to combine the bullet 
and wave aspects in a single picture, we must say, as 
before, that light behaves like waves while travelling 
through empty space, but like bullets as soon as it en- 
counters matter. 


When we adopt the particle picture, we are, in effect, 
interpreting the energy of free vibrations of any specified 
frequency as energy of photons of the same frequency, but 
we must be careful not to identify individual free vibra- 
tions with individual photons. The energy of a free 
vibration in our laboratory extends through the whole of 
the laboratory, and on imagining the walls of the labora- 
tory to recede to an infinite distance, we find that the 
energy of a free vibration in space extends through the 
whole of space. A mathematical theorem shews that the 
energy of any isolated disturbance in space can be re- 
garded as the sums of the energies of a number of free 
vibrations, each extending through the whole of the 
available space. This is true no matter how restricted the 
area of the disturbance may be, or how large the space 
may be; inside the area of the disturbance, the different 
vibrations are cumulative in their effects; outside it, they 
destroy one another by interference. It is the energies 
which reside in such restricted areas, not the energies of 
the separate free vibrations, that must be identified with 
the photons. 

We have not yet found any reason why the energy of 
photons should occur only in complete quanta; we shall 
only understand the atomic aspect of radiation through a 
study of the properties of matter, to which we now turn. 

Atomic Spectra 

This problem is most naturally approached through a 
study of the complex spectra emitted by atoms of the 
chemical elements. We have seen that striking a piano 
string in any way whatever causes it to emit sound-waves of 
various distinct frequencies, which are in the simple ratio 


1:2:3:4:.... For instance, when the fundamental fre- 
quency is 256, the other frequencies are 512, 768, 1024, 
and so on all the integral multiples of 256 in succession. 
In precisely the same way a mass of incandescent 
hydrogen, or of any other chemical element, emits light- 
waves of various distinct frequencies, which can be 
measured with great accuracy in a spectroscope. These 
frequencies are not found to stand in any such simple 
ratio as 1:2:3:4:...; indeed, for a long time, no 
relation whatever could be discovered between them. 
Finally minor regularities began to appear. The fre- 
quencies of the three most conspicuous lines in the spec- 
trum of hydrogen HQJ, H, HT were found to be in the 
ratio 20 : 27 : 32. At first it was conjectured that these 
vibrations might be the 20th, 27th and 32nd members of a 
series of harmonics similar to those of a piano string, but 
it soon became apparent that the vibrations of the hydro- 
gen atom were far more complicated than this. Ritz 
made a great advance in 1908, when he shewed that all 
the intricate frequencies of the light emitted by any single 
substance were connected in a very simple way. He 
found that there exist a number of fundamental fre- 
quencies v^ *% v c > such that the frequencies of the 
emitted light are the differences between them, namely 
v a v*i *>b ?, v* PC, and so on. Even these funda- 
mental frequencies do not stand in any such simple 
proportion as the 1:2:3:4:... of a piano wire, al- 
though Balmer and others found that for the hydrogen 
spectrum they stood in the ratio 

L.I. L.I. 
1*'2 2 '3 2 *4 2 ""' 

which is not very much more complicated. 


The first step towards the interpretation of spectra must 
obviously be the assigning of meanings to the frequencies 
?a, n, PC, ... etc. This may seem a simple matter, but 
actually it has presented a problem of very great diffi- 
culty. The clue to its solution was found to lie in a sug- 
gestion which had originally been made by Bohr on 
theoretical grounds, and was subsequently confirmed 
experimentally by Franck and Hertz: An atom can only exist 
in certain distinct states possessing different clearly defined amounts 
of energy. When it passes from one state to another of lower 
energy, the liberated energy forms a single photon. 

If we know the amounts of energy in these various 
states, Planck's original quantum law will at once tell us 
the frequency of the photon emitted at the passage of the 
atom from one state to another. For if W a denotes the 
energy of the atom before the photon was emitted, and 
Wi> the energy afterwards, the amount of energy liberated 
is W a Wb, and this must be equal to hv> where v is the 
frequency of the single photon emitted. 

The frequency of this photon is accordingly given by 

_ W _ Wi 

h T"' 

and we notice at once that the fundamental frequencies 
?, v*9 ... of Ritz are merely the energies W a , W^ ... of the 
distinct states postulated by Bohr, all divided by h. Thus 
the problem of interpreting atomic spectra reduces to 
that of assigning meanings to W a , W^ ..., the values of 
the energy of the atom in the various distinct states in 
which it can exist. 

When Bohr attacked this problem in 1913, he adopted 
the then current view that the proton and electron of the 


hydrogen atom were minute particles charged with elec- 
tricity, in which case the mutual attraction of the opposite 
kinds of electricity would cause the electron to describe an 
orbit round the more massive proton, much as a planet 
describes its orbit round the sun. In those days the orbit 
of a planet round the sun was supposed to be determined 
by the inverse-square law of gravitational attraction, so 
that it was natural to expect the orbit of the electron to be 
determined by the similar law of electrical attraction. 
This law would compel the electron to describe a circle 
or an ellipse round the proton, but the law of itself gave 
no definiteness of size to the atom, and so could not limit 
the atom to distinct states with different clearly defined 
amounts of energy; it permitted the atom to have any 
amount of energy. 

Bohr accordingly found it necessary to suppose that the 
orbit not only conformed to this law, but to certain other 
laws as well. These other laws were of the nature of re- 
strictions; they restricted the electron to one or other of 
a number of definite clearly defined distances from the 
proton. It was rather as though a number of grooves, 
some circular and some elliptical in shape, were cut in 
the space round the proton. The electron had to stay 
in its groove, but its speed of motion was continually 
changed as it was accelerated or retarded by the electrical 
attraction of the proton, just as the speed of a planet's 
motion is continually changed by the gravitational pull 
of the sun. Physicists still describe these orbits as Bohr 
orbits. An electron might go on describing the same 
Bohr orbit for ever, or it might suddenly fall from one 
orbit to another of smaller dimensions. When such a fall 
occurred, the system lost a certain amount of energy. 


Assuming that this reappeared as a single photon of 
radiation, Bohr was able to calculate the frequencies of 
the different kinds of photons which could be emitted, in 
the way already explained. The frequencies calculated 
for the hydrogen atom agreed very closely, and probably 
perfectly, with those actually observed in the light emitted 
by incandescent hydrogen. This, however, was only a 
fragment of a far larger problem, and when more com- 
plicated spectra of other substances were discussed in the 
same way, Bohr's theory was found to lead to a less per- 
fect agreement with experiment. In certain cases, it 
quite obviously gave a wrong result, and no conceivable 
modification seemed capable of bringing it into line with 
the facts of observation. 

Observable* and Unobservables 

At this stage Heisenberg introduced a new method of 
looking at the whole problem, which has proved bril- 
liantly successful. In brief Bohr's theory had pictured an 
atom as consisting of particles which pushed and pulled 
one another about in space and time; it represented a last 
brave but unsuccessful attempt to force nature into a 
mechanical setting, and to depict the atom as existing in 
space and time. The difficulties it encountered seem to 
shew the imperfections of both these concepts. Bohr had 
pictured the atom as a mechanical structure, but was 
finally compelled to suppose that at intervals it evaded 
the limitations of this picture, when it passed from one 
orbit to another in a wholly non-mechanical way. He 
had tried to force the electron into space and time, yet 
was finally compelled to postulate jumps which shewed 
, no continuity in space-time. 


Heisenberg did not fetter himself with either mechan- 
ical pictures or space-time representations. A priori, as 
we have seen, there are very great odds against our being 
able to form any kind of visual picture of the fundamen- 
tal processes of nature. Heisenberg was not prepared to 
handicap his investigation at the outset by assuming a 
picture of any kind whatever to be possible. Just as the 
visual picture of light as waves in an ether had brought 
confusion into optical theory, so he thought that a picture 
of an atom as a structure of electrified particles was bound 
to bring confusion into atomic physics, as indeed it 
obviously was doing. 

In place of Bohr's picture concept of an atom, he intro- 
duced a new set of ideas which followed naturally on the 
changes in scientific outlook which were described in 
Chapter m, and can perhaps best be approached through 
a consideration of these changes. These had, in effect, 
amounted to the dismissal of three concepts from the 
scheme of science absolute space, absolute time and 
the luminiferous ether. Einstein's successes made it clear 
that these three dismissals had started science on the right 
road, and by travelling farther along the same road, 
Heisenberg was now able to bring order into atomic 
physics. Just as the theory of relativity had removed a 
whole massi of inconsistencies and contradictions from 
large-scale physics and astronomy, creating hardly a 
single new difficulty in their place, so Heisenberg's new 
line of thought has performed a similar service for atomic 

It was not mere accident that selected the three above- 
mentioned entities for dismissal. Our mental activities 
are stimulated by sense-impressions which originate 


beyond our senses; to account for these, we invent an 
external world of objects and entities, but everything 
beyond our senses is pure inference. The inferred entities 
are of many kinds, but fall into two distinct categories, 
which we may label as "observables" and "unobserv- 
ables". In brief the distinction is that the observables 
produce a direct effect on our senses, or on the instru- 
ments in the laboratory, whereas the unobservables affect 
our senses and instruments only indirectly, through the 
intervention of observables. A typical observable is a 
photon; a typical unobservable is the ether. Both types 
of entities may have quantities associated with them, so 
that we have "observable" quantities and "unobserv- 
able" quantities. An observable quantity admits of di- 
rect instrumental observation, whereas an unobservable 
quantity can only be a matter for abstract calculation. 
Typical of the former is the wave-length of a photon; 
typical of the latter is the rigidity of the ether. 

The universe of the scientist may be expressed dia- 
grammatically somewhat as follows: 



Instrumental effects 



All the items of the last two categories are purely 
inferential, but the type of inference is of course different 
in the two categories. The observables certainly repre- 


sent something objective, because they affect the senses of 
everyone in the same way, and affect instruments which 
are independent of our individual senses, but the very 
existence of the unobservables is in doubt because they 
do not affect our senses or instruments at all; unobserv- 
ables may represent nothing more than bad guesses. In 
brief, the properties of the observables are inferential, but 
the very existences of the unobservables are inferential. 
We may elaborate our scheme by setting against each 
category the principal items which belong to it, when it 
will stand somewhat as follows: 


Sense-data Sights, Sounds, Smells, Tastes and 


Instrumental effects Light, Photographic action, Elec- 
tric currents, etc. 

Observables Events at hand (impact of photons), 

Individual space, Individual time 

Unobservables Distant events, Objects, Ether, 

Absolute space, Absolute time 

This represents the universe as it appears to a scientist 
who explores it with the help of instrumental resources; 
primitive mart would of course short-circuit the item 
"instrumental effects" and pass directly from observ- 
ables to sense-data, in the way shewn in our previous 
ci i fMrr^tTTH T 

The observable ingredients of the external world are 
those which directly affect either our instruments or our 
senses. At first sight these may seem to be of a vast 
number of kinds; actually there is only one the impact 
of photons. It is obvious that the imprints on photo- 
graphic plates, which play so large a part in modern 


experimental science, are the result solely of the impact 
of photons, and that all optical and photometric effects 
must be the same. It is less immediately obvious how 
effects such as galvanometer deflections, which measure 
the passage of an electric current, or thermometer read- 
ings, which measure temperature, or the pressure on an 
ear-drum which registers the arrival of sound-waves, can 
be caused by the impact of photons. Yet they are; 
neither a physical instrument nor a sense-organ can 
exhibit an effect unless energy is in some way transferred 
to it, and all energy which is transferred from one object 
to another consists of photons. We are not of course 
speaking of photons in the limited sense of bullets of light, 
but in the more general sense of bullets of energy, which 
we reach by extending the concept of light to all possible 
wave-lengths and frequencies. In brief, all instrumental 
effects and sense-impressions depend on the transfer of 
energy, and all transfer of energy is by photons. So great 
a simplification may seem almost too good to be true, 
but it -is not pure gain to the scientist, as we shall find 
that it imposes very severe restrictions on his exploration 
of the universe (p. 230). 

In addition to the impact of photons, individual space 
and time may properly figure in our list of observables, as 
the mental framework in which the arrival of the various 
photons is set. When we try to arrange and classify the 
photons which impinge on our senses, we find that the 
mental creation of a three-dimensional space and a one- 
dimensional time instantly introduces complete and per- 
fect order. 

Out beyond the observables i.e. still farther away 
from our senses and instruments come the unobserv- 


ables. These are mere pictures, images or models, which 
science has imagined to exist in reality merely because 
they seemed capable of inciting the observables to pro- 
duce the effects actually observed. We have, however, 
no guarantee that these effects cannot be produced in 
other ways. To establish that the supposed unobservables 
exist, it would be necessary to shew that nothing else 
could produce the observed effects. 

The three supposed entities which were dismissed from 
science in the previous chapter absolute space, abso- 
lute time, and the ether were all drawn from the list 
of unobservables, as was of course inevitable. Their fate 
naturally raises the question of the status of the two 
unobservables which still remain in our scientific scheme 
distant events and objects. Are these also mere bad 
guesses, made in our hasty efforts to depict, in the light of 
inadequate knowledge, an external world of which no 
picture is, in all probability, possible? 

It may at first seem surprising that material objects 
such as electrons, protons and atoms should figure in 
our list of unobservables; yet it is here they must be 
placed. An eventless electron or proton could never dis- 
close its existence to us, and a single electron or proton 
must necessarily be eventless. The simplest kind of event 
can affect our senses which needs the juxtaposition of at 
least two such objects. When the two objects are of the 
same kind, the event is an encounter of electrons or 
protons such as can be observed in certain favourable 
cases. When they are of different kinds, they constitute 
a hydrogen atom, and the event is the emission (or 
absorption) of any one of the many types of photons which 
figure in the hydrogen spectrum. The mere orbital 


motion of an electron round a proton, which figured so 
largely in Bohr's theory, is not an observable event. It 
emitted no light, and so could not affect our senses. 

..We must then regard electrons and protons merely as 
unobservable sources of events which are themselves 
observable. The millions of electrons and protons in the 
sun exist only as inferences, created to explain the stream 
of photons which fall on our eyes and skin all day long. 

The Stream of Radiation 

Heisenberg, holding the views already explained, refused 
to concern himself with the unobservable electrons and 
protons in distant atoms, and concentrated on the ob- 
servable photons which came from them. These form a 
mixed bag of distinct kinds, which can be distinguished 
primarily by their frequencies. The bag is less like the 
bag of animals killed by a sportsman, than like the bag of 
tickets taken by a ticket collector on a railroad. For the 
principle of Ritz shews that each photon has two funda- 
mental frequencies associated with it (its own frequency 
being a measure of the distance between the two), just as 
each railway ticket has two railway stations associated 
with it e.g. Aberdeen to Birmingham. Bohr's theory 
had pictured these two frequencies as those of motions in 
fixed orbits, and imagined the emission of a photon to be 
the result of the passage of an electron from one orbit to 
another, just as we may picture the giving up of a rail- 
way ticket to be the result of a journey from one railway 
station to another. Heisenberg did not tie himself to any 
definite picture as to the origin of the photons; he was 
concerned only with the stream of light. In terms of our 
analogy, he did not try to picture in detail the move- 


ments of the trains on the railroad, but studied the work- 
ings of the system as a whole, as put in evidence by the 
stream of tickets forwarded to headquarters by the ticket 
collectors. For headquarters were our senses, so that the 
tickets were observables, but engines, coaches and pas- 
sengers were all unobservables. 

Headquarters could gain a great variety of statistical 
knowledge from such a collection of tickets; they could 
discover, for instance, the total number of passengers 
leaving Aberdeen, the total number arriving at Aberdeen, 
the total money taken at Aberdeen booking-office, the 
total number of miles travelled by passengers starting 
from Aberdeen, and so on for every station on the railway. 
If they wished to co-ordinate aU this knowledge, they 
would probably begin by tabulating the numbers of 
tickets issued from every station to every other station. 
If we denote the various stations Aberdeen, Birming- 
ham, Carlisle, Dundee, Edinburgh, and so on, by A, B, 
C 9 D, E, ..., the knowledge could be put in the form: 

A B - 23 
B > C = 72 
A >C = 13, etc. 

It could all be very concisely recorded in a table of 
double entry, as, for instance: 

From A B C D E 






84 22 





28 43 











and soon 


A square table of double entry provides the obvious way 
of tabulating quantities, each of which is associated with 
two others. In mathematics, such a table is called a 
matrix, and the entries 103, 23, 13, ... are described as 
the elements of the matrix. 

The first entry 103 in our particular table would mean 
that headquarters received 103 tickets from Aberdeen to 
Aberdeen; we may if we like interpret these as platform 
tickets, bought at Aberdeen and finished with at Aber- 
deen. The entry 23 to the right of this means that 23 
people travelled from Aberdeen to Birmingham, while 
the entry 23 in the line below means that an equal num- 
ber made the same journey reversed. This may or may 
not accord with the facts of railway travel, but it does 
with the original problem of physics. For this is idealised 
until all conditions become as simple as possible; in 
particular, the gas which emits the photons is supposed to 
be in a steady state. This requires that it shall be in 
equilibrium with its own radiation, and so absorb as 
many photons of each kind as it emits, except for an 
insignificant number which escape to affect our instru- 
ments. Thus as many atoms pass from one state to 
another, emitting photons, as pass in the reverse di- 
rection, absorbing photons. The result is that in the 
matrix which specifies the numbers of photons, the 
corresponding elements are equal, and the matrix is 
"symmetrical" in the sense that each vertical column 
contains just the same entries as the corresponding hori- 
zontal line. 

Our matrix has so far merely specified the numbers 
or, perhaps better, the relative proportions of the 
different kinds of photons, but this does not contain all 


the "observable" knowledge at our disposal. A railway 
ticket has printed on it the names of two places, and also 
the fare, which gives an indication of the distance between 
them. In the same way every photon has indelibly and 
unalterably stamped on it a quantity,, its frequency, 
which tells us the distance between the two fundamental 
frequencies associated with it. 

Even this does not exhaust our observable knowledge. 
It contains our knowledge of the relative numbers of pho- 
tons, but not of the manner of their oscillation. A large 
mass of experimental evidence shews that the oscillations 
of all photons are like those of a pure musical tone; they 
can be compared to the up-and-down motion of a point 
on the rim of a fly-wheel which goes round-and-round 
with absolute evenness. 

In mathematics, such oscillations are described as 
"simple harmonic". The changes they involve are pro- 
portional to those of the quantity cos (2in>t + c), where v 
is the frequency of the oscillation, and e fixes its phase 
(p. 121). Yet such a formula is too explicit for our 
present problem, in which the phase does not permit of 

It is, however, very easy to avoid over-precision of this 
type. A well-known theorem of Demoivre tells us that, 
if is any quantity whatever, 

e = cos + i sin 9, 

where e is the ordinary exponential function (so that ** 
is the quantity e or 2-71828... raised to the power of #), 
cos 9 and sin are the ordinary trigonometrical functions, 
and t stands for the square root of 1. Any algebraic 
quantity C can be represented in the form A + iB, where 


A and B are ordinary numerical quantities, and De- 
moivre's theorem now shews that 

+ B* [cos (9 + ) + i sin (6 + )], 

where depends on the ratio of A to B. If we now assign 
to the specific value 2irvt> we find that the first term 
V A* + B 2 cos (9 + e) exactly represents the needed 
oscillation. We may accordingly represent the oscilla- 
tion by Ge**** 9 with the understanding that the parts 
multiplied by i are to be ignored; in using such an ex- 
pression, we make no claim to know the phase e, because 
we do not divide C into its constituent parts A and iB. 
This is a useful and very common artifice in the mathe- 
matical theory of oscillations and vibrations of all kinds. 
Subject to this understanding, the whole of our observ- 
able knowledge of the stream of light received from a 
mass of, say, hydrogen can be embodied in a single matrix 
of the form 

Cace** i ( v a- v c')* 9 and so on, 

in which v has been replaced by its known value v* v^ 
etc., where *>, ^, ... are the fundamental frequencies of 
Ritz. We do not know the values of v a , v^ ... separately; 
but only their differences, which alone occur in the 


Atomic Structure 

Having embodied all our observable knowledge of the 
light emitted by, say, hydrogen in such a matrix, 
Heisenberg found the clue to his next step in Bohr's earlier 
investigation which we have already described. Bohr 


had pictured the hydrogen atom as consisting of an 
electron describing an orbit around a proton, and had 
obtained agreement with observation by supposing that 
certain "quantum restrictions" only permitted orbits 
whose diameters were proportional to the squares of the 
integral numbers. The large outer orbits were nearer 
together relatively than the small inner orbits, and the 
largest orbits of all could be regarded as continuous with 
one another, because their distances apart were insig- 
nificant compared with their total dimensions. Here we 
need no longer think of an electron which passes from one 
orbit to the next as jumping; we may think of its motion 
as continuous, and of the change in its energy as a con- 
tinuous change. 

So long as the electron remained in these orbits, Bohr's 
quantum restrictions were found to have no restrictive 
effect whatever on its motion, so that Bohr's picture of 
the atom coincided exactly with the older mechanical 
picture. As Bohr's picture predicted an emission of 
radiation which agreed with that actually observed, it 
follows that the old mechanical picture did the same in 
this extreme case of atoms of infinite diameter. 

In the case of orbits of very large, but not actually in- 
finite, diameter, Bohr shewed that the quantum restric- 
tions only came into play to a minute extent, and that the 
agreement extended to this case also. This is known as 
Bohr's "correspondence principle". 

Thus the old mechanical picture of nature predicted 
the radiation from very large atoms with accuracy, al- 
though it was known to fail badly for ordinary atoms of 
small diameter. Heisenberg set himself the problem of 
modifying the old picture so that it would remain true to 


observation over its whole range i.e. for small atoms 
as well as for large. 

As he had resolved to leave unobservables alone, the 
only material available for the construction of the new 
picture consisted of the matrix just described, and quan- 
tities directly reducible from it. Thus the line of attack 
was obvious first to try to reconstruct the old picture, 
which was valid for very large atoms, in terms of this 
material, and then to try to extend the picture to cover 
atoms of all sizes. 

Each element in the matrix is of course proportional 
to the intensity of the corresponding spectral line. If we 
halve the amount of gas by which the light is emitted, 
we halve the intensity of each spectral line, and so must 
halve each element of the matrix. If we reduce each 
element by dividing it by a suitable factor, we shall obtain 
a "reduced" matrix which will represent the average 
light emitted by one single atom. 

We have seen how the motion of a particle can always 
be represented as compounded of a great number of 
oscillations. In the special case in which the atoms are 
very large, Heisenberg found that the "reduced" matrix 
specifies the position of the point electron of Bohr's pic- 
ture, its separate elements representing the separate 
oscillations, the compounding of which gives the motion 
of the electron. 

We must not of course expect a similar interpretation 
when the atom is small. Let us however assume that, 
even in this case, the hydrogen atom consists of two con- 
stituents, which we may still call the proton and electron, 
without knowing in the least what these words now 


As the matrix was found to specify the position of the 
electron in a large atom, Heisenberg imagined that it 
must also specify some sort of position of the electron 
when the atom was small, although, as we have seen, the 
electron must no longer be compared to a point, but 
rather to a complete railway system. The elements of the 
matrix must no longer be taken to represent oscillations 
which can be merely added together; they rather cor- 
respond to the journeys of the various trains on the 

It may seem strange that an electron which is de- 
scribing a large orbit in an atom can be pictured as a 
particle, while an electron which is describing a small 
orbit can be pictured as nothing less than a complete 
railway system. We must, however, remember that 
Heisenberg's matrix is not concerned with a single photon 
originating in a single atom, but rather with a beam of 
millions of photons of different kinds originating in 
innumerable atoms in different states. For this reason, 
it could not be expected to give us a picture of the state 
of any single individual atom, but only a composite 
photograph of all the atoms. We may if we like say that 
the reduced matrix gives us a picture of a "statistical 
atom" whose properties and qualities are the average of 
the properties and qualities of all the actual atoms con- 
cerned in the emission of the light. 

Each element in the Heisenberg matrix changes with 
the time in a calculable way. on replacing each element 
by its rate of change, we obtain a second matrix, which 
will give a sort of representation of the changes occurring 
in the statistical atom somewhat like the speeds of 
the trains on our railway system; for large atoms, it 


simply represents the speed of motion of the particle 
electron* If we multiply every element by the mass of 
an electron, we obtain a matrix which represents the 
momentum of the electron. 

Heisenberg s relation 

Let us denote this last matrix by p, and the original 
reduced matrix, which corresponds to the position of the 
electron, by q. Heisenberg found himself able to con- 
struct a new picture of atomic workings in terms of the 
matrices p and q, and this is found to agree exactly with 
observation throughout. 

This picture does not involve the matrices p and q sep- 
arately, but only their product. If p and q were simple 
quantities, such as 6 and 8, the product pq would have an 
exact meaning, such as 48. With things as they are, the 
product pq has no meaning as yet, and we are free to 
assign to it any meaning we like. Actually a mathe- 
matical theory of matrices had been in existence long 
before Heisenberg, and this had assigned a conventional 
meaning to the product p q, p multiplied by , and also a 
conventional meaning to qp, q multiplied by p. Ifp and q 
were mere quantities such as 6 and 8, the product pq 
would of course be the same thing as qp, but when they 
are matrices, this need not be so. Indeed, on the con- 
ventional mathematical interpretations of pq and qp, it is 
not so; pq qp has a definite meaning, being itself a 
matrix which is usually different from zero. 

In the extreme case of an atom of very large radius, 
Dirac has shewn that pq qp becomes identical with a 
quantity which occurs in ordinary dynamical theory, and 
is known as a Poisson-bracket (for our discussion it does 


not matter what these words mean). Ordinary dynamics 
tells us that the Poisson-bracket must retain a constant 
value throughout the whole motion, and we have already 
seen that ordinary dynamics predicts the radiation from 
an atom of very large size with accuracy. Thus a true 
picture of the radiative processes of an atom of very large 
size can be obtained by supposing that pq qp retains a 
constant value. 

This picture contains only the "observable" matrix q, 
and the matrix p which can be directly derived from it. 
It can obviously be extended to small atoms, since every 
term in it has a clearly defined meaning for small atoms. 
The picture is perfectly precise and clear-cut except for 
the evaluation of the constant quantity on the right of 
the equation; the only question is whether it is true to 
observational facts. Now Heisenberg and a great num- 
ber of other workers in this field have jointly shewn that 
the whole problem of atomic spectra is solved not a 
single old difficulty remaining outstanding, nor a single 
new one appearing by supposing that the activities 
of the electron conform in every case to the simple law 

In this equation is itself a matrix, of the kind known 
as a unit diagonal matrix, namely 




and so on 

in which all the diagonal terms are equal to unity, while 
all the rest are equal to zero. Thus the meaning of 


Heisenberg's equation is that, in the matrix pq qp 9 all 
those elements which do not occur in the diagonal are 
equal to zero (as they would be if p and q were ordinary 

numbers or quantities), while all the rest are equal to . 

Here h is again Planck's constant which pervades the 
whole of small-scale nature. The letter i stands for 
V 1, which again pervades all atomic physics, pos- 
sibly as representing a transition from some sort of real 
time to the time of our observation (p. 100); TT stands 
as usual for the number 3- 14159... (the circumference of a 
circle in terms of its diameter), while somewhat re- 
freshingly, after all the other symbols 2 merely denotes 
the ordinary 2 of arithmetic, twice one. 

The central fact in the situation is that pq is not equal 
to qp. There is nothing mystical or surprising in this, 
since p and q are not single quantities, but arrays of 
quantities, and the product pq is defined in such a way as 
to make it different from qp at the very outset. Yet the 
quantities which enter into the arrays p and q are inter- 
locked in a curious way, so that p and q may almost be 
treated as single quantities rather than as arrays of inde- 
pendent quantities. This appears very clearly in the 
limiting case of atoms of very large dimensions, in which, 
as we have seen, we may think of p and q as the co- 
ordinate and momentum of a point-electron. 

A simple illustration may explain the nature of this 
interlocking. Any pair of independent numbers, say 2, 3, 
forms an array of two numbers, but out of these we may 
form the algebraic combination 2 + 3V 1, which may 
be treated as a single number it is for instance a root of 


the quadratic equation x* 4* + 13 = although 
its constituents 2, 3 do not lose their identities. If we take 
P and Q, to be the numbers whose constituents are 2, 3 
and 4, 5 respectively, the product PQ has the value 

PQ = (2 + 3V r ^l)(4 -j- 5\/~) - - 7 + 22 \/^"l, 

which is of the same nature (- 7, 22) as the original 

Here of course the product PQ appears to be precisely 
equal to QP. Yet there is an ambiguity here, since we 
have taken no account of the fact that there are two 
distinct square roots of 1, of equal values but of 
opposite signs. Let us denote these by i and j, and further 
agree that any product PQ is to be obtained by putting 
V 1 equal to i in the first factor, and equal to j in the 
second. With this definition of a product, PQ is not the 
same thing as QP; in fact we find that 

PQ - QP = (2 + 30(4 + 5j) - (4 + 5f)(2 + 3/) 
of which the value is 2 (i j) or 4i. 

In some such way as this, we may treat/?, q almost, but 
not quite, as single quantities, q being analogous to the 
co-ordinate and p to the momentum of a point-electron. 
To the extent to which this analogy holds, we find that 
pq is analogous to the action (p. 123) of the electron 
throughout the description of a complete orbit* measured 
in one way, while qp is analogous to this same action 
measured in a different way. We now see that the equa- 
tion of Heisenberg, 


* More precisely, 2*pq and 2rqp are analogous. 


merely expresses a relation between two different quanti- 
ties, each of which becomes analogous to the action in 
an electron-orbit in the limiting case in which this orbit 
is of very large dimensions. 

Although this equation has so far only been found true 
for an "average" or statistical atom (p. 183), yet, as so 
often happens in science, its validity appears to extend far 
beyond the special conditions which led to its discovery. 
The state of any dynamical system can be specified by a 
number of co-ordinates (q) 9 and its motion by the values 
of the momenta (p) associated with these co-ordinates. 
With any such pair of corresponding values assigned to 
p and ?, Heisenbergfs relation appears to be confirmed by 
observation, although we must remember that the ob- 
servations can never be made on single atoms, but only 
on statistical assemblies. 

This generalisation implies that the equation must hold 
for individual atoms, as well as for statistical atoms. This 
is necessary to explain why a mass of gas gives a spectrum 
of sharp lines. It also shews that each individual electron, 
when inside an atom, has the complexity of a whole rail- 
way system rather than of a simple moving point; the 
picture of an electron as a point in space and time fails, 
completely and finally. 

To take a further instance, the atoms of a solid body are 
in effect a vast number of oscillators vibrating around 
positions of equilibrium, the energy of their oscillations 
being merely the heat-energy of the solid. Heisenberg's 
relation shews that such an oscillator can never be com- 
pletely devoid of energy, its oscillations necessarily 
possessing or f or f or any such number (n + ) of 
quanta of energy to. Thus a solid body can never lose all 


its heat; even at the absolute zero of temperature, each of 
its internal oscillations possesses half a quantum of energy. 
Surprising though this conclusion may be, it is strikingly 
confirmed by investigations on the specific heats of solids 
at low temperatures. 

The relation can also be applied to the rotations of 
atoms and molecules, when it shews that the angular 
momentum must always be one or the other of certain 
definite and calculable multiples of h. Transitions from 
one of these values of the angular momentum to another 
produce the lines observed in band spectra, and the 
calculated frequencies are confirmed by observation. 

Perhaps, however, the most interesting application of 
the equation is to ordinary radiation in empty space. 
We have seen (p. 166) how the energy of such radiation 
can be regarded as made up of the energies of a number 
of separate free vibrations. Heisenberg's relation shews 
that the total energy of the free vibrations of any given 
frequency v must be an integral multiple of hv, which is 
precisely the supposition on which Planck originally 
based his quantum theory of radiation. In other words, 
the relation shews that all radiation can be regarded as 
consisting of indivisible photons each having energy hv\ 
it brings atomicity into the wave picture of radiation, and 
this, in combination with the considerations of p. 164, 
establishes the equivalence of the wave picture and the 
particle picture. 

All this suggests that Heisenberg's relation must be the 
expression of some quite fundamental law which, so far as 
we can at present see, must hold throughout the whole of 
nature. We shall discuss possible interpretations of this 
law below (p. 208). 


Transition to Newtonian Mechanics 

The quantities of action which occur in the large-scale 
events of everyday life are of course enormously large 
multiples of h. For instance, the action for a complete 
oscillation of the pendulum of a grandfather clock is about 

400,000 seconds X ergs, 
while the value of Planck's constant is only 

h - 0-000000,000000,000000,000000,006555 seconds X 

The products pq and qp which occur in Heisenberg's 
equation are each of the nature of action, and for tangible 
bodies they will be such enormous multiples of A, that the 
single h on the right-hand side may be neglected in com- 
parison. When this is done, the equation takes the form 

P<1 = <lP> 

and merely tells us what was accepted without hesitation 
for the whole of nature, until challenged by Heisenberg. 
This shews very clearly how the new theory of Heisen- 
berg gradually fades into the ordinary mechanical theory 
of Galileo and Newton as we pass from atomic structures 
to objects of tangible size. Primitive man did not regard 
a river as a collection of molecules of water but as a con- 
tinuous stream, and his more sophisticated descendants, 
still treating water as continuous, developed the science 
of hydraulics. This was only suited for dealing with 
molecules in vast crowds; it gives accurate results for 
rivers in which billions of molecules are involved, but fails 
for single molecules. In the same way the Newtonian 
mechanics was only suited for dealing with processes in 


which the unit of action h occurred in vast crowds; it gave 
accurate results for the motion in large-scale processes, 
where billions of units of action were involved, but failed 
for sub-atomic processes which involved only single units. 
Newtonian mechanics is the limit to which Heisenberg's 
picture of nature proceeds when the units of action be- 
come so numerous that they can be treated as a crowd. 

We may also compare Heisenberg's equation with the 
earlier quantum restrictions of Bohr, of which it is of 
course the more accurate successor. It used to be sup- 
posed that these restrictions were somehow added to the 
Ne\vtonian laws; large systems were subject only to the 
Newtonian laws, \vhile atomic systems were subject to 
these laws and to the quantum restrictions in addition. 
Heisenberg^s equation is occasionally discussed in a 
similar manner, as though the laws of nature allowed less 
liberty to little systems than to big. Actually this is not the 
case. The information that little systems obey Heisen- 
berg's law only corresponds to what we have already 
tacitly assumed about the big systems in supposing that 
for them p q is the same thing as qp. We have perfectly 
unwittingly assumed this in supposing that big objects 
can be represented in time and space. The fact that pq is 
not the same thing as qp for sub-atomic nature casts 
doubt on whether sub-atomic nature can be represented 
in time and space at all, a question to which we shall 
return later (p. 252). 

Somewhere in Heisenberg's equation the innermost 
nature of atomic structure must Ke hidden, if we could 
but read the riddle aright. As the equation does not bear 
any obvious interpretation on its face, our best procedure 
will be to try to construct a kind of model system which 


shall conform to the laws expressed in the equation. If 
our attempt succeeds, the model will not necessarily, or 
even probably, be identical with any real structure in 
nature, but is likely nevertheless to throw some light on 
the nature of the atom, for it would be surprising if two 
distinct systems, both governed by the same equation, did 
not have some properties or characteristics in common. 



While Professor Heisenberg of Leipzig was following up 
the train of thought described in the preceding chapter, 
Prince Louis de Broglie of Paris and Professor Schrodinger 
of Berlin were engaged on an independent attack on the 
problem of the structure of matter. Between them, they 
devised an alternative explanation of the origin of the 
spectra of chemical substances, which at first sight seems 
to bear but little relation to that of Heisenberg. Sub- 
sequently Schrodinger himself, as well as Born and 
Wiener, shewed that the two sets of ideas not only led to 
the same results, namely those actually observed in na- 
ture, but were fundamentally identical. Schrodinger had 
in effect obtained a solution of Heisenberg's equation 
which admitted of physical representation, and so pro- 
vided us with a sort of model of the electron. This model 
has proved capable of interpreting all the results actually 
observed by spectroscopists. We shall first explain the 
work of de Broglie and Schrodinger from this standpoint, 
although it will be understood that in the first instance its 
authors achieved it entirely independently of Heisen- 
bergfs ideas. Later we shall discuss the more physical 
concepts of de Broglie and Schrodinger in a less mathe- 
matical manner; some readers may prefer to pass directly 
to this discussion (p. 203). 



The Wave Picture 

If any quantity r increases n times as rapidly as a second 
quantity q, then we describe n as "the differential co- 
efficient of r with respect to (f\ which we write as ~(r). 


Any other quantity, such as the product qr, will also have 
a differential co-efficient with respect to q, which we write 

as | for). 

The product qr has a double cause of change, namely 
the changes in q and the changes in r. on account of the 
first cause, qr changes r times as fast as q; on account of 
the second, q times as fast as r. Adding these changes 
together, we obtain the relation 

We notice that r enters just once in each term of this 
equation, so that it can be written in the alternative form 

7 f *~ 

in which we regard as an operator, this meaning that 

everything which comes after it must undergo differentia- 
tion with respect to q. We also regard q itself as an oper- 
ator, meaning that everything that comes after it is to be 
multiplied by #, this latter being of course the meaning 
in ordinary algebra. As our equation is true whatever r 
is, we may write our knowledge in the form 

d d 


in which every symbol is treated as an operator, including 
unity on the right-hand side. 

If we think of a number, then double it and then halve 
it, the final result will of course be the number we first 
thought of. In the same way, we may think of any 
quantity, perform upon it all the operations directed by 
the combination on the left-hand side of the above equa- 
tion, and our equation shews that the result will always 
be the quantity we first thought of. If we multiply both 

sides of the equation by -., it reads 

O - = 

2mdq q q 2iridq 2m 

which, we see at once, is of the same general form as 
Heisenberg's relation. Indeed, the two relations become 
identical if we take 


P 2m dq 

When we say that x = 3 is a solution of * 2 + 2 = 11, 
we mean that the substitution of 3 for x reduces the equa- 
tion to a truism. In the same sense the value we have 
just written down for p reduces Heisenberg's equation to a 
truism, so that in a certain sense we may say that it is a 
solution of this equation. We must not say it is the solu- 
tion, or the only solution, any more than that x = 3 is the 
only solution of x 2 + 2 = 11, but it is one solution, and so 
will shew us something of the meaning of the equation. 
In more physical language, it shews us how to construct a 
model or picture, which may be only one of many possible 
models or pictures, and yet may perhaps tell us some- 
thing of the physical meaning of the equation. 


Let us first use it to examine what Heisenberg's equa- 
tion means when applied to the motion of a particle, such 
as an electron. We need not suppose that the particle is 
so minute as to be point-like, but we shall suppose that its 
position can be specified by the position of a point in 
space, just as the position of a cricket ball can be specified 
by the position of its centre, or that of a train by the foot- 
plate of its engine. Then we can specify its position (as 
on p. 86) by an array of three co-ordinates x, y, , which 
we may regard as giving the distances through which the 
particle has moved in three directions at right angles to 
one another as, for instance, vertical, north-south and 
east-west. The speed of motion of the particle will be 
specified by a similar array w, z>, 20, and its momentum 
by yet a third array a, b, c, in which, however, we know 
that a is the same thing as mu, where m is the mass of the 
particle. In this simple case, the "solution 35 we have 
already obtained for Heisenberg's equation assumes the 

form h // 

mu = Af (C), 

2iri dx 

and there will of course be similar values for mv and mw, 
the system of three equations thus being symmetrical in 
the three co-ordinates *, j>, z. We have, however, seen 
(p. 99) that any true picture of nature must be sym- 
metrical in the four co-ordinates x 9 y, z and ict of the con- 
tinuum. This indicates that our system of three equations 
is incomplete; there must be a fourth equation cor- 
responding to the co-ordinate ict. This is easily found 

tobe h d 

me* = : -y (D). 

t at 


If these equations seem meaningless, this is only what 
was to be expected. Heisenberg began by conjecturing 
that it would prove impossible to construct an intelligible 
picture or model of an electron. The fact that our equa- 
tion the first step towards such a model already 
proves to be unintelligible, suggests that he may have 
been right. If the equations had led us to a simple model, 
such as a tiny hard sphere, Heisenberg would have stood 
convicted of unwarranted pessimism. 

We can, nevertheless, infuse a little more physical 
meaning into our equations by remembering that both 
sides are operators, and so are hungry for something on 
which to operate. If each member of the first equation 
(C) is given a symbol ^ on which to operate, it becomes 

cty _ 2wimu . 


which is a very familiar equation. Solving it, we find that 
^ must be of the form 

where C is a constant. 

The formulae given on pp. 179, 180 shew that this rep- 
resents a train of regular waves. As x changes, the value 
of ^ fluctuates between the values +C and C, repeat- 
ing itself at regular intervals k/mu. It is beyond the scope 
of this book to enter upon either a rigorous or a com- 
prehensive discussion of mathematical formulae. The 
present discussion is neither, but shews that a momentum 
mu is in some way related to a train of regular waves of 
wave -length h/mu, or h divided by the momentum in 


Photons have already provided an instance of this, for 
we have seen that the wave-length of a photon is equal 
to h divided by its momentum. There is experimental 
evidence that the relation is equally true for electrons. 

In the experiments of G. P. Thomson, a shower of 
electrons, all moving with the same speed and in the 
same direction, like a regiment of soldiers marching in 
perfect order, was allowed to strike on a very thin film 
of metal* An older physics would have predicted that 
each electron would fight its individual way, as best it 
could, through the atoms of the film and their interstices, 
so that the current of electricity would emerge on the 
other side as a disordered mob of electrons, the indi- 
viduals moving at different speeds and in different 
directions. Instead of this, the experiments shewed that 
they form a perfectly regular wave pattern such as is 
shewn in fig. 2 of the frontispiece. Their scramble 
through the interstices of the solid has not introduced dis- 
order into their formation but a new kind of order. It 
has changed the quality of what order there was, and 
given it wave-like characteristics. 

It is found possible to measure the precise wave- 
lengths of these waves. Their wave pattern proves to be 
identical with that which is formed by X-rays of a certain 
known wave-length, so that this must be the wave-length 
of the shower of electrons also. Now the wave-lengths 
obtained in this way are found to obey the theoretical 
law exactly invariably the wave-length is h divided by 
the momentum of each electron in the shower. It is 
important to understand that this wave-length has 
nothing to do with the spacing of the atoms in the metal 



Experiments on showers of positively charged protons 
have given similar results. 

Thus we may proceed with some confidence that we 
are on the right road; our picture, however unintelligible 
it may be to us, is true to reality. 

The wave-length of the electron waves is determined by 
the speed of travel of the shower of electrons, and this 
shews that the waves cannot have any objective existence 
when the shower is travelling through empty space. For 
under these conditions "speed of travel" has no meaning 
at all this of course is the main message of the theory 
of relativity. As soon as a shower of particles encounters 
matter of any kind or an electric field, this provides a 
frame of reference against which the speed of motion can 
be measured, and the expressions "speed of motion" and 
"momentum of a particle" acquire a definite meaning. 
There is no reason why the waves should not be real now, 
but we shall soon see that even in this case they must 
not be supposed to possess any material or substantial 

We can discuss the fourth equation of our group, 
equation (D) on p. 196, in precisely the same way. If we 
provide a quantity ^ for each side to operate on, we get 
an equation of the same type as before, of which the 
solution is found to be 

This represents vibrations which repeat themselves at 
regular intervals h/mc 2 of time. Again we cannot enter 
on a rigorous or comprehensive mathematical discussion, 
but we see that the existence of a particle of mass is in some 
way associated with vibrations of frequency mc*/h. 


The theory of relativity tells us that a particle of mass 
m is a storehouse of energy of amount me 2 . Thus the 
frequency of vibration is equal to the energy divided by h. 

Again photons provide an instance of this, their fre- 
quency being equal to their energy divided by h. 

The theory of relativity shews that the mass of a moving 
particle depends on the speed of motion of the particle, 
being proportional to 

the factor we have already encountered in Chapter m 
(p. 89). Thus a moving particle has a greater mass than 
one at rest, and so has also greater energy; actually the 
excess is precisely the kinetic energy of its motion. It 
follows that a moving particle is associated with more 
rapid vibrations than one at rest; its motion increases the 
frequency of vibration. 

It might seem at first that these vibrations must be 
subjective, like the waves just discussed, because they 
depend on the speed of motion of the particle, and we 
have no objective framework against which to estimate 
this speed. Actually this is not the case. Motion with a 
speed u increases the rate of oscillation by the factor men- 
tioned above, but also, as we saw on p. 91, it increases the 
unit of time in precisely the same ratio, so that the two 
effects cancel out. Thus no reason has so far appeared 
why these vibrations should not be real. If we picture 
the electron as oscillating me 2 /h times a second, the wave- 
pattern of an electron is very simply explained as result- 
ing from the interaction of this oscillating system with the 
solid surface on which it falls. Calculation shews that the 


wave-lengths disclosed by actual experiment would be 
produced by the electron-structure vibrating at the rate 
of 1-24 X 10 20 (124 million million million) complete 
oscillations a second. Thus we have a far more wonder- 
ful and surprising picture of the electron than that which 
exhibits it as a tiny billiard ball charged with electricity, 
although it must be remembered that we are still in the 
dark as to how far these vibrations are real, and how far, 
like the waves, they are mere mathematical fictions. 
Theory and observation agree in suggesting that protons 
may also be pictured as performing vibrations, real or 
fictitious, at the even greater rate of 229,000 million 
million million complete oscillations a second. 

We have seen that both waves in space and vibrations 
in time are associated with the motion of a particle. We 
can combine the two effects in the single formula 

this value of ^ satisfying equations (C) and (D) simul- 

It represents waves of wave-length h/mu 9 travelling in 
the direction of # at a speed c*/u . This speed again is one 
with which the theory of relativity has made us familiar; 
it is the speed of propagation of local time (p. 91). Thus 
we see that a particle of mass m moving at a speed u is 
in some way related to such a train of waves. 

A motion which is performed with varying speed is 
related to a more complicated system of waves. As an 
illustration, let us consider the special case of an electron 
moving in a field of electric force. To simplify the prob- 
lem we shall suppose that its speed always remains grnall 
in comparison with that of light. 


As we have seen, its motion must conform to the laws of 
ordinary mechanics as well as to Heisenberg's restriction. 
Ordinary mechanics tells us that a moving electron pos- 
sesses kinetic energy of amount \m (u 2 + v* + w*)\ and 
that it moves in such a way that the sum of this and its 
potential energy, which we may call V, retains a constant 
value throughout the motion. If we denote this constant 
value by , we know from ordinary mechanics that 

\m (u* + v* + w*)=E-V ...... (E) 

throughout the motion, while Heisenberg's relation adds 
the further restrictions which we can represent in our 

model by replacing mu by . , and so on. The com- 

2717 dx 

bined information leads to the equation 

In this equation, as before, every term is to be regarded 
as an operator. Let us again supply a symbol ^ for the 
operators to act on, without for the moment pausing to 
enquire what ^ is. Our equation now becomes 

The value of ^ will still oscillate with the time, so that 
mathematicians will recognise the equation as expressing 
the propagation of waves. The wave-length is no longer 
definite; it varies from point to point, being proportional 
to 1 / V V, just as the wave-length of light varies as 
it passes through a refracting substance. We see that an 
electron moving in an electric field is in some way related 


to waves like those of light moving through a refracting 


De Broglie Waves 

The equation we have just obtained is generally known 
as Schrodinger's wave equation, because Schrodinger 
first obtained it by a distinct method of his own. This 
method did not involve passing through the particle con- 
cept of an electron at all. Like Heisenberg, Schrodinger 
was convinced that Bohr's earlier theory had failed be- 
cause it envisaged the electron and proton in the too 
precise and too concrete form of small charged particles. 
Before Schrodinger came on the scene at all, this failure 
had reminded de Broglie of the earlier, and very similar, 
failure of the corpuscular theory of light. A theory which 
had pictured light as a shower of minute corpuscles had 
explained shadows and other simple large-scale proper- 
ties of light, but only a wave picture had been found 
capable of explaining its more subtle small-scale prop- 
erties. In the same way the picture of matter as a collec- 
tion of minute particles, namely electrons and protons, 
explained some but not all of its properties, and these 
were mainly the large-scale properties. De Broglie sus- 
pected that a wave picture might be needed to explain 
the remainder. 

Whatever an electron may be, it must be supposed to 
conform, like the rest of nature, to the theory of relativity. 
This shews that it is meaningless to assign to an electron 
properties which can be specified in space alone; their 
description must involve time as well. This may seem a 
small clue to go on, but in actual fact it is found to restrict 
the structure and behaviour of the unknown object some- 
what drastically. We find that if the electron conforms to 


the theory of relativity, it must be possible to picture its 
structure mathematically as a system of waves. 

For if the structure of an electron at rest is specified in 
terms of x, y, z and t, then when the same electron is mov- 
ing with a speed u in the direction of x, its specification 
will be the same except that x must be replaced by x ut 

and t must be replaced by the local time t ^ This 


is of course the content of the Lorentz transformation, 
except that we have disregarded certain small changes 
which only become important when the electron is mov- 
ing with a speed comparable with that of light. 

The change of x into x ut is a necessary consequence 
of the motion of the electron; x ut retains its value if 
the electron moves with a speed u in the direction of x. 

But the change of t into t ^ implies motion of a dif- 


ferent type; for this quantity to retain its value, an 
unknown something must move with a speed c*/u in the 
direction of x. We may picture this as the propagation 
of some kind of disturbance, or a system of waves, moving 
in the same direction as the electron at a speed c */u, the 
speed at which local time is propagated. 

A group of waves, each of which was travelling at pre- 
cisely the same speed, would of course itself travel at the 
same speed as its individual waves. A flash of light pro- 
vides an obvious instance of this: it travels at the same 
speed as the waves of light which compose it. But clearly 
we cannot imagine an electron represented by such a 
group, since waves travelling at a uniform speed c*/u 
would immediately run away from the electron, which 
only travels at the slower speed u. 


Let us now consider the motion of a group of waves, 
anywhere and of any kind, which travel at different 
speeds, but momentarily extend through only a small 
region of space a storm at sea will help to fix our ideas. 
The front of the disturbance will of course forge ahead at 
the speed of its fastest wave, but its tail will travel only at 
the speed of the slowest. Thus the head and tail will 
continually increase their distance apart, so that the 
group will spread, while the centre of the group, always 
lying between the two, will move at a speed intermediate 
between those of the fastest and slowest waves. 

There is one case in which these conclusions do not 
follow, namely when each constituent wave is already 
spread through the whole of space. This is not incom- 
patible with the whole system forming a group of finite, or 
even small extent, because the outlying waves may neu- 
tralise one another completely by destructive interference 
(p. 166). Indeed the mathematical theorem to which we 
have already referred has shewn that any disturbance, no 
matter how restricted, can always be regarded as made up 
of constituent waves, each of which extends through 
space. A group of waves which is restricted to a small 
extent of space in this way is known as a "wave packet". 

Analysis shews that a wave packet will in general 
spread, but may do so only slowly. When it spreads so 
slowly as to remain a compact structure through an 
appreciable time, we may properly speak of its speed of 
motion, and we find that this need no longer be inter- 
mediate between the speeds of the fastest and slowest 
waves which enter into its composition. For instance, a 
wave packet may itself travel with a speed u, although 
each of its constituent waves may have a speed of travel 


near to some quite different speed, such as c*/u. This is of 
course exactly what we want if we are to explain the 
electron as a group of waves. The condition that the 
speeds of the group and of its individual waves shall be 
related in this particular way is quite simple; it is that the 
frequency of a wave travelling at a speed c z /u shall be 
proportional to 

which is exactly the relation we have already obtained for 
the waves we discussed on p. 200. 

Thus we can identify the de Broglie waves we are now 
discussing with those previously discussed, the wave- 
length and period of vibration being given by the for- 
mulae already mentioned, and so proving to be the same 
for electrons, protons and photons. 

A detailed study confirms that this system of waves will 
not run away from the electron. Individual waves are 
continually neutralising one another before and behind, 
and reinforcing one another in the intermediate regions, 
and they do this in just such a way as to build up a per- 
manent structure which moves with precisely the speed 
of the electron. Except for the fact that the electron- 
waves are purely mathematical, their action proves to be 
similar to what we see in the ordinary bow wave of a boat. 
Although ripples on the surface of the sea may travel 
faster or slower than the boat, this complete system of 
ripples does not run away from the boat, but progresses 
steadily at exactly the speed of the boat. 

The speed of light c is necessarily greater than u, the 
speed of motion of the electron. The speed c*/u of the 


waves is greater than the speed of light in just the same 
proportion, so that the waves not only travel faster than 
the electron, but faster than light itself. For instance, 
if the electron is travelling at a quarter the speed of light, 
the individual waves travel at four times the speed of 
light; if the electron is a slow one, travelling at only a 
thousandth part of the speed of light, its waves travel a 
thousand times as fast as light, and so a million times as 
fast as the electron. 

This of course only gives us a mathematical picture of 
a kind with which mathematicians are very familiar; they 
are accustomed to regarding all kinds of changes as pro- 
duced by successions of waves, merely as a convenient 
means of description. 

We have seen that the wave-lengths and periods of 
electrons, protons and photons are all given by the same 
formulae, namely 

momentum X wave-length = A, 
energy X period = h. 

It is certainly very remarkable that these formulae are 
the same for three such dissimilar objects as electrons, 
protons and photons. one possible explanation is of 
course that these apparently diverse objects may in the 
last resort be of die same nature, at any rate so far as their 
oscillations, as expressed by their frequencies and wave- 
lengths, are concerned. The fundamental distinction 
that electrons and protons carry electric charges, whereas 
photons do not, is enough in itself to account for many of 
the differences of their properties. Both cany energy and 
so possess mass. For a charged particle to carry a finite 
amount of energy, it must move more slowly than light, 


whereas for a photon to carry a finite amount of energy, 
it must move at precisely the speed of light. This explains 
why photons always travel with the speed of light, while 
electrons and protons travel more slowly. Again, elec- 
trons and protons interact with one another through the 
attractions and repulsions of their electric charges, 
whereas photons, having no charges, cannot interact with 
one another at all. Thus there is considerable justifica- 
tion for regarding photons as being of the same nature as 
electrons and protons, but without electric charges. 

Yet the facts admit of a far wider interpretation than 
this. The formulae in question are direct consequences 
of Heisenberg's relation, and the whole of the available 
observational evidence indicates that this relation is true 
throughout all nature; its validity certainly extends be- 
yond electrons, photons and protons (p. 188). 

Hence it seems probable that the derived formulae also 
are valid throughout all nature, the cases of photons, 
electrons and protons only providing illustrations of 
a quite general truth. In other words, the formulae 
may perhaps express some general property of space and 
time, rather than of special phenomena or objects in 
space and time. Energy for instance may be merely 
another way of regarding frequency or time, and mo- 
mentum only another way of regarding wave-length or 
space. Just as we can regard light almost indifferently as 
either waves or particles, so it may be that we can regard 
space almost indifferently as either extension or momen- 
tum, and time as either oscillations or energy. 

There may seem to be two distinct conjectures here; 
actually there is only one. For the theory of relativity 
shews that space is related to momentum in the same way 


in which time is related to energy. When the imaginary 
observer of the theory of relativity changes his speed of 
motion through space, he changes space into time in the 
sense already explained; and in precisely the same sense, 
and to the same extent, he changes momentum into en- 
ergy. In the space-time continuum, momentum and en- 
ergy are merged into one, like space and time themselves. 

Thus the conservation of energy may admit of inter- 
pretation also as a conservation of oscillations; the total 
number of oscillations taking place throughout the uni- 
verse in unit time may remain constant, and this may give 
an absolute measure of time. Similarly the conservation 
of momentum may admit of interpretation as a conser- 
vation of wave-number (number of waves per unit of 
length), and this may give an absolute scale of length. 

Modern science has taken a great deal away from the 
nature of nineteenth-century science, and may be ex- 
pected to supply something to fill the gaps it has created. 
The concept of relativity seems to qualify as one of the 
new factors; possibly the concept just mentioned may be 
another. The mere circumstance that we have to con- 
sider such possibilities seriously shews how far removed 
the science of to-day is from that of thirty years ago; if, as 
we hope, present day science is one stage on from the 
world of appearance, as represented by the older science 
and the "common-sense" view of nature, towards the 
world of reality, the same circumstance may also suggest 
how wide is the gulf between appearance and reality. 

In any event, the frequencies of electrons and protons 
ought to possess the same kind of reality as the corre- 
sponding quantities for photons. Now we know some- 
thing, at least, as to the reality of these latter frequencies. 


If we turn a dynamo at such a rate that a coil passes 
through a magnetic field 50 times every second, we shall 
obtain an alternating current with a frequency of 50 
cycles a second. Through whatever magnetic leakage 
there may be, electric waves will travel out into the sur- 
rounding space, and these waves will also have a fre- 
quency of 50 cycles a second. If we picture the radiation 
as consisting of photons, then the frequency of the pho- 
tons will still be 50, and this frequency must be just as real 
in time as that of the turbines which drive the dynamo. 

It is much the same with wave-length of photons. 
When electric waves are sufficiently long, we can map 
them out with a spark-gap. We may find, for instance, 
that we have to walk 50 feet to pass from one crest to the 
next. If we picture the radiation as photons, we must 
suppose their wave-length to be 50 feet, and this wave- 
length would seem to be just as real a length in space as 
the 50 feet we have to walk in order to measure it. 

This may seem to suggest that the wave-lengths of pho- 
tons are real, although we have just seen that the wave- 
lengths of electrons cannot be. We shall soon see that 
there is no real contradiction; a want of reality pervades 
all and everything, creeping in from a quite unexpected 
direction a direction, at any rate, which must seem 
very surprising to a mind brought up to think in terms of 
the objective concepts of the older physics. 

The Nature of Electron Waves 

We have obtained a complete mathematical specification 
of electron waves, but this tells us nothing as to the true 
nature of the waves themselves. Out of the waves which 
we find to be connected with the electrons and protons 


of the atom, mathematical theory can reconstruct the 
wave-lengths of the photons which the atom ought to 
emit, and finds a perfect agreement with observation. 
This tells us something about the wave-lengths, but tells 
us nothing about the waves except their lengths. We 
have little but conjecture to help us discover of what the 
waves actually consist. 

It was at first conjectured that they consisted of elec- 
tricity. The most directly observable property of the 
electron is that it carries a charge of electricity of un- 
varying amount it seems to be deprived of most of its 
other supposed properties, such as minuteness, hardness, 
sphericity, by the wave-mechanics. Thus when it was first 
suspected that the electron had a structure, it was natural 
to think of this as a structure of electricity. Yet there are two 
distinct reasons why this concept cannot be maintained. 

In the first place, it is a universal property of every 
kind of wave to scatter through space. We may, for 
instance, picture a proton at any one instant as a packet 
of waves, occupying a diameter of a hundred millionth 
part of a centimetre, which is roughly the diameter of the 
hydrogen atom, but the waves will rapidly spread so as to 
occupy more space than this. Ehrenfest has calculated 
that such a bundle of waves would double its linear di- 
mensions in a ten million millionth part of a second, so 
that obviously such a system of waves would soon grow 
too big to shew the spatial properties of a proton. A 
smaller bundle of waves would expand even more rapidly. 

Mathematical theory shews that it is quite impossible 
to devise a system of waves which shall not scatter at all. 
Suppose, however, that we could devise a system of waves 
which would not scatter to any appreciable extent, while 


a proton or electron was pursuing an undisturbed path 
through empty space. Even so, the waves must scatter 
as soon as the particle interacts with other matter; we 
have direct experimental evidence of this in the wave 
patterns they form on a photographic plate. Thus, if the 
wave structure of a proton or an electron merely repre- 
sented its electric charge, this would be scattered as soon 
as the particle encountered matter. Yet observation 
shews that this does not occur; electrons and protons 
maintain their identity, and preserve their charges intact. 
The second objection is even more directly fatal. Let 
us consider what will happen to the waves of one electron 
when it meets another electron, and the two exert electric 
forces on one another. Again the motion of each electron 
must conform both to the old mechanics and to Heisen- 
berg*s conditions. The old mechanics tells us that the 
total energy, which consists of the sum of the kinetic 
energies of the two moving particles with the potential 
energy added on, retains a constant value which we may 
again denote by E. This is expressed by the equation 

im( 2 + * 2 + a; 2 ) + m V 2 + v'* + w'*) = E - V 9 

where the accented symbols m\ u' 9 v' 9 w' refer to the 
second particle. We can now represent Heisenbergfs 


further conditions by replacing m'u' by -^ , etc., and 

2m 5x 

we obtain, in place of our previous wave-equation (F) of 
p. 202, the new equation 

dx* dy* dz* dx'* dy'* d 

...... (O). 


This equation still represents the propagation of waves, 
but no longer in the ordinary space of three dimensions, 
having x,y, z as co-ordinates. The waves are propagated 
in a six-dimensional space having #, j?, , #', j?', z f for co- 
ordinates. In the same way, if a million electrons met, 
their waves would be propagated in a space of three 
million dimensions. Such a space can only be regarded 
as a mathematical fiction, and as we cannot suppose 
waves to be more real than the space through which they 
are propagated, the waves must be of the same nature. 

We have seen that the waves of a single electron are 
propagated in a space of only three-dimensions. We 
might be tempted to identify this with the space of every- 
day life, and conclude that the waves were real, were it 
not for the possibility of this electron meeting a second 
electron. When two electrons meet, they meet on an 
equal footing, so that neither of their sets of waves can 
claim a greater reality than the other. If we are asked to 
say which set of waves is real, we can only perform a 
judgment of Solomon, and declare that both are fictitious. 
And this makes it impossible that the waves should con- 
sist of electricity, or indeed of anything else which exists 
in our ordinary everyday space. 

Yet before leaving this as a final statement of truth, we 
shall do well to consider precisely what we mean by 


Photon Waves 

The waves of a photon are of course the ordinary light- 
waves of the undulatory theory. If the light is of fre- 
quency ?, the equation which governs their propagation is 

/d* + d* + d* \ , + 47rV , __ 
\dx 2 dy* dz 2 ) c* 


in which ^ is any component of electric or magnetic force 
or any linear combination of components. This can be 
shewn to be precisely identical mutatis mutandis with 
Schrodinger's equation (F) (p. 202) for the propagation 
of the electron waves of a single electron. 

But when two photons meet, the equation of wave- 
propagation is not identical mutatis mutandis with Schro- 
dinger's equation (G). In fact it is still equation (H). 
The reasons for the divergence here is of course that 
photons do not interact when they meet, whereas electrons 
do; this again can be traced back to the fact that photons 
do not carry electric charges as electrons do. Thus, al- 
though the waves of a million electrons need a space of 
three million dimensions for their proper representation, 
a space of three dimensions suffices for the waves of a 
million photons. Now we may properly identify this 
latter space with the space of our everyday life. For this 
is the space in which we see the sun, moon and stars, 
which again is the space in which the photons from the 
sun, moon and stars travel, and ultimately reach us. 
This is precisely the space of equation (H). 

Thus we may say that photon waves can be represented 
in our ordinary everyday space this indeed is the 
definition of this space but that electron waves cannot 
be so represented. 

Let us, however, proceed somewhat further with our 
comparison or contrast of electron waves with photon 
waves. on p. 159 we discussed an experiment in which 
monochromatic light was passed through two pin-holes 
and made to form a pattern of light and dark bands 
by interference. If the pattern is formed on a sensitive 
photographic plate, chemical action will occur at places 


where the pattern is light, but none where it is dark. Let 
us now picture our monochromatic light as a shower of 
photons, all having the same wave-length and so also the 
same momentum. We have seen that photons do not 
admit of localisation at single points of space; they are 
merely free vibrations of the laboratory (p. 166) or else 
the waves of such vibrations combined into wave packets 
(p. 205). 

Suppose, however, that we press the bullet aspect of 
radiation to the illegitimate extreme of supposing that 
each photon can be localised at a particular point of space. 
Then, to make our picture consistent, we must suppose 
that no photons fall on our screen where the interference 
pattern is absolutely dark; they all fall where it is light. 
Indeed, we must suppose that the number which falls on 
any small area of the screen is precisely proportional to 
the total illumination of the screen. We can even dis- 
pense with the screen altogether, and speak of the number 
of photons in a small volume of empty space; this num- 
ber will of course be proportional to the total amount of 
light-energy in the volume of space in question. To see 
that this is a legitimate and necessary extension of our 
ideas, we need only imagine a small screen placed at the 
far end the end away from the light of the small 
piece of space in question. This will catch the photons, 
much as a fish-net catches the fish swimming into it, and 
it is easy to see that the light-energy per unit volume of 
space is exactly proportional to the number of photons 
in this unit volume. 

Electrical theory, however, teaches us to regard energy 
as being spread continuously through space, not con- 
centrated in the isolated points which happen to be 


occupied by photons. How are we to reconcile this with 
our supposition that the energy is merely the aggre- 
gate energy of individual photons occurring at isolated 

The ordinary theory of gases (p. 156) points the way. 
It shews us a gas as a number of bullet-like projectiles 
its molecules. The mass of the gas is concentrated wholly 
in the few points of space which happen to be occupied by 
molecules. Nevertheless when we speak of the density of 
the gas, we suddenly change our picture; we form, so to 
speak, an out-of-focus picture in our minds in which the 
separate molecules are blurred into a continuous cloud, 
and what we describe as the density of the gas is merely 
the density of this blurred cloud, which we see spread 
continuously through space. 

When we put this picture back into sharp focus, we 
see the separate molecules again. We see] that the true 
density of matter varies abruptly from point to point it 
is large here, where there happens to be a molecule, and 
zero at an adjacent point where there is no molecule. 
Yet our fohner conception of a density which varied con- 
tinuously from point to point still retains a perfectly pre- 
cise and clear-cut meaning. It is this: if we take a tiny 
fragment of space surrounding a point P, the chance of 
our finding a molecule inside it is exactly proportional to 
the density at P. 

So, when we picture a beam of light as a shower of 
bullet-like photons, we must suppose that the density of 
light-energy at each point of space gives a measure of the 
chance of our finding a photon there. In ordinary elec- 
trical theory the density of light-energy is shewn to be 
E 2 + H 2 , where E and Hare the electrical and magnetic 


forces measured in suitable units.* Thus in our photon- 
picture of light, we may interpret E and H as quantities 
which between them give a measure of the probability of 
our finding a photon at a particular spot in space. 

If we take Schrodinger's equation which governs the 
propagation of electron waves, and make the changes ap- 
propriate to the transition from an electron to a photon, 
we obtain, as we have seen, the equation of propagation 
of electric disturbances, of which waves of light form a 
special case. Thus Schrodinger's quantity ^ must in some 
way be analogous to the quantities which specify electric 
disturbance i.e. to electric and magnetic force. These 
latter provide us with a measure of the probability of 
finding a photon at a particular spot when we picture 
photons as being localised at points of space, and in the 
same way $ must provide a measure of the probability 
of finding an electron at a particular spot when we 
picture electrons as being localised at points. 

Mathematical theory discloses the exact relation be- 
tween # and the probability in question. The propaga- 
tion of electric disturbance is determined by the equation 

where E and H are the electric and magnetic force,t i is 
the square root of 1, aodjij* jz> ji are symbols whose 
exact meaning does not matter at the moment. They are 

* In the more ordinary units, the density of energy is of course 

t Actually E f iH stands for the six-vector X + to, *" + % + nr 
X -f ao, T -f #, -f i% and the equation must be read as a vector 
equation. Hie vanishing of the four components of the vector on the left 
then give Maxwell's eight equations exactly. 


a sort of geometrical square roots of 1, being unit 
vectors drawn in the directions of the axes of x 9 y, a and r 
respectively, where r = id. 

We may compare this with the equation for the propa- 
gation of electron waves, which Dirac has reduced to 
the form 

Here ^ is the symbol which appears in Schrodinger's 
equation, and m is the mass of the electron, this intro- 
ducing a new term which did not appear in the photon 
equation above. Again E^ E% E^ Ei are symbols whose 
exact meaning does not matter at the moment. Like 
ji9J*>j*9J* they are square roots of 1, but being matrices 
they no longer admit of simple geometrical interpretation. 
This re-affirms the fact already noticed (p. 214), that 
photon waves can be represented in space and time, 
whereas electron waves cannot. 

A comparison of the two equations immediately sug- 
gests that ^ for an electron is the analogue of E + iH for 
a photon. We have already seen (p. 179) that ^ will have 
an imaginary part as well as a real part, so that we may 
replace ^ by ^i + t^* Then ty\ is analogous to E and fa 
to T, so that fa 2 + fa 2 is analogous to E 2 + H* which, as 
we saw, gives a measure of the chance of finding a photon 
at a particular spot of space, when we picture photons 
as existing at spots in space. 

Waves of Probability 

Reasons of this kind led Born to conjecture that, if we try 
to locate electrons at spots in space, the value of fa 2 + fa* 


at any particular spot in space will give a measure of the 
probability of our finding an electron there. Such a 
statistical interpretation is in keeping with the statistical 
origin of the wave-mechanics. As Heisenberg's relation 
was obtained from statistical data, our working model 
of this relation is a priori likely to be only of statistical 

We can test the truth of the foregoing conjecture in 
various ways, and it emerges triumphantly from all. 

To fix our ideas, let us return to the experiments we 
have already discussed, in which a current of electricity 
passes through a thin metal film, and emerges on the 
other side with the attributes of waves. We have to sup- 
pose, as on p. 200, that the oscillation-characteristic was 
inherent in the electrons of the current from the outset, 
and that as soon as the stream impinged on the film these 
oscillations gave rise to waves. The stream accordingly 
emerged from the film with the characteristics of a group 
of waves. There would be crests and troughs giving rise 
to places of great wave-intensity, and quiescent regions 
of small or zero wave-intensity. If we picture the current 
as a shower of point-electrons, then there would be many 
electrons at places where the wave-intensity was great, 
few electrons where it was small, and no electrons at all 
at places of complete quiescence, and the same would 
remain true after the current had emerged from the film. 
This exactly explains the pattern formed on the photo- 
graphic plate. 

Now let us gradually reduce the strength of our electric 
current until it almost vanishes, and let us consider what 
happens in an interval of time so short that only one 
electron passes. Broken electrons and fractions of elec- 


trons are never found in nature, so that to keep our pic- 
ture consistent with the known facts of nature, we must 
suppose that our single electron is not broken up by the 
experiment, but retains its identity, and emerges from the 
experiment, as it went in, a single particle charged with 
electricity. Thus it can only interact with a single particle 
of the sensitised plate at one single point. It cannot form 
a complete pattern; only a shower of electrons can do this. 
The light and dark of the pattern previously formed by a 
whole shower have, however, given a sort of graphical 
representation of the probabilities of any individual 
electron striking particular spots. Thus when our solitary 
electron comes along, there is no chance at all of its 
striking a spot where the previous pattern was dark; a 
million million electrons have already taken their chances 
of hitting such a spot, and not a single one succeeded in 
doing so, so that the chance that this isolated traveller will 
do so is nil. But there is a finite chance that the electron 
will strike any area which was bright in the previous pat- 
tern, and this chance is proportional to the total bright- 
ness of the area in question. Before performing the 
experiment with the single electron, we can only say that 
the chances of such and such a result are so and so, the 
various probabilities being determined by the waves 
specified by the Schrodinger equation. 

In this way electron waves become reduced to mere 
diagrammatic representations of probabilities, and this 
explains at once why they need a space having three times 
as many dimensions as there are electrons. When, how- 
ever, there is only one electron, or a shower of electrons 
whose motion is indistinguishable from one another, only 
three dimensions are needed for our diagram, and it is 


natural to think of this as constructed in ordinary space, 
In this sense we may think of the Schrodinger waves of a 
single electron as existing in ordinary space, although we 
must always remember that they are mere mathematical 
waves, and possess no physical reality. 

Ordinary material waves, such as waves of sound or 
ripples on the surface of water, spread their energy about 
until finally it is diffused through the whole of the space 
accessible to the waves. The total amount of energy 
remains the same throughout, the process of wave-propa- 
gation merely altering its distribution in space. 

The conjecture we are now considering endows elec- 
tron waves with an analogous property. on passing from 
material waves to electron waves, we replace energy by 
the chance of finding an electron. Just as the total energy 
of the material wave remained constant throughout, so of 
course does the total probability in the electron wave, 
since the aggregate of all the probabilities at the different 
points of space must always be exactly equal to the total 
number of electrons. 

When a material wave meets a surface of another sub- 
stance, part of it may be reflected and part of it trans- 
mitted, the total energy of the two new waves being 
exactly equal to that of the old. In the same way, when 
an electron meets a material surface, its probability wave 
breaks up into two parts, a transmitted wave and a re- 
flected wave, which represent the probabilities of the 
electron being transmitted and reflected respectively. 
Just as the total energy remains unaffected in a material 
wave, so does the total probability here. This circum- 
stance endows the Schrodinger waves with many of the 
attributes of material waves, although it naturally does 


not provide the slightest justification for supposing that 
they are material waves. 

It may at first seem strange that a probability should be 
propagated in so distinct a wave-like form, with a defi- 
nite wave-length and period of oscillation. It seems less 
strange when we notice that a probability field must con- 
form to the theory of relativity and so may be treated 
precisely in the way in which we treated the supposed 
structure of an electron on p. 203. With this in mind, we 
see at once that probabilities must necessarily be propa- 
gated in waves, the whole group of waves travelling 
with exactly the known speed of die electron. 

We do not of course solve the whole problem of the 
nature of these waves by describing them as waves of 
probability; before our task is finished, we must specify 
the meaning of probability with far greater precision 
than has yet been done. 

The mention of a probability in ordinary life implies 
that our knowledge is in some way imperfect. We speak, 
let us say, of the "probability" of a good channel crossing 
as we travel in the train to Dover, although we should not 
do so if we knew the state of the sea. one man may say, 
"The sea is only rough one day in three this month, so 
that the odds are two to one that we shall have a good 
crossing". Another may say, "The odds are better than 
that, for the weather forecast predicts a smooth sea, and 
it is right 95 times out of 100". A third may say, "It is 
practically certain, for I saw this morning's telegram at 
the Meteorological Office saying there was a smooth sea 
and no wind". These estimates of probability are all 
different, and yet they can all be correct; this is possible 
because probability involves two things, a future event 


and present knowledge. We have interpreted the system 
of Schrbdinger waves as giving a definite estimate of the 
probability of a future event, and must proceed to in- 
quire: Relative to what present knowledge is this a true 
estimate of probability? 

Let us first notice that as our knowledge increases, the 
probability of a smooth sea or any other event continually 
approximates to either zero or unity; it gradually changes 
into a certainty, either one way or another. The meteoro- 
logical expert who is armed with the latest telegrams need 
hardly use the word probability at all; he can say with 
practical certainty either that the sea will be smooth, or 
that it will be rough. 

In physics we may need to speak of the probability of a 
future event, or of the result of an experiment, for either 
of two reasons it may be that we have inadequate 
knowledge of present conditions, or it may be that even 
when present conditions are fully known, there is still 
uncertainty about the future in other words, the prin- 
ciple of the uniformity of nature may fail. 

Subjective Probability 

on the former alternative, the probability of a specified 
future happening, or result to an experiment, is a sub- 
jective probability; if we have different amounts of 
knowledge as to present conditions, your estimate of the 
probability may be different from mine, and yet we may 
both be right. on the latter alternative, the probability 
is wholly objective; even nature herself does not know the 
result of the experiment until after it has happened. The 
question "What is the probability of a specified result?** 
admits of only one answer. 


The former was the only alternative which the nine- 
teenth-century physicist would have admitted at all. He 
would have dismissed the second as preposterous, as 
indeed the scientific layman still does. Unaccustomed to 
thinking beyond the apparent determinism of the natural 
events which make up his everyday experience, he un- 
consciously assumes that a similar determinism must 
permeate nature even down to its most small-scale opera- 
tions, and denounces any alternative as illogical or 
contrary to the laws of nature often with much heat 
and emotion. 

Let us again concentrate on the particular instance of 
the single electron shot at the thin metal film. The lay- 
man in science, like the physicist of the old school, would 
probably say its path would be determined by the ob- 
stacles it met in the film. If he played billiards, he would 
know that when one ball hits another a slight difference 
in the path before impact may make a very great differ- 
ence in the path after impact. So he might argue that, 
as we could not know the exact circumstances of the 
collision between the electron and the atom or atoms 
which deflected it, we could not know the final path of 
the electron and so could only speak of the "probabilities" 
of one path or another. 

This interpretation of probability will not stand 
scrutiny, since the wave-length underlying the pattern 
does not depend on the spacing of atoms in the film, but 
solely on the speed of the electron. 

Objective Probability 

The second alternative, which postulates indeterminacy in 
nature, although foreign to nineteenth-century thought, 


has a far longer history than the former. When the 
more intelligent of our remote ancestors said that the 
prospects of a fair crossing depended on the whims of 
Poseidon and Boreas, they did little more than personify 
nature and attribute indeterminacy to her. Even up to 
the time of Newton, the concept of indeterminacy played 
a large part in science. The experiment of letting a 
shower of electrons fall on a metal film has its optical 
counterpart in the falling of a beam of light on the surface 
of a transparent substance. Some of the light is reflected, 
and some transmitted, so that when moonlight falls on the 
surface of the sea, some enters our eyes and we see a re- 
flection of the moon, while the remainder lights up the 
depths of the sea so that the fishes too can see the moon. 
If we picture the moonlight as a shower of photons, then 
clearly some photons must undergo reflection at the sur- 
face of the water, while others do not. If, however, the 
beam is reduced to a single photon, then since photons are 
indivisible the whole beam must go either one way or the 
other, and we shall only be able to speak of the "proba- 
bility" of its being reflected or transmitted. 

Newton, who regarded a beam of light as a shower of 
bullet-like corpuscles, encountered a similar difficulty, 
and met it by imagining that the molecules which formed 
the surface of the water suffered from "alternate fits of 
easy transmission and of easy reflection". 

It is not clear what precise degree of absence of deter- 
minism may have been implied in this. Whatever it was, 
it left science when the corpuscular theory of light gave 
place to the undulatory, only to reappear in the pres- 
ent picture which envisages a beam of light as a shower 
of indivisible protons. Like its predecessor, the light- 


corpuscle, a photon may follow one path or another, but 
cannot distribute itself over two paths, and once again its 
choice becomes, to all appearances, a matter of proba- 

Instances of a similar apparent want of determinism 
have recently appeared in other departments of physics. 
A conspicuous instance is provided by radio-active trans- 
formation. In 1903 Rutherford and Soddy found that 
radio-active substances disintegrate in a way they de- 
scribed as "spontaneous" the rate of decay cannot be 
expedited or retarded by any known physical process. 
Each year a certain fraction of all the atoms of radium in 
the world disintegrate into simpler atoms, the individual 
atoms being to all appearances selected by pure chance 
and nothing else. If anything else could select them, it 
ought to be possible to concentrate the selecting agency 
on one special sample of radium and expedite its dis- 
integration. So far no such selecting agency has been 
discovered, and theoretical considerations make it highly 
probable that, apart from extreme heat such as cannot be 
produced on earth, none such can exist. 

In 1917 a theoretical investigation by Einstein seemed 
to shew that spontaneous processes of this kind must 
pervade the whole of nature. He began by supposing 
that atoms could only exist in certain distinct states 
the suppositions which had previously been made by 
Bohr (p. 53), and subsequently received experimental 
confirmation in the experiments of Franck and Hertz 
and that they absorbed or emitted energy by complete 
photons in passing from one state to another. He then 
shewed that ordinary temperature radiation (p. 150) 
could be interpreted as an assemblage of photons pro- 


duced in this way, but only on certain conditions. Many 
of the photons observed in the radiation could be ac- 
counted for by the interaction between the radiation 
itself and the atoms of the substance, but Einstein shewed 
that a residue remained, which could only be accounted 
for on the supposition that the atoms fell spontaneously 
from one of their possible states to another. Thus even 
the familiar everyday phenomena of temperature radia- 
tion seem to call for some sort of action which is incon- 
sistent with a strict determinism. 

Einstein is of the opinion that these particular phe- 
nomena are consistent neither with indeterminism nor 
with causality as at present understood. He says:* 

"Indeterminism is quite an illogical concept. ... If I say 
that the average life-span of a radioactive atom is such and 
such, that is a statement which expresses a certain order 
(Gesetzlichkeii). But this idea does not of itself involve the idea 
of causation. We call it the law of averages; but not every 
such law need have a causal significance. At the same time 
if I say that the average life-span of such an atom is indeter- 
mined in the sense of being not caused, then I am talking 
nonsense. . . . 

"When Aristotle and the scholastics defined what they 
meant by a cause, the idea of objective experiment in the 
scientific sense had not yet arisen. Therefore they were con- 
tent with defining the metaphysical concept of cause. And 
the same is true of Kant. Newton himself seems to have 
realized that this pre-scientific formulation of the causal prin- 
ciple would prove insufficient for modern physics. . . . Now 
I believe that events in nature are controlled by a much stricter 
and more closely binding law than we suspect to-day, when we 
speak of one event being the cause of another. Our concept 
here is confined to one happening within one time-section. 

* Where is Science going? by Max Planck (1933), pp. 202, 203. 


It is dissected from the whole process. Our present rough way 
of applying the causal principle is quite superficial. We are 
like a child who judges a poem by the rhyme and knows 
nothing of the rhythmic pattern. Or we are like a juvenile 
learner at the piano, just relating one note to that which im- 
mediately precedes or follows. To an extent this may be very 
well when one is dealing with very simple and primitive com- 
positions; but it will not do for the interpretation of a Bach 
Fugue. Quantum physics has presented us with very com- 
plex processes and to meet them we must further enlarge and 
refine our concept of causality". 

Professor Weyl of Gottingen, writing on the meta- 
physical implications of science, expresses the same 

"These considerations force upon us the impression that 
the law of causality as a principle of natural science is one 
incapable of formulation in a few words, and is not a self- 
contained exact law. Its content can in fact only be made 
dear in connection with a complete phenomenological descrip- 
tion of how reality constitutes itself from the immediate data 
of consciousness". 

Even though a certain measure of indeterminism may 
appear necessary to explain certain small-scale phe- 
nomena, the principle of the uniformity of nature still 
prevails so long as nature is only studied in appreciable 
amounts. Even the tiniest bit of matter we can perceive 
through our senses contains billions of atoms, and if each 
of these is free to go to the right or left as it pleases, the 
laws of probability secure that, as far as our senses can 
tell, half will go each way. For this reason, our everyday 
experience will never shew us any violations of the so- 

* The Open World (1932), p. 43. 


called law of the uniformity of nature, and the man whose 
thoughts are guided only by intuition or instinct, or who 
holds to the common-sense view of nature, is certain to 
be a determinist. 

We shall return to a discussion of this question after 
obtaining further evidence in the next chapter. 


We have seen that our whole knowledge of the external 
world of physics may be pictured as arising from the im- 
pact of photons of energy either on our sense organs or 
on our physical instruments. As these photons occur in 
such profusion and variety, it might have been hoped 
that they would give us an almost perfect knowledge of 
the outer world. 

Yet, as a means of acquiring knowledge, photons suffer 
from one very serious limitation. They are indivisible; no 
experiment has ever revealed a fraction of a photon or 
given any reason for supposing that energy can be either 
emitted or absorbed in fractions of photons. Thus the 
only means which are at our disposal for the study of 
physical nature suffer from a certain coarse-grairiedness. 

This is of little consequence as regards direct study by 
our senses, since these are even more coarse-grained. 
Each sense has its perceptions limited by a certain 
"threshold of sensation", and if the stimulus of a physical 
effect falls below this, the sense in question registers 
nothing at all. We cannot experience the sweetness of a 
single molecule of sugar, nor the smell of a single molecule 
of musk; neither can we hear a bell at more than a certain 
limit of distance, nor see a star which is below a certain 
limit of faintness. In general we cannot experience a 
single photon; thousands at least are necessary to attain 
the threshold of sensation. 



Our physical instruments have in a sense a similar 
"threshold of sensation", this being the arrival of a sin- 
gle complete photon. Like all other physical structures, 
they accept energy and momentum only by complete 


The Uncertainty Principle 

Thus the most refined piece of information we can obtain 
about any piece of the universe is that conveyed by the 
arrival of a single photon. This transfers to our instru- 
ment energy and momentum which it has brought with 
it from the fragment of the external world in which it 
originated. Now just as a shot gives a backward kick to 
the gun from which it is fired, so a photon gives a back- 
ward kick to the atom which sends it out, and through 
this atom to the fragment of the universe which we are 
trying to study. Thus it may give us accurate news of the 
universe as it was, but the kick it gave to the universe in 
leaving it to bring us news makes the news out of date 
before it reaches us; we receive news only of an old uni- 
verse which has already passed away. 

It might be thought that as photons carry all possible 
amounts of momentum ranging from zero upwards, we 
could obtain as accurate information as we pleased by 
employing photons of small momentum. This is, in the 
abstract, true; in practise it only shifts the difficulty. For 
photons of small momentum have such long periods of 
oscillation that we cannot fix the instant to which their 
information refers with any great precision; it is like try- 
ing to time a hundred yards race with a grandfather clock 
that only ticks seconds. 

Thus we are confronted with the dilemma that one 
kind of photons are so energetic that they give the uni- 


verse a violent kick before leaving it, and so give us in- 
exact information about its present condition, while the 
other kind are so slow in telling their story when 
they arrive that they cannot give us exact informa- 
tion in respect of time. Intermediate kinds fail jin both 

Science has found no way out of this dilemma. on the 
contrary, it has proved that there is no way out. What 
is known as Heisenberg's "principle of indeterminacy" 
or "uncertainty principle" shews that so long as we can 
only explore nature by complete photons, there is no 
hope of obtaining information which is perfectly exact 
with respect to both time and space. Exactness in either 
direction is obtained at the price of inexactness in the 
other; we can only prevent the shoe pinching at one place 
by letting it pinch at another. 

Exact mathematical discussion shews that, as we try 
one kind of photon after another, the product of the two 
errors can never fall below a certain minimum value. If 
the experiments are designed and performed with perfect 
skill, the product of the two errors is the same for all kinds 
of photons, and is equal to this mtefrniim value. 

For instance, to obtain a complete knowledge of the 
motion of a particle, we need two data the exact 
instant at which the particle passes an assigned landmark 
in our apparatus, and the exact speed with which it is 
moving as it passes this landmark. If we agree to measure 
the speed of our particle in terms of its momentum, then 
we find that the product of the errors in position and mo- 
mentum can never be less than Planck's constant h. We 
have already noticed how this quantity dominates the 
whole of atomic physics. We come upon it here as speci- 


fying the coarseness of the probe, the photon, with which 
we are trying to penetrate the outer world. 

In centimetre-gramme-second units, the value of h is 
6-55 X 10" 27 and the mass of an electron is 9 X lO" 27 . 
Thus the product of the uncertainties in the position and 
speed of an electron, measured in the same units, is 0-73. 
For instance, if, by letting it make a flash on a screen, or 
by any other means, I discover that an electron is within 
a hundredth of a centimetre of a certain point, then the 
speed of its motion will necessarily be uncertain to at 
least 73 centimetres a second the rate of a slow walk. 

We have so far pictured an electron as a particle, but 
we can also picture it in terms of Schrodinger waves. If 
the two pictures specify the same object, we ought of 
course to be able to derive precisely the same "principle 
of uncertainty" from the wave picture as we have already 
derived from the particle picture. We shall now see that 
this can be done. 

When we regard an electron as a system of waves, their 
wave-length depends on the speed of the electron in the 
way already described. Thus the problem of measuring 
this speed with exactness reduces to that of specifying a 
wave-length with exactness. Abstract mathematics shews 
that this cannot be done unless we have an infinite num- 
ber of waves at our disposal; with fewer waves the con- 
cept of wave-length has no exact meaning. If we have 
a Tnillion waves at our disposal, we can measure their 
wave-length to within something like a millionth part of 
its amount, but to speak of measuring it more accurately 
than this is meaningless. 

We can illustrate this point of pure mathematics by the 
difficulties which arise when we try to measure the wave- 


length of a finite train of waves in the laboratory. For 
simplicity let us suppose they are wireless waves, and that 
we allow them to fall on an ordinary wireless receiving 
set, which can be tuned to any wave-length we please. 
Any train of waves will set up disturbance by resonance 
over a wave-band of finite width. As we lengthen the 
train of waves, the interference with neighbouring wave- 
lengths diminishes, but it does not completely disappear 
until the train of waves is made infinitely long. only 
then can we say that the waves have a clearly defined 

It follows that we cannot specify the speed of motion 
of an electron with perfect precision unless it is repre- 
sented by an infinitely long train of waves. But since we 
have seen that the waves represent the probabilities of 
finding the electron in different positions in space, the 
electron may be anywhere along the whole length of the 
train, and an infinitely long train of waves implies an 
uncertainty of infinite amount as to the position of the 

Let us now pass to the other extreme, and imagine an 
infinitely short train of waves to pass over the receiving 
set. The set sees nothing in such a train of waves but a 
mere sudden disturbance, which disappears the instant it 
has come into being. As every wireless expert knows, 
such a disturbance affects all wave-lengths indiscrimi- 
nately, and so cannot be said itself to have any definite 
wave-length. An infinitely short train of waves of this 
kind represents an electron whose position can be speci- 
fied with precision, but we see that its wave-length, and 
so also the momentum and speed of its motion, are com- 
pletely indefinite. 


A mathematical discussion of intermediate cases leads 
to exactly the principle of indeterminacy already ex- 
plained. Greater precision in momentum implies greater 
uncertainty in position, and vice-versa; the product of the 
two uncertainties can never be less than Planck's constant 
h, and, under the most favourable circumstances, is 
exactly equal to h. 

Interpretation of the Wave Picture 

It is not surprising that the particle picture and the wave 
picture lead to the same "principle of uncertainty"; there 
would have been something wrong had they not done so. 
Yet they lead to this principle in very different ways. 
When we use the particle picture of an electron, the 
uncertainty refers to the knowledge of nature we obtain 
through experiments on nature. When we use the wave 
picture, we find that the uncertainty is inherent in the 
picture itself. In brief, the particle picture tells us that 
our knowledge of an electron is indeterminate; the wave 
picture that the electron itself is indeterminate, regardless 
of whether experiments are performed upon it or not. 

Yet the content of the uncertainty principle must be 
exactly the same in the two cases. There is only one way 
of making it so; we must suppose that the wave picture 
provides a representation, not of objective nature, but 
only of our knowledge of nature. 

Earlier in our book we saw nineteenth-century science 
trying to explore nature as the explorer explores the desert 
from an aeroplane. The uncertainty principle makes it 
clear that nature cannot be explored in this detached 
way; we can only explore it by tramping over it and 
disturbing it; and our vision of nature includes the clouds 


of dust we ourselves kick up. We may make clouds of 
different kinds, but the uncertainty principle shews that 
there is no way of crossing the desert without raising a 
cloud of some kind or other to obstruct our view. The 
wave picture depicts the blurred view of nature that we 
see through these dust clouds, so that, as we shall shortly 
see, there are as many wave pictures as there are ways of 
raising a dust cloud. 

If we turn our thoughts back to the origins of the wave 
picture, we can see why all this must be. This picture was 
introduced to provide us with a sort of working model 
of Heisenberg's equation, and this equation was con- 
cerned solely with observables that is to say, not with 
objective nature but with our observation of nature. 
Heisenberg attacked the enigma of the physical universe 
by giving up the main enigma the nature of the objec- 
tive universe as insoluble, and concentrating on the 
minor puzzle of co-ordinating our observations of the 
universe. Thus it is not surprising that the wave picture 
which finally emerged should prove to be concerned 
solely with our knowledge of the universe as obtained 
through our observations. 

Electron Waves as Waves of Probability 

This interpretation of the wave picture explains a great 
deal that would otherwise seem very mysterious, and 
gives greater precision to the discussions of our previous 

We there described the waves of the wave picture as 
"waves of probability", but were unable to assign any 
precise meaning to the term. Let us now suppose that we 
perform an experiment to find the speed and position, of a* 


moving electron. An experiment of one type may fix its 
position with great accuracy but its speed with great 
uncertainty; the electron so observed appears in the wave 
picture as a short train of waves. An experiment of an- 
other type may fix the speed with great accuracy but the 
position with great inaccuracy; the electron is now repre- 
sented by a long train of waves. The same electron may 
be represented by two different wave pictures, not be- 
cause it is itself different in the two cases, but because 
our knowledge of it is different in the two cases. Thus the 
waves represent subjective probabilities. 

Suppose our experiments tell us that an electron is at 
such-or-such a point in space, subject to a certain inde- 
terminacy, and is moving with such-or-such a speed, 
again subject to a certain indeterminacy. 

We may represent our lack of precise knowledge as to 
the position of the electron by substituting a fog for the 
latter; our knowledge is that the electron is somewhere 
inside the fog. If we knew the precise speed of the 
electron, we could let the fog move forward at this precise 
speed, and the electron would always lie inside the fog. 
We cannot, however, know the precise speed, but only 
know that it lies within certain limits, as for instance 
between 50 and 55 miles a second. To represent this, 
we must regard our fog as being made up by the super- 
position of a number of separate individual fogs, and let 
these individual fogs move forward at the various speeds 
within these limits one at 50 miles a second, another at 
51 miles a second and so on. We shall then know that at 
every instant the electron lies somewhere within the 
fogginess produced by all these fogs. Let us notice that 
the area of fogginess continually increases; this means that 


our knowledge of the position of the electron continually 
gets more and more vague. This is faithfully represented 
in the wave picture, because it is a general property of 
waves to spread. The electron itself retains its identity 
throughout, but the fog representing not the electron, 
but our knowledge of it must perforce continue to 
spread until it ultimately pervades all space. 

For many purposes the weight of a massive body may 
be supposed concentrated in a single point, which we call 
the centre of gravity of the body. So also, for many 
purposes, we may suppose the whole electron to be con- 
centrated at the centre of gravity of the fog. For instance, 
Ehrenfest has shewn that, when the waves of the fog travel 
as directed by Schrodinger's equation, the centre of 
gravity of the fog will describe precisely the same curved 
path in an electric field as a single point-electron would 

Let us now imagine that our moving electron meets a 
metal film. Each small fragment of our fog will break up 
into a system of waves, in the way we have already 
explained, and the aggregate of all these systems of waves 
constitutes the wave pattern of the electron, such as we 
see in fig. 2 of the frontispiece. This new wave pattern is 
more extended in space than the original fog, because our 
uncertainty as to the position and motion of the electron 
has been increased by a further uncertainty as to its 
conduct in passing through the metal film. 

Light-waves as Waves of Probability 

We can discuss light and light-waves in a precisely 
similar manner. We can picture light as consisting of 
photons, and these photons have wave pictures just as 


electrons have; they are neither more nor less than the 
ordinary waves of the undulatory theory of light. When 
we picture photons as localised at points, we have already 
seen (p. 217) that these waves must be interpreted as 
waves of probability the probability of finding the 
photon at a given spot. It is rather surprising to discover 
that we must take the further step of regarding them as 
mere diagrammatic representations of our knowledge as 
to the whereabouts of photons, yet a simple instance, 
which has been discussed by Einstein and Ehrenfest, will 
shew that such is the case. 

A photon which meets the surface of a transparent 
substance may be either reflected or transmitted. For the 
sake of simplicity, let us suppose that the chances of the 
two events are equal. This means that when the wave- 
system of the photon falls on the reflecting surface, it will 
divide into two beams of equal intensity, one reflected and 
one transmitted. These are of course beams of ordinary 
everyday light. After a few seconds interval, these two 
beams may be a million miles apart, which means that 
on our present knowledge we cannot fix the position of 
our photon to within a million miles. 

A new experiment will, however, clear up some at least 
of our uncertainty. Let us calculate the path of the re- 
flected ray by geometrical means, and place across it a 
screen which will register a spot of light if the photon 
strikes it. Up to this moment, our knowledge of the 
doings of the photon has been represented by two beams 
of light-waves; one is just about to fall on the screen, the 
other is a million miles away. We watch the screen. If it 
does not light up, we know that the photon has chosen the 
other path, and is a million miles away. If it does light 


up, we see that the photon has chosen this particular path, 
and this new knowledge completely transforms the system 
of waves. We now know for certain that the photon is not 
at any point on the distant beam, because it is here, and 
the whole system of distant waves disappears for ever; it 
is completely annihilated. on the other hand, the beam 
of light we are watching becomes contracted to a point 
the point which lighted up on our screen. 

It is at first very startling and not a little puzzling to 
reflect that by the mere act of watching a screen here we 
can annihilate light-waves a million miles away. Old- 
fashioned physics told us that light-waves were waves of 
energy, so that the act of looking at a screen here has 
apparently destroyed energy a million miles away. Even 
if the total energy is conserved, our action has removed 
the energy from there to here, and this at infinite speed, 
although we used to be told that energy could not travel 
faster than light. 

The paradox disappears as soon as we treat the light- 
waves as waves' of probability, their extension in space 
defining the uncertainties of our knowledge. The waves 
are no longer waves of energy, but of the chance of find- 
ing energy. When there are billions of billions of photons 
in the field, the total measure of the chance of finding 
energy is, for all practical purposes, the same thing as the 
measure of the energy we shall find, and we need not 
trouble to distinguish between the two. When there is 
only one photon involved, the distinction becomes im- 
portant. What is transferred now is not energy, but the 
chance of finding energy, which in turn depends on our 
knowledge as to the whereabouts of the energy. This 
may well be transferred, not with the speed of light 


which is finite, but with the speed of thought, which is 

As this is one of the most difficult parts of the new 
quantum theory, let us try to illustrate it by a very pro- 
saic illustration. Suppose I am anxious to meet my 
relative John Smith, who is owing me a sum of money, 
and that all I know of him for certain is that he left his 
home in London three days ago for an unspecified desti- 
nation. My knowledge as to the whereabouts of John 
Smith is represented by a fog which extends over all those 
parts of the earth's surface which are within three days' 
travel of London. I next find that a passenger named 
John Smith sailed on the Majestic three days ago for New 
York, and the fog becomes particularly dense in mid- 
Atlantic, three days out from land. I hurry to a cable 
office to communicate with the Majestic in the hope of 
getting a reply which will inform me, with the speed of 
light, whether my relative is in mid-Atlantic or not. But, 
on my way, I run into John Smith himself. This simple 
act not only concentrates all the fog into one spot in space, 
namely that at which my relative is standing; it also 
abolishes the fog in the Atlantic, and does this far more 
promptly than a wireless message, travelling with the 
speed of light, could do. It can do this because the fog 
is not a material fog, such as delays shipping; it consists of 
knowledge knowledge about John Smith. 

So, in the last resort, the waves which we describe as 
light-waves, and those other waves which we interpret 
as the waves of an electron and a proton, also consist of 
knowledge knowledge about photons, electrons and 
protons respectively. We can see now why modern 
science does not need the old material ether, millions of 


times more dense than lead, for light-waves to travel 


The Waves of the Hydrogen Atom 

Let us now revert to electron waves, and consider what 
happens when an electron, originally moving freely 
through space, combines with a proton to form a hydro- 
gen atom. If we knew its original path with fair accuracy, 
the electron may be represented by a fairly compact 
packet of waves when it first comes under the attraction 
of the proton. In accordance with the principles already 
explained, this wave packet will describe a curved path 
round the proton (p. 238), and will continually increase 
in extent as it does so (p. 238). The general principle 
that waves continually spread shews that there is only one 
end possible the wave packet must ultimately fill the 
whole of space. Throughout all these changes, as also 
when the final state is reached, the electron waves will 
conform to Schrodinger's equation. This equation has 
many solutions, which will of course represent many 
different kinds of waves. Some will represent permanent 
unchanging systems of waves, and these are found to 
specify the possible permanent states of the hydrogen 

It is a comparatively simple problem to discover all the 
solutions of this type. Such solutions are found to exist 
for certain values of the energy E in equation (F) of p. 202, 
and for no others. Thus the hydrogen atom can only 
exist permanently in certain discrete states specified by 
their different amounts of energy precisely as was first 
postulated by Bohr, and was subsequently confirmed by 
the experiments of Franck and Hertz. These amounts of 
energy are of course easily calculated, and the lines of the 


hydrogen spectrum are found to correspond exactly to 
transitions from one of them to another, so that the tri- 
umph of wave-mechanics is complete. 

It may be of interest to try to understand something 
of the geometrical disposition of these waves. Bohr's 
earlier theory, as we have already seen (p. 53), supposed 
the hydrogen atom to consist of a charged particle an 
electron describing a circular orbit round another 
charged particle the nucleus. The electron was sup- 
posed to be confined to orbits possessing certain definite 
amounts of energy, and so also definite diameters; it was 
as though certain grooves were cut in space, and the 
electron was compelled to run round and round in the 
same groove except on the comparatively rare occasions 
when it jumped from one groove to another. 

Wave-mechanics also finds that the electron-nucleus 
combination can only have these particular amounts of 
energy. But the electron is no longer a particle; it is a 
system of waves running round and round the nucleus, 
somewhat as waves might run round and round in a 
circular trough of water, except for the quite important 
difference that troughs of water have clearly defined 
boundaries, whereas these waves have not. If, notwith- 
standing this, we like to imagine the waves imprisoned in 
troughs, then we find that these troughs must be of cer- 
tain definite diameters such that the complete circum- 
ference of a trough may be occupied by one, two, three or 
any other exact number of complete waves, but never by 
a fractional number of waves. This condition makes the 
diameters of the troughs very nearly the same as the 
diameters of the orbits which Bohr had previously calcu- 
lated from his simpler theory, namely, 1, 4, 9, 16, ... times 


the diameter of the normal atom in its normal state of 
lowest energy. 

With a view to understanding this, let us consider a 
very simply hypothetical hydrogen atom which ad- 
mittedly has not much relation to actual facts. Let us 
suppose that its electron is in some way constrained 
rather as imagined in Bohr's first theory perpetually 
to move round the nucleus in a circle of always the same 
radius a. If the atom is beyond the reach of our experi- 
ments, as for instance an atom in Sirius, our knowledge of 
the position of its electron will consist of the single fact 
that this is at a distance a from the nucleus. We cannot 
know which point of its orbit it will occupy at any instant, 
and neither can we know the orientation of this orbit in 
space. Thus our knowledge of the position of the electron 
is represented by a thin shell of fog, forming a sphere of 
radius a surrounding the nucleus. 

If we suppose that the electron is constrained to move 
in one or other of a number of circular orbits, having 
radii a, i, c, ... , our knowledge will be represented by a 
number of thin shells of fog having radii 0, b, c, ... 

Let us now go one step farther in the direction of 
reality. An electron which is describing a circular orbit 
of radius a may be deflected, without loss or gain of 
energy, into an elliptical orbit, in describing which it will 
be alternately inside and outside its original circular orbit. 
Its distance from the nucleus will range between a(l + e) 
and a(\ *), where e is the quantity known as the ec- 
centricity of the ellipse. As this can have any value 
between and 1, the distance of the electron from the 
nucleus may be anything between and 2a. Thus if we 
know nothing about the electron except that it is describ- 


ing an orbit of specified energy, our knowledge will be 
represented by a fog, which will extend through the whole 
of a sphere of radius 2a, but no farther, and will be par- 
ticularly dense at a distance a from the nucleus. If the 
electron can describe an orbit having any one of a num- 
ber of specified energies, our knowledge will be repre- 
sented by the superposition of a number of fogs lying 
inside the spheres of corresponding radii. 

Such a system of fogs does not differ very widely from 
the probability diagram furnished by the wave-mechanics 
for the actual hydrogen atom, except for one very im- 
portant point of difference. An electron moving in an 
orbit of specified energy can never move to more than a 
certain distance from the nucleus, whereas the probability 
diagram of the wave-mechanics extends throughout the 
whole of space. In brief, this diagram tells us that there 
is always a finite probability that the electron will reach 
points which would be entirely beyond its reach, because 
of insufficient energy, if it were an ordinary charged 
particle moving in space and time. This illustrates a 
complication which is not peculiar to the hydrogen 
spectrum, nor even to the more general problem of atomic 
structure, but permeates the whole of the new quantum 
theory. It seems as though, if an electron waits long 
enough, it will always be able to violate the law of con- 
servation of energy by reaching places which its energy 
does not entitle it to reach. Gamow has suggested that 
the disintegration of radio-active nuclei may be due to 
this cause. 

This makes it clear that if the conservation of energy is 
to remain in our picture of nature, we must attach a wider 
meaning to energy than we have done hitherto. We have 


so far thought of the energy of an electron as due to the 
position of a particle in space and its motion through 
space, and this in spite of our having seen that an electron, 
when inside an atom, cannot be represented in space and 
time. There is no obvious difficulty in taking a wider 
view of energy, and imagining that the electron can draw 
on energy which also cannot be represented in time and 
space. Heisenberg's equation has already shewn that the 
real electron has a greater complexity than mere position 
in space; if we know that its total energy has a certain 
value, and try from this to find its position in space, the 
complexity of its wave pattern shews us how many 
answers we may receive to our question. We may regard 
this wave pattern as representing the projection into 
space and time of all the configurations which have a 
given total energy. 

Objective and Subjective Waves 

Yet clearly this wave pattern is wholly objective, and at 
first this may seem to be at variance with our earlier 
statements that a wave pattern represented subjective 
knowledge. But it is easy to see that there is no conflict, 
and that our former interpretation of the electron waves 
as diagrammatic representations of subjective knowledge 
leads to precisely the result we have just reached. For, 
although we may at first have fairly precise knowledge 
which limits the position of an electron to a small region 
of space, yet uncertainties increase with the passage of 
time, so that our knowledge of this position gets con- 
tinually vaguer; the electron waves spread. Finally, after 
a time which we may treat as infinite, they fill all space. 
And as the position from which an electron started an 


infinity of time ago can have no possible influence on the 
positions it may occupy now, the present electron waves 
are entirely independent of our knowledge, and so form 
an objective system. 

We can illustrate this very simply by considering an 
electron in an otherwise empty universe. The wave- 
equation for such an electron is formally similar to the 
equation which governs the flow of heat in a solid of in- 
finite extent. Thus our uncertainty of knowledge, which 
was at first localised within a small volume of space, 
spreads like heat through a solid. Just as the solid finally 
reaches a state of uniform temperature, which is inde- 
pendent of the spot from which the heat started, so the 
system of waves in wave-mechanics finally attains uni- 
form intensity everywhere throughout space, and this 
independently of the earlier movements of the electron. 
This merely means that, no matter where the electron 
started, all positions for it are equally likely after an 
infinite time. In fact we could only predict its position 
after infinite time if we had known its original position 
and speed with infinite precision, and the uncertainty 
principle keeps this pair of data for ever beyond our reach. 
This final system of waves is of course wholly objective; 
it cannot represent subjective knowledge for the simple 
reason that we no longer have any. Or, to put the same 
thing in another way, it represents the fact that our 
knowledge is nil. 

Another very simple solution of the wave-equation is 
of the form 


and this represents waves advancing in the direction 
/, m, n, with speed V. This may be taken to represent an 
electron, or a current of electricity, advancing in the 
direction I, m, n, with a speed c 2 /V. In choosing this 
solution to represent any particular electron, we assume 
we know that the electron is travelling with components 
of velocity which are precisely lc*/V 9 etc. The uncer- 
tainty principle now tells us that we must pay for this 
precision in our knowledge of the momentum of the elec- 
tron by accepting an infinite uncertainty in the position of 
the electron, which of course is precisely what the solu- 
tion also tells us, since the waves are of uniform intensity 
throughout space. Here again we have a wholly objec- 
tive system of waves. 

Such systems of waves, extending uniformly through 
the whole of space, provide the only strictly objective 
representations of electrons or electric currents. Com- 
binations of them give the concentrated wave packets 
which we observe and designate as electrons the 
electrons of our observation. But if we ask mathematics 
to tell us how to build up such a wave packet i.e. to 
provide us with the wave specification of a single electron 
moving freely in space there is no answer. Or rather, 
strictly speaking, the answer takes the form of a further 
question "Tell me first how much you know about the 
electron, and I will answer your question. If you know 
nothing, my only answer will be that I too know nothing". 
All the observational knowledge which has been put into 
the quantum theory and wave-mechanics proves to be of 
no avail to elicit what an electron is objectively. 

The situation changes as soon as we put a second body, 
as for instance a proton, into this space. Since the two 


particles attract one another, the electron, quite apart 
from any detailed knowledge on our part, is more likely 
to be in the neighbourhood of the proton than elsewhere. 
The wave-system now wraps itself symmetrically round 
the proton, displaying the various possibilities and 
relative likelihoods that we have already discussed. We 
now have an objective system of waves, and it appears 
that an electron can only be objectively specified when 
it is anchored to a proton or other material frame of 
reference; otherwise it merely fills all space uniformly. 
An objective electron localised in an empty universe is as 
meaningless as objective time or rather we can attach 
to either as many meanings as we please. 

Thus, if our wave picture of nature is to be wholly 
objective, it must contain no reference to isolated elec- 
trons or protons, but only to such combinations of these 
as can produce events which can affect our senses. on 
limiting our picture to these, we obtain a system of waves 
which is completely objective, in the sense that we must 
imagine them existing whether we experiment to discover 
their existence or not. The waves do not admit of repre- 
sentation in space and time, and so cannot be said to 
possess any physical reality. 

Yet, in spite of this want of physical reality, this wave 
picture is in many respects more true to nature, and so is 
presumably more fundamental, than the particle picture 
which depicts nature as concrete objects existing in space 
and time. This is especially true of the more refined 
problems of atomic structure and spectral lines. Just as, 
in optics, the ray picture gives a rough approximation, 
but a wave picture is needed to exhibit the finer details of 
phenomena, so here the particle picture will often give a 


rough approximation to a truth which the wave picture 
explains with perfect precision. The following illustra- 
tions may serve to typify a whole mass of highly technical 

The spectra of the alkali metals consist entirely of 
"doubled 53 lines pairs of lines which are quite distinct 
and yet very close together such as the well-known 
D lines of sodium. Two Dutch physicists, Uhlenbeck and 
Goudsmit, tried to explain this doubling by supposing 
that the electrons of the particle picture spun round on 
their axes as they described their orbits in the atom. This 
gave the electrons slightly different energies according as 
they spun in the same direction as their orbital rotation, 
like the earth, moon and planets, or in the opposite 
direction, like the outermost satellites of Jupiter and 
Saturn, and the single satellite of Neptune. Further, the 
amounts of spin required to account for the doubling of 
the spectral lines agreed exactly with that needed to ac- 
count for the Zeeman effect the rearrangement of 
spectral lines that occurs when the incandescent gas is 
placed between the poles of a powerful magnet. 

Thus the particle picture could be made true to nature 
by supposing the electrons to be spinning. Nevertheless 
such spins seemed very artificial until it was found, a few 
weeks later, that they were a necessary consequence of the 
wave picture. Just as light-waves admit of different 
kinds of polarisation, which we can represent in the 
particle picture of light by different spins of the photons 
(p. 158), so electron waves admit of different kinds of 
polarisation, which we may represent in the particle 
picture of matter by different spins of the electrons. 

Yet this does not place the two pictures quite on the 


same footing. For we see that if the wave picture of 
matter is fundamental, all the electrons in the particle 
picture must necessarily be spinning. on the other 
hand, if the particle picture is fundamental i.e. if 
nature really consists of particles, and the waves merely 
provide a diagrammatic representation of our imperfect 
knowledge of their positions there is no obvious reason 
why the particles should spin at all. Both pictures explain 
the facts, but the explanation of the particle picture 
appears artificial, while that of the wave picture appears 
natural, and indeed inevitable. 

At a later stage the experiments of Stern and Gerlach 
provided what amounts to an experimentum crucis, enabling 
us to decide between the particle picture and the wave 
picture. In the particle picture, the spin of the electrons 
turned each atom into a small magnet,.so that if a shower 
of atoms is passed between the poles of a fixed magnet, the 
atoms ought to be affected differently, according to the 
directions in which the axes of the spin were pointing, and 
the shower of parallel-moving atoms would be spread out 
into a broad continuous band. The wave picture, on the 
other hand, predicts that the shower would be divided 
into two quite distinct showers, corresponding to the two 
directions of polarisation of the electron waves. The 
experiments quite definitely confirmed the predictions of 

The failure of the particle picture in this and similar 
cases is of great interest. For the particle picture implies 
the possibility, and the wave picture the impossibility, of 
representation in space and time. So long as we were 
concerned only with the simplest constituents of nature, 
electrons, protons and photons, the two pictures appeared 


to possess equal validity. As soon as we pass to the more 
complex structure of the atom, the wave picture acquires 
a definite pre-eminence. Thus the wave picture begins to 
appear as the true picture of reality, and the particle 
picture merely as a clumsy approximation to the truth, an 
approximation obtained by trying to force into a frame- 
work of space and time a structure which does not admit 
of representation in space and time. 

This implies that our interpretation of the wave picture 
as a diagram of the probabilities of finding particles at 
various spots can no longer be regarded as final. For 
obviously the true picture of nature must admit of a direct 
interpretation, without reference to a less perfect picture. 
The particle representation has served its purpose when 
it has led us to the wave picture, and may henceforth be 
disregarded as mere scaffolding. 

Thus it is through the wave picture of matter that we 
must approach reality, and the abandonment of a space- 
time representation of nature would seem to be the first 
step on the journey. 

It is a difficult step to take. Our knowledge of the 
external world is brought us by photons which travel in 
a setting of space and time, with the result that from our 
earliest days we have thought of objective nature itself as 
also existing in space and time. Our thoughts have 
become space-time bound, and can get no grip on con- 
cepts outside space and time. Thus no progress has been 
made along the new road as yet, and we are still com- 
pelled to discuss nature in terms of the partial pictures 
of waves (incomprehensible) and of particles (inac- 



As the new theory of quanta and the theory of wave- 
mechanics are believed to agree exactly with observa- 
tion, and so perhaps to contain the final mathematical 
truth about nature, they ought to be capable of throwing 
some light on the question of determinism. 

We have seen how our knowledge about nature can be 
visualised, part by part, in a number of different pictures, 
although no single picture enables us to visualise the 
whole truth at once. Of these partial pictures, there are 
for instance the picture which depicts electrons as parti- 
cles and that which depicts them as waves. There are 
again the corresponding pictures for light, one depicting 
it as waves and the other as photons. 

Let us consider the wave picture first. This assumes its 
simplest form in the case of radiation. The wave-equa- 
tions become the ordinary equations of Maxwell for the 
propagation of electric action, and are to all appearances 
completely deterministic. That is to say, if we know elec- 
tric conditions at any one instant, we are able, through 
these equations, to determine these conditions throughout 
all future time. The wave-equation of an electron implies 
an exactly similar determinism. If we know the value of 
$ throughout space at any one instant, these equations en- 
able us to calculate its value through all subsequent time. 

Yet this does not mean that nature is completely deter- 
ministic, since, on the only interpretation we have yet 
been able to devise, both the $ of Schrodinger's equation 
and the electric forces of the Maxwell equations are not 
determined by nature but by our knowledge of nature. 
If the distinction between nature and our knowledge of 


nature were to disappear in any special case, as it does, 
for instance, when we are discussing an assemblage of an 
immense number of photons, then of course the wave- 
equations would shew that there was complete determin- 
ism in respect of the special phenomenon in question. 
Other cases of this kind are the two final steady states 
we have just discussed, but in these our knowledge is not 
perfect but nil. In both cases there is unmistakable deter- 
minism, but it is of a very trivial kind. Objectively it is 
expressed in the sentence: "When a piece of the universe 
can change no more, its future course is unalterably 
determined"; subjectively by the equally useless sen- 
tence: "If we begin by knowing nothing, and perform no 
new experiments, we shall continue to know nothing 
throughout all time". 

It is often overlooked that the wholly deterministic 
wave-equation does not, and cannot, take the whole of 
nature for its province. Although this is hard to realise, 
wave-mechanics has no more knowledge of the existence 
of separate atoms than the undulatory theory of light has 
of the existence of separate photons. The original analy- 
sis of Heisenberg, let us recollect, was not concerned with 
a succession of photons emitted from a number of distinct 
atoms, but with a stream of radiation of origin unknown 
and unspecified. Out of this emerged entities p and q 
which were made analogous to the momentum and co- 
ordinate of an electron in an atom by the crude device 
of adjusting a constant so as to secure agreement when 
the atom was of very large radius. But the atom in which 
this electron moved was no real atom such as could exist 
in nature; it was rather a sort of statistical atom a com- 
posite photograph of all the atoms in the world which 


conformed to certain specified conditions as much an 
abstraction as the "economic man 5 * of the political 
economist. Wave-mechanics tries to give a concrete 
picture of this atom, but inevitably this is still a com- 
posite photograph, and because of this it extends over the 
whole plate and is very blurred. 

Just because the wave-mechanics deals only with prob- 
abilities and statistical assemblies, its apparent deter- 
minism may be only another way of expressing the law of 
averages. The determinism may be of a purely statistical 
kind, like that relied on by an Insurance Company, or the 
Bank at Monte Carlo. 

This being so, there is no assignable reason why the 
apparent determinism of the wave-equation should not 
conceal a complete objective indeterminism. In the 
mathematical problem known as the "random walk", we 
imagine that a traveller walks 20 miles a day, but with 
no causal relation between the directions of his walks 
on successive days we can, for instance, imagine his 
throwing a stick up in the air at random every morning, 
and letting the direction of its fall determine the direction 
of his walk for the day. A mathematical formula can of 
course be obtained to exhibit the chances of his being at 
various points at successive nightfalls. If we now reduce 
the unit of time from a day to a second, so that his every 
step is indeterminate, we find that the probabilities 
spread out in waves, much as in Schrodinger's equation; 
the spread of the waves conforms to a strict determinism, 
although the underlying physical cause is a complete 

In the same way the apparent determinism of the wave 
picture may conceal any amount of true objective in- 


determinism in matters of detail, such as appears to be 
necessary to account for radioactive disintegration and 
the atomic jumps by which Einstein obtains the statistical 
law of black-body radiation. 

Let us now turn to the particle picture. We have 
already seen that this shews no determinism, and, as 
Bohr has pointed out, it is impossible that it should. We 
cannot include determinism in our picture of nature 
unless we have an experimental technique for discover- 
ing that it exists in nature. Now this requires that if we 
picture nature in terms of particles whether photons or 
electrons and protons existing in time and space, we 
must possess a means of discovering the positions and 
velocities of these particles with complete accuracy. This 
is precisely what the uncertainty principle denies us. 
Thus a picture which represents nature as consisting of 
particles in time and space cannot at the same time 
exhibit determinism. 

This brings us to the point of cleavage between the 
old classical theories and the new quantum theory. The 
classical theory represented nature as situated wholly in 
time and space, and at the same time governed by a strict 
determinism. The newer theories, which alone agree 
completely with observation, shew that we can retain 
either the space-time representation of the older pictures 
of nature or the strict determinism, but never both. Deter- 
minism and representation in space and time are like the 
old man and the old woman of the string-hygrometer; 
when one comes in, the other goes out. Heisenberg has 
exhibited the contrast between the present view and the 
old in the following scheme:* 

* The Physical Principles oj the Quantum Theory, p. 65. 


Classical Theory Quantum Theory 

Causal relationships of phe- Either: 

nomena described in terms of Phenomena described in terms 
space and time of space and time (but uncer- 

tainty principle) 


Causal relationship expressed 
by mathematical laws (but 
physical description of phe- 
nomena in space and time 

The kind of indeterminacy implied in the first alterna- 
tive on the right is a natural consequence of the atomicity 
of the particle picture. We cannot foretell the future 
because we can never know the present with complete 

It is very unfortunate that the same word "indeter- 
minacy" is so often used to express both this and the 
indeterminism of quite different type which may be in- 
herent in nature itself; here we cannot foretell the future 
because nature herself does not know what is going to 

In addition to the indeterminacy of the former kind, 
which necessarily occurs when we picture nature as 
particles, we have seen that our picture of nature may 
also contain indeterminacy of the second kind which 
ought thus to be present whether we picture it as waves 
or particles. 

Yet it may be argued that if we go far enough back, we 
must come upon a cause at last'; the direction in which the 
travellers' stick fell was not really undetermined, but de- 
pended on the force of his throw, which in turn depended 


on whether he was feeling vigorous, and this in turn on 
whether the journey of the day before had lain through 
easy or fatiguing country, and so on indefinitely. If we 
are to introduce such considerations into our description 
of nature, it will perhaps take some such form as the 

We set out to build a conjectural picture of the external 
world, the only rule of the game being that this picture is 
to account for our sense-impressions, exactly and down to 
the smallest detail, and yet is to be objective in the sense 
of not explaining merely the sense-impressions of a single 
individual. Each sense-impression is caused by a transfer 
of energy from the external world to the nerve terminals 
of our bodies. This transfer is invariably by photons, 
which, in the new science as in the old, can be adequately 
represented as travelling in space and time. Thus we 
naturally begin our conjectural picture of nature by con- 
structing a mental framework of space and time, against 
which to draw our picture. Going one stage farther, we 
find that the photons which cause our sense-impressions 
originate in events. We now find that if our picture is to 
be objective in the sense just explained, these cannot be 
represented as localised in space and time separately, but 
they can still be localised in the blend of space and time 
we describe as space-time. So long, then, as we do not 
insist on dividing space-time up into space and time 
separately, the framework remains adequate for the 
picture. But these events are the interactions of material 
objects, electrons and protons and their combinations, 
and we find that these cannot be adequately depicted as 
existing in space and time. Thus our space-time frame- 
work proves inadequate for the representation of the 


whole of nature; it is suited to form a framework for but 
little more than our sense-impressions, which is precisely 
the purpose it was originally constructed to serve. We are 
thus led conjecturally to think of space and time as a sort 
of outer surface of nature, like the surface of a deep 
flowing stream. The events which affect our senses are 
like ripples on the surface of this stream, but their origins 
the material objects throw roots deep down into the 
stream. When we say a brick is three-dimensional we 
mean merely that we can only establish contact with it, 
through our senses, in three dimensions of space. Ripples 
come from the brick to us in three dimensions of space, 
but this in no way limits the real existence of the brick 
to these three dimensions. 

Two surface-ripples may appear exactly similar, and 
may yet be caused by very different happenings down in 
the depths of the stream, so that the similarity of their 
appearance provides no guarantee that they will behave 
in the same way. For this reason we cannot expect the 
ripple-phenomena on the surface of the stream to shew a 
strict determinism, nor to conform otherwise than statis- 
tically to the law which we describe as the "uniformity of 
nature". The fact that the surface-phenomena of space- 
time shew a want of determinism leaves the question of 
whether real objective nature is deterministic or not com- 
pletely open. 

Space-time is not the framework of the world of nature, 
but of the world of our sense-perceptions, and when we 
represent objects beyond our senses in space-time, their 
apparent absence of determinism may be merely the 
price we pay for trying to force the real world of nature 
into too cramped a framework. So, when birds fly 



The province of atomic physics is to discuss the nature of 
particular events, and it has been very successful in shew- 
ing us how it is that certain kinds of events occur, while 
others do not. Yet this can give us but little information 
as to what is happening to the universe as a whole. An- 
other branch of physics, known as thermodynamics, takes 
this problem in hand; it does not concern itself with 
individual events separately, but studies events in crowds, 
statistically. Its province is to discuss the general trend 
of events, with a view to predicting how the universe as 
a whole will change with the passage of time. 

The science of thermodynamics had its origin in 
severely practical problems relating to the efficiency of 
engines, but it was sqon extended to cover the operations 
of nature as a whole. All this happened in the days when 
nature was assumed, without question, to be mechanical 
and deterministic. In what follows, we shall not treat 
nature as mechanical, but for the moment we shall 
treat it as though it were strictly deterministic. 

on a deterministic view of nature, the universe never 
has any choice; its final state is inherent in its present 
state, just as this present state was inherent in its state 
at its creation. It must inevitably move along a single 
road to a predestined end, like a train rolling along a 
single-track line, on which there are no junctions of any 




The province of atomic physics is to discuss the nature of 
particular events, and it has been very successful in shew- 
ing us how it is that certain kinds of events occur, while 
others do not. Yet this can give us but little information 
as to what is happening to the universe as a whole. An- 
other branch of physics, known as thermodynamics, takes 
this problem in hand; it does not concern itself with 
individual events separately, but studies events in crowds, 
statistically. Its province is to discuss the general trend 
of events, with a view to predicting how the universe as 
a whole will change with the passage of time. 

The science of thermodynamics had its origin in 
severely practical problems relating to the efficiency of 
engines, but it was soon extended to cover the operations 
of nature as a whole. All this happened in the days when 
nature was assumed, without question, to be mechanical 
and deterministic. In what follows, we shall not treat 
nature as mechanical, but for the moment we shall 
treat it as though it were strictly deterministic. 

on a deterministic view of nature, the universe never 
has any choice; its final state is inherent in its present 
state, just as this present state was inherent in its state 
at its creation. It must inevitably move along a single 
road to a predestined end, like a train rolling along a 
single-track line, on which there are no junctions of any 



kind. Thus if a super-experimentalist could discover the 
exact position and the exact speed of motion of every 
particle in the universe at any single instant, a super- 
mathematician would be able to deduce the whole past 
and the whole future of the universe from these data. 

Experimental physics has not yet been able to provide 
such data, and the uncertainty principle shews that it 
never will be. Yet a super-mathematician, who had un- 
limited time at his disposal, might calculate out all the 
different pasts and futures which would result from all 
conceivable sets of data in other words from all con- 
ceivable present states. 

He might commence his labors by making a diagram 
in which to map out all possible states of the universe, 
just as all points in England are mapped out in an 
ordinary geographical map. He could start from any 
particular point in this diagram and trace out, by mathe- 
matical calculation, the whole future of a universe which 
started from the state represented by this point. He 
could represent this future by a line through the point, 
which would run through his diagram much as a railway 
line is represented by a line running across the map of 
England. He could take point after point in his diagram 
in turn, and represent the development of a universe 
which started from each point by a line, until his whole 
diagram was filled with lines. These lines would repre- 
sent all the lines of development which were possible for 
the universe. If the universe was strictly deterministic, as 
we have so far supposed, the diagram would look like the 
map of a country covered with single-track lines of rail- 
way, with no junctions of any kind. If, on the other hand, 
strict determinism does not prevail in the universe, there 


may be any number of junctions and connecting tracks 
between the different lines. 

Let us imagine that a perfect diagram of this kind is 
at our disposal, as it would be in theory at least if 
we had a perfect knowledge of the laws of nature. No 
matter how perfect the diagram is, we are still unable to 
gain a detailed knowledge of our future from it, because 
we do not know our present position on the map. This 
makes it impossible to identify the particular track on 
which we are travelling, so that we can neither say what 
part of the diagram it will traverse next nor where it will 
end. Yet it may be possible to discover in what kind of 
country it ends, and this is the information we really 
want. It is information of this kind that the science of 
thermodynamics can provide. 

Imagine that we suddenly waken up from a state of 
unconsciousness to discover we are on a British railway. 
We have no means of knowing where our journey will 
end. Yet if we have a physical map of Great Britain with 
us, we may notice that only a few hundred acres out of 
55 million lie as much as 4000 feet above sea-level. 
Although we cannot say where our journey will end, there 
are obviously very long odds that it will end at a height 
of something less than 4000 feet above sea-level. If a 
barometer in our compartment indicates that we are 
already as much as 4000 feet above sea-level, then there 
are very long odds that the general trend of our journey 
will be downhill. 

It is to considerations of this kind, rather than to exact 
knowledge, that we must turn for guidance in our efforts 
to study the evolution and final end of the universe. As 
certain knowledge is beyond our reach, we must be 


guided entirely by probabilities. Yet the odds we en- 
counter in calculating these probabilities prove always to 
be so immense that we may, for all practical purposes, 
treat long odds as certainties. Because the number of 
particles electrons and protons in the universe is of 
the order of 10 79 , we find that high powers of 10 79 enter 
into all our odds, and, this being so, we need not trouble 
to differentiate too carefully between probabilities of such 
a kind and certainties. 


Thermodynamics is much concerned with a quantity 
known as "entropy". This plays much the same part in 
our diagram of the universe as height played in our 
imaginary railway map of Great Britain, except that 
small entropy corresponds to great height, and vice-versa; 
thus entropy does not correspond so much to height above 
the level of the sea, as to depth below the top of the 
highest mountain. The highest mountain in Great 
Britain rises to 4400 ft. above sea-level, and as most of 
Great Britain is only a little above sea-level, most of it is 
at a depth, in this sense of the word, of nearly 4400 ft. 
the maximum depth possible. In the same way, we find 
that most of the configurations which figure in our map of 
the universe are at the maximum entropy possible all, 
indeed, except for minute regions whose sizes are pro- 
portional to inverse powers of 10 79 . 

At the moment we cannot justify this statement because 
we have not yet defined " entropy". And there is no need 
to justify it, because the best definition of "entropy" 
makes the statement true of itself and automatically. It 
is convenient to define "maximum entropy" as specifying 


the condition which is commonest in our map of the 
universe, and then, having done this, to define entropy in 
general in such a way that the more common condition is 
always of higher entropy than the less common. Thus 
we define entropy to be a measure of the "commonness" 
of a given state in our map.* 

With this definition we find that, just because the 
numerical factors involved are so immense, conditions of 
"maximum 55 entropy are not only more common, but 
incomparably more common, than those whose entropy is 
less, and so it is all down the ladder. Because of this, 
it is practically certain that each state of the universe will 
be succeeded by a state of higher entropy than itself, so 
that the universe will "evolve" through a succession of 
states of ever-increasing entropy, until it finally reaches a 
state of maximum entropy. Beyond this it cannot go; it 
must come to rest not in the sense that every atom in it 
will have come to rest (for maximum entropy does not 
involve this), but rather in the sense that its general 
characteristics cannot change any more. 

Yet if someone asserts that this will not happen, and 
that the universe will move to a state of lower entropy than 
the present, we cannot prove him wrong. He is entitled 
to his opinion, either as a speculation or as a pious hope. 
All we can say is that the odds against his dream coming 
true involve a very high power of 10 79 in his disfavour. 

Thermodynamics is accustomed to disregard all such 
infinitesimal chances and forlorn hopes, and announces 
its laws as certainties. We must nevertheless always bear 
in mind that there is a small risk of failure attached to 

* If W is the "commonness" of a certain state, the mathematician 
defines the entropy of this state as k log W, where is the gas-constant. 


every such law. The famous "second law of thermo- 
dynamics" asserts that the entropy of a natural system 
always increases, until a final state is attained in which the 
entropy can increase no further; a fuller statement of the 
law would be that the chances of the entropy doing other- 
wise are negligibly small. 

The Final State of Maximum Entropy 

We now see that the question of discovering the final state 
of the universe is merely that of discovering how far the 
entropy of the universe can increase without violating the 
physical laws which govern the motions of its smallest 
parts. There was no need to take the physical properties 
of matter into account in defining entropy, but we must 
do so before we can discover the state in which the en- 
tropy is a maximum. 

The process is usually very complicated, but two simple 
instances may illustrate the general characteristics of a 
state of maximum entropy. They do not refer to the 
universe as a whole, but merely to minute portions which 
have been selected for their simplicity and familiarity. 

Let us pour some red ink into water, and leave the ink 
and the water to diffuse into one another. We know, 
before the event occurs, that the final state will be one in 
which they are uniformly mixed to form a homogeneous 
pinkish fluid, and as this state of uniform mixture is 
invariably the. final state, we know that it must be the 
state of maximum entropy. 

Again, let us put a kettle of cold water over a hot fire. 
We know, before we perform the experiment, that the 
final state will be one in which all the water is turned into 
steam. This also must be a state of maximum entropy. 


Just as the red ink diffused itself equally through all parts 
of the water in attaining a state of maximum entropy, so 
the heat of the fire tends to diffuse itself equally through 
coal, kettle, and water. 

These instances have shewn us two final states in which 
the entropy is a maximum. They illustrate a very wide 
and very general principle the final state of maximum 
entropy avoids concentration, whether of special sub- 
stances (as with the ink) or of energy (as with the heat 
of the fire) . The "commonest state" is one in which both 
substance and energy are uniformly diffused, just as the 
commonest state in which we find a concert audience is 
that in which tall people and short, dark and fair, and 
so on, are uniformly diffused. 

General considerations of this kind can tell us some- 
thing at least as to the final end of the universe, but they 
cannot indicate the road by which it will be reached. All 
they can tell us is that the road is practically certain to be 
one of increasing entropy throughout; and the better we 
understand entropy, the more this statement will convey 
to us. It is not impossible for the entropy to decrease, but 
it is almost infinitely improbable that it should do so. 

For instance, when the ink and water have once be- 
come thoroughly mixed, the state of maximum entropy 
has been attained; the ink-water mixture cannot change 
its general characteristics any further without a decrease 
of entropy. Yet the molecules of ink and water still jostle 
one another about, and change places as they do so. It is 
quite conceivable that their random motions should take 
diem into a configuration in which all the ink molecules 
are found at one end of the vessel, and all the water 
molecules at the other. The entropy of such a configu- 


ration is far below the maximum possible, so that the 
odds against the molecules of ink and water assuming such 
a configuration are immense. Yet it is important to 
notice that no law of nature prohibits it. Indeed, if we 
had an infinite number of vessels of ink and water, the 
unexpected would be bound to happen in a few of them 
just as, if an enormous number of hands of bridge are 
played, there are bound to bd*a few deals in which each 
player gets one complete suit, in spite of the immense a 
priori odds against such an event occurring in a single 
individual case. The event is bound to occur either if an 
enormous number of players play bridge for a short time 
or if a single party of players play for an enormous time. 
In the same way we may say that a complete separation of 
the ink and water is bound to occur, either if we have an 
infinite, number of vessels containing the mixture, or if a 
single vessel exists for an infinite time. 

Similar considerations apply to our other miniature 
universe of fire, kettle and water; the water in the kettle 
may freeze as the result of being put over a hot fire. To 
prove this, we need only notice that there is a possible 
state of this group of objects in which the water exists 
in the form of ice, and the fire is even hotter than before 
because there is less heat in the kettle and its contents. 
If we map out all the configurations of the system, this 
particular configuration must appear on the map, so that 
we cannot know for certain that it will not be the end 
of the journey. We know, however, that when we put a 
kettle of ice on the fire the normal event is for it to turn 
into a kettle of water. This shews that the entropy of the 
water-configuration is higher than that of the ice-con- 
figuration, and this in turn shews that although it is 


possible for a kettle of water to freeze when placed over 
a hot fire, it is almost infinitely improbable that it will 
do so on any single occasion. If even the most credible 
of witnesses told us a kettle of water had frozen when he 
put it on a hot fire, we should not believe him, although 
there is nothing in the laws of nature to prohibit such an 
occurrence; indeed these very laws assure us that the 
event must occasionally happen. Yet such occasions must 
from the nature of things be so very rare, that we should 
think it far more likely that our informant had gone 
crazy, had been deceived, or was lying, than that he had 
been present at one of them. 

These examples have both illustrated cases in which the 
individual atoms and molecules are left to perform ran- 
dom motions under the play of blind forces. If the atoms 
and molecules receive any kind of guidance, the result 
may be very different. Suppose that, instead of ink, we 
pour oil into our water. We no longer expect the final 
result to be a uniform mixture; we know that we shall 
find all the oil on top and all the water below. An 
arrangement which is inconceivably improbable for ink 
and water is found .to be the most probable of all for oil 
and water; indeed, exact calculation confirms that a state 
of practically complete separation is the state of maximum 
entropy in the case now under consideration. The reason 
for the change is that the force of gravity differentiates 
between the molecules of oil and of water. When we say 
that oil is of lower specific gravity than water, we mean in 
effect that the earth's attraction draws particles of water 
downward with a force greater than it exerts on equal- 
sized particles of oil. Because it continually drags these 
latter particles down with a smaller force, it encourages 


them to move upwards through the water. When we mix 
oil and water, we are not handing over their molecules 
to be the playthings of a blind chance, but rather to a 
chance over-ridden by the selective action of gravitation. 
There is blind interplay of the molecules of oil between 
themselves and of the molecules of water between them- 
selves, but the cross interplay is controlled by gravitation. 
Suppose, for instance, that we divide our vessel into two 
equal divisions, each holding a pint, by a horizontal 
membrane with a small pinhole in it. Let us mix a 
pint of oil and a pint of water as thoroughly as possible, 
and fill our vessel on both sides of the membrane with 
this quart of mixed liquid. After a sufficient time, we 
shall of course find that all the oil has passed into the 
upper half, while all the water has passed into the lower 
half; our careful mixing has been undone, and this by 
very simple means. Whenever a particle of oil in the 
lower half met a particle of water in the upper half at the 
pinhole the only place at which they could meet the 
force of gravity urged them to change places, and such 
interchanges have continually increased the amount of oil 
in the upper half and that of water in the lower half, until 
complete separation has been effected. 

The Sorting Demon of Maxwell 

If we performed a similar experiment with our previous 
mixture of water and red ink, no such action would take 
place in the ordinary course of nature, since gravity makes 
no distinction between liquids of the same specific gravity. 
Yet suppose an intelligent being of microscopic size were 
placed at the pinhole, armed with a tiny shutter with 
which he could close the aperture when he wished, and 


was given instructions to open it only for molecules of 
ink passing upwards or for molecules of water passing 
downwards in brief his task would be to perform a 
selective action like that which gravity performs for oil 
and water. It is clear that after a long enough time the 
ink and water will be as thoroughly separated as the oil 
and water had previously been, although this time the 
separation would have been produced not by gravity 
but by intelligence. 

The intelligent microscopic being we have just de- 
scribed was introduced into science by the Cambridge 
physicist Clerk Maxwell, and is generally described as 
"Maxwell's demon". The demon, we must notice, in no 
way sets himself in opposition to the laws of mechanics. 
We do not know how often he finds it necessary to open 
and close his microscopic shutter. The natural motions 
of the molecules may conceivably be such that he finds no 
occasion to close it at all. Then everything will go on 
precisely as though the demon had not been called on to 
help, the ink and water separating out under their own 
natural random motions. Yet the odds against such an 
occurrence are unthinkably large. As each separate 
molecule comes into view, the demon must ask himself the 
question "To act or not to act?", and then put his de- 
cision into practice. A prolonged run of decisions all in 
the same sense will be as improbable as a prolonged run of 
heads or of tails when we spin a coin. Thus it is exceed- 
ingly unlikely that our demon will find that no action is 
needed time after time; the normal event will be that he 
will need to open and close his shutter millions of times. 
Even so, he expends no energy in so doing, and each time 
the shutter is closed against a molecule, we may reflect 


that had the path of the molecule in question been a 
hair's-breadth to right or left, it would have bounced off 
the membrane without the demon touching his shutter. 

Although the demon does not interfere with the opera- 
tion of the laws of nature, yet he exercises a selective 
effect, and by this alone he can cause any system to pass 
to a state of lower entropy. Natural forces, left to their 
own blind interplay, are practically certain to increase 
the entropy, but it is the play of the laws of probability 
rather than of the laws of nature that produces this result. 
The demon has not been told to circumvent the laws of 
nature, but the laws of probability; he can so to speak 
load the dice from moment to moment, and obtain any 
result he wants provided this does not violate the laws 
of nature the conservation of mass, of energy, and so 
forth. When red ink and water are mixed, he cannot 
increase the total amount of either or both; all he can do 
is to disentangle them, as one might sort out a heap of red 
and white beads, or again as a railway shunter divides up 
a goods train by moving the switches in different ways for 
different wagons. When a kettle of water is placed over 
a fire, he cannot add to the total amount of heat, but he 
can, if he wishes, increase the heat of the fire by subtract- 
ing heat from the kettle. His accomplishments are 
limited to robbing Peter to pay Paul, whereas unaided 
nature would leave Peter and Paul to fight it out or 
perhaps to toss up for it time after time. 

Quite general considerations shew that the universe as 
a whole has a very long way to go before coming any- 
where near its final state of maximum entropy. In this 
final state, concentrations of radiation and of temperature 
will equally have disappeared, so that radiation will be 


distributed uniformly throughout space, and the tempera- 
ture will be everywhere the same. At present, the density 
of radiant energy out in the farthest depths of space cor- 
responds to a temperature of less than one degree above 
absolute zero; in the interstellar spaces of the galactic 
system, to three or four degrees only; near the earth's 
orbit to about 280 degrees; at the sun's surface to about 
6000 degrees; at the sun's centre to perhaps 40 or 50 
million degrees. The universe can always increase its 
entropy by equalising these temperatures; as for instance 
by letting energy flow from the sun's hot centre to its 
cooler surface, by letting it then stream out into space, 
past the earth's orbit, into the cold and dark of interstellar 
and intergalactic space. There can be no end to the in- 
crease of entropy until these regions are all at the same 
temperature, with radiant energy diffused uniformly 
throughout space. Then, and then only, will the universe 
have reached its final state, a state in which the tempera- 
ture will everywhere have fallen too low for life to exist 
the perfect quiet and perfect darkness of eternal night. 

The Activities of Life 

A general survey of the universe as a whole suggests that 
it is rapidly moving towards such an end. The sun is 
dying, pouring out some 250 million tons of its substance 
in the form of radiation each minute, thereby lowering 
its own heat and raising that of empty space. Other stars 
tell the same story; we find no evidence of sorting demons 
sitting on their surfaces to turn the heat back into their 
hot interiors. Yet a being from another universe who 
scrutinised this earth of ours with sufficient care might 
notice signs which led him to wonder whether there 


might not be local exceptions to the general increase of 
entropy. For instance, regarded from the purely physical 
point of view, gold is a fairly ordinary metal; natural laws 
shew it no favour nor special treatment. Yet our visitor 
might notice that the world's total supply of gold, which 
had originally been fairly uniformly scattered throughout 
parts of the earth's crust, tended to become highly con- 
centrated in a few small regions, in a way which would 
seem to set the demands of the second law of thermo- 
dynamics utterly at defiance. Again, the law does not 
approve of fires occurring at all, although it admits that 
accidents will happen. It insists, however, that these 
accidental violations are most likely to occur when the 
weather is hot and dry. Yet our observer would not only 
detect innumerable fires on earth, but would notice that 
they occurred most frequently when it was cold and 
damp; he would see more in those parts of the surface of 
the earth which were covered with winter snow than in 
those which were parched with equatorial or summer 
heat. on the other hand, he might notice that small 
accumulations of ice were especially in evidence when the 
weather was hot and sultry. 

The odds against all these events occurring in the 
normal course of a nature which had not been tampered 
with would be of the same order as the odds against a 
kettle of water freezing when placed on a hot fire. Thus 
no picture of nature can claim to be complete, unless it 
contains some means by which the statistical laws of 
nature may be evaded if not throughout the whole of 
nature, at least in chosen spots on our own earth. Our 
visitor might perhaps conjecturally attribute these eva- 
sions to the activities of innumerable sorting demons. 


A statistical survey of the more violent offences com- 
mitted against the second law of thermodynamics would 
shew that the hotbeds of crime are precisely those places 
we describe as centres of civilisation. Inanimate matter 
obeys the law implicitly; what we describe as life succeeds 
in evading it in varying degrees. In fact it would seem 
reasonable to define life as being characterised by a 
capacity for evading this law. It probably cannot evade 
the laws of atomic physics, which are believed to apply 
as much to the atoms of a brain as to the atoms of a brick, 
but it seems able to evade the statistical laws of probabil- 
ity. The higher the type of life, the greater is its capacity 
for evasion. And the observed evasions so closely re- 
semble the results that would be produced by an army 
of sorting demons, that it would seem permissible to 
conjecture that life operates in some similar way. 

So long as nature was believed to be mechanistic, and 
therefore deterministic, such a conjecture was hardly per- 
missible the sorting demons would have interfered with 
the predestined course of nature. 

on the other hand, modern physics can adduce no 
such objection to the conjecture; the only determinism 
of which it is at all sure is of a merely statistical kind. 
We still see the actions of vast crowds of molecules or 
particles conforming to determinism this is of course 
the determinism we observe in our everyday life, the basis 
of the so-called law of the uniformity of nature. But no 
determinism has so far been discovered in the motions of 
the separate individuals; on the contrary, the phenomena 
of radio-activity and radiation rather suggests that these 
do not move as they are pushed and pulled by inexorable 
forces; so long as we picture them in time and space, their 


future appears to be undetermined and uncertain at every 
step. They may go one way or another if nothing in- 
tervenes to direct their paths; they are not controlled by 
pre-determined forces, but only by the statistical laws 
of probability. If an unknown something intervenes to 
guide them, they may transfer their allegiance from the 
laws of probability to the guiding something, as the 
molecules of the oil-water mixture did to the force of 
gravitation. There seems no longer to be any reason why 
this something should not be similar to the action of sort- 
ing demons, the volitions of intelligent minds loading the 
dice in their own favour, and so influencing, so to speak, 
the motions of the molecules when they are in doubt 
which path to take provided always that volitions 
and molecules are not too dissimilar in their nature for 
such interaction to be possible. 

Space-time and Nature 

We can also look at the matter in the alternative way de- 
scribed on p. 257. We have just been picturing nature as 
an assemblage of particles set in a framework of space and 
time. Yet we have seen elsewhere that such a framework 
is not suited for the arrangement of the whole external 
world, but only for the photons by which it sends messages 
to our senses. Because these messages arrive in a frame- 
work of space-time, we must not conclude that the whole 
external world exists within the confines of the same 
framework. Our observational knowledge of the outer 
world is limited by the aperture of our senses, and these 
form blinkers which prevent our seeing beyond space 
and time just as our telescope may prevent us seeing 
more than a small angle of the sky. But the events we see 


in space and time may have their origin outside space and 
time just as the curve in the tail of the brilliant comet 
we see in our telescope may have its origin in the sun 
which lies outside the field of the telescope. The recent 
developments of theoretical physics suggest that this may 
be the case with many of the phenomena of physics. 
It has proved impossible to find any description of elec- 
trons and protons in space and time such as shall fully 
account for the phenomena originating in them. 

This has led us to think of space-time as a sort of 
surface-layer of the universe; the sources of events appear 
in this space-time surface in the form of material protons 
and electrons, but they have their roots in a deeper 
stratum. Thus although no causality may be discernible 
while we limit our vision to the surface of things, yet if we 
could take the whole of reality into view, we might see 
cause and effect inter-related, events following clearly 
specified laws, and not occurring merely as illustrations of 
the laws of probability. The gardener plants a dozen 
trees which, so far as he can see, are exactly similar; he 
has been told that only fifty per cent, of trees of this 
species thrive, and this is confirmed when he finds that, 
out of his dozen trees, six thrive and six fail. Yet he does 
not attribute their different fates to the laws of proba- 
bility, but to happenings in the soil beyond his vision. 
He digs down and finds wire-worms at the roots of the 
six failures. The wire-worms play much the same sort of 
part as we have imagined the sorting demon to play in 
our space-time picture, or as we have conjectured that 
our volitions and intelligences might play. Residing be- 
yond the stratum of time and space, they can influence 
events, which also have their roots outside time and space, 


and so exercise some control over the happenings in time 
and space. 

Conjectures of the kind bring us to a region of thought 
in which human predilections are deeply concerned. 
Some who are eager to find a place for virtue, beauty 
and other "values" in the scheme of things are very 
ready to hail any evidence of indeterminism in nature as 
almost affording a proof of human free-will. Others re- 
fuse to admit the possibility of indeterminism even in 
nature, and insist that we, like all nature, are mere cogs of 
a machine which is running down to its predetermined 

Apart from extremists, a number of moderate men still 
adopt an attitude of extreme caution, and even suspicion, 
towards any attempt to reconcile human free-will with 
the scheme of physical science. Many quote recent 
investigations in physiology and psychology as providing 
evidence, not against the possibility of free-will, but 
against its probability. Others regard the present 
situation in physics as a mere transitory phase. For 
instance, Planck, who has given much thought to this 
question, writes, with reference to the impact of quan- 
tum ideas on the fundamental laws of physics: * 

"Some essential modification seems to be inevitable; but 
I firmly believe, in company with most physicists, that the 
quantum hypothesis will eventually find its exact expression 
in certain equations which will be a more exact formulation 
of the law of causality 9 *, 

and is prepared to extend the operation of this law to 
human activities: 

* Where is Science going? by Max Planck (1933), pp. 143, 155. 


"The principle of causality must be held to extend even 
to the highest achievements of the human soul. We must 
admit that the mind of each one of our greatest geniuses 
Aristotle, Kant or Leonardo, Goethe or Beethoven, Dante 
or Shakespeare even at the moment of its highest flights 
of thought or in the most profound inner workings of the 
soul, was subject to the caused fiat and was an instrument 
in the hands of an almighty law which governs the world". 

Einstein is reported as expressing similar opinions:* 

"I am entirely in agreement with our friend Planck in re- 
gard to the stand which he has taken on this principle. He 
admits the impossibility of applying the causal principle to the 
inner processes of atomic physics under the present state of 
affairs; but he has set himself definitely against the thesis that 
from this Unbrauchbarkeit or inapplicability we are to conclude 
that the process of causation does not exist in external reality. 
Planck has really not taken up any definite standpoint here. 
He has only contradicted the emphatic assertions of some 
quantum theorists and I agree fully with him. And when you 
mention people who speak of such a thing as free will in nature 
it is difficult for me to find a suitable reply. The idea is of 
course preposterous. . . . 

"Honestly I cannot understand what people mean when 
they talk about freedom of the human will". 

Weyl, on the other hand, after explaining how the 
limits to determinism, if any, will be found by passing 
along the road from the large scale phenomena of astron- 
omy and physics, which necessarily appear deterministic 
(p. 230), to the small scale phenomena at the far end of 
the road, continues: f 

"We firmly believe today that we have touched these limits 
in quantum mechanics. . . . 

* L.c. pp. 210, 211. 

t The Open World, pp. 35, 43. 


"At the same time 'fate' as expressed in the natural laws 
appears to be so weakened by our analysis that only through 
misunderstanding can it be placed in opposition to free will". 

I do not think that either the facts of physical science 
or their interpretation within the legitimate province of 
physical science are in dispute among men of science; 
on the contrary, I believe we are all in agreement. Dif- 
ferences only arise when physicists take to speculation 
either about the future progress of science (as Planck does 
in the above quotation), or about the ultimate problem of 
human free-will, which of course lies beyond the province 
of physics. The famous dictum of Schopenhauer "Man 
can do what he wills, but cannot will what he wills" 
contains two distinct statements. The latter has to do 
with happenings on the mind side of the mind-body 
bridge, which are not the concern of physics. The former 
is the concern of physics. In brief, it was believed to be 
in conflict with nineteenth-century physics, but is not in 
conflict with the physics of to-day; whether it will be 
in conflict with the physics of to-morrow remains to be 

Nevertheless, the most we can say is that crevices have 
begun to appear in what used to be considered the im- 
pregnable closed cycle of physical science. Whether the 
volitions of the human mind can pass through these and 
affect the operations of nature must in the last resort 
depend on whether the two are sufficiently alike to in- 
teract a keyhole is useless unless we have a key of the 
same nature as the lock. It may still be, as Descartes 
maintained, that mind is too dissimilar from matter ever 
to be able to influence it. 


Mind and Matter 

A century after Descartes, we find Berkeley maintaining 
that we had no right to say that matter was different from 
mind. With no knowledge of matter except such as 
comes to us through the perceptions of our minds, what 
warrant can there possibly be for supposing the two are of 
unlike natures ? Matter outside our minds produces ideas 
inside our minds; causes must be of like nature to their 
effects, and "after all, there is nothing like an idea but an 
idea". Thus Berkeley argued that matter must be of the 
same general nature as an idea, like the matter we see in a 
dream. To say that mind cannot influence matter now 
becomes as absurd as to say that mind cannot influence 

A later school of philosophy shewed how this argument 
could be turned against its author. Even if matter and 
mind were of similar nature, how did we know they were 
of the nature of mind rather than of matter? The science 
of that time claimed to know a great deal about matter, 
but admittedly knew very little about mind; thus it was 
said that the scientific picture of matter must also portray 
mind and its operations. And this picture was that to 
which we have so often referred a jumble of mechan- 
ical atoms moving blindly along their pre-arranged paths 
to predestined ends. 

The logic of this argument stands, but not the premised 
picture of matter. In so far as science now draws any 
picture at all of matter, it is one which seems in every 
way closer in mind. 

To some extent this must be the case. The old science 
which pictured nature as a crowd of blindly wandering 


atoms, claimed that it was depicting a completely objec- 
tive universe, entirely outside of, and detached from, the 
mind which perceived it. Modern science makes no such 
claim, frankly admitting that its subject of study is pri- 
marily our observation of nature, and not nature itself. 
The new picture of nature must then inevitably involve 
mind as well as matter the mind which perceives and 
the matter which is perceived and so must be more 
mental in character than the fallacious picture which 
preceded it. 

Yet the essence of the present situation in physics is not 
that something mental has come into the new picture of 
nature, so much as that nothing non-mental has survived 
from the old picture. As we have watched the gradual 
metamorphosis of the old picture into the new, we have 
not seen the addition of mind to matter so much as the 
complete disappearance of matter, at least of the kind 
out of which the older physics constructed its objective 

The Einstein-Heisenberg policy of concentrating on 
observables might well have been adopted in the first 
instance as a mere matter of scientific technique; it was an 
obvious precaution to make as few assumptions as possible 
about unobservables, and so lessen the risk of unjustified 
assumptions and wasted work. 

Such a policy might have resulted in either of two 
ways. If the phenomena of observation were evidence of 
an objective nature existing in its own right, the proce- 
dure might have been expected first to co-ordinate the ob- 
servables and then to throw some light on the real nature 
of the unobservables behind them. If, on the other hand, 
nature was largely or wholly subjective, the procedure 


might have been expected to disclose this fact. Actually 
the result has come nearer to the latter alternative than 
to the former. The observables do not appear to owe 
their existence to our supposed unobservables existing in 
the reality behind them so much as to our conscious 
minds observing them from in front. Electrons, protons, 
and their varied arrangements seem as unable to provide 
true primary qualities as were the older mass, motion and 
extension in space of Locke and Descartes; the theory of 
quanta seems to dethrone the former as effectively as the 
theory of relativity dethroned the latter. Thus the pro- 
cedure of concentrating on observables appears to be 
leading to results different from those which might have 
been anticipated if the unobservables had existed in their 
own right; it seems to lend a new meaning to the dictum 
"Esse est percipi" of the philosophy of an earlier age. 

Such considerations as these undoubtedly introduce a 
markedly subjective tinge into all discussion of the present 
situation in mathematical physics. We must, however, 
be on our guard against taking a wrong turning at this 
point of our discussion. Even if our assumed unobserv- 
ables electrons and protons should prove to be 
wholly subjective, this would not prove that all nature is 
subjective. Our unobservables are at best mere guesses. 
These particular unobservables may have been bad 
guesses, mere creations of our own imagining, but this 
does not shew that others might not have been good 
guesses. Perhaps the proper interpretation of the situa- 
tion is merely that we must look for new unobservables. 
It is not difficult to know where to look. We have 
already seen that the particle picture, which treats matter 
as consisting of electrons and protons, fails, in some re- 


spects, to represent the true properties of matter; the 
wave picture, on the other hand, is nowhere known to 
fail, and so may provide the true gateway to reality 
(p. 252). Now the waves of this picture are of course 
unobservables; it may be that a study of these, rather than 
of electrons and protons, will lead us to the true objective 
reality behind appearances. Our attempt to relate these 
waves to particles introduced subjectivity, but this may 
have entered from the particle side and not from the wave 
side of the attempted relation. We have so far interpreted 
waves as specifying the probabilities of particles existing 
at points in space-time; we may equally well interpret 
them as specifying the probabilities of happenings at 
points in space-time the spot of light on the screen, the 
blackening of the photographic plate by the impact of the 
supposed "electron". 

Indeed, if these waves are to lead us to an objective 
reality, we must associate them with happenings rather 
than with particles, since we have already seen (pp. 199, 
249) that they have no objective existence for particles to 
which nothing is happening beyond bare existence. The 
quantum theory seizes the photons from a source of radia- 
tion at the moment of their emergence into space-time, 
analyses them and tries to refer them to the motion of an 
assumed electron (p. 183) under the electric attraction of 
a nucleus. Yet this procedure leads to no objective speci- 
fication for the assumed electron when it is away from the 
electric field; we found that we could only make our pic- 
ture of matter objective by leaving isolated electrons and 
protons out of it altogether; these seem to acquire objec- 
tive reality only when combined to form an atom, and so 
to produce events (p. 248), just as our individual space 


and time are found only to acquire objective reality when 
they are combined to form a four-dimensional space-time. 

It may be objected that, as nothing is put into the 
theory except our knowledge of radiation, we can hardly 
expect a positive knowledge of objects to emerge. Yet if 
the electron and proton had permitted of separate objec- 
tive specifications, we might have expected to be able to 
distinguish the two ingredients separately in the specifi- 
cation of the combination. It has not proved possible to 
do this; the quantum theory does not encourage us to 
regard the combination as the juxtaposition of two parti- 
cles, but merely as a source of radiation issuing into 

We cannot explain the situation away by saying that 
the uncertainty principle makes objective specifications 
impossible; this is putting the cart before the horse. The 
impossibility of objective specifications is inherent in the 
wave picture, so that if, as we now suppose, the wave 
picture is fundamental, the uncertainty principle is the 
consequence, and not the cause, of this impossibility. 

This inevitably raises doubts as to whether the isolated 
electron and proton have any existence at all in reality. 
The theory of relativity raises the same doubts, although 
in a somewhat different form. For, after experimental 
physics has reduced the supposed matter to its ultimate 
constituents, electrons and protons, the theory of rela- 
tivity finds it necessary to carry the resolution further. 
According to the older physics, a particle of matter was 
characterised by continued existence in time. The theory 
of relativity represents this continued existence by a con- 
tinuous line in space-time, and then resolves this line 
into its points, each of which represents an "event" the 


existence of the particle at a single instant of time. Space- 
time is warped at every point, and in particular at the 
points along this line. Yet the warping at these points 
does not differ in essential character from that elsewhere. 
If the particle had no extension in space beyond that of a 
mere point, we might find a sharp edge or ridge of warp- 
ing, but we have seen that we cannot assign to the 
elementary particles either a definite localisation or a 
sharply defined boundary; the wave picture of a particle, 
whatever else it may be, is never a point. Thus the 
"world-line" of a particle is, strictly speaking, not a line 
at all, but is a continuous and unbounded curved region, 
and must logically be separated into small curved spots 
the particle resolves itself into events. Most of these 
events are unobservable; it is only when two particles 
meet or come near to one another that we have an ob- 
servable event which can affect our senses. We have no 
knowledge of the existence of the particle between times, 
so that observation only warrants us in regarding its 
existence as a succession of isolated events. 

It may be objected that all nature goes on as though 
these particles had a real existence, and this provides pre- 
sumptive evidence that they have. A similar argument 
might of course be adduced to prove the real existence of 
photons; we have seen that the evidence for their existence 
is of the same general type as that for electrons (p. 154), 
and that a large part of nature can be explained by sup- 
posing photons to have a real existence (p. 155). Indeed 
it is easy to imagine beings in intergalactic space, where 
matter is rare, endowed with electric senses in place of our 
material senses, who would regard photons as the primary 
constituent of reality and matter as something outside the 


general course of nature. Yet we have seen that photons 
are merely combinations of free vibrations, so that if the 
wave picture is fundamental, photons cannot be said to 
have a real existence of the kind which we used to attrib- 
ute to electrons. And, if photons must be dismissed 'from 
the realm of reality, it is hard to find any reason for 
retaining electrons and protons. 

It becomes important at this stage to make a clear dis- 
tinction between existence and identifiable existence. 
For instance, the pounds, shillings and pence of our bank 
accounts have a real existence, but not an identifiable 
existence; we cannot say they are Bank of England notes 
of numbers so and so. In physics it is the same with 
energy; it would for instance be meaningless and silly to 
say that the energy which is now lighting my room is 
identical with that with which Samson pulled down the 
pillars of the house in Gaza; energy has no identifiable 
existence. Again it is the same with electrons. When 
two dogs A y B engage in a dog-fight, two damaged dogs 
C, D emerge of slightly altered appearance; but it is al- 
ways possible to say, for instance, that C was A and D was 
B. But when two electrons meet in an encounter, this is 
not the case; the identification is not only impossible in 
practise but is meaningless in theory. The mere assump- 
tion that it is possible leads to difficulties and wrong re- 
sults in physics. Yet when electrons and protons are 
combined to form an atom, this atom appears to retain an 
identifiable existence, at least through long periods of 
time. It is not meaningless to say to-day that certain 
atoms of gold formed part of Cleopatra's crown, but it is 
meaningless to say that certain electrons formed part of 
the pearl she drank. No doubt it is often convenient to 


regard events as strung on to electrons and protons, like 
beads on a thread, but the manner of stringing is merely 
a matter of subjective choice; I may string them in one 
way, and you in another, and both ways are equally 
valid. Thus the events must be treated as the funda- 
mental objective constituents, and we must no longer 
think of the universe as consisting of solid pieces of matter 
which persist in time, and move about in space. 

on some such grounds as these it is possible to conjec- 
ture, with Leibnitz, that matter as ordinarily understood, 
the matter of solid objects and hard particles, has no 
existence in reality, and only appears to exist through 
our observing non-material things in a confused way 
through the bias of our human spectacles. Events and 
not particles constitute the true objective reality, so that a 
piece of matter becomes, in Bertrand RusselPs words, 

"not a persistent thing with varying states, but a system of 
inter-related events. The old solidity is gone, and with it the 
characteristics that, to the materialist, made matter seem more 
real than fleeting thoughts 55 . 

This at once takes all force out of the popular objection 
that mind and matter are so unlike that all interaction is 
impossible. With matter resolved into events, the objec- 
tion is no longer tenable. We see the territory on both 
sides of the mind-body bridge occupied by events, and as 
Bertrand Russell says:* 

"The events that happen in our minds are part of the 
course of nature, and we do not know that the events which 
happen elsewhere are of a totally different kind". 

There is then no longer any reason, on these grounds, 
why the two should not interact. This of course brings us 

* Outline oj Philosophy, p. 311. 


to something which is very like Berkeley's famous argu- 
ment, clad in modern dress, and supported by scientific 
knowledge. It obviously follows that, to quote Russell 

"the world presented for our belief by a philosophy based 
upon modern science is in many ways less alien to ourselves 
than the world of matter as conceived in former centuries". 

The Mathematical Pattern 
Einstein has written: f 

"In every important advance the physicist finds that the 
fundamental laws are simplified more and more as experi- 
mental research advances. He is astonished to notice how 
sublime order emerges from what appeared to be chaos. 
And this cannot be traced back to the workings of his own 
mind but is due to a quality that is inherent in the world of 

Weyl has made a similar comment, writing: J 

"The astonishing thing is not that there exist natural laws, 
but that the further the analysis proceeds, the finer the details, 
the finer the elements to which the phenomena are reduced, 
the simpler and not the more complicated, as one would 
originally expect the fundamental relations become and the 
more exactly do they describe the actual occurrences". 

We have had ample evidence of this tendency toward 
simplicity in the present book. We have seen Hero's 
simple synthesis of the two laws of Euclid gradually ex- 
panding in scope until it embraces almost all the activities 
of the universe, and yet maintaining its original simplicity 
of mathematical form throughout. Phenomenal nature 

*!,.*. p. 311. 

t Introduction to Where is Science going? p. 13. 

J The Open World, p. 41. 


is reduced to an array of events in the four-dimensional 
continuum, and the arrangement of these events proves 
to be of an exceedingly simple mathematical kind. The 
discovery of the pattern underlying the arrangement 
might have been expected to suggest some reason why this 
special arrangement prevailed rather than another. It is 
as though we had set out to study the fundamental texture 
of a picture and had found this to consist of regularly 
spaced dots, as in a half-tone print. We are not con- 
cerned with the meaning of the picture as a whole, which 
may be moral or aesthetic or anything else; this is not the 
province of science. We are concerned only with the 
fundamental texture of the picture, which might con- 
ceivably have told us something as to its physical nature, 
something for instance as to the substance on which the 
picture was printed. But science has so far been unable to 
discover anything about the dots except the exceeding 
simplicity of their arrangement. 

This simplicity is of a mathematical kind; it seems to 
admit of a very simple mathematical interpretation and 
of no other, as though, in Boyle's phase, mathematics is 
the alphabet of the language in which nature is written. 
The words of this language may or may not be mental in 
their meanings; the immediate point is that, even in the 
alphabet, we can discover no reality different in kind 
from that we associate with a mere mental concept 
These mental concepts are not of the kind that we associ- 
ate with the work of the engineer or the poet or the moral- 
ist, but with the thinker who works with pure thought 
alone as his raw material, the mathematician at work in 
his study. 

Space provides an obvious instance of this. The con- 


cept of a finite space reduces the science of astronomy to 
law and order, just as the concept of a finite surface for 
the earth reduces the science of geography to law and 
order. It is easy to make a model of the earth's surface. 
We merely take any spherical object, and its surface 
the transition from matter to something which is not 
matter gives us our model. But we cannot make a 
model of a finite space in the same way, because we 
cannot imagine a layer of transition from space to some- 
thing which is not space. Anyone who mentions the 
finiteness of space in his writings or lectures is besieged 
with questions as to what lies beyond the finite space. It 
is impossible, we are told, to think of finite space as a 
physical reality. If we try to do so, we are at once asked 
what is outside the space. What can there be except 
more space? and so on ad infinitum, which proves that 
space cannot be finite. 

If we give up trying to attach any sort of reality to 
finite space except that of a purely mental concept, our 
way immediately becomes clear. Our everyday thoughts 
are never concerned with more than a finite part of space, 
so that finite space as a framework for mental processes 
is farm liar to us all. 

It seems likely that to bring law and order into the 
phenomena of nature, we shall further have to suppose 
that the finite space is expanding, and this raises similar 
questions. What can space expand into, except more 
space? Yet if it does so, the space which expands cannot 
be the whole of space, and so on as before, whence it 
follows that the whole of space cannot be expanding. 
Thus we cannot attribute any reality to the space of the 
universe, except again as a mental concept; any attempt 


to assign a degree of reality different from this to space 
leads only to confusion and contradictions. 

It may be urged that this does not prove anything new, 
since we already know that space cannot have any objec- 
tive reality except as one constituent of the continuum. 
But similar considerations apply to the continuum itself, 
the one entity in which science absorbs all others, and to 
which alone an objective reality seems possible. We find 
that we must picture this also as limited, so that unless 
we treat this also as a mere mental concept, we are con- 
fronted with the question as to what lies beyond the 
limits. Yet when we so treat it we find we have reduced 
the whole of nature to a mental concept, since the 
texture of nature is nothing but the texture of the space- 
time continuum. 

Some may dissent from Einstein's view that this sim- 
plicity of pattern is inherent in the world of perception, 
and may claim that it is due to the way in which our 
minds perceive. A thoroughgoing Kantian would argue 
that our minds act as lawgivers to nature, prescribing to 
the external world the ways in which its phenomena shall 
be perceived by us. The fact that only unbent pennies 
are found in an automatic machine does not prove that 
the outer world consists of unbent pennies, but merely that 
the machine has a selective mechanism which will only 
accept unbent pennies. In this same way our minds may 
have a selective action for simple mathematical laws. 

on such a view our supposed laws of nature become a 
mere specification of our own mental processes, telling us 
little or possibly nothing about nature, but certainly 
something about ourselves. Yet, if so, what precisely do 
they tell us ? That our minds run naturally and inevitably 


to matrices, tensors, four-dimensional geometry, and all 
the various square roots of minus one? Every schoolboy 
will dismiss such a suggestion as grotesque, and the 
physicist will certainly concur. If our minds had been 
thrusting mathematical properties on to nature, we 
should have designed a more readily intelligible nature 
than that described in the present book; we may feel just 
as sure that the repellently difficult matrices and tensors 
and the brain-racking constructions of four-dimensional 
geometry come to us from the external world, as the 
child is sure about the pin which runs into its finger. And 
if this is so, the same must be true of the simplicity of 
arrangement of events in the continuum. 

Moreover, if the mathematicians merely impress their 
own mathematical laws on to nature, why cannot the 
artist, the poet or the moralist do the same and meet with 
equal success? Why is not the artist able to say "the 
sunset will now turn a little more green, or purple; this is 
necessary to keep it quite perfect as the light decreases" 
or "the star will appear at the centre of the crescent 
formed by the new moon, for this is the most aesthetic 
arrangement of a star and a crescent" ? We know that 
such predictions are worthless. The cloud on the western 
horizon does not produce the sunset hues by conforming 
to the canons of art, but by moving in accordance with 
certain concepts of pure mathematics, and the only way 
to discover the future of the sunset is to solve the mathe- 
matical problem of finding which order of events makes 
an interval in the continuum continually a minimum. 

Finally, if our mathematical minds mould nature to 
their own laws now, why did they not do so before the 
twentieth century? It can hardly be supposed that the 


inherent qualities of the human mind underwent a 
revolutionary change when Planck published his famous 
paper in 1 900. If the new knowledge expresses a property 
of the human mind rather than of nature, surely some 
learned metaphysician might have foreseen that only a 
mathematical picture could ever be successful, and in so 
doing have saved science all the misguided effort of trying 
to draw pictures of other kinds. For three centuries 
science had projected mechanical ideas on to nature, and 
made havoc of a large part of nature by so doing. 
Twentieth-century science, projecting the ideas of pure 
mathematics on to nature, finds that they fit as perfectly, 
and as uniquely, as Cinderella's slipper fitted her foot. 
We can hardly explain this away by saying that we have 
merely shaped the foot to fit the slipper, for so many other 
slippers were tried first and no amount of ingenuity could 
get the foot into them. 

The fact that the mathematical picture fits nature 
must, I think, be conceded to be a new discovery of 
science, embodying new knowledge of nature such as 
could not have been predicted by any sort of general 
argument. If we could translate our knowledge from the 
language of phenomena into the language of reality, the 
word "mathematical" would, I think, have some sort of 
translation in the latter language; it would not drop away 
as having represented a mere form of apprehending 
phenomena. And if this is so, it would seem to suggest 
that reality must have something of a mental nature 

about it. 

The Road to Ultimate Reality 

Yet the fact that the search for a physical reality under- 
lying the mathematical description of nature has so far 


fatted does not of course imply that the search must for 
ever fail. We must admit it as conceivable that the 
further advances of science may yet clothe our present 
mathematical abstractions in new dresses of physical 
reality, and possibly even of material substance. It is 
not easy to imagine how formulae in which V 1 plays 
such a prominent part can admit of such an interpreta- 
tion, yet with the surprising and kaleidoscopic changes of 
recent years still fresh in our minds, we cannot disregard 
the possibility. It is, however, so far out of the range of 
our vision at the present moment, that it is idle to specu- 
late as to what the new dress may be. It may perchance 
restore mechanical properties to nature, space-time may 
prove to be a real substantial island floating in something 
which is not space-time, and so on nothing can be 
ruled out as impossible. 

Or it may be that no such substantial or material dress 
will ever be found, and that our knowledge of the uni- 
verse will for ever remain similar in kind to our present 
knowledge, a knowledge of our perceptions expressed as a 
group of mathematical formulae stamped with the stamp 
of the pure mathematician the kind of formulae which 
result from the operation of thought working within its 
own sphere. In such an event, there may or may not be a 
non-mental reality behind the form; if there is, it will be 
beyond our scientific capacity to imagine. 

All these possibilities are in the field, since all refer to 
the future and the unknown. Our positive knowledge of 
the road along which science is travelling is confined to 
that which lies behind it. We cannot say how much 
farther, if at all, the road extends in front,, or what the 
far end of it is like; at best we can only guess. 


Some may think that the most plausible conjecture is 
that the end of the road will be like what is at the half-way 
house, or perhaps more so. We have already described 
recent progress in physical science as resulting from a con- 
tinuous emancipation from the purely human point of 
view. Our last impression of nature, before we began to 
take our human spectacles off, was of an ocean of mecha- 
nism surrounding us on all sides. As we gradually discard 
our spectacles, we see mechanical concepts continually 
giving plaice to mental. If from the nature of things we can 
never discard them entirely, we may yet conjecture that 
the effect of doing so would be the total disappearance of 
matter and mechanism, mind reigning supreme and alone. 

Others may think it more likely that the pendulum will 
swing back in time. 

Broadly speaking, the two conjectures are those of the 
idealist and realist or, if we prefer, the mentalist and 
materialist views of nature. So far the pendulum 
shews no signs of swinging back, and the law and order 
which we find in the universe are most easily described 
and also, I think, most easily explained in the language 
of idealism. Thus, subject to the reservations already 
mentioned, we may say that present-day science is favour- 
able to idealism. In brief, idealism has always main- 
tained that, as the beginning of the road by which we 
explore nature is mental, the chances are that the end also 
will be mental. To this present-day science adds that, 
at the farthest point she has so far reached, much, and 
possibly all, that was not mental has disappeared, and 
nothing new has come in that is not mental. Yet who 
shall say what we may find awaiting us round the next 


o-particles, 147 

Absolute time and space, 94 ff., 1 09, 
141, 173, 175 

velocity, 94 
Action, 123 ff., 128 

at a distance, 111 ff., 118 

principle of least, 122 ff., 124 ff. 
Activities of life, 273 ff., 274 
Alkali metals, spectra of, 250 
Analysis of light, 20, 28, 30 
Animism, 33, 224 
Anthropomorphic views of nature, 

33, 43, 225 

Aristarchus of Samos, 48 
Aristotle, 122, 227 
Atomic physics, 52 
Atomism, 15 ff. 
Atoms, 15, 16, 180 ff., 242 

nucleus of, 17 

spectra of, 166 ff. 

structure of, 17, 180 ff., 245 

0-particles, 147 

Berkeley (Bishop), 15, 281, 289 
Black-body radiation, 150, 156, 157 
Bohr, N., 5, 53, 54, 169, 170, 242, 


correspondence principle, 181 
quantum restrictions, 169, 181, 

191, 244 
theory of spectra, 53, 54, 168, 

170, 176, 203, 243 
Born, M., 193, 218 
Boyle, R., 290 
Bradley, F. H., 4, 40, 68, 109, 143 

-/-radiation, 155 

Causality, 36, 227, 228, 257, 278 ff. 
Change, meaning of, 108 
Common-sense, 41, 114 

view of nature, 1, 42, 114, 229 
Compton, A. H., 153 
Continuum, 100, 101, 292 

curvature o 117, 128, 129 ff., 

Copernicus, 48, 49 

Cornford, F. M., 73 

Corpuscular theory of light, 22, 

23 ff., 83, 122, 159, 225, 286 
Correspondence principle of Bohr, 


Cosmic radiation, 155 
Coulomb's law, 100 
Curvature, of continuum, 117, 128, 

129 ff., 292 
of space, 132,135 

Dalton, J., 16 

de Broglie, L., 193 ff., 203 

waves, 203, 206 ff. 
Democritus, 15 

Demon, sorting, 270, 271, 275, 277 
Descartes, causality, 38 

mind and matter, 37, 75, 281 

primary qualities, 14, 75, 283 

space and ether, 75, 97 

vision, 11 
de Sitter, W., 138 
Determinism, 36, 43, 227, 253 ff., 


Differential co-efficient, 194 
Dirac, P. A. M., 43, 44, 56, 184, 218 

Eddington, A. S., 136 
Ehrenfest, P., 211, 238, 239 
Einstein, A., 5, 42, 56, 65, 84, 93, 
94, 97, 118, 144, 226, 227, 239, 
279, 282 

determinism, 227, 279 
free will, 279 
gravitation, 117 ff. 
quanta, 152, 239 
radiation, 227 
relativity, 3, 50, 51, 84, 93, 101, 

117, 138 

unitary field-theory, 126 
universe, 127 ff., 138, 144 
Electron, 17, 21, 147, 148, 286, 287, 

frequency of, 200, 207 




Electron, polarisation of, 250 
spinning, 250 

wave properties o 54, 66, 199, 
207, 210 ff., 220, 221, 236, 241, 
248, 284 
Entropy, 264 ff. 
Ether, luminiferous, 75, 76, 77, 

171, 175, 242 
Euclid, 119, 126 
Events, 11, 101, 102, 261 ff., 284 
Evolution, astronomical, 137, 138 

in nature, 107, 142 
Expanding Universe, 130 ff., 291 

Faraday, M., 84 
Format's principle, 120 
Fitzgerald, G. F., 89, 90 
Fitzgerald-Lorentz contraction, 89 
Force (mechanical), 34, 35 
Franck and Hertz, 53, 168, 226, 242 
Free vibrations, 162, 164, 166, 189, 


Free will, 36, 278 
Frequency, 151 
Fresnel, 23 

Galileo, 34, 123, 190 

Gamow, 245 

Gases, theory of, 16, 146, 155, 156 

Generalised theory of relativity, 

50 ff., 117 

Gerlach, Stern and, 251 
Gilbert, W. S., 90 
Goudsmit, 250 
Gravitation (Einstein), 50, 51, 117 

(Newton), 49, 51, 100, 116 
Greek science, 34, 48, 73, 74 
Greek views of nature, 3"frrvsmj 34 

atomism, 15 

time and space, 73 

Harmonics, 163, 167 
Heisenberg, W., 3, 5, 42, 170 ff., 

193, 196, 197, 203, 256, 282 
Helium atom, 17, 54, 147 
Hero of Alexandria, 119, 126, 289 
Hertz (Franck and), 53, 168, 226, 


Hesiod, 74 
Homer, 33, 73, 74 
Huygens, 35 

Hydrogen atom, spectrum, 52, 54, 

167, 181 

structure, 17, 52, 242, 243 
structure (wave-mechanics), 242, 

Idealism, 15, 68, 296 
Indeterminacy, 230 ff., 257 

Heisenberg's principle of, 232. 


Indeterminism, 227, 228, 257 
Intensity of radiation, 156 
Interference (of light), 120, 215 
Interval, in continuum, 101, 102, 
104 ff. 

principle of least, 125, 293 

Jowett, B., 73 

Kant, L, causality, 227 

epistemology, 292 

space, 73, 97, 129 
Kelvin, Lord, 60 
Kepler, J., 49 
Kinetic theory, of gases, 16, 146, 

of radiation, 155 

Laplace, P. S., 58 

Lavoisier, 16 

Leibnitz, 38, 288 

Lemaitre, G., 130, 137, 144 

Leverrier, U.J.J.,50 

Life, .activities of, 273, 274 

Light, nature of, 19, 22, 24 ff., 63, 

particle-theory of, 23, 24 ff., 83, 

122, 189, 225 . 
quanta of Einstein, 152 
reflection of, 18, 119 
undulatory theory of, 19, 23ff. 

63, 83, 120, 122, 160, 241 
Local time of Lorentz, 91, 93, 94 
Locke, primary qualities, 14, 283 
Lorentz, H. A., 86, 87, 88, 89, 90, 

91, 92, 93, 100 
Lorentz-transformation, 86, 91, 97, 


Magnetic Induction, 86, 91, 92 
Materialism, 31, 64, 260 



Matrix (in mathematics), 178, 182, 


Matter, 11, 12, 112, 288 
primary qualities of, 13, 284 
reality of, 288 
secondary qualities of, 13 
structure of, 16 flf., 112, 146 
Maupertuis, 123 
Maximum Entropy, states of, 266, 

Maxwell, J. C., electromagnetic 

theory, 84, 86, 253 
sorting demon of, 270, 271, 274, 


theory of gases, 16, 146, 155, 156 
Mechanical views of nature, 35, 43, 
64, 170, 171, 190, 191, 294, 

Mercator projection, 114 ff., 127 
Mercury, motion of planets, 50 
Michelson-Morley experiment, 

80 ff., 93, 139 
Mind-body bridge, 12, 280 
Minkowski, H., 97, 99, 100 
Molecule, 15, 16 
Monochromatic light, 20 
Morley (see Michelson) 

Nature, mechanical views of, 35, 

43, 64, 170, 171 
objective, 1, 4, 67, 249, 282 ff., 

subjective, 2, 4, 65, 67, 68, 283 ff., 


uniformity of, 7, 35, 229 
Nebular motions, 136 
Newton, Isaac, 43, 97, 119, 225 
astronomy, 35, 49 
determinism, 225, 227 
gravitation, 49, 51, 100, 116 
mechanics, 14, 35, 43, 44, 123, 

190, 191 
optics, 23, 225 
relativity, 77, 85, 88 

Observables and Unobservables, 

170, 282, 283, 284 
Osculations of a solid, 188 ff. 

Particle picture, indeterminacy of, 
255, 256, 257 

Particle picture, of electron, 251. 
284, 285, 286 

of radiation, 22, 122, 159, 166 
Pearson,|Karl, 56 
Phase of vibrations, 121, 180 
Philosophical theories, idealist, 15, 
67, 68, 296 

materialist, 31, 64, 260 

mentalist, 15, 296 

objective, 1, 67, 68, 282 

realist, 68, 296 

subjective, 2, 67, 68, 282 
Photography, 151 
Photon, 25 ff., 52, 83, 152, 173, 174, 
214 ff., 286, 287 

frequency and wave-length of, 

waves of, 210, 214, 215, 238, 241 
Planck, M., 150, 278, 279, 280, 294 

constant of, 190, 232 ff. 

determinism, 278, 279 

free will, 278 

law of radiation, 156 

quanta, 150 
Plato, change, 109 ff. 

space, 74, 140, 141, 144 

time, 109, 110 
PoincarS, H., 42 
Poisson-brackets, 184 ff. 
Polarisation, of electrons, 250 

of radiation, 158, 250 
Primary qualities, 13, 14, 15, 20 ff., 

75, 283 
Probability, 223 ff., 252 

objective, 224 

subjective, 223, 237 

waves of, 62, 218, 236, 252 
Proton, 17, 21 

frequency of, 201, 207 

wave properties of, 54, 66, 198, 

Pythagoras, 48 

Quantum theory, 52, 54, 150, 189, 

257, 278, 279 
Quotations t 
Bradley, F. H., 4, 40, 68, 109 ff, 


Cornford, F. M., 73 
Descartes, 38 ff. 
Dirac, P. A. M., 44, 56 



Quotations (cont.): 
Einstein, A., 56, 227 ff., 279, 


Heisenberg, W., 3, 256 ff. 
Huygcns, 35 
Jowett, B., 73 
Newton, I., 43 fit, 77, 85 
Pearson, Karl, 56 
Planck, M., 278 ff. 
Plato, 74, 144 ff. 
Russell, Bertrand, 288, 289 
Schopenhauer, 280 
Sidgwick, H., 96 
Weyl, H., 228, 279, 289 
Whitehead, A. N., 40 ff. 

Radiation, 25, 26, 149, 154 
black-body, 150, 156, 157, 227, 

Radioactivity, 226, 227, 245, 256 

Radium, 226, 227 

Rainbow, analogy of, 2, 109 
mechanism of 3, 13, 20, 47 

Random walk, problem of the, 255 

Realism, 68, 296 

Reflection of light, 19, 119, 150 

Relativity, 1, 3, 14, 15, 50, 93, 285 
generalised (gravitation), 50, 51, 

mass, momentum and energy, 
152, 199, 200, 209 

Newtonian, 77, 85, 88 

restricted, 93, 117 
Retina of eye, 28, 71 
Ritz, principle of, 167, 168 
Rotations of atoms and molecules, 


Russell, Bertrand, 288, 289 
Rutherford, Lord, 226 

Schopenhauer, 280 
Schrodinger, E., 193, 203 
equation of, 203, 217, 218, 220, 

238, 255 

equation of (atom), 242 
equation of (particle), 247 
Secondary qualities, 13 
Sensation, threshold of, 230 
Sense-impressions, 6, 11, 21, 276, 

Senses, operation of, 8, 21, 27, 29 

Shadows, 23 

Soddy, F., 226 

Sorting demon of Maxwell, 270 ff 

Space, 96, 139, 140, 214, 290 ff. 

curvature of, 113 

Euclidean, 113ff. 

rudimentary views of, 70, 96, 214 
Space-like intervals, 105 
Space-time, 101, 252 

representation in, 252 
Spectra, atomic, 52, 166 ff., 180 
Spectroscope, 20, 31 
Spuming electron, 250 
Spontaneous disintegration (radio- 
active), 226 

Statistical atom, 183, 188 
Stefan's law, 157 
Stern and Gerlach, 251 
Sunlight, nature of, 18, 23, 26 ff. 

Temperature radiation, 150, 156, 

157, 227, 256 
Thermodynamics, 261 ff. 
Thomson, G. P., 198 
Thomson, J. J., 147 
Threshold of sensation, 230 
Time-like interval, 105 
Time, rudimentary views of, 70 ff., 

synchronisation, 78 

Uhlenbeck, 250 

Uncertainty principle of Heisen- 
berg, 232, 235, 285 

Undulatory theory of light, 19, 
23 ff., 63, 83, 120, 122, 160, 

Uniformity of Nature, 7, 35, 229 
Unitary field theory (Einstein), 

Universe (Einstein), 127 ff., 138, 

expanding, 130ff.,291 
Unobservables, 170, 174, 175, 263, 
283, 284 

Vision, act of, 11, 28 

Wave-equation, of electron, 202, 

217 ff., 247 
of photon, 213 ff. 



Wave-length, 20, 26 

of electron, 198, 199 

of photon, 213 ff. 
Wave-mechanics, 54, 193 
Wave packet, 205, 242, 248 
Wave picture, of particles, 194, 
235 ff., 247, 284 

of photons, 210, 213 ff., 238, 241 
Weyl, H., 228, 279, 289 
Whitehead, A. N. t 40 ff. 

White light, 18 

Wiener, N., 193 

Wireless transmission, 26, 30, 31, 

World-line, 101 ff., 286 

X-radiation, 151, 155 

Zeeman effect, 250 
Zero-interval, 105