(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Lectures and essays"

THE LIBRARY 
OF 

THE UNIVERSITY 

OF CALIFORNIA 

LOS ANGELES 



LECTURES AND ESSAYS 








< 



London, PuttosTuJ, fyJfaaff 



LECTURES AND ESSAYS 



BY THE LATE 



WILLIAM KINGDON CLIFFORD, F.R.S. 

LATE PROFESSOR OF APPLIED MATHEMATICS AND MECHANICS 

IN UNIVERSITY COLLEGE, LONDON J AND SOMETIME 

FELLOW OF TRINITY COLLEGE, CAMBRIDGE 



EDITED BY 

LESLIE STEPHEN 

AND 

SIR FREDERICK POLLOCK 



" La verite est toute pour tous." PAUL-LOUIS COURIER 



IN TWO VOLUMES 
VOL. I 



iLontJon 
MACMILLAN AND CO., LIMITED 

NEW YORK : THE MACMILLAN COMPANY 
1901 



First Edition, a Vols. &vo. ifyg. 
Second Edition, i. Vol. Crown Zvo. 1886. 
Third Edition, Eversley Series, 2 Vols. Globe Zvo. 



Eh 

Sciu 



% 



CONTENTS 



INTRODUCTION 

PART PACK 

I. BlOGRAPHICAI I 

II. SELECTIONS FROM LETTERS, ETC. . . . 56 



LECTURES AND ESSAYS 

ON SOME OF THE CONDITIONS OF MENTAL DEVELOP- 
MENT. , . ,_ v . ^ . ,, . 79 
ON THEORIES OF THE PHYSICAL FORCES . . .120 
ON THE AIMS AND INSTRUMENTS OF SCIENTIFIC 

THOUGHT . ...=.'. . . . 139 

ATOMS. . . . ,..., , .. , < :. . . . 181 

THE FIRST AND THE LAST CATASTROPHE . . . 222 

THE UNSEEN UNIVERSE 268 

THE PHILOSOPHY OF THE PURE SCIENCES . . 301 



if 



INTRODUCTION 

PART I 
BIOGRAPHICAL 1 

IT is an open secret to the few who know it, 
but a mystery and a stumbling-block to the 
many, that Science and Poetry are own sisters ; 
insomuch that in those branches of scientific in- 
quiry which are most abstract, most formal, and 
most remote from the grasp of the ordinary 
sensible imagination, a higher power of 
imagination akin to the creative insight of 
the poet is most needed and most fruitful of 
lasting work. This living and constructive 
energy projects itself out into the world at the 
same time that it assimilates the surrounding 
world to itself. When it is joined with quick 
perception and delicate sympathies, it can work 
the miracle of piercing the barrier that separates 

1 Written in 1879. A few sentences have now (1886) been 
added. Some verbal alterations, mostly rendered necessary by the 
lapse of time, will explain themselves. F. P. 

VOL. I B 



2 INTRODUCTION 

one mind from another, and becomes a personal 
charm. It can be known only in its operation, 
and is by its very nature incommunicable and 
indescribable. Yet this faculty, when a man is 
gifted with it, seems to gather up the best of 
his life, so that the man always transcends 
every work shapen and sent forth by him ; his 
presence is full of it, and it lightens the air his 
friends breathe ; it commands not verbal assent 
to propositions or intellectual acquiescence in 
arguments, but the conviction of being in the 
sphere of a vital force for which nature must 
make room. Therefore when, being happy in 
that we knew and saw these things, and have 
received the imperishable gifts, we must un- 
happily speak of the friend who gave them as 
having passed from us, it becomes nothing less 
than a duty to attempt the impossible task, to 
describe that which admits of no description, 
and communicate that for which words are but 
blundering messengers. And perhaps it may 
not be in vain ; for a voice which is in itself 
weak may strengthen the kindred notes that 
vibrate in other memories touched by the same 
power, and those we know to be very many. 
For this power, when it works for fellowship 
and not ambition, wins for its wearer the love 
of all sorts and conditions of men, and this was 
marked in Clifford by all who had to do with 
him even a little. More than this, our words 



BIOGRAPHICAL 3 

may peradventure strike further, though by no 
force or skill of their own, and stir some new 
accord in imaginations favourably attuned for 
the impulse. The discourses and writings 
collected in this book will indeed testify to the 
intellectual grasp and acuteness that went to 
the making of them. Clifford's earnestness and 
simplicity, 'too, are fairly enough presented to 
the reader, and the clearness of his expression 
is such that any comment by way of mere ex- 
planation would be impertinent. But of the 
winning felicity of his manner, the varied and 
flexible play of his thought, the almost bound- 
less range of his human interests and sym- 
pathies, his writing tells at least, so it seems 
to those who really knew him nothing or very 
little. To say a word or two in remembrance 
of one's friend is but natural ; and in these 
days excuse is hardly needed for saying it in 
public. But here this is the least part of the 
matter in hand. Personal desires and aims are 
merged in the higher responsibility of telling 
the world that it has lost a man of genius ; a 
responsibility which must be accepted even with 
the knowledge that it cannot be adequately 
discharged. 

Not many weeks had passed of my first 
year at Trinity when it began to be noised 
about that among the new minor scholars there 
was a young man of extraordinary mathematical 



4 INTRODUCTION 

powers, and eccentric in appearance, habits, and 
opinions. He was reputed, and at the time 
with truth, an ardent High Churchman. I 
think it was then a more remarkable thing at 
Cambridge than it would be now, the evangelical 
tradition of Simeon and his school being still 
prevalent This was the first I heard of Clifford ; 
and for some two years he continued to be 
nothing more to me than a name and a some- 
what enigmatic person. In the course of our 
third year circumstances brought us together : 
it is difficult to remember the beginnings of a 
friendship that seems as if it must always have 
been, but to the best of my recollection there 
was nothing very sudden or rapid in our closer 
approach. I should assign about six months 
as the interval filled by the transition from 
acquaintance to intimacy. At an early stage 
in my knowledge of him I remember being 
struck by the daring versatility of his talk. 
Even then there was no subject on which he 
was not ready with something in point, generally 
of an unexpected kind ; and his unsurpassed 
power of mathematical exposition was already 
longing to find exercise. I shall be pardoned 
for giving a concrete instance which may be in 
itself trivial. In the analytical treatment of 
statics there occurs a proposition called Ivory's 
Theorem concerning the attractions of an ellip- 
soid. The text -books demonstrate it by a 



BIOGRAPHICAL 5 

formidable apparatus of co-ordinates and in- 
tegrals, such as we were wont to call a grind. 
On a certain day in the Long Vacation of 1 866, 
which Clifford and I spent at Cambridge, I was 
not a little exercised by the theorem in question, 
as I suppose many students have been before and 
since. The chain of symbolic proof seemed 
artificial and dead ; it compelled the under- 
standing but failed to satisfy the reason. After 
reading and learning the proposition one still 
failed to see what it was all about. Being out 
for a walk with Clifford, I opened my per- 
plexities to him ; I think I can recall the very 
spot. What he said I do not remember in 
detail, which is not surprising, as I have had no 
occasion to remember anything about Ivory's 
Theorem these twelve years. But I know that 
as he spoke he appeared not to be working out 
a question, but simply telling what he saw. 
Without any diagram or symbolic aid he 
described the geometrical conditions on which 
the solution depended, and they seemed to 
stand out visibly in space. There was no 
longer consequences to be deduced, but real 
and evident facts which only required to be 
seen. And this one instance, fixed in my 
memory as the first that came to my know- 
ledge, represents both Clifford's theory of what 
teaching ought to be, and his constant way of 
carrying it out in his discourses and conversa- 



6 INTRODUCTION 

tion on mathematical and scientific subjects. 
So whole and complete was the vision that for 
the time the only strange thing was that any- 
body should fail to see it in the same way. 
When one endeavoured to call it up again, and 
not till then, it became clear that the magic of 
genius had been at work, and that the common 
sight had been raised to that higher perception 
by the power which makes and transforms 
ideas, th^ conquering and masterful quality of 
the human mind which Goethe called in one 
Word das Damonische. 

A soul eager for new mastery and ever 
looking forward cares little to dwell upon the 
past ; and Clifford was not much apt to speak 
of his own earlier life, or indeed of himself at 
all. Hence I am indebted to his wife and to 
other friends for what little I am able to say of 
the time before I knew him. William Kingdon 
Clifford was born at Exeter on May 4, 1845 ; 
his father was a well-known and active citizen, 
and filled the office of justice of the peace. 
His mother he lost early in life ; he inherited 
from her probably some of his genius, and 
almost certainly the deep-seated constitutional 
weakness, ill paired with restless activity of 
nerve and brain, which was the cause of his 
premature loss. He was educated at Exeter 
till 1 860, when he was sent to King's College, 
London, not without distinction already won in 



BIOGRAPHICAL 7 

the University Local Examinations. At school 
he showed little taste for the ordinary games, 
but made himself proficient in gymnastics ; a 
pursuit which at Cambridge he carried out, in 
fellowship with a few like-minded companions, 
not only into the performance of the most 
difficult feats habitual to the gymnasium, but 
into the invention of other new and adventurous 
ones. But (as he once said himself of Dr. 
Whewell) his nature was to touch nothing 
without leaving some stamp of invention upon 
it. His accomplishments of this kind were the 
only ones in which he ever manifested pride. 
When he took his degree there was a paragraph 
in Bell's Life pointing out, for the rebuke of those 
who might suppose manly exercises incompatible 
with intellectual distinction, that the Second 
Wrangler, Mr. Clifford, was also one of the most 
daring athletes of the University. This paragraph 
gave him far more lively pleasure than any of 
the more serious and academical marks of 
approval which he had earned. In 1 869 he 
wrote from Cambridge : " I am at present in 
a very heaven of joy because my corkscrew 
was encored last night at the assault of arms : 
it consists in running at a fixed 'upright pole 
which you seize with both hands and spin 
round and round descending in a corkscrew 
fashion." In after years he did not keep up 
his gymnastic practice with anything like 



regularity; but he was with great difficulty 
induced to accept the necessity of completely 
abandoning it when it was known to be posi- 
tively injurious to his health. A friend who 
was his companion in gymnastics writes to 
me : " His neatness and dexterity were un- 
usually great, but the most remarkable thing 
was his great strength as compared with his 
weight, as shown in some exercises. At one 
time he could pull up on the bar with either 
hand, which is well known to be one of the 
greatest feats of strength. His nerve at 
dangerous heights was extraordinary. I am 
appalled now to think 'that he climbed up and 
sat on the cross bars of the weathercock on a 
church tower, and when by way of doing some- 
thing worse I went up and hung by my toes to 
the bars he did the same." 

At King's College Clifford's peculiar mathe- 
matical abilities came to the front, but not so 
as to exclude attention to other subjects. He 
was at various times and in various ways marked 
out for honourable mention in classics, modern 
history, and English literature. His knowledge 
of the classics, though he did not cultivate the 
niceties of scholarship, was certainly as sound 
and extensive as that of many professedly 
classical students ; and, like all his knowledge, 
it was vital. If he made use of it for quota- 
tion or otherwise, it was not because the 



BIOGRAPHICAL 9 

passage or circumstance was classical, but 
because it was the thing he wanted to illustrate 
his own thought. Of history he knew a good 
deal ; he was fond of historical reading 
throughout his life, and had a ready corr.mand 
of parallels and analogies between widely 
remote times and countries, sometimes too 
ingenious to bear criticism. I doubt if he 
studied historical works critically ; it seems to 
me that he regarded history in a poetical rather 
than a scientific spirit, seeing events in a series 
of vivid pictures which had the force of present 
realities as each came in turn before the mind's 
eye. Thus he threw himself into the past with 
a dramatic interest and looked on the civilised 
world as a field where the destinies of man are 
fought out in a secular contest between the 
powers of good and evil, rather than as a scene 
of the development and interaction of infinite 
and infinitely complex motives. This indeed, 
in a meagre and far cruder form, is essentially 
the popular view ; the sort of history upon 
which most people are still brought up divides 
men, actions, and institutions into good and 
bad according to the writer's present notions 
of what might and ought to be, and distributes 
blessing and cursing without more ado. 1 Only 
Clifford, accepting to some extent the popular 
or pictorial way of looking at history, took on 

1 As children learning history say " But was he a good man ? " 



io INTRODUCTION 

most questions the unpopular side, and so 
found himself in collision with current opinions. 
He had a fair general knowledge of English 
literature (by which I mean considerably more 
than is yet supposed necessary for an English- 
man's education), with a preference for modern 
poetry, and especially for such as gave expres- 
sion to his own ideas. Milton's prose had also 
a special attraction for him. I do not think 
he cared much for the use of language as a 
fine art, though he had a great appreciation of 
arrangement and composition. His own style, 
always admirably clear and often eloquent, was 
never elaborate ; for we cannot fairly count the 
studied ornament of his College declamations, 
which were not only produced while he was 
an undergraduate, but for an occasion which 
justified some special aiming at rhetorical effect. 
Much of his best work was actually spoken 
before it was written. He gave most of his 
public lectures with no visible preparation 
beyond very short notes, and the outline 
seemed to be filled in without effort or hesita- 
tion. Afterwards he would revise the lecture 
from a shorthand-writer's report, or sometimes 
write down from memory almost exactly what 
he had said. It fell out now and then, how- 
ever, that neither of these things was done ; 
and in such cases there is now no record of 
the lecture at all. Once or twice he tried 



BIOGRAPHICAL 11 

writing part of the lecture beforehand, but 
found it only an embarrassment in the delivery. 
I believe the only one wholly put in writing in 
the first instance was " Ethics of Religion," 
which he was unable to deliver himself. I 
cannot find anything showing early aptitude 
for acquiring languages ; but that he had it 
and was fond of exercising it in later life is 
certain. One practical reason for it was the 
desire of being able to read mathematical 
papers in foreign journals ; but this would not 
account for his taking up Spanish, of which he 
acquired a competent knowledge in the course 
of a tour to the Pyrenees. When he was at 
Algiers in 1876 he began Arabic, and made 
progress enough to follow in a general way 
a course of lessons given in that language. 
He read modern Greek fluently, and at one 
time he was curious about Sanskrit. He even 
spent some time on hieroglyphics. A new 
language is a riddle before it is conquered, a 
power in the hand afterwards : to Clifford 
every riddle was a challenge, and every chance 
of new power a divine opportunity to bd seized. 
Hence he was likewise interested in the various 
modes of conveying and expressing language 
invented for special purposes, such as the 
Morse alphabet and shorthand. One of his 
ideas about education was that children might 
learn these things at an early age, perhaps in 



12 INTRODUCTION 

play, so as to grow up no less familiar with 
them than with common printing and writing. 
I have forgotten to mention his command of 
French and German, the former of which he 
knew very well, and the latter quite sufficiently ; 
I think his German reading was mostly in the 
direction of philosophy and mathematics. 

In 1863 Clifford came up with a minor 
scholarship to Trinity College, Cambridge ; in 
his third year (to continue for the present on 
the line of his literary accomplishments) he 
won the College declamation prize 1 with a 
very brilliant discourse on Sir W. Raleigh, 
partly cast in the form of quasi - dramatic 
dialogues, and accordingly had to deliver the 
annual oration at the Commemoration of 
Benefactors in December. His subject was a 
panegyric of the late Master of the College, 
Dr. Whewell, whose death was then recent It 
was treated in an original and unexpected 
manner, Dr. Whewell's claim to admiration 
and emulation being put on the ground of his 
intellectual life exemplifying in an eminent 
degree the active and creating faculty. 
" Thought is powerless except it make some- 
thing outside of itself: the thought which 
conquers the world is not contemplative but 

1 He was bracketed with Mr. C. A. Elliott for the first prize ; 
but (I now forget for what reason) the office of delivering the 
Oration fell to Clifford alone. 



BIOGRAPHICAL 13 

active. And it is this that I am asking you 
to worship to-day." Taking this oration as a 
whole, it must be considered as a tour de force, 
giving glimpses and undetermined promises of 
speculative power. But there occurred in it an 
apologue which caught the attention of some 
good judges at the time, and so well illustrates 
the fanciful and sportive side of Clifford's mind 
that I shall here transcribe it. 

" Once upon a time much longer than six 
thousand years ago the Trilobites were the 
only people that had eyes ; and they were only 
just beginning to have them, and some even 
of the Trilobites had as yet no signs of coming 
sight. So that the utmost they could know 
was that they were living in darkness, and that 
perhaps there was such a thing as light. But at 
last one of them got so far advanced that when 
he happened to come to the top of the water 
in the daytime he saw the sun. So he went 
down and told the others that in general the 
world was light, but there was one great light 
which caused it all. Then they killed him for 
disturbing the commonwealth ; but they con- 
sidered it impious to doubt that in general the 
world was light, and that there was one great 
light which caused it all. And they had great 
disputes about the manner in which they had 
come to know this. Afterwards another of 
them got so far advanced that when he 



, 4 INTRODUCTION 

happened to come to the top of the water in 
the night-time he saw the stars. So he went 
down and told the others that in general the 
world was dark, but that nevertheless there 
was a great number of little lights in it. Then 
they killed him for maintaining false doctrines : 
but from that time there was a division amongst 
them, and all the Trilobites were split into two 
parties, some maintaining one thing and some 
the other, until such time as so many of them 
had learned to see that there could be no doubt 
about the matter." 

The interpretation was barely indicated on 
this occasion ; but it is worked out in another 
Cambridge MS. of somewhat later date, 1 in 
which the apologue stands first as a kind of 
text It was nothing less than a theory of the 
intellectual growth of mankind ; and the posi- 
tion was that, as the physical senses have been 
gradually developed out of confused and un- 
certain impressions, so a set of intellectual 
senses or insights are still in course of develop- 
ment, the operation of which may ultimately be 
expected to be as certain and immediate as our 
ordinary sense-perceptions. 

This theory may be traced in the discourse 

1 It has now (1886) been ascertained that this MS., which was 
found among Clifford's papers fairly written out, but without title 
or indication of date, was used for a lecture delivered to a military 
audience at Woolwich in 1869. Still the ideas distinctly belong 
to an early and tentative stage. 



BIOGRAPHICAL 15 

" On some of the Conditions of Mental Develop- 
ment," delivered in March 1868, which stands 
first in the present collection ; and for that 
reason I make special mention of it. Other- 
wise it was only one inventive experiment 
among many. I should far exceed my limits 
if I were to attempt any account of the various 
forms of speculation, physical, metaphysical, 
social, and ethical, through which Clifford 
ranged in the first few years after his degree. 
Not that he was constantly changing his 
opinions, as a superficial observer might have 
thought ; he was seeking for definite principles, 
and of set purpose made his search various and 
widespread. He had a singular power of taking 
up any theory that seemed to him at all worth 
investigating, realising it, working it out, and 
making it completely his own for the time being, 
and yet all the while consciously holding it as 
an experiment, and being perfectly ready to 
give it up when found wanting. 

Clifford's mathematical course at Cambridge 
was a struggle between the exigencies of the 
Tripos and his native bent for independent 
reading and research going far beyond the sub- 
jects of the examination ; and the Tripos had 
very much the worst of it. If there was any 
faculty in which he was entirely wanting, it was 
the examination-faculty. On this subject I am 
not competent to speak with certainty, but it is 



,6 INTRODUCTION 

my belief that, from the point of view to which 
the class-list is an end in itself, Clifford omitted 
most of the things he ought to have read, and 
read everything he ought not to have read. 
Nevertheless his powers of original work carried 
him so far that he came out Second Wrangler 
in the Tripos of 1867, and was also Second 
Smith's Prizeman. I am fortunately able to 
quote on this head the statement of one of our 
first living analysts, Professor Sylvester : 

" Like the late Dr. Whewell, Professor 
Clerk Maxwell, and Sir William Thomson, 
Mr. Clifford was Second Wrangler at the 
University of Cambridge. I believe there is 
little doubt that he might easily have been first 
of his year had he chosen to devote himself ex- 
clusively to the University curriculum instead 
of pursuing his studies, while still an under- 
graduate, in a more extended field, and with 
a view rather to self-culture than to the acquisi- 
tion of immediate honour or emolument." 

This pursuit of knowledge for its own sake, 
and without even such regard to collateral in- 
terests as most people would think a matter of 
common prudence, was the leading character of 
Clifford's work throughout his life. The dis- 
covery of truth was for him an end in itself, 
and the proclamation of it, or of whatever 
seemed to lead to it, a duty of primary and 
paramount obligation. This had something to 



BIOGRAPHICAL 17 

do with the fascination of his teaching ; he 
never seemed to be imposing dogmas on his 
hearers, but to be leading them into the enjoy- 
ment of a common possession. He did not 
tell them that knowledge was priceless and 
truth beautiful ; he made them feel it. He gave 
them not formulas, but ideas. Again I can 
appeal to a witness of undoubted authority. 
The following words were written in 1871 by 
a man who was in no way given to unmeasured 
expression of his mind, and who was as eminent 
in mathematical physics as the author of the 
statement I have already cited is in pure mathe- 
matics I mean Clerk Maxwell : 

" The peculiarity of Mr. Clifford's researches, 
which in my opinion points him out as the 
right man for a chair of mathematical science, 
is that they tend not to the elaboration of 
abstruse theorems by ingenious calculations, 
but to the elucidation of scientific ideas by the 
concentration upon them of clear and steady 
thought. The pupils of such a teacher not 
only obtain clearer views of the subjects taught, 
but are encouraged to cultivate in themselves 
that power of thought which is so liable to be 
neglected amidst the appliances of education." 

I shall not attempt to enter in more detail 
on the amount and character of Clifford's sub- 
sequent contributions to mathematical science. 
But in an introduction to his philosophical 

VOL. i c 



,8 INTRODUCTION 

writings it is fitting to call attention to the 
manner in which he brought mathematical con- 
ceptions to bear upon philosophy. He took 
much pleasure in the speculative constructions 
of imaginary or non-Euclidean systems of space- 
relations which have been achieved by Conti- 
nental geometers, partly because they afforded 
a congenial field for the combined exercise of 
scientific intuition and unbridled fancy. He 
liked talking about imaginary geometry, as a 
matter of pure amusement, to any one interested 
in it. But at the same time he attached a 
serious import to it He was the first in 
this country, as Helmholtz in Germany, to call 
attention to the philosophical importance of 
these new ideas with regard to the question of 
the nature and origin of geometrical knowledge. 
His opinion on this point is briefly expressed 
in the lectures " On the Philosophy of the Pure 
Sciences." He intended to recast and expand 
these, and doubtless would have amplified this 
particular discussion. It will be seen that he 
considered Kant's position in the matter of 
"transcendental aesthetic" to be wholly un- 
assailable if it was once admitted that geo- 
metrical knowledge is really exact and universal. 
The ordinary arguments for the derivative nature 
of axioms appeared to him ingenious but hope- 
less attempts to escape from this fatal admission. 
And it may be said in general terms that he 



BIOGRAPHICAL 19 

had a much fuller appreciation of the merit 
and the necessity of Kant's work than most 
adherents of the English school of psychology. 
Of course I do not include Professor Huxley, 
whose testimony to Kant in his little book on 
Hume is as unmistakable as it is weighty. 

Few words will suffice to set down the re- 
maining facts of Clifford's life, or what we are 
accustomed to call facts because they can be 
dated and made equally known to everybody, 
as if that made them somehow more real than 
the passages and events which in truth decide 
the issues of life and fix the courses of a man's 
work. In 1868 he was elected a Fellow of 
Trinity College, and after spending rather more 
than two years at Cambridge, he was in 1871 
appointed to the Professorship of Applied 
Mathematics at University College, London. 
Meanwhile he had taken part in the English 
Eclipse expedition of 1870 : his letters of that 
time show keen enjoyment of the new ex- 
perience of men and cities, and of the natural 
beauty of the Mediterranean coasts, which he 
was to visit again, as fate would have it, only 
on the sad and fruitless errand of attempting 
to recover strength when it was too late. In 
June 1874 he was elected a Fellow of the 
Royal Society ; he might have been proposed 
at a much earlier time, but had then declined, 
turning it off with the remark that he did not 



20 INTRODUCTION 

want to be respectable yet And such was the 
absence in him of anything like vanity or self- 
assertion, that when his scruples were overcome, 
and his election took place, he was the last 
person from whom his friends heard of it. I 
did not know it myself till several months later. 
On April 7, 1875, he married Lucy, daughter 
of Mr. John Lane, formerly of Barbados. This 
was the occasion of the only voluntary leave of 
absence he ever took from his lectures at 
University College, when he characteristically 
informed his class that he was obliged to be 
absent on important business which would prob- 
ably not occur again. Clifford's house was 
thenceforward (as, indeed, his rooms, both at 
Cambridge and in London, had already been) 
the meeting -point of a numerous body of 
friends, in which almost every possible variety 
of taste and opinion was represented, and many 
of whom had nothing else in common. The 
scientific element had naturally a certain pre- 
dominance ; and with Clifford, as with other 
men, a close friendship implied, as a rule, some 
sort of general coincidence in sentiments and 
aims, personal and intellectual concord being 
apt to go together. But he cared for sympathy, 
not for agreement ; coincidence in actual results 
was indifferent to him. He wrote of a very 
near and dear friend (G. Crotch, of St. John's 
College, Cambridge), whose death preceded his 



BIOGRAPHICAL 21 

own by some years : " We never agreed upon 
results, but we always used the same method 
with the same object." Much more would it 
be a mistake to suppose that Clifford was a 
scientific fanatic who reserved his social 
qualities for such persons as happened to 
accept his theories, or that he could not be at 
his ease and make the charm of his presence 
felt among those who did not care for theories 
at all. It was possible to take offence at certain 
passages in his writings, but impossible not to 
like the man ; and some of those to whom 
Clifford's published opinions were naturally 
most repugnant, but who had the opportunity 
of personal intercourse with him, were by no 
means the last to express their sympathy and 
anxiety when the threatenings of the disease 
which carried him off became apparent. This 
charm remained with him to his very last days ; 
even when he was in an enfeebled and almost 
prostrate condition there were those who con- 
ceived for him and his, upon sudden and casual 
acquaintance, an affection and good-will which 
bore such fruit of kindly deeds as men usually 
look for only from the devotion ripened by 
long familiarity. Something of this was due 
to the extreme openness and candour of his 
conversation ; something to the quickness with 
which he read the feelings of others, and the 
delicacy and gentleness with which he adapted 



22 INTRODUCTION 

himself to them ; something, perhaps most, to 
a certain undefinable simplicity in which the 
whole man seemed to be revealed, and the 
whole moral beauty of his character to be 
grounded. It was by this simplicity, one may 
suppose, that he was endeared from his early 
days to children. He always took delight in 
being with them, and appeared to have a special 
gift of holding their attention. That he did 
not live to teach his own children is deeply to 
be regretted not only for their sake, but in 
the interest of education as a science and an 
art. What he could do for the amusement of 
children (and of all persons healthy enough not 
to be ashamed of childishness) was shown to 
the world in his contributions to a collection of 
fairy tales called The Little People. One of 
these ("The Giant's Shoes") is one of the 
choicest pieces of pure nonsense ever put 
together ; and he doubtless enjoyed writing it 
as much as any child could enjoy hearing it. 
A children's party was one of Clifford's greatest 
pleasures. At one such party he kept a wax- 
work show, children doing duty for the figures ; 
but he reproached himself for several days after- 
wards because he had forgotten to wind up the 
Siamese twins. He seemed to have an in- 
exhaustible store of merriment at all times : 
not merely a keen perception of the ludicrous, 
but an ever fresh gaiety and gladness in the 



BIOGRAPHICAL 23 

common pleasures of life. His laughter was 
free and clear like a child's, and as little re- 
strained by any consideration of conventional 
gravity. And he carried his mirth and humour 
into all departments of life, by no means ex- 
cepting philosophy. When he came home 
from the meetings of the Metaphysical Society 
(attending which was one of his greatest 
pleasures, and most reluctantly given up when 
going abroad after sunset was forbidden him), 
he would repeat the discussion almost at length, 
giving not only the matter but the manner of 
what had been said by every speaker, and now 
and then making his report extremely comic 
by a touch of plausible fiction. There was an 
irresistible affectation of innocence in his manner 
of telling an absurd story, as if the drollery of 
it were an accident with which he had nothing 
to do. It was hardly possible to be depressed 
in his company : and this was so not only in 
his best days, but as long as he had strength 
to sustain conversation at all. The charm of 
his countenance and talk banished for the time 
the anxiety we felt for him (only too justly) 
whenever we were not with him. 

On the intellectual side this character of 
simplicity manifested itself in the absolute 
straightforwardness of everything he said and 
did ; and this, being joined to subtlety and a 
wide range of vision, became in speculation 



24 INTRODUCTION 

and discussion a very formidable power. If 
there was anything for which he had no tolera- 
tion, and with which he would enter into no 
compromise, it was insincerity in thought, word, 
or deed. He expressed his own opinions 
plainly and strongly because he held it the 
duty of every man so to do ; he could not 
discuss great subjects in a half-hearted fashion 
under a system of mutual conventions. As for 
considerations of policy or expediency that 
seemed to interfere in any way with the 
downright speaking of truth for the truth's 
sake, he was simply incapable of entertaining 
them. " A question of right and wrong," he 
once wrote to me, " knows neither time, place, 
nor expediency." Being always frank, he was 
at times indiscreet ; but consummate discretion 
has never yet been recognised as a necessary 
or even a very appropriate element of moral 
heroism. This must be borne in mind in esti- 
mating such passages of his writings as, judged 
by the ordinary rules] of literary etiquette, may 
seem harsh and violent 

Personal enmity was a thing impossible to 
Clifford. Once he wrote : " A great misfortune 

has fallen upon me ; I shook hands with . 

I believe if all the murderers and all the priests 
and all the liars in the world were united into 
one man, and he came suddenly upon me round 
a corner and said, ' How do you do ?' in a 



BIOGRAPHICAL 25 

smiling way, I could hot be rude to him upon 
the instant." And it was the bare truth. 
Neither did he ever make an enemy that I 
know of ; I do not count one or two blundering 
attacks which, however far they might go beyond 
the fair bounds of controversy or satire, were 
made by people who only guessed at the man 
from a superficial inspection of his writings, 
and were incapable of understanding either. 
Yet he carried about with him as deadly a foe 
as could have been wished him by any of those 
who fear and hate the light he strove so man- 
fully to spread abroad. This was the perilous 
excess in his own frame of nervous energy over 
constitutional strength and endurance. He was 
able to call upon himself, with a facility which 
in the result was fatal, for the expenditure of 
power in ways and to an extent which only a 
strong constitution could have permanently 
supported ; and here the constitution was feeble. 
He tried experiments on himself when he ought 
to have been taking precautions. He thought, 
I believe, that he was really training his body 
to versatility and disregard of circumstances, 
and fancied himself to be making investments 
when he was in fact living on his capital. At 
Cambridge he would constantly sit up most of 
the night working or talking. In London it 
was not very different, and once or twice he 
wrote the whole night through ; and this without 



36 INTRODUCTION 

any proportionate reduction of his occupations 
in more usual hours. The paper on "The 
Unseen Universe " was composed in this way, 
except a page or two at the beginning, at a 
single sitting which lasted from a quarter to 
ten in the evening till nine o'clock the following 
morning. So, too, was the article on Virchow's 
address. But Clifford's rashness extended 
much further than this one particular. He 
could not be induced, or only with the utmost 
difficulty, to pay even moderate attention to 
the cautions and observances which are 
commonly and aptly described as taking care 
of one's self. Had he been asked if it was 
wrong to neglect the conditions of health in 
one's own person, as well as to approve or 
tolerate their neglect on a larger scale, he 
would certainly have answered yes. But to be 
careful about himself was a thing that never 
occurred to him. Even when, in the spring 
of 1 876, distinct and grave indications of 
pulmonary disease were noted, his advisers and 
friends could hardly persuade him that there 
was anything more serious than could be set 
right by two or three weeks' rest in the country. 
Here, however, there came into play something 
more than incredulity or indifference ; the spirit 
of the worker and inventor rebelled against 
thus being baffled. His repugnance was like 
that of a wounded soldier who thinks himself 



BIOGRAPHICAL 27 

dishonoured if he quits the field while his limbs 
can bear him. Reluctantly and almost in- 
dignantly he accepted six months' leave of 
absence, and spent the summer of that year in 
a journey to Algiers and the south of Spain. 
He came back recruited for the time, and was 
allowed to winter in England on pledges of 
special care and avoidance of exposure. These 
were in the main observed, and so matters went 
on for a year and a half more, as it seemed 
with fair prospects of ultimate recovery and 
tolerably secure enjoyment of life. What 
mischief was already done could not be undone ; 
but the spread of it seemed in a way to be 
permanently arrested. But in the early months 
of 1878 there came a sudden change for the 
worse. His father's death, which happened at 
this time, was a grievous blow, and the conjunc- 
tion of this with exciting literary work, done 
under pressure of time, threw upon him a strain 
which he was wholly unable to resist. The 
essay on Virchow's address, which closes the 
present collection, is both in my opinion and 
in that of other and more competent judges 
one of Clifford's best and most mature perform- 
ances. But it was produced at a fearful cost, 
we have already seen in what manner. A few 
days after the MS. had left his hands he 
received a peremptory warning that he was in 
a state of such imminent danger that he must 



28 INTRODUCTION 

give up all work and leave England forthwith. 
This time the warning was too stern to admit 
of doubt or even delay. Yet, while the neces- 
sary preparations were in hand, he would not 
leave his official duties until he actually broke 
down in the attempt to complete a lecture. 
He was now suffering, not from any inroad of 
specific local disease, but from a rapid and 
alarming collapse of general strength which 
made it seem doubtful if he could live many 
weeks. But his constitutional frailty was 
accompanied withal by a wonderful power of 
rallying from prostration ; and one could not 
help entertaining a dim hope, even to the last, 
that this vitality was somehow the deepest 
thing in his nature, and would in the long run 
win the day. In April that year, Clifford and 
his wife left England for the Mediterranean ; 
the accounts they sent home were various and 
often anxious ; but after voyages and short 
halts which embraced Gibraltar, Venice, and 
Malta, they rested for some weeks at Monte 
Generoso, and there for the first time there was 
the appearance of steady improvement setting 
in. From this place Clifford wrote long letters 
with his own hand, full of his usual spirit and 
manifold interest in everything about him. I 
may mention here that his letters were the 
more valuable because they were always spon- 
taneous and could seldom be counted on before- 



BIOGRAPHICAL 29 

hand. He wrote quickly and easily ; and yet 
for some obscure reason letter-writing, especially 
as a matter of business, was beyond measure 
irksome and difficult to him. He would rather 
take almost any trouble than answer a letter, 
and the painfulness of answering was at its 
height when (as pretty often happened) old 
acquaintances applied to him for testimonials. 
For in this case it was aggravated by the utter 
impossibility of lending himself to the petty 
exaggerations and dissimulations which custom 
allows to pass current for such purposes, and 
which are almost thought to be required by 
civility. One such application, from a man he 
had known before but had lost sight of, vexed 
him extremely ; he did not know what to do 
with it, for he could honestly have certified only 
as to the past, and he carried the letter about 
with him till it was ragged, being newly vexed 
every time he saw it. There were many 
letters of friends which he regretted to the last 
not having answered. Several received in the 
last months or weeks of his life he intended to 
answer if he had ever become strong enough. 
Yet now and then he would write unsought to 
some one he was intimate with, and throw him- 
self completely into his letter ; and then his 
descriptions were so full of life and colour that 
they might well be taken as models by any one 
minded to study the art of correspondence, not 



30 INTRODUCTION 

uncommonly alleged to be lost since the intro- 
duction of cheap and rapid communications. 
Such letters he sent to England from Spain and 
Sicily in 1870, and from Algiers in 1876. 
Some of them are printed farther on. 

In August 1878, there being signs of im- 
provement, and a warm climate not being 
judged necessary or very desirable at that 
season, leave was given for a short return to 
England. Clifford came home looking very ill 
and feeble to ordinary observation, but much 
better to those who had seen him before he 
started. He was incapable of continuous exer- 
tion of any kind, but much of the old animation 
had come back, and his conversation had lost 
nothing of its vigour and brilliancy. The 
object of the summer journey had been rest 
and freedom from care above all things : now 
it was planned that with the first days of 
autumn he should again go in search of condi- 
tions which might be not only rest-giving but 
curative. But all plans were cut short by a 
relapse which took place late in September, 
induced by fatigue. From that day the fight 
was a losing one, though fought with such 
tenacity of life that sometimes the inevitable 
end seemed as if it might yet be put far off. 
Clifford's patience, cheerfulness, unselfishness, 
and continued interest in his friends and in 
what was going on in the world, were unbroken 



BIOGRAPHICAL 31 

and unabated through all that heavy time. 
Far be it from me, as it was far from him, to 
grudge to any man or woman the hope or 
comfort that may be found in sincere expecta- 
tion of a better life to come. But (let this be 
set down and remembered, plainly and openly, 
for the instruction and rebuke of those who 
fancy that their dogmas have a monopoly of 
happiness, and will not face the fact that there 
are true men, ay and women, to whom the 
dignity of manhood and the fellowship of this 
life, undazzled by the magic of any revelation, 
unholpen of any promises holding out aught as 
higher or more enduring than the fruition of 
human love and the fulfilment of human duties, 
are sufficient to bear the weight of both life 
and death. Here was a man who utterly 
dismissed from his thoughts, as being unprofit- 
able or worse, all speculations on a future or 
unseen world ; a man to whom life was holy 
and precious, a thing not to be despised, but to 
be used with joyfulness ; a soul full of life and 
light, ever longing for activity, ever counting 
what was achieved as not worthy to be reckoned 
in comparison of what was left to do. And this 
is the witness of his ending, that as never man 
loved life more, so never man feared death less. 
He fulfilled well and truly that great saying of 
Spinoza, often in his mind and on his lips : Homo 
liber de nulla re minus quam de morte cogitat. 



32 INTRODUCTION 

One last stand was made, too late to be 
permanently successful (if ever it could have so 
far availed), but yet not wholly in vain. At 
the opening of the year 1 879 Clifford's remnant 
of strength was visibly diminishing. The peril 
of attempting a journey was great, but no peril 
could be greater than that which he already lay 
in. Medicine had no new thing to recommend, 
and almost nothing to forbid : a last experiment 
could only be tried. Clifford sailed for Madeira, 
his friends hardly expecting him to live out the 
voyage. Of the friendship and devotion that 
accompanied and tended him there it is not 
fitting that I should speak. So it was, how- 
ever, that he arrived safely in the island, and 
some weeks were added to his life. The 
change from the bitterest of recent English 
winters to the fair and temperate air of 
Madeira had no power to restore the waning 
forces ; but it enabled him to spend his last 
days in ease and comparative enjoyment. He 
could once more look on the glories of a bounti- 
ful world, and breathe under a free sky. Some- 
thing of spirit and even of strength revived ; 
his powers of conversation, which had been 
restrained by mere physical weakness in his 
last days in England, returned to some extent, 
and in that short time, with all the disadvantages 
of a stranger and an invalid, he made new 
friends : one such (though in spirit not a 



BIOGRAPHICAL 33 

stranger before) of whose friendship even he 
might have been proud. There was a glimmer 
of hope, faint, uncertain, but perceptible ; there 
was a possibility that if amendment once began, 
it might go further than we had dared to 
speculate upon. But it was not to be. In 
the last days of February we learnt that his 
condition was hopeless; on the 3rd of March the 
end came. For a week he had known that it 
might come at any moment, and looked to it 
steadfastly. So calmly had he received the 
warning which conveyed this knowledge that it 
seemed at the instant as if he did not understand 
it. He gave careful and exact directions as to 
the disposal of his works, which are partly 
carried out in this volume, and have been sub- 
stantially fulfilled as to his mathematical 
remains also. His work was, indeed, the only 
thing personal to himself that he took much 
thought for ; and that not because it was his 
own possession, but because he felt that it was 
his own to do and to make a possession for 
others. He loved it for the work's and the 
truth's sake, not for his own. More than 
this, his interest in the outer world, his affec- 
tion for his friends and his pleasure in their 
pleasures, did not desert him to the very 
last. He still followed the course of events, 
and asked for public news on the morning 
of his death : so strongly did he hold fast his 
VOL. I D 



34 INTRODUCTION 

part in the common weal and in active social 
life. 

It has been mentioned how unwilling Clifford 
was to throw up, even under necessity, his work 
at University College. His friends and col- 
leagues there were equally unwilling to lose 
him ; and when it became evident that he 
could never permanently resume his lectures, 
they still cast about for means to retain him as 
one of their number. In 1879 the Senate, in 
reviewing the whole question of the teaching of 
mathematics and physics, recommended that 
Clifford should "remain in possession of his 
chair, and that if, against the expectation, but 
in accordance with the most earnest desire of 
his colleagues, he should so far recover health 
as to be able to lecture, he should be invited to 
lecture upon special subjects in mathematics, to 
which he could bring his own rare qualities of 
mind without being subjected to any strain of 
constant necessary work." This recommenda- 
tion only awaited the assent of the Council to 
take effect, and that assent would almost 
certainly have been given ; but before the 
matter could be submitted to the Council it 
was known that the time of expectation was 
over, and desire quelled by the final certainty 
of loss. 

The essays here brought together represent, 
with few if any exceptions, the general view of 



BIOGRAPHICAL 35 

the world and human knowledge which Clifford 
had definitely arrived at in his later years. I do 
not mean that he had got a fixed set of results 
and meant to rest in them ; he admitted no 
finality of that sort. But he did believe very 
decidedly that the difference between right and 
wrong method is everywhere important, and 
that there is only one right method for all de- 
partments of knowledge. He held that meta- 
physical and theological problems ought to be 
discussed with exactly the same freedom from 
preconceived conclusions and fearlessness of 
consequences as any other problems. And he 
further held that, as the frank application of the 
right method of search to the physical sciences 
has put them on a footing of steady progress, 
though they differ in the amount and certainty 
of the knowledge already won in their respective 
fields, so the like effects might be expected 
when philosophical speculation was taken in 
hand by the light of science and with scientific 
impartiality and earnestness. For the popular 
or unscientific rhetoric which frequently assumes 
the name of philosophy Clifford had as much 
contempt as he permitted himself to feel for 
anything. Once he said of an acquaintance 
who was believed to be undertaking something 
in this kind : " He is writing a book on meta- 
physics, and is really cut out for it ; the clear- 
ness with which he thinks he understands 



3 6 INTRODUCTION 

things and his total inability to express what 
little he knows will make his fortune as a 
philosopher." But he never accepted, and I do 
not think he was ever tempted to accept, the 
doctrine that all metaphysical inquiries ought 
to be put aside as unprofitable. Indeed he 
went beyond most English psychologists, 
though in a general way he must be classed 
with the English school, in his estimate of the 
possibility of constructing a definite meta- 
physical system on scientific principles. With 
regard to the application of his philosophical 
ideas to theological conceptions, it may perhaps 
be said that he aimed at doing for dogmatic 
and natural theology something like what the 
Tubingen school in Germany have done for 
historical theology, namely, bringing them to the 
light of unbiassed common sense, including 
therein as an important element the healthy 
moral sense of civilised men. Whether Clifford 
had any feeling that his line of work was com- 
plementary to the historical criticism of dogmas 
I cannot say : but so it was that he paid no 
special attention to the historical side of these 
questions, either because it did not particularly 
interest him, or because he thought it outside 
his competence. In ethics, on the other hand, 
he attached the utmost importance to the 
historical facts of moral culture as affording the 
key of the speculative position and indicating 



BIOGRAPHICAL 37 

the profitable directions of inquiry. And it 
may be noted as an instance of the freshness 
and openness of his mind that the importance 
of this point of view, set forth in " The Scientific 
Basis of Morals" and the papers following it, 
was perceived by him only after he left Cam- 
bridge. The main points of the last-named 
essay were stated by Clifford himself in a letter 
written when he had nearly finished it. He 
described it as " showing that moral maxims 
are ultimately of the same nature as the maxims 
of any other craft : if you want to live together 
successfully, you must do so-and-so. . . . That 
conscience is developed out of experience by 
healthy natural processes. . . . That responsi- 
bility is founded on such order as we can 
observe, and not upon such disorder as we can 
conjecture." This is quite a different line from 
that which his speculations on the nature of 
duty were wont to take at Cambridge, both in 
the conversations I remember, and in various 
MS. fragments of that period which are now 
before me. 

A letter of the autumn of 1874, written by 
Clifford to his wife during their engagement, 
bears upon his practical conception of ethics 
and is otherwise interesting. " At the Savile 
I found C, who had just done dinner, but sat 
down while I ate mine, and we solved the 
universe with great delight until A. came in 



38 INTRODUCTION 

and wanted to take him off to explain coins to 
somebody. Of course I would not let him go. 
. . . We walked about in the New Road 
solving more universe. He says the people in 
the middle ages had a closer connection between 
theory and practice ; a fellow would get a 
practical idea into his head, be cock-sure it 
was right, and then get up and snort and just 
have it carried through. Nowadays we don't 
have prophets with the same fire and fervour 
and insight. To which it may be said that 
our problems are infinitely more complex, and 
that we can't be so cock-sure of the right thing 
to do. He quoted the statesmanship of the 
great emperors, e.g. Frederic II.; and some of 
the saints, as St. Francis and St. Catherine of 
Siena. Still there is room for some earnest 
person to go and preach around in a simple 
way the main straightforward rules that society 
has unconsciously worked out and that are 
floating in the air ; to do as well as possible 
what one can do best ; to work for the im- 
provement of the social organisation ; to seek 
earnestly after truth and only to accept pro- 
visionally opinions one has not inquired into ; 
to regard men as comrades in work and their 
freedom as a sacred thing ; in fact, to recognise 
the enormous and fearful difference between 
truth and falsehood, right and wrong, and how 
truth and right are to be got at by free inquiry 



BIOGRAPHICAL 39 

and the love of our comrades for their own 
sakes and nobody else's. Mazzini has done a 
great deal in this direction, and formed the 
conception of the world as a great workshop 
where we all have to do our best to make 
something good and beautiful with the help of 
the others. Such a preaching to the people of 
the ideas taught by the great Rabbis was (as 
near as we can make out) the sort of work that 
Christ did ; but he differed from the Rabbis 
and resembled all other Jew prophets in not 
being able to stand priests." 

It will not be amiss to go back to the time 
when we left Clifford celebrating the late Master 
of Trinity in parables, and to take up more 
continuously than we have yet done the growth 
of his philosophic ideas. Before he took his 
degree, and I think for some little time after, 
he was (as before mentioned) a High Churchman ; 
but there was an intellectual and speculative 
activity about his belief which made it impossible 
that it should remain permanently at that stage. 
On the one hand he acquired a far more ac- 
curate knowledge of Catholic theology than is 
often met with in England even among those 
who discuss theological questions ; he was pretty 
well read in St. Thomas Aquinas, and would 
maintain the Catholic position on most points 
with extreme ingenuity, not unfrequently adding 
scientific arguments and analogies of his own. 



4 o INTRODUCTION 

On the other hand, believing from the first 
in the unity or at least the harmony of all 
truth, he never slackened in the pursuit of 
scientific knowledge and ideas. For a while 
he experimented in schemes for the juxta- 
position of science and dogma. Religious 
beliefs he regarded as outside the region of 
scientific proof, even when they can be made 
highly probable by reasoning; for, as he 
observes in a MS. fragment of this time, they 
are received and held not as probable but as 
certain. And he actually defined superstition 
as "a belief held on religious or theological 
grounds, but capable of scientific proof or 
disproof." He also held that there was a 
special theological faculty or insight, analogous 
to the scientific, poetic, and artistic faculty ; 
and that the persons in whom this genius is 
exceptionally developed are the founders of 
new religions and religious orders. He seems 
to have been always and equally dissatisfied 
with attempts at proving theological pro- 
positions, especially in the usual manner of 
Protestant divinity, and with the theological 
version of natural history commonly called 
Natural Theology. There are indications in 
his note -books of that which might have 
become, under other conditions, a spiritual 
vocabulary no whit less original than William 
Blake's. Underlying all these experiments and 



BIOGRAPHICAL 41 

endeavours there was a permanent element of 
active intellectual faith by which Clifford was 
akin to a philosophic scholar in most external 
respects exceedingly unlike him, Mark Pattison. 
This faith is summed up by Pattison in a 
saying not known to Clifford, I think, in its 
terms, but wholly after his heart : " The learning 
of true propositions, dogmatically delivered, is 
not science." When or how Clifford first came 
to a clear perception that his position of quasi- 
scientific Catholicism was untenable I do not 
exactly know ; but I know that the discovery 
cost him an intellectual and moral struggle, 
of which traces may be found here and there 
in his essays. It is not the case, however, 
that there was any violent reaction or rushing 
to an opposite extreme. Some time elapsed 
before his philosophical opinions assumed their 
final consistency ; and in truth what took 
place was not a reaction, but the fuller develop- 
ment of principles which had been part of 
his thoughts ever since he began to think 
for himself. 

Meanwhile he was eagerly assimilating the 
ideas which had been established as an assured 
possession of biological science by Mr. Darwin, 
and the kindred ones already at an earlier time 
applied and still being applied to the framing 
of a constructive science of psychology, and to 
the systematic grouping and gathering together 



4* INTRODUCTION 

of human knowledge, by Mr. Herbert Spencer ; 
who had, in Clifford's own words, " formed the 
conception of evolution as the subject of general 
propositions applicable to all natural processes." 
Clifford was not content with merely giving his 
assent to the doctrine of evolution : he seized 
on it as a living spring of action, a principle 
to be worked out, practised upon, used to win 
victories over nature, and to put new vigour 
into speculation. For two or three years the 
knot of Cambridge friends of whom Clifford 
was the leading spirit were carried away by 
a wave of Darwinian enthusiasm : we seemed 
to ride triumphant on an ocean of new life and 
boundless possibilities. Natural Selection was 
to be the master-key of the universe ; we ex- 
pected it to solve all riddles and reconcile all 
contradictions. Among other things it was to 
give us a new system of ethics, combining the 
exactness of the utilitarian with the poetical 
ideals of the transcendentalist. We were not 
only to believe joyfully in the survival of the 
fittest, but to take an active and conscious 
part in making ourselves fitter. At one time 
Clifford held that it was worth our while to 
practise variation of set purpose ; not only to 
avoid being the slaves of custom, but to eschew 
fixed habits of every kind, and to try the 
greatest possible number of experiments in 
living to increase the chances of a really 



BIOGRAPHICAL 43 

valuable one occurring and being selected for 
preservation. So much of this theory as he 
ever gave to the world will be found in the 
discourse " On Some Conditions of Mental 
Development " ; and I do not know that he 
would ever have deliberately committed himself 
to anything more than is there propounded. 
One practical deduction was that education 
ought to be directed not to mere instruction, 
but to making people think and act for them- 
selves ; and this Clifford held to be of special 
importance in the case of women, where the 
cultivation of independent power is too com- 
monly neglected or even purposely discouraged. 
" It seems to me," he once wrote, " that the 
thing that is wanting in the education of women 
is not the acquaintance with any facts, but 
accurate and scientific habits of thought, and 
the courage to think that true which appears 
to be unlikely. And for supplying this want 
there is a special advantage in geometry, namely 
that it does not require study of a physically 
laborious kind, but rather that rapid intuition 
which women certainly possess ; so that it is 
fit to become a scientific pursuit for them." 

The duty of independence and spontaneous 
activity conceived by Clifford as being revealed 
by the philosophy of evolution was reinforced 
from another side by the reading of Mazzini ; 
and the result was a conception of freedom 



44 INTRODUCTION 

as the one aim and ideal of man. This freedom 
was a sort of transfigured blending of all powers 
of activity and progress ; it included republi- 
canism as opposed to the compulsory aspect 
of government and traditional authority in 
general, but was otherwise not bound to any 
particular theory in politics. Indeed it forbade 
binding one's self irrevocably to any theory 
whatever ; and the one commandment of freedom 
was thus expressed, Thou shalt live and not 
formulise. That alone was right which was 
done of one's own inner conviction and mere 
motion ; that was lifeless and evil which was 
done out of obedience to any external authority. 
"There is one thing in the world," Clifford 
wrote about this time, " more wicked than the 
desire to command, and that is the will to 
obey." Now this doctrine of individual and 
independent morality may look on the face 
of it anarchical, and therefore it may be worth 
while to observe that the Catholic doctrine of 
the duty of following conscience is essentially 
at one with it. The conscience may or may 
not be rightly informed. It may be wrongly 
informed without one's own fault, as in the 
case of invincible ignorance, or with it, as in 
the case of culpable ignorance or perversity. 
But even in this last case we are told that 
the sin of doing an absolutely wrong thing in 
obedience to the voice of conscience, however 



BIOGRAPHICAL 45 

misguided, is infinitely less than the sin of 
doing the absolutely right thing against one's 
conscience. The conscience must be rightly 
informed before a completely right action is 
possible. 1 Again, Fichte treats the sense of 
will and duty (from which he deduces not 
only morality but the existence of other men 
and of the world, in fact all knowledge and 
reality whatever) as absolutely personal and 
individual. Clifford's early doctrine of freedom 
was ardent and immature ; but whoever should 
call it immoral would find himself committed 
to applying the same language to some of 
the greatest moralists of the world. The social 
theory of morality stated and partly worked 
out in the ethical portion of Clifford's essays 
is quite independent of this earlier phase. At 
the same time it is not necessarily inconsistent 
with it ; for the determination of social morality 
is apart from the assignment of motives for 
individual morality, and leaves untouched the 
cultivation of individual perfection. Clifford, 
however, does in his later writings freely and 
distinctly recognise the validity of the social, 
or, as he sometimes calls it, the tribal judgment, 

1 See the authorities collected in Dr. Newman's Letter to the 
Duke of Norfolk, pp. 65, 66: " Secundum sententiam, et certam, 
asserentem esse peccatum discordare a conscientia erronea, in- 
vincibili aut vincibili, tenet D. Thomas, quern sequuntur omnes 
Scholastici. " " In no manner is it lawful to act against conscience, 
even though a law or a superior commands it." Some writers 
even say that this opinion is dejidt. 



46 INTRODUCTION 

on the moral character of individual acts re- 
garded aS an external quality ; and there was 
a time when he would probably have hesitated 
to allow this. 

In a note-book of Clifford's later Cambridge 
time there are some speculations on the com- 
pensating intellectual pleasures that help to 
break the shock of parting with old beliefs. 
I make an extract from one of these pages. 
"Whosoever has learnt either a language or 
the bicycle can testify to the wonderful sudden 
step from troublesome acquirement to the 
mastery of new powers, whose mere exercise 
is delightful, while it multiplies at once the 
intensity and the objects of our pleasures. 
This, I say, is especially and exceptionally true 
of the pleasures of perception. Every time 
that analysis strips from nature the gilding 
that we prized, she is forging thereout a new 
picture more glorious than before, to be suddenly 
revealed by the advent of a new sense whereby 
we see it a new creation, at sight of which 
the sons of God shall have cause to shout 
for joy. 

"What now shall I say of this new-grown 
perception of Law, which finds the infinite in 
a speck of dust, and the acts of eternity in 
every second of time? Why, that it kills 
our sense of the beautiful, and takes all the 
romance out of nature. And moreover that 



BIOGRAPHICAL 47 

it is nothing more than a combining and re- 
organising of our old experiences, never can 
give us anything really new, must progress in 
the same monotonous way for ever. But wait 
a moment. What if this combining and 
organising is to become first habitual, then 
organic and unconscious, so that the sense of 
law becomes a direct perception ? Shall we 
not then be really seeing something new ? 
Shall there not be a new revelation of a great 
and more perfect cosmos, a universe freshborn, 
a new heaven and a new earth ? Mors janua 
vita ; by death to this world we enter upon 
a new life in the next. A new Elysium opens 
to our eager feet, through whose wide fields 
we shall run with glee, stopping only to stare 
with delight and to cry, ' See there, how beauti- 
ful ! ' for the question, ' Why ? ' shall be very 
far off, and for a time shall lose its meaning." 

" For a time ? It may well be that the 
new world also shall die. Doubtless there 
shall by and by be laws as far transcending 
those we know as they do the simplest obser- 
vation. The new incarnation may need a 
second passion ; but evermore beyond it is 
the Easter glory." 

Even at the time of these half-poetical 
meditations I think Clifford must have felt 
them to be too poetical for scientific use. 
Later in life, as we have seen above and 



4 8 INTRODUCTION 

may see in the Essays, he chose to make sure 
of a solid foundation in experience at the 
cost of sacrificing ornament and rhetoric, and 
his admiration of Mazzini became compatible 
with practical empiricism in politics. " On the 
whole I feel confirmed," he wrote in a letter, 
" that the English distrust .of general principles 
in a very complex affair like politics is a sound 
scientific instinct, and that for some time we 
must go blundering on, finding out by ex- 
perience what things are to be let alone and 
what not." 

The command, " thou shalt not formulise," 
was expressed in an amusing shape in a review 
of Problems of Life and Mind, published in 
1 874. " Rules of philosophising are admirable 
things if two conditions are satisfied : first, 
you must philosophise before you make your 
rules ; secondly, you should publish them with 
a fond and fervent hope that no philosophiser 
will attend to them." 

As to Clifford's ideas on metaphysics proper 
I have not much to say beyond what is dis- 
closed in the Essays themselves. His interest 
in philosophy grew up rapidly after he took 
his degree, as is generally the case with men 
who have any bent that way. I remember 
many long talks with him on metaphysical 
questions, but not much of the substance of 
them. One evening in the Long Vacation of 



BIOGRAPHICAL 49 

1868, when we were up for the Fellowship 
examination, we discussed the Absolute for 
some couple of hours, and at last defined it 
to our own exceeding content as that which 
is in necessary relation to itself. Probably 
we laughed at our definition the next morning, 
or soon after ; but I am still of opinion that, 
as definitions of the Absolute go, this will do 
quite as well as any other. Clifford's philo- 
sophical reading was rather select than wide. 
He had a high admiration for Berkeley, next 
only to Hume, and even more, perhaps, for 
the Ethics of Spinoza. The interpretation of 
Spinoza's philosophy which I have put forward 
on one or two occasions was common to 
Clifford and myself, and on that subject (as, 
indeed, on everything we discussed together) 
I owe very much to him. He was to have 
lectured on Spinoza at the London Institution 
in 1877, but his health would not allow it. 
There is little doubt that this would have 
been one of his most brilliant and original 
discourses. Students of Spinoza will easily 
trace the connection between his theory of 
mind and matter and the doctrine set forth 
in Clifford's Essays on " Body and Mind," and 
' The Nature of Things-in-themselves." This 
was arrived at, to the best of my recollection, 
in 1871 or 1872; certainly before 1874, in 
which year the last-mentioned paper was read 
VOL. I E 



5 o INTRODUCTION 

at a meeting of the Metaphysical Society. 
Briefly put, the conception is that mind is 
the one ultimate reality ; not mind as we 
know it in the complex forms of conscious 
feeling and thought, but the simpler elements 
out of which thought and feeling are built 
up. The hypothetical ultimate element of 
mind, or atom of mind-stuff, precisely corre- 
sponds to the hypothetical atom of matter, 
being the ultimate fact of which the material 
atom is the phenomenon. Matter and the 
sensible universe are the relations between 
particular organisms, that is, mind organised 
into consciousness, and the rest of the world. 
This leads to results which would in a loose 
and popular sense be called materialist. But 
the theory must, as a metaphysical theory, 
be reckoned on the idealist side. To speak 
technically, it is an idealist monism. Indeed 
it is a very subtle form of idealism, and by 
no means easy of apprehension at first sight. 
Nevertheless there are distinct signs of a con- 
vergence towards it on the part of recent 
inquirers who have handled philosophical prob- 
lems in a scientific spirit, and particularly those 
who have studied psychology on the physio- 
logical side. Perhaps we shall be told that 
this proves the doctrine to be materialism in 
disguise ; but it is hardly worth while to dispute 
about names while more serious things remain 



BIOGRAPHICAL 51 

for discussion. And the idea does require 
much more working out ; involving, as it does, 
extensive restatement and rearrangement of 
metaphysical problems. It raises not only 
several questions, but preliminary (and really 
fundamental) problems as to what questions 
are reasonable. For instance, it may be asked 
why, on this hypothesis, mind should become 
conscious at a particular degree of complexity, 
or be conscious at all. I should myself say 
that I do not know and do not expect ever 
to know, and I believe Clifford would have 
said the same. But I can conceive some one 
taking up the theory and trying to make it 
carry further refinements and explanations. 
Again, a more subtle objection, but in my 
opinion a fallacious one, would be that it is 
not really a monism but a dualism, putting 
mind (as the undetermined mind-stuff} and 
consciousness in place of the old-fashioned 
matter and mind. This, however, is not the 
place to pursue the subject ; and I do not 
think the outline of the hypothesis can be made 
clearer by any explanation of mine than Clifford 
has already made it Looking back on this 
brilliant piece of speculation after seven years, 
I suppose my sight is more impartial. I alter 
nothing of what I wrote in the first edition, 
but feel bound in sincerity to add that I cannot 
now accept mind-stuff. The atom of mind- 



52 INTRODUCTION 

stuff is a " thing in itself" : Clifford so described 
it. But the purpose of modern philosophy is 
to abolish things in themselves. Kant proved 
them unknowable : the inevitable step onward 
is to cast them out as illusions, though Kant 
would not take it. By no amount of ingenious 
manipulation can psychology henceforth be 
made to serve instead of metaphysics. Mind 
per se, or mind-stuff, abstracted by Clifford's 
or any like method from the intelligible world, 
is no more intelligible than matter per se. 
We have simplified a scientific statement, not 
solved a philosophical problem. 

After all I have wished to speak of the man 
rather than his opinions ; but the speculative 
interests I shared with him, being in a manner 
part of himself, have claimed their due, and 
perhaps obtained rather more. Let us now 
gather up a few matters of personal habit and 
character which have not yet been noticed. 
The predominance of light as a figure and a 
symbol in Clifford's writing will be remarked : 
he associates it with the right and all things 
good so constantly and naturally that it is one 
of the marks of his style. He had physically 
a great love of light, and chose to write, when 
he could, in a clear and spacious room, with the 
windows quite free of curtains. Though he was 
not for most ordinary purposes a business-like 
man, and was careless of his own attire, he was 



BIOGRAPHICAL 53 

neat and exact in his literary work. He would 
not allow books to be misused or carelessly cut, 
and his own MS. was very fair, regular, and 
free from erasures. He was careful about 
punctuation, and insisted on having his own 
way in it, and he especially disliked superfluous 
commas. At the same time he was fond of 
handicraft, and his thoughts often ran upon 
mechanical invention. He speculated much on 
the practicability of constructing a flying machine, 
and began experiments at sundry times, which, 
however, never led to anything definite. Indeed 
it is pretty obvious that if a successful flying 
machine is ever made (and there is no impossi- 
bility in it), the inventor will be some one who 
combines theoretical knowledge of mechanics 
with familiar knowledge of machinery and the 
strength of materials and ready command of 
the various resources of engineering. At one 
time the notion of the flying machine turned 
Clifford's attention to kites, and this led to a 
ludicrous accident. It was in the Long Vacation 
of 1877, when Clifford and his wife were Mrs. 
Crawshay's guests in Wales. A kite of unusual 
dimensions, with tail in proportion, had been 
made ready for a flight which was to exceed 
everything achieved by kites before. It was to 
be flown with a great length of string, and it 
cost a morning's work to lay out the string in 
a field so that the kite might rise easily when 



54 INTRODUCTION 

started. Having accomplished this, the party 
went in to luncheon, and were presently called 
out by the announcement that a flock of sheep 
had been turned into the field. Clifford rushed 
out to prevent the disaster, but it was too late. 
Shepherd and sheep were caught as in a snare, 
and when they were extricated the string was 
left hopelessly entangled. Another piece of 
engineering undertaken at the same time and 
place was the construction of a duck-pond for 
the benefit of a family of ducklings who fre- 
quented a narrow ditch by the roadside. The 
little stream that trickled in the ditch was 
dammed according to the rules of art, and in 
course of time a complete pond was formed, 
and the ducks were happy for a season : till 
one day some over-zealous minister of local 
authority, conceiving the pond, as it was 
supposed, to be an encroachment on the high- 
way, restored the ancient state of things with a 
few strokes of the spade. Clifford regretted the 
duck-pond even more than the kite. Other 
amusing and characteristic anecdotes might be 
added ; but I forbear. 

No enumeration of tastes and occupations 
can adequately represent the variety and flexi- 
bility of Clifford's intellect, and still less the 
tender, imaginative, poetical side of his mind. 
Now and then he wrote verses in which this 
partly found expression. They were mostly of 



BIOGRAPHICAL 55 

a private or occasional nature, or else too 
fragmentary for publication. One very graceful 
song is to be found in the volume of fairy tales 
already spoken of. But the real expression of 
Clifford's varied and fascinating qualities was 
in his whole daily life and conversation, per- 
ceived and felt at every moment in his words 
and looks, and for that very reason impossible 
to describe. Nor can portraits go very far to 
supply that part of it which fell to the sight ; 
for the attractive animation and brightness of 
his countenance depended on very slight, subtle, 
and rapidly succeeding changes. His com- 
plexion was fair ; his figure slight, but well- 
knit and agile ; the hands small, and, for a man, 
singularly slender and finely formed. The 
features were of a massive and irregular type 
which may be called Socratic ; in a bust they 
might have looked stern, in the living face they 
had an aspect not only of intellectual beauty 
but of goodwill and gentle playfulness. But I 
began with declaring my task impossible, and 
at the end I feel still more keenly that all 
words fall short of what I would convey. The 
part has fallen to me of doing to a loved and 
honoured friend such honour as I could : the 
will at least will be accepted. 

Purpureos spargam flores . . et fungar inani 
munere. 



PART II 
SELECTIONS FROM LETTERS, ETC. 

THE following is a selection from letters written 
by Clifford at various times, partly to my mother 
and partly to myself. I begin with some philo- 
sophical passages. 

[To F. Pollock.} 

"Trinity College, Cambridge, April 2, 1870. 

" Several new ideas have come to me lately : 
first I have procured Lobatschewsky, tudes 
Gfomttrigues sur la Thtorie des Paralleles . . . 
a small tract, of which Gauss, therein quoted, 
says, c L'auteur a trait6 la matiere en main de 
maitre et avec le veritable esprit geomtrique. 
Je crois devoir appeler votre attention sur ce 
livre, dont la lecture ne peut manquer de vous 
causer le plus vif plaisir.' It is quite simple, 
merely Euclid without the vicious assumption, 
but the way the things come out of one another 
is quite lovely. . . . 



SELECTIONS FROM LETTERS, ETC. 57 

" I am a dogmatic nihilist, and shall say the 
brain is conscious if I like." (This in reply to 
some verbal criticism of mine.) " Only I do 
not say it in the same sense as that in which I 
say that / am conscious. It seems to me that 
not even Vogt, however you fix it, can talk 
about matter for scientific purposes except as a 
phenomenon ; that in saying the brain is con- 
scious or, better, that you are conscious, I only 
affirm a correlation of two phenomena, and am 
as ideal as I can be ; that, consequently, a true 
idealism does not want to be stated, and, con- 
versely, an idealism that requires to be stated 
must have something wrong about it. In the 
same way to say that there is God apart from 
the universe is to say that the universe is not 
God, or that there is no real God at all ; it may 
be all right, but it is atheism. And an idealism 
which can be denied by any significant aggrega- 
tion of words is no true idealism." 

The following is on the recent edition of 
Hume by Messrs. Green and Grose : 

[To F. Pollock^ 

"Exeter, September II, 1874. 

"... I hope you have seen Sidgwick's 
remarks (I think in the Academy] ; 1 he points 
out that to prove Hume insufficient is not to do 

1 May 30, 1874, vol. v. p. 608. 



58 INTRODUCTION 

much in the present day. It should, I think, 
be brought out clearly that if we pay attention 
only to the scientific or empirical school, the 
theory of consciousness and its relation to the 
nervous system has progressed in exactly the 
same way as any other scientific theory ; that 
no position once gained has ever been lost, and 
that each investigator has been able to say ' I 
don't know ' of the questions which lay beyond 
him without at all imperilling his own con- 
clusions. Green, for instance, points out that 
Hume has no complete theory of the object^ 
which is of course a very complex thing from 
the subjective point of view, because of the 
mixture of association and symbolic substitution 
in it ; and in fact I suppose this piece of work 
has not yet been satisfactorily done. But it 
seems merely perverse to say that the scientific 
method is a wrong one, because there is yet 
something for it to do ; and to find fault with 
Hume for the omission is like blaming Newton 
for not including Maxwell's Electricity in the 
Prtncipta" 

The following suggestions on education 
were sent from Algiers in June 1876 : 

[To F. Pollock.} 

"... I have a scheme which has been com- 
municated in part to Macmillan, and which 



SELECTIONS FROM LETTERS, ETC. 59 

grows like a snowball. It is founded on 
Pleasant Pages, the book I was taught out of; 
which is a series of ten minutes' lessons on the 
Pestalozzian plan of making the kids find out 
things for themselves : history of naughty boys 
on Monday, animals on Tuesday, bricks on 
Wednesday, Black Prince on Thursday, and so 
on. In the book it was very well done, by a 
man who had a genius for it. If you go to see 
Macmillan in Bedford Street he will show you 
the book, which he got on my recommendation 
he is also himself newly interested in the 
question. His partner Jack read part of it and 
was struck. Well, I first want that brought up 
to to-day, both in choice of subject and in 
accuracy ; adding, e.g. a series of object lessons 
on man (papa, mamma, house, street, clothes, 
shop, policeman, c wild and field '). Then I 
want it taught on the Russian system, in 
different languages on successive days ; no 
direct teaching of language until there are facts 
enough to make Grimm's law intelligible, for 
which English, German, and the Latin element 
in French would be enough ; no grammar at 
all till very late, and then as analysis of 
sentences and introductory to logic. This is 
the difficult part ; it would require a French 
and German teacher, both trained and com- 
petent, besides the English one. So far as the 
book is concerned, it would of course be easy to 



60 INTRODUCTION 

print it in the three languages. Lastly, I have 
bought twelve volumes of the Bibliotheque 
Nationale for three francs Rabelais, five 
volumes, and Montesquieu, Pascal, Diderot, and 
Vauvenargues. They are twenty-five centimes 
each, admirable for the pocket and of course 
you know them. There are two or three 
hundred volumes. Whereupon we must of 
course get the same thing done for English 
literature, and the setting forth of all literature 
in English (e.g. I have Les Maximes (tEpictete), 
but more particularly we must get published 
excellent little manuals at twopence or three- 
pence for the use of Board and other primary 
schools. I do not even know that penny 
schoolbooks would not be a successful move 
the size of a Daily News, say, printed by the 
million in a Walter press, folded and sewed by 
machinery to about the size of the Bibliotheque. 

" A Daily News would just make one of 
these volumes. Fancy the Penstes of Pascal, 
with the notes of Voltaire, Fontenelle, and 
Condorcet, a good life at the beginning, etc., 
all well printed on a sheet of the Daily News ! 
But of such a size could be made a very good 
elementary schoolbook of arithmetic, geometry, 
animals, plants, physics, etc. rather larger 
than Macmillan's primers, but of the same 
sort." 

The remaining letters and extracts are 



SELECTIONS FROM LETTERS, ETC. 61 

chiefly descriptive, and will be given without 
further remark, except such brief note of dates 
and circumstances as may seem necessary. 

[To Lady Pollock] 

"Cambridge, September 26, 1871. 

"... My ideal theory is quite different 
from yours. In the case of persons I worship 
the actual thing always ; this is the only way 
to be trusted. The one advantage of having 
indestructible family relations is that, whatever 
you do and whatever anybody thinks of you, 
there are always one or two people who will 
love you exactly as much as (if not more than) 
if you were blameless and universally respected. 
I used to recognise an exception, viz. that in 
certain cases what had been a person might 
cease to be one, and become a thing, towards 
which one could have no moral relations, and 
which might be set aside by safe means, or used 
as the occasion served. But the more people 
I know and the better I know each, the further 
off this possibility seems to be. I want to take 
up my cross and follow the true Christ, 
humanity ; to accept the facts as they are, 
however bitter or severe, to be a student and 
a lover, but never a lawgiver. But then besides 
this I do look for an ideal which is at some 
time to be created or awakened out of potenti- 



62 INTRODUCTION 

alities like the lady that Phantastes set free 
from the block of marble. Meanwhile I chip 
various blocks, and generally set free something ; 
not hitherto I think quite the right one ; when 
I do she will probably go straight off to some- 
body else. All this, by the way, is only theory ; 
my practice is just like other people's." 

[To Lady Pollock.} 

"Florence, December 1870. 

(Clifford was one of the English Eclipse 
expedition : the Psyche, with the expedition on 
board, struck on a rock near Catania. All 
hands and the instruments were saved, the ship 
was lost.) 

"No ink, no paper, no nothing Florence, 
Thursday 5th. The above 1 you guess. After 
that somehow to Catania, some in boats and 
some in holy carts of the country, all over 
saints in bright shawls well, if ever a ship- 
wreck was nicely and comfortably managed, 
without any fuss but I can't speak calmly 
about it because I am so angry at the idiots 
who failed to save the dear ship alas ! my 
heart's in the waters close by Polyphemus's 
eye, which we put out. At Catania, orange 
groves and telescopes ; thence to camp at 
Augusta ; Jonadab, son of Rechab, great fun, 

1 A grotesque fancy sketch of the shipwreck. 



SELECTIONS FROM LETTERS, ETC. 63 

natives kept off camp by a white cord ; 200 
always to see us wash in the morning a per- 
formance which never lost its charm only five 
seconds totality free from cloud, found polarisa- 
tion on moon's disk, agree with Pickering, other 
people successful. Then by Catania to Messina, 
no steamers, kept five days, Mediterranean 
stormy, we also at last to Naples, very bad 
night, everybody ill but me, and I have been 
out of sorts ever since. Called on Mrs. 
Somerville, and came on to Rome after seeing 
Pompeii. At Rome 2-^ days, pictures, statues, 
Coliseum by moonlight. Both of us sneezed 
awfully next morning. The shops are in the 
streets where the Tiber left them nice for 
purchasing but not so convenient for walking 
about. This morning arrive in Florence 
Pitti palace spent all my money, and shall 
get stranded between Cologne and Ostend 
unless I can live on one egg every other day, 
and thereout suck no small advantage, be 
better off in Paris. Addio." 

[To Lady Pollock] 

"Sunday, July 2, 1876. 

"This comes from Oran in the west of 
Algeria, a sad place, with too many Spaniards 
in it. We came here yesterday after a long 
and tiresome journey from Blidah, near Algiers. 



64 INTRODUCTION 

The train is somewhat amusing because the 
carriages are open at the ends and you can sit 
in the air as if it was a tram-car. You have 
then to be careful not to let the very large 
grasshoppers eat you up. Playfair, the English 
Consul at Algiers, told us to go to Bougie to 
see the gorge of the Chabet ; so we got a 
Murray's Guide and started off obediently. It 
was the steamer that had brought us from 
Marseilles, and the captain, who is very fond 
of us, gave us the ladies' cabin all to ourselves. 
There was on board a little Frenchman who 
had observed us in a restaurant at Algiers. 
He made great love to us, and said he wanted to 
marry an Englishwoman, but we think he lied 
a good deal about his town and country house, 
and his carriage and his good family. How- 
ever, he woke us up in time for the diligence at 
Bougie, and there is no harm in him, though 
indeed very little else. All this expedition 
was undertaken for the sake of the road from 
Bougie to Setif, and it was well worth it 
There is a narrow rent made by the stream 
which winds in and out for miles among the 
hills ; these are splendidly wooded, and rise to 
an enormous height on either side, while the 
torrent roars away down below. The road is 
cut in one side of the gorge. The cochon who 
drove the diligence tried every ruse to get us 
inside, that he might have a friend of his on 



SELECTIONS FROM LETTERS, ETC. 65 

the front seat ; but we stuck to our places till 
the scenery was finished, and then a great rain 
came and drenched both of them well. Setif 
is a complete French town, stuck in the middle 
of an African plain with its cafes and boulevards, 
just as if it had never lived anywhere else. We 
saw more Arabs there than anywhere else, and 
the native market pleased us much. On the 
way back we travelled with an Arab who had 
a gazelle in a basket which he was taking to 
somebody at Bougie ; he said you might buy 
them occasionally in the market at Se"tif for 
twenty-five francs ; we pitied the sweet little 
thing, which baaed like a sheep and struggled 
hard to get out, but he was pacified with some 
bread and some flowers which I had picked, 
and went to sleep with his head on my arm. 
On waking up he saw Lucy's straw hat near 
him and tried to eat it. We saw the most 
exquisite masses of maiden-hair fern, as large 
as the side of a room (the masses I mean, not 
the fern), where the streams came down near the 
side of the road. Our little Frenchman was 
still at Bougie and came back with us in the 
boat. The next day but one we had an amus- 
ing experience in the Jardin d'Acclimatation. 
We were taking coffee in an Arab cafe, and 
there was a boy there with an instrument of 
two strings, whose sounding board was made of 
bladder stretched over the shell of a tortoise 
VOL. I F 



66 INTRODUCTION 

quite the Apollo. We asked him to play 
something to us, and then a flute painted red 
and blue was given to an old man who had 
been smoking quite still. I couldn't make out 
the music because the little Frenchman kept 
on chattering ; but the old man gradually 
became excited ; he had been sitting European 
fashion with his feet on the ground, but one of 
his great toes got restive and then all the others, 
until his shoe was too much for that foot ; so 
he dropped the shoe and laid the foot on his 
knee, where it could wriggle comfortably. 
Then the other foot became excited and went 
through the same process. When his agony 
grew still more intense, he put one foot down 
and bent the shoe about with it to get more 
resistance. All this time the upper part of his 
body, except the fingers playing on the pipe, 
was perfectly still, and his face had a rapt 
expression. Meanwhile a pipe of kif had been 
got ready and was handed round, and a whiff 
of that seemed to calm him. I tried it also, 
and it brought the tears into my eyes, I was so 
nearly suffocated. I went to a lecture of the 
Arabic course which is given at Algiers in the 
Museum. It consisted in the translation of an 
article from a ' Constantinople paper, passages 
from which were written up on a black board, 
read out, and translated. The point of interest 
was the quotation from a passage in the Koran 



SELECTIONS FROM LETTERS, ETC. 67 

in support of the constitution, to the effect that 
'the Government shall not be absolute but 
consultative.' The lecturer said that absolutism 
was a Turkish institution, not Arabic, and that 
the Caliphate had been a sort of republic, with 
a president elected for life. Also that when a 
certain Caliph boasted that he had never 
swerved from the path of justice, a soldier 
looked up and said ' Inshallah ! (or words to 
that effect, meaning, By Jove !) our swords 
would have speedily brought you back.' This 
appears interesting if true. Already a Parisian 
scent is sold in the Moorish bazaars as a per- 
fume of the Sultana Valide. 

" We felt very much injured at only seeing 
two monkeys in the woods at La Chiffa the 
day before yesterday, but there were some 
green parrots on the bushes near the railway. 

" To-morrow we go by a Spanish boat to 
Almeira, and thence by diligence or another 
boat to Malaga. The Spanish boat will be 
nasty, but it is only twelve hours or so. I am 
very much better, and shall be glad of a rest 
at Granada after this gadding about. 

" P. S. I wrote to Fred about the education 
of our infants. I am very glad we have both 
begun with girls, because it will be so good for 
the other children to have an elder sister. How 
very fond those kids will be of each other and 
of Fred and me ! because girls always like their 



68 INTRODUCTION 

fathers best, you know. I have thought of a 
way to make them read and write shorthand 
by means of little sticks (not to whop them 
with but to put together on a table and make 
the shorthand signs). Ask G. whether she 
thinks they had better learn to sing on the sol- 
fa system ; it is very amusing and seems to me 
more adapted for children than the other. Of 
course I can teach them to stand on their 
heads. 

" We have seen the Spanish boat, which is 
called La Encarnacion, and that rightly ; for it 
is the incarnation of everything bad." 

[The Encarnacion aforesaid more than justi- 
fied the worst expectations : the engines broke 
down at sea, nobody on board was competent 
to repair them, and the ship lay helpless till a 
vessel was hailed which had a French engineer 
on board.] 

[To F. Pollock.} 

"Malaga, Saturday, July 15, 1876. 

"... As for this country, I think it re- 
quires to be colonised by the white man. 
The savages would gradually die out in his 
presence. The mark of a degraded race is 
clear upon their faces ; only the children have 
a look of honesty and intelligence, a fact which 
is also observed in the case of the negro, and is 
a case of Von Bar's law, that the development 



SELECTIONS FROM LETTERS, ETC. 69 

of the individual is an epitome of that of the 
race. It is instructive also to contrast the 
politeness fossilised in their language with the 
brutal coarseness of their present manners, of 
which I may some time tell you what I will 
not soil paper with. I think it possible that 
one Spaniard may have told me the truth : he 
had lost so many teeth that he left out all his 
consonants, and I could not understand a word 
he said. When we went on board the Rosario 
at 1 1 P.M. the boatman stood in the way to 
keep us from the ladder, and threatened us for 
the sake of another peseta over the regular 
charge. The steward tried to cheat me over 
the passage -money, but I appealed to the 
authorities who came on board at Malaga and 
got the money back (there are many strangers 
here). Then he made another grab in the 
matter of our breakfasts, in the face of a tariff 
hung up in the cabin. It is tiring to have to 
think that every man you meet is ready to be 
your enemy out of pure cussedness. I don't 
understand why one is expected to be polite 
and reticent about the distinction between the 
Hebrew piety and Roman universalism attri- 
buted to Jesus and Paul, and the ecclesiastical 
system which is only powerful over men's lives 
in Spain, the middle and south of Italy, and 
Greece countries where the population con- 
sists chiefly of habitual thieves and liars, who 



7 o INTRODUCTION 

are willing opportunely to become assassins for 
a small sum. I suppose it frightens people to 
be told that historical Christianity as a social 
system invariably makes men wicked when it 
has full swing. Then I think the sooner they 
are well frightened the better." 



[To F. Pollock.} 

" Washington Irving Hotel, Granada, 
August 3, 1876. 

" You are quite right, and one ought not to 
despair of the Republic. These folks are kind 
and rather pleasant when one is en rapport with 
them, and they have a deal of small talk. We 
found a jolly old couple one morning when we 
were coming back from a hot walk in the Vega 
of Almeira (vega = cultivated plain surrounding 
a town which feeds it) ; we asked for some 
milk, which they had not, but they gave us a 
rifresco of syrup and cold water, not at all bad, 
and the old woman showed Lucy all over her 
house while the man smoked a cigarette with 
me. Lucy's passport is the baby's portrait, 
with which she gains the hearts of all the 
women and most of the men. What made it 
more surprising was that they took us for Jews. 
Wilkinson, our Consul at Malaga, who has been 
here with his wife and daughter (awfully nice 



SELECTIONS FROM LETTERS, ETC. 71 

people and cheered us up no end), says that 
the country people are better than those in the 
towns. 

"... But although we have been nearly a 
fortnight at Granada, only one murder has been 
even attempted, so far as I know, within a 
hundred yards of the hotel. A. had been mak- 
ing love to B.'s wife, and so she was instructed to 
walkwithhim one eveningunderthese lovely trees. 
She took occasion to borrow his sword-stick, and 
stuck him in the back with it while her husband 
fired at his head with a revolver. One ball grazed 
his temple, and another went in at his cheek 
and out of his mouth, carrying away some 
teeth and lip. He came round to the Spanish 
hotel opposite and was tied up on the door- 
step ; they dared not let him come in because 
the police are so troublesome about these affairs. 
The defence was that A. was a Republican, and 
had been a Protestant ; so you see B.'s love of 
order was such that he did not think jealousy 
a sufficient justification. Wilkinson had just 
received a report of the last quarter of 1875 '> 
in those three months there had been only a 
few more than 400 murder cases in the whole 
province of Granada. The hot weather seems 
to try them ; a paragraph in the Malaga paper, 
headed ' Estadistico Criminal de Domingo, 30,' 
gives 1 5 cases of shooting and stabbing last 
Sunday in Malaga, but only five appear to have 



72 INTRODUCTION 

been fatal. This is not assassination, but is 
merely an accompaniment of their somewhat 
boisterous conviviality ; they get drunk together 
and then draw their knives and go in for a 
hacking match. It is not even quarrelling in 
all cases ; in Granada the other day three men 
shut themselves up and fought till they were 
all dead. They might, to be sure, have dis- 
liked each other mutually all round, but I am 
inclined to think it was a party of pleasure 
rather than of business. They do not attack 
strangers in this way (i.e. with knives and 
revolvers), unless, of course, there is a reason 
for it ; but when anything offends their delicate 
sense of propriety one cannot expect them not 
to show it a little. Thus they threw stones in 
Seville and Cordova at a lady who is now stay- 
ing here, because she went into the street by 
herself, and they do not approve of that. I am 
afraid my Norfolk jacket hurts their feelings 
in some way, but they have been very forbear- 
ing, and have only stoned me once, and then 
did not hit me. Another time a shopkeeper 
set his dog at me, but although this was rather 
alarming, with temperature 92 in the shade, it 
must have been meant as a joke, for Spanish 
dogs only bite cripples of their own species 
except, indeed, the great mastiffs that are kept 
to bait bulls that won't fight. Of course one 
is not so insular as to think there is only one 



SELECTIONS FROM LETTERS, ETC. 73 

way of giving a welcome to the stranger ; and 
the ' 'eave 'arf a brick at 'im ' method is im- 
proved by variety. What generally happens 
is this : the grown people stop suddenly at the 
sight of you, and wheel round, staring with 
open mouths until you are out of sight ; while 
the children, less weighted with the cares of this 
world, form a merry party and follow at your 
heels. When you go into a shop to buy any- 
thing, they crowd round the door so that it is 
rather difficult to get out. The beggars come 
inside and pull you by the arm while you are 
talking to the shopman. I have invented a 
mode of dealing with the crowd of children ; 
it is to sit on a chair in the shop door and 
tickle their noses with the end of my cane. I 
fear that universal sense of personal dignity 
which is so characteristic of this country is in 
some way injured by my familiarity ; the more 
so as it cannot be resented, for the other end 
of my cane is loaded, and I do not try it on in 
a macadamised street. Anyhow they go a 
little way off. In Malaga the people seemed 
more accustomed to the sight of strangers, and 
contented themselves with shouting abusive 
epithets. . . . Everybody says there will be a 
revolution before long. ... If Castelar returns 
to power, I hope among other little reforms 
that he will prevent the post-office officials from 
stealing letters for the sake of the stamps on 



74 INTRODUCTION 

them ; it is a great interruption to business 
and must be a laborious way of earning money. 
One of them was caught in Malaga because 
a packet of letters which he had thrown into 
the sea was accidentally fished up ; but 
he was shielded from punishment by the 
authorities. 

"We are very happy here, with a Swiss 
cook and an Italian landlord. There are some 
English, Germans, and Italians staying over the 
way, and in a few minutes we can be among 
the memorials of a better time. I am too tired 
now to talk about the Alhambra, but it seems 
to me to want that touch of barbarism which 
hangs about all Gothic buildings. One thinks 
in a Cathedral that since somebody has chosen 
to make it it is no doubt a very fine thing in 
its way ; but that, being a sane man, one would 
not make anything like it for any reasonable 
purpose. But the Alhambra gives one the 
feeling that one would wish to build something 
very like it, mutatis mutandis, and the more like 
it the more reasonable the purpose was. More- 
over, I think it must be beautiful, if anything 
ever was ; but then I have no taste." 

Clifford's verses, as has been said, were 
mostly fragmentary or intimate. Two songs, 
however, may here be given, of which one is 
unpublished elsewhere. 



SELECTIONS FROM LETTERS, ETC. 75 



Song from " The Little People? 

THIS is the song that Daisy sang j and it is 
about a water-lily bud that saw a reflection of 
herself in the surface of the water while she was 
under it. 

You grow through the water apace, lily ; 

You'll soon be as tall as the pond, 
There is fresh hope high in your face, lily, 
Your white face so firm and so fond. 
Ah, lily, white lily, 

What can you see 
Growing to meet lily 
Graciously ? 

There's a face looks down from the sky, lily ; 

It grows to me dim from above. 
If I ever can reach me so high, lily, 
I shall kiss ah ! the face of my love. 
Ah, lily, white lily, 
That can I see, 
Giving me light, lily, 
Lovingly. 

The lily-bud met with her mate, ah me ! 

And her flower came through to the air, 
And her bright face floated in state, ah me ! 
But the shadow-love never was there ! 
Ah, lily, great lily, 

Queenly and free, 
Float out your fate, lily, 
Friendlessly. 



INTRODUCTION 



Verses sent to George Eliot with a Copy of 
" The Little People? J 

Baby drew a little house, 

Drew it all askew ; 
Mother saw the crooked door 

And the window too. 

Mother-heart, whose wide embrace 

Holds the hearts of men, 
Grows with all our growing hopes, 

Gives them birth again. 

Listen to this baby-talk ; 

'Tisn't wise or clear ; 
But what baby-sense it has 

Is for you to hear. 

The bibliographical sketch of Clifford's work 
which formed part of this Introduction in the 
first edition is considered to have served its 
turn, and is not now reproduced. The editors 
have not received any later information capable 
of giving definite results. 

1 Now (1886) first printed. 



LECTURES AND ESSAYS 



ON SOME OF THE CONDITIONS OF 
MENTAL DEVELOPMENT 1 

IF you will carefully consider what it is that 
you have done most often during this day, I 
think you can hardly avoid being drawn to this 
conclusion : that you have really done nothing 
else from morning to night but change your 
mind. You began by waking up. Now that 
act of waking is itself a passage of the mind 
from an unconscious to a conscious state, which 
is about the greatest change that the mind can 
undergo. Your first idea upon waking was 
probably that you were going to rest for some 
time longer ; but this rapidly passed away, and 
was changed into a desire for action, which 
again transformed itself into volition, and pro- 
duced the physical act of getting up. From 
this arose a series of new sensations ; that is to 
say, a change of mind from the state of not 
perceiving or feeling these things to the state 
of feeling them. And so afterwards. Did you 

1 Discourse delivered at the Royal Institution, March 6, 1868. 



8o LECTURES AND ESSAYS 

perform any deliberate action ? There was the 
change of mind from indecision to decision, from 
decided desire to volition, from volition to act. 
Did you perform an impulsive action ? Here 
there is the more sudden and conspicuous 
change marked by the word impulsive ; as if 
your mind were a shuttlecock, which has its 
entire state of motion suddenly changed by the 
impulse of the battledore : conceive the shuttle- 
cock descending quite regularly with a gentle 
corkscrew motion the battledore intervenes 
instantaneously the shuttlecock flies off in a 
totally unexpected direction, having apparently 
no relation to its previous motion ; and you 
will see how very apt and expressive a simile 
you use when you speak of certain people as 
having an impulsive temperament. Have you 
felt happy or miserable ? It was a change in 
your way of looking at things in general ; a 
transition, as Spinoza says, from a lower to a 
higher state of perfection, or vice versa. In a 
word, whatever you have done, or felt, or thought, 
you will find upon reflection that you could not 
possibly be conscious of anything else than a 
change of mind. 

But then, you will be inclined to say, this 
change is only a small thing after all. It does 
not penetrate beyond the surface of the mind, 
so to speak. Your character, the general atti- 
tude which you take up with regard to circum- 



CONDITIONS OF MENTAL DEVELOPMENT 81 

stances outside, remains the same throughout 
the day : even for great numbers of days. You 
can distinguish between individual people to 
such an extent that you have a general idea of 
how a given person will act when placed in 
given circumstances. Now for this to be the 
case, it is clear that each person must have 
retained his individual character for a consider- 
able period, so as to enable you to take note of 
his behaviour in different cases, to frame some 
sort of general rules about it, and from them to 
calculate what he would do in any supposed 
given case. But is it true that this character 
or mark by which you know one person from 
another is absolutely fixed and unvarying ? Do 
you not speak of the character of a child growing 
into that of a man : of a man in new circum- 
stances being quite a different person from what 
he was before? Is it not regarded as the 
greatest stroke of art in a novelist that he 
should be able not merely to draw a character 
at any given time, but also to sketch the growth 
of it through the changing circumstances of life ? 
In fact, if you consider a little further, you will 
see that it is not even true that a character 
remains the same for a single day : every cir- 
cumstance, however trivial, that in any way 
affects the mind, leaves its mark, infinitely small 
it may be, imperceptible in itself, but yet more 
indelible than the stone-carved hieroglyphics of 
VOL. I G 



82 LECTURES AND ESSAYS 

Egypt. And the sum of all these marks is 
precisely what we call the character, which is 
thus itself a history of the entire previous life of 
the individual ; which is therefore continually 
being added to, continually growing, continually 
in a state of change. 

Let me illustrate this relation by the example 
of the motion of a planet. People knew, ages 
and ages ago, that a planet was a thing con- 
stantly moving about from one place to another ; 
and they made continual attempts to discover 
the character of its motion, so that by observ- 
ing the general way in which it went on, they 
might be able to tell where it would be at any 
particular time. And they invented most in- 
genious and complicated ways of expressing 
this character : 

" Cycle on epicycle, orb on orb," 

till a certain very profane king of Portugal, who 
was learning astronomy, said that if he had been 
present at the making of the Solar System, he 
would have tendered some good advice. But 
the fact was that they were all wrong, and the 
real case was by no means so complicated as 
they supposed it to be. Kepler was the first 
to discover what was the real character of a 
planetary orbit ; and he did this in the case of 
the planet Mars. He found that this planet 
moved in an ellipse or oval curve round the 



CONDITIONS OF MENTAL DEVELOPMENT 83 

sun which was situated rather askew near the 
middle. But upon further observation, this was 
found to be not quite exact ; the orbit itself is 
revolving slowly round the sun, it is getting elon- 
gated and then flattened in turns, and even the 
plane in which the motion takes place sways 
slowly from side to side of its mean position. 
Thus you see that although the elliptic character 
of the motion does represent it with consider- 
able exactness for a long time together, yet this 
character itself must be regarded as incessantly 
in a state of gradual change. But the great 
point of the comparison to aid in the concep- 
tion of which, in fact, I have used the compari- 
son at all is this : that for no two seconds 
together does any possible ellipse accurately 
represent the orbit. It is impossible for the 
planet to move a single inch on its way, without 
the oval having slightly turned round, become 
slightly elongated or shortened, and swayed 
slightly out of its plane ; so that the oval which 
accurately represented the motion at one end 
of the inch would not accurately represent the 
motion at the other end. The application is 
obvious. In like manner it is true that the 
character which will roughly represent the law 
of a man's actions for some considerable time, 
will not accurately represent that law for two 
seconds together. No action can take place in 
accordance with the character without modify- 



84 LECTURES AND ESSAYS 

ing the character itself; just as no motion of a 
planet could take place along its orbit without 
a simultaneous change in the orbit itself. 

But I will go even further. Historians are 
accustomed to say that at any given point of a 
nation's history there is a certain general type 
which prevails among the various changes of 
character which different men undergo. There 
is some kind of law, they say, which regulates 
the slow growth of each character from child- 
hood to age ; so that if you compared together 
all the biographies you would find a sort of 
family likeness suggesting that some common 
force had acted upon them all to make these 
changes. This force they call the Spirit of the 
Age. The spirit, then, which determines all 
the changes of character that take place, which 
is, therefore, more persistent than character 
itself, is this, at last, a thing absolutely fixed, 
permanent, free from fluctuations ? No : for 
the entire history of humanity is an account of 
its continual changes. It tells how there were 
great waves of change which spread from 
country to country, and swept over whole 
continents, and passed away ; to be succeeded 
by similar waves. No history can be philo- 
sophical which does not trace the origin and 
course of these : things far more important 
than all the kings and rulers and battles and 
dates which some people imagine to be history. 



CONDITIONS OF MENTAL DEVELOPMENT 85 

To recapitulate. The mind is changing so 
constantly that we only know it by its changes. 
The law of these changes, which we call char- 
acter, is also a thing which is continually 
changing, though more slowly. And that law 
of force which governs all the changes of 
character in a given people at a given time, 
which we call the Spirit of the Age, this also 
changes, though more slowly still. 

Now it is a belief which, whether true or 
not, we are all of us constantly acting upon, 
that these changes have some kind of fixed 
relation to the surrounding circumstances. In 
every part of our conduct towards other people 
we proceed constantly upon the assumption 
that what they will do is to a certain extent, 
and in some way or other, dependent upon 
what we do. If I want a man to treat me 
with kindness and respect, I have to behave in 
a certain way towards him. If I want to pro- 
duce a more special and defined effect, I have 
recourse to threats or promises. And even if 
I want to produce a certain change of mind in 
myself, I proceed upon the same assumption 
that in some way or other, and to a certain 
extent, I am dependent on the surrounding 
circumstances. People tie knots in their hand- 
kerchiefs to make themselves remember things ; 
they also read definite books with a view of 
putting themselves into definite mental states 



86 LECTURES AND ESSAYS 

or moods ; and attempts are constantly made 
to produce even a further and more permanent 
effect, to effect an alteration in character. 
What else is the meaning of schools, prisons, 
reformatories, and the like? Some have actually 
gone further than this : there have not been 
wanting enterprising and far-seeing statesmen 
who have attempted to control and direct the 
Spirit of the Age. Now in all these cases in 
which we use means to an end, we are clearly 
proceeding on the assumption that there is 
some fixed relation of cause and effect, in virtue 
of which the means we adopt may be ante- 
cedently expected to bring about the end we 
are in pursuit of. We are all along assuming, 
in fact, that changes of mind are connected by 
some fixed laws or relations with surrounding 
circumstances. Now this being so, since every 
mind is thus continually changing its character 
for better or worse, and since the character of 
a race or nation is subject to the same constant 
change ; since also these changes are connected 
in some definite manner with surrounding 
circumstances ; the question naturally presents 
itself, What is that attitude of mind which is 
likely to change for the better ? All the in- 
dividuals of a race are changing in character, 
all changing in different directions, with every 
possible degree of divergence ; also the average 
character itself, the Spirit of the Age, is either 



CONDITIONS OF MENTAL DEVELOPMENT 87 

changing in some one definite direction, or 
tending to split into two different characters : 
an individual, therefore, may be going with the 
race or dropping out of it ; a portion of the 
race may be going right or wrong. Let us 
suppose that some portion of the race is going 
right and improving : the question is, In what 
way are we to distinguish that individual who 
is improving with the race, from the others who 
are either dropping out of the march altogether 
or going wrong ? 

Now what I have proposed to myself to do 
to-night is this, merely to suggest a method 
by which this question may ultimately be 
answered. I shall also endeavour afterwards 
to point out what I conceive to be one or two 
results of this method : but this part will be of 
minor importance ; the results depend upon my 
application of the method, can be only partially 
true, and may be wholly false ; the method 
itself I believe to be altogether a true one, and 
one which must ultimately lead to the correct 
results. 

It consists in observing and making use of 
a certain analogy, namely, the analogy between 
the mind and the visible forms of organic life. 
You know that every animal and every plant 
is constantly going through a series of changes. 
The flower closes at night and opens in the 
morning ; trees are bare in winter and covered 



88 LECTURES AND ESSAYS 

with leaves in summer; while the growth of 
every organism from birth to maturity cannot 
fail to strike you as a forcible illustration of the 
gradual change of character in the human mind. 
In fact, it is the peculiarity of living things not 
merely that they change under the influence of 
surrounding circumstances, but that any change 
which takes place in them is not lost but re- 
tained, and, as it were, built into the organism 
to serve as the foundation for future actions. 
If you cause any distortion in the growth of a 
tree and make it crooked, whatever you may 
do afterwards to make the tree straight, the 
mark of your distortion is there ; it is absolutely 
indelible ; it has become part of the tree's nature, 
and will even be transmitted in some small de- 
gree to the seeds. Suppose, however, that you 
take a piece of inanimate matter a lump of 
gold, say, which is yellow and quite hard you 
melt it, and it becomes liquid and green. Here 
an enormous change has been produced ; but 
let it cool ; it returns to the solid and yellow 
condition, and looks precisely as before there 
is no trace whatever of the actions that have 
been going on. No one can tell by examining 
a piece of gold how often it has been melted 
and cooled in geologic ages by changes of the 
earth's crust, or even in the last year by the 
hand of man. Any one who cuts down an oak 
can tell by the rings in its trunk how many 



CONDITIONS OF MENTAL DEVELOPMENT 89 

times winter has frozen it into widowhood and 
summer has warmed it into life. A living 
being must always contain within itself the 
history not merely of its own existence but of 
all its ancestors. Seeing then that in its con- 
tinual changes and in the preservation of the 
records of those changes every organism re- 
sembles the mind, so that to this extent they 
belong to the same order of phenomena, may 
we not reasonably suppose that the laws of 
change are alike, if not identical, in the two 
cases ? This is of course a mere supposition, 
not deducible from anything which we have yet 
observed, which requires therefore to be tested 
by facts. I shall endeavour to show that the 
supposition is well founded ; that such laws of 
change as have been observed in animals and 
plants do equally hold good in the case of the 
mind. I shall then endeavour to find out what 
we mean by higher and lower in the two cases, 
and to show, in fact, that we mean much the 
same thing. Supposing all this to have been 
done, the question will have been stated in a 
form which it is possible to answer. I shall 
then make an attempt to give part of the 
answer to it. 

In investigating the laws of change of 
organic beings I shall make use of what is 
called the Evolution -hypothesis, which, as 
applied to this subject, is much the same thing 



90 LECTURES AND ESSAYS 

as the Darwinian theory, though it is not by 
any means tied down to the special views of 
Mr. Darwin. But I shall use this merely as 
an hypothesis ; and the validity of the method 
of investigation which I have suggested is 
entirely independent of the truth of that 
hypothesis. If you will pardon me for a short 
time, I should like to illustrate somewhat 
further what I mean by this. 

When Kepler found out what was the form 
of the orbit described by the planet Mars, he 
thought that the planet was driven by some 
force which acted in the direction in which the 
planet was going. I have known people who 
learned a certain amount of astronomy for 
nautical purposes, whose ideas were very 
similar to those of Kepler. They thought 
that the sun's rotation was what caused the 
planets to revolve about him, just as if you spin 
a teaspoon in the middle of a cup of tea, it 
makes the bubbles go round and round. But 
Newton discovered that the real state of the 
case was far different. If you fasten a ball on 
to the end of an elastic string, and then swing 
it round and round, you can make the ball 
describe an orbit very similar to that of the 
planet, so that your hand is not quite in the 
centre of it Now here the pulling force does 
not act in the direction in which the ball is 
going, but always in the direction of your hand, 



CONDITIONS OF MENTAL DEVELOPMENT 91 

and yet the ball revolves about your hand and 
never actually comes to it. Newton supposed 
that the case of the planet was similar to that 
of the ball ; that it was always pulled in the 
direction of the sun, and that this attraction or 
pulling of the sun produced the revolution of 
the planet, in the same way that the traction 
or pulling of the elastic string produces the 
revolution of the ball. What there is between 
the sun and the planet that makes each of them 
pull the other, Newton did not know ; nobody 
knows to this day ; and all we are now able to 
assert positively is that the known motion of the 
planet is precisely what would be produced if 
it were fastened to the sun by an elastic string, 
having a certain law of elasticity. Now observe 
the nature of this discovery, the greatest in its 
consequences that has ever yet been made in 
physical science : 

I. It begins with an hypothesis, by suppos- 
ing that there is an analogy between the motion 
of a planet and the motion of a ball at the end 
of a string. 

II. Science becomes independent of the 
hypothesis, for we merely use it to investigate 
the properties of the motion, and do not trouble 
ourselves further about the cause of it. 

I will take another example. It has been 
supposed for a long time that light consists of 
waves transmitted through an extremely thin 



92 LECTURES AND ESSAYS 

ethereal jelly that pervades all space ; it is 
easy to see the very rapid tremor which 
spreads through a jelly when you strike it at 
one point From this hypothesis we can 
deduce laws of the propagation of light, and of 
the way in which different rays interfere with 
one another, and the laws so deduced are 
abundantly confirmed by experiment. But 
here also science kicks down the ladder by 
which she has risen. In order to explain the 
phenomena of light it is not necessary to 
assume anything more than a periodical oscil- 
lation between two states at any given point 
of space. What the two states are nobody 
knows ; and the only thing we can assert with 
any degree of probability is that they are not 
states of merely mechanical displacement like 
the tremor of a jelly ; for the phenomena of 
fluorescence appear to negative this supposi- 
tion. Here again, then, the same two remarks 
may be made. The scientific discovery appears 
first as the hypothesis of an analogy; and science 
tends to become independent of the hypothesis. 
The theory of heat is another example. If 
you hold one end of a poker in the fire, the 
other end becomes hot, even though it is not 
exposed to the rays of the fire. Fourier, in 
trying to find the laws of this spread of heat 
from one part of a body to another part, made 
the hypothesis that heat was a fluid which 



CONDITIONS OF MENTAL DEVELOPMENT 93 

flowed from the hot end into the cold as water 
flows through a pipe. From this hypothesis 
the laws of conduction were deduced ; but in 
the process it was found that the very same 
laws would flow from other hypotheses. In 
fact, whatever can be explained by the motion 
of a fluid can be equally well explained either 
by the attraction of particles or by the strains 
of a solid substance ; the very same mathe- 
matical calculations result from the three 
distinct hypotheses ; and science, though com- 
pletely independent of all three, may yet choose 
one of them as serving to link together different 
trains of physical inquiry. 

Now the same two remarks which may be 
made in all these cases apply equally to the 
evolution -hypothesis. It is grounded on a 
supposed analogy between the growth of a 
species and the growth of an individual. It 
supposes, for instance, that the race of crabs 
has gone through much the same sort of 
changes as every crab goes through now, in the 
course of its formation in the egg ; changes 
represented by its pristine shape utterly unlike 
what it afterwards attains, and by its gradual 
metamorphosis and formation of shell and 
claws. By this analogy the laws of change are 
suggested, and these are afterwards checked 
and corrected by the facts. But as before, 
science tends to become independent of hypo- 



94 LECTURES AND ESSAYS 

thesis. The laws of change are established for 
present and finitely distant times ; but they 
give us no positive information about the origin 
of things. So, therefore, if I make use of this 
hypothesis to represent to you the laws of 
change that are deduced from it, you will see 
that the truth of those laws and the conclusions 
which may be drawn from them are in no 
way dependent on the truth of the hypothesis. 

There are certain errors current about the 
nature of the evolution-theory which I wish 
particularly to guard against. In the first place 
it is very commonly supposed that all existing 
animals can be arranged in one continuous 
chain, from the highest to the lowest ; that the 
transition is gradual all through, and that nature 
makes no jumps. This idea was worked out 
into a system of classification by Linnaeus, and 
survived among naturalists until the time of 
Cuvier. " They were bent," says Agassiz, 
" upon establishing one continual uniform series 
to embrace all animals, between the links of 
which it was supposed there were no unequal 
intervals." ..." They called their system la 
chatne des etres." The holders of the Darwinian 
theory are then supposed to believe that all 
these forms grew out of one another, beginning 
with the lowest and ending with the highest ; 
so that any one animal of the series has in the 
course of its evolution passed through all the 




. JfcDTOTtolji*- , * 

f * ReptJia. 
VERTEBRATA 

l vibia. \ Pisces 



Amphibia. 



Pteropoda Cephalopoda. 

* ^Gasteropoda 

dicecia. 
Gasurcpoda. 
nwrioecia 

MOLLU SCA 



Arachnida. 



Artjiculata 



ANN U,' L O S A 



A'nnuloida 
Echitwdernfaa. ** % 



JMolluscoida 
'l\Poly* 



. Gregarinida 



/ Sponaida Jnfiiseria 
Hydroxaa. * 



CCELENTE RA TA 



96 LECTURES AND ESSAYS 

lower forms. And as the species is thus sup- 
posed to have grown up through the chain, and 
the lower species to be continually growing into 
the higher, so it is imagined that every individual 
creature, in the course of its production, passes 
through the lower adult forms ; that a chicken, 
for instance, while it is being formed in the egg, 
becomes in succession a snail, an insect, a fish, 
and a reptile, before it becomes a bird. Now 
that all these ideas are entirely wrong, I need 
hardly remind you ; and I have mentioned 
them in order that there may be no mistake 
about the theory which I am using as an 
analogy. So far is it from being possible to 
arrange existing organisms in a single line or 
chain, that they cannot be adequately repre- 
sented even in the manner which is attempted 
in the preceding diagram, taken from Spencer's 
Principles of Biology ', vol. i. p. 303. 

In the next place, no existing organism 
could possibly grow into any other. What is 
really supposed is this : that if you went back 
a million years or so, and made a picture like 
this one, representing the forms that existed 
then, no single spot which is covered in one 
figure would be covered in the other ; but the 
general arrangement would be very similar, 
except that all the groups would be nearer to 
the centre or radiant point, and therefore nearer 
to each other. And if you made a third 



CONDITIONS OF MENTAL DEVELOPMENT 97 

picture, representing the state of things another 
million years or so further back, then they 
would be still nearer together ; and at a 
distance of time too vast to be represented, 
they would all converge into this radiant point. 
So the theory is that at that stupendous distance 
of time all species were alike, mere specks of 
jelly ; that they gradually diverged from each 
other and got more and more different, till at 
last they attained the almost infinite variety 
that we now find. If you will imagine a tree 
with spreading branches, like an oak ; then the 
outside leaves at any time may be taken to 
represent all the existing species at a given 
time. It is quite impossible to arrange them 
in any serial order. As the tree grows, the 
outer leaves diverge, and get further from the 
trunk and from each other ; and two extremities 
that have once diverged never converge and 
grow together again. But even this simile is 
insufficient ; for species may diverge in a far 
greater variety of directions than the branches 
of a tree. Space has not dimensions enough 
to represent the true state of the case. 

Von Baer's doctrine of development is illus- 
trated by the same figure. If you took embryos 
of polypes, and snails, and cuttle-fish, and 
insects, and crabs, and fish, and frogs, and if 
you could watch their gradual growth into 
these several animals : at first they would be 
VOL. I H 



98 LECTURES AND ESSAYS 

all absolutely alike and indistinguishable. 
Then, after a little while, you would find that 
they might be sorted off into these four great 
classes. Afterwards these groups might be 
divided into smaller groups, representing orders ; 
then these into families and genera ; last of all 
would appear those differences which would 
separate them into species. 

The evolution -hypothesis, then, represents a 
race of animals or plants as a thing slowly 
changing : and it also represents these changes 
as connected by fixed laws with the action of 
the surrounding circumstances, or, as it is 
customary to say, the environment. Now the 
action of the environment on a race is of two 
kinds, direct and indirect That part which is 
called direct action is very easily understood. 
There is no difficulty in seeing how changes of 
climate might produce changes in the colour of 
the skin, or how new conditions which neces- 
sitated the greater use of any organ would lead 
to the increase of that organ, as we know that 
muscles may be made to swell with exercise ; 
and changes thus made habitual would in time 
be inherited. But the indirect action of the 
environment, which is called natural selection, 
is still more important. The mode of its opera- 
tion may be seen from an example. There are 
two butterflies in South America, nearly resem- 
bling one another in form, but one of which 



CONDITIONS OF MENTAL DEVELOPMENT 99 

has a very sweet taste and is liked by the birds, 
while the other is bitter and distasteful to them. 
Now suppose that, for some reason or other, 
sweet butterflies were occasionally produced 
with markings similar to the bitter ones, these, 
being mistaken by the birds for bitter ones, 
would run less chance of being eaten, and there- 
fore more chance of surviving and leaving off- 
spring. If this peculiarity of marking is at all 
inheritable, then the number of sweet butterflies 
with bitter marks will in the next generation 
be greater in proportion to the whole number 
than before ; and, as this process goes on, the 
sweet butterflies which retain their distinguish- 
ing marks will be all weeded out by the birds, 
and the entire species will have copied the 
markings of the bitter species. This has 
actually taken place : the one species has 
mimicked the markings of the other. Here we 
see the working of Natural Selection. Any 
variation in an individual which gives him an 
advantage in the struggle for life is more likely 
to be transmitted to offspring than any other 
variation, because the individual is more likely 
to survive ; so that nature gradually weeds out 
all those forms which are not suited to the 
environment, and thus tends to produce equili- 
brium between the species and its surrounding 
circumstances. Changes, then, are produced in 
a species by the selection of advantageous 



ioo LECTURES AND ESSAYS 

changes which happen to be made in in- 
dividuals. Now there are three kinds of 
change that are produced in individuals : change 
of size, or growth ; change of structure, that is 
to say, change in the shape and arrangement 
of the parts, as when the cartilaginous skeleton 
of an infant becomes hardened into bone ; and 
change of function, that is to say, change in the 
use which is made of any part of the organism. 
I have one or two remarks to make about the 
first of these, namely, growth, or change of size. 
Every organism is continually taking in matter 
through the external surface to feed the inside. 
A certain quantity of this is needed to make 
up for the waste that is continually going on. 
But let us suppose, to begin with, that an 
organism has more surface than it absolutely 
wants to make up for waste, then a certain 
portion of the assimilated matter, or food, will 
remain over, and the organism will increase in 
size. But, you say, if this is all that is meant 
by growth why does it not go on for ever ? 
The explanation is very simple. I take this 
cube, which has six sides, each a square inch ; 
let us suppose it to represent an animal, and 
imagine, to begin with, that two of the sides by 
themselves are capable of feeding the whole 
mass, then the nutrition taken in by the other 
four sides is left over, and the mass must 
increase in size. Imagine it now grown to 



CONDITIONS OF MENTAL DEVELOPMENT 101 

twice the linear dimensions, that is to say, to a 
cube every side of which is two inches. The 
mass to be fed is now eight times what it was, 
while the surface is only four times as great ; 
of the twenty -four square inches of surface 
sixteen are taken up with feeding the mass, 
while only eight, or one-third, are left to supply 
the materials for growth. Still there is an 
overplus, and the organism will grow. Let it 
now acquire three times its original height and 
breadth and thickness, the mass is twenty-seven 
times as great, and the surface only nine times : 
that is to say, while there are twenty-seven 
cubic inches to be fed, there are just fifty-four 
square inches to feed them. There is no longer 
any overplus ; the organism will stop growing. 
And it is a general rule that, in any case, when 
a thing grows its mass increases much faster 
than its surface. However much, therefore, the 
feeding power of the surface may be in excess 
to begin with, the mass must inevitably catch 
it up, and the growth will stop. 

Now the changes of an individual mind may 
be reduced to the same three types : 

Growth. 

Change of structure. 

Change of function. 

First, then, what is the growth of the mind ? 
It is the acquisition of new knowledge ; not 
merely of that which is required to make up for 



loa LECTURES AND ESSAYS 

our wonderful power of forgetting, for oblivion 
is really a far more marvellous thing than 
memory ; but of a certain overplus which goes 
to increase the entire mass of our mental 
experiences. Now I do not know whether 
there is any race between surface and mass 
here as in the case of an organism ; but it is 
certainly true that whereas in childhood the 
amount we forget is very little, and our powers 
of acquisition preponderate immensely over our 
powers of oblivion ; as we grow up, the powers 
of oblivion gain rapidly upon the acquisitive 
ones, and finally catch them up ; the growth 
ceases as soon as this balance is attained. So 
that in this first law, you see, there is an entire 
analogy between the two cases. 

In the next place, the mind experiences 
changes of structure ; that is to say, changes in 
the shape and arrangement of its parts. Ideas 
which were only feebly connected become 
aggregated into a close and compact whole. 
The ideas of several different qualities, for 
instance, which we never thought of as connected 
with each other, are brought together by the 
qualities being found to exist in the same object. 
In this way we form conceptions of things, 
which gradually get so compact that we cannot 
even in thought separate them into their com- 
ponent parts. Portions of our knowledge which 
we held as distinct are connected together 



CONDITIONS OF MENTAL DEVELOPMENT 103 

by scientific theories ; images which were 
scattered all about are bound up into living 
bundles by the artist, and so we find them 
rearranged. 

Lastly, changes of function take place. 
Everybody knows how the mental faculties open 
out and become visible as a child grows up. 
Men acquire faculties by practice. And without 
any conscious seeking, you must know how 
often we wake up as it were and find ourselves 
gifted with new powers. We have found evi- 
dence then of the existence of our three types 
of change, growth, structure, and function. 

The actions therefore which go on between 
the environment and the individual may be 
reduced to the same three types in the case of 
the mind as in the case of any visible organism. 
Being somewhat encouraged by this result, let 
us go back to our original question. What is 
that attitude of mind which is likely to change 
for the better ? What is the meaning of better ? 

Although it is quite impossible to arrange 
all existing organisms in a serial chain, yet we 
certainly have a general notion of higher and 
lower. A bird we regard as higher than a fish, 
and a dog is higher than a snake. And if we 
return to our illustration of the tree, we shall 
see that at every point, at any given time, there 
is a definite direction of development. So that 
though we might not be able to say which of 



io 4 LECTURES AND ESSAYS 

two co-existing organisms was the higher, yet, 
by comparing a species with itself at a slightly 
later time, we might say whether it had de- 
generated or improved. Now by examining 
various cases, we shall find that there are six 
marks of improvement : 

The parts of the organism get more different. 

The parts of the organism get more con- 
nected. 

The organism gets more different from the 
environment 

The organism gets more connected with the 
environment 

The organism gets more different from other 
individuals. 

The organism gets more connected with 
other individuals. 

The processes in fact which result in develop- 
ment are made up of differentiation and integra- 
tion ; differentiation means the making things 
to be different, integration means the binding 
them together into a whole ; these are applied 
to the parts of the organism, the organism and 
surrounding nature, the organism and other 
organisms. Differentiation of parts is illustrated 
by the figure on the following page. [Spencer's 
Principles of Biology, vol. ii. p. 187.] 

Integration of parts means the connected 
play of them ; so that one being affected the 
rest are affected. Differentiation from the 



CONDITIONS OF MENTAL DEVELOPMENT 105 

environment takes place in weight, composition, 
and temperature. A polype is little else than 




sea-water, which it inhabits ; a fish is several 
degrees of temperature above it, and made of 
quite different materials ; till at last a mammal 
is 70 or 80 above the surrounding matter, 



io6 LECTURES AND ESSAYS 

and made of still more different materials. 
Integration with the environment means close 
correspondence with it ; actions of the environ- 
ment are followed by corresponding actions 
of the animal. Differentiation from other 
organisms means individuality ; integration 
with them sociality. 

In a similar way we have a sort of general 
notion of higher and lower stages of mental 
development. I will endeavour to show that 
this general notion resolves itself into a measure 
of the extent to which the same six processes 
have gone on, namely : 

Separation of parts, 

Connection of parts, 

Separation from the environment, 

Closer correspondence with the environment, 

Separation from other individuals, 

Sociality. 

The only conception we can form of a purely 
unconscious state is one in which all is exactly 
alike, or rather, in which there is no difference. 

There is not one thing with another, 
But Evil saith to Good : My brother, 

My brother, I am one with thee : 
They shall not strive nor cry for ever : 
No man shall choose between them : never 

Shall this thing end and that thing be. 

The first indication of consciousness is a 
perception of difference. The child's eyes 



CONDITIONS OF MENTAL DEVELOPMENT 107 

follow the light. Immediately this colourless, 
homogeneous universe splits up into two parts, 
the light part and the dark part. A line is 
drawn across it, it is made heterogeneous, and 
the first thing that exists is a distinction. Then 
other lines are drawn ; appearance is separated 
into white, black, blue, red, and so on. This is 
the first process, the differentiation of the parts 
of consciousness. But by and by a number of 
these lines of distinction are found to enclose 
a definite space ; they assume relations to one 
another ; the lines white, round, light, capable 
of being thrown at people, include the con- 
ception of a ball ; this gains coherence, becomes 
one, a thing, holding itself together not only 
separated from the rest of consciousness, but 
connected in itself into a distinct whole, in- 
tegrated. Here we have the second process. 
And throughout our lives the same two pro- 
cesses go hand in hand ; whatever we perceive 
is a line of demarcation between two different 
things ; we can be conscious of nothing but a 
separation, a change in passing from one thing 
to another. And these different lines of de- 
marcation are constantly connecting themselves 
together, marking out portions of our conscious- 
ness as complete wholes, and making them 
cohere. Just as a sculptor clears away from 
a block of marble now this piece and now that, 
making every time a separation between what 



io8 LECTURES AND ESSAYS 

is to be kept and what is to be chipped off, till 
at last all these chippings manifest the connec- 
tion that ran through them, and the finished 
statue stands out as a complete whole, a positive 
thing made up of contradictory negations : so 
is a conception formed in the mind. 

And this conception, when it is thus made 
into a whole, integrated, by an act of the mind, 
what does it immediately appear to be ? Why, 
something outside of ourselves, a real thing, 
different from us. This is the third process, 
the process of differentiation from the environ- 
ment. This is beautifully described by Cuvier, 
who pictures the first man wandering about in 
ecstasies at the discovery of so many new parts 
of himself; till gradually he learns that they 
are not himself, but things outside. This 
notion, then, of a thing being real, existing 
external to ourselves, is due to the active power 
of the mind which regards it as one, which 
binds together all its boundaries. And this 
goes on as long as we live. Constantly we 
frame to ourselves more complicated combina- 
tions of ideas, and by giving them unity make 
them real. And, at the same time, the con- 
verse process is equally active. While more 
and more of our ideas are put outside of us 
and made real, our minds are continually growing 
more and more into accordance with the nature 
of external things ; our ideas become truer, 



CONDITIONS OF MENTAL DEVELOPMENT 109 

more conformable to the facts ; and at the same 
time they answer more surely and completely 
to changes in the environment ; a new experi- 
ence is more rapidly and more completely 
connected with the sum of previous experiences. 
But there is more than this. The action of 
these two laws taken together does in fact 
amount to the creation of new senses. Men of 
science, for example, have to deal with extremely 
abstract and general conceptions. By constant 
use and familiarity, these, and the relations 
between them, become just as real and external 
as the ordinary objects of experience ; and the 
perception of new relations among them is so 
rapid, the correspondence of the mind to 
external circumstances so great, that a real 
scientific sense is developed, by which things 
are perceived as immediately and truly as I see 
you now. Poets and painters and musicians 
also are so accustomed to put outside of them 
the idea of beauty, that it becomes a real 
external existence, a thing which they see with 
spiritual eyes, and then describe to you, but by 
no means create, any more than we seem to 
create these ideas of table and forms and light, 
which we put together long ago. There is no 
scientific discoverer, no poet, no painter, no 
musician, who will not tell you that he found 
ready-made his discovery or poem or picture 
that it came to him from outside, and that he 



no LECTURES AND ESSAYS 

did not consciously create it from within. And 
there is reason to think that these senses or 
insights are things which actually increase 
among mankind. It is certain, at least, that 
the scientific sense is immensely more developed 
now than it was three hundred years ago ; and 
though it may be impossible to find any 
absolute standard of art, yet it is acknowledged 
that a number of minds which are subject to 
artistic training will tend to arrange themselves 
under certain great groups, and that the 
members of each group will give an independent 
yet consentient testimony about artistic ques- 
tions. And this arrangement into schools, and 
the definiteness of the conclusions reached in 
each, are on the increase, so that here, it would 
seem, are actually two new senses, the scientific 
and the artistic, which the mind is now in the 
process of forming for itself. There are two 
remaining marks of development : differentiation 
from surrounding minds, which is the growth of 
individuality ; and closer correspondence with 
them, wider sympathies, more perfect under- 
standing of others. These, you will instantly 
admit, are precisely the twin characteristics of 
a man of genius. He is clearly distinct from 
the people that surround him, that is how you 
recognise him ; but then this very distinction 
must be such as to bind him still closer to them, 
extend and intensify his sympathies, make him 



CONDITIONS OF MENTAL DEVELOPMENT m 

want their wants, rejoice over their joys, be cast 
down by their sorrows. Just as the throat is 
a complicated thing, quite different from the 
rest of the body, but yet is always ready to cry 
when any other part is hurt. 

We have thus got a tolerably definite notion 
of what mental development means. It is a 
process of simultaneous differentiation and 
integration which goes on in the parts of con- 
sciousness, between the mind and external things, 
between the mind and other minds. And the 
question I want answered is, What attitude of 
mind tends to further these processes ? 

I have now done all that it was my business 
to do, namely, I have stated the question in a 
form in which it is possible to answer it. 
There is no doubt that by a careful study of 
the operations of nature we shall be able to 
find out what actions of an organism are 
favourable to its higher development. Having 
formulated these into a law, we shall be able 
to interpret this law with reference to the mind. 

But now I am going to venture on a partial 
answer to this question. What I am going to 
say is mere speculation, and requires to be 
verified by facts. 

The changes which take place in an organism 
are of two kinds. Some are produced by the 
direct action of things outside, and these are to 
a*great extent similar to the changes which we 



U2 LECTURES AND ESSAYS 

observe in inanimate things. When a tree is 
bent over by the wind and gets ultimately fixed 
in this position, the change is in no way different 
from that which takes place when we bend a 
wire and it does not entirely return to its former 
straightness. Other changes are produced by 
the spontaneous action of that store of force 
which by the process of growth is necessarily 
accumulated within the organism. Such are 
all those apparently disconnected motions 
which make up the great distinction between 
living things and dead. Now my speculation 
is, that advantageous permanent changes are 
always produced by the spontaneous action of 
the organism, and not by the direct action of 
the environment This, I think, is most clear 
when we take an extreme case. Let us suppose 
a race of animals that never had any changes 
produced by their spontaneous activity. The 
race must at a certain time have a definite 
amount of plasticity, that is, a definite power of 
adapting itself to altered circumstances by 
changing in accordance with them. Every 
permanent effect of the environment upon them 
is a crystallisation of some part which before 
was plastic ; for the part must have been 
plastic for the effect to be produced at all ; 
and as the effect is permanent, the part has to 
that extent lost in plasticity. As this goes on, 
the race of animals will bind up in itself more 



CONDITIONS OF MENTAL DEVELOPMENT 113 

and more of its history, but will in that process 
lose the capability of change which it once had ; 
at last it will be quite fixed, crystallised, in- 
capable of change. Then it must inevitably 
die out in time ; for the environment must 
change sooner or later, and then the race, in- 
capable of changing in accordance with it, must 
be killed off. On the other hand, any addition 
to the organism which is made by its spon- 
taneous activity is an addition of something 
which has not yet been acted upon by the 
environment, which is therefore plastic, capable 
of indefinite modification, in fact, an increase 
of power. The bending of a tree by the wind 
is a positive disadvantage to it if the wind 
should ever happen to blow from the other 
side. But when a plant, for no apparent 
reason, grows long hairs to its seed the 
material for which may have been accidentally 
supplied by the environment, while its use in 
this way is a spontaneous action of the plant 
this is a definite increase of power ; for the 
new organ may be modified in any conceivable 
way to suit the exigencies of the environment, 
may cling to the sides of beasts, and so 
help the distribution of the seed, or effect the 
same object by being caught by the wind. 
Activity, in fact, is the first condition of de- 
velopment. A very good example of this 
occurs in Professor Huxley's lizards, of which 
VOL. I I 



U4 LECTURES AND ESSAYS 

you heard two or three weeks ago. 1 About 
the time marked by the Primary strata it 
appears that there was a race of lizards, thirty 
feet high, that walked on their hind legs, 
balancing themselves by their long tails, and 
having three toes like birds. This race di- 
verged in three directions. Some of them 
yielded to the immediate promptings of the 
environment, found it convenient to go on all 
fours and eat fish ; they became crocodiles. 
Others took to exercising their forelegs vio- 
lently, developed three long fingers, and became 
birds. The rest were for a long while un- 
decided whether they would use their arms or 
their legs most ; at length they diverged, and 
some became pterodactylesand others kangaroos. 
For Mr. Seeley, of Cambridge, has discovered 
marsupial bones in pterodactyles ; that is to 
say, bones like those which were supposed 
peculiar to the order of mammals to which the 
kangaroo belongs. 

Assuming now that this law is true, and 
that the development of an organism proceeds 
from its activities rather than its passivities, 
let us apply it to the mind. What, in fact, 
are the conditions which must be satisfied by 
a mind in process of upward development, so 
far as this law gives them ? 

1 [" On the animals which are most nearly intermediate between 
birds and reptiles," Roy. Inst. Proc. I'. 1869, p. 278.] 



CONDITIONS OF MENTAL DEVELOPMENT 115 

They are two ; one positive, the other 
negative. The positive condition is that the 
mind should act rather than assimilate, that its 
attitude should be one of creation rather than 
of acquisition. If scientific, it must not rest in 
the contemplation of existing theories, or the 
learning of facts by rote ; it must act, create, 
make fresh powers, discover new facts and laws. 
And, if the analogy is true, it must create 
things not immediately useful. I am here put- 
ting in a word for those abstruse mathematical 
researches which are so often abused for having 
no obvious physical application. The fact is 
that the most useful parts of science have been 
investigated for the sake of truth, and not for 
their usefulness. A new branch of mathematics, 
which has sprung up in the last twenty years, 
was denounced by the Astronomer-Royal before 
the University of Cambridge as doomed to be 
forgotten, on account of its uselessness. Now 
it turns out that the reason why we cannot 
go further in our investigations of molecular 
action is that we do not know enough of this 
branch of mathematics. If the mind is artistic, 
it must not sit down in hopeless awe before the 
monuments of the great masters, as if heights 
so lofty could have no heaven beyond them. 
Still less must it tremble before the conven- 
tionalism of one age, when its .mission may be 
to form the whole life of the age succeeding. 



ii6 LECTURES AND ESSAYS 

No amount of erudition or technical skill or 
critical power can absolve the mind from the 
necessity of creating, if it would grow. And 
the power of creation is not a matter of static 
ability, so that one man absolutely can do these 
things and another man absolutely cannot ; it is 
a matter of habits and desires. The results of 
things follow not from their state but from their 
tendency. The first condition then of mental 
development is that the attitude of the mind 
should be creative rather than acquisitive: or, as 
it has been well said, that intellectual food should 
go to form mental muscle and not mental fat. 

The negative condition is plasticity : the 
avoidance of all crystallisation as is immediately 
suggested by the environment. A mind that 
would grow must let no ideas become per- 
manent except such as lead to action. Towards 
all others it must maintain an attitude of 
absolute receptivity ; admitting all, being 
modified by all, but permanently biassed by 
none. To become crystallised, fixed in opinion 
and mode of thought, is to lose the great char- 
acteristic of life, by which it is distinguished 
from inanimate nature : the power of adapting 
itself to circumstances. 

This is true even more of the race. There 
are nations in the East so enslaved by custom 
that they seem to have lost all power of change 
except the capability of being destroyed. 



CONDITIONS OF MENTAL DEVELOPMENT 117 

Propriety, in fact, is the crystallisation of a 
race. And if we consider that a race, in pro- 
portion as it is plastic and capable of change, 
may be regarded as young and vigorous, while 
a race which is fixed, persistent in form, unable 
to change, is as surely effete, worn out, in peril 
of extinction ; we shall see, I think, the im- 
mense importance to a nation of checking the 
growth of conventionalities. It is quite possible 
for conventional rules of action and conventional 
habits of thought to get such power that pro- 
gress is impossible, and the nation only fit to 
be improved away. In the face of such a 
danger it is not right to be proper. 

NOTE. The following letter, published in 
the Pall Mall Gazette of June 24, 1 868, should 
be read in connection with this Discourse. 

" Sir I ask for a portion of your space to 
say something about a lecture, ' On some of 
the Conditions of Mental Development,' which I 
delivered at the Royal Institution in March last. 

"In that lecture I attempted to state and 
partially answer the question, 'What is that 
attitude of mind which is most likely to change 
for the better ? ' I proposed to do this by 
applying the hypothesis of the variability of 
species to the present condition of the human 
race. I put forward also for this purpose a 
certain biological law, viz. that permanent 



Ii8 LECTURES AND ESSAYS 

advantageous changes in an organism are due 
to its spontaneous activity, and not to the 
direct action of the environment. 

" In the short account of the evolution- 
hypothesis which I prefixed, I followed Mr. 
Herbert Spencer's Principles of Biology, not 
knowing, at the time, how much of the theory 
was due to him personally, but imagining that 
the greater part of it was the work of previous 
biologists. On this account I omitted to make 
such references to my special sources of informa- 
tion as I should otherwise have made. I was 
also ignorant of the developments and applica- 
tions of the theory which he has made in his 
other works, in which a great portion of my 
remarks had been anticipated. These omissions 
I desire now to rectify. 

" Mr. Spencer's theory is to the ideas which 
preceded it even more than the theory of 
gravitation was to the guesses of Hooke and 
the facts of Kepler. 

" Finding only a vague notion of progress 
from lower to higher, he has affixed the specific 
meaning to the word higher of which I gave an 
account, defining the processes by which this 
progress is effected. He has, moreover, formed 
the conception of evolution as the subject of 
general propositions applicable to all natural 
processes, a conception which serves as the 
basis of a complete system of philosophy. 



CONDITIONS OF MENTAL DEVELOPMENT 119 

In particular, he has applied this theory to the 
evolution of mind, developing the complete 
accordance between the laws of mental growth 
and of the growth of other organic functions. 
In fact, even if the two points which I put 
forward as my own viz. the formal application 
of the biological method to a certain special 
problem, and the biological law which serves as 
a partial solution of it have not before been 
explicitly developed (and of this I am not sure), 
yet they are consequences so immediate of the 
general theory that in any case the credit of 
them should entirely belong to the philosopher 
on whose domains I have unwittingly trespassed. 
The mistake, of course, affects me only, and 
could in no way injure the fame of one whose 
philosophical position is so high and so assured. 
" I may perhaps be excused for anticipating 
here what I hope to say more at length at 
another time, 1 that in my belief the further 
deductions to be made from this theory, with 
reference to modern controversies, will lead to 
results at once more conservative, and in a 
certain sense more progressive, than is com- 
monly supposed. 

" I remain, Sir, yours, etc., 

" W. K. CLIFFORD." 

1 This intention was never carried out, so far as the editors 
are aware. 



ON THEORIES OF THE PHYSICAL 
FORCES l 

[REFERRING to the passage in Faust, 

" Geschrieben steht : Im Anfang war das Wort. 
Hier stock" ich schon ! Wer hilft mir welter fort ? 
Ich kann das Wort so hoch unmoglich schatzen, 
Ich muss es anders iibersetzen, 
Wenn ich vom Geiste recht erleuchtet bin. 
Geschrieben steht : Im Anfang war der Sinn. 
Bedenke wohl die erste Zeile, 
Dass deine Feder sich nicht iibereile ! 
1st es der Sinn, der alles wirkt und schafft ? 
Es sollte stehn : Im Anfang war die Kraft ! 
Doch, auch indem ich dieses niederschreibe, 
Schon warnt mich was, dass ich dabei nicht bleibe. 
Mir hilft der Geist ! Auf einmal seh' ich Rath, 
Und schreibe getrost : Im Anfang war die That ! " 

the speaker regarded it as a description of four 
views or stages of opinion through which a 
man looking for himself on the face of things 

1 Discourse delivered at the Royal Institution, February 18, 
1870. This discourse is reprinted as it stands in the Proceedings 
of the Royal Institution. The opening paragraphs, being reported 
in the third person and apparently abridged, are enclosed in square 
brackets. 



THEORIES OF THE PHYSICAL FORCES 121 

is likely to pass ; through which also successive 
generations of the men who look for themselves 
on the face of things are likely to pass. He 
considered that by far the larger portion of 
scientific thought at the present day is in the 
third stage that, namely, in which Force is 
regarded as the great fact that lies at the bottom 
of all things ; but that this is so far from being 
the final one, that even now the fourth stage is 
on its heels. In the fourth stage the concep- 
tion of Force disappears, and whatever happens 
is regarded as a deed. The object of the dis- 
course was to explain the nature of this transi- 
tion, and to introduce certain conceptions which 
might serve to prepare the way for it. 

There are, then, to be considered two different 
answers to the question, " What is it that lies 
at the bottom of things ? " The two answers 
correspond to two different ways of stating the 
question ; namely, first, " Why do things hap- 
pen ? " and, secondly, " What is it precisely that 
does happen ? " The speaker maintained that 
the first question is external to the province of 
science altogether, and science has nothing to 
do with it ; but that the second is exactly the 
question to which science is always trying to 
find the answer. It may be doubted whether 
the first question is within the province of 
human knowledge at all. For it is as necessary 
that a question should mean something^ in order 



122 LECTURES AND ESSAYS 

to be a real question, as that an answer should 
mean something, in order to be a real answer. 
And it is quite possible to put words together 
with a note of interrogation after them without 
asking any real question thereby. Whether the 
phrase, " Why do things happen ? " as applied 
to physical phenomena, is a phrase of this kind 
or no, is not here to be considered. But that 
to the scientific inquirer there is not any "why" 
at all, and that if he ever uses the word it is 
always in the sense of what, the speaker regarded 
as certain. In order to show what sort of way 
an exact knowledge of the facts would super- 
sede the inquiry after the cause of them, he then 
made use of the hypothesis of continuity ; show- 
ing, in the following manner, that it involves 
such an interdependence of the facts of the uni- 
verse as forbids us to speak of one fact or set of 
facts as the cause of another fact or set of facts.] 

The hypothesis of the continuity of space and 
time is explained, and the alternative hypothesis 
is formulated. 

From the hypothesis of the complete continuity 
of time-changes, a knowledge of the entire history 
of a single particle is shown to be involved in a 
complete knowledge of its state at any moment. 

Things frequently move. Some things move 
faster than others. Even the same thing moves 
faster at one time than it does at another time. 
When you say that you are walking four miles 



THEORIES OF THE PHYSICAL FORCES 123 

an hour, you do not mean that you actually 
walk exactly four miles in any particular hour ; 
you mean that if anybody did walk for an 
hour, keeping all the time exactly at the rate 
at which you are walking, he would in that 
hour walk four miles. But now suppose that 
you start walking four miles an hour, and 
gradually quicken your pace, until you are 
walking six miles an hour. Then this question 
may be asked : Suppose that anybody chose a 
particular number between four and six, say 
four and five-eighths, is it perfectly certain that 
at some instant or other during that interval 
you were walking at the rate of four miles and 
five-eighths in the hour ? Or, to put it more 
accurately, suppose that we have a vessel con- 
taining four pints of water exactly, and that 
somebody adds to it a casual quantity of water 
less than two pints. Then is it perfectly certain 
that between these two times, when you were 
walking at four miles an hour, and when you 
were walking six miles an hour, there was some 
particular instant at which you were walking 
exactly as many miles and fractions of a mile 
an hour as there are pints and fractions of a 
pint of water in the vessel ? The hypothesis of 
continuity says that the answer to this question 
is yes ; and this is the answer which everybody 
gives no wad ays ; which everybody has given mostly 
since the invention of the differential calculus. 



I2 4 LECTURES AND ESSAYS 

But this is a question of fact, and not of 
calculation. Let us, therefore, try and imagine 
what the contrary hypothesis would be like. 

You know what a " wheel of life " is. There 
is a cylinder with slits in its side, which can be 
spun round rapidly ; and you look through the 
slits at the pictures opposite. The result is 
that you see the pictures moving ; moreover, 
you see them move faster or slower according 
as you turn the cylinder faster or slower. This 
is what you see, and what appears to happen ; 
but now let us consider what actually does 
happen. I remember in particular a picture of 
a man rolling a ball down an inclined plane 
towards you ; he was standing at the farther 
edge of the inclined plane, as it were behind a 
counter, and he picked up the balls one by one 
and rolled them towards you. But now when 
you took out the strips of paper on which the 
pictures were drawn, you found that they were 
really pictures of this man and his ball in a 
graduated series of positions. Each picture, of 
course, was perfectly still in itself, a mere draw- 
ing on the paper. The first one represented 
him with his hand below the counter, just pick- 
ing up the ball ; in the next, he had the ball in 
his hand, drawn back ready to roll down ; in 
the next, the hand was thrown forwards with 
the ball in it ; in the next, the ball had just 
left his hand and rolled a little way down ; in 



THEORIES OF THE PHYSICAL FORCES 125 

the next farther, and so on. Now, these pic- 
tures being put in the inside of the cylinder 
which is turning round, come opposite you one 
by one. But you do not look directly at them ; 
there are slits interposed. The effect of that 
is, that if you look straight at a certain portion 
of the opposite picture you can only see it for 
a very small interval of time ; that, namely, 
during which the slit is passing in front of your 
eye. Now let us carefully examine what hap- 
pens. When the slit passes, it goes so quickly 
that you get, as it were, almost an instantaneous 
photograph on your eye of the opposite picture; 
say of the man with his hand below the counter. 
Then this is effaced, and you see absolutely 
nothing until the next slit passes. But by the 
time the next slit comes, another picture has 
got opposite to you ; so that you get an 
instantaneous photograph this time of the man 
with his hand drawn back and the ball in it. 
Then this in its turn is effaced, for a time you 
see nothing, and then you are given an in- 
stantaneous glimpse of the hand thrown forward. 
In this way, what you really see is darkness 
relieved by regularly-recurring glimpses of the 
figure in different positions. Now, this experi- 
ence that you get is obviously consistent with 
the hypothesis that the man goes on moving 
all the time when he is hidden from you ; so 
as to be in exactly that series of positions when 



126 LECTURES AND ESSAYS 

you do catch a glimpse of him. And, in fact, 
you do instinctively, by an inevitable habit, 
admit this hypothesis, not merely into your 
mind as a speculation, but into your very sensa- 
tion as an observed phenomenon. You simply 
see the man move ; and, except for a certain 
weariness in the eyes, there is nothing to dis- 
tinguish this perception of movement from any 
other perception of movement. At the same 
time we do know very distinctly, and beyond 
the shadow of a doubt, that there is no con- 
tinuity in the picture at all : that, in fact, you 
do not see the same picture twice following, but 
a new one every time till the cycle is completed ; 
and that the picture never is in any position in- 
termediate between two successive ones of those 
which you see. Here then is an apparently 
continuous motion which is really discontinuous ; 
and moreover there is an apparently continuous 
perception of it which is really discontinuous 
that is, it seems to be gradually changed, while 
it really goes by little jumps. 

I suppose very few people have looked at 
this toy without wondering whether it is not 
actually and truly a wheel of life, without any 
joke at all. I mean, that it is very natural for 
the question to present itself, Do I ever really 
see anything move ? May not all my appar- 
ently continuous perceptions be ultimately made 
up of little jumps, which I run together by this 



THEORIES OF THE PHYSICAL FORCES 127 

same inevitable instinct ? There is another 
way in which this is sometimes suggested. If 
you move your hand quickly, you can see a 
continuous line of light, because the image of 
every position of your hand lingers a little while 
upon the retina. But now, if you do this in a 
room lighted only by an electric spark which is 
not going very fast, so that the general result 
is darkness broken by nearly instantaneous 
flashes at regular intervals ; then, instead of 
seeing a continuous line of light, you will see a 
distinct series of different hands, perhaps about 
an inch apart, if the electric spark is going very 
slowly, and you move your hand very quickly. 
But now make the spark go quicker, or your 
hand slower; the distances between these several 
hands will gradually diminish, till you do not 
know how the continuous line of light is re- 
stored. And the question inevitably presents 
itself is not every case of apparently con- 
tinuous perception really a case of successive 
distinct images very close together? 

That is to say, for instance, if I move my 
hand so in front of me, and apparently see it 
take up in succession every possible position on 
its path between the two extreme positions ; 
do I really see this, or do I only see my hand 
in a certain very large number of distinct 
positions, and not at all in the intervening 
spaces ? 



i8 LECTURES AND ESSAYS 

I have no doubt whatever myself, that the 
latter alternative is the true one, and that the 
wheel of life is really an illustration and type 
of every moment of our existence. But I am 
not going to give my reasons for this opinion, 
because it is quite a different question from the 
one I am trying to get at. The question, 
namely, is this. What I see, or fancy I see, is 
quite consistent with the hypothesis that my 
hand really does go on moving continuously all 
the time, and takes up an infinite number of 
positions between the two extreme ones. But 
if this hypothesis is not true, what is true ? and 
how are we to imagine any other state of 
things than that supposed by the hypothesis of 
continuity ? 

I draw here two rows of points. The upper 
row of points is to represent a series of positions 
in space which it is conceivable that a certain 
thing might take up. The lower row of points 
is to represent a series of instants in time at 
which it is conceivable that the same thing 
might exist. Suppose now that at the instant 
of time represented by the first point of the 
lower row, the thing held the position in space 
represented by the first point of the upper row. 
Suppose that it only existed there for that in- 
stant, and then disappeared utterly, so that at 
these succeeding instants where the lower points 
have no points directly above them the thing 



THEORIES OF THE PHYSICAL FORCES 129 

is nowhere at all. Lastly, suppose that at this 
instant of time which has a space-dot above it, 
the thing existed in that space-position ; and 
so on all through, the thing only existing at 
those instants whose representative points have 
a space-dot exactly above them, and being then 
in the space -position signified by such dots. 
Then we may call this a discontinuous motion ; 
a motion because the thing is in different places 
at different times, though it is not at all times 
that it exists at all ; and a discontinuous motion 
because the thing passes from one position to 
another distant from it without going through 
any intermediate position. 

Now imagine that in each of these two 
series the dots are very close together indeed, 
and very great in number ; so that, however 
small one made them on the paper, the lines 
would look as if they were continuous lines. 
And let the thing be a white speck travelling 
along the upper line in the manner I have 
described ; namely, existing only when there 
is one dot exactly over another ; only that as 
the lower dots represent instants of time, we 
may make some definite supposition and 
assume that one inch of them represents a 
second. 

Then it is clear that if the dots were taken 
close enough together, and enough of them, the 
appearance would be precisely what we ordi- 
VOL. I K 



130 LECTURES AND ESSAYS 

narily see when a white speck moves along a 
line. That is to say, we have got some sort 
of representation of what we might have to 
suppose, if we did not assume the truth of the 
law of continuity. 

You must here notice in particular that I 
suppose the series of positions denoted by the 
upper dots to be all the positions that are 
between the two end ones ; that is, I suppose 
the path from one of these end ones to the 
other to be made up of a series of discrete 
positions. And similarly I suppose the series 
of instants denoted by the lower dots to be all 
the time that elapses between the two end ones ; 
that is, I suppose the interval of time to contain 
a perfectly definite number of instants, these 
being further indivisible. Or we may say that 
on this alternative hypothesis space and time 
are discontinuous ; that is, they are in separate 
parts which do not hold together. Now I must 
beg you to remember for a little while what 
the hypothesis of continuity is not, for I shall 
have to refer to this point again subsequently. 
In this kind of jumping motion that we have 
been imagining, the rate of motion of a thing 
could only be measured by the size of one of 
its jumps ; that is, by the number of positions 
it passed over between two existences compared 
with the number of instants passed over. And 
this rate might obviously change by jumps as 



THEORIES OF THE PHYSICAL FORCES 131 

violent and sudden as those of the thing itself ; 
at any instant when the thing was non-existent 
its rate would be non-existent, and whenever 
the thing came into existence its rate would 
suddenly have a value depending on how far 
off its last position was. In this case, there- 
fore, our question about the intermediate rate 
whether between walking four miles an hour 
and walking six miles an hour you must 
necessarily walk at all intermediate paces 
must be answered in the negative. Now then, 
at last, let us investigate some consequences of 
supposing that motion is really continuous as 
it seems to be. 

First, how to measure the rate at which a 
thing is moving? This was done experi- 
mentally by Galileo in the case of falling 
bodies, and I shall have to speak again of the 
results which he obtained. But at present I 
want to speak not of an experimental method 
of finding the rate, but of a theoretical method 
of representing it, invented by Newton, and 
called the curve of velocities. 

Suppose that a point N is going along the 
line O Y, sometimes fast and sometimes slow ; 
and that a point M is going along the line O x 
always at the same rate. Also somebody 
always holds a stick N P so as to move with 
the point N, and be horizontal ; and somebody 
holds a stick M P so as to move with the 



132 LECTURES AND ESSAYS 

point M, and be vertical ; and a third person 
keeps a pencil pressed in the corner where 
the two sticks cross at P. Then 
. when the points M and N move, 
Jf the point P will move too ; and 
its motion will depend on that 
of the two other points. For 
instance, if the point N moves 
always exactly as fast as the 
point M, then the point P will go along the line 
O P midway between the lines o X O Y. If N 
moves twice as fast as M always, the 
point P will go along a line nearer N 
O Y ; and if N moves only half as 
fast as M, then P will go along a line 
nearer o x. And in general, the faster 
N moves, the more the line will be tilted up ; 
and if the rate at which N goes is changeable, 
the direction of P's motion will be changeable, 
and P will then describe a curve, which will be 
very steep when N is going fast, and more flat 
when N is going slow. So that the steepness 
of this curve is now a visible measure of the 
rate at which N is going, and the curvature of 
it is a visible expression of the fact that the 
rate is changeable. Now the hypothesis of 
continuity in the motion of N asserts not 
merely that N itself moves without any jumps, 
but that the rate at which N is going changes 
gradually without any jumps, and consequently 



THEORIES OF THE PHYSICAL FORCES 133 

that the direction of P's motion changes gradu- 
ally ; or that the curve described by P cannot 
have a sharp point like this. But it asserts a 
great deal more besides this, which I shall now 
endeavour to explain. Let us imagine a new 
point N 1? so moving that whenever the old N is 
going at four inches a second, N x shall be four 
inches from O ; and when N is going at two 
inches a second, N a shall be two inches from 
O, and so on, the distance of N x from being 
always exactly as far as N would go in a second 
if it went at the rate at which it was moving at 
that instant. Then the distance O N : measures 
the rate at which N is going, or the velocity of 
N. If, for example, there was a thing like a 
thermometer hung up in a train, so that the 
height of the mercury always indicated how 
fast the train was going ; when the train was 
going 17 miles an hour, the mercury stood at 
17 inches, and so on ; then the top of the 
mercury would behave towards the train exactly 
as I want the point NJ to behave towards the 
point N. It is to indicate by its height how 
fast N is going. 

If, then, the velocity of N is changeable, the 
point N X will move up and down ; and the rate 
at which N x moves up or down is clearly the 
rate at which the velocity of N is increasing or 
diminishing. This rate at which the velocity 
of N changes is called its acceleration. To 



134 LECTURES AND ESSAYS 

return to our gauge instead of a train, if in the 
course of a minute it went up from 1 7 to 19, 
the train would be said to have an acceleration 
of two miles an hour per minute. 

Now I shall take another point N 2 , which is 
to behave towards N x exactly as N t behaves 
towards N ; namely, the distance of N 2 from O 2 
is to be always equal to the number of inches 
which N X is going in a second. And then I 
shall take a point N 3 , related in just this same 
way to N 2 , and so on, until I come to a point 
that does not move at all ; and that I might 
never come to, so that I should have to go on 
taking new points for ever. But suppose now 
that I have got this series of points, and that 
they are all moving together. Then first of all 
there is my point N, which moves anyhow. 
Next there is N I} such that O l N x is the velocity 
of N, or the rate of change of N'S position. 
Next there is N 2 , such that O 2 N 2 is the acceler- 
ation of N, or the rate of change of the rate 
of change of N'S position. Then again O 3 N 3 
is the change of the acceleration of N, or 
the rate of change of the rate of change of 
the rate of change of N'S position, and so 
on. We may, if we like, agree to call the 
velocity of N the change of the first order, the 
velocity of N x the change of the second order, 
and so on. 

Then the hypothesis of the perfect continuity 



THEORIES OF THE PHYSICAL FORCES 135 

of N'S motion asserts that all these points move 
continuously without any jumps. Now, a jump 
made by any one of these points, being a finite 
change made in no time, would be a change 
made at an infinite rate ; the next point, there- 
fore, and all after it, would go right away from 
O, and disappear altogether. We may thus 
express the law of continuity also in this form ; 
that there is no infinite change of any order. 

Now, observe further that the rate at which 
anything is going is a property of the thing at 
that instant, and exists whether the thing goes 
any more or not. If I drop a marble on the 
floor, it goes faster and faster till it gets there, 
and then stops ; but at the instant when it hit 
the floor it was going at a perfectly definite rate, 
which can be calculated, though it did not 
actually go any more. 

In the same way the configuration of all 
these points which depend on the point N is a 
property of its motion at any given instant, 
quite independent of the continuance of that 
motion. I want you to take particular notice 
of this fact, that as the point N moves about, 
the whole set of points connected with it moves 
too ; and that you may regard them as con- 
nected by some machine, which you may stop 
at any moment to contemplate the simultane- 
ous positions of all these points ; and that this set 
of simultaneous positions belongs just simply 



136 LECTURES AND ESSAYS 

to that one position of the point N, and there- 
fore to one instant of time. 

Now I am going to state to you dogmatic- 
ally a certain mathematical theorem, called 
Taylor's theorem ; whereby you will see the 
very remarkable consequences of this hypothesis 
that we have made. 

Namely, there is a certain rule whereby when 
the positions of all these points are known for 
any particular instant of time, then their posi- 
tions at any other instant of time may be 
calculated from these ; and it is impossible 
that they should have at that other instant any 
other positions than those so calculated. Pro- 
vided always that there is no infinite change of 
any order ; that is to say, that no one of the 
points has taken a sudden jump and sent all 
the points after it away to an infinite distance 
from O at any instant between the one for 
which the positions are given and the one for 
which they are calculated. 

Remember that the positions of all the de- 
rivative points are mere properties of the motion 
of the point N at any instant ; that in fact we 
must know them all in order to know com- 
pletely the state of the point N at that instant. 
And then observe the result that we have 
arrived at. From the knowledge of the com- 
plete state at any instant of a thing whose 
motion obeys the law of continuity, we can 



THEORIES OF THE PHYSICAL FORCES 137 

calculate where it was at any past time, and 
where it will be at any future time. Now 
the hypothesis of continuity, of which we have 
only got disjointed fragments hitherto, is this ; 
that the motion of every particle of the whole 
universe is entirely continuous. It follows from 
this hypothesis that the state at this moment 
of any detached fragment say a particle of 
matter at the tip of my tongue is an infallible 
record of the eternal past, an infallible prediction 
of the eternal future. 

This is not the same as the statement that 
a complete knowledge of the position and 
velocity of every body in the universe at a 
given moment would suffice to determine the 
position at any previous or subsequent moment. 
That depends on an entirely different hypothesis, 
and relates to the whole, while this proposition 
that I am now expounding relates to every 
several part however small. Now reflect upon 
the fact that for a single particle quite irre- 
spective of everything else the history of 
eternity is contained in every second of time ; 
and then try if you can find room in this one 
stifling eternal fact for any secondary causes 
and the question why ? Why does the moon 
go round the earth ? When the Solar system 
was nebulous, anybody who knew all about some 
one particle of nebulous vapour might have 
predicted that it would at this moment form 



138 LECTURES AND ESSAYS 

part of the moon's mass, and be rotating about 
the earth exactly as it does. But why with 
an acceleration inversely as the square of the 
distance ? There is no why ; the fact is prob- 
ably equivalent to saying that the continuous 
motion of one body is such as not to interfere 
with the continuous motion of another. If 
once so, then always ; the cause is only the 
fact that at some moment the thing is so, 
or rather, the facts of one time are not the 
cause of the facts of another, but the facts of 
all time are included in one statement, and 
rigorously bound up together. 

Parallel, however, with this hypothesis of 
temporal continuity, there is another hypo- 
thesis, not so universally held, of a continuity in 
space ; for which indeed I hope to make more 
room presently. And out of this it appears 
that as the history of eternity is written in 
every second of time, so the state of the uni- 
verse is written in every point of space. 






ON THE AIMS AND INSTRUMENTS 
OF SCIENTIFIC THOUGHT 1 

IT may have occurred (and very naturally too) 
to such as have had the curiosity to read the 
title of this lecture, that it must necessarily be 
a very dry and difficult subject ; interesting to 
very few, intelligible to still fewer, and, above 
all, utterly incapable of adequate treatment 
within the limits of a discourse like this. It is 
quite true that a complete setting-forth of my 
subject would require a comprehensive treatise 
on logic, with incidental discussion of the main 
questions of metaphysics ; that it would deal 
with ideas demanding close study for their 
apprehension, and investigations requiring a 
peculiar taste to relish them. It is not my in- 
tention now to present you with such a treatise. 
The British Association, like the world in 
general, contains three classes of persons. In 
the first place, it contains scientific thinkers ; 

1 A Lecture delivered before the members of the British Associa- 
tion, at Brighton, on August 19, 1872. 



140 LECTURES AND ESSAYS 

that is to say, persons whose thoughts have 
very frequently the characters which I shall 
presently describe. Secondly, it contains persons 
who are engaged in work upon what are called 
scientific subjects, but who in general do not, 
and are not expected to, think about these 
subjects in a scientific manner. Lastly, it 
contains persons who suppose that their work 
and their thoughts are unscientific, but who 
would like to know something about the busi- 
ness of the other two classes aforesaid. Now, 
to any one who belonging to one of these classes 
considers either of the other two, it will be 
apparent that there is a certain gulf between 
him and them ; that he does not quite under- 
stand them, nor they him ; and that an oppor- 
tunity for sympathy and comradeship is lost 
through this want of understanding. It is this 
gulf that I desire to bridge over, to the best of 
my power. That the scientific thinker may 
consider his business in relation to the great 
life of mankind ; that the noble army of 
practical workers may recognise their fellowship 
with the outer world, and the spirit which must 
guide both ; that this so-called outer world may 
see in the work of science only the putting in 
evidence of all that is excellent in its own work, 
may feel that the kingdom of science is 
within it : these are the objects of the present 
discourse. And they compel me to choose 



AIMS OF SCIENTIFIC THOUGHT 141 

such portions of my vast subject as shall be 
intelligible to all, while they ought at least to 
command an interest universal, personal, and 
profound. 

In the first place, then, what is meant by 
scientific thought ? You may have heard some 
of it expressed in the various Sections this 
morning. You have probably also heard ex- 
pressed in the same places a great deal of 
unscientific thought ; notwithstanding that it 
was about mechanical energy, or about hydro- 
carbons, or about eocene deposits, or about 
malacopterygii. For scientific thought does 
not mean thought about scientific subjects with 
long names. There are no scientific subjects. 
The subject of science is the human universe ; 
that is to say, everything that is, or has been, 
or may be related to man. Let us then, taking 
several topics in succession, endeavour to make 
out in what cases thought about them is 
scientific, and in what cases not. 

Ancient astronomers observed that the 
relative motions of the sun and moon recurred 
all over again in the same order about every 
nineteen years. They were thus enabled to 
predict the time at which eclipses would take 
place. A calculator at one of our observatories 
can do a great deal more than this. Like 
them, he makes use of past experience to 
predict the future ; but he knows of a great 



142 LECTURES AND ESSAYS 

number of other cycles besides that one of the 
nineteen years, and takes account of all of them ; 
and he can tell about the solar eclipse of six 
years hence exactly when it will be visible, and 
how much of the sun's surface will be covered 
at each place, and, to a second, at what time of 
day it will begin and finish there. This pre- 
diction involves technical skill of the highest 
order ; but it does not involve scientific thought, 
as any astronomer will tell you. 

By such calculations the places of the planet 
Uranus at different times of the year had been 
predicted and set down. The predictions were 
not fulfilled. Then arose Adams, and from 
these errors in the prediction he calculated the 
place of an entirely new planet, that had never 
yet been suspected ; and you all know how the 
new planet was actually found in that place. 
Now this prediction does involve scientific 
thought, as any one who has studied it will tell 
you. 

Here then are two cases of thought about 
the same subject, both predicting events by the 
application of previous experience, yet we say 
one is technical and the other scientific. 

Now let us take an example from the 
building of bridges and roofs. When an 
opening is to be spanned over by a material 
construction, which must bear a certain weight 
without bending enough to injure itself, there 



AIMS OF SCIENTIFIC THOUGHT 143 

are two forms in which this construction can 
be made, the arch and the chain. Every part 
of an arch is compressed or pushed by the other 
parts ; every part of a chain is in a state of 
tension, or is pulled by the other parts. In 
many cases these forms are united. A girder 
consists of two main pieces or booms, of which 
the upper one acts as an arch and is compressed, 
while the lower one acts as a chain and is 
pulled ; and this is true even when both the 
pieces are quite straight. They are enabled to 
act in this way by being tied together, or 
braced, as it is called, by cross pieces, which 
you must often have seen. Now suppose that 
any good practical engineer makes a bridge or 
roof upon some approved pattern which has 
been made before. He designs the size and 
shape of it to suit the opening which has to be 
spanned ; selects his material according to the 
locality; assigns the strength which must be 
given to the several parts of the structure 
according to the load which it will have to 
bear. There is a great deal of thought in the 
making of this design, whose success is predicted 
by the application of previous experience ; it 
requires technical skill of a very high order ; 
but it is not scientific thought. On the other 
hand, Mr. Fleeming Jenkin l designs a roof 

1 On Braced Arches and Suspension Bridges. Edinburgh : 
Neill, 1870. 



144 LECTURES AND ESSAYS 

consisting of two arches braced together, instead 
of an arch and a chain braced together ; and 
although this form is quite different from any 
known structure, yet before it is built he assigns 
with accuracy the amount of material that must 
be put into every part of the structure in order 
to make it bear the required load, and this 
prediction may be trusted with perfect security. 
What is the natural comment on this? Why, 
that Mr. Fleeming Jenkin is a scientific engineer. 

Now it seems to me that the difference 
between scientific and merely technical thought, 
not only in these but in all other instances 
which I have considered, is just this : Both of 
them make use of experience to direct human 
action ; but while technical thought or skill 
enables a man to deal with the same circum- 
stances that he has met with before, scientific 
thought enables him to deal with different cir- 
! cumstances that he has never met with before. 
But how can experience of one thing enable us 
to deal with another quite different thing ? To 
answer this question we shall have to consider 
more closely the nature of scientific thought. 

Let us take another example. You know 
that if you make a dot on a piece of paper, and 
then hold a piece of Iceland spar over it, you 
will see not one dot but two. A mineralogist, 
by measuring the angles of a crystal, can tell 
you whether or no it possesses this property 



AIMS OF SCIENTIFIC THOUGHT 145 

without looking through it. He requires no 
scientific thought to do that. But Sir William 
Rowan Hamilton, the late Astronomer-Royal 
of Ireland, knowing these facts and also the 
explanation of them which Fresnel had given, 
thought about the subject, and he predicted 
that by looking through certain crystals in a 
particular direction we should see not two dots 
but a continuous circle. Mr. Lloyd made the 
experiment, and saw the circle, a result which 
had never been even suspected. This has 
always been considered one of the most signal 
instances of scientific thought in the domain of 
physics. It is most distinctly an application of 
experience gained under certain circumstances 
to entirely different circumstances. 

Now suppose that the night before coming 
down to Brighton you had dreamed of a 
railway accident caused by the engine getting 
frightened at a flock of sheep and jumping 
suddenly back over all the carriages ; the result 
of which was that your head was unfortunately 
cut off, so that you had to put it in your hat- 
box and take it back home to be mended. 
There are, I fear, many persons even at this 
day, who would tell you that after such a dream 
it was unwise to travel by railway to Brighton. 
This is a proposal that you should take experi- 
ence gained while you are asleep, when you 
have no common sense, experience about a 
VOL. I L 



i 4 6 LECTURES AND ESSAYS 

phantom railway, and apply it to guide you 
when you are awake and have common sense, 
in your dealings with a real railway. And yet 
this proposal is not dictated by scientific thought. 
Now let us take the great example of Bio- 
logy. I pass over the process of classification, 
which itself requires a great deal of scientific 
thought ; in particular when a naturalist who 
has studied and monographed a fauna or a 
flora rather than a family is able at once to 
pick out the distinguishing characters required 
for the subdivision of an order quite new to 
him. Suppose that we possess all this minute 
and comprehensive knowledge of plants and 
animals and intermediate organisms, their 
affinities and differences, their structures and 
functions ; a vast body of experience, collected 
by incalculable labour and devotion. Then 
comes Mr. Herbert Spencer : he takes that ex- 
perience of life which is not human, which is 
apparently stationary, going on in exactly the 
same way from year to year, and he applies 
that to tell us how to deal with the changing 
characters of human nature and human society. 
How is it that experience of this sort, vast as 
it is, can guide us in a matter so different from 
itself? How does scientific thought, applied 
to the development of a kangaroo foetus or the 
movement of the sap in exogens, make predic- 
tion possible for the first time in that most 



AIMS OF SCIENTIFIC THOUGHT 147 

important of all sciences, the relations of man 
with man ? 

In the dark or unscientific ages men had 
another way of applying experience to altered 
circumstances. They believed, for example, 
that the plant called Jew's-ear, which does bear 
a certain resemblance to the human ear, was a 
useful cure for diseases of that organ. This 
doctrine of " signatures," as it was called, exer- 
cised an enormous influence on the medicine of 
the time. I need hardly tell you that it is 
hopelessly unscientific ; yet it agrees with those 
other examples that we have been considering 
in this particular ; that it applies experience 
about the shape of a plant which is one cir- 
cumstance connected with it to dealings with 
its medicinal properties, which are other and 
different circumstances. Again, suppose that 
you had been frightened by a thunder-storm 
on land, or your heart had failed you in a storm 
at sea ; if any one then told you that in 
consequence of this you should always cultivate 
an unpleasant sensation in the pit of your 
stomach, till you took delight in it, that you 
should regulate your sane and sober life by the 
sensations of a moment of unreasoning terror : 
this advice would not be an example of scientific 
thought. Yet it would be an application of 
past experience to new and different circum- 
stances. 



148 LECTURES AND ESSAYS 

But you will already have observed what is 
the additional clause that we must add to our 
definition in order to describe scientific thought 
and that only. The step between experience 
about animals and dealings with changing 
humanity is the law of evolution. The step 
from errors in the calculated places of Uranus 
to the existence of Neptune is the law of 
gravitation. The step from the observed be- 
haviour of crystals to conical refraction is 
made up of laws of light and geometry. The 
step from old bridges to new ones is the laws 
of elasticity and the strength of materials. 

The step, then, from past experience to new 
circumstances must be made in accordance with 
an observed uniformity in the order of events. 
This uniformity has held good in the past in 
certain places ; if it should also hold good in 
the future and in other places, then, being com- 
bined with our experience of the past, it enables 
us to predict the future, and to know what is 
going on elsewhere ; so that we are able to 
regulate our conduct in accordance with this 
knowledge. 

The aim of scientific thought, then, is to 
apply past experience to new circumstances ; 
the instrument is an observed uniformity in the 
course of events. By the use of this instru- 
ment it gives us information transcending our 
experience, it enables us to infer things that we 



AIMS OF SCIENTIFIC THOUGHT 149 

have not seen from things that we have seen ; 
and the evidence for the truth of that infor- 
mation depends on our supposing that the 
uniformity holds good beyond our experience. 
I now want to consider this uniformity a little 
more closely ; to show how the character of 
scientific thought and the force of its inferences 
depend upon the character of the uniformity of 
Nature. I cannot of course tell you all that is 
known of this character without writing an 
encyclopaedia ; but I shall confine myself to 
two points of it about which it seems to me 
that just now there is something to be said. I 
want to find out what we mean when we say 
that the uniformity of Nature is exact ; and 
what we mean when we say that it is reasonable. 
When a student is first introduced to those 
sciences which have come under the dominion 
of mathematics, a new and wonderful aspect of 
Nature bursts upon his view. He has been 
accustomed to regard things as essentially more 
or less vague. All the facts that he has hitherto 
known have been expressed qualitatively, with 
a little allowance for error on either side. Things 
which are let go fall to the ground. A very 
observant man may know also that they fall 
faster as they go along. But our student is 
shown that, after falling for one second in a 
vacuum, a body is going at the rate of thirty- 
two feet per second, that after falling for two 



i$o LECTURES AND ESSAYS 

seconds it is going twice as fast, after going 
two and a half seconds two and a half times as 
fast If he makes the experiment, and finds a 
single inch per second too much or too little in 
the rate, one of two things must have happened : 
either the law of falling bodies has been wrongly 
stated, or the experiment is not accurate there 
is some mistake. He finds reason to think 
that the latter is always the case ; the more 
carefully he goes to work, the more of the error 
turns out to belong to the experiment. Again, 
he may know that water consists of two gases, 
oxygen and hydrogen, combined ; but he now 
learns that two pints of steam at a temperature 
of 150 Centigrade will always make two pints 
of hydrogen and one pint of oxygen at the 
same temperature, all of them being pressed as 
much as the atmosphere is pressed. If he 
makes the experiment and gets rather more or 
less than a pint of oxygen, is the law disproved ? 
No ; the steam was impure, or there was some 
mistake. Myriads of analyses attest the law of 
combining volumes ; the more carefully they 
are made, the more nearly they coincide with 
it The aspects of the faces of a crystal are 
connected together by a geometrical law, by 
which, four of them being given, the rest can 
be found. The place of a planet at a given 
time is calculated by the law of gravitation ; if 
it is half a second wrong, the fault is in the 



AIMS OF SCIENTIFIC THOUGHT 151 

instrument, the observer, the clock, or the law ; 
now, the more observations are made, the more 
of this fault is brought home to the instrument, 
the observer, and the clock. It is no wonder, 
then, that our student, contemplating these and 
many like instances, should be led to say, " I 
have been short-sighted ; but I have now put 
on the spectacles of science which Nature had 
prepared for my eyes ; I see that things have 
definite outlines, that the world is ruled by 
exact and rigid mathematical laws ; xal <rv, 
0eo9, 7ecoytteTpe4<?." It is our business to con- 
sider whether he is right in so concluding. Is 
the uniformity of Nature absolutely exact, or 
only more exact than our experiments ? 

At this point we have to make a very im- 
portant distinction. There are two ways in 
which a law may be inaccurate. The first way 
is exemplified by that law of Galileo which I 
mentioned just now : that a body falling in 
vacuo acquires equal increase in velocity in 
equal times. No matter how many feet per 
second it is going, after an interval of a second 
it will be going thirty-two more feet per second. 
We now know that this rate of increase is not 
exactly the same at different heights, that it 
depends upon the distance of the body from 
the centre of the earth ; so that the law is only 
approximate ; instead of the increase of velocity 
being exactly equal in equal times, it itself 



152 LECTURES AND ESSAYS 

increases very slowly as the body falls. We 
know also that this variation of the law from 
the truth is too small to be perceived by direct 
observation on the change of velocity. But 
suppose we have invented means for observing 
this, and have verified that the increase of 
velocity is inversely as the squared distance 
from the earth's centre. Still the law is not 
accurate ; for the earth does not attract ac- 
curately towards her centre, and the direction 
of attraction is continually varying with the 
motion of the sea ; the body will not even fall 
in a straight line. The sun and the planets, 
too, especially the moon, will produce deviations ; 
yet the sum of all these errors will escape our 
new process of observation, by being a great 
deal smaller than the necessary errors of that 
observation. But when these again have been 
allowed for, there is still the influence of the 
stars. In this case, however, we only give up 
one exact law for another. It may still be 
held that if the effect of every particle of matter 
in the universe on the falling body were calcu- 
lated according to the law of gravitation, the 
body would move exactly as this calculation 
required. And if it were objected that the 
body must be slightly magnetic or diamagnetic, 
while there are magnets not an infinite way off; 
that a very minute repulsion, even at sensible 
distances, accompanies the attraction ; it might 



AIMS OF SCIENTIFIC THOUGHT 153 

be replied that these phenomena are themselves 
subject to exact laws, and that when all the 
laws have been taken into account, the actual 
motion will exactly correspond with the calcu- 
lated motion. 

I suppose there is hardly a physical student 
(unless he has specially considered the matter) 
who would not at once assent to the statement 
I have just made ; that if we knew all about it, 
Nature would be found universally subject to 
exact numerical laws. But let us just consider 
for another moment what this means. 

The word " exact " has a practical and a 
theoretical meaning. When a grocer weighs 
you out a certain quantity of sugar very care- 
fully, and says it is exactly a pound, he means 
that the difference between the mass of the 
sugar and that of the pound weight he employs 
is too small to be detected by his scales. If a 
chemist had made a special investigation, wish- 
ing to be as accurate as he could, and told you 
this was exactly a pound of sugar, he would 
mean that the mass of the sugar differed from 
that of a certain standard piece of platinum by 
a quantity too small to be detected by his 
means of weighing, which are a thousandfold 
more accurate than the grocer's. But what 
would a mathematician mean, if he made the 
same statement ? He would mean this. 
Suppose the mass of the standard pound to be 



154 LECTURES AND ESSAYS 

represented by a length, say a foot, measured 
on a certain line ; so that half a pound would 
be represented by six inches, and so on. And 
let the difference between the mass of the sugar 
and that of the standard pound be drawn upon 
the same line to the same scale. Then, if that 
difference were magnified an infinite number of 
times, it would still be invisible. This is the 
theoretical meaning of exactness ; the practical 
meaning is only very close approximation ; 
how close, depends upon the circumstances. 
The knowledge then of an exact law in the 
theoretical sense would be equivalent to an 
infinite observation. I do not say that such 
knowledge is impossible to man ; but I do say 
that it would be absolutely different in kind 
from any knowledge that we possess at present 
I shall be told, no doubt, that we do possess 
a great deal of knowledge of this kind, in the 
form of geometry and mechanics ; and that it 
is just the example of these sciences that has 
led men to look for exactness in other quarters. 
If this had been said to me in the last century, 
I should not have known what to reply. But 
it happens that about the beginning of the 
present century the foundations of geometry 
were criticised independently by two mathe- 
maticians, Lobatschewsky l and the immortal 

1 Geometrische Untersuchvngen sur Theorie der Parallellinien 
Berlin. 1840. Translated by Hottel. Gauthier-Villars, 1866. 



AIMS OF SCIENTIFIC THOUGHT 155 

Gauss ; l whose results have been extended and 
generalised more recently by Riemann 2 and 
Helmholtz. 3 And the conclusion to which 
these investigations lead is that, although the 
assumptions which were very properly made by 
the ancient geometers are practically exact 
that is to say, more exact than experiment 
can be for such finite things as we have to 
deal with, and such portions of space as we can 
reach ; yet the truth of them for very much 
larger things, or very much smaller things, or 
parts of space which are at present beyond our 
reach, is a matter to be decided by experiment, 
when its powers are considerably increased. I 
want to make as clear as possible the real state 
of this question at present, because it is often 
supposed to be a question of words or meta- 
physics, whereas it is a very distinct and simple 
question of fact. I am supposed to know then 
that the three angles of a rectilinear triangle 
are exactly equal to two right angles. Now 
suppose that three points are taken in space, 
distant from one another as far as the Sun is 
from a Centauri, and that the shortest distances 
between these points are drawn so as to form 
a triangle. And suppose the angles of this 

1 Letter to Schumacher, Nov. 28, 1846 (refers to 1792). 

2 Ueber die Hypothesen -wekhe der Geometrie zu Grunde liegen. 
Gottingen, Abhandl., 1866-67. Translated by Hotiel in Annali 
di Matematica, Milan, vol. iii. 

3 The Axioms of Geometry, Academy, vol. i. p. 128 (a popular 
exposition). [And see now his article in Mind, No. III.] 



IJ6 LECTURES AND ESSAYS 

triangle to be very accurately measured and 
added together ; this can at present be done so 
accurately that the error shall certainly be less 
than one minute, less therefore than the five- 
thousandth part of a right angle. Then I do 
not know that this sum would differ at all from 
two right angles ; but also I do not know that 
the difference would be less than ten degrees, 
or the ninth part of a right angle. 1 And I have 
reasons for not knowing. 

This example is exceedingly important as 
showing the connection between exactness and 
universality. It is found that the deviation if 
it exists must be nearly proportional to the 
area of the triangle. So that the error in the 
case of a triangle whose sides are a mile long 
would be obtained by dividing that in the case 
I have just been considering by four hundred 
quadrillions ; the result must be a quantity in- 
conceivably small, which no experiment could 
detect. But between this inconceivably small 
error and no error at all, there is fixed an 
enormous gulf ; the gulf between practical and 
theoretical exactness, and, what is even more 
important, the gulf between what is practically 
universal and what is theoretically universal. 
I say that a law is practically universal which 

1 Assuming that parallax observations prove the deviation less 
than half a second for a triangle whose vertex is at the star and base 
a diameter of the earth's orbit 



AIMS OF SCIENTIFIC THOUGHT 157 

is more exact than experiment for all cases 
that might be got at by such experiments as 
we can make. We assume this kind of univer- 
sality, and we find that it pays us to assume 
it. But a law would be theoretically universal 
if it were true of all cases whatever ; and this 
is what we do not know of any law at all. 

I said there were two ways in which a law 
might be inexact. There is a law of gases 
which asserts that when you compress a perfect 
gas the pressure of the gas increases exactly in 
the proportion in which the volume diminishes. 
Exactly ; that is to say, the law is more accurate 
than the experiment, and experiments are 
corrected by means of the law. But it so 
happens that this law has been explained ; we 
know precisely what it is that happens when a 
gas is compressed. We know that a gas 
consists of a vast number of separate molecules, 
rushing about in all directions with all manner 
of velocities, but so that the mean velocity of 
the molecules of air in this room, for example, 
is about twenty miles a minute. The pressure 
of the gas on any surface with which it is in 
contact is nothing more than the impact of 
these small particles upon it. On any surface 
large enough to be seen there are millions of 
these impacts in a second. If the space in 
which the gas is confined be diminished, the 
average rate at which the impacts take place 



158 LECTURES AND ESSAYS 

will be increased in the same proportion ; and 
because of the enormous number of them, the 
actual rate is always exceedingly close to the 
average. But the law is one of statistics ; its 
accuracy depends on the enormous numbers 
involved ; and so, from the nature of the case, 
its exactness cannot be theoretical or absolute. : 

Nearly all the laws of gases have received 
these statistical explanations ; electric and 
magnetic attraction and repulsion have been 
treated in a similar manner ; and an hypothesis 
of this sort has been suggested even for the law 
of gravity. On the other hand the manner in 
which the molecules of a gas interfere with each 
other proves that they repel one another in- 
versely as the fifth power of the distance ; x so 
that we here find at the basis of a statistical 
explanation a law which has the form of 
theoretical exactness. Which of these forms is 
to win ? It seems to me again that we do not 
know, and that the recognition of our ignorance 
is the surest way to get rid of it. 

The world in general has made just the 
remark that I have attributed to a fresh student 
of the applied sciences. As the discoveries of 
Galileo, Kepler, Newton, Dalton, Cavendish, 
Gauss, displayed ever new phenomena following 
mathematical laws, the theoretical exactness of 

1 [This statement of the law has since been abandoned : see 
"The Unseen Universe," below.] 



AIMS OF SCIENTIFIC THOUGHT 159 

the physical universe was taken for granted. 
Now, when people are hopelessly ignorant of a 
thing, they quarrel about the source of their 
knowledge. Accordingly many maintained 
that we know these exact laws by intuition. 
These said always one true thing, that we did 
not know them from experience. Others said 
that they were really given in the facts, and 
adopted ingenious ways of hiding the gulf 
between the two. Others again deduced from 
transcendental considerations sometimes the laws 
themselves, and sometimes what through im- 
perfect information they supposed to be the 
laws. But more serious consequences arose 
when these conceptions derived from Physics 
were carried over into the field of Biology. 
Sharp lines of division were made between 
kingdoms and classes and orders ; an animal 
was described as a miracle to the vegetable 
world ; specific differences which are practically 
permanent within the range of history were 
regarded as permanent through all time ; a 
sharp line was drawn between organic and 
inorganic matter. Further investigation, how- 
ever, has shown that accuracy had been pre- 
maturely attributed to the science, and has filled 
up all the gulfs and gaps that hasty observers 
had invented. The animal and vegetable 
kingdoms have a debateable ground between 
them, occupied by beings that have the char- 



160 LECTURES AND ESSAYS 

acters of both and yet belong distinctly to 
neither. Classes and orders shade into one 
another all along their common boundary. 
Specific differences turn out to be the work of 
time. The line dividing organic matter from 
inorganic, if drawn to-day, must be moved 
to-morrow to another place ; and the chemist 
will tell you that the distinction has now no 
place in his science except in a technical sense 
for the convenience of studying carbon com- 
pounds by themselves. In Geology the same 
tendency gave birth to the doctrine of distinct 
periods, marked out by the character of the 
strata deposited in them all over the sea ; 
a doctrine than which, perhaps, no ancient 
cosmogony has been further from the truth, 
or done more harm to the progress of 
science. Refuted many years ago by Mr. 
Herbert Spencer, 1 it has now fairly yielded to 
an attack from all sides at once, and may be 
left in peace. 

When then we say that the uniformity which 
we observe in the course of events is exact and 
universal, we mean no more than this : that we 
are able to state general rules which are far 
more exact than direct experiment, and which 
apply to all cases that we are at present likely 
to come across. It is important to notice, 

1 "Illogical Geology," in Essays, vol. i. Originally published 
in 1859. 



AIMS OF SCIENTIFIC THOUGHT 161 

however, the effect of such exactness as we 
observe upon the nature of inference. When a 
telegram arrived stating that Dr. Livingstone 
had been found by Mr. Stanley, what was the 
process by which you inferred the finding of 
Dr. Livingstone from the appearance of the 
telegram ? You assumed over and over again 
the existence of uniformity in nature. That 
the newspapers had behaved as they generally 
do in regard to telegraphic messages ; that the 
clerks had followed the known laws of the 
action of clerks ; that electricity had behaved 
in the cable exactly as it behaves in the 
laboratory ; that the actions of Mr. Stanley 
were related to his motives by the same 
uniformities that affect the actions of other 
men ; that Dr. Livingstone's handwriting con- 
formed to the curious rule by which an ordinary 
man's handwriting may be recognised as having 
persistent characteristics even at different periods 
of his life. But you had a right to be much 
more sure about some of these inferences than 
about others. The law of electricity was known 
with practical exactness, and the conclusions 
derived from it were the surest things of all. 
The law about the handwriting, belonging to 
a portion of physiology which is unconnected 
with consciousness, was known with less, but 
still with considerable accuracy. But the laws 
of human action in which consciousness is 
VOL. I M 



162 LECTURES AND ESSAYS 

concerned are still so far from being completely 
analysed and reduced to an exact form that 
the inferences which you made by their help 
were felt to have only a provisional force. It 
is possible that by and by, when psychology 
has made enormous advances and become an 
exact science, we may be able to give to 
testimony the sort of weight which we give to 
the inferences of physical science. It will then 
be possible to conceive a case which will show 
how completely the whole process of inference 
depends on our assumption of uniformity. 
Suppose that testimony, having reached the 
ideal force I have imagined, were to assert that 
a certain river runs uphill. You could infer 
nothing at all. The arm of inference would be 
paralysed, and the sword of truth broken in its 
grasp ; and reason could only sit down and 
wait until recovery restored her limb, and 
further experience gave her new weapons. 

I want in the next place to consider what 
we mean when we say that the uniformity 
which we have observed in the course of events 
is reasonable as well as exact. 

No doubt the first form of this idea was sug- 
gested by the marvellous adaptation of certain 
natural structures to special functions. The 
first impression of those who studied comparative 
anatomy was that every part of the animal 
frame was fitted with extraordinary complete- 



AIMS OF SCIENTIFIC THOUGHT 163 

ness for the work that it had to do. I say 
extraordinary, because at the time the most 
familiar examples of this adaptation were 
manufactures produced by human ingenuity ; 
and the completeness and minuteness of natural 
adaptations were seen to be far in advance of 
these. The mechanism of limbs and joints 
was seen to be adapted, far better than any 
existing ironwork, to those motions and com- 
binations of motion which were most useful to 
the particular organisms. The beautiful and 
complicated apparatus of sensation caught up 
indications from the surrounding medium, 
sorted them, analysed them, and transmitted 
the results to the brain in a manner with which, 
at the time I am speaking of, no artificial 
contrivance could compete. Hence the belief 
grew amongst physiologists that every structure 
which they found must have its function and 
subserve some useful purpose ; a belief which 
was not without its foundation in fact, and 
which certainly (as Dr. Whewell remarks) has 
done admirable service in promoting the growth 
of physiology. Like all beliefs found successful 
in one subject, it was carried over into another, 
of which a notable example is given in the 
speculations of Count Rumford about the 
physical properties of water. Pure water attains 
its greatest density at a temperature of about 
39^ Fahrenheit ; it expands and becomes 



1 64 LECTURES AND ESSAYS 

lighter whether it is cooled or heated, so as to 
alter that temperature. Hence it was concluded 
that water in this state must be at the bottom 
of the sea, and that by such means the sea was 
kept from freezing all through ; as it was 
supposed must happen if the greatest density 
had been that of ice. Here then was a sub- 
stance whose properties were eminently adapted 
to secure an end essential to the maintenance 
of life upon the earth. In short, men came to 
the conclusion that the order of nature was 
reasonable in the sense that everything was 
adapted to some good end. 

Further consideration, however, has led men 
out of that conclusion in two different ways. 
First, it was seen that the facts of the case had 
been wrongly stated. Cases were found of 
wonderfully complicated structures that served 
no purpose at all ; like the teeth of that whale 
of which you heard in Section D the other 
day, or of the Dugong, which has a horny 
palate covering them all up and used instead of 
them ; like the eyes of the unborn mole, that 
are never used, though perfect as those of a 
mouse until the skull opening closes up, cutting 
them off from the brain, when they dry up and 
become incapable of use ; like the outsides of 
your own ears, which are absolutely of no use 
to you. And when human contrivances were 
more advanced it became clear that the natural 



AIMS OF SCIENTIFIC THOUGHT 165 

adaptations were subject to criticism. The eye, 
regarded as an optical instrument of human 
manufacture, was thus described by Helmholtz 
the physiologist who learned physics for the 
sake of his physiology, and mathematics for the 
sake of his physics, and is now in the first 
rank of all three. He said, "If an optician 
sent me that as an instrument, I should send it 
back to him with grave reproaches for the care- 
lessness of his work, and demand the return of 
my money." 

The extensions of the doctrine into Physics 
were found to be still more at fault That 
remarkable property of pure water, which was 
to have kept the sea from freezing, does not 
belong to salt water, of which the sea itself is 
composed. It was found, in fact, that the idea 
of a reasonable adaptation of means to ends, 
useful as it had been in its proper sphere, could 
yet not be called universal, or applied to the 
order of nature as a whole. 

Secondly, this idea has given way because it 
has been superseded by a higher and more 
general idea of what is reasonable, which has 
the advantage of being applicable to a large 
portion of physical phenomena besides. Both 
the adaptation and the non-adaptation which 
occur in organic structures have been explained, 
The scientific thought of Dr. Darwin, of Mr. 
Herbert Spencer, and of Mr. Wallace, has 



166 LECTURES AND ESSAYS 

described that hitherto unknown process of 
adaptation as consisting of perfectly well-known 
and familiar processes. There are two kinds of 
these : the direct processes, in which the physical 
changes required to produce a structure are 
worked out by the very actions for which that 
structure becomes adapted as the backbone or 
notochord has been modified from generation 
to generation by the bendings which it has 
undergone ; and the indirect processes included 
under the head of Natural Selection the 
reproduction of children slightly different from 
their parents, and the survival of those which are 
best fitted to hold their own in the struggle for 
existence. Naturalists might give you some 
idea of the rate at which we are getting ex- 
planations of the evolution of all parts of animals 
and plants the growth of the skeleton, of the 
nervous system and its mind, of leaf and flower. 
But what then do we mean by explanation ? 

We were considering just now an explana- 
tion of a law of gases the law according to 
which pressure increases in the same proportion 
in which volume diminishes. The explanation 
consisted in supposing that a gas is made up of 
a vast number of minute particles always flying 
about and striking against one another, and 
then showing that the rate of impact of such a 
crowd of particles on the sides of the vessel 
containing them would vary exactly as the 



AIMS OF SCIENTIFIC THOUGHT 167 

pressure is found to vary. Suppose the vessel 
to have parallel sides, and that there is only one 
particle rushing backwards and forwards between 
them ; then it is clear that if we bring the sides 
together to half the distance, the particle will 
hit each of them twice as often, or the pressure 
will be doubled. Now it turns out that this 
would be just as true for millions of particles as 
for one, and when they are flying in all directions 
instead of only in one direction and its opposite. 
Observe now ; it is a perfectly well-known and 
familiar thing that a body should strike against 
an opposing surface and bound off again ; and 
it is a mere everyday occurrence that what has 
only half so far to go should be back in half the 
time ; but that pressure should be strictly pro- 
portional to density is a comparatively strange, 
unfamiliar phenomenon. The explanation de- 
scribes the unknown and unfamiliar as being 
made up of the known and the familiar ; and 
this, it seems to me, is the true meaning of 
explanation. 1 

Here is another instance. If small pieces 
of camphor are dropped into water, they will 
begin to spin round and swim about in a most 
marvellous way. Mr. Tomlinson gave, I believe, 

1 This view differs from those of Mr. J. S. Mill and Mr. Herbert 
Spencer in requiring every explanation to contain an addition 
to our knowledge about the thing explained. Both these writers 
regard subsumption under a general law as a species of explana- 
tion. See also Ferrier's Remains, vol. ii. p. 436. 



168 LECTURES AND ESSAYS 

the explanation of this. We must observe, to 
begin with, that every liquid has a skin which 
holds it ; you can see that to be true in the case 
of a drop, which looks as if it were held in a 
bag. But the tension of this skin is greater in 
some liquids than in others ; and it is greater 
in camphor and water than in pure water. 
When the camphor is dropped into water it 
begins to dissolve and get surrounded with 
camphor and water instead of water. If the 
fragment of camphor were exactly symmetrical, 
nothing more would happen ; the tension would 
be greater in its immediate neighbourhood, 
but no motion would follow. The camphor, 
however, is irregular in shape ; it dissolves 
more on one side than the other ; and con- 
sequently gets pulled about, because the tension 
of the skin is greater where the camphor is most 
dissolved. Now it is probable that this is not 
nearly so satisfactory an explanation to you as 
it was to me when I was first told of it ; and 
for this reason. By that time I was already 
perfectly familiar with the notion of a skin upon 
the surface of liquids, and I had been taught by 
means of it to work out problems in capillarity. 
The explanation was therefore a description of 
the unknown phenomenon which I did not 
know how to deal with as made up of known 
phenomena which I did know how to deal with. 
But to many of you possibly the liquid skin 



AIMS OF SCIENTIFIC THOUGHT 169 

may seem quite as strange and unaccountable 
as the motion of camphor on water. 

And this brings me to consider the source 
of the pleasure we derive from an explanation. 
By known and familiar I mean that which we 
know how to deal with, either by action in the 
ordinary sense, or by active thought. When 
therefore that which we do not know how to 
deal with is described as made up of things that 
we do know how to deal with, we have that 
sense of increased power which is the basis of 
all higher pleasures. Of course we may after- 
wards by association come to take pleasure in 
explanation for its own sake. Are we then to 
say that the observed order of events is reason- 
able, in the sense that all of it admits of 
explanation ? That a process may be capable 
of explanation, it must break up into simpler 
constituents which are already familiar to us. 
Now, first, the process may itself be simple, and 
not break up ; secondly, it may break up into 
elements which are as unfamiliar and impractic- 
able as the original process. 

It is an explanation of the moon's motion 
to say that she is a falling body, only she is 
going so fast and is so far off that she falls 
quite round to the other side of the earth, 
instead of hitting it ; and so goes on for ever. 
But it is no explanation to say that a body 
falls because of gravitation. That means that 



170 LECTURES AND ESSAYS 

the motion of the body may be resolved into 
a motion of every one of its particles towards 
every one of the particles of the earth, with an 
acceleration inversely as the square of the 
distance between them. But this attraction of 
two particles must always, I think, be less 
familiar than the original falling body, however 
early the children of the future begin to read 
their Newton. Can the attraction itself be 
explained ? Le Sage said that there is an 
everlasting hail of innumerable small ether- 
particles from all sides, and that the two 
material particles shield each other from this 
and so get pushed together. This is an explana- 
tion ; it may or may not be a true one. The 
attraction may be an ultimate simple fact ; or it 
may be made up of simpler facts utterly unlike 
anything that we know at present ; and in either 
of these cases there is no explanation. We have 
no right to conclude, then, that the order of 
events is always capable of being explained. 

There is yet another way in which it is said 
that Nature is reasonable ; namely, inasmuch 
as every effect has a cause. What do we mean 
by this ? 

In asking this question, we have entered 
upon an appalling task. The word represented 
by " cause " has sixty-four meanings in Plato 
and forty-eight in Aristotle. These were men 
who liked to know as near as might be what 



AIMS OF SCIENTIFIC THOUGHT 171 

they meant ; but how many meanings it has had 
in the writings of the myriads of people who 
have not tried to know what they meant by it 
will, I hope, never be counted. It would not 
only be the height of presumption in me to 
attempt to fix the meaning of a word which 
has been used by so grave authority in so many 
and various senses ; but it would seem a thank- 
less task to do that once more which has been 
done so often at sundry times and in divers 
manners before. And yet without this we 
cannot determine what we mean by saying 
that the order of nature is reasonable. I shall 
evade the difficulty by telling you Mr. Grote's 
opinion. 1 You come to a scarecrow and ask, 
what is the cause of this ? You find that a 
man made it to frighten the birds. You go 
away and say to yourself, "Everything resembles 
this scarecrow. Everything has a purpose." 
And from that day the word " cause " means 
for you what Aristotle meant by " final cause." 
Or you go into a hairdresser's shop, and wonder 
what turns the wheel to which the rotatory 
brush is attached. On investigating other 
parts of the premises, you find a man working 
away at a handle. Then you go away and 
say, " Everything is like that wheel. If I in- 
vestigated enough, I should always find a man 
at a handle." And the man at the handle, or 
1 Plato, vol. ii. (Phaedo). 



172 LECTURES AND ESSAYS 

whatever corresponds to him, is from henceforth 
known to you as " cause." 

And so generally. When you have made 
out any sequence of events to your entire 
satisfaction, so that you know all about it, the 
laws involved being so familiar that you seem 
to see how the beginning must have been 
followed by the end, then you apply that as a 
simile to all other events whatever, and your 
idea of cause is determined by it. Only when 
a case arises, as it always must, to which the 
simile will not apply, you do not confess to 
yourself that it was only a simile and need not 
apply to everything, but you say, " The cause 
of that event is a mystery which must remain 
for ever unknown to me." On equally just 
grounds the nervous system of my umbrella is 
a mystery which must remain for ever unknown 
to me. My umbrella has no nervous system ; 
and the event to which your simile did not 
apply has no cause in your sense of the word. 
When we say then that every effect has a 
cause, we mean that every event is connected 
with something in a way that might make 
somebody call that the cause of it. But I, at 
least, have never yet seen any single meaning 
of the word that could be fairly applied to the 
whole order of nature. 

From this remark I cannot even except an 
attempt recently made by Mr. Bain to give the 



AIMS OF SCIENTIFIC THOUGHT 173 

word a universal meaning, though I desire to 
speak of that attempt with the greatest respect. 
Mr. Bain l wishes to make the word " cause " 
hang on in some way to what we call the law 
of energy ; but though I speak with great 
diffidence I do think a careful consideration 
will show that the introduction of this word 
" cause " can only bring confusion into a matter 
which is distinct and clear enough to those who 
have taken the trouble to understand what 
energy means. It would be impossible to 
explain that this evening ; but I may mention 
that " energy " is a technical term out of 
mathematical physics, which requires of most 
men a good deal of careful study to understand 
it accurately. 

Let us pass on to consider, with all the 
reverence which it demands, another opinion 
held by great numbers of the philosophers who 
have lived in the Brightening Ages of Europe ; 
the opinion that at the basis of the natural 
order there is something which we can know 
to be unreasonable, to evade the processes of 
human thought. The opinion is set forth first 
by Kant, so far as I know, in the form of his 
famous doctrine of the antinomies or contra- 
dictions, a later form 2 of which I will endeavour 



1 Inductive Logic, chap. iv. 

2 That of Mr. Herbert Spencer, First Principles. I believe 
Kant himself would have admitted that the antinomies do not 



174 LECTURES AND ESSAYS 

to explain to you. It is said, then, that space 
must either be infinite or have a boundary. 
Now you cannot conceive infinite space ; and 
you cannot conceive that there should be any 
end to it Here, then, are two things, one of 
which must be true, while each of them is in- 
conceivable ; so that our thoughts about space 
are hedged in, as it were, by a contradiction. 
Again, it is said that matter must either be 
infinitely divisible, or must consist of small 
particles incapable of further division. Now 
you cannot conceive a piece of matter divided 
into an infinite number of parts, while, on the 
other hand, you cannot conceive a piece of 
matter, however small, which absolutely cannot 
be divided into two pieces ; for, however great 
the forces are which join the parts of it together, 
you can imagine stronger forces able to tear it 
in pieces. Here, again, there are two state- 
ments, one of which must be true, while each 
of them is separately inconceivable ; so that 
our thoughts about matter also are hedged in 
by a contradiction. There are several other 
cases of the same thing, but I have selected 
these two as instructive examples. And the 
conclusion to which philosophers were led by 
the contemplation of them was that on every 



exist for the empiricist. [Much less does he say that either of a 
pair of antinomies must be true. The real Kantian position is 
that both assertions are illegitimate.] 



AIMS OF SCIENTIFIC THOUGHT 175 

side, when we approach the limits of existence, 
a contradiction must stare us in the face. The 
doctrine has been developed and extended by 
the great successors of Kant ; and this un- 
reasonable, or unknowable, which is also called 
the absolute and the unconditioned, has been 
set forth in various ways as that which we 
know to be the true basis of all things. As I 
said before, I approach this doctrine with all 
the reverence which should be felt for that 
which has guided the thoughts of so many of 
the wisest of mankind. Nevertheless I shall 
endeavour to show that in these cases of 
supposed contradiction there is always some- 
thing which we do not know now, but of which 
we cannot be sure that we shall be ignorant 
next year. The doctrine is an attempt to 
found a positive statement upon this ignorance, 
which can hardly be regarded as justifiable. 
Spinoza said, "A free man thinks of nothing 
so little as of death ; " it seems to me we may 
parallel this maxim in the case of thought, and 
say, " A wise man only remembers his ignorance 
in order to destroy it." A boundary is that 
which divides two adjacent portions of space. 
The question, then, " Has space (in general) a 
boundary ? " involves a contradiction in terms, 
and is, therefore, unmeaning. But the question, 
" Does space contain a finite number of cubic 
miles, or an infinite number ? " is a perfectly 



176 LECTURES AND ESSAYS 

intelligible and reasonable question which re- 
mains to be answered by experiment. 1 The 
surface of the sea would still contain a finite 
number of square miles, if there were no land 
to bound it Whether or fno the space in 
which we live is of this nature remains to be 
seen. If its extent is finite, we may quite 
possibly be able to assign that extent next 
year ; if, on the other hand, it has no end, it is 
true that the knowledge of that fact would be 
quite different from any knowledge we at 
present possess, but we have no right to say 
that such knowledge is impossible. Either the 
question will be settled once for all, or the 
extent of space will be shown to be greater 
than a quantity which will increase from year 
to year with the improvement of our sources 
of knowledge. Either alternative is perfectly 
conceivable, and there is no contradiction. 
Observe especially that the supposed contra- 
diction arises from the assumption of theoretical 
exactness in the laws of geometry. The other 
case that I mentioned has a very similar origin. 
The idea of a piece of matter the parts of which 
are held together by forces, and are capable of 
being torn asunder by greater forces, is entirely 
derived from the large pieces of matter which 
we have to deal with. We do not know 

' 1 The very important distinction between unboundedntss and 
infinite extent is made by Riemann, loc. cit. 



AIMS OF SCIENTIFIC THOUGHT 177 

whether this idea applies in any sense even to 
the molecules of gases ; still less can we apply 
it to the atoms of which they are composed. 
The word force is used of two phenomena : the 
pressure, which when two bodies are in contact 
connects the motion of each with the position 
of the other ; and attraction or repulsion, 
that is to say, a change of velocity in one body 
depending on the position of some other body 
which is not in contact with it. We do not 
know that there is anything corresponding to 
either of these phenomena in the case of a 
molecule. A meaning can, however, be given 
to the question of the divisibility of matter in 
this way. We may ask if there is any piece of 
matter so small that its properties as matter 
depend upon its remaining all in one piece. 
This question is reasonable ; but we cannot 
answer it at present, though we are not at all 
sure that we shall be equally ignorant next 
year. If there is no such piece of matter, no 
such limit to the division which shall leave it 
matter, the knowledge of that fact would be 
different from any of our present knowledge ; 
but we have no right to say that it is impossible. 
If, on the other hand, there is a limit, it is quite 
possible that we may have measured it by the 
time the Association meets at Bradford. Again, 
when we are told that the infinite extent of 
space, for example, is something that we cannot 
VOL. I N 



i;8 LECTURES AND ESSAYS 

conceive at present, we may reply that this is 
only natural, since our experience has never 
yet supplied us with the means of conceiving 
such things. But then we cannot be sure that 
the facts will not make us learn to conceive 
them ; in which case they will cease to be 
inconceivable. In fact, the putting of limits to 
human conception must always involve the 
assumption that our previous experience is 
universally valid in a theoretical sense ; an 
assumption which we have already seen reason 
to reject. Now you will see that our considera- 
tion of this opinion has led us to the true sense 
of the assertion that the Order of Nature is 
reasonable. If you will allow me to define a 
reasonable question as one which is asked in 
terms of ideas justified by previous experience, 
without itself contradicting that experience, 
then we may say, as the result of our investi- 
gation, that to every reasonable question there 
is an intelligible answer which either we or 
posterity may know. 

We have, then, come somehow to the follow- 
ing conclusions. By scientific thought we mean 
the application of past experience to new circum- 
stances by means of an observed order of events. 
By saying that this order of events is exact we 
mean that it is exact enough to correct experi- 
ments by, but we do not mean that it is theo- 
retically or absolutely exact, because we do 



AIMS OF SCIENTIFIC THOUGHT 179 

not know. The process of inference we found 
to be in itself an assumption of uniformity, and 
we found that, as the known exactness of the 
uniformity became greater, the stringency of 
the inference increased. By saying that the 
order of events is reasonable we do not mean 
that everything has a purpose, or that every- 
thing can be explained, or that everything has 
a cause ; for neither of these is true. But we 
mean that to every reasonable question there 
is an intelligible answer, which either we or 
posterity may know by the exercise of scientific 
thought. 

For I specially wish you not to go away 
with the idea that the exercise of scientific 
thought is properly confined to the subjects 
from which my illustrations have been chiefly 
drawn to-night. When the Roman jurists 
applied their experience of Roman citizens to 
dealings between citizens and aliens, showing 
by the difference of their actions that they re- 
garded the circumstances as essentially different, 
they laid the foundations of that great structure 
which has guided the social progress of Europe. 
That procedure was an instance of strictly 
scientific thought. When a poet finds that he 
has to move a strange new world which his 
predecessors have not moved ; when, neverthe- 
less, he catches fire from their flashes, arms 
from their armoury, sustentation from their 



i8o LECTURES AND ESSAYS 

footprints, the procedure by which he applies 
old experience to new circumstances is nothing 
greater or less than scientific thought. When 
the moralist, studying the conditions of society 
and the ideas of right and wrong which have 
come down to us from a time when war was 
the normal condition of man and success in war 
the only chance of survival, evolves from them 
the conditions and ideas which must accompany 
a time of peace, when the comradeship of equals 
is the condition of national success ; the process 
by which he does this is scientific thought and 
nothing else. Remember, then, that it is the 
guide of action ; that the truth which it arrives 
at is not that which we can ideally contemplate 
without error, but that which we may act upon 
without fear ; and you cannot fail to see that 
scientific thought is not an accompaniment or 
condition of human progress, but human pro- 
gress itself. And for this reason the question 
what its characters are, of which I have so 
inadequately endeavoured to give you some 
glimpse, is the question of all questions for the 
human race. 



ATOMS l 

IF I were to wet my finger and then rub it 
along the edge of this glass, I should no doubt 
persuade the glass to give out a certain musical 
note. So also if I were to sing to that glass 
the same note loud enough, I should get the 
glass to answer me back with a note. 

I want you to remember that fact, because 
it is of capital importance for the arguments 
we shall have to consider to-night. The very 
same note which I can get the tumbler to give 
out by agitating it, by rubbing the edge, that 
same note I can also get the tumbler to answer 
back to me when I sing to it. Now, remember- 
ing that, please to conceive a rather complicated 
thing that I am now going to try to describe 
to you. The same property that belongs to 
the glass belongs also to a bell which is made 
out of metal. If that bell is agitated by being 
struck, or in any other way, it will give out the 
same sound that it will answer back if you sing 

1 Sunday Lecture Society, January 7, 1872 ; Hulme Town 
Hall, Manchester, November 20, 1872. 



i8z LECTURES AND ESSAYS 

that sound to it ; but if you sing a different 
sound to it then it will not answer. 

Now suppose that I have several of these 
metal bells which answer to quite different 
notes, and that they are all fastened to a set 
of elastic stalks which spring out of a certain 
centre to which they are fastened. All these 
bell, then, are not only fastened to these stalks, 
but they are held there in such a way that they 
can spin round upon the points to which they 
are fastened. 

And then the centre to which these elastic 
stalks are fastened or suspended, you may 
imagine as able to move in all manner of 
directions, and that the whole structure made 
up of these bells and stalks and centre is able 
to spin round any axis whatever. We must 
also suppose that there is surrounding this 
structure a certain framework. We will sup- 
pose the framework to be made of some elastic 
material, so that it is able to be pressed in to 
a certain extent Suppose that framework is 
made of whalebone, if you like. This structure 
I am going for the present to call an " atom." 
I do not mean to say that atoms are made of a 
structure like that. I do not mean to say that 
there is anything in an atom which is in the 
shape of a bell ; and I do not mean to say 
that there is anything analogous to an elastic 
stalk in it. But what I mean is this that an 



ATOMS 183 

atom is something that is capable of vibrating 
at certain definite rates ; also that it is capable 
of other motions of its parts besides those 
vibrations at certain definite rates ; and also 
that it is capable of spinning round about any 
axis. Now by the framework which I suppose 
to be put round that structure made out of 
bells and elastic stalks, I mean this that sup- 
posing you had two such structures, then you 
cannot put them closer together than a certain 
distance, but they will begin to resist being put 
close together after you have put them as near 
as that, and they will push each other away if 
you attempt to put them closer. That is all I 
mean then. You must only suppose that that 
structure is described, and that set of ideas is 
put together, just for the sake of giving us some 
definite notion of a thing which has similar 
properties to that structure. But you must 
not suppose that there is any special part of an 
atom which has got a bell-like form, or any 
part like an elastic stalk made out of whale- 
bone. 

Now having got the idea of such a compli- 
cated structure, which is capable, as we said, of 
vibratory motion, and of other sorts of motion, 
I am going on to explain what is the belief of 
those people who have studied the subject 
about the composition of the air which fills this 
room. The air which fills this room is what is 



i3 4 LECTURES AND ESSAYS 

called a gas ; but it is not a simple gas ; it is 
a mixture of two different gases, oxygen and 
nitrogen. What is believed about this air is 
that it consists of quite distinct portions or little 
masses of air that is, of little masses each of 
which is either oxygen or nitrogen ; and that 
these little masses are perpetually flying about 
in all directions. The number of them in this 
room is so great that it strains the powers of 
our numerical system to count them. They 
are flying about in all directions and mostly in 
straight lines, except where they get quite near 
to one another, and then they rebound and fly 
off in other directions. Part of these little 
masses which compose the air are of one sort 
they are called oxygen. All those little masses 
which are called oxygen are alike ; they are of 
the same weight ; they have the same rates of 
vibration ; and they go about on the average 
at a certain rate. The other part of these 
little masses is called nitrogen, and they have a 
different weight ; but the weight of all the 
nitrogen masses is the same, as nearly as we 
can make out They have again the same 
rates of vibration ; but the rates of vibration 
that belong to them are different from the rates 
of vibration that belong to the oxygen masses ; 
and the nitrogen masses go about on the aver- 
age at a certain rate, but this rate is different 
from the average rate at which the oxygen 



ATOMS 185 

masses go about. So then, taking up that 
structure which I endeavoured to describe to 
you at first, we should represent the state of 
the air in this room as being made up of such 
a lot of compound atoms of those structures of 
bells and stalks, with frameworks round them, 
that I described to you, being thrown about in 
all directions with great rapidity, and continu- 
ally impinging against one another, each fly- 
ing off in a different direction, so that they 
would go mostly in straight lines (you must 
suppose them for a moment not to fall down 
towards the earth), excepting where they come 
near enough for their two frameworks to be in 
contact, and then their frameworks throw them 
off in different directions : that is a conception 
of the state of things which actually takes place 
inside of gas. 

Now, the conception which scientific men 
have of the state of things which takes place 
inside of a liquid is different from that. We 
should conceive it in this way : We should 
suppose that a number of these structures are 
put so close together that their frameworks are 
always in contact ; and yet they are moving 
about and rolling among one another, so that 
no one of them keeps the same place for two 
instants together, and any one of them is 
travelling all over the whole space. Inside of 
this glass, where there is a liquid, all the small 



186 LECTURES AND ESSAYS 

particles or molecules are running about among 
one another, and yet none of them goes for any 
appreciable portion of its path in a straight 
line, because there is no distance however small 
that it goes without being in contact with others 
all around it ; and the effect of this contact of 
the others all around it is that they press 
against it and force it out of a straight path. 
So that the path of a particle in a liquid is a 
sort of wavy path ; it goes in and out in all 
directions, and a particle at one part of the 
liquid will, at a certain time, have traversed all 
the different parts one after another. 

The conception of what happens inside of 
a solid body, say a crystal of salt, is different 
again from this. It is supposed that the very 
small particles which constitute that crystal of 
salt do not travel about from one part of the 
crystal to another, but that each one of them 
remains pretty much in the same place. I say 
" pretty much," but not exactly, and the motion 
of it is like this : Suppose one of my structures, 
with its framework round it, to be fastened up 
by elastic strings, so that one string goes to the 
ceiling, and another to the floor, and another to 
each wall, so that it is fastened by all these 
strings. Then if these strings are stretched, 
and a particle is displaced in any way, it will 
just oscillate about its mean position, and will 
not go far away from it ; and if forced away 



%\ ATOMS 187 

from that position it will come back again. 
That is the sort of motion that belongs to a 
particle in the inside of a solid body. A solid 
body, such as a crystal of salt, is made up, just 
as a liquid or a gas is made up, of innumerable 
small particles, but they are so attached to one 
another that each of them can only oscillate 
about its mean position. It is very probable 
that it is also able to spin about any axis in 
that position or near it ; but it is not able to 
leave that position finally, and to go and take 
up another position in the crystal ; it must stop 
in or near about the same position. 

These, then, are the views which are held 
by scientific men at present about what actually 
goes on inside of a gaseous body, or a liquid 
body, or a solid body. In each case the body 
is supposed to be made up of a very large 
number of very small particles ; but in one case 
these particles are very seldom in contact with 
one another, that is, very seldom within range 
of each other's action ; in this case they are 
during the greater part of the time moving 
separately along straight lines. In the case of 
a liquid they are constantly within the range 
of each other's action ; but they do not move 
along straight lines for any appreciable part of 
the time ; they are always changing their 
position relatively to the other particles, and 
one of them gets about from one part of the 



i88 LECTURES AND ESSAYS 

liquid to another. In the case of a solid they 
are always also within the range of each other's 
action, and they are so much within that range 
that they are not able to change their relative 
positions ; and each one of them is obliged to 
remain in very nearly the same position. 

Now what I want to do this evening is to 
explain to you, as far as I can, the reasons 
which have led scientific men to adopt these 
views ; and what I wish especially to impress 
upon you is this, that what is called the " atomic 
theory " that is, what I have just been ex- 
plaining is no longer in the position of a 
theory, but that such of the facts as I have just 
explained to you are really things which are 
definitely known and which are no longer 
suppositions ; that the arguments by which 
scientific men have been led to adopt these 
views are such as, to anybody who fairly 
considers them, justify that person in believing 
that the statements are true. 

Now first of all I want to explain what the 
reasons are why we believe that the air consists 
of separate portions, and that these portions are 
repetitions of the same structures. That is to 
say that in the air we have two structures really, 
each of them a great number of times repeated. 
Take a simple illustration, which is a rather 
easier one to consider. Suppose we take a 
vessel which is filled with oxygen. I want to 



&i ATOMS 189 

show what the reasons are which lead us to 
believe that that gas consists of a certain 
structure which is a great number of times 
repeated, and that between two examples of 
that structure which exist inside of the vessel 
there is a certain empty space which does not 
contain any oxygen. That oxygen gas con- 
tained in the vessel is made up of small 
particles which are not close together, and 
each of these particles has a certain structure, 
which structure also belongs to the rest of the 
particles. This argument is rather a difficult 
one, and I shall ask you therefore to follow it 
as closely as possible, because it is an extremely 
complicated argument to follow out the first 
time that it is presented to you. 

I want to consider again the case of this 
finger-glass. You must often have tried that 
experiment that a glass will give out when it 
is agitated the same note which it will return 
when it is sung to. Well, now, suppose that I 
have got this room filled with a certain number 
of such atomic structures as I have endeavoured 
to describe that is to say, of sets of bells, the 
bells answering to certain given notes. Each 
of these little structures is exactly alike, that is 
to say, it contains just the bells corresponding 
to the same notes. Well, now, suppose that 
you sing to a glass or to a bell, there are three 
things that may happen. First, you may sing 



190 LECTURES AND ESSAYS 

a note which does not belong to the bell at all. 
In that case the bell will not answer ; it will 
not be affected or agitated by your singing that 
note, but it will remain quite still. Next, if 
you sing a note that belongs to the bell, but if 
you sing it rather low, then the effect of that 
note will be to make the bell move a little, but 
the bell will not move so much as to give back 
the note in an audible form. Thirdly, if you 
sing the note which belongs to the bell loud 
enough, then you will so far agitate the bell that 
it will give back the note to you again. Now 
exactly that same property belongs to a 
stretched string or the string of a piano. You 
know that if you sing a certain note in a room 
where there is a piano, the string belonging to 
that note will answer you if you sing loud 
enough. The other strings won't answer at all. 
If you don't sing loud enough the string will be 
affected, but not enough to answer you. Now 
let us imagine a screen of piano strings, all of 
exactly the same length, of the same material, 
and stretched equally, and that this screen of 
strings is put across the room ; that I am at 
one end and that you are at another, and that 
I proceed to sing notes straight up the scale. 
While I sing notes which are different from that 
note which belongs to the screen of strings, they 
will pass through the screen without being 
altered, because the agitation of the air which 



8V ATOMS 191 

I produce will not affect the strings. But that 
note will be heard quite well at the other side 
of the screen. You must remember that when 
the air carries a sound it vibrates at a certain 
rate belonging to the sound. I make the air 
vibrate by singing a particular note, and if that 
rate of vibration corresponds to the strings the 
air will pass on part of its vibration to the 
strings, and so make the strings move. But if 
the rate of vibration is not the one that corre- 
sponds to the strings, then the air will not pass 
on any of its vibrations to the strings, and 
consequently the sound will be heard equally 
loud after it has passed through the strings. 
Having put the strings of the piano across the 
room, if I sing up the scale, when I come to the 
note which belongs to each of the strings my 
voice will suddenly appear to be deadened, be- 
cause at the moment that the rate of vibration 
which I impress upon the air coincides with that 
belonging to the strings, part of it will be taken 
up in setting the strings in motion. As I pass 
the note, then, which belongs to the strings, 
that note will be deadened. 

Instead of a screen of piano strings let us put 
in a series of sets of bells, three or four belonging 
to each set, so that each set of bells answers to 
three or four notes, and so that all the sets are 
exactly alike. Now suppose that these sets of 
bells are distributed all over the middle part of 



192 LECTURES AND ESSAYS 

the room, and that I sing straight up the scale 
from one note to another until I come to the 
note that corresponds to one of the bells in these 
sets, then that note will appear to be deadened 
at the other end, because part of the vibration 
communicated to the air will be taken up in 
setting those bells in motion. When I come to 
another note which belongs to them, that note 
will also be deadened ; so that a person listening 
at the other end of the room would observe that 
certain notes were deadened, or even had dis- 
appeared altogether. If, however, I sing loud 
enough, I then should set all these bells vibrating. 
What would be heard at the other end of the 
room ? Why, just the chord compounded out 
of those sounds that belonged to the bells, 
because the bells having been set vibrating 
would give out the corresponding notes. So 
you see there are here three facts. When I 
sing a note which does not belong to the bells, 
my voice passes to the end of the room without 
diminution. When I sing a note that does 
belong to the bells, then if it is not loud 
enough it is deadened by passing through the 
screen ; but if it is loud enough it sets the bells 
vibrating, and is heard afterwards. Now just 
notice this consequence. We have supposed a 
screen made out of these structures that I have 
imagined to represent atoms, and when I sing 
through the scale at one end of the room certain 



ATOMS 193 

notes appear to be deadened. If I take away 
half of those structures, what will be the effect ? 
Exactly the same notes will be deadened, but 
they will not be deadened so much ; the notes 
which are picked out of the thinner screen to be 
deadened will be exactly the same notes, but 
the amount of the deadening will not be the same. 
So far we have only been talking about the 
transmission of sound. You know that sound 
consists of certain waves which are passed along 
in the air ; they are called " aerial vibrations." 
We also know that light consists of certain waves 
which are passed along, not in the air, but 
along another medium. I cannot stop at 
present to explain to you what the sort of evi- 
dence is upon which that assertion rests, but it 
is the same sort of evidence as that which I 
shall try to show you belongs to the statement 
about atoms ; that is to say, the " undulatory 
theory," as it is called, of light, the theory that 
light consists of waves transmitted along a 
certain medium, has passed out of the stage of 
being a theory, and has passed into the stage 
of being a demonstrated fact. The difference 
between a theory and a demonstrated fact is 
something like this : If you supposed a man to 
have walked from Chorlton Town Hall down 
here say in ten minutes, the natural conclusion 
would be that he had walked along the Stret- 
ford Road. Now that theory would entirely 
VOL. I O 



194 LECTURES AND ESSAYS 

account for all the facts, but at the same time 
the facts would not be proved by it. But sup- 
pose it happened to be winter time, with snow 
on the road, and that you could trace the man's 
footsteps all along the road, then you would 
know that he had walked along that way. 
The sort of evidence we have to show that 
light does consist of waves transmitted through 
a medium is the sort of evidence that footsteps 
upon the snow make ; it is not a theory merely 
which simply accounts for the facts, but it is a 
theory which can be reasoned back to from the 
facts without any other theory being possible. 
So that you must just for the present take it 
for granted that the arguments in favour of the 
hypothesis that light consists of waves are such 
as to take it out of the region of hypothesis, 
and make it into demonstrated fact. 

Very well, then, light consists of waves 
transmitted along this medium in the same 
way that sound is transmitted along the air. 
The waves are not of the same kind ; but still 
they are waves, and they are transmitted as such; 
and the different colours of light correspond to 
the different lengths of these waves, or to the 
different rates of the vibration of the medium, 
just as the different pitches of sound corre- 
spond to the different lengths of the air-waves 
or to the different rates of the vibration of the 
air. Now, if we take any gas, such as oxygen, 



8 ATOMS 195 

and we pass light through it, we find that that 
gas intercepts, or weakens, certain particular 
colours. If we take any other gas, such as 
hydrogen, and pass light through it, we find 
that that gas intercepts, or weakens, certain 
other particular colours of the light. There 
are two ways in which it can do that: it is 
clear that the undulations, or waves, are made 
weaker, because they happen to coincide with 
the rate of vibration of the gas they are passing 
through. But the gas may vibrate as a whole 
in the same way that the air does when you 
transmit sound. Or the waves may be stopped, 
because the gas consists of a number of small 
structures ; just as my screen, which I imagine 
to consist of structures ; or just as the screen 
of piano strings is made up of the same struc- 
ture many times repeated. Either of these 
suppositions would apparently at first account 
for the fact that certain waves of light are inter- 
cepted by the gas, while others are let through. 
But how is it that we can show one of these 
suppositions is wrong and the other is right ? 
Instead of taking so small a structure as piano 
strings, let us suppose we had got a series of 
fiddles, the strings of all of them being stretched 
exactly in tune. I suppose this case because 
it makes a more complicated structure, for there 
would be two or three notes corresponding in 
each 'fiddle. If you suppose this screen of 



196 LECTURES AND ESSAYS 

fiddles to be hung up and then compressed, 
what will be the effect? The effect of the 
compression will be, if they are all in contact, 
that each fiddle itself will be altered. If the 
fiddles are compressed longways, the strings 
will give lower notes than before, and con- 
sequently the series of notes which will be 
intercepted by that screen will be different from 
the series of notes which were intercepted 
before. But if you have a screen made out of 
fiddles which are at a distance from one another, 
and then if you compress them into a smallerspace 
by merely bringing them nearer together, without 
making them touch, then it is clear that exactly 
the same notes will be intercepted as before ; 
only, as there will be more fiddles in the same 
space, the deadening of the sound will be greater. 
Now when you compress any gas you find 
that it intercepts exactly the same colours of 
light which it intercepted before it was com- 
pressed. It follows, therefore, that the rates of 
vibration which it intercepts depend not upon 
the mass of the gas whose properties are altered 
by the compression, but upon some individual 
parts of it which were at a distance from one 
another before, and which are only brought 
nearer together without being absolutely 
brought into contact so as to squeeze them. 
That is the sort of reasoning by which it is 
made clear that the interception of light, or 



1 ATOMS 197 

particular waves of light, by means of a gas, 
must depend on certain individual structures 
in the gas which are at a distance from one 
another, and which by compression are not 
themselves compressed, but only brought nearer 
to one another. 

There is an extremely interesting con- 
sequence which follows from this reasoning, 
and which was deduced from it by Professor 
Stokes in the year 1851, and which was after- 
wards presented in a more developed form in 
the magnificent researches of Kirchhoff 
namely, the reasoning about the presence of 
certain matter in the sun. If you analyse the 
solar light by passing it through a prism, the 
effect of the prism is to divide it off so as to 
separate the light into the different colours 
which it contains. That line of variously 
coloured light which is produced by the prism 
is, as you know, called the Spectrum. When 
that spectrum is made in a very accurate way, 
so that the parts of it are well defined, it is 
observed to contain certain dark lines. That 
is, there is a certain kind of light which is 
missing in the sunlight ; certain kinds of light, 
as we travel along the scale of lights, are miss- 
ing. Why are they missing? Because there 
is something that the light has passed through 
which intercepts or weakens those kinds of 
light. Now that something which the light 



198 LECTURES AND ESSAYS 

has passed through, how shall we find out 
what it is ? It ought to be the same sort of 
substance which if it were heated would give 
out exactly that kind of light. Now there is 
a certain kind of light which is intercepted 
which makes a group of dark lines in the solar 
spectrum. There are two principal lines which 
together are called the line D ; and it is found 
that exactly that sort of light is emitted by 
sodium when heated hot enough. The con- 
clusion therefore is that that matter which 
intercepts that particular part of the solar light 
is sodium, or that there is sodium somewhere 
between us and the hot portion of the sun 
which sends us the light And other reasons 
lead us to conclude that this sodium is not in 
the atmosphere of the earth, but in the neigh- 
bourhood of the sun that it exists in a gaseous 
state in the sun's atmosphere. And nearly all 
the lines in the solar spectrum have been 
explained in that way, and shown to belong 
to certain substances which we are able to heat 
here, and to show that when they are heated 
they give out exactly the same kind of light 
which they intercepted when the light was first 
given out by the sun and they stood in the 
way. So you see that is a phenomenon 
exactly like the phenomenon presented by the 
finger-glass that we began with. 

Precisely the same light which any gas will 



I ATOMS 199 

give out when it is heated, that same kind of 
light it will stop or much weaken if the light is 
attempted to be passed through it. That means 
that this medium which transmits light, and 
which we call the " luminiferous ether," has a 
certain rate of vibration for every particular 
colour of the spectrum. When that rate of 
vibration coincides with one of the rates of 
vibration of an atom, then it will be stopped 
by that atom, because it will set the atom 
vibrating itself. If therefore you pass light of 
any particular colour through a gas whose 
atoms are capable of the corresponding rate of 
vibration, the light will be cut off by the gas. 
If on the other hand you so far heat the gas 
that the atoms are vibrating strongly enough to 
give out light, it will give out a light of a kind 
which it previously stopped. 

We have reason then for believing that a 
simple gas consists of a great number of atoms ; 
that it consists of very small portions, each 
of which has a complicated structure, but that 
structure is the same for each of them, and 
that these portions are separate, or that there 
is space between them. 

In the next place I want to show you what 
is the evidence upon which we believe that 
these portions of the gas are in motion that 
they are constantly moving. 

If this were a political instead of a scientific 



300 LECTURES AND ESSAYS 

meeting, there would probably be some people 
who would be inclined to disagree with us, 
instead of all being inclined to agree with one 
another ; and these people might have taken 
it into their heads, as has been done in certain 
cases, to stop the meeting by putting a bottle 
of sulphuretted hydrogen in one corner of the 
room and taking the cork out You know that 
after a certain time the whole room would 
contain sulphuretted hydrogen, which is a very 
unpleasant thing to come in contact with. 
Now how is it that that gas which was con- 
tained in a small bottle could get in a short 
time over the whole room unless it was in 
motion ? What we mean by motion is change 
of place. The gas was in one corner, and it is 
afterwards all over the room. There has there- 
fore been motion somewhere, and this motion 
must have been of considerable rapidity, because 
we know that there was the air which filled the 
room beforehand to oppose resistance to that 
motion. We cannot suppose that the sul- 
phuretted hydrogen gas was the only thing 
that was in motion, and that the air was 
not in motion itself, because if we had used 
any other gas we should find that it would 
diffuse itself in exactly the same way. An 
argument just like that applies also to the case 
of a liquid. Suppose this room were a large 
tank entirely filled with water and anybody 



ATOMS 201 

were to drop a little iodine into it, after a 
certain time the whole of the water would be 
found to be tinged of a blue colour. Now that 
drop may be introduced into any part of the 
tank you like, either at the top or bottom, and 
it will always diffuse itself over the whole water. 
There has here again been motion. We cannot 
suppose that the drop which was introduced 
was the only thing that moved about, because 
any other substance would equally have moved 
about. And the water has moved into the 
place where the drop was, because in the place 
where you put the drop there is not so much 
iodine as there was to begin with. Well then 
it is clear that in the case of a gas, these 
particles of which we have shown it to consist 
must be constantly in motion ; and we have 
shown also that a liquid must consist of parts 
that are in motion, because it is able to admit 
the particles of another body among them. 

When we have decided that the particles of 
a gas are in motion, there are two things that 
they may do they may either hit against one 
another, or they may not. Now it is established 
that they do hit against one another, and that 
they do not proceed along straight lines inde- 
pendent of one another. But I cannot at 
present explain to you the whole of the reason- 
ing upon which that conclusion is grounded. 
It is grounded upon some rather hard mathe- 



302 LECTURES AND ESSAYS 

matics. It was shown by Professor Clerk 
Maxwell that a gas cannot be a medium con- 
sisting of small particles moved about in all 
directions in straight lines, which do not inter- 
fere with one another, but which bound off 
from the surfaces which contain this medium. 
Supposing we had a box containing a gas of 
this sort. Well, these particles do not interfere 
with one another, but only rebound when they 
come against the sides of the box ; then that 
portion of the gas will behave not like a gas 
but like a solid body. The peculiarity of 
liquids and gases is that they do not mind 
being bent and having their shape altered. It 
has been shown by Clerk Maxwell that a 
medium whose particles do not interfere with 
one another would behave like a solid body 
and object to be bent. It was a most extra- 
ordinary conclusion to come to, but it is entirely 
borne out by the mathematical formulae. It is 
certain that if there were a medium composed 
of small particles flying about in all directions 
and not interfering with one another, then that 
medium would be to a certain extent solid, that 
is, would resist any bending or change of shape. 
By that means then it is known that these 
particles do run against one another. And 
they come apart again. There were two things 
of course they might do : they might either go 
on in contact, or they might come apart Now 



ATOMS 203 

we know that they come apart for this reason 
we have already considered how two gases 
in contact will diffuse into one another. If 
you were to put a bucket containing carbonic 
acid (which is very heavy) upon the floor of 
this room, it would after a certain time diffuse 
itself over all the room ; you would find carbonic 
acid gas in every part of the room. Graham 
found that if you were to cover over the top of 
that bucket with a very thin cover made out 
of graphite, or blacklead, then the gas would 
diffuse itself over the room pretty nearly as 
fast as before. The graphite acts like a porous 
body, as a sponge does to water, and lets the 
gas get through. The remarkable thing is 
that if the graphite is thin the gas will get 
through nearly as fast as it will if nothing is 
put between to stop it. Graham found out 
another fact. Suppose that bucket to contain 
two very different gases, say a mixture of 
hydrogen and carbonic acid gas. Then the 
hydrogen would come out through the black- 
lead very much faster than the carbonic acid 
gas. It is found by mathematical calculation 
that if you have two gases, which are supposed 
to consist of small particles which are all bang- 
ing about, the gas whose particles are lightest 
will come out quickest ; that a gas which is 
four times as light will come out twice as fast ; 
and a gas nine times as light will come out 



104 LECTURES AND ESSAYS 

three times as fast, and so on. Consequently, 
when you mix two gases together and then 
pass them through a thin piece of blacklead, 
the lightest gas comes out quickest, and is as it 
were sifted from the other. Now suppose we 
put pure hydrogen into a bucket and put 
blacklead on the top, and then see how fast 
the hydrogen comes out. If the particles of 
the hydrogen are different from one another, 
if some are heavier, the lighter ones will come 
out first. Now let us suppose we have got a 
vessel which is divided into two parts by a thin 
wall of blacklead. We will put hydrogen into 
one of these parts and allow it to come through 
this blacklead into the other part ; then if the 
hydrogen contains any molecules or atoms which 
are lighter than the others, those will come 
through first. If we test the hydrogen that 
has come through, we shall find that the atoms, 
as a rule, on one side of this wall are lighter 
than the atoms on the other side. How should 
we find that out? Why we should take these 
two portions of gas, and we should try whether 
one of them would pass through another piece 
of blacklead quicker than the other ; because if 
it did, it would consist of lighter particles. 
Graham found that it did not pass any quicker. 
Supposing you put hydrogen into one half of 
such a vessel, and then allow the gas to diffuse 
itself through the blacklead, the gas on the two 



ATOMS 205 

sides would be found to be of precisely the 
same qualities. Consequently, there has not 
been in this case any sifting of the lighter 
particles from the heavier ones ; and con- 
sequently there could not have been any lighter 
particles to sift, because we know that if there 
were any they would have come through quicker 
than the others. Therefore we are led to the 
conclusion that in any simple gas, such as 
hydrogen or oxygen, all the atoms are, as nearly 
as possible, of the same weight. We have no 
right to conclude that they are exactly of the 
same weight, because there is no experiment 
in the world that enables us to come to an 
exact conclusion of that sort. But we are 
enabled to conclude that, within the limits of 
experiment, all the atoms of a simple gas are 
of the same weight. What follows from that ? 
It follows that when they bang against one 
another, they must come apart again ; for if 
two of them were to go on as one, that one 
would be twice as heavy as the others, and 
would consequently be sifted back. It follows 
therefore that two particles of a gas which bang 
against one another must come apart again, 
because if they were to cling together they 
would form a particle twice as heavy, and so 
this clinging would show itself when the gas 
was passed through the screen of blacklead. 
Now there are certain particles or small 



206 LECTURES AND ESSAYS 

masses of matter which we know to bang against 
one another according to certain laws ; such, 
for example, as billiard balls. The way in 
which different bodies, after hitting together, 
come apart again, depends on the constitution 
of those bodies. The earlier hypothesis about 
the constitution of a gas supposed that the 
particles of them came apart according to the 
same law that billiard balls do ; but that hypo- 
thesis, although it was found to explain a great 
number of phenomena, did not explain them 
all. And it was Professor Clerk Maxwell again 
who found the hypothesis which does explain 
all the rest of the phenomena. He found that 
particles when they come together separate as 
if they repelled one another, or pushed one 
another away ; and as if they did that much 
more strongly when close together than when 
further apart You know that what is called 
the great law of gravitation asserts that all 
bodies pull one another together according to a 
certain rule, and that they pull one another 
more when close than when further apart. Now 
that law differs from the law which Clerk 
Maxwell found out as affecting the repulsion of 
gaseous particles. The law of attraction of 
gravitation is this ; that when you halve the 
distance, you have to multiply the attraction 
four times twice two make four. If you divide 
the distance into three, you must multiply the 



ATOMS 207 

attraction nine times three times three are 
nine. Now in the case of atomic repulsion 
you have got to multiply not twice two, or three 
times three, but five twos together which 
multiplied make 32. If you halve the distance 
between two particles you increase the repulsion 
32 times. So also five threes multiplied to- 
gether make 243 ; and if you divide the dis- 
tance between two particles by three, then you 
increase the repulsion by 243. So you see the 
repulsion increases with enormous rapidity as 
the distance diminishes. That law is expressed 
by saying that the repulsion of two gases is 
inversely as the fifth power of the distance. 
But I must warn you against supposing that 
that law is established in the same sense that 
these other statements that we have been mak- 
ing are established. That law is true provided 
that there is a repulsion between two gaseous 
particles, and that it varies as a power of the 
distance ; it is proved that if there is any law 
of repulsion, and if the law is that it varies as 
some power of the distance, then that power 
cannot be any other than the fifth. It has 
not been shown that the action between the 
two particles is not something perhaps more 
complicated than this, but which on the average 
produces the same results. But still the state- 
ment that the action of gaseous molecules upon 
one another can be entirely explained by the 



208 LECTURES AND ESSAYS 

assumption of a law like that, is the newest 
statement in physics since the law of gravitation 
was discovered. You know that there are 
other actions of matter which apparently take 
place through intervening spaces and which 
always follow the same law as gravitation, such 
as the attraction or repulsion of magnetical or 
electrical particles : those follow the same law 
as gravitation. But here is a law of repulsion 
which follows a different law from that of 
gravitation, and in that lies the extreme interest 
of Professor Clerk Maxwell's investigation. 

The next thing that I want to give you 
reasoning for is again rather a hard thing in 
respect of the reasoning, but the fact is an 
extremely simple and beautiful one. It is this. 
Suppose I have two vessels, say cylinders, with 
stoppers which do not fit upon the top of the 
vessel, but slide up and down inside and yet fit 
exactly. These two vessels are of exactly the 
same size ; one of them contains hydrogen and 
the other contains oxygen. They are to be of 
the same temperature and pressure, that is to 
say they will bear exactly the same weight on 
the top. Very well, these two vessels having 
equal volumes of gas of the same pressure and 
temperature will contain just the same number 
of atoms in each, only the atoms of oxygen 
will be heavier than the atoms of hydrogen. 
Now how is it that we arrive at that result ? I 



ATOMS 209 

shall endeavour to explain the process of 
reasoning. Boyle discovered a law about the 
dependence of the pressure of a gas upon its 
volume which showed that if you squeezed a 
gas into a smaller space it will press so much 
the more as the space has been diminished. If 
the space has been diminished one-half, then 
the pressure is doubled ; if the space is dimin- 
ished to one-third, then the pressure is increased 
to three times what it was before. This holds 
for a varying volume of the same gas. That 
same law would tell us that if we put twice the 
quantity of gas into the same space, we should 
get twice the amount of pressure. Dalton 
made a new statement of that law, which ex- 
presses it in this form, that when you put more 
gas into a vessel which already contains gas, 
the pressure that you get is the sum of the two 
pressures which would be got from the two 
gases separately. You will see directly that 
that is equivalent to the other law. But the 
importance of Dalton's statement of the law is 
this, that it enabled the law to be extended 
from the case of the same gas to the case of 
two different gases. If instead of putting a 
pint of oxygen into a vessel already containing 
a pint, I were to put in a pint of nitrogen, I 
should equally get a double pressure. The 
oxygen and nitrogen, when mixed together, 
would exert the sum of the pressures upon the 
VOL. I P 



2io LECTURES AND ESSAYS 

vessel that the oxygen and nitrogen would 
exert separately. Now the explanation of that 
pressure is this. The pressure of the gas upon 
the sides of the vessel is due to the impact of 
these small particles which are constantly flying 
about and impinging upon the sides of the 
vessel. It is first of all shown mathematically 
that the effect of that impinging would be the 
same as the pressure of the gas. But the 
amount of the pressure could be found if we 
knew how many particles there were in a given 
space, and what was the effect of each one 
when it impinged on the sides of the vessel. 
You see directly why it is that putting twice as 
many particles, which are going at the same 
rate, into the same vessel, we should get twice 
the effect. Although there are just twice as 
many particles to hit the sides of the vessel, 
they are apparently stopped by each other 
when they bound off. But the effect of there 
being more particles is to make them come 
back quicker ; so that altogether the number of 
impacts upon the sides of the vessel is just 
doubled when you double the number of par- 
ticles. Supposing we have got a cubic inch 
of space, then the amount of pressure upon the 
side of that cubic inch depends upon the num- 
ber of particles inside the cube, and upon the 
energy with which each one of them strikes 
against the sides of the vessel. 



ATOMS 2ii 

Again there is a law which connects together 
the pressure of a gas and its temperature. It 
is found that there is a certain absolute zero of 
temperature, and that if you reckon your 
temperature from that, then the pressure of the 
gas is directly proportional to the temperature, 
that twice the temperature will give twice the 
pressure of the same gas, and three times the 
temperature will give three times the pressure 
of the same gas. 

Well now we have just got to remember 
these two rules the law of Boyle, as expressed 
by Dalton, connecting together the pressure of 
a gas and its volume, and this law which con- 
nects together the pressure with the absolute 
temperature. You must remember that it 
has been calculated by mathematics that the 
pressure upon one side of a vessel of a cubic 
inch has been got by multiplying together the 
number of particles into the energy with which 
each of them strikes against the side of the 
vessel. If we keep that same gas in a vessel 
and alter its temperature, then we find that the 
pressure is proportional to the temperature ; but 
since the number of molecules remains the same 
when we double the pressure, we must alter that 
other factor in the pressure, we must double 
the energy with which each of the particles 
attacks the side of the vessel. That is to say, 
when we double the temperature of the gas we 



212 LECTURES AND ESSAYS 

double the energy of each particle ; consequently 
the temperature of the gas is proportional always 
to the energy of its particles. That is the case 
with a single gas. If we mix two gases, what 
happens? They come to exactly the same 
temperature. It is calculated also by mathe- 
matics that the particles of one gas have the 
same effect as those of the other ; that is, the 
light particles go faster to make up for their 
want of weight. If you mix oxygen and 
hydrogen, you find that the particles of hydrogen 
go four times as fast as the particles of oxygen. 
Now we have here a mathematical statement 
that when two gases are mixed together, the 
energy of the two particles is the same ; and 
with any one gas considered by itself that energy 
is proportional to the temperature. Also when 
two gases are mixed together the two tempera- 
tures become equal. If you think over that a 
little, you will see that it proves that whether 
we take the same gas or different gases, the 
energy of the single particles is always pro- 
portional to the temperature of the gas. 

What follows ? If I have two vessels con- 
taining gas at the same pressure and the same 
temperature (suppose that hydrogen is in one 
and oxygen in the other), then I know that 
the temperature of the hydrogen is the same as 
the temperature of the oxygen, and that the 
pressure of the hydrogen is the same as the 



ATOMS 213 

pressure of the oxygen. I also know (because 
the temperatures are equal) that the average 
energy of a particle of the hydrogen is the same 
as that of a particle of the oxygen. Now the 
pressure is made up by multiplying the energy 
by the number of particles in both gases ; and 
as the pressure in both cases is the same, 
therefore the number of particles is the same. 
That is the reasoning ; I am afraid it will seem 
rather complicated at first hearing, but it is this 
sort of reasoning which establishes the fact that 
in two equal volumes of different gases at the 
same temperature and pressure, the number of 
particles is the same. 

Now there is an exceedingly interesting 
conclusion which was arrived at very early in 
the theory of gases, and calculated by Mr. 
Joule. It is found that the pressure of a gas 
upon the sides of a vessel may be represented 
quite fairly in this way. Let us divide the 
particles of gas into three companies or bands. 
Suppose I have a cubical vessel in which one 
of these companies is to go forward and back- 
ward, another right and left, and the other to 
go up and down. If we make those three 
companies of particles to go in their several 
directions, then the effect upon the sides of the 
vessel will not be altered ; there will be the 
same impact and pressure. It was also found 
out that the effect of this pressure would not be 



214 LECTURES AND ESSAYS 

altered if we combined together all the particles 
forming one company into one mass, and made 
them impinge with the same velocity upon the 
sides of the vessel. The effect of the pressure 
would be just the same. Now we know what 
the weight of a gas is, and we know what the 
pressure is that it produces, and we want to 
find the velocity it is moving at on the average. 
We can find out at what velocity a certain 
weight has to move in order to produce a 
certain definite impact Therefore we have 
merely to take the weight of the gas, divide it 
by three, and to find how fast that has to move 
in order to produce the pressure, and that will 
give us the average rate at which the gas is 
moving. By that means Mr. Joule calculated 
that in air of ordinary temperature and pressure 
the velocity is about 500 metres per second, 
nearly five miles in sixteen seconds, or nearly 
twenty miles a minute about sixty times the 
rate of an ordinary train. 

The average velocity of the particles of gas 
is about i^ times as great as the velocity of 
sound. You can easily remember the velocity of 
sound in air at freezing point it is 3 3 3 metres 
per second ; so that about i \ times, really 
1.432 of that would be the average velocity of 
a particle of air. At the ordinary temperature 
60 degrees Fahrenheit the velocity would, 
of course, be greater. 



ATOMS 215 

Let us consider how much we have estab- 
lished so far about these small particles of which 
we find that the gas consists. We have so far 
been treating mainly of gases. We find that a 
gas, such as the air in this room, consists of 
small particles, which are separate with spaces 
between them. They are as a matter of fact 
of two different types, oxygen and nitrogen. 
All the particles of oxygen contain the same 
structure, and the rates of internal vibration are 
the same for all these particles. It is also 
compounded of particles of nitrogen which have 
different rates of internal vibration. We have 
shown that these particles are moving about 
constantly. We have shown that they impinge 
against and interfere with one another's motion ; 
and we have shown that they come apart again. 
We have shown that in vessels of the same size 
containing two different gases of the same 
pressure and temperature there is the same 
number of those two different sorts of particles. 
We have shown also that the average velocity 
of these particles in the air of this room is about 
twenty miles a minute. 

There is one other point of very great 
interest to which I want to call your attention. 
The word " atom," as you know, has a Greek 
origin ; it means that which is not divided. 
Various people have given it the meaning of 
that which cannot be divided ; but if there is 



2i6 LECTURES AND ESSAYS 

anything which cannot be divided we do not 
know it, because we know nothing about 
possibilities or impossibilities, only about what 
has or has not taken place. Let us then take 
the word in the sense in which it can be applied 
to a scientific investigation. An atom means 
something which is not divided in certain cases 
that we are considering. Now these atoms I 
have been talking about may be called physical 
atoms, because they are not divided under those 
circumstances that are considered in physics. 
These atoms are not divided under the ordinary 
alteration of temperature and pressure of gas, 
and variation of heat ; they are not in general 
divided by the application of electricity to the 
gas, unless the stream is very strong. But 
there is a science which deals with operations 
by which these atoms which we have been 
considering can be divided into two parts, and 
in which therefore they are no longer atoms. 
That science is chemistry. The chemist there- 
fore will not consent to call these little particles 
that we are speaking of by the name of atoms, 
because he knows that there are certain processes 
to which he can subject them which will divide 
them into parts, and then they cease to be 
things which have not been divided. I will give 
you an instance of that. The atoms of oxygen 
which exist in enormous numbers in this room 
consist of two portions, which are of exactly 



ATOMS 217 

the same structure. Every molecule, as the 
chemist would call it, travelling in this room, 
is made up of two portions which are exactly 
alike in their structure. It is a complicated 
structure ; but that structure is double. It is 
like the human body one side is like the other 
side. How do we know that ? We know it in 
this way. Suppose that I take a vessel which 
is divided into two parts by a division which I 
can take away. One of these parts is twice as 
large as the other part, and will contain twice 
as much gas. Into that part which is twice as 
big as the other I put hydrogen ; into the other 
I put oxygen. Suppose that one contains a 
quart and the other a pint ; then I have a quart 
of hydrogen and a pint of oxygen in this vessel. 
Now I will take away the division so that they 
can permeate one another, and then if the vessel 
is strong enough I pass an electric spark 
through them. The result will be an explosion 
inside the vessel ; it will not break if it is 
strong enough ; but the quart of hydrogen and 
the pint of oxygen will be converted into steam ; 
they will combine together to form steam. If 
I choose to cool down that steam until it is 
just as hot as the two gases were before I passed 
the electric spark through them, then I shall 
find that at the same pressure there will only 
be a quart of steam. Now let us remember 
what it was that we established about two equal 



2i8 LECTURES AND ESSAYS 

volumes of different gases at the same tempera- 
ture and pressure. First of all, we had a quart 
of hydrogen with a pint of oxygen. We know 
that that quart of hydrogen contains twice as 
many hydrogen molecules as the pint of oxygen 
contains of oxygen molecules. Let us take 
particular numbers. Suppose instead of a 
quart or a pint we take a smaller quantity, 
and say that there are 100 hydrogen and 50 
oxygen molecules. Well, after the cooling has 
taken place, I should find a volume of steam 
which was equal to the volume of hydrogen, 
that is, I should find 100 steam molecules. 
Now these steam molecules are made up of 
hydrogen and oxygen molecules. I have got 
therefore 100 things which are all exactly alike, 
made up of 100 things and 50 things 100 
hydrogen and 50 oxygen, making 100 steam 
molecules. Now since the I oo steam molecules 
are exactly alike, we have those 50 oxygen 
molecules distributed over the whole of these 
steam molecules. Therefore, unless the oxygen 
contains something which is common to the 
hydrogen also, it is clear that each of those 50 
molecules of oxygen must have been divided 
into two, because you cannot put 50 horses into 
100 stables, so that there shall be exactly the 
same amount of horse in each stable ; but you 
can divide 50 pairs of horses among 100 stables. 
There we have the supposition that there is 






ATOMS 219 

nothing common to the oxygen and hydrogen, 
that there is no structure that belongs to each 
of them. Now that supposition is made by 
a great majority of chemists. Sir Benjamin 
Brodie, however, has made a supposition that 
there is a structure in hydrogen which is also 
common to certain other elements. He has 
himself, for particular reasons, restricted that 
supposition to the belief that hydrogen is 
contained as a whole in many of the other 
elements. Let us make that further supposition 
and it will not alter our case at all. We have 
then 100 hydrogen and 50 oxygen molecules, 
but there is something common to the two. 
Well, this something we will call X. Of this 
we have to make 100 equal portions. Now 
that cannot be the case unless that structure 
occurred twice as often in each molecule of 
oxygen as in each molecule of hydrogen. 
Consequently, whether the oxygen molecule 
contains something common to hydrogen or 
not, it is equally true that the oxygen molecule 
must contain the same thing repeated twice 
over ; it must be divisible into two parts which 
are exactly alike. 

Similar reasoning applies to a great number 
of other elements ; to all those which are said 
to have an even number of atomicities. But 
with regard to those which are said to have an 
odd number, although many of these also are 



aao LECTURES AND ESSAYS 

supposed to be double, yet the evidence in 
favour of that supposition is of a different 
kind ; and we must regard the supposition as 
still a theory and not yet a demonstrated fact. 

Now I have spoken so far only of gases. 
I must for one or two moments refer to some 
calculations of Sir William Thomson, which 
are of exceeding interest as showing us what 
is the proximity of the molecules in liquids 
and in solids. By four different modes of 
argument derived from different parts of science, 
and pointing mainly to the same conclusion, 
he has shown that the distance between two 
molecules in a drop of water is such that there 
are between five hundred millions and five 
thousand millions of them in an inch. He 
expresses that result in this way that if you 
were to magnify a drop of water to the size of 
the earth, then the coarseness of the graining 
of it would be something between that of 
cricket -balls and small shot. Or we may 
express it in this rather striking way. You 
know that the best microscopes can be made 
to magnify from 6000 to 8000 times. A 
microscope which would magnify that result 
as much again would show the molecular 
structure of water. 

There is another scientific theory analogous 
to this one which leads us to hope that some 
time we shall know more about these molecules. 



ATOMS 221 

You know that since the time that we have 
known all about the motions of the solar 
system, people have speculated about the origin 
of it ; and a theory started by Laplace and 
worked out by other people has, like the theory 
of luminiferous ether, been taken out of the 
rank of hypothesis into that of fact. We know 
the rough outlines of the history of the solar 
system, and there are hopes that when we 
know the structure and properties of a molecule, 
what its internal motions are and what are the 
parts and shape of it, somebody may be able 
to form a theory as to how that was built up 
and what it was built out of. It is obvious 
that until we know the shape and structure of 
it, nobody will be able to form such a theory. 
But we can look forward to the time when the 
structure and motions in the inside of a molecule 
will be so well known that some future Kant 
or Laplace will be able to make an hypothesis 
about the history and formation of matter. 1 

1 The mathematical development of this subject is due to 
Clausius and Maxwell. Reference to the chief papers will be 
found at the beginning of Maxwell's Memoir, ' ' On the Dynamical 
Theory of Gases," Phil. Trans. 1867. 



THE FIRST AND THE LAST 
CATASTROPHE 1 

A CRITICISM ON SOME RECENT SPECULATIONS 
ABOUT THE DURATION OF THE UNIVERSE 

I PROPOSE in this lecture to consider specula- 
tions of quite recent days about the beginning 
and the end of the world. The world is a 
very interesting thing, and I suppose that from 
the earliest times that men began to form any 
coherent idea of it at all, they began to guess 
in some way or other how it was that it all 
began, and how it was all going to end. But 
there is one peculiarity about these speculations 
which I wish now to consider, that makes them 
quite different from the early guesses of which 
we read in many ancient books. These modern 
speculations are attempts to find out how things 
began, and how they are to end, by consider- 
ation of the way in which they are going on 
now. And it is just that character of these 

1 Sunday Lecture Society. April 12, 1874 ; afterwards revised 
for publication. 



FIRST AND LAST CATASTROPHE 223 

speculations that gives them their interest for 
you and for me ; for we have only to consider 
these questions from the scientific point of view. 
By the scientific point of view I mean one 
which attempts to apply past experience to 
new circumstances according to an observed 
order of nature. So that we shall only con- 
sider the way in which things began, and the 
way in which they are to end, in so far as we 
seem able to draw inferences about the questions 
from facts which we know about the way in 
which things are going on now. And, in fact, 
the great interest of the subject to me lies in 
the amount of illustration which it offers of the 
degree of knowledge which we have now 
attained of the way in which the universe is 
going on. 

The first of these speculations is one set 
forth by Professor Clerk Maxwell, in a lecture on 
Molecules delivered before the British Associa- 
tion at Bradford. Now, this argument of his 
which he put before the British Association at 
Bradford depends entirely upon the modern 
theory of the molecular constitution of matter. 
I think this the more important, because a great 
number of people appear to have been led to 
the conclusion that this theory is very similar 
to the guesses which we find in ancient writers 
Democritus and Lucretius. It so happens 
that these ancient writers did hold a view of 



224 LECTURES AND ESSAYS 

the constitution of things which in many strik- 
ing respects agrees with the view which we hold 
in modern times. This parallelism has been 
brought recently before the public by Professor 
Tyndall in his excellent address at Belfast. 
And it is perhaps on account of the parallelism, 
which he pointed out at that place, between 
the theories held amongst the ancients and the 
theory held amongst the moderns, that many 
people who are acquainted with classic literature 
have thought that a knowledge of the views of 
Democritus and Lucretius would enable them 
to understand and criticise the modern theory 
of matter. That, however, is a mistake. The 
difference between the two is mainly this : the 
atomic theory of Democritus was a guess, and 
no more than a guess. Everybody around 
him was guessing about the origin of things, 
and they guessed in a great number of ways ; 
but he happened to make a guess which was 
more near the right thing than any of the 
others. This view was right in its main 
hypothesis that all things are made up of 
elementary parts, and that the different 
properties of different things depend rather 
upon difference of arrangement than upon 
ultimate difference in the substance of which 
they are composed. Although this was con- 
tained in the atomic theory of Democritus, as 
expounded by Lucretius, yet it will be found 



FIRST AND LAST CATASTROPHE 225 

by any one who examines further the con- 
sequences which are drawn from it that it very 
soon diverges from the truth of things, as we 
might naturally expect it would. On the 
contrary, the view of the constitution of matter 
which is held by scientific men in the present 
day is not a guess at all. 

In the first place I will endeavour to explain 
what are the main points in this theory. First 
of all we must take the simplest form of matter, 
which turns out to be a gas such, for example, 
as the air in this room. The belief of scientific 
men in the present day is that this air is not 
a continuous thing, that it does not fill the 
whole of the space in the room, but is made up 
of an enormous number of exceedingly small 
particles. There are two sorts of particles : 
one sort of particle is oxygen, and another sort 
of particle nitrogen. All the particles of 
oxygen are as near as possible alike in these 
two respects ; first in weight, and secondly in 
certain peculiarities of mechanical structure. 
These small molecules are not at rest in the 
room, but are flying about in all directions with 
a mean velocity of seventeen miles a minute. 
They do not fly far in one direction ; but any 
particular molecule, after going over an in- 
credibly short distance the measure of which 
has been made meets another, not exactly 
plump, but a little on one side, so that they 
VOL. I Q 



226 LECTURES AND ESSAYS 

behave to one another somewhat in the same 
way as two people do who are dancing Sir 
Roger de Coverley; they join hands, swing 
round, and then fly away in different directions. 
All these molecules are constantly changing 
the direction of each other's motion ; they are 
flying about with very different velocities, 
although, as I have said, their mean velocity 
is about seventeen miles a minute. If the 
velocities were all marked off on a scale, they 
would be found distributed about the mean 
velocity just as shots are distributed about a 
mark. If a great many shots are fired at a 
target, the hits will be found thickest at the 
bull's-eye, and they will gradually diminish as 
we go away from that, according to a certain 
law which is called the law of error. It was 
first stated clearly by Laplace ; and it is one 
of the most remarkable consequences of theory 
that the molecules of a gas have their velocities 
distributed amongst them precisely according 
to this law of error. In the case of a liquid, 
it is believed that the state of things is quite 
different. We said that in the gas the mole- 
cules are moved in straight lines, and that it is 
only during a small portion of their motion 
that they are deflected by other molecules ; but in 
a liquid we may say that the molecules go about 
as if they were dancing the grand chain in the 
Lancers. Every molecule after parting com- 



FIRST AND LAST CATASTROPHE 227 

pany with one finds another, and so is constantly 
going about in a curved path, and never sent 
quite clear away from the sphere of action of 
the surrounding molecules. But, notwithstand- 
ing that, all molecules in a liquid are constantly 
changing their places, and it is for that reason 
that diffusion takes place in the liquid. Take 
a large tank of water and drop a little iodine 
into it, and you will find after a certain time 
all the water turned slightly blue. That is 
because all the iodine molecules have changed 
like the others and spread themselves over the 
whole of the tank. Because, however, you 
cannot see this, except where you use different 
colours, you must not suppose that it does not 
take place where the colours are the same. 
In every liquid all the molecules are running 
about and continually changing and mixing 
themselves up in fresh forms. In the case of 
a solid quite a different thing takes place. In 
a solid every molecule has a place which it 
keeps ; that is to say, it is not at rest any more 
than a molecule of a liquid or a gas, but it has 
a certain mean position which it is always 
vibrating about and keeping fairly near to, and 
it is kept from losing that position by the 
action of the surrounding molecules. These 
are the main points of the theory of the con- 
stitution of matter as at present believed. v^I*;? 
It differs from the theory of Democritus in 



228 LECTURES AND ESSAYS 

this way. There is no doubt that in the first 
origin of it, when it was suggested to the 
mind of Daniel Bernouilli as an explanation 
of the pressure of gases, and to the mind 
of Dalton as an explanation of chemical 
reactions, it was a guess ; that is to say, it 
was a supposition which would explain these 
facts of physics and chemistry, but which 
was not known to be true. Some theories 
are still in that position ; other theories 
are known to be true, because they can be 
argued back to from the facts. In order to 
make out that your supposition is true, it is 
necessary to show, not merely that that parti- 
cular supposition will explain the facts, but also 
that no other one will. Now, by the efforts of 
Clausius and Clerk Maxwell, the molecular 
theory of matter has been put in this other 
position. Namely, instead of saying, Let us 
suppose such and such things are true, and 
then deducing from that supposition what the 
consequences ought to be, and showing that 
these consequences are just the facts which we 
observe instead of doing that, I say, we make 
certain experiments ; we show that certain facts 
are undoubtedly true, and from these facts we 
go back by a direct chain of logical reasoning, 
which there is no way of getting out of, to the 
statement that all matter is made up of separate 
pieces or molecules, and that in matter of a 



FIRST AND LAST CATASTROPHE 229 

given kind, in oxygen, or in hydrogen, or in 
nitrogen, these molecules are of very nearly the 
same weight, and have certain mechanical pro- 
perties which are common to all of them. In 
order to show you something of the kind of 
evidence for that statement, I must mention 
another theory which, as it seems to me, is in 
the same position ; namely, the doctrine of the 
luminiferous ether, or that wonderful substance 
which is distributed all over space, and which 
carries light and radiant heat. By means of 
certain experiments upon interference of light 
we can show, not by any hypothesis, not by 
any guess at all, but by a pure interpretation of 
the experiment that in every ray of light there 
is some change or other, whatever it is, which 
is periodic in time and in space. By saying it 
is periodic in time, I mean that, at a given 
point of the ray of light, this change increases 
up to a certain instant, then decreases, then 
increases in the opposite direction, and then 
decreases again, and so on alternately. That 
is shown by experiments of interference ; it is 
not a theory which will explain the facts, but 
it is a fact which is got out of observation. 
By saying that this phenomenon is periodic in 
space, I mean that, if at any given instant you 
could examine the ray of light, you would find 
that some change or disturbance, whatever it 
is, has taken place all along it in different 



230 LECTURES AND ESSAYS 

degrees. It vanishes at certain points, and 
between these it increases gradually to a 
maximum on one side and the other alternately. 
That is to say, in travelling along a ray of light 
there is a certain change (which can be observed 
by experiments, by operating upon a ray of 
light with other rays of light) which goes 
through a periodic variation in amount. The 
height of the sea, as you know if you travel 
along it, goes through certain periodic changes ; 
it increases and decreases, and increases and 
decreases again at definite intervals. And if 
you take the case of waves travelling over the 
sea, and place yourself at a given point, or 
mark a point by putting a cork upon the 
surface, you will find that the cork will rise up 
and down ; that is to say, there will be a change 
or displacement of the cork's position, which is 
periodic in time, which increases and decreases, 
then increases in the opposite direction, and 
decreases again. Now this fact, which is 
established by experiment, and which is not a 
guess at all the fact that light is a phenomenon 
periodic in time and space is what we call the 
wave theory of light. The word " theory " here 
does not mean a guess ; it means an organised 
account of the facts, such that from it you may 
deduce results which are applicable to future 
experiments, the like of which have not yet 
been made. But we can see more than this. 



FIRST AND LAST CATASTROPHE 231 

So far we say that light consists of waves, 
merely in the sense that it consists of some 
phenomenon or other which is periodic in time 
and in place ; but we know that a ray of light 
or heat is capable of doing work. Radiant 
heat, for example, striking on a body, will 
warm it and enable it to do work by expansion ; 
therefore this periodic phenomenon which takes 
place in the ray of light is something or other 
which possesses mechanical energy, which is 
capable of doing work. We may make it, if 
you like, a mere matter of definition, and say : 
Any change which possesses energy is a motion 
of matter ; and this is perhaps the most in- 
telligible definition of matter that we can frame. 
In that sense, and in that sense only, it is a 
matter of demonstration, and not a matter of 
guess, that light consists of the periodic motion 
of matter, of something which is between the 
luminous object and our eyes. 

But that something is not matter in the 
ordinary sense of the term ; it is not made up 
of such molecules as gases and liquids and 
solids are made up of. This last statement 
again is no guess, but a proved fact. There 
are people who ask : Why is it necessary to 
suppose a luminiferous ether to be anything 
else except molecules of matter in space, in 
order to carry light about ? The answer is a 
very simple one. In order that separate mole- 



2 3 2 LECTURES AND ESSAYS 

cules may carry about a disturbance, it is 
necessary that they should travel at least as 
fast as the disturbance travels. Now we know, 
by means that I shall afterwards come to, that 
the molecules of gas travel at a very ordinary 
rate about twenty times as fast as a good 
train. But, on the contrary, we know by the 
most certain of all evidence, by five or six 
different means, that the velocity of light is 
200,000 miles a second. By that very simple 
consideration we are able to tell that it is quite 
impossible for light to be carried by the mole- 
cules of ordinary matter, and that it wants 
something else that lies between those mole- 
cules to carry the light. Now, remembering 
the evidence which we have for the existence 
of this ether, let us consider another piece of 
evidence ; let us now consider what evidence 
we have that the molecules of a gas are 
separate from one another and have something 
between them. We find out, by experiment 
again, that the different colours of light depend 
upon the various rapidity of these waves, depend 
upon the size and upon the length of the waves 
that travel through the ether, and that when 
we send light through glass or any transparent 
medium except a vacuum, the waves of different 
lengths travel with different velocities. That 
is the case with the sea ; we find that long 
waves travel faster than short ones. In much 



FIRST AND LAST CATASTROPHE 233 

the same way, when light comes out of a 
vacuum and impinges upon any transparent 
medium, say upon glass, we find that the rate 
of transmission of all the light is diminished ; 
that it goes slower when it gets inside of a 
material body ; and that this change is greater 
in the case of small waves than of large ones. 
The small waves correspond to blue light, and 
the large waves correspond to red light. The 
waves of red light are not made to travel so 
slowly as the waves of blue light ; but, as in 
the case of waves travelling over the sea, when 
light moves in the interior of a transparent 
body the largest waves travel most quickly. 
Well, then, by using such a body as will 
separate out the different colours a prism 
we are able to affirm what are the constituents 
of the light which strikes upon it. The light 
that comes from the sun is made up of waves 
of various lengths ; but, making it pass through 
a prism, we can separate it out into a spectrum, 
and in that way we find a band of light instead 
of a spot coming from the sun, and to every 
band in the spectrum corresponds a wave of a 
certain definite length and definite time in 
vibration. Now we come to a very singular 
phenomenon. If you take a gas such as 
chlorine and interpose it in the path of that 
light, you will find that certain particular rays 
of the spectrum are absorbed, while others are 



234 LECTURES AND ESSAYS 

not. How is it that certain particular rates of 
vibration can be absorbed by this chlorine gas, 
while others are not ? That happens in this 
way that the chlorine gas consists of a great 
number of very small structures, each of which 
is capable of vibrating internally. Each of 
these structures is complicated, and is capable 
of a change of relative position amongst its 
parts of a vibratory character. We know that 
molecules are capable of such internal vibrations 
for this reason, that if we heat any solid body 
sufficiently it will in time give out light ; that 
is to say, the molecules are got into such a state 
of vibration that they start the ether vibrating, 
and they start the ether vibrating at the same 
rate at which they vibrate themselves. So that 
what we learn from the absorption of certain 
particular rays of light by chlorine gas is that 
the molecules of that gas are structures which 
have certain natural rates of vibration which 
they absorb, precisely those rates of vibration 
which belong to the molecules naturally. If 
you sing a certain note to a string of a piano, 
that string if in tune will vibrate. If, therefore, 
a screen of such strings were put across a room, 
and you sang a note on one side, a person on the 
other side would hear the note very weakly or 
not at all, because it would be absorbed by the 
strings ; but if you sang another note, not one 
to which the strings naturally vibrated, then it 



FIRST AND LAST CATASTROPHE 235 

would pass through, and would not be eaten 
up by setting the strings vibrating. Now this 
question arises. Let us put the molecules aside 
for a moment. Suppose we do not know of 
their existence, and say : Is this rate of vibra- 
tion which naturally belongs to the gas a thing 
which belongs to it as a whole, or does it belong 
to the separate parts of it ? You might suppose 
that it belongs to the gas as a whole. A jar 
of water, if you shake it, has a perfectly definite 
time in which it oscillates, and that is very 
easily measured. That time of oscillation 
belongs to the jar of water as a whole. It 
depends upon the weight of the water and the 
shape of the jar. But now, by a very certain 
method, we know that the time of vibration 
which corresponds to a certain definite gas does 
not belong to it as a whole, but belongs to the 
separate parts of it for this reason, that if you 
squeeze the gas you do not alter the time of 
vibration. Let us suppose that we have a great 
number of fiddles in a room which are all in 
contact, and have strings accurately tuned to 
vibrate to certain notes. If you sang one of 
those notes all the fiddles would answer ; but 
if you compress them you clearly put them all 
out of tune. They are all in contact, and they 
will not answer to the note with the same 
precision as before. But if you have a room 
which is full of fiddles, placed at a certain 



236 LECTURES AND ESSAYS 

distance from one another, then if you bring 
them within shorter distances of o: : another, 
so that they still do not touch, they will not be 
put out of tune they will answer exactly to 
the same note as before. We see, therefore, 
that since compression of a gas within certain 
limits does not alter the rate of vibration which 
belongs to it, that rate of vibration cannot 
belong to the body of gas as a whole, but it 
must belong to the individual parts of it. Now, 
by such reasoning as this it seems to me that 
the modern theory of the constitution of matter 
is put upon a basis which is absolutely in- 
dependent of hypothesis. The theory is simply 
an organised statement of the facts ; a state- 
ment, that is, which is rather different from the 
experiments, being made out from them in just 
such a way as to be most convenient for rinding 
out from them what will be the results of other 
experiments. That is all we mean at present 
by scientific theory. 

Upon this theory Professor Clerk Maxwell 
founded a certain argument in his lecture before 
the British Association at Bradford. It is a 
consequence of the molecular theory, as I said 
before, that all the molecules of a certain given 
substance, say oxygen, are as near as possible 
alike in two respects first in weight, and 
secondly in their times of vibration. Professor 
Clerk Maxwell's argument was this. He first 



FIRST AND LAST CATASTROPHE 237 

of all said that the theory required us to believe, 
not that these molecules were as near as may 
be alike, but that they were exactly alike in 
these two respects at least the argument 
appeared to me to require that Then he said 
all the oxygen we know of, whatever processes 
it has gone through whether it is got out of 
the atmosphere, or out of some oxide of iron 
of carbon, or whether it belongs to the sun or 
the fixed stars, or the planets or the nebulae 
all this oxygen is alike. And all these mole- 
cules of oxygen we find upon the earth must 
have existed unaltered, or appreciably unaltered, 
during the whole of the time the earth has 
been evolved. Whatever vicissitudes they 
have gone through, however many times they 
have entered into combination with iron or 
carbon and been carried down beneath the 
crust of the earth, or set free and sent up again 
through the atmosphere, they have remained 
steadfast to their original form unaltered, the 
monuments of what they were when the world 
began. Professor Clerk Maxwell argues that 
things which are unalterable, and are exactly 
alike, cannot have been formed by any natural 
process. Moreover, being exactly alike, they 
cannot have existed for ever, and therefore 
they must have been made. As Sir John 
Herschel said, " They bear the stamp of the 
manufactured article." 



238 LECTURES AND ESSAYS 

Into these further deductions I do not pro- 
pose to enter at all. I confine myself strictly 
to the first of the deductions which Professor 
Clerk Maxwell made from the molecular theory. 
He said that because these molecules are ex- 
actly alike, and because they have not been in 
the least altered since the beginning of time, 
therefore they cannot have been produced by 
any process of evolution. It is just that ques- 
tion which I want to discuss. I want to con- 
sider whether the evidence we have to prove 
that these molecules are exactly alike is 
sufficient to make it impossible that they can 
have been produced by any process of evolution. 

The position that this evidence is not 
sufficient is evidently by far the easier to 
defend ; because the negative is proverbially 
hard to prove ; and if any one should prove 
that a process of evolution was impossible, it 
would be an entirely unique thing in science 
and philosophy. In fact, we may see from 
this example precisely how great is the influence 
of authority in matters of science. If there is 
any name among contemporary natural philo- 
sophers to whom is due the reverence of all 
true students of science, it is that of Professor 
Clerk Maxwell. But if any one not possessing 
his great authority had put forward an argu- 
ment, founded apparently upon a scientific 
basis, in which there occurred assumptions 



FIRST AND LAST CATASTROPHE 239 

about what things can and what things cannot 
have existed from eternity, and about the exact 
similarity of two or more things established by 
experiment, we should say : " Past eternity ; 
absolute exactness ; this won't do ; " and we 
should pass on to another book. The experi- 
ence of all scientific culture for all ages during 
which it has been a light to men has shown us 
that we never do get at any conclusions of that 
sort. We do not get at conclusions about 
infinite time or infinite exactness. We get at 
conclusions which are as nearly true as experi- 
ment can show, and sometimes which are a 
great deal more correct than direct experiment 
can be, so that we are able actually to correct 
one experiment by deductions from another ; 
but we never get at conclusions which we have 
a right to say are absolutely exact ; so that 
even if we find a man of the highest powers 
saying that he had reason to believe a certain 
statement to be exactly true, or that he believed 
a certain thing to have existed from the begin- 
ning exactly as it is now, we must say : " It 
is quite possible that a man of so great eminence 
may have found out something which is entirely 
different from the whole of our previous know- 
ledge, and the thing must be inquired into. But, 
notwithstanding that, it remains a fact that this 
piece of knowledge will be absolutely of a differ- 
ent kind from anything that we knew before." 



240 LECTURES AND ESSAYS 

Now let us examine the evidence by which 
we know that the molecules of the same gas 
are as near as may be alike in weight and in 
rates of vibration. There were experiments 
made by Dr. Graham, late Master of the Mint, 
upon the rate at which different gases were 
mixed together. He found that if he divided 
a vessel by a thin partition made of blacklead 
or graphite, and put different gases on the two 
opposite sides, they would mix together nearly 
as fast as though there was nothing between 
them. The difference was that the plate of 
graphite made it more easy to measure the 
rate of mixture ; and Dr. Graham made 
measurements and came to conclusions which 
are exactly such as are required by the mole- 
cular theory. It is found by a process of 
mathematical calculation that the rate of 
diffusion of different gases depends upon the 
weight of the molecules. A molecule of oxygen 
is sixteen times as heavy as a molecule of 
hydrogen, and it is found upon experiment 
that hydrogen goes through a septum or wall 
of graphite four times as fast as oxygen does. 
Four times four are sixteen. We express that 
rule in mathematics by saying that the rate of 
diffusion of gas is inversely as the square root 
of the mass of its molecules. If one molecule 
is thirty-six times as heavy as another the 
molecule of chlorine is nearly that multiple of 



FIRST AND LAST CATASTROPHE 241 

hydrogen it will diffuse itself at one-sixth of 
the rate. 

This rule is a deduction from the molecular 
theory, and it is found, like innumerable other 
such deductions, to come right in practice. But 
now observe what is the consequence of this. 
Suppose that, instead of taking one gas and 
making it diffuse itself through a wall, we take 
a mixture of two gases. Suppose we put 
oxygen and hydrogen into one side of a vessel 
which is divided into two parts by a wall of 
graphite, and we exhaust the air from the other 
side, then the hydrogen will go through this 
wall four times as fast as the oxygen will. 
Consequently, as soon as the other side is full 
there will be a great deal more hydrogen in it 
than oxygen that is to say, we shall have 
sifted the oxygen from the hydrogen, not com- 
pletely, but in a great measure, precisely as by 
means of a screen we can sift large coals from 
small ones. Now let us suppose that when we 
have oxygen gas unmixed with any other the 
molecules are of two sorts and of two different 
weights. Then you see that if we make that 
gas pass through a porous wall, the lighter 
particles would pass through first, and we 
should get two different specimens of oxygen 
gas, in one of which the molecules would be 
lighter than in the other. The properties of 
one of these specimens of oxygen gas would 
VOL. I R 



242 LECTURES AND ESSAYS 

necessarily be different from those of the other, 
and that difference might be found by very 
easy processes. If there were any perceptible 
difference between the average weight of the 
molecules on the two sides of the septum, 
there would be no difficulty in finding that out. 
No such difference has ever been observed. 
If we put any single gas into a vessel, and we 
filter it through a septum of blacklead into 
another vessel, we find no difference between 
the gas on one side of the wall and the gas on 
the other side. That is to say, if there is any 
difference it is too small to be perceived by our 
present means of observation. It is upon that 
sort of evidence that the statement rests that 
the molecules of a given gas are all very nearly 
of the same weight. Why do I say very 
nearly? Because evidence of that sort can 
never prove that they are exactly of the same 
weight. The means of measurement we have 
may be exceedingly correct, but a certain limit 
must always be allowed for deviation ; and if 
the deviation of molecules of oxygen from a 
certain standard of weight were very small, 
and restricted within small limits, it would be 
quite possible for our experiments to give us 
the results which they do now. Suppose, for 
example, the variation in the size of the oxygen 
atoms were as great as that in the weight of 
different men, then it would be very difficult 



FIRST AND LAST CATASTROPHE 243 

indeed to tell by such a process of sifting what 
that difference was, or in fact to establish that 
it existed at all. But, on the other hand, if we 
suppose the forces which originally caused all 
those molecules to be so nearly alike as they 
are to be constantly acting and setting the thing 
right as soon as by any sort of experiment we 
set it wrong, then the small oxygen atoms on 
one side would be made up to their right size, and 
it would be impossible to test the difference by 
any experiment which was not quicker than the 
processes by which they were made right again. 
There is another reason why we are obliged 
to regard that experiment as only approximate, 
and as not giving us any exact results. There 
is very strong evidence, although it is not con- 
clusive, that in a given gas say in a vessel 
full of carbonic acid the molecules are not 
all of the same weight. If we compress the 
gas, we find that when in the state of a 
perfect gas, or nearly so, the pressure increases 
just in the ratio that the volume diminishes. 
That law is entirely explained by means of the 
molecular theory. It is what ought to exist 
if the molecular theory is true. If we compress 
the gas further, we find that the pressure is 
smaller than it ought to be according to this 
law. This can be explained in two ways. First 
of all we may suppose that the molecules are 
so crowded that the time during which they 



244 LECTURES AND ESSAYS 

are sufficiently near to attract each other 
sensibly becomes too large a proportion of the 
whole time to be neglected ; and this will 
account for the change in the law. There 
is, however, another explanation. We may 
suppose, for illustration, that two molecules 
approach one another, and that the speed at 
which one is going relatively to the other is 
very small, and then that they so direct one 
another that they get caught together, and go 
on circling, making only one molecule. This, 
on scientific principles, will account for our 
fact, that the pressure in a gas which is near 
a liquid state is too small that instead of the 
molecules going about singly, some are hung 
together in couples and some in larger numbers, 
and making still larger molecules. This sup- 
position is confirmed very strikingly by the 
spectroscope. If we take the case of chlorine 
gas, we find that it changes colour that it 
gets darker as it approaches the liquid condi- 
tion. This change of colour means that there 
is a change in the rate of vibration which belongs 
to its component parts ; and it is a very simple 
mechanical deduction that the larger molecules 
will, as a rule, have a slower rate of vibration 
than the smaller ones very much in the same 
way as a short string gives a higher note than 
a long one. The colour of chlorine changes 
just in the way we should expect if the mole- 



FIRST AND LAST CATASTROPHE 245 

cules, instead of going about separately, were 
hanging together in couples ; and the same 
thing is true of a great number,, of the metals. 
Mr. Lockyer, in his admirable researches, has 
shown that several of the metals and metalloids 
have various spectra, according to the tempera- 
ture and the pressure to which they are exposed ; 
and he has made it exceedingly probable that 
these various spectra that is, the rates of 
vibration of the molecules depend upon the 
molecules being actually of different sizes. 
Dr. Roscoe has a few months ago shown an 
entirely new spectrum of the metal sodium, 
whereby it appears that this metal exists in a 
gaseous state in four different degrees of 
aggregation as a simple molecule, and as 
three or four or eight molecules together. 
Every increase in the complication of the 
molecules every extra molecule you hang on 
to the aggregate that goes about together 
will make a difference in the rate of the vibra- 
tion of that system, and so will make a difference 
in the colour of the substance. 

So then we have an evidence of an entirely 
extraneous character that in a given gas the 
actual molecules that exist are not all of the 
same weight. Any experiment which failed 
to detect this would fail to detect any smaller 
difference. And here also we can see a reason 
why, although a difference in the size of the 



246 LECTURES AND ESSAYS 

molecules does exist, yet we do not find that 
out by sifting. Suppose you take oxygen gas 
consisting of single molecules and double mole- 
cules, and you sift it through a plate ; the 
single molecules get through first, but, when 
they get through, some of them join themselves 
together as double molecules ; and although 
more double molecules are left on the other 
side, yet some of them break up and make 
single molecules ; so the process of sifting, 
which ought to give you single molecules on 
the one side and double on the other, merely 
gives you a mixture of single and double on 
both sides ; because the reasons which origin- 
ally decided that there should be just those 
two forms are always at work and continually 
setting things right. 

Now let us take the other point in which 
molecules are very nearly alike namely, that 
they have very nearly the same rate of vibra- 
tion. The metal sodium in the common salt 
upon the earth has two rates of vibration ; it 
sounds two notes, as it were, which are very 
near to each other. They form the well-known 
double line D in the yellow part of the spectrum. 
These two bright yellow lines are very easy to 
observe. They occur in the spectra of a great 
number of stars. They occur in the solar 
spectrum as dark lines, showing that there is 
sodium in the outer rim of the sun, which is 



FIRST AND LAST CATASTROPHE 247 

stopping and shutting off the light of the 
bright parts behind. All these lines of sodium 
are just in the same position in the spectrum, 
showing that the rates of vibration of all these 
molecules of sodium all over the universe, so 
far as we know, are as near as possible alike. 
That implies a similarity of molecular structure, 
which is a great deal more delicate than any 
mere test of weight. You may weigh two 
fiddles until you are tired, and you will never 
find out whether they are in tune ; the one 
test is a great deal more delicate than the 
other. Let us see how delicate this test is. 
Lord Rayleigh has remarked that there is a 
natural limit for the precise position of a given 
line in the spectrum, and for this reason. If a 
body which is emitting a sound comes towards 
you, you will find that the pitch of the sound 
is altered. Suppose that omnibuses run every 
ten minutes in the streets, and you walk in a 
direction opposite to that in which they are 
coming, you will obviously pass more omni- 
buses in an hour than if you walked in an 
opposite direction. If a body emitting light 
is coming towards you, you will find more 
waves in a certain direction than if it were 
going from you ; consequently, if you are 
approaching a body emitting light, the waves 
will come at shorter intervals, the vibration 
will be of shorter period, and the light will be 



248 LECTURES AND ESSAYS 

higher up in the spectrum it will be more 
blue. If you are going away from the body, 
then the rate is slower, the light is lower down 
on the spectrum, and consequently more red. 
By means of such variations in the positions of 
certain known lines, the actual rate of approach 
of certain fixed stars to the earth has been 
measured, and the rate of going away of certain 
other fixed stars has also been measured. 
Suppose we have a gas which is glowing in 
a state of incandescence, all the molecules are 
giving out light at a certain specified rate of 
vibration ; but some of these are coming 
towards us at a rate much greater than seven- 
teen miles a minute, because the temperature 
is higher when the gas is glowing, and others 
are also going away at a much higher rate than 
that. The consequence is, that instead of hav- 
ing one sharply defined line on the spectrum, 
instead of having light of exactly one bright 
colour, we have light which varies between 
certain limits. If the actual rate of the vibra- 
tion of the molecules of the gas were marked 
down upon the spectrum, we should not get 
that single bright line there, but we should get 
a bright band overlapping it on each side. 
Lord Rayleigh calculated that, in the most 
favourable circumstances, the breadth of this 
band would not be less than one-hundredth of 
the distance between the sodium lines. It is 



FIRST AND LAST CATASTROPHE 249 

precisely upon that experiment that the evidence 
of the exact similarity of molecules rests. We 
see, therefore, from the nature of the experi- 
ment, that we should get exactly the same 
results if the rates of vibration of all the 
molecules were not exactly equal, but varied 
within certain very small limits. If, for 
example, the rates of vibration varied in the 
same way as the heads of different men, then 
we should get very much what we get now 
from the experiment. 

From the evidence of these two facts, then 
the evidence that molecules are of the same 
weight and degree of vibration all that we 
can conclude is that whatever differences there 
are in their weights, and whatever differences 
there are in their degrees of vibration, these 
differences are too small to be found out by 
our present modes of measurement. And that 
is precisely all that we can conclude in every 
similar question of science. 

Now, how does this apply to the question 
whether it is possible for molecules to have 
been evolved by natural processes ? I do not 
understand myself how, even supposing we 
knew that they were exactly alike, we could 
infer for certain that they had not been 
evolved ; because there is only one case of 
evolution that we know anything at all about 
and that we know very little about yet 



250 LECTURES AND ESSAYS 

namely, the evolution of organised beings. The 
processes by which that evolution takes place 
are long, cumbrous, and wasteful processes of 
natural selection and hereditary descent. They 
are processes which act slowly, which take a 
great lapse of ages to produce their natural 
effects. But it seems to me quite possible to 
conceive, in our entire ignorance of the subject, 
that there may be other processes of evolution 
which result in a definite number of forms 
those of the chemical elements just as these 
processes of the evolution of organised beings 
have resulted [in a greater number of forms. 
All that we know of the ether shows that its 
actions are of a rapidity very much exceeding 
anything we know of the motions of visible 
matter. It is a possible thing, for example, 
that mechanical conditions should exist accord- 
ing to which all bodies must be made of regular 
solids, that molecules should all have flat sides, 
and that these sides should all be of the same 
shape. I suppose that it is just conceivable 
that it might be impossible for a molecule to 
exist with two of its faces different. In that 
case we know there would be just five shapes 
for a molecule to exist in, and these would 
be produced by a process of evolution. The 
various forms of matter that chemists call 
elements seem to be related one to another 
very much in that sort of way ; that is, as if 



FIRST AND LAST CATASTROPHE 251 

they rose out of mechanical conditions which 
only rendered it possible for a certain definite 
number of forms to exist, and which, whenever 
any molecule deviates slightly from one of 
these forms, would immediately operate to set 
it right again. I do not know at all we have 
nothing definite to go upon what the shape 
of a molecule is, or what is the nature of the 
vibration it undergoes, or what its condition is 
compared with the ether ; and in our absolute 
ignorance it would be impossible to make any 
conception of the mode in which it grew up. 
When we know as much about the shape of a 
molecule as we do about the solar system, for 
example, we may be as sure of its mode of 
evolution as we are of the way in which the 
solar system came about ; but in our present 
ignorance all we have to do is to show that 
such experiments as we can make do not give 
us evidence that it is absolutely impossible for 
molecules of matter to have been evolved out 
of ether by natural processes. 

The evidence which tells us that the mole- 
cules of a given substance are alike is only ap- 
proximate. The theory leaves room for certain 
small deviations ; and consequently if there 
are any conditions at work in the nature of the 
ether which render it impossible for other forms 
of matter than those we know of to exist, the 
great probability is that when by any process 



252 LECTURES AND ESSAYS 

we contrive to sift molecules of one kind from 
molecules of another, these very conditions at 
once bring them back and restore to us a mass 
of gas consisting of molecules whose average 
type is a normal one. 

Now I want to consider a speculation of an 
entirely different character. A remark was 
made about thirty years ago by Sir William 
Thomson upon the nature of certain problems 
in the conduction of heat. These problems had 
been solved by Fourier many years before in a 
beautiful treatise. The theory was that if you 
knew the degree of warmth of a body, then you 
could find what would happen to it afterwards ; 
you would find how the body would gradually 
cool. Suppose you put the end of a poker in 
the fire and make it red hot, that end is very 
much hotter than the other end ; but if you 
take it out and let it cool, you will find that 
heat is travelling from the hot end to the cool 
end ; and the amount of this travelling, and the 
temperature at either end of the poker, can be 
calculated with great accuracy. This comes 
out of Fourier's theory. Now suppose you try 
to go backwards in time, and take the poker at 
any instant when it is about half cool, and say : 
" Does this equation give me the means of 
finding out what was happening before this 
time, in so far as the present state of things has 
been produced by cooling ? " You will find the 



FIRST AND LAST CATASTROPHE 253 

equation will give you an account of the state of 
the poker before the time when it came into 
your hands, with great accuracy up to a certain 
point ; but beyond that point it refuses to give 
you any more information, and it begins to 
talk nonsense. It is in the nature of a problem 
of the conduction of heat that it allows you to 
trace the forward history of it to any extent 
you like ; but it will not allow you to trace the 
history of it backward beyond a certain point. 
There is another case in which a similar thing 
happens. There is an experiment in that 
excellent manual, the Boy's Own Book, which 
tells you that if you half fill a glass with beer, 
and put some paper on it, and then pour in 
water carefully, and draw the paper out without 
disturbing the two liquids, the water will rest 
on the beer. The problem then is to drink the 
beer without drinking the water, and it is 
accomplished by means of a straw. Let us 
suppose these two liquids resting in contact ; 
we shall find they begin to mix ; and it is 
possible to write down an equation exactly of 
the same form as the equation for the conduction 
of heat, which would tell you how much water 
had passed into the beer at any given time after 
the mixture began. So that, given the water 
and the beer half mixed, you could trace forward 
the process of mixing, and measure it with 
accuracy, and give a perfect account of it ; but 



254 LECTURES AND ESSAYS 

if you attempt to trace that back you will have 
a point where the equation will stop, and will 
begin to talk nonsense. That is the point where 
you took away the paper, and allowed the 
mixing to begin. If we apply that same con- 
sideration to the case of the poker, and try to 
trace back its history, you will find that the 
point where the equation begins to talk nonsense 
is the point where you took it out of the fire. 
The mathematical theory supposes that the 
process of conduction of heat has gone on in a 
quiet manner, according to certain defined laws, 
and that if at any time there was a catastrophe, 
an event not included in the laws of the con- 
duction of heat, then the equation could give 
you no account of it. There is another thing 
which is of the same kind namely, the trans- 
mission of fluid friction. If you take your tea 
in your cup, and stir it round with a spoon, it will 
not go on circulating round for ever, but will 
come to a stop ; and the reason is that there is 
a certain friction of the liquid against the sides 
of the cup, and of the different parts of the 
liquid with one another. The friction of the 
different parts of a liquid or a gas is precisely 
a matter of mixing. The particles which are 
going fast, and are in the middle, not having 
been stopped by the side, get mixed ; and the 
particles at the side going slow get mixed with 
the particles in the middle. This process of 



FIRST AND LAST CATASTROPHE 255 

mixing can be calculated, and it leads to an 
equation of exactly the same sort as that which 
applies to the conduction of heat. We have, 
therefore, in these problems a natural process 
which consists in mixing things together, and 
this always has the property that you can go 
on mixing them for ever without coming to 
anything impossible ; but if you attempt to 
trace the history of the thing backward, you 
must always come to a state which could not 
have been produced by mixing namely, a 
state of complete separation. 

Upon this remark of Sir William Thomson's, 
the true consequences of which you will find 
correctly stated in Mr. Balfour Stewart's book 
on the Conservation of Energy, a most singular 
doctrine has been founded. These writers have 
been speaking of a particular problem on which 
they were employed at the moment. Sir 
William Thomson was speaking of the conduc- 
tion of heat, and he said this heat problem leads 
you back to a state which could not have been 
produced by the conduction of heat. And so 
Professor Clerk Maxwell, speaking of the same 
problem, and also of the diffusion of gases, said 
there was evidence of a limit in past time to the 
existing order of things, when something else 
than mixing took place. But a most eminent 
man, who has done a great deal of service to 
mankind, Professor Stanley Jevons, in his very 



256 LECTURES AND ESSAYS 

admirable book, the Principles of Science^ which 
is simply marvellous for the number of examples 
illustrating logical principles which he has drawn 
from all kinds of regions of science, and for the 
small number of mistakes that occur in it, 
takes this remark of Sir W. Thomson's, and 
takes out two very important words, and puts 
in two other very important words. He says : 
" We have here evidence of a limit of a state of 
things which could not have been produced by 
the previous state of things according to the 
known laws of nature." It is not according to 
the known laws of nature, it is according to the 
known laws of conduction of heat, that Sir 
William Thomson is speaking ; and from this 
we may see the fallacy of concluding that if we 
consider the case of the whole universe we 
should be able, supposing we had paper and 
ink enough, to write down an equation which 
would enable us to make out the history of the 
world forward as far forward as we liked to 
go ; but if we attempted to calculate the history 
of the world backward, we should come to a 
point where the equation would begin to talk 
nonsense we should come to a state of things 
which could not have been produced from any 
previous state of things by any known natural 
laws. You will see at once that that is an 
entirely different statement. The same doctrine 
has been used by Mr. Murphy, in a very able 



FIRST AND LAST CATASTROPHE 257 

book, the Scientific Basis of Faith, to build upon 
it an enormous superstructure I think the 
restoration of the Irish Church was one of the 
results of it. But this doctrine is founded, as 
I think, upon a pure misconception. It is 
founded entirely upon forgetfulness of the con- 
dition under which the remark was originally 
made. All these physical writers, knowing 
what they were writing about, simply drew such 
conclusions from the facts which were before 
them as could be reasonably drawn. They say : 
" Here is a state of things which could not have 
been produced by the circumstances we are at 
present investigating." Then your speculator 
comes ; he reads a sentence, and says : " Here 
is an opportunity for me to have my fling." 
And he has his fling, and makes a purely base- 
less theory about the necessary origin of the 
present order of nature at some definite point 
of time which might be calculated. But, if we 
consider the matter, we shall see that this is 
not in any way a consequence of the theory of 
the conduction of heat ; because the conduction 
of heat is not the only process that goes on in 
the universe. 

If we apply that theory to the case of the 
earth, we find that at present there is evidence 
of a certain distribution of temperature in the 
interior of it ; there is a certain rate at which 
the temperature increases as we go down ; and 

VOL. I S 



258 LECTURES AND ESSAYS 

9 

no doubt, if we made further investigations, we 
should find that if we went deeper an accurate 
law would be found, according to which the 
temperature increases in the interior. 

Now, assuming this to be so, taking this as 
the basis of our problem, we might endeavour 
to find out what was the history of the earth in 
past times, and when it began cooling down. 
That is exactly what Sir William Thomson has 
done. When we attempt it, we find that there 
is a definite point to which we can go, and 
beyond which our equation talks nonsense. 
But we do not conclude that at that point the 
laws of nature began to be what they are ; we 
only conclude that the earth began to solidify. 
Now solidification is not a process of the con- 
duction of heat, and so the thing cannot be 
given by our equation. That point is given 
definitely as a point of time, not with great 
accuracy, but still as near as we can expect to 
get it with such means of measuring as we 
have ; and Sir William Thomson has calculated 
that the earth must have solidified at some time 
between a hundred millions and two hundred 
millions of years ago ; and there we arrive at 
the beginning of the present state of things 
the process of cooling the earth which is going 
on now. Before that it was cooling as a liquid, 
and in passing from the liquid to the solid state 
there was a catastrophe which introduced a new 



FIRST AND LAST CATASTROPHE 259 

rate of cooling. So that by means of that law 
we do come to a time when the earth began to 
assume its present state. We do not find the 
time of the commencement of the universe, but 
simply of the present structure of the earth. 
If we went farther back we might make more 
calculations and find how long the earth had 
been in a liquid state. We should come to 
another catastrophe, and say not that at that 
time the universe began to exist, but that the 
present earth passed from the gaseous to the 
liquid state. And if we went farther back still 
we should probably find the earth falling 
together out of a great ring of matter surround- 
ing the sun and distributed over its orbit. The 
same thing is true of every body of matter : if 
we trace its history back, we come to a certain 
time at which a catastrophe took place ; and 
if we were to trace back the history of all the 
bodies of the universe in that way, we should 
continually see them separating up into smaller 
parts. What they have actually done is to fall 
together and get solid. If we could reverse the 
process we should see them separating and 
getting fluid ; and, as a limit to that, at an 
indefinite distance in past time, we should find 
that all these bodies would be resolved into 
molecules, and all these would be flying away 
from each other. There would be no limit to 
that process, and we could trace it as far back 



26o LECTURES AND ESSAYS 

as ever we liked to trace it. So that on the 
assumption a very large assumption that the 
present constitution of the laws of geometry and 
mechanics has held good during the whole of 
past time, we should be led to the conclusion 
that at an inconceivably long time ago the 
universe did consist of ultimate molecules, all 
separate from one another, and approaching one 
another. Then they would meet together and 
form a great number of small, hot bodies. 
Then you would have the process of cooling 
going on in these bodies, exactly as we find it 
going on now. But you will observe that we 
have no evidence of such a catastrophe as 
implies a beginning of the laws of nature. We 
do not come to something of which we cannot 
make any further calculation ; we find that 
however far we like to go back, we approximate 
to a certain state of things, but never actually 
get to it. 

Here, then, we have a doctrine about the 
beginning of things. First, we have a pro- 
bability, about as great as science can make it, 
of the beginning of the present state of things 
on the earth, and of the fitness of the earth for 
habitation ; and then we have a probability 
about the beginning of the universe as a whole 
which is so small that it is better put in this 
form, that we do not know anything at all 
about it The reason why I say that we do 



FIRST AND LAST CATASTROPHE 261 

not know anything at all of the beginning of 
the universe is that we have no reason whatever 
for believing that the known laws of geometry 
and mechanics are exactly and absolutely 
true at present, or that they have been even 
approximately true for any period of time 
further than we have direct evidence of. The 
evidence we have of them is founded on 
experience ; and we should have exactly the 
same experience of them now, if those laws 
were not exactly and absolutely true, but were 
only so nearly true that we could not observe 
the difference. So that in making the assump- 
tion that we may argue upon the absolute 
uniformity of nature, and suppose these laws to 
have remained exactly as they are, we are 
assuming something we know nothing about. 
My conclusion then is that we do know, with 
great probability, of the beginning of the 
habitability of the earth about one hundred or 
two hundred millions of years back, but that of 
a beginning of the universe we know nothing 
at all. 

Now let us consider what we can find out 
about the end of things. The life which exists 
upon the earth is made by the sun's action, 
and it depends upon the sun for its continuance. 
We know that the sun is wearing out, that it is 
cooling ; and although this heat which it loses 
day by day is made up in some measure, 



262 LECTURES AND ESSAYS 

perhaps completely at present, by the contrac- 
tion of its mass, yet that process cannot go on 
for ever. There is only a certain amount of 
energy in the present constitution of the sun ; 
and when that has been used up, the sun 
cannot go on giving out any more heat. Sup- 
posing, therefore, the earth remains in her 
present orbit about the sun, seeing that the 
sun must be cooled down at some time, we 
shall all be frozen out. On the other hand, we 
have no reason to believe that the orbit of the 
earth about the sun is an absolutely stable 
thing. It has been maintained for a long time 
that there is a certain resisting medium which 
the planets have to move through ; and it may 
be argued that in time all the planets must be 
gradually made to move in smaller orbits, and 
so to fall in towards the sun. But, on the 
other hand, the evidence upon which this 
assertion was based, the movement of Encke's 
comet and others, has been recently entirely 
overturned by Professor Tait. He supposes 
that these comets consist of bodies of meteors. 
Now it was proved a long time ago that a mass 
of small bodies travelling together in an orbit 
about a central body will always tend to fall in 
towards it, and that is the case with the rings 
of Saturn. So that, in fact, the movement of 
Encke's comet is entirely accounted for on the 
supposition that it is a swarm of meteors, with- 



FIRST AND LAST CATASTROPHE 263 

out regarding the assumption of a resisting 
medium. On the other hand, it seems exceed- 
ingly natural to suppose that some matter in a 
very thin state is diffused about the planetary 
spaces. Then we have another consideration, 
just as the sun and moon make tides upon 
the sea, so the planets make tides upon 
the sun. Consider the tide which the earth 
makes upon the sun. Instead of being a great 
wave lifting the mass of the sun up directly 
under the earth, it is carried forward by the 
sun's rotation ; the result is that the earth, 
instead of being attracted to the sun's centre, 
is attracted to a point before the centre. The 
immediate tendency is to accelerate the earth's 
motion, and the final effect of this upon the 
planet is to make its orbit larger. That planet 
disturbing all the other planets, the consequence 
is that we have the earth gradually going away 
from the sun, instead of falling into it. 1 

In any case, all we know is that the sun is 
going out If we fall into the sun then we 
shall be fried ; if we go away from the sun, or 
the sun goes out, then we shall be frozen. So 
that, so far as the earth is concerned, we have 
no means of determining what will be the 
character of the end, but we know that one of 



1 I learn from Sir W. Thomson that the ultimate effect of tidal 
deformation on a number of bodies is to reduce them to two, which 
move as if they were rigidly connected. 



264 LECTURES AND ESSAYS 

these two things must take place in time. But 
in regard to the whole universe, if we were to 
travel forward as we have travelled backward 
in time, and consider things as falling together, 
we should come finally to a great central mass, 
all in one piece, which would send out waves 
of heat through a perfectly empty ether, and 
gradually cool itself down. As this mass got 
cool it would be deprived of all life and motion ; 
it would be just a mere enormous frozen block 
in the middle of the ether. But that conclusion, 
which is like the one that we discussed about 
the beginning of the world, is one which we 
have no right whatever to rest upon. It 
depends upon the same assumption that the 
laws of geometry and mechanics are exactly 
and absolutely true ; and that they will con- 
tinue exactly and absolutely true for ever and 
ever. Such an assumption we have no right 
whatever to make. We may therefore, I think, 
conclude about the end of things that, so far as 
the earth is concerned, an end of life upon it is 
as probable as science can make anything ; but 
that in regard to the universe we have no 
right to draw any conclusion at all. 

So far, we have considered simply the 
material existence of the earth ; but of course 
our greatest interest lies not so much with the 
material life upon it, the organised beings, as 
with another fact which goes along with 



FIRST AND LAST CATASTROPHE 265 

that, and which is an entirely different 
one the fact of the consciousness that exists 
upon the earth. We find very good reason 
indeed to believe that this consciousness in the 
case of any organism is itself a very complex 
thing, and that it corresponds part for part to 
the action of the nervous system, and more 
particularly of the brain of that organised thing. 
There are some whom such evidence has led to 
the conclusion that the destruction which we 
have seen reason to think probable of all 
organised beings upon the earth will lead also 
to the final destruction of the consciousness 
that goes with them. Upon this point I know 
there is great difference of opinion amongst 
those who have a right to speak. But to those 
who do see the cogency of the evidences of 
modern physiology and modern psychology in 
this direction it is a very serious thing to con- 
sider that not only the earth itself and all that 
beautiful face of nature we see, but also the 
living things upon it, and all the consciousness 
of men, and the ideas of society, which have 
grown up upon the surface, must come to an 
end. We who hold that belief must just face 
the fact and make the best of it ; and I think we 
are helped in this by the words of that Jew 
philosopher, who was himself a worthy crown 
to the splendid achievements of his race in the 
cause of progress during the Middle Ages, 



266 LECTURES AND ESSAYS 

Benedict Spinoza. He said : " The free man 
thinks of nothing so little as of death, and his 
wisdom is a meditation not of death but of 
life." Our interest lies with so much of the 
past as may serve to guide our actions in the 
present, and to intensify our pious allegiance to 
the fathers who have gone before us and the 
brethren who are with us ; and our interest lies 
with so much of the future as we may hope will 
be appreciably affected by our good actions 
now. Beyond that, as it seems to me, we do 
not know, and we ought not to care. Do I 
seem to say : " Let us eat and drink, for to- 
morrow we die ? " Far from it ; on the contrary 
I say : " Let us take hands and help, for this 
day we are alive together." 

The following note was afterwards published 
by the author (Fortnightly Review, vol. xvii. p. 
793) : 

The passage referred to from the Principles 
of Science is as follows (vol. ii. p. 438) : 

" For a certain negative value of the time 
the formulae give impossible values, indicating 
that there was some initial distribution of heat 
which could not have resulted, according to 
known laws of nature, from any previous dis- 
tribution." 

The words italicised are here inserted into a 
sentence from Tait's Thermo-dynamics, p. 38. 



FIRST AND LAST CATASTROPHE 267 

Had the words conduction of heat been used 
instead of nature, the sentence would have 
remained correct, but would not have led to 
the alarming inference that 

" The theory of heat places us in the 
dilemma either of believing in creation at some 
assignable date in the past, or else of suppos- 
ing that some inexplicable change in the work- 
ing of natural laws then took place." 

It has been pointed out by Mr. Higgins that 
the ultimate effect of tides in the sun caused 
by the earth's attraction will be precisely similar 
to that of a resisting medium that is, will 
diminish the orbit of the earth and increase its 
velocity ; and that I was wrong in supposing 
the contrary effect. It results that the earth 
will certainly fall into the sun ; but whether 
before or after the sun has cooled down so 
much as not to be able to support life on this 
planet remains undetermined. The final con- 
clusion remains therefore as before that there 
must be an end, but whether by heat or by 
cold we cannot tell. 



THE UNSEEN UNIVERSE 1 

THE primary motive of this treatise is indicated 
by its second title : " Physical Speculations on 
a Future State." A sketch of the beliefs and 
yearnings of many different folk in regard to a 
life after death leads up to an attempt to find 
room for it within the limits of those physical 
doctrines of continuity and the conservation of 
energy which are regarded as the established 
truths of science. In this attempt it is necessary 
to discuss the ultimate constitution of matter 
and its relation to the ether. When, by a 
singular inconsequence in writers possessing 
such power in their right minds of sound 
scientific reasoning, room has been found for a 
future life in the manner indicated above, it is 
discovered that there is room for a great deal 
more. Accordingly some of the main doctrines 
of the Christian religion are interpreted in relation 
to the authors' hypothesis, and placed in their 

1 "The Unseen Universe; or, Physical Speculations on a 
Future State." London : Macmillan and Co. 1875. [Fort- 
nightly Review, June 1875.] 



THE UNSEEN UNIVERSE 269 

appropriate niches. It will perhaps be con- 
venient, therefore, if we consider these three 
things in their order : first, the desire for a 
future life ; secondly, the physical speculations 
that make room for it ; and lastly, that system, 
the seemingly innocent dried carcase of which 
is to be smuggled into our house at the same 
time, that it may peradventure find means of 
resurrection. 

I. 

It is often said that the universal longing 
for immortality among all kinds and conditions 
of men is a presumption that there is some 
future life in which this longing shall be satis- 
fied. Let us endeavour, therefore, to find out 
in what this longing for immortality actually 
consists ; whether the existence of it, when its 
nature is understood, can be explained on 
grounds which do not require it to have any 
objective fulfilment other than the life and the 
memory of those who come after us ; and what 
relation it bears to the equally widespread 
dream or vision of a spiritual world peopled by 
supernatural or monstrous beings, ghosts and 
gods and goblins. 

First, let us notice that all the words used 
to describe this immortality that is longed for 
are negative words : m-mortality, end-tess life, 
z#-finite existence. Endless life is an incon- 



270 LECTURES AND ESSAYS 

ceivable thing, for an endless time would be 
necessary to form an idea of it. Now it is 
only by a stretch of language that we can be 
said to desire that which is inconceivable. No 
doubt many persons say that they are smitten 
with an insatiable longing for the unattainable 
and ineffable ; but this means that they feel 
generally dissatisfied and do not at all know 
what they want. Longing for deathlessness 
means simply shrinking from death. However 
or whenever we who live endeavour to realise 
an end to this healthy life of action in ourselves 
or in our brethren the effort is a painful one ; 
and the mind, in so far as it is healthy, tries to 
put it off and avoid it. The state of one who 
really wishes for death is firmly linked in our 
thoughts with the extreme of misery and 
wretchedness and disease ; and, in so far as it 
can be realised, we seem to feel that such an 
one is fit to die. In those cases of ripe old 
age not hastened by disease, where the physical 
structure is actually worn out, having finished 
its work right honestly and well ; where the 
love of life is worn out also, and the grave 
appears as a bed of rest to the tired limbs, and 
death as a mere quiet sleep from thought ; 
there also, in so far as we are able to realise 
the state of the aged and to put ourselves in 
his place, death seems to be normal and natural, 
a thing to be neither sought nor shunned. But 



THE UNSEEN UNIVERSE 271 

such putting of ourselves in the place of one to 
whom death is no evil must in all cases be 
imperfect. I cannot, in my present life and 
motion, clearly conceive myself in so parlous a 
state that no hope of better things should make 
me shrink from the end of all. However 
vividly I recall the feelings of pain and weak- 
ness, it is the life and energy of my present self 
that pictures them ; and this life and energy 
cannot help raising at the same time combative 
instincts of resistance to pain and weakness, 
whose very nature it is to demand that the sun 
shall not go down upon Gibeon until they have 
slain the Amalekites. Nor can I really and 
truly put myself in the place of the worn-out 
old man whose consciousness may some day 
have a memory of mine. No force of imagina- 
tion that I can bring to bear will avail to cast 
out the youth of that very imagination which 
endeavours to depict its latter days ; no 
thoughts of final and supreme fatigue can help 
suggesting refreshment and new rising after sleep. 
If, then, we do not want to die now, nor 
next year, nor the year after that, nor at any 
time that we can clearly imagine ; what is this 
but to say that we want to live for ever, in the 
only meaning of the words that we can at all 
realise ? It is not that there is any positive 
attraction in the shadowy vistas of eternity, for 
the effort to contemplate even any very long 



272 LECTURES AND ESSAYS 

time is weariness and vexation of spirit ; it is 
that our present life, in so far as it is healthy, 
rebels once for all against its own final and 
complete destruction. And forasmuch as so 
many and so mighty generations have in time 
past ended in death their noble and brave 
battle with the elements, that we also and our 
brethren can in nowise hope to escape their 
fate, therefore we are sorely driven to find some 
way by which at least the image of that ending 
shall be avoided and set aside. As the fruit 
of this search two methods have been found 
and practised among men. By one method 
we detach ourselves from the individual body 
and its actions which accompany our con- 
sciousness, to identify ourselves with something 
wider and greater that shall live when we as 
units shall have done with living that shall 
work on with new hands when we, its worn-out 
limbs, have entered into rest. The soldier who 
rushes on death does not know it as extinction ; 
in thought he lives and marches on with the 
army, and leaves with it his corpse upon the 
battlefield. The martyr cannot think of his 
own end because he lives in the truth he has 
proclaimed ; with it and with mankind he 
grows into greatness through ever new victories 
over falsehood and wrong. But there is another 
way. Since when men have died such orderly, 
natural, and healthy activity as we have known 



THE UNSEEN UNIVERSE 273 

in them and valued their lives for has plainly 
ceased, we may fashion another life for them, 
not orderly, not natural, not healthy, but 
monstrous or super- natural ; whose cloudy 
semblance shall be eked out with the dreams 
of uneasy sleep or the crazes of a mind diseased. 
And it is to this that the universal shrinking 
of men from death, which is called a yearning 
for immortality, is alleged to bear witness. 

But whence now does it really come, and 
what is the true lesson of it ? Surely it is a 
necessary condition of life that has desires at 
all that these desires should be towards life and 
not away from it ; seeing how cheap and easy 
a thing is destruction on all hands, and how 
hard it is for race or unit to hold fast in the 
great struggle for existence. Surely our way 
is paved with the bones of those who have 
loved life and movement too little, and lost it 
before their time. If we could think of death 
without shrinking it would only mean that this 
world was no place for us, and that we should 
make haste to be gone to make room for our 
betters. And therefore that love of action 
which would put death out of sight is to be 
counted good, as a holy and healthy thing (one 
word whose meanings have become unduly 
severed), necessary to the life of men, serving 
to knit them together and to advance them in 
the right. Not only is it right and good thus 
VOL. I T 



274 LECTURES AND ESSAYS 

to cover over and dismiss the thought of our 
own personal end, to keep in mind and heart 
always the good things that shall be done, 
rather than ourselves who shall or shall not 
have the doing of them ; but also to our friends 
and loved ones we shall give the most worthy 
honour and tribute if we never say nor remember 
that they are dead, but contrariwise that they 
have lived ; that hereby the brotherly force and 
flow of their action and work may be carried 
over the gulfs of death and made immortal in 
the true and healthy life which they worthily 
had and used. It is only when the bloody 
hands of one who has fought against the light 
and the right are folded and powerless for 
further crime, that it is most kind and merciful 
to bury him and say, " The dog is dead." 

But for you noble and great ones, who have 
loved and laboured yourselves not for your- 
selves but for the universal folk, in your time 
not for your time only but for the coming 
generations, for you there shall be life as broad 
and far-reaching as your love, for you life- 
giving action to the utmost reach of the great 
wave whose crest you sometimes were. 

II. 

Believing that every finite intelligence must 
be " conditioned in time and space," and there- 



THE UNSEEN UNIVERSE 275 

fore must have an " organ of memory " and a 
"power of varied action," and consequently 
must be associated with a physical organism, 
recognising also that the world, as it is known 
at present, is made up of material molecules 
and of ether, our authors frankly admit that 
no room is here to be found either for ghosts 
of the dead, or "superior intelligences," or 
bogies of any kind whatever. But modifying 
a hypothesis of Sir W. Thomson's about the 
ultimate form of atoms and their relation to the 
ether, they find in a second ether the material 
wherewith to refashion all these marvels which 
advancing knowledge had banished from the 
realm of reality. We may here, then, review 
with advantage for a short time the state of 
that borderland between the known and the 
unknown in physical science to which this in- 
genious hypothesis belongs ; with the view of 
inquiring what measure of probability is to be 
attached to the modification of it which our 
authors propose. 

Imagine a ring of indiarubber, made by 
joining together the ends of a cylindrical piece 
(like a lead pencil before it is cut), to be put 
upon a round stick which it will just fit with a 
little stretching. Let the stick be now pulled 
through the ring while the latter is kept in its 
place by being pulled the other way on the 
outside. The indiarubber has then what is 



276 LECTURES AND ESSAYS 

called vortex -motion. Before the ends were 
joined together, while it was straight, it might 
have been made to turn round without chang- 
ing position by rolling it between the hands. 
Just the same motion of rotation it has on the 
stick, only that the ends are now joined together. 
All the inside surface of the ring is going one 
way namely, the way the stick is pulled ; and 
all the outside is going the other way. Such 
a vortex -ring is made by the smoker who 
purses his lips into a round hole and sends out 
a puff of smoke. The outside of the ring is 
kept back by the friction of his lips while the 
inside is going forwards ; thus a rotation is set 
up all round the smoke-ring as it travels out 
into the air. If we half immerse a teaspoon 
in our tea and draw it across the surface, we 
may see two little eddies formed at the edges 
of the spoon. These eddies are really united 
by a sort of rope of fluid underneath the surface, 
which follows the shape of the spoon, and 
which has throughout the same motion of 
rotation that the indiarubber ring had when 
the stick was drawn through it ; except that 
in this case only half a ring is formed, being 
cut off, as it were, by the surface of the liquid. 
In all these cases vortex-motion is produced by 
friction, and would be ultimately destroyed by 
friction. But, by way of an approximation to 
the study of water, men had been led to the 



THE UNSEEN UNIVERSE 277 

conception of a perfect liquid ; that is, a liquid 
absolutely free from friction, or (which is the 
same thing) offering no resistance to change of 
shape, or the sliding of one part over another. 
Water at rest behaves just as such a liquid 
would behave ; but water in motion is altogether 
a different thing. Helmholtz found, by a 
wonderfully beautiful calculation, that in a per- 
fect liquid where there is no friction it is 
impossible for vortex-motion to be generated 
or destroyed ; in any part of the liquid where 
there is no vortex-motion no mechanical action 
can possibly start it ; but where it once exists 
there it is for ever, and no mechanical action 
can possibly stop it. A vortex-ring may move 
from place to place ; but it carries with it the 
liquid of which it is composed, never leaving 
any particle behind, and never taking up any 
particle from the surrounding liquid. If we 
tried to cut it through with a knife it would 
thin out like a stream of treacle, and the thinner 
it got the faster it would go round ; so that if 
we multiplied together the number of revolu- 
tions in a second, and the number of square 
millimetres in the cross-section of the vortex- 
ring, we should always get the same pro- 
duct, not only in all parts of the ring, but 
through all time. Any portion of liquid which 
is rotating must form part of a vortex -ring, 
either returning into itself, after no matter how 



278 LECTURES AND ESSAYS 

many knots and convolutions, or having its two 
ends cut off at the surface of the liquid. That 
such more complex forms of vortex-motion may 
exist is easily shown by making knots (to be 
left loose) in a piece of string, and then join- 
ing the ends : motion of rotation may be given 
to any part of it by rolling it between two 
fingers, and will be carried all over it. Such a 
knotted vortex-ring is figured on the cover of 
the " Unseen Universe " for a fitting device. 

Thus far Helmholtz, examining into the 
consequences of supposing that a fiction, serv- 
ing to represent the actual properties of liquids 
at rest, holds good also in the case of motion. 
Here steps in Sir William Thomson with a 
brilliant conjecture. The ultimate atom of 
matter is required to be indestructible, to have 
a definite mass, and definite rates of vibration. 
A vortex-ring in a perfect liquid is indestructible, 
has a definite mass, and definite rates of vibra- 
tion. Why should not the atom be a vortex- 
ring in a perfect liquid ? If the whole of space 
were filled with an incompressible frictionless 
fluid in which vortex -rings once existed, at 
least some of the known phenomena of matter 
would be produced. Why should it not be 
possible in this way to explain them all ? 

The answer to this question is only to be 
got at by examining further into the con- 
sequences of the fundamental supposition, until 



THE UNSEEN UNIVERSE 279 

either the desired explanation of all phenomena 
is reached or some clear discordance with 
observed results shows that the whole hypo- 
thesis is untenable. To this task, with splendid 
energy and insight, Sir William Thomson has 
applied himself; arriving at results which, if 
they are not the foundation of the final theory 
of matter, are at least imperishable stones in 
the tower of dynamical science. 

Independently, however, of these results in 
the theory of the motion of perfect liquids, and 
independently of the final success of the hypo- 
thesis itself, it has led to two very important 
ideas of physical explanation. First, there is 
the idea that matter differs from ether only in 
being another state or mode of motion of the 
same stuff; which suggests the hope that we 
may by and by get to know something about 
the method of evolution of atoms, and the 
reason why there are so many kinds of them 
and no more. It must not be supposed that 
in Sir W. Thomson's hypothesis the part of 
the ether is played simply by the universal 
frictionless fluid. Such a fluid, by the defini- 
tion of it, offers no resistance to a change of 
shape of any part of it ; but the actual ether 
which fills space is so elastic that the slightest 
possible distortion produced by the vibration of 
a single atom sends a shudder through it with 
inconceivable rapidity for billions and billions 



28o LECTURES AND ESSAYS 

of miles. This shudder is Light. To account 
for such elasticity it has to be supposed that 
even where there are no material molecules the 
universal fluid is full of vortex-motion, but that 
the vortices are smaller and more closely packed 
than those of matter, forming altogether a more 
finely grained structure. So that the difference 
between matter and ether is reduced to a mere 
difference in the size and arrangement of the 
component vortex-rings. Now, whatever may 
turn out to be the ultimate nature of the ether 
and of molecules, we know that to some extent 
at least they obey the same dynamic laws, and 
that they act upon one another in accordance 
with these laws. Until, therefore, it is absolutely 
disproved, it must remain the simplest and most 
probable assumption that they are finally made 
of the same stuff that the material molecule 
is some kind of knot or coagulation of ether. 

Secondly, this hypothesis has accustomed 
us to the very important idea that the hardness, 
resistance, or elasticity of solid matter may be 
explained by the very rapid motion of some- 
thing which is infinitely soft and yielding. 
This general view Sir William Thomson has 
illustrated by exceedingly beautiful experiments. 
One striking form is the complete enclosure of 
a gyroscope in a flat cylindrical box, with a 
sharp projecting edge, so that the motion of 
the contained wheel can only be perceived by 



THE UNSEEN UNIVERSE 281 

the curious resistance to rotation of the box ; 
which will balance itself on its edge on a piece 
of glass, and only tremble and stand firm when 
it is struck a violent blow with the hand. So 
also, if a chain hanging straight down be rapidly 
spun round, it becomes stiff and stark like a 
rigid rod. And, lastly, a solid suspended in 
the centre of a globe of water will, when the 
water is made to revolve rapidly, oscillate about 
its mean position as if it were fastened by a 
spring. All these things make one inclined to 
look to the rapid motion of something soft for 
explanation of hardness and stiffness ; and the 
value of this explanation does not depend upon the 
ultimate success of the hypothesis of vortex-atoms. 
But these things being admitted, it may 
perhaps not be too great a presumption in us 
to make some criticisms on the hypothesis itself. 
A true explanation describes the previous un- 
known in terms of the known ; thus light is de- 
scribed as a vibration, and such properties of light 
as are also properties of vibrations are thereby 
explained. Now a perfect liquid is not a known 
thing, but a pure fiction. The imperfect liquids 
which approximate to it, and from which the 
conception is derived, consist of a vast number 
of small particles perpetually interfering with one 
another's motion. This molecular structure not 
only explains the fact that they behave like 
perfect liquids when at rest, but also makes it 



282 LECTURES AND ESSAYS 

necessary that they should not behave like 
perfect liquids when in motion. Thus a liquid 
is not an ultimate conception, but is explained 
it is known to be made up of molecules ; 
and the explanation requires that it should not 
be frictionless. The liquid of Sir William 
Thomson's hypothesis is continuous, infinitely 
divisible, not made of molecules at all, and it is 
absolutely frictionless. This is as much a mere 
mathematical fiction as the attracting and repel- 
ling points of Boscovitch. 

The authors of the " Unseen Universe " 
modify the hypothesis in such a way as to dis- 
pose of this objection. They regard the atoms 
as not absolutely indestructible, but only very 
long-lived. Consequently it is not necessary 
for them that the universal liquid should be 
quite perfect, but only that its viscosity or 
friction should be exceedingly small small 
enough to let the atoms keep going for billions 
of years when they are once started, with no 
appreciable change in their properties during the 
short time in which we can observe them. Thus, 
instead of a fiction, we have indeed a known 
thing, an imperfect liquid, by which to explain 
the molecules that are wanted to explain the 
properties of water. Can we, then, explain 
this universal imperfect liquid ? Certainly ; it 
consists of molecules inconceivably smaller than 
those of ordinary matter. But how to explain 



THE UNSEEN UNIVERSE 283 

the molecules ? Why, clearly, they are vortex- 
rings in a liquid of still finer grain and less 
viscosity. Molecules, liquid, molecules, liquid, 
alternately for ever ; each term of the infinite 
series being fully explained by the next follow- 
ing. Could anything be more satisfactory ? 

It is, moreover, to be observed that known 
facts about the ether and about atoms do lead 
us a very great way towards a conception of 
their relative structure. The experimental dis- 
coveries and the geometric insight of Faraday, 
and the application to these of mathematical 
analysis by Thomson, Helmholtz, and above all 
by Clerk Maxwell, have shown that the ether 
which was required for the theory of light is 
capable also of explaining magnetic and electric 
phenomena. Whatever that motion is which is 
periodically reversed in a ray of light, we have 
very strong evidence to show that the same 
motion is continuous along an electric current. 
This stream makes vortex-motion all round it, 
as if it were a stick drawn through indiarubber 
rings; and the vortex -rings are Faraday's 
" lines of magnetic force." The direction in 
which a small magnet will point indicates at 
any place the axis of rotation of the ether : 
thus, except in the neighbourhood of magnets 
or batteries, the ether in this country is all 
rotating in a plane rather tilted up on the north 
side. According to Maxwell's provisional con- 



284 LECTURES AND ESSAYS 

ception, we may suppose that this rotation 
belongs to soft balls, all spinning the same way, 
and separated by smaller " idle wheels," which 
turn in the opposite direction. It is a con- 
tinuous stream of these idle wheels that 
constitutes an electric current. Now there is 
great reason to believe that every material 
atom carries upon it a small electric current, 
if it does not wholly consist of this current. 
For, in the first place, every particle of a 
magnet is itself a magnet. Now, when a piece 
of iron is magnetised, there are two possible 
suppositions : either every particle is made into 
a magnet as it stands, having had no previous 
magnetism ; or else all the particles were 
originally magnets which neutralise one another 
because they were turned in all manner of 
directions, but which by the process of mag- 
netising have been made to approximate to 
the same direction. The latter supposition is 
conclusively picked out by experiment as the 
true one. Thus it seems that the molecule of 
iron is a magnet. If, however, the magnetism 
of the molecules were so much increased that 
they held each other tight, and so could not be 
turned round by ordinary magnetising forces, it 
is shown that effects would be produced like those 
of diamagnetism. Faraday gave reasons for 
believing that all bodies are either ferromagnetic 
or diamagnetic. Next, the theory of Ampere, 



THE UNSEEN UNIVERSE 285 

confirmed by many subsequent experiments 
and calculations, makes all magnetism to 
depend upon small electric currents. But 
magnetism is an affair of molecules ; if the 
molecules are groups of atoms we find in this 
way good reason to suppose that all atoms 
carry upon them electric currents. 

Three important sets of phenomena are 
(among many others) still unexplained the 
action of molecules upon one another, the 
action of transparent bodies on light, and 
gravitation. The precise law of action of mole- 
cules on one another is in fact unknown, the 
inverse fifth power of the distance, proposed by 
Maxwell, having been given up on the evidence 
of later experiments. The study of the mutual 
action of free small magnets in space offers 
mathematical difficulties which at present pre- 
vent us from saying whether a great number 
of these magnets would have such known pro- 
perties of gases as depend upon the law of 
mutual action of molecules. Transparent 
bodies act upon light as if the ether in their 
interior were somewhat less elastic than the 
ether outside them. It is possible that this 
change of elasticity may be explained by the 
electric field surrounding their molecules, 
although the most powerful fields that we can 
produce have not yet been observed to have 
any such effect. There is something left for 



286 LECTURES AND ESSAYS 

gravitation. In the theories of electric and 
magnetic action the motion of the " idle wheels," 
except in actual currents, is neglected in com- 
parison with that of the revolving soft spheres. 
It is, perhaps, conceivable that in some way or 
other an explanation may be found in them for 
the relatively weaker force of gravitation. If 
and what an if! these three explanations 
were made out, we might reasonably suppose 
not merely that an atom carries an electric 
current, but that it is nothing else. We should 
thus be led to find an atom, not in the rota- 
tional motion of a vortex-ring, but in irrotational 
motion round a re-entering channel. It might 
well be that such motion, to be permanent, 
must have some definite relation to the size of 
the rotating spheres and their interstices, so 
that only certain kinds of atoms could survive. 
In this way we may get an explanation of the 
definite number of chemical elements, and of 
the fact that all the molecules of each are as 
near alike as we can judge. 

The position is this. We know, with great 
probability, that wherever there is an atom 
there is a small electric current. Very many 
of the properties of atoms are explained by 
means of this current : we have vague hopes 
that all the rest will likewise be explained. If 
these hopes should be realised, we shall say 
that an atom is a small current. If not, we 



THE UNSEEN UNIVERSE 287 

shall have to say that it is a small current and 
something else besides. 

Of course, after all this, there is room for 
vortex-motion or other such hypothesis to ex- 
plain the observed properties of the ether ; but 
in the last resort all these questions of physical 
speculation abut upon a metaphysical question. 
We are describing phenomena in terms of 
phenomena ; the objects we observe are groups 
of perceptions, and exist only in our minds ; 
the molecules and ether, in terms of which we 
describe them, are only still more complex 
mental images. Is there anything that is not 
in our minds of which these things are pictures 
or symbols ? and if so, what ? 

Our authors reply that matter and energy 
possess this external reality, because they can- 
not be created or destroyed by us ; the quantity 
of each is fixed and invariable. The argument 
is better than most that belong to this question, 
but it will not hold water for a moment. Every 
quantitative relation among phenomena can be 
put into a form which asserts the constancy of 
some quantity which can be calculated from 
the phenomena. " Gravitation is inversely as 
the square of the distance for the same two 
bodies " ; this may be also said in the form, 
"gravitation multiplied by the square of the 
distance is constant for the same two bodies." 
" Pressure varies as density, in a perfect gas at 



288 LECTURES AND ESSAYS 

the same temperature," may be also expressed, 
" pressure divided by density is constant in a 
perfect gas at the same temperature." But 
this does not make the quotient of pressure by 
density to be an external reality transcending 
phenomena. It is entirely beside the question, 
as we may see in another way. A dream is a 
succession of phenomena having no external 
reality to correspond to them. Do we never 
dream of things that we cannot destroy? 

So the fact that matter, as a phenomenon, is 
not to be increased or diminished in quantity, 
has nothing to say to the question about the 
existence of something which is not matter, not 
phenomenon at all, but of which matter is the 
symbol or representative. The answer to this 
question is only to be found in the theory of 
sensation ; which tells us not merely that there 
is a non-phenomenal counterpart of the material 
or phenomenal world, but also in some measure 
what it is made of. Namely, the reality cor- 
responding to our perception of the motion of 
matter is an element of the complex thing we 
call feeling. What we might perceive as a 
plexus of nerve-disturbances is really in itself a 
feeling ; and the succession of feelings which 
constitutes a man's consciousness is the reality 
which produces in our minds the perception of 
the motions of his brain. These elements of 
feeling have relations of ne.rtness or contiguity 



THE UNSEEN UNIVERSE 289 

in space, which are exemplified by the sight- 
perceptions of contiguous points ; and relations 
of succession in time, which are exemplified by 
all perceptions. Out of these two relations the 
future theorist has to build up the world as 
best he may. Two things may, perhaps, help 
him. There are many lines of mathematical 
thought which indicate that distance or quantity 
may come to be expressed in terms of position 
in the wide sense of the analysis situs. And 
the theory of space-curvature hints at a possi- 
bility of describing matter and motion in terms 
of extension only. 

So much for the vortex-atom, its relation to 
the present state of science, and the prospects 
of physical speculation. We propose now to 
follow our authors farther ; to examine their 
hypothesis of a second ether, and to see what 
good it can do them. 

There are four ways of accounting for the 
too small number of stars of low magnitudes 
without assuming that light is absorbed by the 
ether. In the first place, the calculation as- 
sumes that stars are distributed with approxi- 
mate uniformity over infinite space. So far is 
this from being true, that we know the vast 
majority of stars that we can see to belong to 
a single system, of which the nebulae also are 
members, and which occupies a finite portion 
of space. It is very probable that around and 

VOL. I U 



290 LECTURES AND ESSAYS 

beyond this, to distances vaster even than its 
vast dimensions, there are regions nearly devoid 
of stars. If other such systems do anywhere 
exist, they may well be too far off to be seen 
at all. The method of Struve has, indeed, been 
beautifully applied by Mr. Charles S. Peirce to 
the richer materials now at hand with the view 
of determining approximately the shape of the 
solar galaxy and the mode of distribution of 
stars in it. Secondly, a great amount of light 
must be stopped by the dark bodies of burnt- 
out suns. Thirdly, space contains gaseous 
matter in a state of extreme diffusion not too 
rare, however, to produce an effect in distances 
so enormous as we have here to consider. 
Lastly, the possible curvature and finite extent 
of space have been suggested by Zollner as an 
escape from the reasoning of Olbers and 
Struve. Of these four the first is undoubtedly 
the true account of the matter, and will supply 
us with trustworthy knowledge of the contents 
of surrounding space. 

But if the ether did absorb light what would 
this mean ? Vibratory motion of solids, which 
is really a molecular disturbance, is absorbed 
by being transformed into other kinds of mole- 
cular motion, and so may finally be transferred 
to the ether. There is no reason why vibratory 
motion of the ether should not be transformed 
into other kinds of ethereal motion ; in fact, 



THE UNSEEN UNIVERSE 291 

there is no reason why it should not go to the 
making of atoms. Of course there is equally 
no reason why it should ; but we present this 
speculation to anybody who wants the universe 
to go on for ever. 

Apart from this, however, the laws of motion 
and the conservation of energy are very general 
propositions which are as nearly true as we can 
make out for gross bodies, and which, being 
tentatively applied to certain motions of mole- 
cules and the ether, are found to fit. There is 
nothing to tell us that they are absolutely ex- 
act in any particular case, or that they are 
everywhere and always true. If it were shown 
conclusively that energy was lost from the 
ether, it would not at all follow that it was 
handed on to anything else. The right state- 
ment might be that the conservation of energy 
was only a very near approximation to the 
facts. 

It is perhaps hardly necessary to say that 
the experiment of Tait and Balfour Stewart, 
who found that a disc was heated by rapid 
rotation in vacuo, though of the first importance 
in itself, by no means bears upon the question 
of the internal friction of the ether. That a 
molecule in travelling through the ether should 
be made to vibrate is just what we might 
expect ; the only wonder is that it gets through 
with so little resistance. But this is a transfer 



292 LECTURES AND ESSAYS 

of energy of translation of a molecule into 
energy of vibration ; a task to which one ether 
is entirely competent. 

Far greater, indeed, is the work which the 
second ether has to perform : nothing less than 
the fashioning of a " spiritual body." While 
our consciousness proceeds part passu with 
molecular disturbance in our brains, this mole- 
cular disturbance agitates the first ether, which 
transfers a part of its energy to the second. 
Thus is gradually elaborated an organism in 
that second or unseen universe, with whose 
motions our consciousness is as much connected 
as it is with our material bodies. When the 
marvellous structure of the brain decays, and it 
can no more receive or send messages, then the 
spiritual body is replete with energy, and starts 
off through the unseen, taking consciousness 
with it, but leaving its molecules behind. 
Having grown with the growth of our mortal 
frame, and preserving in its structure a record 
of all that has befallen us, it becomes an organ 
of memory, linking the future with the past, 
and securing a personal immortality. 

Can another body, then, avail to stay the 
hand of death, and shall man by a second 
nervous system escape scot free from the ruin 
of the first? We think not. The laws con- 
necting consciousness with changes in the brain 
are very definite and precise, and their necessary 



THE UNSEEN UNIVERSE 293 

consequences are not to be evaded by any such 
means. Consciousness is a complex thing 
made up of elements, a stream of feelings. 
The action of the brain is also a complex thing 
made up of elements, a stream of nerve- 
messages. For every feeling in consciousness 
there is at the same time a nerve-message in 
the brain. This correspondence of feeling to 
nerve-message does not depend on the feeling 
being part of a consciousness, and the nerve- 
message part of the action of a brain. How 
do we know this ? Because the nervous system 
of animals grows more and more simple as we 
go down the scale, and yet there is no break 
that we can point to and say, " above this there 
is consciousness or something like it ; below 
there is nothing like it." Even to those nerve- 
messages which do not form part of the con- 
tinuous action of our brains, there must be 
simultaneous feelings which do not form part 
of our consciousness. Here, then, is a law 
which is true throughout the animal king- 
dom ; nerve-message exists at the same time 
with feeling. Consciousness is not a simple 
thing, but a complex ; it is the combination of 
feelings into a stream. It exists at the same 
time with the combination of nerve-messages 
into a stream. If individual feeling always 
goes with individual nerve-message, if combina- 
tion or stream of feelings always goes with 



294 LECTURES AND ESSAYS 

stream of nerve-messages, does it not follow 
that when the stream of nerve -messages is 
broken up, the stream of feelings will be broken 
up also, will no longer form a consciousness ? 
does it not follow that when the messages 
themselves are broken up, the individual feel- 
ings will be resolved into still simpler elements ? 
The force of this evidence is not to be weakened 
by any number of spiritual bodies. Inexorable 
facts connect our consciousness with this body 
that we know ; and that not merely as a whole, 
but the parts of it are connected severally with 
parts of our brain - action. If there is any 
similar connection with a spiritual body, it only 
follows that the spiritual body must die at the 
same time with the natural one. 

Consider a mountain rill. It runs down in 
the sunshine, and its water evaporates ; yet it 
is fed by thousands of tiny tributaries, and the 
stream flows on. The water may be changed 
again and again, yet still there is the same 
stream. It widens over plains, or is prisoned 
and fouled by towns ; always the same stream ; 
but at last 

"even the weariest river 
Winds somewhere safe to sea." 

When that happens no drop of the water is 
lost, but the stream is dead. 



THE UNSEEN UNIVERSE 295 

3ttS Vf,' 

III. 

Our authors " assume, as absolutely self- 
evident, the existence of a Deity who is the 
Creator of all things." They must both have 
had enough to do with examinations to be 
aware that " it is evident " means " I do not 
know how to prove." The creation, however, 
was not necessarily a direct process ; the great 
likeness of atoms gives them the " stamp of the 
manufactured article," and so they must have 
been made by intelligent agency, but this may 
have been the agency of finite and conditioned 
beings. As such beings would have bodies 
made of one or other of the ethers, this form of 
the argument escapes at least one difficulty of 
the more common form, which may be stated 
as follows : " Because atoms are exactly alike 
and apparently indestructible, they must at 
one time have come into existence out of 
nothing. This can only have been effected by 
the agency of a conscious mind not associated 
with a material organism." Forasmuch as the 
momentous character of the issue is apt to 
blind us to the logic of such arguments as these, 
it may not be useless to offer for consideration 
the following parody : " Because the sea is salt 
and will put out a fire, there must at one time 
have been a large fire lighted at the bottom of 



296 LECTURES AND ESSAYS 

it. This can only have been effected by the 
agency of the whale who lives in the middle of 
Sahara." But let us return to our finite in- 
telligences having ethereal bodies, who made 
the atomic vortex-rings out of ether. With 
such a machinery it seems a needless simplifica- 
tion to adopt Prout's hypothesis, and suppose 
that the sixty-three elements are compounded 
of one simpler form of matter. Rather let us 
contemplate the reposeful picture of the uni- 
versal divan, where these intelligent beings 
whiled away the tedium of eternity by blowing 
smoke-rings from sixty-three different kinds of 
mouths. We may suppose, if we like, that the in- 
telligent beings were all alike, and each had sixty- 
three mouths ; or that each was so constituted in 
his physical or moral nature that he could or 
would pull only sixty-three faces. How lofty 
must have been the existence of such a makerand 
master of grimace ! How fertile of resource is 
the theologic method, when it once has clay for 
its wheel ! 

As the permanence of matter proves the 
existence of an external reality, a substance in 
which all things consist, so the conservation of 
energy points to a principle of motion, coming 
out of the unconditioned, entering into the 
visible universe and obeying its laws, to pass 
back finally into the unseen world. But, 
further, the fact that organisms large enough to 



THE UNSEEN UNIVERSE 297 

be visible have not yet under the conditions of 
the laboratory been produced from inorganic 
matter, shows that life is a great mystery, pene- 
trating into the depths of the arcana of the 
universe, proceeding from substance and energy 
and yet not identical with either. The reader 
will see what this points to. It is clear that 
the good old gods of our race sun, sky, 
thunder, and beauty are to be replaced by 
philosophic abstractions substance, energy, 
and life, under the patronage respectively of 
the persons of the Christian Trinity. But why 
are we to stay here? Is not neurility, the 
universal function of nerves, as much a special 
and distinct form of life as life is a distinct 
form of energy ? And over against these 
physical principles, absolutely separate and 
distinct from them, stands Consciousness, which 
cannot be left out of a fair estimate of the 
world. It would seem fitting that the presi- 
dency and patronage of the nerves should be 
assigned to the modern Isis as her portion. 
While if, as Von Hartmann says, Conscious- 
ness is the great mistake of the universe, it will 
not unsuitably fall to the care of the devil. 
In this way we shall save the odd number 
(numero deus impare gaudet\ and give a 
certain historical completeness to our repre- 
sentation. 

But why does a material so plastic present 



298 LECTURES AND ESSAYS 

itself in this identical shape ? Why this 
particular trinity of the great Ptah, Horus the 
Son, and Kneph the Wind-god, retained and 
refurbished by bishops of Alexandria and 
Carthage out of the wrecks of Egyptian super- 
stition ? Not because it is contained in the 
unseen universe, but because we were born in a 
particular place. If you, however, choose to 
find one thing in the chain of ethers, we may 
quite lawfully find another. If there is room 
in the unseen universe for the harmless pan- 
theistic deities which our authors have put there, 
room may also be found for the goddess Kali, 
with her obscene rites and human sacrifices, or 
for any intermediate between these. Here is 
the clay : make your images to your heart's 
desire ! 

When Mohammed was conquering Arabia, 
a certain tribe offered to submit if they should 
be spared the tribute and service in the holy 
war, and if they might keep their idol Lat for a 
year. The prophet agreed, and began to 
dictate to his scribe the terms of the treaty. 
When it came to the permission of idolatry he 
paused and looked on the ground. The envoys 
were impatient, and repeated the article. Then 
arose Omar, and turned upon them furious. 
" You have soiled the heart of the Prophet," he 
said ; " may God fill your hearts with fire ! " 
" I refuse the treaty," said Mohammed, looking 



THE UNSEEN UNIVERSE 299 

up. " Let us keep Lat only six months, then," 
pleaded the envoys. " Not another hour," said 
the Prophet ; and he drove them out and 
subdued them. 

" Only for another half-century let us keep 
our hells and heavens and gods." It is a 
piteous plea ; and it has soiled the heart of 
these prophets, great ones and blessed, giving 
light to their generation, and dear in particular 
to our mind and heart. These sickly dreams 
of hysterical women and half-starved men, what 
have they to do with the sturdy strength of a 
wide-eyed hero who fears no foe with pen or 
club ? This sleepless vengeance of fire upon 
them that have not seen and have not believed, 
what has it to do with the gentle patience of 
the investigator that shines through every page 
of this book, that will ask only consideration 
and not belief for anything that has not with 
infinite pains been solidly established ? That 
which you keep in your hearts, my brothers, is 
the slender remnant of a system which has 
made its red mark on history, and still lives to 
threaten mankind. The grotesque forms of its 
intellectual belief have survived the discredit of 
its moral teaching. Of this what the kings 
could bear with, the nations have cut down ; 
and what the nations left, the right heart of 
man by man revolts against day by day. You 
have stretched out your hands to save the dregs 



300 LECTURES AND ESSAYS 

of the sifted sediment of a residuum. Take heed 
lest you have given soil and shelter to the seed 
of that awful plague which has destroyed two 
civilisations, and but barely failed to slay such 
promise of good as is now struggling to live 
among men. 



THE PHILOSOPHY OF THE PURE 
SCIENCES l 

I. STATEMENT OF THE QUESTION 

ON entering this room and looking rapidly 
round, what do I see ? I see a theatre, with a 
gallery, and with an arrangement of seats in 
tiers. I see people sitting upon these seats, 
people with heads more or less round, with 
bodies of a certain shape ; sitting in various 
positions. Above I see a roof with a skylight, 
and a round disc evidently capable of vertical 
motion. Below I see the solid floor supporting 
us all. In front of me I see a table, and my 
hands resting upon it. In the midst of all 
these things I see a void space, which I can 
walk about in if I like. The different things 
I have mentioned I see at various distances 
from one another, and from me ; and (now that 
the door is shut) I see that they completely 
enclose this void space, and hedge it in. My 

1 Lectures delivered at the Royal Institution in March 1873. 



302 LECTURES AND ESSAYS 

view is not made of patches here and there, 
but is a continuous boundary going all round 
the void space I have mentioned. All this I 
see to exist at the same time ; but some of you 
are not sitting quite still, and I see you move ; 
that is to say, I see you pass from one position 
into another by going through an infinite series 
of intermediate positions. Moreover, when I 
put my hands on the table, I feel a hard flat 
horizontal surface at rest, covered with cloth. 

Have I spoken correctly in making these 
assertions ? Yes, you will say, this is on the 
whole just what I ought to have seen and felt 
under the circumstances. With the exception 
of one or two points expressed in too technical 
a form, this is just the sort of language that a 
witness might use in describing any ordinary 
event, without invalidating his testimony. You 
would not say at once, " This is absurd ; the 
man must not be listened to any longer." And 
if, having been precisely in my situation, you 
wished to describe facts with the view of draw- 
ing inferences from them even important 
inferences you would make all these state- 
ments as matter of your own direct personal 
experience ; and if need were, you would even 
testify to them in a court of law. 

And yet I think we shall find on a little re- 
flection that not one of these statements can by 
any possibility have been strictly true. 



PHILOSOPHY OF THE PURE SCIENCES 303 

" I see a theatre." I do not ; the utmost I 
can possibly see is two distinct curved pictures 
of a theatre. Upon the two retinas of my eyes 
there are made pictures of the scene before me, 
exactly as pictures are made upon the ground 
glass in a photographer's camera. The sensa- 
tion of sight which I get comes to me at any 
rate through those two pictures ; and it cannot 
tell me any more, or contain in itself any more, 
than is in those two pictures. Now the 
pictures are not solid ; each of them is simply 
a curved surface variously illuminated at 
various parts. Whereas, therefore, I think I 
see a solid scene, having depth, and relief, and 
distance in it, reflection tells me that I see 
nothing of the kind ; but only (at the most) 
two distinct surfaces, having no depth and no 
relief, and only a kind of distance which is 
quite different from that of the solid figures 
before me. You will say, probably, that this 
is only a quibble on two senses of the word 
" see." Whether it is so or not makes no 
difference to our subsequent argument ; and 
yet I think you will admit that the latter sense, 
in which I do not see the solid things, is the 
more correct one. For the question is not 
about what is there, but about what I see. 
Now exactly the same sensation can be pro- 
duced in me by two slightly different pictures 
placed in a stereoscope I say exactly the 



3 o 4 LECTURES AND ESSAYS 

same ; because if I had sufficiently accurate 
coloured photographs of this room properly 
illuminated, the rays of light converging on 
every part of each of my retinas might be made 
exactly the same as they are now ; and the 
sensation would therefore not only appear to 
be the same but would actually be the same. 
I should think I saw a solid scene; and I 
should not be seeing one. Now to see, and to 
see what is actually there, are two different 
things. 

Again, " I see people with heads more or 
less round." I cannot see your heads ; I can 
only see your faces. I must have imagined the 
rest. But just consider what it is that I have 
imagined. It is merely that besides what I do 
see I have added something that I might see 
by going round to the other side ? No, there 
is more than that. The complete sensation 
which I have of a human head when I look at 
one is not merely something which I do not see 
now, but something which I never could see by 
any possibility. I have the sensation of a solid 
object, and not of a series of pictures of a solid 
object. Although that sensation may be really 
constructed out of a countless number of possible 
pictures, yet it is not like any of them. I im- 
agine to myself, and seem to see the other side of 
things, not as it would look if viewed from beyond 
them, but as it would look if viewed from here. 



PHILOSOPHY OF THE PURE SCIENCES 305 

I seem to see the back of your head, not as it 
would look if I got behind you, but as if I saw 
it through your face from the spot where I am 
standing ; and that, you know, is impossible. 

I seem to see all these objects as exist- 
ing together. But really as a matter of fact 
I move my eyes about and see a succes- 
sion of small pictures very rapidly changed. 
Each of my eyes has six muscles which pull it 
about, and if I knew which of these muscles 
were moving, and how fast, at any moment, 
I should get information about the direction 
in which my eye was looking at the time. 
Now it is only a very small part of the scene 
before me that I can really see distinctly at 
once ; so that I have really seen a panorama, 
and not the one large picture that I imagined ; 
and yet while looking at the small portion 
which I can really see distinctly, I think I see 
distinctly the whole room. 

Again, I seem to see that in some directions, 
at least, this void space in the middle is com- 
pletely bounded the surface of the floor, for 
example, which bounds it, appears to be com- 
pletely filled up and continuous, to have no 
breaks in it. And when you move I seem to 
see you go continuously from one position to 
another through an infinite series of intermediate 
positions. Now, quite apart from the question 
whether these conclusions are true or not, it 

VOL. I X 



306 LECTURES AND ESSAYS 

can be made out distinctly that I could not 
possibly see either the surface of a thing, or a 
motion, as continuous ; for the sensitive portion 
of my retina, which receives impressions, is not 
itself a continuous surface, but consists of an 
enormously large but still finite number of 
nerve filaments distributed in a sort of network. 
And the messages that go along my nerves do 
not consist in any continuous action, but in a 
series of distinct waves succeeding one another 
at very small but still finite intervals. All I 
can possibly have seen therefore at any moment 
is a picture made of a very large number of 
very small patches, exceedingly near to one 
another, but not actually touching. And all 
I can have seen as time passed is a succession 
of such distinct pictures coming rapidly after 
one another. You know that precisely as the 
stereoscope is made to imitate the property of 
my two eyes out of which I imagine solid 
things, so another instrument has been con- 
structed to imitate that property of my nerves 
out of what I imagine continuous motion. The 
instrument is called the Zoetrope, or Wheel of 
Life. It presents to you a succession of distinct 
pictures coming after one another at small in- 
tervals ; and the impression produced by that 
series is precisely the impression of one thing 
in continuous motion. 

Let us now put shortly together what we 



PHILOSOPHY OF THE PURE SCIENCES 307 

have said about this sensation of sight I shall 
use the word mosaic to represent a few discon- 
nected patches which a painter might put down 
with a view of remembering a scene he had no 
time to sketch. Then, I seem to see a large 
collection of solid objects in continuous motion. 
The utmost I can really see is a panorama 
painted in mosaic and shown in a wheel of life. 
I do not know that my direct perception 
amounts to so much ; but it cannot possibly 
amount to more. What it really does amount 
to must be reserved for subsequent discussion. 
At any rate I must have imagined the rest. 

Lastly, when I put my hands on the table, 
I feel a hard, flat, horizontal surface at rest, 
covered with cloth. Now there are three 
things that really happen. First, there is a 
definite kind of irritation of certain organs of 
my skin, called papillae. It is that irritation 
that makes me say cloth. Secondly, certain of 
my muscles are in a state of compression, and 
they tell me that. Thirdly, I make a certain 
muscular effort which is not followed by motion. 
This is all that I can really feel ; but those 
three things do not constitute a hard, flat, 
horizontal surface covered with cloth. As 
before, I must have imagined the rest 

Do not suppose that I am advocating any 
change in our common language about sensation. 
I do not want anybody to say, for instance, 



308 LECTURES AND ESSAYS 

instead of, " I saw you yesterday on the other 
side of the street," " I saw a series of panoramic 
pictures in a sort of mosaic, of such a nature 
that the imaginations I constructed out of them 
were not wholly unlike the imaginations I have 
constructed out of similar series of panoramic 
pictures seen by me on previous occasions when 
you were present." This would be clumsy, and 
it would not be sufficient. And yet I cannot 
help thinking that in certain assemblies, when 
some of those who are present are in an exalted 
state of emotional expectation, and the lights 
are low, even this roundabout way of putting 
things might be, to say the least, a salutary 
exercise. 

But the conclusion I want you to draw from 
all this that we have been saying is that there 
are really two distinct parts in every sensation 
that we get. There is a message that comes 
to us somehow ; but this message is not all 
that we apparently see and hear and feel. In 
every sensation there is, besides the actual 
message, something that we imagine and add 
to the message. This is sometimes expressed 
by saying that there is a part which comes 
from the external world and a part which is 
supplied by the mind. But however we ex- 
press it, the fact to be remembered is that not 
the whole of a sensation is immediate experi- 
ence (where by immediate experience I mean 



PHILOSOPHY OF THE PURE SCIENCES 309 

the actual message whatever it is that comes 
to us) ; but that this experience is supplemented 
by something else which is not in it. And thus 
you may see that it is a perfectly real question, 
" Where does this supplement come from ? " 
This question has been before philosophers for 
a very long time ; and it is this question that 
we have to discuss. 

But first of all we must inquire a little 
further into the nature of the supplement by 
which we fill in our experience. When I fill 
in my experience of this room in the way that I 
have described, I do not do so at random, but 
according to certain rules. And in fact I 
generally fill it in right; that is to say, from 
the imaginations that I have built up I can 
deduce by rules certain other experiences which 
would follow from actions of a definite sort. 
When I seem to see a solid floor, I conclude 
that if I went there I could feel it as I do the 
table. And upon trial these conclusions in 
general turn out right. I cannot therefore have 
filled in my experience at random, but accord- 
ing to certain rules. Let us now consider 
what are a few of these rules. 

In the first place, out of pictures I have 
imagined solid things. Out of space of two 
dimensions, as we call it, I have made space 
of three dimensions, and I imagine these solid 
things as existing in it ; that is to say, as having 



310 LECTURES AND ESSAYS 

certain relations of distance to one another. 
Now these relations of distance are always so 
filled in as to fulfil a code of rules, some called 
common notions, and some called definitions, 
and some called postulates, and some assumed 
without warning, but all somehow contained in 
Euclid's Elements of Geometry. For example, 
I sometimes imagine that I see two lines in a 
position which I call parallel. Parallelism is 
impossible on the curved pictures of my retina; 
so this is part of the filling in. Now when- 
ever I imagine that I see a quadrilateral figure 
whose opposite sides are parallel, I always fill 
them in so that the opposite sides are also 
equal. This equality is also a part of the filling 
in, and relates to possible perceptions other 
than the one immediately present. From this 
example, then, you can see that the funda- 
mental axioms and definitions of geometry are 
really certain rules according to which we 
supplement or fill in our experience. 

Now here is a rather more complicated ex- 
ample. If I see a train going along and a 
man moving inside of it, I fill in the motion of 
the train as continuous out of a series of dis- 
tinct pictures of it ; and so also I fill in the 
motion of the man relatively to the train as 
continuous. I imagine all motions, therefore, 
according to the rule of continuity ; that is, 
between the distinct pictures which I see, I 



PHILOSOPHY OF THE PURE SCIENCES 311 

insert an infinite number of intermediate 
pictures. Moreover, both of these motions are 
imagined in accordance with the laws of 
geometry ; that is to say, they are imagined so 
that the relations of distance at any instant obey 
those laws. But now I may, if I like, consider, 
besides the motion of the train and the motion 
of the man relative to it, the motion of the man 
relative to me, as if there were no train ; and 
this like the other motions is part of the filling 
in. But I always fill this in in such a way that 
the three motions of the train by itself, of the 
man by himself, and of the man relatively to 
the train satisfy certain rules, by which one 
can be found when the other two are given. 
These rules are called the laws of kinematic, or 
of the pure science of motion. 

Then we may say, to begin with, that we 
supplement our experience in accordance with 
certain rules ; and that some of these rules are 
the foundations of the pure sciences of Space 
and Motion. 

Instead of Space and Motion, many people 
would like to say Space and Time. But in re- 
gard to the special matter that we are consider- 
ing, it seems to me, for reasons which I do not 
wish to give at present, to be more correct to 
say that we imagine time by putting together 
space and motion, than that we imagine motion 
by putting together space and time. 



312 LECTURES AND ESSAYS 

There are other rules, besides those of space 
and motion, according to which we fill in our 
experience. One of these rules I may call the 
continuity of things. I can see this table, and 
feel it, and hear a sound when I strike it. The 
table is an imagination by which I fill in a 
great variety of different experiences. It is 
what I call a thing. Now, if I come into this 
room again, and have any experience of the 
table, I shall fill it in in such a way as to imply 
that the same variety of experiences might be 
combined again ; that is, I shall imagine the 
thing to be persistent. But this rule will not 
apply universally, and I do not always observe 
it Because I have seen a tree without leaves 
in the winter, I do not in the summer fill in my 
experience of the trunk with imagination of 
leafless branches above. But I do fill in the 
two experiences with an imagination of an 
infinite series of gradual intermediate changes. 
Some people divide this rule into two the 
persistence of substance and the continuity of 
qualities. I prefer to make one rule, and to 
call it the continuity of things. Things that 
is to say, combinations of possible experience 
are not persistent, but they change continuously 
in the imagination by which we fill up that ex- 
perience. Or we may say that experience at 
any one time is always so filled in as to aggre- 
gate together the possible perceptions implied 



PHILOSOPHY OF THE PURE SCIENCES 313 

by the result into groups which we call things ; 
and that experience of a period of time is 
always so filled in that things change only in a 
continuous manner. 

Another rule of the supplement which we 
imagine is that which provides that these 
changes of things shall take place according to 
a certain uniformity. The simplest case of this 
is when the same experience is repeated, and 
we fill up the changes subsequent to the second 
experience so that they shall be the same as 
those subsequent to the first. It is not neces- 
sary that the experience should be actually 
repeated ; it may only be filled up in the same 
way. The uniformity, however, which is in- 
volved in this law is a much more complicated 
thing than this simple case. I can only say 
here that experience is filled up always so 
as to make the imagined history of things 
exhibit some uniformity ; but the definiteness 
of this varies in different individuals and at 
different times. Some people prefer to call 
this the law of causation, and to say that we 
always supplement our experiences in such a 
way that every event has a cause or causes 
which determine it, and effects which flow 
from it. 

Now all this filling up that we have been 
considering happens directly in the sensations 
that I get from day to day, just as I get them. 



314 LECTURES AND ESSAYS 

(It is convenient to use the word sensation as 
meaning the whole phenomenon, not only the 
immediate experience, but also the supplement.) 
But if I want to talk to you about them, or if, 
advancing upon that practice, I talk to myself 
about them, then I am obliged to use language, 
or to represent them by signs ; and this requires 
me to group them in a new manner. I have 
to make imaginations not of things, but of whole 
series of things, of relations of these to one 
another, and combinations of the relations. I 
have to construct, in fact, what I shall call for 
shortness the apparatus of thought the means 
by which I talk to myself. For there seems 
reason to think that the conceptions which 
correspond to general terms names of a class, 
or of an abstract relation are first rendered 
necessary by the language which expresses 
them. 1 But however that may be, this new 
world of conceptions is not made wholly at 
random, but satisfies certain laws. For ex- 
ample, in order to describe a certain group of 
things, I introduce the very complicated concep- 
tion six, and say there are six of them. Now, 
whenever this is done in the case of two groups, 
giving rise to the conceptions six and three, it 
is possible to apply the same process to the 



l See this view ably defended in Professor Max Mullers 
Lectures, delivered at the Royal Institution in April 1873, and 
since published in Prater's Magazine. 



PHILOSOPHY OF THE PURE SCIENCES 315 

group compounded of those two, and it always 
gives rise to the conception nine. Here, then, 
is a law of combination to which the world of 
conceptions has to conform. And another is 
this : If every individual which belongs to the 
class A belongs also to the class B, and if every 
individual which belongs to the class B belongs 
also to the class C, then always every individual 
which belongs to the class A belongs also to 
the class C. Rules like these which regulate 
the world of conceptions, built out of our 
sensations, are also said to belong to the pure 
sciences ; and the two examples which I have 
chosen belong respectively to the sciences of 
Number and Logic. 

There may be other kinds of rules according 
to which experience is supplemented and sensa- 
tions are built up into conceptions ; but I am 
not aware of any more kinds, and perhaps those 
that I have mentioned will be sufficient for our 
purpose. I will just state again the names of the 
sciences which consist in these three groups : 

The rules about Space and Motion constitute 
the pure sciences of Geometry and Kinematic. 

The rules about Things and Uniformity 
have been said to belong to a pure science of 
Nature. 

The rules about Numbers and Classes con- 
stitute the pure sciences of Arithmetic and 
Formal Logic. 



316 LECTURES AND ESSAYS 

But for the present let us confine our atten- 
tion to the first group of rules, those which 
relate to space and motion. There is one other 
property of them which we have to consider, 
besides the fact that our experience is filled up 
in accordance with them. I have already 
mentioned this property, but only in passing. 
It is that in general this filling in of experience 
is right: and that, so far as these rules are 
concerned, it is not only right in general, but 
always right. That is to say, if from the sensa- 
tion which is made by the filled-up experience 
we predict certain other perceptions as con- 
sequent upon our actions, these predictions will 
actually be fulfilled. To take the example we 
considered before, I always imagine a parallelo- 
gram so that its opposite sides are equal. Now 
the conclusion from this is that if I go to the 
parallelogram and apply one of the sides to 
the other, I shall not perceive any difference. 
The rule by which I supplement my perception 
is also a true statement about objects ; it is 
capable of a certain kind of verification, and it 
always stands this test. 

Here, however, I could use the word equal 
only in its practical sense, in which two things 
are equal when I cannot perceive their differ- 
ence ; not in its theoretical sense, in which 
two things are equal when they have no 
difference at all. But there has been for ages 



PHILOSOPHY OF THE PURE SCIENCES 317 

a conviction in the minds of men that these 
rules about space are true objectively in the 
exact or theoretical sense, and under all 
possible circumstances. If two -straight lines 
are drawn perpendicular to the same plane, 
geometers would have told you for more than 
two thousand years that these straight lines may 
be prolonged for ever and ever without getting 
the least bit nearer to one another or further 
away from one another ; and that they were 
perfectly certain of this. They knew for 
certain that the sum of the angles of a triangle, 
no matter how big or how small it was, . or 
where it was situated, must always be exactly 
equal to two right angles, neither more nor less. 
And those who were philosophers as well as 
geometers knew more than this. They knew not 
only that the thing was true, but that it could not 
possibly have been otherwise ; that it was neces- 
sarily true. And this means, apparently, not 
merely that I know that it must be, but that I 
know that you must know that it must be. 

The case of arithmetical propositions is 
perhaps more easily comprehended in this 
respect. Everybody knows that six things and 
three things make nine things at all possible 
times and places ; you cannot help seeing not 
only that they do always without exception 
make nine things, but that they must do so, 
and that the world could not have been con- 



3i8 LECTURES AND ESSAYS 

structed otherwise. For to those ingenious 
speculations which suppose that in some other 
planet there may always be a tenth thing in- 
evitably suggested upon the union of the six 
and the three, so that they cannot be added 
together without making ten ; to these, I say, 
it may be replied that the words number and 
thing, if used at all, must have different mean- 
ings in that planet. The reply is important, 
and I shall return to it in a subsequent lecture. 
Locke and Hume gave explanations of the 
existence of two of these general rules which I 
have put into my second group. Locke ex- 
plained the notion of substance, the notion that 
a thing means something more than an aggre- 
gate of possible perceptions, by the fact that we 
are accustomed to get these perceptions all 
together ; by this custom they are welded or 
linked together, and our imagination of the 
thing is then this connected structure of per- 
ceptions, which is called up as a whole when- 
ever one or more of the component perceptions 
is called up. Having thus by custom formed 
the complete sensation which we have of the 
thing, we suppose that this is a message, like 
the actual perceptions, and comes from some- 
thing outside. That something is the substance. 
Locke did not admit that this supposition is 
right, and that the linking together of messages 
is really itself a message ; but still he thought 



PHILOSOPHY OF THE PURE SCIENCES 319 

there was something outside to correspond to 
this linking. Hume explained in the same 
way the rule of causation. He said we get it 
from being accustomed to perceive one event 
following another ; so that these two percep- 
tions got linked together, and when one of 
them occurs alone, we fill it in with the other 
one. And then, regarding this link, produced 
only by custom, as if it were a message from 
somewhere, like the simple perceptions, we give 
it the name of causation. 

These explanations agree in saying that the 
supplement of experience is made up of past 
experience, together with links which bind to- 
gether perceptions that have been accustomed 
to occur together. This fact, that perceptions 
and feelings which have frequently occurred 
together get linked, so that one calls up the 
other, is called the law of Association, and has 
been made the basis of scientific Psychology. 
According to these explanations of Locke and 
Hume (which extended to the other two groups 
of rules) all the knowledge we have that the 
rules are right, or may be objectively verified, 
is really derived from experience ; only it is 
past experience, which we have had so often 
and got so accustomed to that it is now really 
a part of ourselves. 

But Kant, after being staggered for some 
time by Hume's explanation, at length said, 



320 LECTURES AND ESSAYS 

" It is impossible that all your knowledge can 
have come from experience. For you know 
that the axioms of mathematics are absolutely 
and universally true, and no experience can 
possibly have told you this. However often 
you may have found the angles of a triangle 
amount to two right angles, however accustomed 
you may have got to this experience, you have 
no right to know that the angles of every 
possible triangle are equal to two right angles, 
nor indeed that those of any one triangle are 
absolutely and exactly so equal. Now you do 
know this, and you cannot deny it. You have 
therefore some knowledge which could not 
possibly be derived from experience ; it must 
therefore have come in some other way ; or 
there is some other source of knowledge besides 
experience." 

At that time there was no answer whatever 
to this. For men did think that they knew at 
least the absolute universality if not the neces- 
sity of the mathematical axioms. To any one 
who admitted the necessity, the argument was 
even stronger ; for it was clear that no experi- 
ence could make any approach to supply 
knowledge of this quality. But if a man felt 
absolutely sure that two straight lines per- 
pendicular to the same line would never meet, 
however far produced, he could not maintain 
against Kant that all knowledge is derived 



PHILOSOPHY OF THE PURE SCIENCES 321 

from experience. He was obliged to admit 
the existence of knowledge a priori, that is, 
knowledge lying ready in the mind from the 
first, antecedent to all experience. 

But now here is a difficulty to be explained. 
How is it possible that I can have knowledge 
about objects which is prior to all experience 
of objects, and which transcends the bounds of 
possible experience? 

First of all, what do I mean by objects ? 
In the answer to this question lies really Kant's 
solution of the problem, and I shall endeavour 
to make this clear by a comparison. 

If a man had on a pair of green spectacles, 
he would see everything green. And if he 
found out this property of his spectacles, he 
might say with absolute certainty that while he 
had those spectacles on everything that he saw 
without exception would be green. 

" Everything that he saw ; " that is to say, 
all objects of sight to him. But here it is clear 
that the word object is relative ; it means a 
representation that he gets, and has nothing to 
do with the thing in itself. And the assertion 
that everything is green would not be an 
assertion about the things in themselves, but 
about the representations of them which came 
to him. The colour of these representations 
would depend partly on the things outside and 
partly on his spectacles. It would vary for 
VOL. l Y 



322 LECTURES AND ESSAYS 

different things, but there would always be 
green in it. 

Let us modify this example a little. I 
know for certain that the colour of every object 
in the universe is made up of colours that lie 
within the range of the visible spectrum. This 
is apparently a universal statement, and yet I 
know it to be true of things which it is im- 
possible that I should ever see. How is this ? 
Why, simply, that my eyes are only affected 
by light which lies within the range of the 
visible spectrum. Now I say that this case is 
only a little modified from the previous one. 
The green glass lets in a certain range of light ; 
the range is very little increased when you take 
it away. Only in the second case it happens 
that we are all actually wearing very nearly the 
same spectacles. That universal statement 
which I made is true not only of objects as 
they appear to me, but also of objects as 
they appear to you. It is a statement about 
objects ; that is, about certain representations 
which we perceive. It may therefore so far 
have its origin in the things of which these are 
representations, or it may have its origin in us. 
And we happen to know that in this case it 
is not a statement about external things, but 
about our eyes. 

Admitting, then, that the objects of our 
sensations are representations made to us ; that 



PHILOSOPHY OF THE PURE SCIENCES 323 

their character must therefore be partly 
dependent upon our own character ; what 
properties of these objects should we naturally 
suppose to have this origin, to be derived from 
the constitution of our minds ? Why, clearly, 
those which are necessary and universal ; for 
only such properties can be so derived, and 
there is no other way in which they can be 
known to be universal. 

Accordingly, Kant supposes that Space and 
Time are necessary forms of perception, imposed 
upon it by the perceiving mind ; that things are 
in space and time as they appear to us, and not 
in themselves ; and that consequently the state- 
ment that all things exist in space and time is a 
statement about the nature of our perception 
and not about the things perceived. 

The word corresponding to experience 
(Erfahrung) is used by Kant nearly in the 
sense in which I have used sensation, to mean 
the whole phenomenon consisting of the bare 
message and also of the filling-in, the complete 
representation which we get of objects. But it 
is not apparently confined to this ; it means 
not merely the sensations which I get, but the 
sensations which I talk about. Giving to the 
word this sense for the present, we may say 
that in his theory the form, the general char- 
acter, of experience is imposed upon it by two 
faculties which we all possess : Intuition and 



324 LECTURES AND ESSAYS 

Understanding. Intuition has necessarily the 
forms of Space and Time ; but we are not to 
say that those properties of space which are 
expressed in the geometrical axioms are all 
necessitated by the forms of intuition ; for it is 
the understanding that supplies us with the 
pure notions of quantity, quality, relation, and 
modality. It is not always easy to separate 
the parts played by these two faculties in 
supplying the general rules to which experience 
conforms ; but it appears, for example, that 
the three dimensions of space are given by pure 
intuition itself, while the equality of the opposite 
sides of a parallelogram is only given by help 
of the understanding. It is not to our purpose 
to investigate the difference between these two 
faculties, or even to remember that Kant made 
a distinction between them. All that is im- 
portant for us is the theory that those general 
statements upon which the pure sciences are 
founded, although really true of objects, that is 
of representations made to me, are in fact state- 
ments about me and not about the things in 
themselves : just as my general statement about 
the colours of things was really a statement 
about my own eyes and not about the things. 
And it is just because these statements are 
about me that I know them to be not only 
universally, but always necessarily true about 
the objects I perceive ; for it is always the 



PHILOSOPHY OF THE PURE SCIENCES 325 

same me that perceives them or at any rate it 
is a me possessing always the same faculties of 
representation. 

Now observe what it is that this theory does 
with general statements ; what is the means by 
which it gets rid of them for it does get rid 
of them. It makes them into particular state- 
ments. Instead of being statements about all 
possible places and times and things, they are 
made out to be statements about me, and about 
other men in so far as they have the same 
faculties that I have. I want you to notice 
this transformation particularly, because I shall 
afterwards endeavour to establish a similar 
transformation, though in rather a different 
manner. 

In the next place, observe that the question 
which was proposed by the Critical Philosophy 
is a perfectly real and important question. It 
is this : " Are there any properties of objects 
in general which are really due to me and to 
the way in which I perceive them, and which 
do not belong to the things themselves ? " But 
it seems to me that the method by which Kant 
attempted to answer this question was not the 
right method. It consisted in finding what are 
those characters of experience which we know 
to be necessary and universal ; and concluding 
that these are characters of me. It requires, 
therefore, some infallible way of judging what 



326 LECTURES AND ESSAYS 

characters are necessary and universal. Now, 
unfortunately, as I hope to show you, judgments 
of this kind may very possibly be mistaken. 
If you went up to our man with the green 
spectacles, and argued with him that since he 
knew for certain that everything was green, 
whereas no experience could tell him so, this 
greenness must be somewhere in the apparatus 
by which he perceived things ; there would be 
just one weakness in the argument. He might 
be mistaken in thinking he knew that every- 
thing was green. But the proper thing to do, 
as it appears to me, would be to take him to a 
looking-glass and show him that these spectacles 
were actually upon his nose. And so also in 
the general question which is proposed by 
the Critical Philosophy. The answer to that 
question must be sought not in the subjective 
method, in the conviction of universality and 
necessity, but in the physiological method, in 
the study of the physical facts that accompany 
sensation, and of the physical properties of the 
nervous system. The materials for this valid 
criticism of knowledge did not exist in Kant's 
time. I believe that they do exist at present 
to such an extent at least as to indicate 
the nature of the results which that criticism 
is to furnish. 

The Kantian theory of universal truths was 
largely, though not completely, accepted by 



PHILOSOPHY OF THE PURE SCIENCES 327 

Whewell, and applied with considerable detail 
in his Philosophy of the Inductive Sciences. 
It is necessary to mention him here, not on 
account of any important modification that he 
introduced into the theory, but because the form 
into which he put it has had great influence in 
directing the attention of scientific students to 
the philosophy of science ; and because by in- 
telligent controversy he contributed very much 
to the clearing up and development of an 
opinion which we have next to consider that 
of Mr. John Stuart Mill. I can best, I think, 
set this opinion before you, if I have permission 
to quote a short passage. 

" To these arguments (of Dr. Whewell, con- 
tending that the axioms could not be known 
by experience) ... a satisfactory answer will, 
I conceive, be found, if we advert to one of the 
characteristic properties of geometrical forms 
their capacity of being painted in the imagina- 
tion with a distinctness equal to reality : in 
other words, the exact resemblance of our ideas 
of form to the sensations which suggest them. 
This, in the first place, enables us to make (at 
least with a little practice) mental pictures of all 
possible combinations of lines and angles, which 
resemble the realities quite as well as any which 
we could make on paper ; and in the next place, 
make those pictures just as fit subjects of geo- 



328 LECTURES AND ESSAYS 

metrical experimentation as the realities them- 
selves ; inasmuch as pictures, if sufficiently 
accurate, exhibit of course all the properties 
which would be manifested by the realities at 
one given instant, and on simple inspection ; 
and in geometry we are concerned only with 
such properties, and not with that which 
pictures could not exhibit, the mutual action 
of bodies upon one another. The foundations 
of geometry would therefore be laid in direct 
experience, even if the experiments (which in 
this case consist merely in attentive contempla- 
tion) were practised solely upon what we call 
our ideas, that is, upon the diagrams in our 
minds, and not upon outward objects. For in 
all systems of experimentation we take some 
objects to serve as representatives of all which 
resemble them ; and in the present case the 
conditions which qualify a real object to be the 
representative of its class are completely fulfilled 
by an object existing only in our fancy. 
Without denying, therefore, the possibility of 
satisfying ourselves that two straight lines 
cannot enclose a space, by merely thinking of 
straight lines without actually looking at them, 
I contend that we do not believe this truth on 
the ground of the imaginary intuition simply, 
but because we know that the imaginary lines 
exactly resemble real ones, and that we may 
conclude from them to real ones with quite as 



PHILOSOPHY OF THE PURE SCIENCES 329 

much certainty as we could conclude from one 
real line to another. The conclusion, therefore, 
is still an induction from observation. And 
we should not be authorised to substitute 
observation of the image in our mind for 
observation of the reality, if we had not learnt 
by long -continued experience that the pro- 
perties of the reality are faithfully represented in 
the image ; just as we should be scientifically 
warranted in describing an animal which we 
had never seen from a picture made of it with 
a daguerreotype ; but not until we had learnt 
by ample experience that observation of such a 
picture is precisely equivalent to observation of 
the original. 

"These considerations also remove the 
objection arising from the impossibility of our 
ocularly following the lines in their prolonga- 
tion to infinity. For though, in order actually 
to see that two given lines never meet, it would 
be necessary to follow them to infinity ; yet 
without doing so we may know that if they 
ever do meet, or if, after diverging from one 
another, they begin again to approach, this 
must take place not at an infinite, but at a 
finite distance. Supposing, therefore, such to 
be the case, we can transport ourselves thither 
in imagination, and can frame a mental image 
of the appearance which one or both of the 
lines must present at that point, which we may 



3JO LECTURES AND ESSAYS 

rely on as being precisely similar to the reality. 
Now, whether we fix our contemplation upon this 
imaginary picture, or call to mind the generali- 
sations we have had occasion to make from 
former ocular observation, we learn by the 
evidence of experience that a line which, after 
diverging from another straight line, begins to 
approach to it, produces the impression on our 
senses, which we describe by the expression ' a 
bent line,' not by the expression ' a straight 
line.'" Logic, Book ii. chap. v. s. 5. 

Upon this argument I have one very simple 
remark to make. That "characteristic pro- 
perty of geometrical forms " is derived from 
experience ; we have " learnt by long-con- 
tinued experience that the properties of the 
reality are faithfully represented in the image." 
Experience could only tell us this of realities 
and of images both of which we have ex- 
perienced. I must know both of two things 
to know that one faithfully represents the other. 
Experience then tells me that my mental 
images of geometrical figures are faithful repre- 
sentations of those realities which are within 
the bounds of experience. But what is to tell 
me that they are faithful representations of 
realities that are beyond the bounds of ex- 
perience? Surely no experience can tell me 
that. 



PHILOSOPHY OF THE PURE SCIENCES 331 

Again, our notion of straiglit is a combina- 
tion of several properties, an aggregate of im- 
pressions on our senses, which holds together 
within the limits of experience. But what is 
to tell us that these impressions hold together 
beyond the limits of experience ? 

It seems to me, then, that in admitting the 
universality of certain statements Mr. Mill 
knows something which on his own principles 
he has no right to know. 

In the following section Mr. Mill deals with 
the supposed necessity of these truths. Taking 
this to mean the inconceivability of the nega- 
tion of them, he explains it in somewhat the 
same way as Hume explained the idea of 
cause, namely, by means of the law of associa- 
tion. But that which in Locke and Hume had 
been merely a special explanation of particular 
phenomena has in the meantime grown into 
an extensive and most successful science of 
Psychology. It began, as you remember, in 
the form of a link between two impressions 
that occur frequently together. Perhaps the 
most important step was Hartley's idea of 
" mental chemistry " ; that the result of two 
linked impressions might not put in evidence 
either of the components any more than water 
exhibits to us the hydrogen and the oxygen 
which it contains. In the hands of James 
Mill and Mr. Bain this mode of explanation 



332 LECTURES AND ESSAYS 

has been applied with marked success to a vast 
number of mental phenomena ; so that when 
Mr. Mill makes use of it to account for the 
inconceivability of that which has not yet been 
experienced, he is backed by an enormous mass 
of similar and most successful explanations. 

This view, that the supplementary part of 
our sensations is an accumulation of past 
experience, has been further defended by Mr. 
Bain in many excellent books. But there is 
one respect in which the doctrines of Mr. Mill 
and Mr. Bain differ very importantly from the 
one which we have next to consider that of 
Mr. Herbert Spencer. He also believes that 
the whole of our knowledge comes from ex- 
perience ; but while in the former view this 
experience is our own, and has been acquired 
during the lifetime of the individual, in the 
latter it is not the experience of you or me, 
but of all our ancestors. The perceptions, not 
only of former generations of men, but of those 
lower organisms from which they were originally 
derived, beginning even with the first molecule 
that was complex enough to preserve records 
of its own changes ; all these have been built 
into the organism, have determined its character, 
and have been handed down to us by hereditary 
descent. The effect of this upon Kant's doctrine 
may be best displayed by another quotation : 

" The universal law that, other things equal, 



PHILOSOPHY OF THE PURE SCIENCES 333 

the cohesion of psychical states is proportionate 
to the frequency with which they have followed 
one another in experience, supplies an explana- 
tion of the so-called ' forms of thought,' as soon 
as it is supplemented by the law that habitual 
psychical successions entail some hereditary 
tendency to such successions, which, under per- 
sistent conditions, will become cumulative in 
generation after generation. We saw that the 
establishment of those compound reflex actions 
called instincts is comprehensible on the prin- 
ciple that inner relations are, by perpetual 
repetition, organised into correspondence with 
outer relations. We have now to observe that 
the establishment of those consolidated, those 
indissoluble, those instinctive mental relations 
constituting our ideas of Space and Time, is 
comprehensible on the same principle. . . . 

"In the sense, then, that there exist in the 
nervous system certain pre-established relations 
answering to relations in the environment, 
there is a truth in the doctrine of ' forms of in- 
tuition ' not the truth which its defenders 
suppose, but a parallel truth. Corresponding 
to absolute external relations, there are 
established in the structure of the nervous 
system absolute internal relations relations 
that are potentially present before birth in the 
shape of definite nervous connections ; that are 
antecedent to, and independent of, individual 



334 LECTURES AND ESSAYS 

experiences ; and that are automatically dis- 
closed along with the first cognitions. And, as 
here understood, it is not only these funda- 
mental relations which are thus pre-determined ; 
but also hosts of other relations of a more or 
less constant kind, which are congenitally 
represented by more or less complete nervous 
connections. But these pre-determined internal 
relations, though independent of the experiences 
of the individual, are not independent of ex- 
periences in general : they have been determined 
by the experiences of preceding organisms. The 
corollary here drawn from the general argument 
is that the human brain is an organised register 
of infinitely numerous experiences received 
during the evolution of life, or rather, during 
the evolution of that series of organisms through 
which the human organism has been reached. 
The effects of the most uniform and frequent 
of these experiences have been successively 
bequeathed, principal and interest ; and have 
slowly mounted to that high intelligence which 
lies latent in the brain of the infant which 
the infant in after-life exercises and perhaps 
strengthens or further complicates and which, 
with minute additions, it bequeaths to future 
generations. And thus it happens that the 
European inherits from twenty to thirty cubic 
inches more brain than the Papuan. Thus it 
happens that faculties, as of music, which 



PHILOSOPHY OF THE PURE SCIENCES 335 

scarcely exist in some inferior human races, 
become congenital in superior ones. Thus it 
happens that out of savages unable to count up 
to the number of their fingers, and speaking 
a language containing only nouns and verbs, 
arise at length our Newtons and our Shake- 
speares." Principles of Psychology, 208, vol. 
i. pp. 466, 470. 

This doctrine of Mr. Spencer's is what I 
believe to be really the truth about the matter ; 
and I shall have to return to it again by and 
by. But I have a remark to make here. It 
seems to me that the Kantian dilemma about 
universal propositions is just as valid now, in 
spite of these explanations, as it was in his 
time. How am I to know that the angles of a 
triangle are exactly equal to two right angles 
under all possible circumstances ; not only in 
those regions of space where the solar system 
has been, but everywhere else ? The accumu- 
lated experience of all my ancestors for a 
hundred and fifty million years is no more 
competent to tell me tJiat than my own experi- 
ence of the last five minutes. Either I have 
some source of knowledge other than experi- 
ence, and I must admit the existence of a 
priori truths, independent of experience ; or I 
cannot know that any universal statement is 
true. Now the doctrine of evolution itself for- 
bids me to admit any transcendental source of 



336 LECTURES AND ESSAYS 

knowledge ; so that I am driven to conclude 
in regard to every apparently universal state- 
ment, either that it is not really universal, but 
a particular statement about my nervous 
system, about my apparatus of thought ; or that 
I do not know that it is true. And to this 
conclusion, by a detailed examination of various 
apparently universal statements, I shall in sub- 
sequent lectures endeavour to lead you. 

II. KNOWLEDGE AND FEELING 

The following fragment appears to represent what was the con- 
clusion of the series of Lectures as they were delivered in 
March 1873. It was found among Professor Clifford's 
papers without any external indication of its proper con- 
text ; and as the Lectures now stand after the author's re- 
vision, it seems to come in better as an appendix to the 
first of them. Clifford himself regarded it apparently (note 
to the Third Lecture in Nineteenth Century, March 
1879) as superseded by his article on "The Nature of 
Things-in-themselves " ; but it contains critical remarks 
and illustrations which are not there, and it has seemed 
best to the editors to let it stand in this place. 

IN order to consider at this point what it is 
that we have arrived at, we must call to mind 
the point from which we started. We said 
that the whole of our sensations could not 
possibly be a message from outside, but that 
some part at least of them must be a supple- 
ment or filling-in of this message, added by 
ourselves. A theory came before us that of 
Mr. Herbert Spencer according to which this 



PHILOSOPHY OF THE PURE SCIENCES 337 

filling-in was accounted for as the product of 
past experience, which had taken effect on the 
brains of our ancestors and produced certain 
changes in them. These changes have gradually 
moulded the structure of the nervous system 
which was handed on to us by hereditary 
descent. There was one obstacle to our accept- 
ance of that theory as a sufficient account of 
the matter ; namely, that we apparently had 
some knowledge which could not possibly have 
been got in that way knowledge that certain 
general statements are absolutely and universally 
true. This obstacle I shall endeavour to re- 
move, by showing that such general statements 
may be divided into two classes ; of which 
those in the first class may for all we know be 
false, while those in the second class are general 
statements only in form, and really are judg- 
ments about the apparatus of thought. If this 
be so, we are at liberty to accept the view that 
all human knowledge is derived from experi- 
ence ; and that of the two factors in sensation, 
that supplement which we provide of ourselves 
is a giving out again of what has originally be- 
longed to the other factor, to experience proper, 
But here a doubt suggests itself which appears 
exactly to reverse all that we have done. We 
said there were two factors of experience : that 
all of it could not be direct message ; and we 
have come to the conclusion that the two factors 
VOL. I Z 



338 LECTURES AND ESSAYS 

are really of the same kind. But we did not 
show that any of it was direct message from 
outside ; we only showed that some portions of 
it were not Suppose it is all supplement, and 
there is no message at all ! In that case our 
two factors will indeed be reduced to one ; but 
in what sense can we say that our knowledge 
is derived from experience ? It will of course 
be derived from experience in the large sense, 
that is, from sensation ; but in the sense in 
which we have used the term, as meaning that 
part of sensation which is not supplied by our- 
selves, there will be' no experience for us to 
derive knowledge from. This question then is 
an extremely important one ; for if we have to 
admit that there is no real message from with- 
out, all the sciences will become pure sciences, 
all knowledge will be a priori knowledge ; and 
we may construct the universe by sitting down 
and thinking about it. It is this question then 
that I propose to consider for a short time, a 
time very much too short for the consideration 
of it, but perhaps long enough to let me indicate 
in some way the kind of answer which is given 
by an extension of that Physiological Method 
which we began by using. 

We traced the message of sight to the retina 
of the eye, saying that the only direct message 
possible is contained in the picture there drawn. 
But we may go a little farther. The picture 



PHILOSOPHY OF THE PURE SCIENCES 339 

consists in an aggregate of forms and colours 
having a certain mode of connection. It is 
carried inwards by the optic nerve ; but in 
order to be so carried, it has to undergo a still 
further transformation. The optic nerve is a 
great bundle of telegraph wires, each carrying 
its own message undisturbed by the rest. Each 
wire only tells what is happening at a particular 
point of the retina ; that is to say, what colour 
and what intensity the light impinging on the 
point has. Now in order to tell the colour and 
intensity, it appears that it must consist of three 
distinct strands ; for it has been made out that 
every sensation of colour is composed of three 
simple sensations combined in a certain propor- 
tion, this proportion varying from colour to 
colour. Does then the optic nerve carry the 
picture itself as a message ? It is clear that it 
cannot ; but it may take an account of every 
point in it, and of their relations of contiguity ; 
that is, it carries an aggregate of elementary 
messages, which has a point-for-point connec- 
tion with the picture, of such a nature as to 
retain the relations of nextness or contiguity. 
But the point to notice is that two messages 
carried by the optic nerve differ only as two 
chords played upon the same organ, or as two 
books written in the same alphabet ; they are 
combinations or connected aggregates of the 
same elementary messages, selected and fastened 



340 LECTURES AND ESSAYS 

together in different ways. The difference is a 
matter of arrangement and building up ; not a 
difference of the elements that are built up. 
This very important step in the theory of 
sensation was made by Helmholtz, following in 
the steps of Miiller, equally in the case of sight 
and sound. It was he who made out clearly 
that the special nerves of the senses had not 
absolutely special functions of transmitting their 
particular sensation as a whole, but that the 
difference consisted in the various ways of 
combining together the same elementary nerve- 
message. Where, then, are these messages 
taken ? They are taken to the gray corpuscles 
within the brain ; and apparently each nerve 
goes to its own corpuscle, and sets it in com- 
motion with the message. Finally we get this 
result : that the presence of a picture on the 
retina involves the commotion of a certain 
number of gray corpuscles ; the selection of 
which and the amount of excitement given to 
each are determined by the picture. And the 
same thing happens for every other kind of 
sensation. Now the direct knowledge that we 
get can only be knowledge of this commotion 
in the gray matter. For we can tap the tele- 
graph, so to speak, and transmit a false message 
by it ; and it is found that if the optic nerve 
be excited either by pressure of the eye or by 
an electric shock, the sensation of sight is pro- 



PHILOSOPHY OF THE PURE SCIENCES 341 

duced, although no light has been present. 
The difference, then, of different sensations is 
made by the difference of the gray corpuscles 
excited ; and the immediate knowledge that is 
given to us by experience can only be know- 
ledge of more or less excitement of certain 
parts of the gray matter. This applies equally 
to touch, taste, smell, muscular action, the 
organic sensations of pain or pleasure. If you 
and I, then, choose to contemplate another 
person, we shall say that the world which he 
directly perceives is really inside his brain, and 
not outside ; but that corresponding to these 
changes that go on in his brain there are 
certain changes going on outside of him, and 
that in many cases there is such a correspond- 
ence of the relations of contiguity in one case 
to the relation of contiguity in the other, that 
conclusions about the outer world may fairly 
be drawn from the world in his brain. 

But now, if instead of considering this other 
person, I consider myself, the case is rather 
altered. I shall conclude by analogy that this 
world which I directly perceive is not really 
outside of me ; that the things which are 
apparently made known to me by my percep- 
tions are really themselves only groups of my 
perceptions ; that the universe which I perceive 
is made up of my feelings ; that in fact it is 
really me. And by analogy also I shall 



342 LECTURES AND ESSAYS 

conclude that there is something besides this, 
different from it ; the changes in which corre- 
spond in a certain way to the changes in my 
universe. Is it then possible for me to know 
what that is ? or is there nothing at all except 
my feelings ? 

If, instead of approaching this question from 
the physiological side, we adopt another point 
of view, it is not unlikely that we shall be led 
to the latter conclusion. If I consider merely 
my own feelings, and ask what evidence they 
give of anything beyond them, it seems to me 
that I must answer, no evidence at all. This 
at least was the answer given by Berkeley in a 
passage which has been quoted here before by 
Professor Huxley, but will bear quoting again : 

" Some truths there are so near and obvious 
to the mind that a man need only open his 
eyes to see them. Such I take this important 
one to be, viz. that all the choir of heaven and 
furniture of the earth, in a word, all those 
bodies which compose the mighty frame of the 
world, have not any subsistence without a 
mind, that their being is to be perceived or 
known ; that consequently so long as they are 
not actually perceived by me, or do not exist 
in my mind or that of any other created spirit, 
they must either have no existence at all, or 
else subsist in the mind of some Eternal Spirit." 
Principles of Human Knowledge, 6. 



PHILOSOPHY OF THE PURE SCIENCES 343 

If I say that such and such things existed 
at some previous time, I mean that if I had 
been there I could have perceived them ; if I 
say that there is hydrogen in the sun, I mean 
that if I could get any of that gas I should be 
able to burn it in oxygen and produce exactly 
the same impressions on my senses as those 
which, in the aggregate, I call water. 

This doctrine, that the essence of things 
consists in my perceiving them, is called 
Idealism. The form of it held by Berkeley, 
however, is not altogether pure. He believed 
that no material external world exists ; but 
only spirits exist, thinking beings whose nature 
consists of conception and volition. Now, 
from this point of view, fairly accepted, you are 
only phenomena of my consciousness as much 
as the rest of the world ; I cannot allow the 
existence of any spirits, but only of one spirit, 
myself. And even this language is hardly 
suitable ; for why should I give myself a class- 
name like spirit when I am really the sum- 
total of the universe ? Notwithstanding this 
failure to reach complete idealism, the doctrine 
of Berkeley, in its positive aspect, is a distinct 
and most important step in philosophy ; it 
established in a security that has never yielded 
to attack the subjective character of the world 
of phenomena ; that this world which I per- 
ceive is my perceptions and nothing more. 



344 LECTURES AND ESSAYS 

Whether there is anything else quite different 
which corresponds to it in a certain way, is 
another question ; Berkeley said there were 
also spirits. 

According to Berkeley, moreover, there 
exists, besides this world of my perceptions, 
a particular spirit, me y that perceives them. 
To get rid of this imaginary soul or substance, 
underlying the succession of my feelings, was 
the work of Hume. Just as an object, in 
Berkeley's theory, is merely a bundle of per- 
ceptions which always occur together, a linked 
aggregate of feelings ; so, said Hume, out of 
the swift current of ideas that succeed one 
another we construct a unity which we call 
Self or Ego. But this, he said, is a pure 
illusion ; and the ego, when analysed, turns 
out to be only the whole complex of my feel- 
ings. This, you see, is a step towards simpli- 
fication ; we had to begin with an external 
thing which is perceived ; then the perception 
or feeling ; then the soul or self which per- 
ceives. With Berkeley we get rid of the 
thing perceived ; it is reduced to a bundle of 
perceptions. With Hume we get rid also of 
the perceiving self ; it is reduced to the whole 
aggregate of feelings, linked together and 
succeeding one another in a certain manner. 

The step made by Mill is a more complete 
definition of the same view, and an explanation 



PHILOSOPHY OF THE PURE SCIENCES 345 

by means of the law of association of the way in 
which we come to believe in an external world. 
He says that objects are completely described 
by the phrase, " permanent possibilities of 
sensation." 

" The Psychological Theory maintains that 
there are associations naturally and even neces- 
sarily generated by the order of our sensations 
and of our reminiscences of sensation, which, 
supposing no intuition of an external world to 
have existed in consciousness, would inevitably 
generate the belief, and would cause it to be 
regarded as an intuition. . . . The conception 
I form of the world existing at any moment 
comprises, along with the sensations I am 
feeling, a countless variety of possibilities of 
sensation : namely, the whole of those which 
past observation tells me that I could, under 
any supposable circumstances, experience at 
this moment, together with an indefinite and 
illimitable multitude of others which though I 
do not know that I could, yet it is possible 
that I might, experience in circumstances not 
known to me. These various possibilities are 
the important thing to me in the world. My 
present sensations are generally of little im- 
portance, and are moreover fugitive : the 
possibilities, on the contrary, are permanent, 
which is the character that mainly distinguishes 



346 LECTURES AND ESSAYS 

our idea of Substance or Matter from our 
notion of sensation. . . . Matter, then, may be 
defined, a Permanent Possibility of Sensation." 1 

In the meanwhile, you observe, the associa- 
tion-theory of the mind had been created ; and 
it is here applied to defend the position of 
Hume. It is worth while to notice now where 
we are. The universe consists of feelings. A 
certain cable of feelings, linked together in a 
particular manner, constitutes me. Similar 
cables constitute you. That is all there is. 
But in the cable of feelings that make up me 
there are certain persistent bundles or strands, 
which occasionally come to the outside ; there 
are similar strands in the cables of which you 
are constituted. These correspond to external 
objects ; we only think them external for the 
reasons assigned. 

Now, when we pass to Mr. Herbert Spencer, 
we come into the presence of another great de- 
partment of science that has not had so strong 
an action upon Mr. Mill ; and that is the 
anatomy of the nervous system. The effect 
of investigations in this subject is to analyse 
all the various kinds of nervous action into 
different combinations of two simple elements ; 
the transmission of messages along nerve- 

1 J. S. Mill, Examination of Sir W. Hamilton's Philosophy, 
pp. 192, 193, 198, 2nd edit. 



PHILOSOPHY OF THE PURE SCIENCES 347 

threads of white matter, and the excitement of 
nerve-cells of gray matter. Apparently all the 
nerve-threads are alike, and all the nerve-cells 
are alike. The only thing that remains to 
produce the very different effects that we 
observe is the variety of ways in which selec- 
tions may be made from the nerve-cells to be 
excited at any moment. The direct effects of 
nerve-action are the effect on muscular tissue 
of contraction or release, and the effect on 
glands of secretion. 

Here, then, were two great branches of 
analysis present to Mr. Spencer : the analysis 
of mental action given by the association-theory, 
which reduced everything to the linking -to- 
gether of feelings, and the analysis of nervous 
action supplied by the histologists. It was his 
business to supply not merely the link between 
the two, but an account of their simultaneous 
evolution. If we find that certain complicated 
forms of mental action always accompany 
certain forms of nervous action ; if each of these 
can be reduced into elements, and the relation 
of each compound to its elements is the same 
the bricks different, but the mode of putting 
them together identical in these two houses 
there is a very strong presumption that the 
element of mental action always accompanies 
the element of nervous action. But this pre- 
sumption is converted into knowledge when we 



348 LECTURES AND ESSAYS 

have an account of their origin. When the 
evolution of the living organism is traced up- 
wards from the simplest forms to the most 
complex, and it is found that the evolution of 
mind proceeds part passu with it, following the 
same laws and passing through the same stages, 
either evolution being expressed as a continual 
building up with the same element, we have 
actual evidence that the one element goes with 
the other. 

Hence, then, is the great advantage of Mr. 
Herbert Spencer in the study of both orders of 
facts. He can make any step in analysis of 
the one help in the analysis of the other. And 
accordingly he has carried both to an extent 
which leaves all previous investigators far be- 
hind. But you will see at once that we must 
look at the question of idealism from the 
physiological point of view. And accordingly 
he considers that there is something different 
from our perceptions, the changes in which 
correspond in a certain way to the changes in 
the worlds we perceive. He thinks, however, 
that we can never know what it is ; and he 
says : 

" We can think of Matter only in terms of 
Mind. We can think of Mind only in terms 
of Matter. When we have pushed our explora- 
tions of the first to the uttermost limit, we are 
referred to the second for a final answer ; and 



PHILOSOPHY OF THE PURE SCIENCES 349 

when we have got the final answer of the 
second, we are referred back to the first for an 
interpretation of it. We find the value of x in 
terms of y ; then we find the value of y in 
terms of x ; and so on we may continue for 
ever without coming nearer to a solution. The 
antithesis of subject and object, never to be 
transcended while consciousness lasts, renders 
impossible all knowledge of that Ultimate 
Reality in which subject and object are 
united." Principles of Psychology, 272 (vol. 
i. p. 627). 

Now, the singular character of this realism 
is that it is defended from the idealistic point 
of view, namely, Mr. Spencer attempts to make 
my feelings give me evidence of something 
which is not included among them. A careful 
study of all his arguments to that effect has 
only convinced me over again that the attempt 
is hopeless. In this respect he differs consider- 
ably from Mr. Shadworth Hodgson, who must 
be regarded as an advance, within the British 
school, in the direction of Berkeley and Hume. 
He accepts the analysis of the individual ego 
or self into a complex of feeling ; and, like 
Hume or Mill, makes the universe to consist of 
feelings variously bound together. But this is 
only one aspect of it and of all contained 
phenomena. Every phenomenon has two 
aspects ; in its subjective aspect it is a feeling 



350 LECTURES AND ESSAYS 

in its objective aspect a quality. But it is not 
necessarily a feeling of my consciousness or of 
your consciousness ; it may be a feeling of the 
general or universal consciousness, which is 
coextensive with all existence. The universal 
consciousness bears the same relation to the 
universal Ego of Schelling or Hegel that the 
stream of feelings does to the soul ; it is an 
analysis of it into elements. 

The important thing here is the conclusion 
that there is only one world, combined with 
the analysis of mental phenomena. The 
German Idealist attempted to construct the 
world out of very abstract ideas, which are the 
most complex of all forms of mental action. 
In this way we did get one world, a mental 
world ; but the bricks of which it was built 
were made by the ingenious piling together of 
houses. I do not think that that process is 
likely to produce serviceable bricks. Now Mr. 
Hodgson's element, feeling, although it seems 
to imply something too complicated, is yet at 
least a step in the way of analysis, an indica- 
tion that analysis is desired. 

Can we now get out of our hobble, and 
arrive at real knowledge derived from external 
experience, from messages and not from im- 
agination? I think we can. But it is necessary 
to say first what is the character of the know- 
ledge we desire. It will be of the nature of 



PHILOSOPHY OF THE PURE SCIENCES 351 

inference, and not of absolute certainty. Now 
inference depends on the assumption of the 
uniformity of nature ; and what does this rest 
on ? We cannot infer that which is the ground 
of all inference ; but although I cannot give 
you a logical reason for believing it, I can give 
you a physical explanation of the fact that we 
all do believe it. We believe a thing when we 
are prepared to act as if it were true. Now, if 
you and I had not habitually acted on the 
assumption of the uniformity of nature from 
the time when we could act at all, we should 
not be here to discuss the question. Nature is 
selecting for survival those individuals and 
races who act as if she were uniform ; and 
hence the gradual spread of that belief over 
the civilised world. 

This uniformity may be merely a uniformity 
of phenomena, a law relating to my feelings. 
So long as I only am concerned, it seems to 
me that the idealist theory is perfectly sufficient. 
It is quite capable of explaining me ; but when 
you come into the question, it is utterly at a 
loss. The distinction between the universal 
and the individual ego seems to me a merely 
useless abstraction that throws dust in our eyes. 
I do believe that you are conscious in the same 
way as I am ; and once that is conceded, the 
whole idealist theory falls to pieces. For there 
are feelings which are not my feelings, which 



35* LECTURES AND ESSAYS 

are entirely outside my consciousness ; so that 
there is at least an external world. But let us 
consider now in what way we infer it ; why do 
I believe that there are feelings which are not 
mine ? Because, as I belong to a gregarious 
race, the greater part of my life consists in act- 
ing upon the supposition that it is true. 

But now further, have I reason for believing 
that the changes in this external world corre- 
spond in any way with the changes in my world 
which I perceive ? I think so. The complex 
of feelings which constitutes you corresponds in 
a definite way with the changes which I might 
perceive in your brain. By inferences that I 
have previously indicated, I conclude that the 
ultimate element into which your feeling can 
be analysed goes with the ultimate element out 
of which the changes of the nerve-matter in 
your brain are built up. But physiological 
action is complicated chemistry in the same 
way that chemistry is complicated mechanics. 
The actions that take place in the brain differ 
in no way from other material actions, except 
in their complexity. Conjoin with this the 
doctrine of Evolution, and you will see evidence 
that the simplest mental change goes always 
with the simplest material change, whether in 
the brain or not. The external world, then, is 
a complex of mental changes ; the ultimate 
elements into which feeling can be analysed ; 



PHILOSOPHY OF THE PURE SCIENCES 353 

so simple that the simplest feeling which we 
can experience is an enormously complex mass 
of them. Some of these are built up into 
sufficiently complicated forms to constitute 
what we call personality, will, consciousness. 
They all succeed one another according to 
certain laws ; and in virtue of these any conscious 
aggregate of them is acted upon by the rest ; 
the changes so produced in it are what we call 
a material world. 

There is thus only one world of elementary 
feelings ; which is perceived by me as my 
material world. And I am not to look for 
those complex forms of mental action called 
intelligence and consciousness, except where 
I can perceive a correspondingly complex 
aggregation of matter. 



III. THE POSTULATES OF THE SCIENCE 

OF SPACE 

IN my first lecture I said that, out of the 
pictures which are all that we can really see, 
we imagine a world of solid things ; and that 
this world is constructed so as to fulfil a certain 
code of rules, some called axioms, and some 
called definitions, and some called postulates, 
and some assumed in the course of demonstra- 
tion, but all laid down in one form or another 
in Euclid's Elements of Geometry. It is this 
VOL. I 2 A 



354 * LECTURES AND ESSAYS 

code of rules that we have to consider to-day. 
I do not, however, propose to take this book 
that I have mentioned, and to examine one 
after another the rules as Euclid has laid them 
down or unconsciously assumed them ; not- 
withstanding that many things might be said 
in favour of such a course. This book has been 
for nearly twenty-two centuries the encourage- 
ment and guide of that scientific thought which 
is one thing with the progress of man from a 
worse to a better state. The encouragement ; 
for it contained a body of knowledge that was 
really known and could be relied on, and that 
moreover was growing in extent and applica- 
tion. For even at the time this book was 
written shortly after the foundation of the 
Alexandrian Museum Mathematic was no 
longer the merely ideal science of the Platonic 
school, but had started on her career of con- 
quest over the whole world of Phenomena. 
The guide ; for the aim of every scientific 
student of every subject was to bring his know- 
ledge of that subject into a form as perfect as 
that which geometry had attained. Far up on 
the great mountain of Truth, which all the 
sciences hope to scale, the foremost of that 
sacred sisterhood was seen, beckoning to the 
rest to follow her. And hence she was called, 
in the dialect of the Pythagoreans, " the purifier 
of the reasonable soul." Being thus in itself at 



PHILOSOPHY OF THE PURE SCIENCES 355 

once the inspiration and the aspiration of 
scientific thought, this Book of Euclid's has 
had a history as chequered as that of human 
progress itself. It embodied and systematised 
the truest results of the search after truth that 
was made by Greek, Egyptian, and Hindu. It 
presided for nearly eight centuries over that 
promise of light and right that was made by 
the civilised Aryan races on the Mediterranean 
shores ; that promise, whose abeyance for nearly 
as long an interval is so full of warning and of 
sadness for ourselves. It went into exile along 
with the intellectual activity and the goodness 
of Europe. It was taught, and commented 
upon, and illustrated, and supplemented, by 
Arab and Nestorian, in the Universities of 
Bagdad and of Cordova. From these it was 
brought back into barbaric Europe by terrified 
students who dared tell hardly any other thing 
of what they had learned among the Saracens. 
Translated from Arabic into Latin, it passed 
into the schools of Europe, spun out with ad- 
ditional cases for every possible variation of 
the figure, and bristling with words which had 
sounded to Greek ears like the babbling of 
birds in a hedge. At length the Greek text 
appeared and was translated ; and, like other 
Greek authors, Euclid became an authority. 
There had not yet arisen in Europe " that fruit- 
ful faculty," as Mr. Winwood Reade calls it, 



356 LECTURES AND ESSAYS 

" with which kindred spirits contemplate each 
other's works ; which not only takes, but gives ; 
which produces from whatever it receives ; 
which embraces to wrestle, and wrestles to em- 
brace." Yet it was coming ; and though that 
criticism of first principles which Aristotle and 
Ptolemy and Galen underwent waited longer 
in Euclid's case than in theirs, it came for him 
at last. What Vesalius was to Galen, what 
Copernicus was to Ptolemy, that was Lobat- 
chewsky to Euclid. There is, indeed, a some- 
what instructive parallel between the last two 
cases. Copernicus and Lobatchewsky were 
both of Slavic origin. Each of them has 
brought about a revolution in scientific ideas so 
great that it can only be compared with that 
wrought by the other. And the reason of the 
transcendent importance of these two changes 
is that they are changes in the conception of 
the Cosmos. Before the time of Copernicus, 
men knew all about the Universe. They could 
tell you in the schools, pat off by heart, all that 
it was, and what it had been, and what it would 
be. There was the flat earth, with the blue 
vault of heaven resting on it like the dome of 
a cathedral, and the bright cold stars stuck into 
it ; while the sun and planets moved in crystal 
spheres between. Or, among the better in- 
formed, the earth was a globe in the centre of the 
universe, heaven a sphere concentric with it ; 



PHILOSOPHY OF THE PURE SCIENCES 357 

intermediate machinery as before. At any 
rate, if there was anything beyond heaven, it 
was a void space that needed no further de- 
scription. The history of all this could be traced 
back to a certain definite time, when it began ; 
behind that was a changeless eternity that 
needed no further history. Its future could be 
predicted in general terms as far forward as a 
certain epoch, about the precise determination 
of which there were, indeed, differences among 
the learned. But after that would come again 
a changeless eternity, which was fully accounted 
for and described. But in any case the Uni- 
verse was a known thing. Now the enormous 
effect of the Copernican system, and of the 
astronomical discoveries that have followed it, 
is that, in place of this knowledge of a little, 
which was called knowledge of the Universe, of 
Eternity and Immensity, we have now got 
knowledge of a great deal more ; but we only 
call it the knowledge of Here and Now. We 
can tell a great deal about the solar system ; 
but, after all, it is our house, and not the city. 
We can tell something about the star-system 
to which our sun belongs ; but, after all, it is 
our star-system, and not the Universe. We 
are talking about Here with the consciousness 
of a There beyond it, which we may know 
some time, but do not at all know now. And 
though the nebular hypothesis tells us a great 



35 LECTURES AND ESSAYS 

deal about the history of the solar system, and 
traces it back for a period compared with which 
the old measure of the duration of the Universe 
from beginning to end is not a second to a 
century, yet we do not call this the history of 
eternity. We may put it all together and call 
it Now, with the consciousness of a Then before 
it, in which things were happening that may 
have left records ; but we have not yet read 
them. This, then, was the change effected by 
Copernicus in the idea of the Universe. But 
there was left another to be made. For the 
laws of space and motion, that we are presently 
going to examine, implied an infinite space and 
an infinite duration, about whose properties 
as space and time everything was accurately 
known. The very constitution of those parts 
of it which are at an infinite distance from us, 
" geometry upon the plane at infinity," is just 
as well known, if the Euclidean assumptions 
are true, as the geometry of any portion of this 
room. In this infinite and thoroughly well- 
known space the Universe is situated during at 
least some portion of an infinite and thoroughly 
well-known time. So that here we have real 
knowledge of something at least that concerns 
the Cosmos ; something that is true throughout 
the Immensities and the Eternities. That 
something Lobatchewsky and his successors 
have taken away. The geometer of to-day 



PHILOSOPHY OF THE PURE SCIENCES 359 

knows nothing about the nature of actually 
existing space at an infinite distance ; he 
knows nothing about the properties of this 
present space in a past or a future eternity. 
He knows, indeed, that the laws assumed by 
Euclid are true with an accuracy that no direct 
experiment can approach, not only in this place 
where we are, but in places at a distance from 
us that no astronomer has conceived ; but he 
knows this as of Here and Now ; beyond his 
range is a There and Then of which he knows 
nothing at present, but may ultimately come to 
know more. So, you see, there is a real parallel 
between the work of Copernicus and his suc- 
cessors on the one hand, and the work of 
Lobatchewsky and his successors on the other. 
In both of these the knowledge of Immensity 
and Eternity is replaced by knowledge of Here 
and Now. And in virtue of these two revolu- 
tions the idea of the Universe, the Macrocosm, 
the All, as subject of human knowledge, and 
therefore of human interest, has fallen to pieces. 
It will now, I think, be clear to you why it 
will not do to take for our present considera- 
tion the postulates of geometry as Euclid has 
laid them down. While they were all certainly 
true, there might be substituted for them some 
other group of equivalent propositions ; and the 
choice of the particular set of statements that 
should be used as the groundwork of the science 



360 LECTURES AND ESSAYS 

was to a certain extent arbitrary, being only 
guided by convenience of exposition. But from 
the moment that the actual truth of these 
assumptions becomes doubtful, they fall of 
themselves into a necessary order and classifica- 
tion ; for we then begin to see which of them 
may be true independently of the others. And 
for the purpose of criticising the evidence for 
them, it is essential that this natural order 
should be taken ; for I think you will see 
presently that any other order would bring 
hopeless confusion into the discussion. 

Space is divided into parts in many ways. 
If we consider any material thing, space is at 
once divided into the part where that thing is 
and the part where it is not. The water in 
this glass, for example, makes a distinction 
between the space where it is and the space 
where it is not. Now, in order to get from one 
of these to the other you must cross the surface 
of the water ; this surface is the boundary of the 
space where the water is which separates it from 
the space where it is not Every thing, con- 
sidered as occupying a portion of space, has a 
surface which separates the space where it is from 
the space where it is not. But, again, a surface 
may be divided into parts in various ways. 
Part of the surface of this water is against the 
air, and part is against the glass. If you travel 
over the surface from one of these parts to the 



PHILOSOPHY OF THE PURE SCIENCES 361 

other, you have to cross the line which divides 
them ; it is this circular edge where water, air, 
and glass meet. Every part of a surface is 
separated from the other parts by a line which 
bounds it. But now suppose, further, that this 
glass had been so constructed that the part 
towards you was blue and the part towards me 
was white, as it is now. Then this line, divid- 
ing two parts of the surface of the water, would 
itself be divided into two parts ; there would 
be a part where it was against the blue glass, 
and a part where it was against the white glass. 
If you travel in thought along that line, so as to 
get from one of these two parts to the other, you 
have to cross a point which separates them, and 
is the boundary between them. Every part of 
a line is separated from the other parts by 
points which bound it. So we may say 
altogether 

The boundary of a solid (i.e. of a part of 
space) is a surface. 

The boundary of a part of a surface is a line. 

The boundaries of a part of a line are 
points. 

And we are only settling the meanings in 
which words are to be used. But here we may 
make an observation which is true of all space 
that we are acquainted with : it is that the 
process ends here. There are no parts of a 
point which are separated from one another by 



362 LECTURES AND ESSAYS 

the next link in the series. This is also in- 
dicated by the reverse process. 

For I shall now suppose this point the last 
thing that we got to to move round the 
tumbler so as to trace out the line, or edge, 
where air, water, and glass meet. In this way 
I get a series of points, one after another ; a 
series of such a nature that, starting from any 
one of them, only two changes are possible that 
will keep it within the series : it must go 
forwards or it must go backwards, and each of 
these is perfectly definite. The line may then 
be regarded as an aggregate of points. Now 
let us imagine, further, a change to take place 
in this line, which is nearly a circle. Let us 
suppose it to contract towards the centre of the 
circle, until it becomes indefinitely small, and 
disappears. In so doing it will trace out the 
upper surface of the water, the part of the 
surface where it is in contact with the air. In 
this way we shall get a series of circles one 
after another a series of such a nature that, 
starting from any one of them, only two changes 
are possible that will keep it within the series : 
it must expand or it must contract. This series, 
therefore, of circles, is just similar to the series 
of points that make one circle ; and just as the 
line is regarded as an aggregate of points, so 
we may regard this surface as an aggregate of 
lines. But this surface is also in another sense 



PHILOSOPHY OF THE PURE SCIENCES 363 

an aggregate of points, in being an aggregate 
of aggregates of points. But, starting from a 
point in the surface, more than two changes are 
possible that will keep it within the surface, for 
it may move in any direction. The surface, 
then, is an aggregate of points of a different 
kind from the line. We speak of the line as a 
point - aggregate of one dimension, because, 
starting from one point, there are only two 
possible directions of change ; so that the line 
can be traced out in one motion. In the same 
way, a surface is a line-aggregate of one dimen- 
sion, because it can be traced out by one motion 
of the line ; but it is a point-aggregate of two 
dimensions, because, in order to build it up of 
points, we have first to aggregate points into a 
line, and then lines into a surface. It requires 
two motions of a point to trace it out. 

Lastly, let us suppose this upper surface of 
the water to move downwards, remaining 
always horizontal till it becomes the under 
surface. In so doing it will trace out the part 
of space occupied by the water. We shall thus 
get a series of surfaces one after another, pre- 
cisely analogous to the series of points which 
make a line, and the series of lines which make 
a surface. The piece of solid space is an aggre- 
gate of surfaces, and an aggregate of the same 
kind as the line is of points ; it is a surface- 
aggregate of one dimension. But at the same 



364 LECTURES AND ESSAYS 

time it is a line-aggregate of two dimensions, 
and a point - aggregate of three dimensions. 
For if you consider a particular line which has 
gone to make this solid, a circle partly con- 
tracted and part of the way down, there are 
more than two opposite changes which it can 
undergo. For it can ascend or descend, or 
expand or contract, or do both together in any 
proportion. It has just as great a variety of 
changes as a point in a surface. And the piece 
of space is called a point-aggregate of three 
dimensions, because it takes three distinct 
motions to get it from a point. We must first 
aggregate points into a line, then lines into a 
surface, then surfaces into a solid. 

At this step it is clear, again, that the process 
must stop in all the space we know of. For it 
is not possible to move that piece of space in 
such a way as to change every point in it 
When we moved our line or our surface, the 
new line or surface contained no point what- 
ever that was in the old one ; we started with 
one aggregate of points, and by moving it we 
got an entirely new aggregate, all the points of 
which were new. But this cannot be done with 
the solid ; so that the process is at an end. 
We arrive, then, at the result that space is of 
three dimensions. 

Is this, then, one of the postulates of the 
science of space ? No ; it is not. The science 



PHILOSOPHY OF THE PURE SCIENCES 365 

of space, as we have it, deals with relations of 
distance existing in a certain space of three 
dimensions, but it does not at all require us to 
assume that no relations of distance are possible 
in aggregates of more than three dimensions. 
The fact that there are only three dimensions 
does regulate the number of books that we write, 
and the parts of the subject that we study : but it 
is not itself a postulate of the science. We in- 
vestigate a certain space of three dimensions, 
on the hypothesis that it has certain elementary 
properties ; and it is the assumptions of these 
elementary properties that are the real postu- 
lates of the science of space. To these I now 
proceed. 

The first of them is concerned with points, 
and with the relation of space to them. We 
spoke of a line as an aggregate of points. 
Now there are two kinds of aggregates, which 
are called respectively continuous and discrete. 
If you consider this line, the boundary of part 
of the surface of the water, you will find yourself 
believing that between any two points of it you 
can put more points of division, and between 
any two of these more again, and so on ; and 
you do not believe there can be any end to the 
process. We may express that by saying you 
believe that between any two points of the line 
there is an infinite number of other points. 
But now here is an aggregate of marbles, which, 



366 LECTURES AND ESSAYS 

regarded as an aggregate, has many characters 
of resemblance with the aggregate of points. 
It is a series of marbles, one after another ; and 
if we take into account the relations of nextness 
or contiguity which they possess, then there are 
only two changes possible from one of them as 
we travel along the series : we must go to the 
next in front, or to the next behind. But yet 
it is not true that between any two of them 
there is an infinite number of other marbles ; 
between these two, for example, there are only 
three. There, then, is a distinction at once 
between the two kinds of aggregates. But 
there is another, which was pointed out by 
Aristotle in his Physics and made the basis of 
a definition of continuity. I have here a row 
of two different kinds of marbles, some white 
and some black. This aggregate is divided 
into two parts, as we formerly supposed the line 
to be. In the case of the line the boundary 
between the two parts is a point which is the 
element of which the line is an aggregate. In 
this case before us, a marble is the element ; 
but here we cannot say that the boundary 
between the two parts is a marble. The 
boundary of the white parts is a white marble, 
and the boundary of the black parts is a black 
marble ; these two adjacent parts have different 
boundaries. Similarly, if instead of arranging 
my marbles in a series, I spread them out on a 



PHILOSOPHY OF THE PURE SCIENCES 367 

surface, I may have this aggregate divided into 
two portions a white portion and a black 
portion ; but the boundary of the white portion 
is a row of white marbles, and the boundary of 
the black portion is a row of black marbles. 
And lastly, if I made a heap of white marbles, 
and put black marbles on the top of them, I 
should have a discrete aggregate of three 
dimensions divided into two parts : the bound- 
ary of the white part would be a layer of white 
marbles, and the boundary of the black part 
would be a layer of black marbles. In all these 
cases of discrete aggregates, when they are 
divided into two parts, the two adjacent parts 
have different boundaries. But if you come to 
consider an aggregate that you believe to be 
continuous, you will see that you think of two 
adjacent parts as having the same boundary. 
What is the boundary between water and air 
here ? Is it water ? No ; for there would still 
have to be a boundary to divide that water from 
the air. For the same reason it cannot be air. 
I do not want you at present to think of the 
actual physical facts by the aid of any mole- 
cular theories ; I want you only to think of 
what appears to be, in order to understand 
clearly a conception that we all have. Suppose 
the things actually in contact If, however 
much we magnified them, they still appeared to 
be thoroughly homogeneous, the water filling 



368 LECTURES AND ESSAYS 

up a certain space, the air an adjacent space ; 
if this held good indefinitely through all degrees 
of conceivable magnifying, then we could not 
say that the surface of the water was a layer of 
water and the surface of air a layer of air ; we 
should have to say that the same surface was 
the surface of both of them, and was itself 
neither one nor the other that this surface 
occupied no space at all. Accordingly, Aristotle 
defined the continuous as that of which two 
adjacent parts have the same boundary ; and 
the discontinuous or discrete as that of which 
two adjacent parts have direct boundaries. 1 

Now the first postulate of the science of 
space is that space is a continuous aggregate 
of points, and not a discrete aggregate. And 
this postulate which I shall call the postulate 
of continuity is really involved in those three 
of the six 2 postulates of Euclid for which 
Robert Simson has retained the name of 
postulate. You will see, on a little reflection, 
that a discrete aggregate of points could not be 

1 Phys. Ausc. V. 3, p. 227, ed. Bekker. Ti 6 <nive\^ ftm 
fifr Strep fx^^"^" TL > ^yw 8' flvai ffwex^i Srav ravrb yfvtjTat ical 
tr TO ina.Ttpov irtpat olt dVroircu, Kal Gxrirep ffrjfjLalvfi roCvofM 
ffw^xn' ToOro 8' oux olbv re Svoly dvroiv dvai TO'IV t<r)(6.TOt.v. 

A little farther on he makes the important remark that on the 
hypothesis of continuity a line is not made up of points in the same 
way that a whole is made up of parts, VI. i, p. 231. 'ASiWrov 
ASiaipiruv etc at ri trwex^*, olov ypa,fjip.i)v tic ffTtyfi&v, ftwep ij 



2 See De Morgan, in Smith's Diet, of Biography and Mythology, 
Art. " Euclid " ; and in the English Cyclopedia, Art. " Axiom." 



PHILOSOPHY OF THE PURE SCIENCES 369 

so arranged that any two of them should be 
relatively situated to one another in exactly the 
same manner, so that any two points might be 
joined by a straight line t which should always 
bear the same definite relation to them. And 
the same difficulty occurs in regard to the 
other two postulates. But perhaps the most 
conclusive way of showing that this postulate 
is really assumed by Euclid is to adduce the 
proposition he proves, that every finite straight 
line may be bisected. Now this could not be 
the case if it consisted of an odd number of 
separate points. As the first of the postulates 
of the science of space, then, we must reckon 
this postulate of Continuity ; according to 
which two adjacent portions of space, or of a 
surface, or of a line, have the same boundary, 
viz. a surface, a line, or a point ; and between 
every two points on a line there is an infinite 
number of intermediate points. 

The next postulate is that of Elementary 
Flatness. You know that if you get hold of a 
small piece of a very large circle, it seems to 
you nearly straight. So, if you were to take 
any curved line, and magnify it very much, 
confining your attention to a small piece of it, 
that piece would seem straighter to you than 
the curve did before it was magnified. At 
least, you can easily conceive a curve possess- 
ing this property, that the more you magnify 
VOL. I 2 B 



370 LECTURES AND ESSAYS 

it, the straighter it gets. Such a curve would 
possess the property of elementary flatness. 
In the same way, if you perceive a portion of 
the surface of a very large sphere, such as the 
earth, it appears to you to be flat. If, then, 
you take a sphere of say a foot diameter, and 
magnify it more and more, you will find that 
the more you magnify it the flatter it gets. 
And you may easily suppose that this process 
would go on indefinitely ; that the curvature 
would become less and less the more the 
surface was magnified. Any curved surface 
which is such that the more you magnify it 
the flatter it gets, is said to possess the property 
of elementary flatness. But if every succeed- 
ing power of our imaginary microscope disclosed 
new wrinkles and inequalities without end, then 
we should say that the surface did not possess 
the property of elementary flatness. 

But how am I to explain how solid space 
can have this property of elementary flatness ? 
Shall I leave it as a mere analogy, and say 
that it is the same kind of property as this of 
the curve and surface, only in three dimensions 
instead of one or two ? I think I can get a 
little nearer to it than that ; at all events I 
will try. 

If we start to go out from a point on a surface, 
there is a certain choice of directions in which 
we may go. These directions make certain 



PHILOSOPHY OF THE PURE SCIENCES 371 

angles with one another. We may suppose 
a certain direction to start with, and then 
gradually alter that by turning it round the 
point : we find thus a single series of directions 
in which we may start from the point. Accord- 
ing to our first postulate, it is a continuous 
series of directions. Now when I speak of a 
direction from the point, I mean a direction of 
starting ; I say nothing about the subsequent 
path. Two different paths may have the same 
direction at starting ; in this case they will 
touch at the point ; and there is an obvious 
difference between two paths which touch and 
two paths which meet and form an angle. 
Here, then, is an aggregate of directions, and 
they can be changed into one another. More- 
over, the changes by which they pass into one 
another have magnitude, they constitute dis- 
tance-relations ; and the amount of change 
necessary to turn one of them into another is 
called the angle between them. It is involved 
in this postulate that we are considering, that 
angles can be compared in respect of magni- 
tude. But this is not all. If we go on changing 
a direction of start, it will, after a certain amount 
of turning, come round into itself again, and be 
the same direction. On every surface which 
has the property of elementary flatness, the 
amount of turning necessary to take a direction 
all round into its first position is the same for 



37 LECTURES AND ESSAYS 

all points of the surface. I will now show you 
a surface which at one point of it has not this 
property. I take this circle of paper from 
which a sector has been cut out, and bend it 
round so as to join the edges ; in this way I 
form a surface which is called a cone. Now on 
all points of this surface but one, the law of 
elementary flatness holds good. At the vertex 
of the cone, however, notwithstanding that there 
is an aggregate of directions in which you may 
start, such that by continuously changing one of 
them you may get it round into its original posi- 
tion, yet the whole amount of change necessary to 
effect this is not the same at the vertex as it is at 
any other point of the surface. And this you 
can see at once when I unroll it ; for only part 
of the directions in the plane have been included 
in the cone. At this point of the cone, then, 
it does not possess the property of elementary 
flatness ; and no amount of magnifying would 
ever make a cone seem flat at its vertex. 

To apply this to solid space, we must notice 
that here also there is a choice of directions in 
which you may go out from any point ; but it 
is a much greater choice than a surface gives 
you. Whereas in a surface the aggregate of 
directions is only of one dimension, in solid 
space it is of two dimensions. But here also 
there are distance-relations, and the aggregate 
of directions may be divided into parts which 



PHILOSOPHY OF THE PURE SCIENCES 373 

have quantity. For example, the directions 
which start from the vertex of this cone are 
divided into those which go inside the cone, 
and those which go outside the cone. The 
part of the aggregate which is inside the cone 
is called a solid angle. Now in those spaces 
of three dimensions which have the property of 
elementary flatness, the whole amount of solid 
angle round one point is equal to the whole 
amount round another point. Although the 
space need not be exactly similar to itself in 
all parts, yet the aggregate of directions round 
one point is exactly similar to the aggregate 
of directions round another point, if the space 
has the property of elementary flatness. 

How does Euclid assume this postulate of 
Elementary Flatness ? In his fourth postulate 
he has expressed it so simply and clearly that 
you will wonder how anybody could make all 
this fuss. He says, " All right angles are equal." 

Why could I not have adopted this at once, 
and saved a great deal of trouble ? Because it 
assumes the knowledge of a surface possessing 
the property of elementary flatness in all its 
points. Unless such a surface is first made 
out to exist, and the definition of a right angle 
is restricted to lines drawn upon it for there 
is no necessity for the word straight in that 
definition the postulate in Euclid's form is 
obviously not true. I can make two lines cross 



374 LECTURES AND ESSAYS ; 

at the vertex of a cone so that the four adjacent 
angles shall be equal, and yet not one of them 
equal to a right angle. 

I pass on to the third postulate of the 
science of space the postulate of Super- 
position. According to this postulate a body 
can be moved about in space without altering 
its size or shape. This seems obvious enough, 
but it is worth while to examine a little closely 
into the meaning of it. We must define what 
we mean by size and by shape. When we say 
that a body can be moved about without 
altering its size, we mean that it can be so 
moved as to keep unaltered the length of all 
the lines in it. This postulate therefore in- 
volves that lines can be compared in respect of 
magnitude, or that they have a length in- 
dependent of position ; precisely as the former 
one involved the comparison of angular magni- 
tudes. And when we say that a body can be 
moved about without altering its shape, we 
mean that it can be so moved as to keep 
unaltered all the angles in it. It is not 
necessary to make mention of the motion of a 
body, although that is the easiest way of 
expressing and of conceiving this postulate ; 
but we may, if we like, express it entirely in 
terms which belong to space, and that we 
should do in this way. Suppose a figure to 
have been constructed in some portion of 



PHILOSOPHY OF THE PURE SCIENCES 375 

space ; say that a triangle has been drawn 
whose sides are the shortest distances between 
its angular points. Then if in any other 
portion of space two points are taken whose 
shortest distance is equal to a side of the 
triangle, and at one of them an angle is made 
equal to one of the angles adjacent to that side, 
and a line of shortest distance drawn equal to 
the corresponding side of the original triangle, 
the distance from the extremity of this to the 
other of the two points will be equal to the 
third side of the original triangle, and the two 
will be equal in all respects ; or generally, if a 
figure has been constructed anywhere, another 
figure, with all its lines and all its angles equal 
to the corresponding lines and angles of the 
first, can be constructed anywhere else. Now 
this is exactly what is meant by the principle 
of superposition employed by Euclid to prove 
the proposition that I have just mentioned. 
And we may state it again in this short form 
All parts of space are exactly alike. 

But this postulate carries with it a most 
important consequence. It enables us to make 
a pair of most fundamental definitions those 
of the plane and of the straight line. In order 
to explain how these come out of it when it is 
granted, and how they cannot be made when 
it is not granted, I must here say something 
more about the nature of the postulate itself, 



376 LECTURES AND ESSAYS 

which might otherwise have been left until we 
come to criticise it. 

We have stated the postulate as referring to 
solid space. But a similar property may exist 
in surfaces. Here, for instance, is part of the 
surface of a sphere. If I draw any figure I 
like upon this, I can suppose it to be moved 
about in any way upon the sphere, without 
alteration of its size or shape. If a figure has 
been drawn on any part of the surface of a 
sphere, a figure equal to it in all respects may 
be drawn on any other part of the surface. 
Now I say that this property belongs to the 
surface itself, is a part of its own internal 
economy, and does not depend in any way 
upon its relation to space of three dimensions. 
For I can pull it about and bend it in all 
manner of ways, so as altogether to alter its 
relation to solid space ; and yet, if I do not 
stretch it or tear it, I make no difference 
whatever in the length of any lines upon it, or 
in the size of any angles upon it. 1 I do not in 
any way alter the figures drawn upon it, or the 
possibility of drawing figures upon it, so far as 

1 This figure was made of linen, starched upon a spherical 
surface, and taken off when dry. That mentioned in the next 
paragraph was similarly stretched upon the irregular surface of the 
head of a bust. For durability these models should be made of 
two thicknesses of linen starched together in such a way that the 
fibres of one bisect the angles between the fibres of the other, and 
the edge should be bound by a thin slip of paper. They will then 
retain their curvature unaltered for a long time. 



PHILOSOPHY OF THE PURE SCIENCES 377 

their relations with the stirface itself are con- 
cerned. This property of the surface, then, 
could be ascertained by people who lived 
entirely in it, and were absolutely ignorant of a 
third dimension. As a point-aggregate of two 
dimensions, it has in itself properties deter- 
mining the distance-relations of the points upon 
it, which are absolutely independent of the 
existence of any points which are not upon it. 

Now here is a surface which has not that 
property. You observe that it is not of the 
same shape all over, and that some parts of it 
are more curved than other parts. If you 
drew a figure upon this surface, and then tried 
to move it about, you would find that it was 
impossible to do so without altering the size 
and shape of the figure. Some parts of it 
would have to expand, some to contract, the 
lengths of the lines could not all be kept the 
same, the angles would not hit off together. 
And this property of the surface that its 
parts are different from one another is a 
property of the surface itself, a part of its 
internal economy, absolutely independent of 
any relations it may have with space outside 
of it. For, as with the other one, I can pull 
it about in all sorts of ways, and, so long as I 
do not stretch it or tear it, I make no alteration 
in the length of lines drawn upon it or in the 
size of the angles. 



378 LECTURES AND ESSAYS 

Here, then, is an intrinsic difference between 
these two surfaces, as surfaces. They are both 
point-aggregates of two dimensions ; but the 
points in them have certain relations of distance 
(distance measured always on the surface), and 
these relations of distance are not the same in 
one case as they are in the other. 

The supposed people living in the surface 
and having no idea of a third dimension might, 
without suspecting that third dimension at all, 
make a very accurate determination of the 
nature of their locus in quo. If the people who 
lived on the surface of the sphere were to 
measure the angles of a triangle, they would find 
them to exceed two right angles by a quantity 
proportional to the area of the triangle. This 
excess of the angles above two right angles, 
being divided by the area of the triangle, would 
be found to give exactly the same quotient at 
all parts of the sphere. That quotient is called 
the curvature of the surface ; and we say that 
a sphere is a surface of uniform curvature. 
But if the people living on this irregular surface 
were to do the same thing, they would not find 
quite the same result. The sum of the angles 
would, indeed, differ from two right angles, but 
sometimes in excess and sometimes in defect, 
according to the part of the surface where they 
were. And though for small triangles in any 
one neighbourhood the excess or defect would 



PHILOSOPHY OF THE PURE SCIENCES 379 

be nearly proportional to the area of the 
triangle, yet the quotient obtained by dividing 
this excess or defect by the area of the triangle 
would vary from one part of the surface to 
another. In other words, the curvature of this 
surface varies from point to point ; it is some- 
times positive, sometimes negative, sometimes 
nothing at all. 

But now comes the important difference. 
When I speak of a triangle, what do I suppose 
the sides of that triangle to be ? 

If I take two points near enough together 
upon a surface, and stretch a string between 
them, that string will take up a certain definite 
position upon the surface, marking the line of 
shortest distance from one point to the other. 
Such a line is called a geodesic line. It is a 
line determined by the intrinsic properties of 
the surface, and not by its relations with ex- 
ternal space. The line would still be the 
shortest line, however the surface were pulled 
about without stretching or tearing. A geodesic 
line may be produced, when a piece of it is 
given ; for we may take one of the points, and, 
keeping the string stretched, make it go round 
in a sort of circle until the other end has turned 
through two right angles. The new position 
will then be a prolongation of the same geodesic 
line. 

In speaking of a triangle, then, I meant a 



380 LECTURES AND ESSAYS 

triangle whose sides are geodesic lines. But 
in the case of a spherical surface or, more 
generally, of a surface of constant curvature 
these geodesic lines have another and most im- 
portant property. They are straight, so far as 
the surface is concerned. On this surface a 
figure may be moved about without altering its 
size or shape. It is possible, therefore, to 
draw a line which shall be of the same shape 
all along and on both sides. That is to say, if 
you take a piece of the surface on one side of 
such a line, you may slide it all along the line 
and it will fit ; and you may turn it round and 
apply it to the other side, and it will fit there 
also. This is Leibnitz's definition of a straight 
line, and, you see, it has no meaning except in 
the case of a surface of constant curvature, a 
surface all parts of which are alike. 

Now let us consider the corresponding 
things in solid space. In this also we may 
have geodesic lines ; namely, lines formed by 
stretching a string between two points. But 
we may also have geodesic surfaces ; and they 
are produced in this manner. Suppose we 
have a point on a surface, and this surface 
possesses the property of elementary flatness. 
Then among all the directions of starting from 
the point, there are some which start in the 
surface, and do not make an angle with it. 
Let all these be prolonged into geodesies ; then 



PHILOSOPHY OF THE PURE SCIENCES 381 

we may imagine one of these geodesies to 
travel round and coincide with all the others in 
turn. In so doing it will trace out a surface 
which is called a geodesic surface. Now in 
the particular case where a space of three 
dimensions has the property of superposition, 
or is all over alike, these geodesic surfaces are 
planes. That is to say, since the space is all 
over alike, these surfaces are also of the same 
shape all over and on both sides ; which is 
Leibnitz's definition of a plane. If you take 
a piece of space on one side of such a plane, 
partly bounded by the plane, you may slide it 
all over the plane and it will fit ; and you may 
turn it round and apply it to the other side, 
and it will fit there also. Now it is clear that 
this definition will have no meaning unless the 
third postulate be granted. So we may say 
that when the postulate of Superposition is 
true, then there are planes and straight lines ; 
and they are defined as being of the same shape 
throughout and on both sides. 

It is found that the whole geometry of a 
space of three dimensions is known when we 
know the curvature of three geodesic surfaces 
at every point. The third postulate requires 
that the curvature of all geodesic surfaces 
should be everywhere equal to the same quantity. 

I pass to the fourth postulate, which I call 
the postulate of Similarity. According to this 



382 LECTURES AND ESSAYS 

postulate, any figure may be magnified or 
diminished in any degree without altering its 
shape. If any figure has been constructed in 
one part of space, it may be reconstructed to 
any scale whatever in any other part of space, 
so that no one of the angles shall be altered, 
though all the lengths of lines will of course be 
altered. This seems to be a sufficiently obvious 
induction from experience ; for we have all 
frequently seen different sizes of the same 
shape ; and it has the advantage of embodying 
the fifth and sixth of Euclid's postulates in a 
single principle, which bears a great resemblance 
in form to that of Superposition, and may be 
used in the same manner. It is easy to show 
that it involves the two postulates of Euclid : 
" Two straight lines cannot enclose a space," 
and " Lines in one plane which never meet 
make equal angles with every other line." 

This fourth postulate is equivalent to the 
assumption that the constant curvature of the 
geodesic surfaces is zero ; or the third and 
fourth may be put together, and we shall then 
say that the three curvatures of space are all of 
them zero at every point. 

The supposition made by Lobatchewsky 
was, that the three first postulates were true, 
but not the fourth. Of the two Euclidean 
postulates included in this, he admitted one, 
viz. that two straight lines cannot enclose a 



PHILOSOPHY OF THE PURE SCIENCES 383 

space, or that two lines which once diverge go 
on diverging for ever. But he left out the 
postulate about parallels, which may be stated 
in this form. If through a point outside of a 
straight line there be drawn another, indefinitely 
produced both ways ; and if we turn this second 
one round so as to make the point of intersec- 
tion travel along the first line, then at the very 
instant that this point of intersection disappears 
at one end it will reappear at the other, and 
there is only one position in which the lines do 
not intersect Lobatchewsky supposed, instead, 
that there was a finite angle through which the 
second line must be turned after the point of 
intersection had disappeared at one end, before 
it reappeared at the other. For all positions 
of the second line within this angle there is 
then no intersection. In the two limiting 
positions, when the lines have just done meet- 
ing at one end, and when they are just going 
to meet at the other, they are called parallel ; 
so that two lines can be drawn through a fixed 
point parallel to a given straight line. The 
angle between these two depends in a certain 
way upon the distance of the point from the 
line. The sum of the angles of a triangle is 
less than two right angles by a quantity pro- 
portional to the area of the triangle. The 
whole of this geometry is worked out in the 
style of Euclid, and the most interesting con- 



384 LECTURES AND ESSAYS 

elusions are arrived at ; particularly in the 
theory of solid space, in which a surface turns 
up which is not plane relatively to that space, 
but which, for purposes of drawing figures upon 
it, is identical with the Euclidean plane. 

It was Riemann, however, who first accom- 
plished the task of analysing all the assump- 
tions of geometry, and showing which of them 
were independent. This very disentangling 
and separation of them is sufficient to deprive 
them for the geometer of their exactness and 
necessity; for the process by which it is 
effected consists in showing the possibility of 
conceiving these suppositions one by one to be 
untrue ; whereby it is clearly made out how 
much is supposed. But it may be worth while 
to state formally the case for and against them. 

When it is maintained that we know these 
postulates to be universally true, in virtue of 
certain deliverances of our consciousness, it is 
implied that these deliverances could not exist, 
except upon the supposition that the postulates 
are true. If it can be shown, then, from ex- 
perience that our consciousness would tell us 
exactly the same things if the postulates are 
not true, the ground of their validity will be 
taken away. But this is a very easy thing to 
show. 

That same faculty which tells you that 
space is continuous tells you that this water is 



PHILOSOPHY OF THE PURE SCIENCES 385 

continuous, and that the motion perceived in a 
wheel of life is continuous. Now we happen 
to know that if we could magnify this water as 
much again as the best microscopes can magnify 
it, we should perceive its granular structure. 
And what happens in a wheel of life is dis- 
covered by stopping the machine. Even apart, 
then, from our knowledge of the way nerves 
act in carrying messages, it appears that we 
have no means of knowing anything more 
about an aggregate than that it is too fine- 
grained for us to perceive its discontinuity, if it 
has any. 

Nor can we, in general, receive a conception 
as positive knowledge which is itself founded 
merely upon inaction. For the conception of 
a continuous thing is of that which looks just 
the same however much you magnify it. We 
may conceive the magnifying to go on to a 
certain extent without change, and then, as it 
were, leave it going on, without taking the 
trouble to doubt about the changes that may 
ensue. 

In regard to the second postulate, we have 
merely to point to the example of polished 
surfaces. The smoothest surface that can be 
made is the one most completely covered with 
the minutest ruts and furrows. Yet geometrical 
constructions can be made with extreme accuracy 
upon such a surface, on the supposition that it 
VOL. I 2 C 



386 LECTURES AND ESSAYS 

is an exact plane. If, therefore, the sharp 
points, edges, and furrows of space are only 
small enough, there will be nothing to hinder 
our conviction of its elementary flatness. It 
has even been remarked by Riemann that we 
must not shrink from this supposition if it is 
found useful in explaining physical phenomena. 

The first two postulates may therefore be 
doubted on the side of the very small. We 
may put the third and fourth together, and 
doubt them on the side of the very great. For 
if the property of elementary flatness exist on 
the average, and the deviations from it being, 
as we have supposed, too small to be perceived, 
then, whatever were the true nature of space, 
we should have exactly the conceptions of it 
which we now have, if only the regions we can 
get at were small in comparison with the 
areas of curvature. If we suppose the curvature 
to vary in an irregular manner, the effect of it 
might be very considerable in a triangle formed 
by the nearest fixed stars ; but if we suppose 
it approximately uniform to the limit of tele- 
scopic reach, it will be restricted to very much 
narrower limits. I cannot perhaps do better 
than conclude by describing to you as well as 
I can what is the nature of things on the 
supposition that the curvature of all space is 
nearly uniform and positive. 

In this case the Universe, as known, becomes 



PHILOSOPHY OF THE PURE SCIENCES 387 

again a valid conception ; for the extent of 
space is a finite number of cubic miles. 1 And 
this comes about in a curious way. If you 
were to start in any direction whatever, and 
move in that direction in a perfect straight line 
according to the definition of Leibnitz ; after 
travelling a most prodigious distance, to which 
the parallactic unit 200,000 times the 
diameter of the earth's orbit would be only a 
few steps, you would arrive at this place. 
Only, if you had started upwards, you would 
appear from below. Now, one of two things 
would be true. Either, when you had got half- 
way on your journey, you came to a place that 
is opposite to this, and which you must have 
gone through, whatever direction you started 
in ; or else all paths you could have taken 
diverge entirely from each other till they meet 
again at this place. In the former case, every 
two straight lines in a plane meet in two points, 
in the latter they meet only in one. Upon 
this supposition of a positive curvature, the 
whole of geometry is far more complete and 
interesting ; the principle of duality, instead of 
half breaking down over metric relations, 
applies to all propositions without exception. 
In fact, I do not mind confessing that I 

1 The assumptions here made about the Zusammenhang of 
space are the simplest ones, but even the finite extent does not 
follow necessarily from uniform positive curvature, as Riemann 
seems to have supposed. 



388 LECTURES AND ESSAYS 

personally have often found relief from the 
dreary infinities of homaloidal space in the 
consoling hope that, after all, this other may 
be the true state of things. 



IV. THE UNIVERSAL STATEMENTS OF 

ARITHMETIC 

WE have now to consider a series of alleged 
universal statements, the truth of which nobody 
has ever doubted. They are statements be- 
longing to arithmetic, to the science of quantity, 
to pure logic, and to a branch of the science of 
space which is of quite recent origin, which 
applies to other objects besides space, and 
is called the analysis of position. I shall 
endeavour to show that the case of these state- 
ments is entirely different from that of the state- 
ments about space which I examined in my last 
lecture. There were four of those statements : 
that the space of three dimensions which we 
perceive is a continuous aggregate of points, 
that it is flat in its smallest parts, that figures 
may be moved in it without alteration of size 
or shape, and that similar figures of different 
sizes may be constructed in it. And the 
conclusion which I endeavoured to establish 
about these statements was that, for all we 
know, any or all of them may be false. In 
regard to the statements we have now to 



PHILOSOPHY OF THE PURE SCIENCES 389 

examine, I shall not maintain a similar doctrine ; 
I shall only maintain that, for all we know, 
there may be times and places where they are 
unmeaning and inapplicable. If I am asked 
what two and two make I shall not reply that 
it depends upon circumstances, and that they 
make sometimes three and sometimes five ; but 
I shall endeavour to show that unless our 
experience had certain definite characters there 
would be no such conception as two, or three, 
or four, and still less such a conception as the 
adding together of two numbers ; and that we 
have no warrant for the absolute universality of 
these definite characters of experience. 

In the first place it is clear that the moment 
we use language at all, we may make state- 
ments which are apparently universal, but which 
really only assign the meaning of words. 
Whenever we have called a thing by two 
names, so that every individual of a certain 
class bears the name A and also the name B, 
then we may affirm the apparently universal 
proposition that every A is B. But it is really 
only the particular proposition that the name 
A has been conventionally settled to have the 
same meaning as the name B. I may, for 
example, enunciate the proposition that all 
depth is profundity, and all profundity is 
depth. This statement appears to be of 
universal generality ; and nobody doubts that 



390 LECTURES AND ESSAYS 

itjis true. But for all that it is not a statement 
of some fact which is true of nature as a whole ; 
it is only a statement about the use of certain 
words in the English language. In this case 
the meaning of the two words is co-extensive ; 
one means exactly as much as, and no more 
than, the other. But if we suppose the word 
crow to mean a black bird having certain 
peculiarities of structure, the statement, "All 
crows are black," is in a similar case. For the 
word black has part of the meaning of the word 
crow ; and the proposition only states this 
connection between the two words. Are the 
propositions of arithmetic, then, mere statements 
about the meanings of words ? No ; but these 
examples will help us to understand them. 
Language is part of the apparatus of thought ; 
it is that by which I am able to talk to myself. 
But it is not all of the apparatus of thought ; 
and just as these apparently general pro- 
positions, "All crows are black," "All depth 
is profundity," are really statements about 
language, so I shall endeavour to show that 
the statements of arithmetic are really state- 
ments about certain other apparatus of thought. 
We know that six and three are nine. 
Wherever we find six things, if we put three 
things to them there are nine things altogether. 
The terms are so simple and so familiar that it 
seems as if there were no more to be said, as if 



PHILOSOPHY OF THE PURE SCIENCES 391 

we could not examine into the nature of these 
statements any further. 

No more there is, if we are obliged to take 
words as they stand, with the complex mean- 
ings which at present belong to them. But the 
real fact is that the meanings of six and three 
are already complex meanings, and are capable 
of being resolved into their elements. This 
resolution is due I believe equally and in- 
dependently to two great living mathema- 
ticians, by whose other achievements this 
country has retained the scientific position 
which Newton won for her at a time of fierce 
competition when no ordinary genius could 
possibly have attained it. The conception of 
number, as represented by that word and also 
by the particular signs, three, six, and so on, 
has been shown to embody in itself a certain 
proposition, upon the repetition of which the 
whole science of arithmetic is based. By means 
of this remark of Cayley and Sylvester, we are 
able to assign the true nature of arithmetical 
propositions, and to pass from thence by an 
obvious analogy to those other cases that we 
have to consider. 

What do I do to find out that a certain set 
of things are six in number ? I count them ; 
and all counting, like the names of numbers, 
belongs first to the fingers. Now this is the 
operation of counting ; I take my fingers in a 



392 LECTURES AND ESSAYS 

certain definite order say I begin with the 
thumb of each hand, and with the right hand. 
Then I lay my fingers in this order upon the 
things to be counted ; or if they are too far 
away, I imagine that I lay them. And I 
observe what finger it is that is laid upon the 
last thing, and call the things by the name of 
this finger. In the present case it is the thumb 
of my left hand ; and if we were savages that 
thumb would be called six. At any rate, if 
the order of my fingers is settled beforehand, 
and known to everybody, I can quite easily 
make the statement, " Here are six things," by 
holding up the thumb of my left hand. 

But, if I have only gone through this process 
once, there is already a great assumption made. 
For, although the order in which I used my 
fingers is fixed, there is nothing at all said 
about the order in which the things are touched 
by them. It is assumed that if the things are 
taken in any other order and applied to my 
fingers, the last one so touched will be the 
thumb of my left hand. If this were not true, 
or were not assumed, the word " number " could 
not have its meaning. There is implied and 
bound up in that word the assumption that a 
group of things comes ultimately to the same 
finger in whatever order they are counted. 
This is the proposition of which I spoke as 
the foundation of the whole science of number. 



PHILOSOPHY OF THE PURE SCIENCES 393 

It -.is involved not only in the general term 
" number," but also in all the particular names 
of numbers ; and not only in these words, but 
in the sign of holding up a finger to indicate 
how many things there are. 

Let us now look in this light at the state- 
ment that six and three are nine. I have 
counted a group of things and come to the 
conclusion that there are six of them. I have 
already said, therefore, that they may be counted 
in any order whatever and will come to the 
same number, six. I have counted another 
distinct group, and come to the conclusion that 
there are three of them. Then I put them 
all together and count them. Now, without 
seeing or knowing any more of the things than 
is implied in the previous statements, I can 
already count them in a certain order with my 
fingers. For I will first suppose the six to be 
counted ; the last of them, by hypothesis, is 
attached in thought to the thumb of my left 
hand. Now I will count the other three ; they 
are then attached, by hypothesis, to the first 
three fingers of my right hand. I can now go 
on counting the aggregate group by attaching 
to these three fingers the successive fingers of 
my left hand ; for thus I shall attach the 
remaining three things to those fingers. I find 
in this way that the last of them comes to the 
fourth finger of my left hand, counting the 



394 LECTURES AND ESSAYS 

thumb as first ; and I know, therefore, that if 
the aggregate group has any number at all, that 
number must be nine. 

But this is an operation performed on my 
fingers ; and the statement that we have 
founded on it must therefore be, at least in 
part, a statement about my counting apparatus. 
We may easily understand what is meant by 
saying that six and three are nine on my fingers, 
independently of any other things than those ; 
this is a particular statement only. The state- 
ment we want to examine is that this is equally 
true of any two distinct groups whatever of six 
things and three things, which appears to be a 
universal statement. Now I say that this 
latter statement can be resolved into two as 
follows : 

1. The particular statement aforesaid : six 
and three are nine on my fingers. 

2. If there is a group of things which can 
be attached to certain of my fingers, one to 
each, and another group of things which can 
be attached to certain other of my fingers, one 
to each, then the compound group can be 
attached to the whole set of my fingers that 
have been used, one to each. 

Now this latter, it seems to me, is a tautology 
or identical proposition, depending merely upon 
the properties of language. The arithmetical 
proposition, then, is resolved or analysed in 



PHILOSOPHY OF THE PURE SCIENCES 395 

this way into two parts a particular statement 
about my counting apparatus, and a particular 
statement about language ; and it is not really 
general at all. But this, it is important to 
notice, is not the complete solution of the 
problem ; there is a certain part of it reserved. 
For I only arrive at the number nine by certain 
definite ways of counting ; I must count the 
six things first and then the three things after 
them. And I only arrive at the result that if 
the aggregate group of things has any number 
at all, that number is nine. It is not yet 
proved that they may be counted in any order 
whatever, and will always come to that number. 
Here, then, we are driven back to consider the 
nature of that fundamental assumption that the 
number of any finite group of distinct things is 
independent of the order of counting. Here is 
a proposition apparently still more general than 
any statement about the sum of two numbers. 
Do I or do I not know that this is true of very 
large numbers ? Consider, for example, the 
molecules of water in this glass. According 
to Sir William Thomson, if a drop of water 
were magnified to the size of the earth it would 
appear coarser-grained than a heap of small 
shot, and finer-grained than a heap of cricket- 
balls. We may therefore soon find that the 
number of molecules in this glass very far 
transcends our powers of conception. Do I 



396 LECTURES AND ESSAYS 

know that if these molecules were counted in 
a certain order, and then counted over again in 
a certain other order, the results of these two 
countings would be the same ? For the opera- 
tions are absolutely impossible in anybody's 
lifetime. Can I know anything about the 
equivalence of two impossible operations, neither 
of which can be conceived except in a symbolic 
way ? And if I do, how is it possible for this 
knowledge to come from experience ? 

I reply that I do know it ; that such know- 
ledge of things as there is in it has come from 
experience ; and that, in fact, it is made up of 
a particular statement and a conventional use of 
words. These views will appear paradoxical ; 
but the justification of them is to be found in 
the analysis of that fundamental assumption 
which lies at the basis of the idea of number. 

In the first place I shall prove this funda- 
mental assumption in the case of the number six 
that is to say, I shall show that it is involved 
in suppositions which are already made before 
there is any question of it. The proposition 
we have to prove is : if a group of distinct 
things comes to six when counted in a certain 
order, it will come to six when counted in any 
other order. I say that the proposition is 
involved in the meaning of the phrase distinct 
things, and may be got out of it by help of a 
particular observation. 



PHILOSOPHY OF THE PURE SCIENCES 397 

What, then, is meant by " a group of distinct 
things " ? That they are all distinct from one 
another, or that any one and any other of them 
make two. That is, if they are attached to 
two of my fingers in a certain order, they can 
also be attached to the same two fingers in the 
other order. Now, for simplicity, let us take 
the letters in the word spring, and count them 
first as they occur in that word and then in 
the alphabetical order. I say that, merely on 
the supposition that they are distinct from one 
another, I can change one order into the other 
while I use the same fingers to attach them to. 

123456 
SPRING 
G P R I N S 
G I R P N S 
G I N P R S 

In the new order I want G to be first ; now 
the letters G and S are by hypothesis distinct, 
they are two letters. I can therefore inter- 
change the fingers to which they are attached 
without using more or fewer fingers than before. 
The same thing is true by hypothesis of I and 
P, and finally of N and R. By these steps, 
then, I have changed one order into the other 
without altering the fingers used in counting 
that is, without altering the number. And 
each of these steps is involved in the meaning 



398 LECTURES AND ESSAYS 

of the words distinct things that is, it is made 
possible by the assumptions which these words 
involve. But now observe further : how do I 
know that I can make enough steps to effect 
the whole change required ? In this way. It 
is given to me in the hypothesis that the things 
have been counted once ; I can therefore go to 
them one by one till I come to the end. But 
as I go to each one I can substitute by this 
process the new one which is wanted in its 
stead in such a way that the required new 
order shall hold good behind me. Thus you 
see that all the steps are involved in the word 
distinct, by the help of an observation on two 
of my fingers ; and that the possibility of a 
sufficient number of them to effect the change 
is involved in the hypothesis that the things 
have been once counted. Here I have two 
distinct statements : the first is that the things 
are distinct, and have been once counted as 
six ; the second is that in another order they 
come to the same. When I examine into the 
meaning of these, I find that they are not 
statements of different facts, but different state- 
ments of the same facts. That one statement 
is true, or that the other statement is true, 
that is a matter of experience ; but that if one 
is true the other is true, that is a matter of 
language. 

I have only spoken, however, of the particular 



PHILOSOPHY OF THE PURE SCIENCES 399 

number six ; how am I to extend these remarks 
to numbers which cannot be counted, like the 
number of molecules in this glass of water ? 
In the first place we all know that cultivated 
races do not count directly with their fingers, 
but with the names of them with the words 
one, two, three, four. Next, this system of 
names has been extended indefinitely, by a 
process to which no end can be conceived. 
But the remarks that we have made about 
finger-counting will hold good in every case in 
which the actual counting can be performed. 
Now in those cases in which this is not true 
in the case of a billion, for example we have 
two statements made, neither of which can be 
adequately represented in thought, but which, 
in so far as they can be represented, are identi- 
cal statements. That there are a billion grains 
of sand in a certain heap, provided they be 
counted in a certain order this is a supposi- 
tion which can only be made symbolically. 
But in so far as it can be made, it is the same 
supposition as that they also come to a billion 
in any other order. Any step towards the 
representation in thought of the one statement 
is the same step towards the representation in 
thought of the other ; and I do not know any 
other way in which two symbolic statements 
can be statements of the same facts. Pure 
water is the same thing as aqua pura ; and 



400 LECTURES AND ESSAYS 

wherever there are seventy thousand million 
tons of pure water there are seventy thousand 
million tons of aqua pura. I know that to be 
true, but it is not a statement of fact : it is a 
statement about language, notwithstanding that 
the language is used to symbolise that which 
cannot be actually represented in thought. So 
when I say of these molecules of water, " If 
they are distinct things, the number of them 
counted in one order is equal to the number of 
them counted in any other order," I make a 
supposition which I cannot realise in thought. 
I cannot possibly call up those molecules two 
and two to observe their distinctness. The 
supposition is only represented symbolically by 
language ; but the statement that follows it is 
the same supposition represented symbolically 
by other language ; and the equivalence of the 
two is, after all, a statement about language 
and not about facts. 

But you will say, I do know that these 
molecules are distinct things; and so I am 
able to make these equivalent statements about 
them. I know that they have a definite number, 
which is the same however they are counted. 

I. Yes, I know that they are distinct things ; 
but only by inference, on the assumption of the 
uniformity of nature ; and about that there is 
more to be said. The distinctness of things- 
the fact that one thing and one thing make two 



PHILOSOPHY OF THE PURE SCIENCES 401 

this belongs to our experience. It is a fact 
that impressions hang together in groups which 
persist as groups, and in virtue of this persistence 
we call them things. So long as our experience 
consists of things, we may build out of it the 
conceptions of number ; and the nature and 
connection of these conceptions are determined 
by the primary sensation of things as individuals. 
Now there can, I think, be no doubt that the 
experience of a hundred or a hundred and fifty 
million years has so modified our nervous 
systems that without total disruption of them 
we cannot cease to aggregate our perceptions 
into more or less persistent groups ; the con- 
tinuity of things has become a form of sense. 
If we were placed in circumstances where these 
aggregations of feeling were not naturally pro- 
duced, where perceptible things were not con- 
tinuous in their changes, we should go on 
perceiving chaos as made of individual things 
for at least some time. But the perception 
would be a false one, and in acting upon it we 
should come to grief. Meanwhile, however, 
the science of number would be perfectly true 
of our perceptions, though practically inappli- 
cable to the world. 

To sum up, then, we carry about with us 

a certain apparatus of counting, which was 

primarily our fingers, but is now extended into 

a series of signs which we can remember in a 

VOL. I 2 D 



402 LECTURES AND ESSAYS 

certain order the names of numbers. Our 
language is so formed as to make us able to 
talk to ourselves about the results of counting. 
The propositions of arithmetic are compounded 
in general of two parts ; a statement about the 
counting apparatus, and a statement about the 
different ways of describing its results. 

But before quite leaving this let us fix our 
attention for a short time on the mode of use 
of the counting apparatus. The operation of 
counting a certain group of things consists in 
assigning one of these numeral words to each 
of them ; in establishing a correspondence 
between two groups, so that to every thing or 
element of the one group is assigned one 
particular thing or element of the other. There 
is here a one-to-one correspondence of two 
aggregates, one of which is carried about as a 
standard ; and the propositions arrived at are 
always of this kind : if a group of things can 
have this correspondence with the standard 
group, then those properties of the standard 
group which are carried over by the corre- 
spondence will belong to the new group. Now 
this establishment of correspondence between 
two aggregates and investigation of the pro- 
perties that are carried over by the correspond- 
ence may be called the central idea of modern 
mathematics ; it runs through the whole of the 
pure science and of its applications. It may 



PHILOSOPHY OF THE PURE SCIENCES 403 

be conceived, therefore, that propositions which 
are apparently as general and certain as those 
we have discussed to-day may be analysed in 
the same manner, and shown to be really state- 
ments about the apparatus of thought 

In my second lecture I endeavoured to 
explain the difference between a discrete and a 
continuous aggregate. In a row of marbles, 
which is a discrete aggregate, we can find 
between any two marbles only a finite number 
of others, and sometimes none at all. But if 
two points are taken on a line, the hypothesis 
of continuity supposes that there is no end to 
the number of intermediate points that we can 
find. Precisely the same difference holds good 
between number and continuous quantity. The 
several marbles, beginning at any one of them, 
may be numbered one, two, three, etc. ; and the 
number attached to each marble will be the 
number of marbles from the starting-point to 
that marble inclusive. If the points on a line 
are regarded as forming a continuous aggregate, 
then lengths measured along the line from an 
arbitrary point on it are called continuous quanti- 
ties. So also, if the instants of time are re- 
garded as forming a continuous aggregate 
that is, if we suppose that between any two 
instants there is no end to the number of inter- 
mediate ones that might be found then 
intervals or lengths of time will be continuous 



404 LECTURES AND ESSAYS 

quantities. And just as we may attach our 
numbers one by one to the marbles which form 
a discrete aggregate, so we may attach continu- 
ous quantities (or shortly quantities} one by 
one to the points which form a continuous 
aggregate. Thus to the point P will be attached 
the quantity or length A P. And we see thus that 

A" j, B 

between any two quantities there may be found 
an infinite number of intermediate quantities, 
while between two numbers there can only be 
found a finite number of intermediate numbers, 
and sometimes none at all. That is to say, 
continuous quantities form a continuous aggre- 
gate, while numbers form a discrete aggregate. 
Thus the science of quantity is a totally different 
thing from the science of number. 

Notwithstanding that this difference was 
clearly perceived by the ancients, attempts have 
constantly been made by the moderns to treat 
the two sciences as one, and to found the science 
of quantity upon the science of number. The 
method is to treat rational fractions as a neces- 
sary extension of numerical division, and then 
to deal with incommensurable quantities by 
way of continual approximation. In the science 
of number, while five-sevenths of fourteen has a 
meaning, namely, ten, five-sevenths of twelve is 



PHILOSOPHY OF THE PURE SCIENCES 405 

nonsense. Let us then treat it as if it were sense, 
and see what comes of it A repetition of this 
process with every impossible operation that 
occurs is supposed to lead in time to continuous 
quantities. The results of such attempts are 
the substitution of algebra for the fifth book of 
Euclid or some equivalent doctrine of continuous 
ratios, and the substitution of the differential 
calculus for the method of fluxions. For my 
own part, I believe this method to be logically 
false and educationally mischievous. For 
reasons too long to give here, I do not believe 
that the provisional use of unmeaning arith- 
metical symbols can ever lead to the science of 
quantity ; and I feel sure that the attempt to 
found it on such abstractions obscures its true 
physical nature. The science of number is 
founded on the hypothesis of the distinctness 
of things ; the science of quantity is founded 
on the totally different hypothesis of continuity. 
Nevertheless, the relations between the two 
sciences are very close and extensive. The 
scale of numbers is used, as we shall see, in 
forming the mental apparatus of the scale of 
quantities, and the fundamental conception of 
equality of ratios is so defined that it can be 
reasoned about in the terms of arithmetic. 1 

1 Defining a fraction as the ratio of two numbers, Euclid's 
definition of proportion is equivalent to the following : Two 
quantity-ratios are equal if every fraction is either le$s than both, 
equal to both, or greater than both of them. 



406 LECTURES AND ESSAYS 

The operations of addition and subtraction of 
quantities are closely analogous to the operations 
of the same name performed on numbers and 
follow the same laws. The composition of 
ratios includes numerical multiplication as a 
particular case, and combines in the same way 
with addition and subtraction. So close and 
far-reaching is this analogy that the processes 
and results of the two sciences are expressed in 
the same language, verbal and symbolical, while 
no confusion is produced by this ambiguity of 
meaning, except in the minds of those who try 
to make familiarity with language do duty for 
knowledge of things. 

Just as in operations of counting there is a 
comparison of some aggregate of discrete things 
with a scale of numbers carried about with us 
as a standard, so in operations of measuring, 
real or ideal, there is comparison of some piece 
of a continuous thing with a scale of quantities. 
We may best understand this scale by the 
example of time. To indicate exactly the time 
elapsed from the beginning of the century to 
some particular instant of to-day, it is necessary 
and sufficient to name the date and point to 
the hands of a clock which was going right and 
was stopped at that instant. This is equivalent 
to saying that the whole quantity of time con- 
sists, first, of a certain number of hours, specified 
by comparison with the scale of numbers 



PHILOSOPHY OF THE PURE SCIENCES 407 

already constructed, and, secondly, of a certain 
part of an hour, which being a continuous 
quantity can only be adequately specified by 
another continuous quantity representing it on 
some definite scale. In the present case this is 
conveniently taken to be the arc of a circle 
described by the point of the minute-hand. 
On the scale in which that whole circumference 
represents an hour, this arc represents the 
portion of an hour which remains to be added. 
With the help of the scale of numbers, then, 
any assigned continuous quantity will serve as a 
standard by which the whole scale of quantities 
may be represented. And when we assert that 
any theorem, e.g. the binomial theorem, is true 
of all quantities whatever, whether of length, of 
time, of weight, or of intensity, we really assert 
two things : first, this theorem is true on the 
standard ; secondly, relations of the measures 
of quantities on the standard are relations of 
the quantities themselves. The first is (in 
regard to the kind of quantity) a particular 
statement ; the second is involved in the mean- 
ing of the words " quantity " and " measure- 
ment." 

But the most familiar and perhaps the most 
natural form of the scale of quantities is that in 
which it is supposed to be marked off on a 
straight line, starting from an arbitrarily 
assumed point which is called the origin. If 



4 o8 LECTURES AND ESSAYS 

we make the four assumptions of Euclidean or 
parabolic geometry, the position of every point 
in space may be specified by three quantities 
marked off on three straight lines at right 
angles to each other, their common point of in- 
tersection being taken as origin, and the direc- 
tion in which each of the quantities is measured 
being also assigned. Namely, these three 
quantities are the distances from the origin to 
the feet of perpendiculars let fall from the point 
to be specified on the three straight lines re- 
spectively. In all space of three dimensions the 
position of a point may be specified in general by 
a set of three quantities ; but two or more points 
may belong to the same set of quantities, or two 
or more sets may specify the same point ; and 
there may be exceptional sets specifying not 
one point, but all the points on a curve or sur- 
face, and exceptional points belonging to an 
infinite number of sets of quantities subject to 
some condition. There are three kinds of 
space of three dimensions in which this specifi- 
cation is unique, one point for one set of 
quantities, one set of quantities for every point, 
and without any exceptional cases. These three 
are the hypothetical space of Euclid, with no 
curvature ; the space of Lobatchewsky, with 
constant negative curvature ; and the space I 
described at the end of my second lecture, 
with constant positive curvature. In only one 



PHILOSOPHY OF THE PURE SCIENCES 409 

of these, the space of Euclid, are the three 
quantities specifying a point actual distances of 
the point from three planes. In this alone 
we have a simple and direct representation of 
the scale of quantities. Now, if we remember 
that the scale of quantities is a mental appa- 
ratus depending only on the first of our four 
assumptions about space, we may see in this 
distinctive property of Euclidean space a prob- 
able origin for the curious opinion that it has 
some a priori probability or even certainty, as 
the true character of the universe we inhabit, 
over and above the observation that within the 
limits of experience that universe does approxi- 
mately conform to its rules. It has even been 
maintained that if our space has curvature, it 
must be contained in a space of more dimen- 
sions and no curvature. I can think of no 
grounds for such an opinion except the property 
of flat spaces which I have just mentioned. 



END OF VOL. I 



Printed by R. & R. CLARK, LIMITED, Edinburgh 



WORKS BY W. K. CLIFFORD. 

Elements of Dynamic : an Introduction to the Study of 
Motion and Rest in Solid and Fluid Bodies. Part I. 
Kinematic. Book I. -1 1 1. Crown 8vo. 73. 6d. 
Book IV. and Appendix. Crown 8vo. 6s. 

Mathematical Papers. Edited by ROBERT TUCKER. 
Introduction by H. J. S. SMITH. 8vo. 305. 

WORKS BY LESLIE STEPHEN. 

Life and Letters of J. R. Green. 8vo. [shortly. 

Samuel Johnson. Crown 8vo. is. 6d. Sewed, is. 
Life of Pope. Crown 8vo. is. 6d. Sewed, is. 
Life of Swift. Crown 8vo. is. 6d. Sewed, is. 

WORKS BY SIR F. POLLOCK, Bart. 

Oxford Lectures and other Discourses. 8vo. 95. 
Essays in Jurisprudence and Ethics. 8vo. IDS. 6d. 
The Land Laws. Crown 8vo. 25. 6d. 
An Introduction to the History of the Science of 

Politics. Crown 8vo. 2s. 6d. 

Leading Cases done into English. Crown 8vo. 35. 6d. 
A First Book of Jurisprudence. Crown 8vo. 6s. 

MACMILLAN AND CO., LTD., LONDON. 



A SELECTION FROM 

THE EVERSLEY SERIES. 

Globe &vo. Cloth. $s. per volume. 

Matthew Arnolds' Works. 

POEMS. 3 vols. ESSAYS IN CRITICISM. First Series. 

ESSAYS IN CRITICISM. Second Series. 

AMERICAN DISCOURSES. LETTERS. 2 vols. 

Dean Church's Mlcellaneous Writings. Collected Edition. 8 vols. 

MISCELLANEOUS ESSAYS. | DANTE ; another Essays. 



ST. ANSELM. | BACON. SPENSER. 

THE OXFORD MOVEMENT. Twelve Years, 1833-1845. 

THE BEGINNING OF THE MIDDLE AGES. Included in this 



Series by permission of Messrs. LONGMANS and Co.). 

OCCASIONAL PAPERS. Selected from The Guardian, The Times, 

and The Saturday Review, 1846-1890. 2 vols. 
Emerson's Collected Works. 6 vols. With Introduction by JOHN MORLKY. 

MISCELLANIES. | ESSAYS. | POEMS. 

ENGLISH TRAITS AND REPRESENTATIVE MEN. 

THE CONDUCT OF LIFE, AND SOCIETY AND SOLITUDE. 

LETTERS AND SOCIAL AIMS. 
Goethe's Prose Maxims. Translated, with Introductions, by T. BAILEY 

SAUNDKRS. 

** The Scientific and A rtistic Maxims were selected by Professor 
Huxley and Lord Leighton respectively. 
Thomas Henry Huxley's Collected Works. 9 vols. 

Vol. I. METHOD AND RESULTS. 

Vol. II. DARWINIANA. 

Vol. III. SCIENCE AND EDUCATION. 

Vol. IV. SCIENCE AND HEBREW TRADITION. 

Vol. V. SCIENCE AND CHRISTIAN TRADITION. 

Vol. VI. HUME. With Helps to the Study of Berkeley. 

Vol. VII. MAN'S PLACE IN NATURE; and other Anthropological 

Vol. VIII. DISCOURSES, BIOLOGICAL AND GEOLOGICAL. 
VoL IX. EVOLUTION AND ETHICS, AND OTHER ESSAYS. 
Modern Greece. By Sir RICHARD C. JKBB, Litt.D., D.C.L., LL.D. 

Second Edition. 

R. H. Button's Collected Essays. 
LITERARY ESSAYS. 
ESSAYS ON SOME OF THE MODERN GUIDES OF ENGLISH 

THOUGHT IN MATTERS OF FAITH. 
THEOLOGICAL ESSAYS. 
CRITICISMS ON CONTEMPORARY THOUGHT AND 

THINKERS. 2 vols. 
ASPECTS OF RELIGIOUS AND SCIENTIFIC THOUGHT. 

Edited by his Niece, ELIZABETH M. ROSCOE. 
Works by Sir John R. Seeley, Litt.D., K.C.M.G. 

THE EXPANSION OF ENGLAND. Two Courses of Lectures. 
LECTURES AND ESSAYS. 

ECCE HOMO. | NATURAL RELIGION. 
INTRODUCTION TO POLITICAL SCIENCE. Two Series of 
Lectures. 

MACMILLAN AND CO., LTD., LONDON. 



ENGINEERING AND MATHEMATICAL 

SCIENCES LIBRARY (213)825-4951 

University of California, Los Angeles 

Please return to the above library NOT LATER THAN DUE DATE 
stamped below. 



' 



v 



OCT2 
NOV 1 6 



PSD 2340 9/77 



UC SOUTHERN REGIONAL LIBRARY FACILITY 



A 000195598 8