changes in the affairs of this earth. I think that superstition has
arisen from a muddle-headed hope that slow work can exclude a Creator
and Will. At any rate there is no proof for it. Some of the processes
have been very slow (they are exceedingly slow to-day), but some jerks
have been rapid enough: revolutionary: catastrophic: and the last
melting of the ice was of these.
And what do you suppose happened in the splendid valleys of Norway?
To-day they are drowned. What recession of the ice filled them more
full? How did man come to occupy the land released? During what
intermission of time, during what generations (few and creative) did
the tall fair race, for a moment wanderers, build their little simple
structure of a religion which we know so little (because what we have
of it is wholly intermixed with the Roman) and of a language which we
know still less (for that also is mixed with the South), but at any
rate of a special culture common to but a few thousands of men. How
came these Scandinavians to copy Roman ships? It all came after the
melting of the ice.
Then I ask myself what men saw, and what they felt as they saw, the
waterfalls. For those marked all Europe also. Glaciers we know to-day.
We have but to imagine them expanded, and the landscape is the same.
Stand on the Maladetta and conceive the field of ice holding not only
the shoulder of the mountain but all the valley below and out to the
plain of France, and you only have the replica of what a man may see
from Mont Blanc. Or stand on an Alpine peak and imagine the sheets of
ice and snow below you, spread, covering every rock for as far as the
eye can reach, and you only have a repetition of what men still see in
Greenland. But we have no modern parallel (save in perhaps half a dozen
places on the whole earth) by which to reconstruct the enormity of the
waterfalls of those days.
For there were not only these swirling waters carving out the great
valleys, there was the thundering of water down over the ledges,
thousand upon thousand. Perhaps they helped to scoop out the smaller
lakes more than did the ice before them. There must have been some such
sight above Grenoble when the great lake which burst seven hundred
years ago was forming. For in the beginning that lake basin in which
Bourg d'Oisans now stands must have been a mass of ice, and then as
the ice melted and as the glacier above it melted all the way up to La
Grave, up to the very shoulders of the Pelvoux, what mighty armies of
water must have roared down to the trench of the Isere! And wherever
there is to-day a gorge (or, at least, in the most of these cuttings)
you must have had the same sight. Their little dwindling descendants
now and then show a trickle of water for our amusement and we are still
astonished. But the grandfathers of these were giants!
They say also that the sea rose. It may have done so. Perhaps it must
have done so. And if it did so what a sight must that also not have
been: the cutting of the straits.
I have read of but one part of the world in which a tradition remains
of such a change, and in that case it may have been an earthquake
rather than a rising of the waters: I mean the Straits of Messina.
Of the water flooding in here there is a legend; but there is none
remaining of the cutting of the chalk between Kent and the Artois; or
of the flooding, if it were flooded, of the channel between the Pillars
of Hercules; or of the slower lap which gradually just covered the
entrances of the Baltic--a fresh-water lake.
And by the way, what made that most amazing issue whereby the Black
Sea feeds the Eastern Mediterranean with a continuous stream? I have
read so many guesses, and they have not satisfied me. It is so long,
so narrow, so artificial, and double at that. Very changed would the
history of the world have been--of the modern world--if Nature had
played some freak of the same sort to join the central Atlantic and the
central Pacific Seas, or if the low sand between the Red Sea and the
Mediterranean had not run dry, or if by some shock the Mediterranean
had poured into the Jordan valley.
There is much else in the melting of the ice. Was it then that Northern
Africa dried up? Was it then that the old watercourses which are now
desert and in which you can still find proof of the habitations of
men, and the stranded beasts and fishes of old rivers, were in full
spate?
Who lived there? What did they in the story of mankind? And did Egypt
when it was already able to build and to carve men out of stone, look
out from the head of the Delta upon a shallow sea?
I think the greater part of the story of the world's landscape has been
lost to us for ever.
ON THE HATRED OF NUMBERS
I knew a man once who was the head of his college and very famous as
a scholar; because, although he had no learning, he was very kind and
witty. But this man had one thing upon which he never jested, and that
was his hatred (which he pretended to be a contempt) of numbers.
He would say, not as a curiosity, still less as an apology, that he was
quite incapable of "doing sums"; and as I met him in early youth the
example has influenced the whole of my life. I have known all my life
what I think many people do not know, or at least not many people of
those engaged in exposition, that a large part of educated people feel
thus about numbers. They neglect them, or hate them, or despise them,
and in any case cannot and will not deal with them.
Now when one learns a thing like that early in life, when one early
discovers that a thing apparently ridiculous is quite common, the mind
sets to work at once to try and find out an explanation. For a very
long time I could pretend to none, but I think I am at last beginning
to see my way in the matter and to grasp why it is that so many people
avoid the prime measure of reality.
On the face of it one would say that the thing was impossible, for one
can do nothing without numbers. The human life is distinguished from
the bestial chiefly by the use and knowledge of numbers. Which reminds
me that I once went to see in Berlin a horse of which it was said by
its owner that it could count up to ten. This was before the war. I do
not know what may have happened since. It was all nonsense; the horse
could not count up to ten. What it could do was to move its foot round,
tapping it along a row of large numbers, and when it got to the number
which the owner wanted it to indicate he spoke to the horse and the
horse stopped moving. But to return from Berlin.
Everything done by a human being all day long, even sleeping, is based
upon numbers: all his observation and all his action. To be the enemy
of numbers is like being the enemy of the air one breathes or of the
necessity of human speech. Why, then, is it that this enmity arises, or
at any rate has arisen to-day, rather late perhaps in modern history,
and especially here in England?
I think it is due to the convergence of a great number of tendencies.
In the first place, numbers always involve _some_ act of thought, just
as progress always involves _some_ action of the body. And people fight
shy of numbers just as they tend to fight shy of exertion. If you have
four people to dinner, and you ask three more, that makes seven. It is
quite a little sum, and may be compared to the exertion of crossing a
room. Still, very often one does not like even to cross the room, but
prefers sitting in one's chair to getting up to take a book.
If you are rich enough to be overdrawn at the bank, and then lucky
enough to get a sum of money larger than your overdraft, and then
honest enough to pay it in, it is your duty to make a little
calculation on the back of the counterfoil, turning a minus into
a plus, and you have to carry pence into shillings and shillings
into pounds. It is quite a little exertion, like going up a flight
of stairs.... I fancy that more than half the people who have bank
accounts shirk that duty. Yet it is a plain duty, for if you do not
know your balance at the bank you may inadvertently smash a stumer, _si
j'ose m'exprimer ainsi_.
There is another cause for this fear of numbers, and that is their
absolute quality. They have no penumbra. You cannot play the fool with
words when they refer to numbers as you can in matters where measure
is not exact. You cannot combine a part of reality with a part of
illusion, which is the very essence of æsthetic enjoyment. Numbers are
thus heartless, and the heart loves to ignore them. If your country
is losing population but gaining trade you may boycott the population
statistics and dwell lovingly upon the trade statistics. But it is
more comfortable still to indulge in a vague vision of your country
thoroughly prosperous all round and happy. You see this cause of the
hatred of numbers coming out very strongly in the case of people who
are losing money. Either they stop keeping accounts or they falsify the
accounts by hiding sheer loss under some fancy name, such as Goodwill,
or Office Furniture, or "Estimated Value." That is an extreme case, but
I am sure that in all cases a part of the dislike of numbers comes from
this absolute, bullying quality they have.
Nor is it wholly an unreasonable attitude, for the functions of the
mind not only include appreciations and creative acts which escape
rigid measure, but indeed those acts are the most important.
For instance, the ratio one to the square root of two is the most
perfect proportion for the dividing of a window, as with a mullion; for
it represents to the eye, and through the eye to the soul, that great
principle of the Mean which governs harmony throughout the world. It
was a proportion which they loved in the true Middle Ages, and I will
show it to you combined with the simple proportions of one to two,
one to three, etc., in many and many a window of the thirteenth and
early fourteenth centuries, in the time when men were still simple.
Nevertheless, an artist may quite justly say that in a particular case
this rigid rule would bind him too much. Something in the way the light
falls, something in the thickness of the walls, or even in the outlook
from the window, makes it fail. It is not to the eye what it is to the
measurement upon the plan, and so the artist is perfectly right to
say that he will here not be bound by an exact framework of number.
It only needs a little exaggeration of such an experience or a little
repetition of it to make a weak mind abjure measure in art altogether.
And there must have been a lot of that abjuration lately to judge by
the funny things one sees.
Numbers also disappoint and annoy in another fashion, which is that
men read into them more than they say and then blame the numbers for
misleading them.
I had a good deal of experience of that during the last two years of
the war. People had got fidgety under the strain, and they did not want
to watch the rate of progress made in wearing down the besieged Boche.
Sometimes they would angrily deny those numbers altogether. So a man
getting into a train for Edinburgh and foolishly expecting it would
take him three hours would after, say, the fifth hour get very much
annoyed with a companion who should be perpetually pulling out a map
and a watch and telling him exactly how far they had come. I remember
that this was particularly the case with the rate of German casualty.
There was nothing very mysterious about this; there was a certain known
margin of error, and therefore a certain known maximum and minimum.
For instance, it could be calculated from all sorts of sources which
confirmed each other--prisoners' reports, published lists, rolls of
honour, the rate of retirement of divisions, the analogy of known
losses upon our side, and so on. But there came a time after which
people were impatient and preferred to believe the whole thing to be
fantastic guesswork, because they were in a hurry, and at the same time
half despairing. There were hundreds and perhaps thousands of expert
soldiers up and down Europe engaged in arriving at these numbers,
their conclusions were centralised in the various staffs; and it was
surprising to see by how little the various estimates differed. I
remember once at Chantilly comparing a set of figures arrived at by one
of the national staffs with another set of figures based upon different
national methods, and remarking that there was not five per cent
difference between them. Yet the ordinary member of the public had by
that time become completely suspicious of all figures.
The German casualties on the Somme are a good example. We do not know
them to a single unit, and shall never so know them. But we know them
within a certain margin of error, and we knew them shortly after the
cessation of the battle within a rather larger margin of error. But
the ordinary member of the public at home was not in a mood for them,
and would not follow them. I got shoals of insulting letters about
them from honest people who were disappointed of a miracle. Yet these
numbers were of supreme importance. For it was the usury of the Somme,
coming after that of Verdun, which began the process of decline upon
the enemy's side, politically and morally even more than militarily.
The plain man could only see that the Somme had not succeeded in
breaking the German line; his expectation was disappointed; he thought
all this talk of numbers beside the mark, and his annoyance with it
took the natural though very illogical form of belabouring the poor
dumb figures that had done him no harm.
I remember very well how, after I had published the approximate figure
of enemy losses of 1916 in "Land and Water" (supplied of course from
Staff figures), the Harmsworth Press fully persuaded the mass of
Englishmen that the German recruitment was _inexhaustible_. The figures
showed a loss of about one-sixth of the total enemy recruiting power.
The general reader could not be bothered with anything so precise.
First he thought I had said the enemy were all gone, then he thought
that the enemy never could be worn down. He disliked numbers, did the
general reader, and every demagogue will do well to play upon that
dislike of numbers. It is one of the main ways of deceiving the silly
populace.
After all these more or less disreputable reasons for the dislike of
numbers one comes to a series of much better reasons, and first among
these undoubtedly is the truth expressed in the epigram, "Lies, Damned
Lies, and Statistics." You can prove anything almost by Statistics, and
therefore if one lives in a time when statistics are a plague (because
there is such facility for gathering them under our modern system of
arbitrary, all-powerful and highly-centralised government) one comes to
suspect their use and even to detest it. Now that suspicion and that
detestation are well founded.
The reason they are well founded is this.
Judgment, the most valuable of all the rational qualities, is
essentially an integration of an almost infinite number of
differentials. The mind is seized with a very large number, an almost
infinite number (or, as mathematicians would say, an indefinitely large
number) of impressions which it combines, and on the combined effect of
which it bases certitude. I have no doubt of an oak tree when I see it,
although I do not look at each leaf nor measure the exact outline even
of one leaf to note within what margin of error it conforms to the type
or form of an oak-leaf. I recognise a voice or a face by integrating
a vast number of small differentials, and no amount of mathematical
or mechanical argument drawn from impressions far _less_ numerous and
not combined by a human mind at all would convince me of error. That
is why we say that a good portrait is always better than a photograph,
and that is why we often say of a photograph that it is not like the
original at all. A photograph is only the record of one selected,
very limited set of impressions, highly restricted in number and in
time, which cannot compare in value with the general conclusion of an
observer using all the human faculties over a considerable time, and
endowed with the organic power of unification.
In the same way a piece of statistics, however accurate, concerns only
one of an infinite number of factors. You get nearer the truth as you
combine one set of statistics with a second, a third, a fourth; but
though you were to have a thousand sets they would not outweigh your
general judgment based on observation.
For instance, a man knows perfectly well what he means by a wet day.
Your statistician may come along with his figures of rainfall and
may prove by them that a very fine day was a wet one. He says, "Half
an inch of rain fell in twenty-four hours." But if the half-inch of
rain all fell in one hour just before the dawn of a hot and cloudless
summer's day he is wrong and you are right. To give even a rough
impression through mere statistics of what common sense calls "a wet
day" you would have to compare many sets of figures. There would be
first of all the inches of rainfall, then the number of hours during
which perceptible rain was falling, then the number of hours during
which the sky was overcast, then the rate of evaporation, then figures
showing what proportion of rainfall came in the waking and in the
sleeping hours, then a table showing how far the rain was intermittent,
for a day during which you have rainfall in the first half and none in
the second is not at all the same thing as a day in which rain falls
every half-hour. And even when you had all those statistics combined,
no sane man would accept them against his own general impression.
There is a parallel here with the very legitimate suspicion people are
beginning to feel toward what is called "scientific" argument in social
and domestic matters. If a man has worked hard all the morning in the
open air and is very hungry he looks forward to a beefsteak and a pint
of beer at luncheon, and if some science fellow comes along with a
tabloid and a little water out of a tank it is no good telling the man
that such a meal will be "scientifically the equivalent" of what he was
expecting. He knows very well that the word "equivalent" here is a lie;
and it is hard lines that the noble title of Science should have been
degraded as it has been in our generation by nonsense of this sort.
They are still at it, but I do not think it will last long, for it is
provoking anger. There are people who come and tell country folk that
the rooms in which they sleep must have a certain "cubical capacity,"
so that a great healthy man as strong as an ox and sleeping as no
townsman can sleep is "scientifically" proved to be in a very parlous
way when he lives as his English fathers have lived before him for a
thousand years. And "I with these mine eyes have seen" an inspector
insisting that the window-space of a room should be not less than
one-tenth or one-quarter or whatever it was of the floor-space, without
noting that the window in one case looked on to a blank wall, and in
another on to the infinite spaces of the sea.
The use of statistics in argument is essentially deduction from
insufficient premises, and though the mass of men who are getting more
and more exasperated with the results of such argument are not yet
conscious of the flaw in their oppressors' reasoning, they only know
instinctively that it is bad reasoning--and they are right.
The whole argument against the abuse of statistics is summed up in the
story of the man who explained what an average was. "If you were struck
dead by lightning at my side and I remained safe and sound, we should
both be on the average half dead."
But there is a deeper final cause for the suspicion of numbers on which
one must touch very carefully, because though it has led to the wildest
extravagance in philosophy, and has weakened the use of the human
reason in modern times, it none the less has a basis of truth.
Numbers, absolute though they be, and the expression of the human
mind's divine capacity for measure, escape us in their ultimate use.
The science of numbers, when you have pursued it far enough, lands
you where every human analysis lands you--in a contradiction or a
mystery. On this account it is that men have been tempted of late to
a monstrous denial of mathematical truth, and that as a part of their
general revolt against reason. But it is true that you approach regions
beyond which what had hitherto been the necessary laws of number cease
to stand. You have it wherever there is a sudden passage from an
indefinitely increasing number to an indefinitely small one; a sudden
passage through what we call "the infinite" to "Zero." And you get it
in the "limit" of any mathematical process.
For instance, as you stretch out an ellipse it ought to get more and
more like a parabola, at any rate at the business end, if I may so
express myself; and it ought gradually at the end of the process to
merge into a parabola. Then, as you go on tilting your plane, you ought
logically to be able to follow the process of the parabola turning into
an hyperbola. Well, the thing does not happen that way. The ellipse
remains an ellipse though its foci get farther and farther apart. It
remains an ellipse right up to the critical moment, when the distant
focus vanishes into infinity, and, then, suddenly, the curve becomes a
parabola, and then in a flash, it turns inside out, splits into two,
and is an hyperbola. And no one will ever be able to tell you what
happened out in the infinities, where the transmogrification took
place.
There is a lot more.... Every operation of subtraction, or of the
addition of opposite signs, is a mystery; for the conception of the
negative in mathematics is a mystery. As for imaginaries, and the dear
old root of minus one, you may call it a convention if you like, but
it cannot be altogether a convention since it is a necessary basis
for arriving at a truth; nor is it a lie, though it certainly dresses
up like one. Its name is _iota_ and I loved it long ago. One might
quote of this formidable being what I think St. Augustine said of some
Bible-story:
"_Non est mendacium sed mysterium._"
You may tell me that by far the greater part of men are not concerned
with these ultimate far goals of bewilderment and of despair to
which the science of numbers will lead them out of their happy
homes. I am not so sure. Nowadays when people are so fond of the
word "sub-conscious" (and I have myself worked the animal as hard
as was possible on meagre rations) we may drag it in here without
extravagance. There is, I think, an instinct in men the least
acquainted with mathematics that it is possible to push them too far;
and though men begin to tremble long before the limits of human reason
are reached--in fact, as a rule, they get the wind up before they reach
the Calculi--yet a god tells them in what direction they are going, and
warns them that too much curiosity is ill for man.
ON THE LAST INFIRMITY
If you were to seek for the most irrational of all appetites, the
one appetite for which you could not give any sort of reason, you
would find it to be the strongest appetite of all: the appetite for
posthumous fame.
Milton made a little fortune (in the literary sense of that word--and
my fellow-hacks will know what sort of fortune _that_ is) by calling it
the last infirmity of noble minds. It is a very true saying, not only
in its direct sense but in its implication. It is not only true that
men who have conquered every other appetite hardly conquer this one; it
is also true that there is something divine about the desire for fame,
infirmity though it be. The mind remains well noble though still fully
possessed of such a desire.