MATHEMATICAL MEMORIES: NEWTON'S BINOMIAL THEOREM
The spider's web is a glorious mathematical problem. I should
enjoy working it out in all its details, were I not afraid of
wearying the reader's attention. Perhaps I have even gone too far
in the little that I have said, in which case I owe him some
compensation: 'Would you like me,' I will ask him, 'would you like
me to tell you how I acquired sufficient algebra to master the
logarithmic systems and how I became a surveyor of Spiders' webs?
Would you? It will give us a rest from natural history.'
I seem to catch a sign of acquiescence. The story of my village
school, visited by the chicks and the porkers, has been received
with some indulgence; why should not my harsh school of solitude
possess its interest as well? Let us try to describe it. And who
knows? Perhaps, in doing so, I shall revive the courage of some
other poor derelict hungering after knowledge.
I was denied the privilege of learning with a master. I should be
wrong to complain. Solitary study has its advantages: it does not
cast you in the official mould; it leaves you all your originality.
Wild fruit, when it ripens, has a different taste from hothouse
produce: it leaves on a discriminating palate a bittersweet flavor
whose virtue is all the greater for the contrast. Yes, if it were
in my power, I would start afresh, face to face with my only
counselor, the book itself, not always a very lucid one; I would
gladly resume my lonely watches, my struggles with the darkness
whence, at last, a glimmer appears as I continue to explore it; I
should retraverse the irksome stages of yore, stimulated by the one
desire that has never failed me, the desire of learning and of
afterwards bestowing my mite of knowledge on others.
When I left the normal school, my stock of mathematics was of the
scantiest. How to extract a square root, how to calculate and
prove the surface of a sphere: these represented to me the
culminating points of the subject. Those terrible logarithms, when
I happened to open a table of them, made my head swim, with their
columns of figures; actual fright, not unmixed with respect,
overwhelmed me on the very threshold of that arithmetical cave. Of
algebra I had no knowledge whatever. I had heard the name; and the
syllables represented to my poor brain the whole whirling legion of
Besides, I felt no inclination to decipher the alarming
hieroglyphics. They made one of those indigestible dishes which we
confidently extol without touching them. I greatly preferred a
fine line of Virgil, whom I was now beginning to understand; and I
should have been surprised indeed had any one told me that, for
long years to come, I should be an enthusiastic student of the
formidable science. Good fortune procured me my first lesson in
algebra, a lesson given and not received, of course.
A young man of about my own age came to me and asked me to teach
him algebra. He was preparing for his examination as a civil
engineer; and he came to me because, ingenuous youth that he was,
he took me for a well of learning. The guileless applicant was
very far out in his reckoning.
His request gave me a shock of surprise, which was forthwith
repressed on reflection: 'I give algebra lessons? ' said I to
myself. 'It would be madness: I don't know anything about the
And I left it at that for a moment or two, thinking hard, drawn now
this way, now that with indecision: 'Shall I accept? Shall I
refuse? ' continued the inner voice.
Pooh, let's accept! An heroic method of learning to swim is to leap
boldly into the sea. Let us hurl ourselves head first into the
algebraical gulf; and perhaps the imminent danger of drowning will
call forth efforts capable of bringing me to land. I know nothing
of what he wants. It makes no difference: let's go ahead and
plunge into the mystery. I shall learn by teaching.
It was a fine courage that drove me full tilt into a province which
I had not yet thought of entering. My twenty-year-old confidence
was an incomparable lever.
'Very well,' I replied. 'Come the day after tomorrow, at five, and
This twenty-four hours' delay concealed a plan. It secured me the
respite of a day, the blessed Thursday, which would give me time to
collect my forces.
Thursday comes. The sky is gray and cold. In this horrid weather,
a grate well filled with coke has its charms. Let's warm ourselves
Well, my boy, you've landed yourself in a nice predicament! How
will you manage tomorrow? With a book, plodding all through the
night, if necessary, you might scrape up something resembling a
lesson, just enough to fill the dread hour more or less. Then you
could see about the next: sufficient for the day is the evil
thereof. But you haven't the book. And it's no use running out to
the bookshop. Algebraical treatises are not current wares. You'll
have to send for one, which will take a fortnight at least. And
I've promised for tomorrow, for tomorrow certain! Another argument
and one that admits of no reply: funds are low; my last pecuniary
resources lie in the corner of a drawer. I count the money: it
amounts to twelve sous, which is not enough.
Must I cry off? Rather not! One resource suggests itself: a highly
improper one, I admit, not far removed indeed from larceny. O
quiet paths of algebra, you are my excuse for this venial sin! Let
me confess the temporary embezzlement.
Life at my college is more or less cloistered. In return for a
modest payment, most of us masters are lodged in the building; and
we take our meals at the principal's table. The science master,
who is the big gun of the staff and lives in the town, has
nevertheless, like ourselves, his own two cells, in addition to a
balcony, or leads, where the chemical preparations give forth their
suffocating gases in the open air. For this reason, he finds it
more convenient to hold his class here during the greater part of
the year. The boys come to these rooms in winter, in front of a
grate stuffed full of coke, like mine, and there find a blackboard,
a pneumatic trough, a mantelpiece covered with glass receivers,
panoplies of bent tubes on the walls, and, lastly, a certain
cupboard in which I remember seeing a row of books, the oracles
consulted by the master in the course of his lessons.
'Among those books,' said I to myself, 'there is sure to be one on
algebra. To ask the owner for the loan of it does not appeal to
me. My amiable colleague would receive me superciliously and laugh
at my ambitious aims. I am sure he would refuse my request.'
The future was to show that my distrust was justified. Narrow
mindedness and petty jealousy prevail everywhere alike.
I decide to help myself to this book, which I should never get by
asking. This is the half-holiday. The science master will not put
in an appearance today; and the key of my room is practically the
same as his. I go, with eyes and ears on the alert. My key does
not quite fit; it sticks a little, then goes in; and an extra
effort makes it turn in the lock. The door opens. I inspect the
cupboard and find that it does contain an algebra book, one of the
big, fat books which men used to write in those days, a book nearly
half a foot thick. My legs give way beneath me. You poor specimen
of a housebreaker, suppose you were caught at it! However, all goes
well. Quick, let's lock the door again and go back to our own
quarters with the pilfered volume.
And now we are together, O mysterious tome, whose Arab name
breathes a strange mustiness of occult lore and claims kindred with
the sciences of almagest and alchemy. What will you show me? Let
us turn the leaves at random. Before fixing one's eyes on a
definite point in the landscape, it is well to take a summary view
of the whole. Page follows swiftly upon page, telling me nothing.
A chapter catches my attention in the middle of the volume; it is
headed, Newton's Binomial Theorem.
The title allures me. What can a binomial theorem be, especially
one whose author is Newton, the great English mathematician who
weighed the worlds? What has the mechanism of the sky to do with
this? Let us read and seek for enlightenment. With my elbows on
the table and my thumbs behind my ears, I concentrate all my
I am seized with astonishment, for I understand! There are a
certain number of letters, general symbols which are grouped in all
manner of ways, taking their places here, there and elsewhere by
turns; there are, as the text tells me, arrangements, permutations
and combinations. Pen in hand, I arrange, permute and combine. It
is a very diverting exercise, upon my word, a game in which the
test of the written result confirms the anticipations of logic and
supplements the shortcomings of one's thinking apparatus.
'It will be plain sailing,' said I to myself, 'if algebra is no
more difficult than this.'
I was to recover from the illusion later, when the binomial
theorem, that light, crisp biscuit, was followed by heavier and
less digestible fare. But, for the moment, I had no foretaste of
the future difficulties, of the pitfall in which one becomes more
and more entangled, the longer one persists in struggling. What a
delightful afternoon that was, before my grate, amid my
permutations and combinations! By the evening, I had nearly
mastered my subject. When the bell rang, at seven, to summon us to
the common meal at the principal's table, I went downstairs puffed
up with the joys of the newly initiated neophyte. I was escorted
on my way by a, b and c, intertwined in cunning garlands.
Next day, my pupil is there. Blackboard and chalk, everything is
ready. Not quite so ready is the master. I bravely broach my
binomial theorem. My hearer becomes interested in the combinations
of letters. Not for a moment does he suspect that I am putting the
cart before the horse and beginning where we ought to have
finished. I relieve the dryness of my explanations with a few
little problems, so many halts at which the mind takes breath
awhile and gathers strength for fresh flights.
We try together. Discreetly, so as to leave him the merit of the
discovery, I shed a little light on the path. The solution is
found. My pupil triumphs; so do I, but silently, in my inner
consciousness, which says:
'You understand, because you succeed in making another understand.'
The hour passed quickly and very pleasantly for both of us. My
young man was contented when he left me; and I no less so, for I
perceived a new and original way of learning things.
The ingenious and easy arrangement of the binomial gave me time to
tackle my algebra book from the proper commencement. In three or
four days, I had rubbed up my weapons. There was nothing to be
said about addition and subtraction: they were so simple as to
force themselves upon one at first sight. Multiplication spoilt
things. There was a certain rule of signs which declared that
minus multiplied by minus made plus. How I toiled over that
wretched paradox! It would seem that the book did not explain this
subject clearly, or rather employed too abstract a method. I read,
reread and meditated in vain: the obscure text retained all its
obscurity. That is the drawback of books in general: they tell you
what is printed in them and nothing more. If you fail to
understand, they never advise you, never suggest an attempt along
another road which might lead you to the light. The merest word
would sometimes be enough to put you on the right track; and that
word the books, hidebound in a regulation phraseology, never give
How greatly preferable is the oral lesson! It goes forward, goes
back, starts afresh, walks around the obstacle and varies the
methods of attack until, at long last, light is shed upon the
darkness. This incomparable beacon of the master's word was what I
lacked; and I went under, without hope of succor, in that
treacherous pool of the rule of signs.
My pupil was bound to suffer the effects. After an attempt at an
explanation in which I made the most of the few gleams that reached
me I asked him:
'Do you understand? '
It was a futile question, but useful for gaining time. Myself not
understanding, I was convinced beforehand that he did not
'No,' he replied, accusing himself, perhaps, in his simple mind, of
possessing a brain incapable of taking in those transcendental
'Let us try another method.'
And I start again this way and that way and yet another way. My
pupil's eyes serve as my thermometer and tell me of the progress of
my efforts. A blink of satisfaction announces my success. I have
struck home, I have found the joint in the armor. The product of
minus multiplied by minus delivers its mysteries to us.
And thus we continued our studies: he, the passive receiver, taking
in the ideas acquired without effort; I, the fierce pioneer,
blasting my rock, the book, with the aid of much sitting up at
night, to extract the diamond, truth. Another and no less arduous
task fell to my share: I had to cut and polish the recondite gem,
to strip it of its ruggedness and present it to my companion's
intelligence under a less forbidding aspect. This diamond cutter's
work, which admitted a little light into the precious stone, was
the favorite occupation of my leisure; and I owe a great deal to
The ultimate result was that my pupil passed his examination. As
for the book borrowed by stealth, I restored it to the shelves and
replaced it by another, which, this time, belonged to me.
At my normal school, I had learnt a little elementary geometry
under a master. From the first few lessons onwards, I rather
enjoyed the subject. I divined in it a guide for one's reasoning
faculties through the thickets of the imagination; I caught a
glimpse of a search after truth that did not involve too much
stumbling on the way, because each step forward rests solidly upon
the step already taken; I suspected geometry to be what it
preeminently is: a school of intellectual fencing.
The truth demonstrated and its application matter little to me;
what rouses my enthusiasm is the process that sets the truth before
us. We start from a brilliantly lighted spot and gradually get
deeper and deeper in the darkness, which, in its turn, becomes
self-illuminated by kindling new lights for a higher ascent. This
progressive march of the known toward the unknown, this
conscientious lantern lighting what follows by the rays of what
comes before: that was my real business.
Geometry was to teach me the logical progression of thought; it was
to tell me how the difficulties are broken up into sections which,
elucidated consecutively, together form a lever capable of moving
the block that resists any direct efforts; lastly, it showed me how
order is engendered, order, the base of clarity. If it has ever
fallen to my lot to write a page or two which the reader has run
over without excessive fatigue, I owe it, in great part, to
geometry, that wonderful teacher of the art of directing one's
thought. True, it does not bestow imagination, a delicate flower
blossoming none knows how and unable to thrive on every soil; but
it arranges what is confused, thins out the dense, calms the
tumultuous, filters the muddy and gives lucidity, a superior
product to all the tropes of rhetoric.
Yes, as a toiler with the pen, I owe much to it. Wherefore my
thoughts readily turn back to those bright hours of my novitiate,
when, retiring to a corner of the garden in recreation time, with a
bit of paper on my knees and a stump of pencil in my fingers, I
used to practice deducing this or that property correctly from an
assemblage of straight lines. The others amused themselves all
around me; I found my delight in the frustum of a pyramid. Perhaps
I should have done better to strengthen the muscles of my thighs by
jumping and leaping, to increase the suppleness of my loins with
gymnastic contortions. I have known some contortionists who have
prospered beyond the thinker.
See me then entering the lists as an instructor of youth, fairly
well acquainted with the elements of geometry. In case of need, I
could handle the land surveyor's stake and chain. There my views
ended. To cube the trunk of a tree, to gauge a cask, to measure
the distance of an inaccessible point appeared to me the highest
pitch to which geometrical knowledge could hope to soar. Were
there loftier flights? I did not even suspect it, when an
unexpected glimpse showed me the puny dimensions of the little
corner which I had cleared in the measureless domain.
At that time, the college in which, two years before, I had made my
first appearance as a teacher, had just halved the size of its
classes and largely increased its staff. The newcomers all lived
in the building, like myself, and we had our meals in common at the
principal's table. We formed a hive where, in our leisure time,
some of us, in our respective cells, worked up the honey of algebra
and geometry, history and physics, Greek and Latin most of all,
sometimes with a view to the class above, sometimes and oftener
with a view to acquiring a degree. The university titles lacked
variety. All my colleagues were bachelors of letters, but nothing
more. They must, if possible, arm themselves a little better to
make their way in the world. We all worked hard and steadily. I
was the youngest of the industrious community and no less eager
than the rest to increase my modest equipment.
Visits between the different rooms were frequent. We would come to
consult one another about a difficulty, or simply to pass the time
of day. I had as a neighbor, in the next cell to mine, a retired
quartermaster who, weary of barrack life, had taken refuge in
education. When in charge of the books of his company he had
become more or less familiar with figures; and it became his
ambition to take a mathematical degree. His cerebrum appears to
have hardened while he was with his regiment. According to my dear
colleagues, those amiable retailers of the misfortunes of others,
he had already twice been plucked. Stubbornly, he returned to his
books and exercises, refusing to be daunted by two reverses.
It was not that he was allured by the beauties of mathematics, far
from it; but the step to which he aspired favored his plans. He
hoped to have his own boarders and dispense butter and vegetables
to lucrative purpose. The lover of study for its own sake and the
persistent trapper hunting a diploma as he would something to put
in his mouth were not made to understand or to see much of each
other. Chance, however, brought us together.
I had often surprised our friend sitting in the evening, by the
light of a candle, with his elbows on the table and his head
between his hands, meditating at great length in front of a big
exercise book crammed with cabalistic signs. From time to time,
when an idea came to him, he would take his pen and hastily put
down a line of writing wherein letters, large and small, were
grouped without any grammatical sense. The letters x and y often
recurred, intermingled with figures. Every row ended with the sign
of equality and a nought. Next came more reflection, with closed
eyes, and a fresh row of letters arranged in a different order and
likewise followed by a nought. Page after page was filled in this
queer fashion, each line winding up with 0.
'What are you doing with all those rows of figures amounting to
zero? ' I asked him one day.
The mathematician gave me a leery look, picked up in barracks. A
sarcastic droop in the corner of his eye showed how he pitied my
ignorance. My colleague of the many noughts did not, however, take
an unfair advantage of his superiority. He told me that he was
working at analytical geometry.
The phrase had a strange effect upon me. I ruminated silently to
this purpose: there was a higher geometry, which you learnt more
particularly with combinations of letters in which x and y played a
prominent part. When my next-door neighbor reflected so long,
clutching his forehead between his hands, he was trying to discover
the hidden meaning of his own hieroglyphics; he saw the ghostly
translation of his sums dancing in space. What did he perceive?
How would the alphabetical signs, arranged first in one and then in
another manner, give an image of the actual things, an image
visible to the eyes of the mind alone? It beat me.
'I shall have to learn analytical geometry some day,' I said.
'Will you help me? '
'I'm quite willing,' he replied, with a smile in which I read his
lack of confidence in my determination.
No matter; we struck a bargain that same evening. We would
together break up the stubble of algebra and analytical geometry,
the foundation of the mathematical degree; we would make common
stock: he would bring long hours of calculation, I my youthful
ardor. We would begin as soon as I had finished with my arts
degree, which was my main preoccupation for the moment.
In those far off days it was the rule to make a little serious
literary study take precedence of science. You were expected to be
familiar with the great minds of antiquity, to converse with Horace
and Virgil, Theocritus and Plato, before touching the poisons of
chemistry or the levers of mechanics. The niceties of thought
could only be the gainers by these preparations. Life's
exigencies, ever harsher as progress afflicts us with its
increasing needs, have changed all that. A fig for correct
language! Business before all!
This modern hurry would have suited my impatience. I confess that
I fumed against the regulation which forced Latin and Greek upon me
before allowing me to open up relations with the sine and cosine.
Today, wiser, ripened by age and experience, I am of a different
opinion. I very much regret that my modest literary studies were
not more carefully conducted and further prolonged. To fill up
this enormous blank a little, I respectfully returned, somewhat
late in life, to those good old books which are usually sold
second-hand with their leaves hardly cut. Venerable pages,
annotated in pencil during the long evenings of my youth, I have
found you again and you are more than ever my friends. You have
taught me that an obligation rests upon whoever wields the pen: he
must have something to say that is capable of interesting us. When
the subject comes within the scope of natural science, the interest
is nearly always assured; the difficulty, the great difficulty, is
to prune it of its thorns and to present it under a prepossessing
aspect. Truth, they say, rises naked from a well. Agreed; but
admit that she is all the better for being decently clothed. She
craves, if not the gaudy furbelows borrowed from rhetoric's
wardrobe, at least a vine leaf. The geometers alone have the right
to refuse her that modest garment; in theorems, plainness suffices.
The others, especially the naturalist, are in duty bound to drape a
gauze tunic more or less elegantly around her waist.
Suppose I say: 'Baptiste, give me my slippers.'
I am expressing myself in plain language, a little poor in
variants. I know exactly what I am saying and my speech is
Others--and they are numerous--contend that this rudimentary method
is the best in all things. They talk science to their readers as
they might talk slippers to Baptiste. Kaffir syntax does not shock
them. Do not speak to them of the value of a well selected term,
set down in its right place, still less of a lilting construction,
sounding rather well. Childish nonsense they call all that; the
fiddling of a short sighted mind!
Perhaps they are right: the Baptiste idiom is a great economizer of
time and trouble. This advantage does not tempt me; it seems to me
that an idea stands out better if expressed in lucid language, with
sober imagery. A suitable phrase, placed in its correct position
and saying without fuss the things we want to say, necessitates a
choice, an often laborious choice. There are drab words, the
commonplaces of colloquial speech; and there are, so to speak,
colored words, which may be compared with the brushstrokes strewing
patches of light over the gray background of a painting. How are
we to find those picturesque words, those striking features which
arrest the attention? How are we to group them into a language
heedful of syntax and not displeasing to the ear?
I was taught nothing of this art. For that matter, is it ever
taught in the schools? I greatly doubt it. If the fire that runs
through our veins, if inspiration do not come to our aid, we shall
flutter the pages of the thesaurus in vain: the word for which we
seek will refuse to come. Then to what masters shall we have
recourse to quicken and develop the humble germ that is latent
within us? To books.
As a boy, I was always an ardent reader; but the niceties of a
well-balanced style hardly interested me: I did not understand
them. A good deal later, when close upon fifteen, I began vaguely
to see that words have a physiognomy of their own. Some pleased me
better than others by the distinctness of their meaning and the
resonance of their rhythm; they produced a clearer image in my
mind; after their fashion, they gave me a picture of the object
described. Colored by its adjective and vivified by its verb, the
name became a living reality: what it said I saw. And thus,
gradually, was the magic of words revealed to me, when the chances
of, my undirected reading placed a few easy standard pages in my
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