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MATHEMATICAL MEMORIES: MY LITTLE TABLE




It is time to start our analytical geometry. He can come now, my
partner, the mathematician: I think I shall understand what he
says. I have already run through my book and noticed that our
subject, whose beautiful precision makes work a recreation,
bristles with no very serious difficulties.

We begin in my room, in front of a blackboard. After a few
evenings, prolonged into the peaceful watches of the night, I
become aware, to my great surprise, that my teacher, the past
master in those hieroglyphics, is really, more often than not, my
pupil. He does not see the combinations of the abscissas and
ordinates very clearly. I make bold to take the chalk in hand
myself, to seize the rudder of our algebraical boat. I comment on
the book, interpret it in my own fashion, expound the text, sound
the reefs until daylight comes and leads us to the haven of the
solution. Besides, the logic is so irresistible, it is all such
easy going and so lucid that often one seems to be remembering
rather than learning.

And so we proceed, with our positions reversed. I dig into the
hard rock, crumble it, loosen it until I make room for thought to
penetrate. My comrade--I can now allow myself to speak of him on
equal terms--my comrade listens, suggests objections, raises
difficulties which we try to solve in unison. The two combined
levers, inserted in the fissure, end by shaking and overturning the
rocky mass.

I no longer see in the corner of the quartermaster's eye the leery
droop that greeted me at the start. Cordial frankness now reigns,
the infectious high spirits imparted by success. Little by little,
dawn breaks, very misty as yet, but laden with promises. We are
both greatly amazed; and my share in the satisfaction is a double
one, for he sees twice over who makes others see. Thus do we pass
half the night, in delightful hours. We cease when sleep begins to
weigh too heavily on our eyelids.

When my comrade returns to his room, does he sleep, careless for
the moment of the shifting scene which we have conjured up? He
confesses to me that he sleeps soundly. This advantage I do not
possess. It is not in my power to pass the sponge over my poor
brain even as I pass it over the blackboard. The network of ideas
remains and forms as it were a moving cobweb in which repose
wriggles and tosses, incapable of finding a stable equilibrium.
When sleep does come at last, it is often but a state of somnolence
which, far from suspending the activity of the mind, actually
maintains and quickens it more than waking would. During this
torpor, in which night has not yet closed upon the brain, I
sometimes solve mathematical difficulties with which I struggled
unsuccessfully the day before. A brilliant beacon, of which I am
hardly conscious, flares in my brain. Then I jump out of bed,
light my lamp again and hasten to jot down my solutions, the
recollection of which I should have lost on awakening. Like
lightning flashes, those gleams vanish as suddenly as they appear.

Whence do they come? Probably from a habit which I acquired very
early in life: to have food always there for my mind, to pour the
never failing oil constantly into the lamp of thought. Would you
succeed in the things of the mind? The infallible method is to be
always thinking of them. This method I practiced more sedulously
than my comrade; and hence, no doubt, arose the interchange of
positions, the disciple turned into the master. It was not,
however, an overwhelming infatuation, a painful obsession; it was
rather a recreation, almost a poetic feast. As our great lyric
writer put it in the preface to his volume, Les Rayons et les
ombres: 'Mathematics play their part in art as well as in science.
There is algebra in astronomy: astronomy is akin to poetry; there
is algebra in music: music is akin to poetry.'

Is this poetic exaggeration? Surely not: Victor Hugo spoke truly.
Algebra, the poem of order, has magnificent flights. I look upon
its formulae, its strophes as superb, without feeling at all
astonished when others do not agree. My colleague's satirical look
came back when I was imprudent enough to confide my
extrageometrical raptures to his ears: 'Nonsense,' said he, 'pure
stuff and nonsense! Let's get on with our tangents.'

The quartermaster was right: the strict severity of our approaching
examination allowed of no such dreamer's outbursts. Was I, on my
side, very wrong? To warm chill calculation by the fire of the
ideal, to lift one's thought above mere formulae, to brighten the
caverns of the abstract with a spark of life: was this not to ease
the effort of penetrating the unknown? Where my comrade plodded
on, scorning my viaticum, I performed a journey of pleasure. If I
had to lean on the rude staff of algebra, I had for my guide that
voice within me, urging me to lofty flights. Study became a joy.

It became still more interesting when, after the angularities of a
combination of straight lines, I learnt to portray the graces of a
curve. How many properties were there of which the compass knew
nothing, how many cunning laws lay contained in embryo within an
equation, the mysterious nut which must be artistically cracked to
extract the rich kernel, the theorem! Take this or that term, place
the + sign before it and forthwith you have the ellipse, the
trajectory of the planets, with its two friendly foci, transmitting
pairs of vectors whose sum is constant; substitute the--sign and
you have the hyperbola with the antagonistic foci, the desperate
curve that dives into space with infinite tentacles, approaching
nearer and nearer to straight lines, the asymptotes, but never
succeeding in meeting them. Suppress that term and you have the
parabola, which vainly seeks in infinity its lost second focus; you
have the trajectory of the bombshell; you have the path of certain
comets which come one day to visit our sun and then flee to depths
whence they never return. Is it not wonderful thus to formulate
the orbit of the worlds? I thought so then and I think so still.

After fifteen months of this exercise, we went up together for our
examination at Montpellier; and both of us received our degrees as
bachelors of mathematical science. My companion was a wreck: I, on
the other hand, had refreshed myself with analytical geometry.

Utterly worn out by his course of conic sections, my chum declares
that he has had enough. In vain I hold out the glittering prospect
of a new degree, that of licentiate of mathematical science, which
would lead us to the splendors of the higher mathematics and
initiate us into the mechanics of the heavens: I cannot prevail
upon him, cannot make him share my audacity. He calls it a mad
scheme, which will exhaust us and come to nothing. Without the
advice of an experienced pilot, with no other compass than a book,
which is not always very clear, because of its laconic adherence to
set terms, our poor bark is bound to be wrecked on the first reef.
One might as well put out to sea in a nutshell and defy the billows
of the vasty deep. He does not use these actual words, but his
gloomy estimate of the extreme difficulties to be encountered is
enough to explain his refusal. I am quite free to go and break my
neck in far countries; he is more prudent and will not follow me.

I suspect another reason, which the deserter does not confess. He
has obtained the title needed for his plans. What does he care for
the rest? Is it worth while to sit up late at night and wear one's
self out in toil for the mere pleasure of learning? He must be a
madman who, without the lure of profit, lends an ear to the
blandishments of knowledge. Let us retreat into our shell, close
our lid to the importunities of the light and lead the life of a
mussel. There lies the secret of happiness.
This philosophy is not mine. My curiosity sees in a stage
accomplished no more than the preparation for a new stage towards
the retreating unknown. My partner, therefore. leaves me.
Henceforth, I am alone, alone and wretched. There is no one left
with whom I can sit up and thresh the subject out in exhilarating
discussion. There is no one near me to understand me, no one who
can even passively oppose his ideas to mine and take part in the
conflict whence the light will spring, even as a spark is born of
the concussion of two flints. When a difficulty arises, steep as a
cliff, there is no friendly shoulder to support me in my attempt to
climb it. Alone, I have to cling to the roughness of the jagged
rock, to fall, often, and pick myself up, covered with bruises, and
renew the assault; alone, I must give my shout of triumph, without
the least echo of encouragement, when, reaching the summit and
broken in the effort, I am at last allowed to see a little way
beyond.

My mathematical campaign will cost me much stubborn thought: I am
aware of this after the first few lines of my book. I am entering
upon the domain of the abstract, rough ground that can only be
cleared by the insistent plow of reflection. The blackboard,
excellent for the curves of analytical geometry studied in my
friend's company, is now neglected. I prefer the exercise book, a
quire of paper bound in a cover. With this confidant, which allows
one to remain seated and rests the muscles of the legs, I can
commune nightly under my lampshade, until a late hour, and keep
going the forge of thought wherein the intractable problem is
softened and hammered into shape.

My study table, the size of a pocket handkerchief, occupied on the
right by the ink stand--a penny bottle--and on the left by the open
exercise book, gives me just the room which I need to wield the
pen. I love that little piece of furniture, one of the first
acquisitions of my early married life. It is easily moved where
you wish: in front of the window, when the sky is cloudy; into the
discreet light of a corner, when the sun is troublesome. In
winter, it allows you to come close to the hearth, where a log is
blazing.

Poor little walnut board, I have been faithful to you for half a
century and more. Ink-stained, cut and scarred with the penknife,
you lend your support today to my prose as you once did to my
equations. This variation in employment leaves you indifferent;
your patient back extends the same welcome to the formulae of
algebra and the formula of thought. I cannot boast this placidity;
I find that the change has not increased my peace of mind; hunting
for ideas troubles the brain even more than hunting for the roots
of an equation.

You would never recognize me, little friend, if you could give a
glance at my gray mane. Where is the cheerful face of former days,
bright with enthusiasm and hope? I have aged, I have aged. And
you, what a falling off, since you came to me from the dealer's,
gleaming and polished and smelling so good with your beeswax! Like
your master, you have wrinkles, often my work, I admit; for how
many times, in my impatience, have I not dug my pen into you, when,
after its dip in the muddy inkpot, the nib refused to write
decently!

One of your corners is broken off; the boards are beginning to come
loose. Inside you, I hear, from time to time, the plane of the
death-watch, who despoils old furniture. From year to year, new
galleries are excavated, endangering your solidity. The old ones
show on the outside in the shape of tiny round holes. A stranger
has seized upon the latter, excellent quarters, obtained without
trouble. I see the impudent intruder run nimbly under my elbow and
penetrate forthwith into the tunnel abandoned by the death-watch.
She is after game, this slender huntress, clad in black, busy
collecting wood lice for her grubs. A whole nation is devouring
you, you old table; I am writing on a swarm of insects! No support
could be more appropriate to my entomological notes.

What will become of you when your master is gone? Will you be
knocked down for a franc, when the family come to apportion my poor
spoils? Will you be turned into a stand for the pitcher beside the
kitchen sink? Will you be the plank on which the cabbages are
shredded? Or will my children, on the contrary, agree and say:

'Let us preserve the relic. It was where he toiled so hard to
teach himself and make himself capable of teaching others; it was
where he so long consumed his strength to find food for us when we
were little. Let us keep the sacred plank.'

I dare not believe in such a future for you. You will pass into
strange hands, O my old friend; you will become a bedside table,
laden with bowl after bowl of linseed tea, until, decrepit, rickety
and broken down, you are chopped up to feed the flames for a brief
moment under the simmering saucepan. You will vanish in smoke to
join my labors in that other smoke, oblivion, the ultimate resting
place of our vain agitations.

But let us return, little table, to our young days; those of your
shining varnish and of my fond illusions. It is Sunday, the day of
rest, that is to say, of continuous work, uninterrupted by my
duties in the school. I greatly prefer Thursday, which is not a
general holiday and more propitious to studious calm. Such as it
is, for all its distractions, the Lord's day gives me a certain
leisure. Let us make the most of it. There are fifty-two Sundays
in the year, making a total that is almost equivalent to the long
vacation.

It so happens that I have a glorious question to wrestle with
today; that of Kepler's three laws, which, when explored by the
calculus, are to show me the fundamental mechanism of the heavenly
bodies. One of them says: 'The area swept out in a given time by
the radius vector of the path of a planet is proportional to the
time taken.'

From this I have to deduce that the force which confines the planet
to its orbit is directed towards the sun. Gently entreated by the
differential and integral calculus, already the formula is
beginning to voice itself. My concentration redoubles, my mind is
set upon seizing the radiant dawn of truth.

Suddenly, in the distance, br-r-r-rum! Br-r-r-rum! Br-r-r-rum! The
noise comes nearer, grows louder. Woe upon me! And plague take the
Pagoda!

Let me explain. I live in a suburb, at the beginning of the Pernes
Road, far from the tumult of the town [of Carpentras where Fabre
was a master at the college]. Twenty yards in front of my house,
some pleasure gardens have been opened, bearing a signboard
inscribed, 'The Pagoda.' Here, on Sunday afternoons, the lads and
lasses from the neighboring farms come to disport themselves in
country dances. To attract custom and push the sale of
refreshments, the proprietor of the ball ends the Sunday hop with a
tombola. Two hours beforehand, he has the prizes carried along the
public roads, preceded by fifes and drums. From a beribboned pole,
borne by a stalwart fellow in a red sash, dangle a plated goblet, a
handkerchief of Lyons silk, a pair of candlesticks and some packets
of cigars. Who would not enter the pleasure gardens, with such a
bait?

'Br-r-r-rum! Br-r-r-rum! Br-r-r-rum!' goes the procession.

It comes just under my window, wheels to the right and marches into
the establishment, a huge wooden booth, hung with evergreens. And
now, if you dislike noise, flee, flee as far as you can. Until
nightfall, the ophicleides will bellow, the fifes tootle and the
cornets bray. How would you deduce the steps of Kepler's laws to
the accompaniment of that noisy orchestra! It is enough to drive
one mad. Let us be off with all speed.

A mile away, I know a flinty waste beloved of the wheatear and the
locust. Here reigns perfect calm; moreover, there are some clumps
of evergreen oak which will lend me their scanty shade. I take my
book, a few sheets of paper and a pencil and fly to this solitude.
What beauteous silence, what exquisite quiet! But the sun is
overwhelming, under the meager cover of the bushes. Cheerily, my
lad! Have at your Kepler's laws in the company of the blue-winged
locusts. You will return home with your problems solved, but with
a blistered skin. An overdose of sun in the neck shall be the
outcome of grasping the law of the areas. One thing makes up for
another.

During the rest of the week, I have my Thursdays and the evenings,
which I employ in study until I drop with sleep. All told I have
no lack of time, despite the drudgery of my college ties. The
great thing is not to be discouraged by the unavoidable
difficulties encountered at the outset. I lose my way easily in
that dense forest overgrown with creepers that have to be cut away
with the axe to obtain a clearing. A fortunate turn or two; and I
once more know where I am. I lose my way again. The stubborn axe
makes its opening without always letting in sufficient light.

The book is just a book, that is to say, a set text, saying not a
word more than it is obliged to, exceedingly learned, I admit, but,
alas, often obscure! The author, it seems, wrote it for himself.
He understood; therefore others must. Poor beginners, left to
yourselves, you manage as best you can! For you, there shall be no
retracing of steps in order to tackle the difficulty in another
way; no circuit easing the arduous road and preparing the passage;
no supplementary aperture to admit a glimmer of daylight.
Incomparably inferior to the spoken word, which begins again with
fresh methods of attack and is ready to vary the paths that lead to
the open, the book says what it says and nothing more. Having
finished its demonstration, whether you understand or no, the
oracle is inexorably dumb. You reread the text and ponder it
obstinately; you pass and repass your shuttle through the woof of
figures. Useless efforts all: the darkness continues. What would
be needed to supply the illuminating ray? Often enough, a trifle,
a mere word; and that word the book will not speak.

Happy is he who is guided by a master's teaching! His progress does
not know the misery of those wearisome breakdowns. What was I to
do before the disheartening wall that every now and then rose up
and barred my road? I followed d'Alembert's precept in his advice
to young mathematical students: 'Have faith and go ahead,' said the
great geometrician.

Faith I had; and I went on pluckily. And it was well for me that I
did, for I often found behind the wall the enlightenment which I
was seeking in front of it. Giving up the bad patch as hopeless, I
would go on and, after I had left it behind, discover the dynamite
capable of blasting it. 'Twas a tiny grain at first, an
insignificant ball rolling and increasing as it went. From one
slope to the other of the theorems, it grew to a heavy mass; and
the mass became a mighty projectile which, flung backwards and
retracing its course, split the darkness and spread it into one
vast sheet of light.

D'Alembert's precept is good and very good, provided you do not
abuse it. Too much precipitation in turning over the intractable
page might expose you to many a disappointment. You must have
fought the difficulty tooth and nail before abandoning it. This
rough skirmishing leads to intellectual vigor.

Twelve months of meditation in the company of my little table at
last won me my degree as a licentiate of mathematical science; and
I was now qualified to perform, half a century later, the eminently
lucrative functions of an inspector of Spiders' webs!





Next: THE BLUEBOTTLE: THE LAYING

Previous: MATHEMATICAL MEMORIES: NEWTON'S BINOMIAL THEOREM



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