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!