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 (1). 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!