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 you.
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 understand either.
"No," he replied, accusing himself, perhaps, in his simple mind, of possessing a brain incapable of taking in those transcendental verities.
"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 pa.s.sive 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 it.
The ultimate result was that my pupil pa.s.sed 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 a.s.semblage 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 cla.s.ses and largely increased its staff. The newcomers all lived in the building, like myself, and we had our meals in common at the princ.i.p.al"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 cla.s.s above, sometimes and oftener with a view to acquiring a degree.
The university t.i.tles 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 pa.s.s 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 b.u.t.ter 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 a.n.a.lytical 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 s.p.a.ce. 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 a.n.a.lytical 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 a.n.a.lytical 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 a.s.sured; 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 understood.
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 way.
CHAPTER XIII. MATHEMATICAL MEMORIES: MY LITTLE TABLE
It is time to start our a.n.a.lytical 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 ma.s.s.
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 pa.s.s 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 pa.s.s the sponge over my poor brain even as I pa.s.s 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...o...b..es: "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 viatic.u.m, 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 compa.s.s 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; subst.i.tute the--sign and you have the hyperbola with the antagonistic foci, the desperate curve that dives into s.p.a.ce 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 bombsh.e.l.l; 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.