Blesok no. 52, January-February, 2007
The French Mathematician
– excerpt from the novel–
I must appear a pathetic figure to that person looking down from that fourth-story window beyond the schoolyard wall. It looks like an artist's studio. Who knows, perhaps I am being sketched with a scrap of charcoal at this very instant—a slight youth with untidy black hair, strong eyebrows, mouth like a sparrow. I have been standing in this corner of the schoolyard since the beginning of morning recess, huddled under an elm, caught in the crisp shadow of its branches. The smell of burning paper fills the yard. Smoke twisting up from a nearby incinerator spreads in a cursive script across the clear winter sky. Arms crossed, fists clenched, fingernails cutting into my palms, I feel sorry for my short, thin shadow which has hardly changed since I came to Louis-le-Grand three years ago. Fifteen last October, I could be mistaken for a twelve-year-old. The boys who started with me have grown noticeably during the past year: Pimples bud from their faces, the down on their cheeks has darkened, their thighs and voices have thickened. I grit my teeth. My recent demotion was due more to lack of physical development than poor performance in a few subjects.
Let them have the present—the future belongs to me. When their idiotic games turn to stone and their laughter vanishes in the wind, my thoughts will live on, not only in this language, but in those which today still lack alphabets. From the time I learned to read under Mother's prodding eyes, I have been intrigued by the magic of ink and sensed that my name would be preserved in print. Is this why my inner voice is so pronounced? Am I projecting myself into the future by means of this inner voice? A message put in a bottle and thrown into the ocean. If I am destined to be written about, my conscious effort here and now might influence how my life will be represented.
– Galois the girl! Galois the girl!
A burst of warm vapor blasts my ear. Before I am able to retaliate the name-caller is sprinting across the yard, laughing wildly, shoes crackling over small pools laced with ice. I hate them all —the students, the teachers, the Director. This school is no better than a prison. Why did Father insist I study in Paris? I was happy at home in Bourg-la-Reine. Mother could have seen to my education. After all, she taught me to read and write, introduced me to Greek and Latin, and later made a point of reading from the Bible. Yes, she was demanding, harsh at times, especially when it came to the Bible, but there was always Father's company to make up for this. My happiest memories are with him: times when we would sing together, recite his own poems, entertain guests with short plays based on the Iliad, Racine, even Shakespeare, who was such a favorite of Father's that he claimed the name was a corruption of the French Jacques-Pierre. We often performed the ghost scene from Hamlet: He was the dead King, and I, the Prince. I became so worked up over the scene that I trembled and tears blurred my vision as I listened to Father. The guests would break out in good-natured laughter and applause. Those were bright days, a world away from this gloomy place.
Four boys are playing a game that involves throwing fingers, adding the number each has thrown, and counting clockwise. The player on whom the counting stops is then slapped on the back of the hand by the others. If they counted in the other direction would the same player be slapped? I see at once, accompanied by a sense of certainty, that if the sum is even and the count begins with the same person in both cases, then the direction affects the outcome; if the sum is odd there is no difference.
Mother's voice suddenly comes to me, harsh and high-pitched. I remember how the sinews in her neck would twitch as she read to me from the Iliad. My favorite was the story of Ajax. How I felt for the poor tragic hero! I understood his dejection and that desperate last act. Often, when refused something, or when I did not get my own way, I would take my wooden sword and run off to the back of the house, where I pretended to be Ajax. On one occasion, buried in the maple's red leaves, I imagined myself hovering over my prostrate body, looking down sadly on the victim of a cruel fate, deriving a strange pleasure in feeling sorry for myself. There is no time for such games in this prison, no opportunity for make-believe, no place for privacy. I sleep with forty others in the dormitory, “wait in a line to wash each morning, eat in the refectory and attend classes from six in the morning until nine at night. Continually in the presence of those I despise, my only respite is the liberating dark when the dormitory lamps are extinguished, and I can follow memory and imagination without interruption.
And now, after a term in the third and final year, this demotion to the second! Confinement in this prison extended by another six months!
Rhetoric! I could have easily passed that useless subject. They call me dunce. I will show them! I played the fool in class to make fools of them. I did poorly in Rhetoric in order to spite the teacher, whose smugness I could not bear. Effective use of language! I refused to follow his rules of grammar and construction. To defy him, I often submitted work written with the rules of my own grammar, constructed sentences that expressed the movement of my own mind, not that of others. I enjoyed these games whose meaning was perfectly clear to me, but which confused him.
Sometimes I would daydream of a private language; magic sounds and symbols that transported me to another world. But the teacher's growl would tear me from such reveries, and he would read my nonsense to the others. The students howled, the teacher grinned. I would bite my lower lip, suppressing my delight, feeling superior to them all because I had set out to reduce them to laughter, to make them look ridiculous. The more they laughed, the stronger my heart pounded in the knowledge of my superiority. Only the foolish and small-minded gaped like that. And then I would imagine the exiled Napoleon, arms crossed, brooding on a rocky beach, waiting for his moment, while the Royalists spent their evenings in frivolous laughter.
Gleaming on the chapel's cross, a crow complains raucously, then flaps to a bough above my head, where it preens its wings. A large feather twirls through the branches and falls at my feet. I admire its shape and point. It would make a fine quill. The ancient Greeks augured the future from birds. What would they have made of this feather?
Over the past year dark feelings have been stirring within me, not only hatred of those around me, but a frustrated desire for something I cannot define, an ambition without a goal, a sense of leaving childhood and moving toward a distant, barely audible calling, which sometimes sounds like nothing more than a faint echo of my own voice, and other times a voice I have never heard before, calling compellingly in a language I do not fully understand. I know I am destined for something, though I do not know exactly what.
Three boys appear from the latrines, glance in the direction of the teachers' quarters, and hurry across the yard. Two of them are m the third year, the other is in the second. This one seems frightened, and he is almost carried along by the others. Another initiation! One of the third-year boys tried to initiate me when I first arrived. He spoke of a secret source of pleasure, a way to overcome sleepless nights, said it was the only way to manhood. I followed him into the stinking latrines. Smiling, swearing the pleasure was beyond words, he unbuttoned his trousers and put his hand inside. I ran out and retched in a corner of the yard. There and then I vowed never to degrade myself through that disgusting practice. Not because the Church condemned it, but because I realized that those who abstained were somehow superior to those who indulged. To abstain meant to overcome instinct with intellect, to live by a higher morality, to raise oneself to the level of the hero. A few days later, the pimply third-year student accosted me in the yard and shouted in my face that I would never grow to be a man. I replied that I would sooner kill myself than follow his example. He gave a green-toothed grin and ran off toward the latrines.
The bell rings chillingly for the next class. Mathematics. I have not studied it before. Mother taught me little more than the basics of arithmetic. I have been in Vernier's class two weeks, doing a course for beginners. Still smarting from the demotion, I cannot motivate myself to work or even to take an interest in the subject. I spent the first few lessons at the back of the room, brooding, sketching Napoleon's profile instead of doing the exercises. Damned demotion! I will show them! The teachers conspired against me. They misrepresented me to Father, who read their reports when I went home for Christmas two months ago.
Called to the study on the afternoon of my arrival, I was struck by how he had aged since my last visit home. His dark hair had turned ashen and there was an unusually somber look about him. Was he working too hard? He was the mayor of Bourg-la-Reine, a position he won during the Hundred Days, and which he continued to hold even after Napoleon's defeat. When the monarchy was restored he took an oath to Louis XVIII without renouncing his strong liberal views and ardent Republican sentiments. Father is neither a hypocrite nor an opportunist: He took that oath in order to keep a monarchist from the position. Apart from his duties as mayor, he also runs the town's boarding school, which he inherited from Grandfather, who obtained it during the Revolution. The school is a profitable business, and I know Father is grooming me for it.
– You have disappointed me, Son, he said in a flat voice, pinching the flesh under his chin. I refused to have you demoted last August because I believed in your ability. I hoped you would prove me right. Seems I was mistaken.
I did not look up from the inkpot on his desk. Were my poor grades responsible for the change that had come over him?
– But your reports show no progress. Quite the opposite. How do you account for this? “Apart from the last few weeks when he has worked a little, and then only from fear of punishment, this student has generally neglected his studies. The strangeness of his character keeps him from his companions.”
– And this: “Though somewhat strange in his manner, the student is very gentle and filled with innocence and good qualities. He never knows a lesson badly—either he hasn't learned it at all, or knows it well.”
– What am I to make of this, Son? Is this why I sent you to Paris?
I felt a pang of remorse. Loving Father more than anybody else, I wanted to please him, to obtain good grades, but I could not overcome the hatred I felt for everyone at school.
– Well, there's no avoiding it this time. I cannot intercede again. You will just have to endure another six months in the second year. You might as well get something out of the demotion. Do something different, maybe a course in mathematics.
Placing his hand on my shoulder, he looked at me tenderly. I wanted to apologize, to promise I would do better, but I fought back an impulse to embrace him and cry in his arms. Noticing the emotion rising to my eyes, he moved to the window overlooking the town square, pulled aside the lace curtain, and gazed at the church at the far end.
– We're living in uncertain times, he said. There's no telling what will happen next year. You can't rely on anything but your own intelligence, Evariste. A good education will serve you well for the rest of your life.
I wanted to see him cheerful again, and if a course of mathematics were needed to dispel that worried look, so be it.
The bell-ringer threatens the stragglers and those reluctant to leave their games. Holding the feather in one palm, I balance it against a thin spread of light in the other.
Should I make an effort? Should I stop playing the fool, at least in mathematics, for Father's sake?
I am now alone in the yard, with a few springy starlings pecking at my cobbled shadow. As though signing a resolution, I use the feather's bony point to scratch my name and date on the back of my hand crimson from the cold: Evariste Galois, 14 February 1827.
– Geometry, announces Vernier.
In three weeks of arithmetic his voice has never risen above a dull drone, and now he introduces the new topic as though it is the title of an epic poem he is about to recite by heart. For the first time since my demotion, I sit up and take note. Geometry. The word has a certain resonance. Or is it just Vernier's enthusiasm coming through? His words strike a chord within me, stir me, make me quiver with wonder. At times, for minutes on end, as he paces and gesticulates, I forget the shadow of the window's grille falling obliquely on my bare hands, forget the anxiety and frustration that has unsettled me during the past year, forget my demotion. During these moments I neither love nor hate, neither fear nor hope. Drawn from the emotional chaos of recent months, I feel at ease in the order and certainty of geometry.
– Euclid's Elements, Vernier continues.
He is about thirty, with black hair fringed across his forehead and spectacles made of circles, lines, and arcs. He wears gray gloves to prevent chalk dust irritating his dermatitis. As he raises the book high above his head, its image is skewed in the lamp's concave reflector.
– The Book of Books, gentlemen. The most influential text ever written, more so than the Bible. It has leaped across centuries, nations, languages, religions. Its relevance is timeless; its nationality, universal; its language, logic. Truth, gentlemen! Absolute, transcendental truth. You have done your exercises in arithmetic: You have added, subtracted, multiplied, divided, and extracted roots. Well and good, all very useful. I would be the last to deny the usefulness of arithmetic. It goes hand in hand with money, and, as we know, coins make the “world go round. It is vital to examiners, bankers, and generals. It determines pass and fail, measures profit and loss, counts life and death.
– But, gentlemen, despite its prevalence in the affairs of the world, arithmetic has nothing to do with the spirit of mathematics. That spirit finds its first expression in geometry. The spirit, gentlemen! Think about it. A spirit not unlike the Holy Spirit, and just as the Holy Spirit cannot be apprehended merely by attending church regularly on Sundays, dropping a coin on a collection plate, saying one hundred Hail Marys and two hundred Our Fathers, so the spirit of mathematics cannot be grasped by the repetitive exercises of arithmetic. Many are called, gentlemen. Some are chosen to serve in schools, and others at higher institutions such as our Polytechnic, but few, perhaps a handful in each generation, are chosen to become initiates, to partake of the mysteries, to serve the spirit in the inner sanctum.
I am entranced by the brown book whose binding has frayed, its gold lettering faded. And when, looking directly at me, Vernier asks “which of us” will be among the chosen few, my heart leaps. I want to know more about Euclid.
Is absolute truth possible in these confusing times, when countless groups claim to have the truth, and the world seems on the brink of chaos? Royalists, Republicans, Bonapartists, Socialists, Saint-Simonists, Anarchists, the reemerging Jesuits—they are all active among students, all seeking to make converts to their particular truth. There seems no end to it, especially when combinations of some groups produce new truths. Perhaps Euclid, having stood the test of time, might dispel the chaos churning inside me and help make sense of the world. My attention is caught by Vernier's gloved hand —the book's truth is too pure to be held by naked flesh.
– Since 1482 more than a thousand editions have appeared in print, he continues, “walking “with measured strides between our desks. We know Euclid compiled the work in thirteen books covering the geometry of triangles, circles, and various quadrilaterals; the theory of proportion; number theory; irrationals; solid geometry. As for geometry, which will be our main concern, we know that he based everything on twenty-three definitions and ten axioms, five of which he called common notions, the other five propositions. From all this, we know the mind of Euclid, but what of the man?
– I put it to you, gentlemen: Does the man matter in the dazzling light of this creation? Beyond the fact that Euclid taught in Alexandria around 300 B.C., probably summoned there by Ptolemy, little else is known about him. We do not know where and when he was born. We do not know his nationality. Some maintain he was not Greek at all, but Egyptian. We do not know how he felt about the gods of the day; whether he was kind or cruel; whether he was fond of wine; or whether he had children. The personality does not matter, gentlemen. For most of us it disappears without trace in four generations, let alone forty. What matters is the idea, the ideal, truth. But the truths in this book are not easy to grasp. There is no royal road to geometry. It will require hard work and concentration. Some of you will fall by the wayside early, but those who persevere will see a world more dazzling than Alexander's conquests, one that withstood the might of Imperial Rome and outlived the shadow of the Dark Ages. A world to which our French mathematicians have contributed greatly since the time of Descartes.
– The call has gone out, gentlemen. Who will heed it? Who will serve the spirit? Who wall deny themselves for the sake of truth?
My hand shoots up involuntarily. The room explodes with laughter, which once again proves their stupidity. They are insensitive to the spirit of mathematics, unable to grasp the subtlety of Vernier's words. Let them laugh. It confirms that I am smarter than them.
How can Vernier tolerate them? He should threaten them with detention. Why is he smiling faintly, rubbing the lenses of his spectacles, squinting as he holds them up to the light? When the laughter finally subsides, he picks up another book and holds it up to the class.
– Euclid has been excellently interpreted and presented by Adrien-Marie Legendre in his Element's of Geometry, a text written for the purpose of teaching the subject in a modern way, and from which I will be drawing heavily. Open your books, gentlemen, and let us commence.
Held firmly in his gloved hand, the chalk scuttles quickly across the blackboard, striking at periods, scratching in underlining, sometimes screeching in its haste. From the back of the room, straining to read the board (my eyesight has always been weak, but it appears to have deteriorated in the past few months), I do my best to keep up with Vernier and copy in my book the definitions, propositions, axioms, and theorems. One idea leads to another, naturally, effortlessly, like the notes that combine to form a pleasing melody. As Vernier erases the restrictive board (the night sky is not wide enough to accommodate the possibilities of geometry), I contemplate the first definition: A point is that which has no part. It seems that a point is and is not. If I represent it on the page, it is no longer a point. If I try to grasp it in thought, it vanishes. I imagine a point moving at a great speed; at that instant it is both a point and a line, a particle and a process. If I grasp the line, I lose sight of the point. If I focus on the point, the line proves an illusion. Astonishing that something so intangible should be the basis of all geometry! In a flash, I see the indivisible point as the seed of creation. Perhaps the universe exploded from the primal point. Perhaps God is the primal point. Perhaps the soul is nothing more than a point.
– What use is geometry? mumbles a student in the front row.
– Use? snaps Vernier.
He takes out a coin and extends it to the student.
– Here! If you cannot see the value of knowledge as an end in itself, I had better pay you for learning.
The student bows without a word.
– Now for a few simple proofs, Vernier glares. Those not completed in class must be done before our next lesson.
The student beside me bites his knuckles over the first problem, but I see the proof at once, as though a long-dormant sense has suddenly been awakened. My heart beats strongly against the edge of the desk, my hand shakes, I am unable to keep my writing on a straight line. I complete the exercise in a few minutes, most of them in my head, while my neighbor sighs, scrapes the floor, and chews his pen in frustration. The bell sounds. Shuffling above Vernier's instructions, the students bustle from the room.
There is a fire in me. I have never been in love, and what little I know has come from books, but it must be something very much like this, a flesh-consuming fire. I want to know more about geometry. I want to ask Vernier whether I might borrow Euclid and Legendre. But how can I explain my insight? Is it intuition? A kind of sixth sense? A feel for geometry? He would not understand.
At the start of the afternoon recreation hour, I hurry to the library. Locating both books, I sit at a window overlooking the courtyard and the Preparatory School, an annex of Louis-le-Grand where students train to become secondary school teachers. I pick up Euclid first, feel the textured cover with my palm, and then flick through the yellowing pages, inhaling their mustiness. I read all that Vernier covered, and continue through more propositions, theorems, and proofs. I read quickly, as though it is a biography or a work of fiction, comprehending everything at once, my mind moving as though it has finally found its element.
I make my way to Rhetoric class as though in a trance, and stare through the lesson without opening my book. The afternoon has turned gray, and grayness now clings to the dreary walls, embraces the potbelly stove bulging with fire, leans against the blackboard, gathers around the overhead lamps. After another three unbearable lessons I hurry back to the library.
As though surprised in an illicit activity, I shut the book with a thud, my index finger between the pages. An elderly fellow with a fleshy smile peers over my shoulder.
– They haven't been touched in years, says the librarian.
He leans over me, his puffy hands on the table like two spotted toads.
– It's the language of the future, he whispers. When Greek and Latin die out, geometry will be the language of the world.
What does he want? He is depriving me of valuable reading time. He straightens up and becomes thoughtful, hairy nose wheezing at each breath. I am struck by the thought that, despite the book's eventual decay, the ideas of geometry will remain intact for another two thousand years and that ideas are like parasites, burrowing into suitable minds, feeding on young thought, moving through time in the vehicle of the human skull. I would gladly give my mind for the sake of these ideas, sacrifice myself for a new theorem. Suddenly, as the librarian is mumbling something, as two boys are chuckling at a corner table, as the night muzzles the window, as a church bell's toll is muffled by the thick dark, I know that I am destined to become a mathematician. Ignoring the librarian's chatter about mildew slowly destroying the library, I spring up from the table. He grasps my forearm and offers me the books, even though they are not for loan, saying they will not be missed for a few days. I embrace them like a youth embracing his first love.