The Affair of the Apricot Millefeuille
~ or ~
The Infernal Infinitesimal
An exploit of the mathematicians Legendre and Fourier, as extracted from the reminiscences of
Charles Fourier
Eminent mathematician and world-respected authority in matters of numerical analysis and shortcrust pastry
Paris
MDCCCXXXVI
It was a bleak early day of the year 181–, and I was fastidiously making my way through a slice of apricot millefeuille that Legendre had foolishly left unguarded in the parlour of our Paris home. The wind was howling with particular abandon, and I licked the cream off my fingers with a satisfaction known only to those who are snug in front of a raging fire and indulging in an illicitly snaffled portion of viennoiserie.
We were at the time engaged in a matter of extreme importance concerning a gentleman by the name of N– B–, whose identity I shall not be so gauche as to divulge. Suffice it to say that he had recently been imprisoned on the island of St Helena after losing a disastrous land battle outside the Belgian town of Waterloo. At Legendre & Fourier, investigative mathematicians for hire, discretion is as guaranteed as the decomposability of arbitrary functions into harmonically related sinusoids.
Our perfect comfort was suddenly disturbed by the sound of knocking. A loud, vigorous rapping whose time domain representation possessed a distinctive sawtooth profile. The noise perturbed my friend Legendre most particularly. He adjusted himself irritably in his stiff-backed chair and scowled at the sheet of paper in front of him. The great mathematician Adrien-Marie Legendre is a tall gentleman of the old school, with a great shock of white hair atop a face like a thunderclap that has just received bad news. Were you to cross him in the street, you could be forgiven for thinking you had encountered a particularly malevolent meringue.
“By Bernoulli’s beard!” he cried. “Another minute or two of intense concentration and I might have made this examination question quite literally unsolvable. As it is, alas, there is a small but non-zero probability that somebody might get it right.”
As he scratched his despondent quill across the sheet of paper I heard Mme. Asymptote, our housekeeper, scurrying up the stairs. Her breathing became more and more laboured as she came nearer and nearer the top, rising in intensity and pitch as her distance from our door narrowed. The volume of her insufflations grew louder and louder with each passing second and yet still she seemed to be wrestling with the ascent. Upwards and upwards she climbed, each exertion more laboured and terrible than the last, and yet still with no end in sight. Eventually I could take the rising tension no longer, and flung the door open to meet her in the stairwell.
“Forgive me for intruding,” she said, through vigorous pants which I had foolishly left hanging in the hallway, “but there is a young man downstairs in a state of extreme agitation.”
Legendre’s imperturbable voice reached us like a blast of cold air. “Is that any reason, Madame, for neglecting to confine your speech exclusively to the harmonic equations corresponding to the sounds you wish to produce?”
Flushing a bright red, Mme. Asymptote reached into the sleeve of her gown and removed the small book of tonal analysis that my friend had prepared for her.
“Must you be so hard on her?” I asked. Legendre abruptly rose from his chair and clasped his hands behind his back. His jaw tightened and three buttons on his waistcoat popped simultaneously. I cringed. Ordinarily his anger was confined to a two-button pop. The last time I recalled seeing a third disengage itself in the same way was on the unfortunate occasion one of his examination students had obtained a pass.
“Two - uh - sine of five x squared,” began Mme. Asymptote, leafing frantically through the pages.
“If I do not maintain certain standards in the running of our agency, my dear Fourier,” Legendre snapped, “then what is to prevent our clients from going elsewhere?”
“Plus five cosine of two - no - three x,” continued the hapless Mme. Asymptote, the floor around her now strewn with torn out pages.
“Mathematical efficiency is to be our watchword, Charles. In this as in all else.” Cowed, I agreed to wait for Mme. Asymptote to finish. As I was about to take another bite from my delightful millefeuille, my friend slapped it violently from my hand.
“Thief!” His eyes flashed dangerously and his peaks of white hair seemed to grow in prominence. “I’ve got a good mind to strip you of your diploma,” he said, and would doubtless have followed through had Mme. Asymptote not interrupted us.
“And another sin of five x, I think, summed over zero to two pi.” She stood there, wiping the sweat from her brow, her face gleaming with exhilaration.
“I see,” said Legendre. “Well if he’s in such a hurry, then what are you waiting for? Show him right in.”
She trotted off, and I busied myself with picking the carpet clean of mistreated pastry. I had just about succeeded in reconstructing a tasty mouthful for myself when the door burst open and a delicately-featured face stared quizzically down at me. I hid my hands behind my back and hastily scrambled to my feet. The intruder disentangled himself from his wet coat and tossed it in my direction. Dropping what meagre crumbs I had been able to collect, I caught it.
“I hope you will forgive my disturbing you, gentlemen, when the hour is so tardy and the weather so drear.”
“Not at all, young man, come in.” Legendre shot me a self-satisfied look which I knew of old as a harbinger of some spectacular act of intellectual prowess. “I deduce,” he said, affecting an air of great nonchalance, “that you are English.”
“My word!” winced our visitor, having thrown himself into one of Legendre’s notoriously uncomfortable wickerwork chairs. “How can you tell?” My friend smirked delightedly.
“Your throwing style, monsieur – there is no other country in Europe where the elbow extends to that precise angle upon release of a damp overcoat.”
“How extraordinary.”
“And then, of course, there is the matter of your hair. It is worn in the English fashion, rather too long for a Frenchman, altogether too pomaded for a German. Oh yes,” said Legendre, settling himself into a chair with great satisfaction. “You can tell a lot by a man’s hair. Such as whether or not he is wearing a hat.”
“Then I hope you have no objection to assisting an Englishman in distress. Or indeed, five.”
“Five?” I started forward at this news and rushed to the door, prepared to count off the number of visitors I could see on my fingers. It did not take me long. “But there is only one of you,” said I.
“I left the others at our lodgings on the Rue —-,” said the man. “I am afraid to say that they are in no fit state to travel.” Legendre pursed his lips.
“Indeed?”
“We arrived in Paris for a long few days of entertainment, monsieur. A soirée en dehors des lads qui sont sur le tour. It is possible that they are currently more entertained than might be either healthful or appropriate”
“I see. And why is it that you seek the help of Legendre and Fourier?”
“My friend, monsieur, he is a poet. Among other things, I should confess that we are all poets.”
Legendre’s ordinarily furrowed brow became yet more heavily lined. A mathematician without equal (or, for that matter, congruence), he holds poets in a contempt that the ordinary citizenry used to reserve for aristocrats with heads. Behind our young guest’s back I implored him to remain civil.
“I bear you no ill will for that confession,” he said icily. “Proceed.”
“I will come straight to the point. In an inspired burst of poetic composition, my friend stumbled upon a piece of mathematics for which he was insufficiently prepared.”
Legendre and I made eye contact, significantly. He exhaled loudly and reached for his leather hold-all.
“By the time any of us had noticed what was afoot, the situation was out of our control. I beg you, messieurs, to come to our aid.”
“Mathematics, my friend, are not to be trifled with. They are for the professional, not the amateur. Great harm can elsewise befall. For example,” he remarked with a touch of malicious humour, “if your own health were at stake, would you not rather trust your wellbeing to a physician rather than a poet?”
“Oh but monsieur, I myself am a student of medicine also. My name is Keats. John Keats.”
Legendre bowed curtly as he reached for his hat.
“Well then, M. Keats, you are clearly a man of many talents. And while I regret to say that I am a man of one singular talent only, I swear before god that you will find it up to the task.”
He stormed past me to the door, knocking the millefeuille which I had so carefully reassembled back on to the carpet.
“Do not stand there dawdling, Fourier. There is mathematics to be put right.”
*
The weather being monstrously inclement, the search for a coach took us a considerable amount of time. It was only by terrifying a horse with his scowl that Legendre was able to hail one down.
“Montmartre”, said Legendre, rapping on the side door with his cane. “Et vite!”
Once we were dryly ensconced, Legendre and Keats sat side by side on the banquette opposite me, their expressions hidden from one another owing to their faces being on the front of their heads rather than the side.
“And what was it exactly that this poet friend of yours attempted to do?” Asked Legendre, smirking at me. I was grateful that Keats could not see his sly expression. “Was it, perhaps, the addition of one number to another?” He tutted solicitously. “Or perhaps it was a brief exercise in multiplication that disturbed him so? We mathematicians know how fiendishly complicated these tasks can be to the entirely untrained and ignorant pen.”
“I don’t think it was anything like that,” said Keats. “In fact, I don’t think I have the French to describe it, but in English we would call it ‘dividing by zero.’”
Legendre turned visibly pale. I likewise gasped in horror.
“What is it? Have I said something wrong?”
Legendre rapped on the roof of the coach.
“Faster,” he cried to the coachman. “We have not a second to lose!”
“Toulouse?” Came the confused voice of the coachman. “I thought you said Montmartre.”
“By Gauss’s goitre,“ cried my friend, leaning out the window. “Hurry! Lives are at stake.”
*
The scene that awaited us at the young man’s lodgings was one that I will remember until my dying day. Never before had I seen anyone try to pair a soft cheese with a young Bordeaux, and the sight of it chilled me to my core.
The remainder of the suite was in a concomitant state of consternation. Two men in an advanced state of inebriation were striking dramatic poses on various items of furniture while a dark-haired man sat astride a crystal chandelier overhead. The fug of smoke, ripe cheese and human exertion was practically visible in the candlelight. Through the haze, I could detect with difficulty an older man slumped over a table in the corner with his head between his hands. My English has never been as good as I would have liked it to be and so it is possible that I misheard him, but it seemed to me that he was mumbling about daffodils.
“May I introduce my friends Shelley and Coleridge,” said Keats, “and that man riding the lighting fixture to a record time at Ascot is Lord Byron.”
“What about that man in the corner?”
“That is William Wordsworth.” He leaned in conspiratorially. “To be honest, it was a mistake inviting him. None of us thought he would come. He’s spent all the holiday moaning about the food and rather harshing our vibe.”
Shelley and Coleridge staggered towards us.
“I see two vast and trunkless legs of stone” said Shelley. “And between them,” he crowed, giggling, “I can see a willy!” Coleridge, meanwhile, gripped Legendre’s lapels firmly between his hands and breathed a disgusting vapour directly into his face.
“You’re not from Porlock, are you? I detest people from Porlock.” With a swift movement of his cane my friend extricated himself from the poet’s grip and thrust him backwards into an awaiting sofa.
“Opium?” he asked Keats. The Englishman nodded. “I thought as much. It explains the behaviour I see around me. But it certainly does not explain that.”
He pointed to the centre of the room with his cane. There, above the table on which I had noticed the substandard wine and cheese pairing, floated what I can only describe as a monstrous puckering. At that one single point, the universe appeared to have folded in on itself and collapsed beneath the weight. All shapes and objects around it were distorted as through a trick mirror, bent to accommodate its fiendish geometry.
Legendre plucked a button from off my coattails and tossed it gently into the room. At the very highest moment of flight, its parabolic trajectory began to waver, and the button pulled as on an invisible thread towards the puckering in the centre of the room. Before our very eyes, it squeezed through the ghastly sphincter and winked out of sight.
I confess to my eternal shame that at this moment I cried out in terror.
“What’s that?” Came a commanding voice from the ceiling. “Who is’t that cries out like an ickle baby poet who has just seen a bunch of daffodils and thinks they’re weally weally adowable?”
In the corner, Wordsworth exhaled loudly and buried his face yet deeper in his hands. He uttered some remark about Westminster Bridge that sadly escaped me.
From his acrobatic pose on the chandelier Byron performed a forward somersault and landed at my side with a flourish.
“What’s this?” he asked as he seized my jowls with a vicelike grip. “A foul monster from the infinite abyss, no doubt, grown fat from feasting on the souls of men. Well I will have you know, demonic fiend from the hereafter, that I am Lord Byron - and I am mad, bad, and dangerous to – no!” He removed a dagger from his belt and would no doubt have stuck it in me had the pull of the black void not sent it flying from his hand and into the depths of space.
“Legendre,” I cried, though scarcely able to breathe. “Help me!”
My friend glanced in my direction as he searched through the papers on the floor, and dispatched Keats to my aid. This latter was able to get my aggressor off me by pointing in the direction of the window where – or so I understood from his English – that an attractive woman was passing by.
Attempting to mould my soft tissues back into the shape they had until recently adopted, I manoeuvred my way around the figure of Shelley, now impersonating a nightingale on the floor, until I had reached my colleague’s side.
“It is quite obvious what has happened here, Fourier.” He held up a sheet of paper on which a sheet of ink had recently been spilt. “The man has been dividing by zero without taking any of the necessary precautions.” I took the dripping paper from him, as he wiped his inky fingers on my cravat.
“What are we to do about it?”
“It is quite simple, my friend. We are going to have to multiply by zero.”
I stared at him. My eyes boggled. My nostrils boggled. Every part of my face that was capable of boggling proceeded to do so with intensity.
“Have you gone mad? That will only make things worse!”
“Will it? Or am I in fact the pretty little daughter of a faërie queen?” He cast his normally grave eyes upon me, and I saw from their uncharacteristically friendly twinkle that something was amiss. The smell of opium in the room was overpowering, and my friend’s slim frame must have rendered him particularly susceptible to its effects.
“Legendre,” I cried. “Snap out of it!”
His countenance took on a benign smile, as of a man attempting to make eye contact with himself. “I’m hungry,” he said. “And you are a tasty fly.” He released the hold-all within which he carried all his mathematical impedimenta, and I watched helplessly as it slid into the wrinkled abyss at the room’s heart.
Coleridge bounded off a chair to squat at his side, and made a noise like a frog.
“Yes! Yes!” cried Legendre. “All must hop for the hour of the frogs!” And assuming a deep squat, he joined Coleridge in bounding about the room.
In a state of frantic terror, I pushed my way over to the window and yanked it open. The fresh air came as a welcome tonic, and I wafted as much of it as I could to the poor intoxicated souls behind me.
As I leaned against the windowsill I suddenly noticed the sheet of paper I held in my hand. Wiping the stray ink away I held it up to the light and scanned the final lines. They ran:
....our spiteful hero
Embarked upon dividing it by zero
“Keats!” I cried across the increasingly disordered room. “What is this?” The young man nimbly hopped between the tiles that flew out of the wall towards the pinched portal at the room’s heart. I could see that he had already lost one shoe to its embrace and was looking frighteningly pale. But I had no time for that. I thrust the paper at him and repeated my interrogatory.
“It’s what Byron was writing. Just as he had finished it the room started to go all puckery.” He crunched his face up to indicate what he meant. “It’s a verse from Don Juan.”
The chandelier smashed onto the floor behind us - the combination of an otherworldly pull and the considerable weight of Lord Byron had proved too much for it - and I was struggling to hear over the crash of glass.
“Don who?”
“Don Juan. It’s this epic poem he’s writing. Been writing it for years.”
“Is it any good?”
“It’s very long,” said Keats sniffily. “Makes you think the man’s compensating for something. After all, it only takes me fourteen lines to satisfy my public.” Keats ducked as a valise went sailing through the room and into the crack between the universe. “Blast,” he said under his breath. “My tuberculosis medication was in there.”
There was no time to waste. Already I could see the pull of the infinite point exerting a considerable force on all the occupants of the room. Coleridge and Shelley, both wispy men rendered gaunt by abuse of opium, were beginning to slide unawares toward it. My friend Legendre, likewise, now under the impression that he was a frog, could not hop fast enough away from it.
Wordsworth joined me at the window. “I should never have come to Paris,” he said wistfully as he climbed out on to the ledge. “Give me the Lake District any day. The food is better and the behaviour more decorous. Now would you be a sport and lower me gently to the ground?” Heedless of his disrespect to my mother country, and in an act of bravery designed to save him from a certain death, I am proud to say that I assisted him vigorously in departing by the window.
I turned my attention now to the elephant in the room. This was my sad intoxicated friend Legendre, who was now using an arm to impersonate a pachyderm of some distinction.
“It’s got a willy!” yelled Shelley with delight, sliding backwards at increasing speed on his stomach. I had to act fast. As any mathematician knows, there is only one truly fool proof remedy for a division by zero – namely, the imposition of a limit and a gradual approach by infinitesimals. It had worked for us before on the terrifying occasion of the Robespierre Integrand, which incident is recounted elsewhere in my remembrances, and I believed it could save us again.
As I had witnessed Legendre do on that occasion, I turned away from the pulsating singularity and searched for something with which to write. A phial of rust-coloured liquid teetered on the edge of the dresser, and I seized it up. Running up to the horizontal Shelley, I sat on his feet to hold him fast and untucked the white shirt from inside his trousers. Dipping my finger in the liquid I began to scribble upon its virgin surface.
“My laudanum!” Cried Coleridge.
“My shirt!” Cried Shelley as he attempted to wrestle it away from me.
“Ribbit!” Cried Legendre, sticking his tongue out as he was drawn to the mouth of the darkling chasm.
“If I were sober, this would make a cracking sonnet,” said Byron, performing a headstand on a bust of Chateaubriand.
Pinning the writhing Shelley’s arms beneath him, I continued my work. Studiously avoiding eye contact with the void, I began redefining the limit on Byron’s division. Instead of zero, I assumed he had divided by a different, arbitrary number, and slowly began to shrink its value.
The wild wind which had until that moment shaken the room to its foundations began suddenly to abate. I could feel myself being eyed suspiciously by forces beyond my ken but I knew I could not stop. Inch by inch, Revolutionary centimetre by centimetre, I cinched my denominator tighter. Down its value plummeted, past one, past one half, down and down further into the fractions, until the previously unapproachable horizon of the nullity began to cautiously make itself felt at the edge of vision.
The room was almost completely still now, the untoward movements had all but ceased, but I could feel deep within my bones that the danger was not yet past.
I was sweating through my jacket, not least because Coleridge had now deduced that I was wasting his most precious intoxicant and was desperately engaged in lapping up the marks I was leaving on Shelley’s shirt. This latter, moreover, was becoming obstreperous, and had succeeded in biting his way through my shoe and causing me no small pain with his incisors. My shouts to Keats to assist me were to no avail, because Byron had knocked him over the head with a Grecian urn.
Despite these intense pressures upon my spirit, I completed the mathematical operation and turned my ire in full upon the weakening eye of infinity at the room’s centre.
“Be gone,” I cried, pausing with satisfaction as I approached the climax of my oration, “I tend the limit to infinity!”
With an almighty clap as of a balloon inflating, the hole in reality unknitted itself. A wave of unbridled energy blew us to the room’s corners, while a flurry of papers and notepads hung in the air like clouds. Everything was quiet. The air felt fresher, too; cleaner, as though a deep-seated layer of filth had been lifted from my lungs. I rushed to my old friend.
“Legendre! It is your fly, old friend. We must take you home to your lily pad.” He stared at me in disgust, a lip curling into a familiar sneer.
“What rubbish you do talk, Fourier. I do believe it is the opium these Englishmen have been taking that has muddled your senses.” I was delighted beyond words to have my friend restored to his familiar misanthropic self, and could hardly resist beaming with delight. He clapped me a good one across the widest part of my face. “Now snap out of it. And stop grinning like a goon.”
He brushed himself off and retrieved his hat and cane from where they had been thrown.
“Well then. It seems like Legendre and Fourier have succeeded once again. Good night M. Keats. You may expect our bill in the morning. I should warn you,” he said with his familiar tight smile, “that it too will contain a fair number of zeros.” He bowed at the prostrate Keats, frowned ill-humouredly at the sight of his fellow poets taking turns at burying their faces in a sweat-sodden shirt, and followed me out of the room.
Once we were out in the street, breathing deep the fresh night air and strolling gently back towards our lodgings, I reached into my coat pocket and retrieved the precious squashed remains of my evening’s luckless pastry.
“I was worried for you, Legendre. I thought for a moment that you might not make it.”
“Don’t be such a buffoon, Fourier.” He prodded me viciously above the kidney with the point of his cane, causing me to drop the final delicious mouthful into the roaring gutter. “If I were gone, who would be left to look after you? Now hurry up, we haven’t got all day.”
*
There the remarkable story of our brush with death would end, had it not been for an unusual missive that was delivered to us by early post some weeks following. Made out to myself, it contained a letter in the hand of Keats. As I opened it up, however, I recoiled as an all-too-familiar sheet of paper tumbled to the floor. I lifted it cautiously, and recognised the same offending parchment that had so nearly caused our undoing.
In the interests of posterity I replicate it here, but urge any future readers who may discover these my notes - upon pain of death - not to read it in full. Else, unbeknownst to you, you may yourselves summoning the power of the infinite to suck the marrow from your own lives. Beware!
At that point the ingratiating waiter
Bent low and reapproached him with his bill
“You ruffian,” cried Juan, “dare inflate a
Modest luncheon where I scarcely ate my fill
To such a sum that I would swear is greater
Than Midas left his kinfolk in his will?”
And seizing up the bill our spiteful hero
Divided him in two – and it by zero.