In the House of Geometers – David Cleden

In the House of Geometers – David Cleden

Cover art
January 2022

Luca heard the voice of god, speaking to him through the arithmos. Everywhere he looked: arithmos! It was there in the black-and-white-patterned tiles stretching the length of the cathedral’s nave; those intricate patterns of triangles-within-triangles, squares-within-squares — a mind-numbing, near-infinite variety of shapes within shapes.

As he stood amongst the other boys in the choir stalls, the silent eloquence of the arithmos overwhelmed him: the way the fluted stone arches reached up into the vaulted gloom! He could call to mind the equations describing those graceful curves, and doing so, felt a shudder of pleasure run through him. It was risky, though. If Ecclesiast Vittori knew of Luca’s illicit understanding, he would beat it out of him. The Church guarded its arithmos knowledge jealously. As with any drug, power lay with those controlling its supply.

Luca let the arithmos sweep through him, befuddling his mind as the transcendent beauty of the mathematics seized him. He realized now that arithmos needed no god to speak on its behalf. It was enough by itself: thrilling him, energizing him like a narcotic.

Concentrate! He must not draw attention to himself. Ecclesiast Vittori would consider any misbehaving boy to be disrespectful of the Church, doubly so in front of such an august congregation.

Luca’s lips moved by rote. His voice was weak and reedy at the best of times and he could no longer tell if it added to those of his fellow choristers. Their singing seemed to wash away his essence, leaving behind only an ethereal voice like the roar of an endlessly breaking wave, speaking to him of a transcending beauty that lay at the heart of geometry and calculus and many other things besides. Arithmos! It drew him on as though he tumbled down some invisible gradient.

Luca felt himself sway.

Ecclesiast Vittori was glaring at him. The rotund man started to move along the line of chorister stalls, but the psalm was ending and Luca sank gratefully onto the wooden bench. Vittori, caught unawares, retreated.

Luca’s eyes strayed again to the fluted stone columns spreading in graceful arcs far above, like the boughs of a fossilized tree. Silver chains hung between them, each commemorating a deceased individual of high rank, yet all Luca could think of was the mathematical formula which described their catenary. He’d seen its equation passed amongst the brighter boys on grubby, crumpled sheets of paper; a dirty secret to be shared and sniggered over after lights-out.

Sweat ran in rivulets down the curve of his spine. He felt faint, and a little sick.

Why did the arithmos affect him so?

Because you are weak, he thought.

He knew that in the Chapel of Cartesia, a brooding, heavily shadowed side-chamber off the nave, there hung a painting by the much-admired third-century artist Flavali; unquestionably the greatest work of his lifetime and perhaps, some said, of all time. It depicted Mother Arithmetica flicking her abacus beads in some complex computation while at her feet earth-bound mortals lived and died, suffered and prospered, according to the whim of her calculations. Visitors sometimes came to the chapel to quietly contemplate the grace and beauty in its artistry. Luca, too, had stood before the vast canvas, scarcely aware of the minutes slipping by.

Yet that was as nothing to the effect arithmos stirred in men’s hearts. Except for a stunted few, there wasn’t a man or woman who didn’t respond to the allure of elaborate geometric patterns, or the intrinsic beauty found in an elegant proof. For some, it was a kind of intellectual ecstasy, but for those without sufficient self-control, it could become an addiction. Arithmos commanded an attraction a hundred times more powerful than that produced by mere canvas and pigment. The brain craved that stimulation as though it were a powerful drug. This was, Luca supposed, simply the way people’s minds worked, intoxicated by the elegance of geometry, algebra, and equation. And it served the Church well; the drip-feed of arithmos in worshipday sermons ensured a congregation’s devotion.

But Luca craved more than the Church would permit, drawn inexorably to greater knowledge the way a flame captivated a moth.

His head filled with a rushing noise. His hands were shaking.

Get a grip!

He noticed a woman in the front row of the congregation staring at him. She was young and her elfin features quite striking, but there was also something hard and glacial in her expression. Luca felt a hotness rising through his body. He glanced away, but each time his eyes slid back, she was still watching him.

Think of the damn patterns, then, he told himself.

Tessellations. (There! He’d used the word, that dirty and perverted expression — and didn’t care that he had, though he would never dare do so aloud.)

It seemed to him the diamond-patterned floor tiles spread like the dappled surface of a monochrome ocean, lapping at the space between the choristers’ stalls and the altar. Geometer shapes, obviously. He saw again how small patterns repeated at larger scales: diamonds arranged to form the sides of endlessly stacked three-dimensional cubes, until the mind’s eye flipped and suddenly these were merely pieces of a much larger hexagonal construction. Perhaps if he could fly like a sparrow up into that vaulted ceiling, he would see yet more patterns writ large.

And, though he knew he should resist the sublime beauty of arithmos, should reject the shame of knowing, he couldn’t save himself. He reached out to embrace it and felt himself falling — or rather, it seemed that the world reached out to him, the patterned floor rising to meet him. Briefly, Luca felt the cool hardness of those tiles against his cheek, and then he was gone to some other place.

The stench that jolted him back to consciousness brought vividly to mind all the very worst kinds of decay and putrescence, as though some evil creature had crawled up his nostrils and died there.

He lay on the cold stone floor of the sacristy. Vittori stood nearby, face contorted with rage, but it was the woman from the congregation who stood over him, recapping a bottle of smelling salts.

“A touch of heat-stroke perhaps,” she announced to the room. “Give me some space to attend him.” At her icy stare, Ecclesiast Vittori frowned and retreated, though he lingered in the passage outside.

“My name is Coriola,” she told Luca, making a play of loosening the tunic around his neck, although the air was cool enough in the sacristy. She leant closer so that they could not be overheard. “So, you have some talent?” She raised an eyebrow. “An affinity for arithmos? That’s unusual for someone so young.”

He felt himself flushing. “I — I don’t know what you mean.”

She smiled, as though seeing right through his lie. “I watched you. You were drinking in the arithmos, weren’t you? You allowed its beauty to swamp your mind. Be careful. I’ve seen untrained minds become addicted to arithmos all too easily. It will take everything from you, if you let it.”

Luca said nothing. Her dark eyes held him prisoner and he found it impossible to look away.

“I wonder — have the Ecclesiasts recognized your talents yet?” She gave a little laugh, bright and brittle. “The Church is adept at subduing its congregations with the soft caress of arithmos. How delightfully ironic if they’ve failed to notice a Bright amongst their own.”

“I obey the True Computations of the Church,” Luca mumbled, the rote words coming easily. “Mother Arithmetica grants us a glimpse of her wondrous grace through her teachings of arithmos.” He wondered if Vittori really was out of hearing range. If there were to be beatings later, better that they were swift and merciful.

“Of course. Of course.” She patted his arm and stroked his hands in a matronly sort of way. Leaning in close again she said, “The Church closely guards arithmos knowledge for one reason only. Its power and influence in this city depend upon it. But that power can be broken. There’s more to arithmos than you’ve ever dreamt of, more than the Church will ever admit to knowing. Knowledge of such sublime elegance and ineffable beauty that it can drive a person to the edge of insanity! Things that might just satisfy the yearning I see inside you.” Her voice became almost inaudible. “Dark equations.

She stepped away and was gone. Luca became aware of something pressed into his palm. A square of card. He had enough presence of mind to slip it into his tunic before Vittori could see.

Later that evening, after his beating had been administered — neither swift nor merciful, as it turned out — he took out the card and held it in a beam of moonlight shining through the high dormitory window.

The House of Geometers.

The address given was in an unfashionable quarter of the city.

He listened for a while to the sound of the other boys; their snores and grunts and soft breaths coming like sighs. Carefully, Luca tore the card into tiny strips, destroying all trace of the printed words.

But the address was seared into his memory, and that was all he needed.

Luca slipped silently from the dormitory under cover of darkness. The Church kept no guard over its orphanages. Any youth ungrateful enough to disappear from its care was of little concern, but he planned to be back in the dormitory before dawn, so that Ecclesiast Vittori would be none the wiser.

By day, the city of Orlondre bustled. Its streets were filled with the clatter of horse-drawn carts bringing fresh goods from the docks to the busy markets. Children skipped and dodged between the tradesmen’s stalls. Bakers baked, butchers butchered, tailors set out their bolts of cloth under bright awnings, and in the richer parts of town, physicians and apothecaries hung out brass name-plates in expectation of business. And on every street-corner (or so it seemed) was a chapel of worship where the faithful could receive communion in the form of some small measure of arithmos. The Church would permit no more than this. How could a city function with a citizenry distracted and intoxicated? If moderation was the key, then it fell to the Ecclesiasts to maintain those limits.

Luca felt strangely out of place as he crossed the darkened central plaza. During the day, crowds gathered to gaze up slack-jawed and dolt-like at the ornate stonework of the Cathedral, drinking in the sublime mathematical beauty of its perfect proportions, the sweep of its alcoves and the patterning of stained windows. It served as a potent symbol of the arithmos wielded by the Church: a reminder of both the affliction and the blessing it brought on its citizens.

He passed on into a poorer quarter of the city, giving wide berth to the beggars on street corners. Many of them were clearly Burned, mumbling meaningless streams of numbers over and over. They paid him scant attention — and those who did raise their gaze had eyes focused on some different plane of reality.

One of his fellow choristers, Andros, had been the first to tempt Luca with arithmos. They had sat together one day in the shady cloisters, Andros boasting of knowing secrets, as the older boys were wont to do. He’d drawn in the dirt with a stick. First a square, then a triangle on each of its sides. (Oh yes, they’d known all the geometer names. All the children did, sniggering at the dirty little secrets they kept when they thought no adults were around). Then upon each triangle, Andros had drawn another square and triangle on each of its sides — squares and triangles, triangles and squares; on and on in a never-ending pattern, the clouds of dust rising as his stick scraped in the dry earth. Luca had felt his eyes bulging. It was the most beautiful thing he had ever seen. All those geometer shapes nestling together and not a single space between them — as though the entire world could be swallowed by this endlessly repeating pattern. What was there to stop it? What boundary could be drawn?

Then he saw the glazed look in the other boy’s eyes as his stick danced and scratched, the flecks of spittle on his lips, and the sheen of sweat on his brow. Only when Luca knocked the stick from his hand did he lay back spent, a vacant grin on his face. “Do you see? Do you feel it?”

Luca had been scared. He had scuffed his feet in the dirt, erasing the patterns before either of them could lose themselves entirely.

“Tessellation,” Andros whispered, and the unfamiliar word sent shivers down his spine.

The illicit thrill of it seemed a distant memory to him now; mundane and tepid arithmos at best. But this woman, Coriola, had promised more. She had promised him dark equations

The address proved hard to find. Three or four times, he must have passed the little archway cut into a high featureless stone wall, mistaking it for some service passageway or tradesmen’s entrance. Eventually he noticed a small brass plaque above the arch, where it was easily missed: ‘Hse of G’.

In the moonlight, he glimpsed a courtyard beyond the end of the passage and knew this had to be the place.

“And who might you be, boy?”

A shadow in the archway moved and Luca took an involuntary step back. A gaunt man rose swiftly to his feet, moonlight glinting from a blade held in one hand.

“Please, sir. Can you tell me — is this the House of Geometers?”

Cold, unfriendly eyes regarded him. “Are you some Ecclesiast toady sent to sniff around and cause trouble?” The knife danced in front of Luca. “Because this a private dwelling, see? And properly god-fearing. Be on your way, boy.”

“A lady came to chapel. Coriola —”

At mention of the name, the man stiffened.

“Stay where I can see you.”

He rapped the hilt of the knife three times on the stones of the arch. The sound rang out into the night’s stillness like a bell chiming.

Several minutes passed. Luca and the gatekeeper remained frozen in place as though a spell might be broken if either moved. Then a small figure appeared, silhouetted against the moon-dappled courtyard beyond. At her gesture, the man subsided back into the shadows. Coriola beckoned Luca forward into a slant of moonlight.

“You came,” she said, as if the matter had never been in any doubt.

Luca swallowed, trying to calm his nerves. “I want to learn more about arithmos.” The truth of that statement was suddenly overwhelming. He did! He wanted to know everything. He wanted to drown in arithmos, let the beauty of equations flow into his soul and fill every fiber of him. He wanted to feel the thrill of new understanding electrifying his senses.

“Do you, now?” Coriola sounded amused. She spoke quietly but there was a hardness in her voice that echoed from the stone archway. “That would mean giving up the only existence you’ve ever known. Arithmos will consume you, if you let it. The addiction has driven many people into madness. Learning is one thing, but the hardest lesson is stopping arithmos from destroying you.”

“Then I’ll learn how to control it!”

She laughed lightly, and Luca felt himself blush. “You’re young and bold and foolish. If you come to the House of Geometers, you cannot change your mind. Not ever. We operate in the shadows, right underneath the noses of the Church. We keep our secrets close. You’ll learn much here, but we are no school. Arithmos is a tool, a thing that rich folk will pay handsomely for in their exclusive clubs. You’ll learn how to peddle arithmos, how to send a connoisseur into paroxysms of ecstasy merely by enlightening them with the beautiful logic of some elegant theorem or via the perfect elucidation of a proof. What the Ecclesiasts preach to their docile congregations on worshipday is a poor relation to what we offer our clients. But we operate in secret and live perilously. This isn’t a decision to be made lightly.” She cocked her head as though examining him critically. “Last chance. Go back to your warm dormitory and erase this place from your thoughts.”

Luca felt as though he stood at a junction. In one direction lay an impoverished life of service to the Church; secure and predictable. In the other… He could not say. What was it that he truly wanted? The very stones of the archway seemed to hold their breath, waiting.

“I choose the path of knowledge.”

“Very well.”

Coriola turned and walked back into the courtyard, and just like that it was decided.

Luca came to the House of Geometers with nothing but the clothes he wore and a burning curiosity. Coriola would not allow him to return to his dormitory. By evening choral-song, doubtless Ecclesiast Vittori would be cursing him for the damnable inconvenience of his absence — but he would not be missed. Just one more lost boy, his place within the Church’s care soon filled by another waif.

Outwardly, the House of Geometers concealed its secrets well. Set around an inner courtyard only accessible via the guarded archway, its many stone-built rooms were tucked away amidst the neighboring buildings. Luca was given a room of his own, high in the eaves: cell-like and unfurnished except for a bed and a chair by the slit of a window. A dozen boys and girls of mixed ages occupied similarly spartan rooms in various wings of the building, but they were not encouraged to socialize outside of mealtimes. He learnt some of their names, but they seemed a strange set of compatriots: either steadfastly studious or intensely withdrawn, as though distracted by higher matters.

“Study,” Coriola instructed him. “As if your life depends on it.”

Each scholar received an individual program of instruction tailored to their ability and sensitivity to arithmos. What one person could absorb with no more than a pleasant buzz of understanding might send another into a paroxysm of helpless babbling. Luca’s personal tutor, Dr Frenkel, was a gangly, red-haired man with a wild, untamable energy and an infectious laugh. He was constantly in motion, striding up and down the tiny room that served as classroom (with Luca the sole pupil), gesticulating wildly as he drove home some abstruse point of number theory, or snatching up chalk to scrawl higher-order equations across the large chalkboard.

Luca would often feel himself grow faint as Dr Frenkel’s lessons progressed. The chalk dust clogged his throat, and such was the beauty of the mathematics spilling from his tutor’s lips that it threatened to overwhelm his mind. He would beg Frenkel to pause so that his clamoring heart could settle.

Simple geometry was done in a trice. Luca came to think of the subject as rather pedestrian: a treat one gives a child, not something fit for the discerning palate. From there, he progressed through trigonometry and on to differential theory. This — at last! — was more potent stuff and he began to struggle. Exponents and logarithmic functions left him sick for days in a delirium of such intense pleasure that he feared his racing heart might give out. Frenkel would take pity on him then, switching to telling stories from his colorful past, of which there were plenty.

One day, during just such a lull, he came and squatted on the edge of Luca’s desk and smiled down at Luca in a melancholy way. “What do you truly want, Luca, my boy? What has brought you to the House of Geometers?”

“I want to understand arithmos! I want to learn everything there is to know!” But instead of rewarding his enthusiasm, Frenkel looked pained.

“You misunderstand. The learning we do here is no more than a means to an end. This is a business, Luca. And all businesses — if they are to succeed — must return a profit.”

At Luca’s puzzled look, he stepped closer and lowered his voice. “Coriola is not to be trusted. We are all cogs in her grand machine. And when those cogs wear down, they must be replaced with fresh ones. She’s shaping you, cutting fresh grooves in your mind to serve her purpose. One day soon, your grasp of arithmos will repay her, perhaps in an original thought or even some elegant reworking of mathematical theory — the kind of thing that’ll command a decent price from the hardened addicts or the gentrified connoisseurs.”

It occurred to him this might be a test. “But I do trust Coriola. She’s doing a worthy thing.”

Frenkel looked somber. “Worthy? Then why do we skulk in the shadows? Why is the House of Geometers never to be spoken of outside these walls? You know as well as I what will happen if the Church discovers the true purpose of this place.” Frenkel patted his arm. “Understand that Coriola comes from a noble family brought low by unprincipled men. When she was just a child, her father was swindled in an unwise business venture. Overnight, the family sank into poverty. Her father, unable to withstand the dishonor he had brought upon them, took his own life. She will stop at nothing to make sure those responsible pay a high price — and her weapon of choice is money, because money buys power.”

When he saw the confusion on Luca’s face, he came and sat next to him. “There is a great deal of profit to be made from arithmos, as the Church understands only too well. Through the profits it has amassed, the House of Geometers has invested in many properties throughout the city, hidden behind false names and secretive accounts. Coriola is building considerable power from the shadows, but she won’t rest until all those who played a part in swindling her father are utterly crushed. And profit is key! We’re all part of her plan now, whether we wish to be or not.”

Luca bridled. “But there’s good in what we do — pushing the limits of our arithmos understanding, searching for new theorems.”

He smiled at that, but there was no warmth to it. “Yes, but ask yourself why, Luca.”

Dr Frenkel sighed and ruffled Luca’s hair. “If you won’t believe me, then let me show you.” He touched his fingers to his lips. “But this must be our secret.”

They heard footsteps approaching, slow and measured. Then the scrape of boots across a stone step, the rustle of garments being gathered in, and finally a grunt as a heavy body lowered onto the stone bench on the other side of the partition.

Luca remained still and quiet, as instructed, hardly daring to breathe. Frenkel, dressed in purloined Ecclesiast garb, was pressed close in the darkness of the confessional, close enough so that Luca could feel the man’s heat radiating off him.

Beyond the partition, a man spoke in a low, rich voice. “By the sanctity of Mother Arithmetica, I come before you to confess my sins in the hope of forgiveness.”

“Then through hope shall you be granted absolution,” Frenkel intoned.

There was a coded signal in this exchange; a tacit understanding passing between them. A moment later a silver coin poked through the slats of the confessional, dropping silently into Frenkel’s cupped hands. It was followed by another, and another; five in total — a not inconsiderable sum. Perhaps Frenkel had been right. Luca was beginning to appreciate how much coin Coriola’s trained scholars could draw in and how wide the network might extend.

Frenkel began to speak in a low voice, reciting some basic tenets of number theory. The unseen man beyond the partition gave an occasional appreciative murmur. Frenkel nudged Luca, indicating that he should stop up his ears with the cloth plugs brought for that very reason. But there was no need; this was just basic fare. Luca basked in a warm tingle of pleasure running through him as he listened to Frenkel’s recitation, certain it was nothing he couldn’t handle.

When Frenkel began to speak of higher-order arithmos, discoursing on hyperbolic series and recounting parametric equations, the murmurs from beyond the partition turned to groans. Frenkel pressed on relentlessly, a steeliness entering his voice as though he himself were fighting to maintain his composure. Luca found the wondrous beauty of the arithmos carrying him along. He really ought to plug his ears before it overwhelmed him, but surely just a little more couldn’t hurt…

The unseen man seemed to be thrashing about on his bench, his groans becoming louder. Luca felt sweat run freely down his body in the oppressive darkness, waves of arithmos breaking against the shores of his mind. Oh, the sweet agony and ecstasy of understanding! Seemingly from far away, he heard the man give a loud cry as if in throes of orgasmic climax, but a black cloud was sweeping through Luca’s brain, darker and more potent than any caused by the mere absence of light. The world shrank to a very distant point, then —

Frenkel must have carried Luca the entire way back to the House of Geometers, because he stirred only as they turned into the archway. Frenkel set him gently on his feet, steadying him with an arm. He looked exhausted and pale, and Luca wondered what his tutor had risked in getting him back. “Safe now,” Frenkel muttered. “But that was a stupid thing you did —”

Some movement made them both glance up at the same instant. Across the courtyard, standing in the shadows, Coriola waited for them. Her eyes were colder than Luca could remember and her face was filled with fury.

The next morning, it was Coriola, not Dr Frenkel, who strode into the cramped classroom and snatched up the chalk.

“Where is Dr Frenkel?”

At first, she wouldn’t answer him, but relented eventually. “He had no business taking you on a commission. You are far from ready. It was a miscalculation on Frenkel’s part — and there is no place for miscalculations in the House of Geometers.”

She began writing out a complex expression on the board, the savageness of her strokes sending a fine sprinkle of chalk dust cascading to the floor.

“But where is he?”

“Flown, Luca. I would have dismissed him for his error, and he knew it and acted accordingly. If he has any sense left, he will vanish into the city’s shadows and never cross my path again.”

Luca had a vague notion that he might search for Dr Frenkel. The man had treated him well, cared for him even, in a way that no one else ever had, and Luca missed their companionship. But in a city as vast as Orlondre, he had no idea where to begin the task. And Coriola kept him busy, working him harder than ever, until gradually the arithmos pushed such thoughts from his mind.

So it was left to Coriola to take him deeper into the universe of arithmos, further than he had ever dared go before: into a land of inverted matrices and laplacian transforms — and much more. Each session filled his mind until it overflowed, often leaving him slumped against the desktop, the blood in his veins thrumming with the beauty of it. He missed Frenkel’s laughter and his boyish energy, though.

“Enough arithmos. The most important lesson I can teach you now is control,” Coriola told him one day. “Learn how to compartmentalize your arithmos knowledge. Separate out the pieces, then build strong walls to contain them. Think of them as wild animals in a zoo. Each must be kept caged and isolated. Learn how to do that and you can master all of arithmos in time — perhaps even push against its boundaries — and still keep your sanity.”

She showed him how. He learned how to build his mental walls. Progress was painfully slow at first, because their foundations had to be strong and deep, and it took discipline. One weakness, one flaw, might lead to catastrophe. If his mental barriers failed, the torrent of arithmos unleashed into his brain would surely drive him mad.

That hardly mattered to Luca. Fresh arithmos knowledge was what he craved.

“Teach me something new,” he pleaded, hearing the addict’s whine in his voice and not caring.

But she would not, insisting they spend more time building his mental walls a little higher, a little stronger.

“You’re not ready.”

“When will I be?”

She just laughed at that, and he felt himself blush.

Weeks passed — or was it months? — to the same drumbeat of daily lessons. Occasionally Coriola absented herself for a day or two on some commission of her own and Luca would grow miserable and withdrawn, impatient for his next lesson.

One morning, Luca awoke not long after dawn to the sound of shouts and the pounding of hammers against the iron doors in the inner courtyard. Then heavy-booted footsteps sounded on the stairs and two thuggish men dragged him out into the courtyard, lining him up next to a dozen other sleepy-eyed youths. Coriola, looking flushed and angry, was held to one side with her arms bound, guarded by two more men. The Senior Ecclesiast who stood before her wore a sly grin which hinted at his enjoyment of this spectacle. Others stood by, ready to deal with any trouble.

“Is this all of them?” the Ecclesiast asked. He turned to Coriola. “You are accused of peddling arithmos from this house, violating not only Church law but the laws of common decency. Furthermore, I have reason to believe that you are corrupting these young minds to your criminal ways, teaching them so that they may tout arithmos on the black market and in the clubs!”

“Those are ridiculous accusations for which you have no proof,” Coriola said calmly. “I run an honest school. I give a few deprived children a better start in life, nothing more.”

A wave of his hand dismissed her statement. “Search the house. You know what to look for.” His gaze scanned along the line of youths, stopping at Luca. He pointed and one of the men pulled him from the line to stand in front of the Ecclesiast.

“Tell the truth, boy. What lessons have you been given in this place?”

His heart thudded. “We… we recite verses from the Ecclesiastical Teachings, sir. A little geography as well. And we learn the line of City Governors stretching back to the time of the Forgotten Wars —”

“Is that so? Then let me hear a verse or two from the Teachings.”

Luca wracked his memory but felt color rise in his cheeks as no words came to him. The Ecclesiast lowered his voice. “Give me the truth and I’ll make sure you are well-treated. The Church shows mercy to its congregation.”

Coriola spoke up. “You’ve chosen a poor example, I’m afraid. Luca is a dullard. His intellect was stunted at birth. He’s one of those rarities who has no appreciation for even the simplest arithmos. If the things you accuse me of were true, what use would I have for such a boy?”

“Oh really?” the Ecclesiast said, his tone mocking. “Let us put it to the test then.” He drew Luca aside and in a conspiratorial voice began to whisper in Luca’s ear. “The square on the hypotenuse must equal the sum of the squares —” When he had finished speaking, he pulled back sharply, searching Luca’s face. Luca understood straight away — he was looking for pupil dilation, perhaps a slackening of the jaw, any indication that Luca had understood and been affected by the arithmos.

Keep control! Build cages for each of these arithmos creatures, then lock the doors!

He kept his expression blank, a vacant gaze fixed on the Ecclesiast as though he were every bit as dull-witted as Coriola suggested.

The Ecclesiast sighed and beckoned again. A shabby figure dressed in little more than rags was dragged into the courtyard. A hooded cloak kept his face in shadows, but it was apparent this was one of the Burned. The man giggled and muttered to himself in a queer, strangled voice. His gaze wandered wildly about the place like some startled bird and one questing hand reached out as if to explore Luca’s face, until the Ecclesiast slapped it away.

Luca had never seen a true Burned man before. There could be no doubt his sanity was damaged beyond salvation — far worse than those brain-addled beggars on street corners, rocking on their heels and talking animatedly to unseen companions.

The Ecclesiast withdrew a crumpled sheet of paper with tight-packed lines of symbols scrawled across it and pressed it into the Burned man’s grasping hands. “Read it. To the boy.”

Luca felt the man’s breath, hot and rank, against his cheek. The Burned man began to read what was written, while everyone else shuffled away out of earshot. His sing-song voice wavered and trilled as he recited complex arithmos expressions. Some were familiar to Luca and he welcomed them as he would old friends. But many were not, and a part of him marveled and rejoiced in this new knowledge, sending him giddy with pleasure. One such equation was at once both astoundingly elegant in its simplicity, yet combined an astonishing range of fundamental expressions. The Burned man read precise descriptions of each term and their relationships, dribbling more than a little as he did so.

Fireworks went off inside Luca’s mind. Sparks danced and shimmied within his skull. He felt hot and itchy as though in the grip of a sudden fever.

Hold it fast! Separate and compartmentalize!

Luca fought against the ecstasy washing through him. He must build those walls high within the compartments of his mind and bar the doors. All the while, he fought to keep his face impassive.

The Ecclesiast waved away the Burned man, his eager gaze searching Luca’s face for a reaction. Coriola’s gaze, too, was focused intently on him.

“Well?” the Ecclesiast demanded.

Luca swallowed, willing himself to appear slow and befuddled. “I could perhaps try to recall a verse from the Second Book of Laments?” he offered. “If that pleases you, sir? The lady does her best to teach me, but… I know I am slow to learn.”

The Ecclesiast muttered an oath and turned away.

Their search found nothing, except for a scattering of religious texts and some well-worn school books. The chalkboards had all been scrubbed clean, as they were at the end of every lesson. Coriola knew better than to make elementary mistakes.

“It would seem,” the Ecclesiast told Coriola stiffly, “that I have been misinformed. However, you can be certain we will be watching this place closely from now on.”

Afterwards, Coriola spoke to Luca quietly. “You did very well, giving no sign of recognition. It’s clear they have nothing linking him to the House of Geometers beyond rumor and supposition.”


Coriola gave a cold little laugh. “You didn’t recognize your former tutor? Well, perhaps it’s no surprise. He is a shadow of what he once was.”

Luca stared at her in horror. That had been Dr Frenkel? The Burned wretch? How could such a transformation be possible? Scarcely a month had passed since his sudden departure. Now that he thought of it though, there had been something familiar in the eyes…

A black thought rose up. “How did he become Burned in such a short time? Was this done to him somehow?”

Coriola held his gaze. A flicker of emotion crossed her face, quickly replaced by coldness. “He brought it upon himself.”

His lessons redoubled under Coriola’s personal tutoring and that suited Luca just fine, even when it drove him to the point of collapse.

Coriola was an exceptionally gifted mathematician. Her store of knowledge was vast, all of it clearly contained within her head, because she never referred to written notes. There were days when he found her tasks near-impossible. She tested him with problems so knotty and convoluted there could be no possible answer. But the next day, with sharp, angry movements that sent chalk-dust flying, she would scratch out answers on the board whose obviousness in hindsight brought a flush of embarrassment to his cheeks. “Show me more,” he begged through the blinding haze of another headache, but she would not.

One night, unable to sleep, he opened a few of those cages in his mind and let that day’s arithmos out to play, as he sometimes liked to do. The next morning though, he was tired and slow-witted and Coriola — guessing the truth — berated him. “Then teach me more,” he pleaded. “I’m ready!”

“You are not ready yet. You’re young and foolish!” she snapped.

“And you are old and slow!”

Coriola became very still, her expression unreadable — and that was somehow worse than the anger he’d expected. Not that it made a difference. He’d be banished to his room now, or his lessons cancelled for the week as punishment.

Instead, Coriola turned back to the chalkboard and began writing out the terms for a statement of equivalence he didn’t recognize. “Then find a proof for this conjecture — if you are half as talented as you clearly think you are.”

And with that, she left.

Luca stayed at the chalkboard late into the night. He missed the evening meal — and thought nothing of it. He worked through a box of chalks as he scratched and erased, scratched and erased, following up idea after idea. He would show her!

Still the problem frustrated him. Just when he thought he saw the glimmer of possibility, it would slip from his mind and wriggle away like an eel in the mud.

At last, exhausted beyond what his body could tolerate, he set his head against the desktop. To the rhythm of a pounding headache deep within his brain, he slept fitfully.

Coriola didn’t appear for their lesson the next day. Or the next.

Luca worked on, at times despairing and at other times exulting in some small breakthrough as he brought forth selected pieces of his arithmos knowledge. He would rise to meet her challenge. He would prove himself to her!

Yet always the answer seemed just beyond reach. He used all the techniques and methods Coriola had shown him and still the proof he sought eluded him. How many times now had he driven himself down a dead-end of reasoning? Each time, he would have to erase a dozen or more steps of logic and begin approaching the proof from a new direction.

On the fourth day, he awoke in the gray, pre-dawn light. He had passed another night consumed by arithmos, his brain too numb to bother returning to his bed. Now he became aware of another person in the room. He raised his head and saw Coriola.

“Enough,” she told him, though he scarcely heard her. There was something pawing at the back of his mind, like a dog seeking his master’s attention. Something important. Something he needed to remember.

How strange he had managed to sleep at all! Exhaustion counted for nothing when the thrall of arithmos seized him. And yet he had slept — and soundly too, at least for a few hours, judging by the candles which had all burned down to a smear of grease. Why had he allowed himself to sleep?

Because he was done.

Mother Arithmetica! He was done.

“Go back to your room, Luca. You’re finished here.”

Finished! Yes! With a final burst of brilliant insight he had found his proof! That final iteration of his work was right there chalked upon the board, spilling onto the stone walls in long lines of equations. Only then had he allowed himself to sleep.

“I have your proof,” he said, struggling to maintain his composure. “The conjecture is solved.”

“Luca, stop this insanity right now. Your mind is too full of arithmos. I see now I was wrong to taunt you with this impossible challenge.”

“Look at the board,” he said quietly. “Study the walls. You’ll find no errors.”

“Luca, there is no proof to the problem I set you! The whole point was to frustrate you. I wanted to punish you by setting a problem long known to be insoluble. That way I hoped you might understand your limitations.”

“LOOK AT THE BOARD!” he bellowed.

Coriola flinched. She cast a quick, nervous glance at the sweeping lines of equations, then looked hurriedly away.

“There is no proof,” she hissed, pushing him back down in his seat. He found he lacked the strength to resist. “It cannot be solved.”

“But I —”

She slapped him and the sting of it stunned him into silence.

“I admit I thought your futile attempts at a solution might prove profitable. There might be a few elegant lines of logic in your attempts, the kind of thing that connoisseurs would pay handsomely to amuse themselves with. Because that’s the real problem, Luca. The supply of arithmos is finite. Clever, eager minds such as those connoisseurs grow weary of the too-familiar concepts. The arithmos loses its luster, its potency. There is no ecstasy to be had from the mundane. Then, every once in a while some mathematical breakthrough is made and it’s the purest form of arithmos experience one can have, to understand something so exquisite for the very first time. And I thought… You’re bright, Luca. The House can make good use of you in time, and I’ll see that you’re well looked after.”

“Like you took care of Frenkel? He didn’t become Burned through his own carelessness, did he?”

She didn’t answer straight way. “Frenkel’s mistakes could have brought about the House’s downfall. What I did was only…” She trailed off.

Luca let his head sink back onto his arms. “There’s no mistake. I’ve made no errors in my working. Look for yourself.” He felt a wave of blackness rising. Sleep hadn’t removed the deep-seated tiredness after all. He was spent.

After a while, when Coriola didn’t answer, his raised his head again. Rag in hand, eyes carefully averted, she was erasing all the scribbles on the walls, the chalkboard already cleaned.


“If what you claim is true, this is too powerful for my purposes. There’s no profit in driving my clients into insanity. Go back to your room —”

He was on her before he even realized what he was doing, knocking her to the floor, pinning her beneath him. “Don’t you understand what you’ve just destroyed? If even half of what you’ve told me is true… And now it’s gone —”

She struggled in his grip. “Stop this fallacy. You’re already an addict, Luca. Do you want to become one of the Burned too?”

“I can control it! You showed me how.”

“No, Luca. Dark equations are beyond anyone’s control But you’re too blind to see that, aren’t you? You’re an arrogant young fool.”

No. She was wrong. And he would show her how wrong she was.

As they struggled, he began to recite lines from the proof Coriola had just erased. They came to him as random, half-remembered things; flashes of insight that made her eyes bulge beneath his grip as he dripped them into her mind. He felt his own vision waver as he fought to recall, driven by the terror that it might all slip from his memory and be lost.

He felt no pity for her. She had brought this on herself, stolen this most precious of things from him. She had wiped it away in smears of chalk-dust. Now he could only clutch at the disjointed memories of it.

And he hated her for what she had tried to destroy.

Coriola had been the one to show him how to control the arithmos, building the cages in his mind. She had coached him and this had saved them both when it mattered most — and now he turned it against her. He spoke the words he knew would fling wide the doors to those compartments. He watched the arithmos run free in her mind, her eyes widening, pupils dilating, and kept whispering what he could recall of his fading proof, flooding her mind with its details — and his too, his brain growing feverish with disjointed thoughts.

She spasmed beneath him. Flecks of spittle foamed between her lips and her eyes rolled back in their sockets.

Unreasoning anger drove him on, repeating the same few lines of his proof over and over until his voice grew hoarse.

At last a silence descended on the room. Coriola lay still, not breathing. Luca got to his feet unsteadily.

He saw now that she had been right. He did only care about the arithmos, even though it would eventually destroy him. But this new proof — a thing so perfect in its elegance, so wondrously beautiful — must not be denied its existence, no matter the cost.

He stepped over Coriola’s body, stooping to pick up a nub of chalk. Before, he had worked in stepwise fashion, tidying each step away in his mind, securing it, before he moved to the next. Now he opened his mind to the entirety of it and it was like stepping out from a darkened room into the blazing noon sun.

With an unsteady hand, he began to scrawl on the walls, unsure if he was recreating what had been so nearly lost or merely writing gibberish. When he finished at last, he couldn’t bring himself to look at the entirety of it. He stood there for what seemed like an hour or more, the slant of light creeping across the floor with the rising sun.

At last, when he could stand no more, he scrubbed the walls clean again, and fled the House of Geometers.

With the passing months, Luca had found concealment in the vast swathes of forest westwards of the city, but the living was hard. Now and then, small mammals blundered into his flimsy snares — for which he was grateful — and there were berries to be scavenged from thickets, and edible roots to be dug up and boiled to a bland paste. But now the days were growing colder and shorter, and the forest possessed a stillness that unnerved him. That quietness amplified the little background noises — animals scurrying in the undergrowth, muted birdsong in the pine trees, whose upper branches shivered in strengthening winds from the east. It was as if the forest held its breath, waiting for… something, like the moment of stillness before the jaws of a trap spring shut.

Luca watched the other man stirring their cook-pot in the camp fire’s smoldering embers. The fire was carefully banked so that it gave out no smoke column which might betray their presence. The figure seemed broken; his cloak tightly wrapped as if to ward against the cold, despite the sun’s warmth this morning.

But better broken than Burned.

Luca had noticed small signs of improvement in his companion these last few days. His night-time ravings had diminished to a gentle burbling, and now there was even a flicker of intellect in the man’s stare, as though something was trying to push through thick layers of confusion.

He wondered again at the impulse that had led him back to Dr Frenkel. Luca had fled the city, fearful of what uses the Ecclesiasts might put him to once they learned of his true abilities. Yet something had compelled him to venture back into the cathedral grounds, and he had stolen the man away under cover of darkness. Frenkel had come willingly, meek and compliant as a child.

What had possessed Luca to do that? At the time, he had convinced himself it was an act of compassion, settling a debt owed to the former tutor who had taught him so much.

Lately, he wondered if there might have been darker motives at work in his subconscious.

Every now and then, snatches of his proof bobbed to the forefront of his mind, like rotting corpses rising to the surface of a lake. He thrust them back into the darkness, of course, but he could feel them circling, ever-present.

Luca took his place by the fire. His traps had been empty again, and the hollow hunger pains in his belly were a constant reminder that they must act soon. A shaking hand reached out for his. He gripped it, making soothing noises until Frenkel settled again.

“Should we go back?” Luca asked quietly.

Frenkel’s lips ceased their convulsive twitching and Luca saw something new in his expression. A gleam of excitement? Fear?

By now, they could have put two hundred miles between themselves and the city. Yet when Luca had climbed to the top of a ridge just that morning, there it was: the dark outline of Orlondre on the horizon, no more than thirty miles distant. He knew why they kept circling back. They wouldn’t find any arithmos scholars amongst the simple folk in the scattered forest villages or the farming communities beyond. No one he could learn from in those quiet backwaters. Yet each day he felt the need for fresh arithmos stirring inside. He had so much still to learn, so many things Coriola or Frenkel might have shown him — but Coriola was dead, and as for Frenkel…

“We could find hiding places,” he told Frenkel. “Burrow deep into Orlondre’s dark underbelly where no Ecclesiast will think to look. And there’ll be no shortage of wealthy benefactors willing to trade for what we can offer.”

That much was true. It occurred to Luca that he might even finish the work Coriola had begun: finding ways to loosen the Church’s vice-like grip on the citizens of Orlondre. And if he could learn how to control his cravings, surely he could teach others. Couldn’t arithmos become a joyous, enriching experience when taken in moderation? Weren’t there better uses it could be put to?

Or was this no more than self-delusion? He’d be returning to the source of his supply just as any hardened addict would. And wasn’t he nurturing some dark kernel of hope that Frenkel might one day be well enough to resume Luca’s lessons?

The choices stretching before him suddenly reminded him of the patterned tiles he had swooned over so long ago: black-and-white, light-and-dark, endlessly repeating.

“Gather your things,” he told Frenkel at last, with a certainty he didn’t feel in his heart.

They would go back, and Luca knew what he must do. He would reclaim the House of Geometers for himself. They would build it back stronger, right under the gaze of the Ecclesiasts.

Whichever way the choices led him after that, well…

Then he would see how well he’d learned his lessons of self-control.

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