Chapter Second Perfect Number
Scene Four: Message from a Partner
The next morning, Denario woke at sunrise, hours before the bank opened. He had finished the last five pages of the duplicate copy of his journal last night by the light of an oil lamp. That had been fun, re-writing and checking his previous math. Now he was left with the problem of what else he could do.
Even though the bank had failed him by destroying his messages, he was inclined to leave his duplicate journal in their care. What choice did he have? The idea of a second copy, after all, was to keep it outside of Oggli in case there was a city fire. Winkel had maintained his backup at a bank branch in the town of North Weird. That way, the Oggli accounting guild records could, in theory, be reassembled from the duplicates scattered in other places. Moreover, Denario’s group theory proof that three is equal to three seemed worth preserving, not to mention the accounts of ancient book keeping systems in the seven valleys. Most of those, no one had seen in action for over a century, sometimes several centuries, and a few of the methods had never before been inspected.
His mind nearly made up, Denario donned his cap and accounting vest. At the last moment, hand on the doorknob, he noticed that his movements felt insubstantial. That made his arms itch. He removed the vest, donned chain mail between a pair of his shirts, and dressed himself again.
Weighted down to a more serious version of himself, he creaked down the stairs. At the front desk, a man lifted his gaze. He closed the ledger he was reading, perhaps out of habit as he expected a complaint.
“Did you request an alarm candle, sir?” The man at the desk was probably innkeeper himself. He had a confident, slightly-overweight bearing about him, not like the young fellow last night who must have been a nephew. As the innkeeper recognized his customer’s vest, he opened his ledger again in a more relaxed fashion.
“No.” It took Denario a moment to remember the concept. Alarms were timed candles. They had horizontal stripes of colored wax in them and burned for a set number of hours. If you wanted an alarm, you set it by pushing a metal nail at the desired time length in the candle and, when it melted to that point, the nail would fall and clank on the metal holder to alert you that it was time to wake up. “I have to confess, I’ve been traveling so long that forgot they were an option.”
The fellow laughed. “Apparently you don’t need one. Are you going for an errand, a bite of breakfast, or just a walk?”
“A light breakfast, I think, and then I want to look around. It’s been half a year since I’ve seen Oupenli. Even that was a glance through the door of the inn and then through my carriage windows.”
“Morning is the safest time, as you must know.” He chortled. “All of the wizards and nobles are sleeping it off. This is when working men get things done.”
They shared the smile between men who knew that, yes, they did get things done. The lords, ladies, and magical folks commanded half of every day. But the early morning hours were ruled by laborers. Moreover, the innkeepers and accountants worked in what had to be among the most rewarding kinds of labor. They weren’t beating mules near to death, nor digging canals. They weren’t peasant farmers working someone else’s land, thank you, nor were they slaves devoted to someone else’s house or business. They had their own businesses, humble as they might be, and they could give orders to their apprentices, wives, sons, and so on. They could even be rude to their customers, although this fellow was not.
“If you’re heading downtown,” the innkeep advised, “steer clear of the first bakery on your right. They use cheap flour that’s mixed with something awful. There’s one on your left that’s fine. A block further, there’s the sign of a cupcake followed by the sign of a slice of pie. Those two are the best.”
Fifteen minutes later, Denario found himself chewing on a honey cake as he hiked along an avenue that approached the embassy tent of Agrabar, bright blue with white and gold trim.
Embassy Row in Oupenli was colorful. Denario had forgotten the gaudiness of the tent cloths and painted wood that lined either side of the alleys. He’d missed the grassy feel of the roads, which were usually not trampled to mud except in the oldest sections of town. Oupenli kept growing. Parts of it were always new.
It was wonderful to have access to a free city sandwiched between the lands of sworn barons and knights of West Ogglia. The place had been traditionally independent for as long as anyone could remember and somehow that made it acceptable. It was home to smugglers of all sorts, who paid bits of cash to the free lances to maintain this state of relatively-stable anarchy. Aside from the knights of no allegiance and the smugglers, there were worse figures like mercenaries, priests of small gods, and nobles from far off places who came to Oupenli to relax in a land where there were no laws and a bit of money was respected as long as it was accompanied by men with armor and pointed weapons.
In the center of it all stood the embassies. Oh, there were a few embassies in Oggli, many of them bigger, but somehow the embassy streets in Oupenli were more fun. There was an adobe fortress from Muntar, home of the great traditions, a stone tower from Baggi, source of sophisticated magic, and there were smaller houses. There was an embassy to Upper Steve located next to Lower Steve, two lands nearly always at war with one another although their embassies in Oupenli usually got along fine. Here, they were too far away to keep up with the declarations of war, the vows of peace, the treaties, alliances, broken treaties, and so on, so the embassies stopped trying. If a reliable traveler from somewhere, like a new ambassador, said that his country was now at war with another country, the staffs of the embassies concerned would reluctantly pretend not to like each other. But they would still invite one another to dinner parties. Even for their newly-arrived ambassadors, the rest of the staff would never go so far as to break off diplomatic relations.
On the way back from the consulate row, along a side street, Denario encountered a wizard who had apparently woken from his drunken slumber. In a silvery, blue robe, he staggered down the middle of his path while everyone gave him a wide berth. He’d lost his staff or he’d left it somewhere. Denario froze and let the bearded fellow meander around him, which the wizard seemed to do on instinct.
Also from a sort of mind-muscle memory, the wizard seemed to decide to sit down and catch his breath. There was nothing to perch on in the middle of the street but, just in time, a chair zipped out from an alley, possibly through a tent flap of the Embassy of Faschnaught, and placed itself under the wizard’s butt before he could tumble.
In a second, he fell asleep.
Denario chuckled. He wasn’t the only one. The residents of the street relaxed. Denario could see the tension easing from the shoulders of shopkeepers. A gate guard smiled. Folks started to go about their normal business, carting boxes, rolling barrels of beer, baking tarts and putting them out to cool, painting portraits, and selling fresh onions. The armed staff in front of the embassies, who had moments before pulled themselves out of the street lest they get turned to chickens, stepped back into their positions and looked tough, which was their job.
Denario continued his stroll with a brief stop at the cooling rack of tarts. After he bought one for a couple pennies, he glanced up to see that another wizard had approached the sleeping one. That seemed nice. Den turned away but, after a moment’s consideration, reverted to his previous spot so he could study the newcomer in grey and black.
It was the bank wizard.
The bank wizard whispered something to the sleeping fellow, who roused enough to respond. Then the wizard glanced around. He spied a nearby pub. With a lively step, he dashed through the door, shouted, and a moment later emerged with a ceramic bottle of wine in his left hand.
“Just reverse time.” He held out a box. It was Denario’s message box. “No one can do it. But I’ve seen you do it. Turn the ashes into papers again.”
“Wine first,” said his silvery-blue associate.
When the bottle hit his hand, he cranked his neck back and took a long pull. It seemed to wake him. His bloodshot eyes snapped open. His back shuddered. As he sat back up, he regarded the box.
“This is bullshit,” he said.
“Excuse me?” The bank wizard seemed more puzzled than offended.
“No one can fix this.” The drunk wizard did something complicated with his left hand, the one not holding the bottle. For Denario, it was strange to see a spell cast without a wand or a staff. Wizards were tool users, generally. This time it looked like the drunkard’s hand caught fire. It got bright. Smoke came from it.
Denario blinked. The smoke was flowing from all directions into the wizard’s left hand. From there, a discharge of magical, white flame and inky soot traveled a hands-width through the air to the bank deposit box. The walls of the box gleamed with the flares of light that emanated from the bottom, amongst the ash of the destroyed documents. In a few seconds, the deed was done. The drunk wizard hauled up the bottled for another long draught.
The bank wizard shook the box. He gazed critically into it. He cocked an eyebrow at this compatriot, who was still taking a long swig. Without a glance, the seated fellow gestured again. There was a puff of white smoke. The box glowed.
At last, the bank wizard smiled. His shoulders relaxed from their tense, hunched-over posture. He poked at something inside the box. He took, from a pocket inside his dark robe, a rectangle of wood. It was the lid. He slid it into a slot at the top of the message bin and pushed it closed.
The bank man took a moment to scan the crowd and see who was watching. Denario ducked behind the shelf of the bakery stall. He didn’t have a particular reason to avoid being seen. It was just something he’d learned from his childhood as a slave and, later, as an apprentice in the guild – sometimes it was better not to let everyone realize how much you knew.
In aid of that, he spun around a corner into the alley behind it. The shelf of tarts blocked him from the view of the street. He finished stuffing his pastry into his mouth. Then he quick-marched toward the bank. Better to beat the wizard there, he thought, and maybe chat with the manager.
At the clock-making shop, half a block from the bank, he checked the time. He was a bit early. Under the Bank of Oggli and Anghrili sign, he tested the door, which was locked. He started to think of math problems to occupy a few minutes. As he stood and tapped his feet, approximating the area under a curved awning, he spotted the youngest bank clerk. The short, thin lad – well, he was only a head taller than Den – had dressed in a white linen shirt and black vest. Someone had oiled his hair, which was combed and trimmed to shoulder length. Except for his pale skin, he had styled himself this morning in the way that Den had done when he’d traveled to Ziegeburg. It was a popular look.
The clerk spotted Denario a second later. He rushed forward and pulled a key from his pocket.
“You’re a bit early, master,” he huffed. “You can have a seat while I get the office ready. It’s my turn, see.”
The inside of the bank was dark but not with the severity of a shuttered room. A high table along the wall between the calculator desks and the vault was illuminated by an oil lamp. The low flame had been left on all night. The clerk lit a taper from it and carried the fire to the candles, sconces, and other lamps. Then he extinguished the overnight wick, which had done its job. He strolled to the south windows and opened the shutters. A clerk apprentice arrived and took over the menial task of opening windows on the north and west sides. Another man came in, dressed in grey, and shouted a hello to everyone. A moment later, two of the calculators entered. They headed straight to their desks and opened their ledgers. One of them complained about the state of his candle. The apprentice got him a different brass holder.
“I don’t suppose you have a copy of the guild math lessons,” said the opening clerk as he circled back to where Denario had taken his seat near a south window.
“Why do you ask?” He did, in fact, have a copy of lesson two in his journal, part of lesson three, and he’d started on lesson nine in a letter destined for Carinde.
“Ah, well, the bank pays my two silvers to the guild for me every month,” the young man said with a sigh. “That gets me library privileges but of course the guild library is thirty miles from this branch. I’ll never see it. Instead, I’ve put in requests for copies of guild lessons. That should be my right. It’s what the fees are mostly for. But I only received Lesson Two, the Algebra System. That means Lesson One, History of Math, and Lesson Three, Fractions, have gotten lost between there and here.”
“That’s odd.” Den had forgotten about the guild fees. There were a lot of them, often met by businesses but just as frequently paid by individual book keepers, clerks, calculators, reckoners, and other professionals who wanted to stay cozy with the Accounting Guild. They paid the guild for protection, of course, but mostly they enjoyed the benefit of the guild copyists who scribed and sent out math courses. Winkel had treated the distribution of teaching materials as a sort of sacred duty. In that, he was like almost any guild master before him.
Filchi and Spioniladro, the current masters, felt differently. They’d argued with Winkel, screamed at him once, in fact, during a sparring match led by Spioniladro. They claimed that the lesson books ‘created competition with guild accountants.’
‘Surely not,” Winkel replied, hands clasped behind his back. The argument had taken place on a fall day. The master had stood next to a window.
‘Some leather worker’s son, somewhere, reads a lesson and thinks he can keep the books for his family business instead of paying us!’
‘A small business can’t afford a guild accountant,’ Winkel had pointed out. ‘All they can spare is the money for lessons. We take it from them. We take it from an awful lot of people. The fees amount to half of the monthly guild income.’
That line of reasoning shut up Spioniladro and Filchi for a while.
“Do you already have the series of accounting lessons?” Denario wondered.
“The bank keeps copies of those.” The young fellow gestured with his left hand. “We have Lesson Two, the Debits and Credits system and Lesson Three, Examples of Recording Transactions, on the apprentice desk. My problem is that the guild took our math lessons fee and didn’t produce the lessons. I’m also concerned because I really would like to take some of the higher math courses.”
“For your career?”
“No. My father paid for me to land a job here at this branch. He’s arranged a marriage for me, too, a nice girl from a cloth merchant family, and she wants me to stay in banking.”
“In that case, why study more?”
“Binomials. Have you seen the Muntar Triangle report that came out two years ago?”
“The one written by Kharaji, you mean. Yes, it shows how the coefficients of a binomial equation are each the sum of the two equation coefficients above. That is, the ones from the previous binomial in the sequence.” It was a badly named paper since the mathematician Kharaji didn’t live in Muntar but had merely published there. Still, the fascination with his work had led to copyists everywhere begging for an original. Copies of the pamphlet had spread around the Complacent Sea. Apparently it had introduced more than one young man to the philosophy of math.
“As you multiply the equation by itself.” The clerk seemed breathless about the idea. “Again and again. It repeats forever, the same pattern. What do you think? Could it be part of the underlying math that forms the world?”
“It is part of that.” It wasn’t necessarily an important part. Denario wasn’t sure. But it was, by definition, a part.
Den got lost in a discussion with the clerk about binomial equations. Together, they wondered if higher-order polynomials would lead to anything practical and whether, really, anyone cared; the equations were so beautiful in their own right. At one point in the conversation, there was a click. They both looked around. Den realized that the bank wizard had arrived and headed straight into his manager’s office. That had been the source of the noise, the closing of the inner sanctum office door. The clerk shrugged and began drawing a binomial equation with his finger on the desk between them.
When the manager emerged, Denario didn’t recognize him. The old fellow’s hair was silver and cut short. His beard had been allowed to grow to the top of his collar, no more. His waistcoat and trousers bore a shade of charcoal grey, cut from the same cloth. He carried a white pine message box and a frown that had weight to it. Hardly noticing the burden of his own scowl, however, he strode up to the desk.
“Is this your correspondence, young man?” he said.
It had been so long since Denario had been addressed like a schoolboy that he looked around in case the gentleman was raising his voice to someone else. Next, he lifted his head to peer inside at what was being offered. The container was a different sort of wood than the one he’d seen yesterday. There were a few papers piled in it, all marked with ink. Some of them had been folded in half and pressed flat. Others had been left to wrinkle and curl. They rested in a sloppy pile, fragile tubes of news, business reports, and dispatches from near or far.
“Probably not,” Denario said. He pulled his hand away. “Those were accidentally burned, weren’t they?”
“I see that you don’t want to check. But I believe this box is yours.”
Gently, Denario took the message bin from the hands of the manager. He placed it on the table between him and the clerk. To his surprise, the clerk did not try to look inside. For a few minutes, they had been hobbyists together, sharing their theories about binomials and the excitement of secret math structures. Now the junior employee seemed reserved, hands in his lap. He leaned back.
Den fumbled with the top, rolled up paper scrap. He teased it apart to discover that it said simply, ‘Pink paper’ although the page was actually a shade of tan or off-white. He set the mystery aside for a moment. A scribbled name had caught his eye. Curo. It was inked onto the plain back of a folded rectangle. Den opened it.
To: Denario, Accountant Certified by Oggli and Anghrili Guild
From: Accountant Liveshamp (?), Partner
We have received your latest installment from Ziegeburg. I have excellent news from the counting house. Unlike your first payment, this one has gone completely to savings. K has done well to hold the Paraventiari account and that covers quite a bit, as you know. B has been gone for three weeks on a survey with the team employed by the Marquis, so that had saved us a great deal on food. Ha ha. We will get a nice stipend on his return. S has worked with me on other accounts. We have lost the farrier account. I'm not sure why. Master F won't talk with me. He might with you.
I'm afraid that M has been a bit of a nuisance. He and S talk about nothing more than missing your presence. However, all is well. We look forward to your return in three weeks at most.
Curo Liveshamp de Oggli
For a moment, Den put his face in his hands. If he had been asked why, he would have had to say that he didn’t know. There was nothing in the letter that was anything but business. For a moment, though, he had envisioned Kroner hunched over his desk at the Paraventiari dock, Buck hiking outdoors in rugged clothes, hauling a rope and a theolodite, Shekel in the counting room with a slate, doing Curo’s work, perhaps Mark and Shekel later running and playing together. Mark was the youngest but tough enough that he had to work to avoid hurting the older apprentices as he tackled-hugged them or tied everyone to their chairs at dinner for a prank.
A bark of laughter escaped Den. He wiped moisture from his eyes and set down Curo’s letter. The fingers of his left hand started to creep toward the white pine box.
With the eyes of three clerks on him, the manager, and the wizard, he pulled out a scrap of parchment. He unfurled it.
“Why does this note say 'red paper'?” Denario gave the oldest two clerks a suspicious glare.
“Oh, um, it was probably supposed to be written on red paper.” The youngest clerk, the one who loved binomials, answered him after waiting for the others. His elder clerks nodded for him to continue. “But that's expensive. We don't keep it in stock.”
“This next one starts out 'pink paper.'”
“Is that your girlfriend? She must be rich. Or is she a business partner of yours, master? These messages came addressed to your bank account number, not to your name.”
“Another one with ‘red paper.’” He sifted through his messages. There was a letter from Carinde that he noticed but it was too likely to be personal. He set it aside. With his other hand, he flipped a paper over. “This one was the earliest, from the date. Next, another one that says ‘pink.’ This last one says 'black paper.'”
He tapped the top of the pile.
“We can't do that.” The bank manager stepped forward. He leaned a bit too close for Denario’s liking. The man had eaten eggs for breakfast. “If we did, you couldn't read it. I don't even know what that means.”
“Right.” In order, they were red, red, pink, pink, pink, black. There was nothing further from her. Whatever Pecunia was trying to tell him by her color choices, it wasn't good. Why couldn't she just come out and say it? Put a plain message in a straightforward code? He felt a flash of irritation but he remembered that she was being observed by the Mayor of Ziegeburg on her end. That she had sent any correspondence at all was dangerous. There were likely to be spies along the way among the other mayors and bank wizards. Those were reasons to send to his account number rather than his name.
He pushed at the other messages in his pile. Many of them had been neatly folded. The dates on them were old. One, in fact, was from Denario de Oggli, himself. He had posted a note to his account describing an idea he’d had for a new way to find prime numbers. The method he’d proposed seemed dubious in hindsight, though, and it was not really an improvement on the Sieve of Eratosthenes method. It was just an idea.
Another note had come from Shekel. It detailed a proposal to navigate the Complacent Sea not by hugging the shoreline but by sighting the stars with an astrolabe. He must have felt it was worth four months’ allowance to convey the concept, which may have been a correct judgment. The math looked good. Kroner had sent three lines to say that he had closed out the books for the Year of Lord John 15 for the Paravientiari docks. The docks had paid a city tax of eight hundred four silver to the marquis and a dock tax of six hundred fifty six silver to the duke.
“All of these are old,” he noticed.
“Excepting the letter you haven’t opened,” the young clerk pointed out.
“There are no dispatches from my business after I was rumored to be dead. That’s what I mean.” He ignored the letter from Carinde and the hint to open it. “I was due home nine weeks ago.”
“Is there anything else that you’d like to deposit with us?” The bank manager stood straighter. “It will make your travel to Oggli go lighter and faster. That is part of what banks are for.”
“In fact, yes.”
He tapped his lip. The clerks and bank wizard waited on him except for one who marched off to greet another customer.
“Since you have recovered my papers, I can trust you with guild business.” He took his accounting bag from where he’d hung it on his chair. A snap of his wrist placed on the table. “We keep duplicate records in counting houses and banks. This would be a good place for a copy of my travel journal.”
Two of the clerks smiled. One was the young fellow so interested in binomial equations.
“It contains the official guild records of thirty-nine recent jobs,” Den concluded, “including those for handling six accounting systems that have never been observed before.”
“Very good, sir.”
“This will occupy a type three box.” The manager saw no need to measure the book binding. “That size runs one silver dollar and ten brassers holding fee per year. As you know, the first year must be paid up front.”
Denario nodded. He had already drafted a note to transfer the money from his re-opened business account.
“May I read it?” the young clerk asked.
“My transfer note?” he held out his paper.
“No, your guild journal. Does it have any of the math lessons in it? I mean, it’s to become part of the library anyway. I’m mostly thinking of the lessons.”
Den paused for a moment as he remembered the long hours at work, the journeys, and the thoughts about math that had gone into creating his small book. In Oggli, he had been interested in the latest math journal he’d read and its attempts to describe motion. In Ziegeburg, he’d continued his thoughts even as he made accounting notes.
“Do you remember,” he asked, “how I said that polynomials might be good for describing curves?”
“Yes, but I haven’t read anything about that before.”
“There’s a description of the process in there.” It was based on ideas about conic sections. An ancient philosopher, Apollonius of Petard, had written the definitive text but his math had been based on polar coordinates, string measurements, and axioms. A new fellow, coincidentally named Apollon of Muntar, had transformed the calculations into a modern coordinate system.
“And there is, in fact, part of a math lesson that I prepared for a student who I met in the barbarian lands.”
“A barbarian student!” All of the men who had been standing around the table leaned back for a moment. Den noticed that they were a well-dressed bunch. Every one of them had neat, oil-slicked hair, some down to their shoulders, some farther down their backs but mostly the same. They wore clean tunics or shirts, all roughly a shade of eggshell white. They had blackened their shoes and boots with polish.
“Ha!” Everyone had a good laugh. Den shied away from telling them that the student had been a girl as well. It would be too much for them to accept.