Chapter Baker's Dozen
Scene Three: A Math Tutorial
The inside of the animal doctor's barn looked dark even in candle light. Denario spent two evenings in the place as he worked for Jake and Hadewig Kaufmann. The barn’s decorations were odd. The inhabitants were, too. Aside from the knocker-fox, there were two pigs, a milk cow, a calf, a rather elderly dog, four goats, and a colony of flying frogs. There were bowls of water nailed to the walls of the barn. Many of the bowls rested high up near the rafters. All of them needed refilling daily. The job had to be done a certain way, too, or the flying frogs would die. Jake handled most of it by himself but Denario helped with the ladder and filled the occasional bowl under Jake's watchful eye.
The flying frogs were obviously magical but otherwise they weren't impressive. When they puffed themselves up and stretched their warty, green wings, they almost reached the size of ordinary bullfrogs. They needed to swim in a bath at least once an hour. That meant they couldn't fly far from the barn. Jake said there was only one spring on the hill and he'd built his house next to it. So the flying frogs couldn't find enough water to live elsewhere in the countryside. Even with the barn door open in good weather, most of the frogs had the sense not to leave.
Last fall, Jake had followed a female as it tried to migrate. In less than a mile, the frog had reached the edge of the dense background magic. There, her wings shrunk. She fell out of the sky. The fall nearly killed her.
“They're odd creatures,” Jake said as he added water to the bowls. “Right now, they and the knocker-fox are the only magical animals I've got but I've enjoyed raising them. I'm building a separate barn for baby griffins. The Oggli and Angrili Wizards' Guild is willing to pay me handsomely to try to farm them.”
“Do griffins really hatch from jeweled eggs?”
“Oh yes. Nowadays, people are hunting their nests a bit too fiercely. Griffins are getting scarce. It'll be good to raise some in captivity.”
Something in Jake's manner made Denario think that he had already planned to 'lose' one or two griffins to the wilds. The wizards would get the eggs and the profits, of course, but only so that Jake could continue his work creating more griffins.
The knocker-fox, as its name implied, was possessed by a friendly human ghost. Jake thought this one had been a wizard who had borrowed the fox's body, gotten over-involved in acting like an animal, and foolishly let his real body die.
“Early on, I tried a lot of names with him,” said Jake. He gave the fox a pet as if it were a dog. “He responds well to Dedric. I think that must be close to his human name.”
“Poor fellow,” murmured Denario. He started to pet the animal but hesitated, a bit frightened by its apparent smile.
“Not really.” Jake pointed to a small, black form thrashing in the straw on the floor. “He gets a lot to eat. In fact, he almost ate Hadewig's cat last year in a fight about who got to have the fallen tadpoles. But they've come to terms. Dedric gets first pick.”
The thing that was writhing on the barn floor turned out to be a tadpole. The fox took an interest in it. The slimy creature must have jumped from its bowl. Maybe it had tried to fly. It was a rather large tadpole with legs and wings. When it saw the fox coming it tried to burrow into the straw. The fox put a paw down and stopped it in a rather human manner. Then it sniffed.
A pale orange cat appeared from the shadows. It turned its head away as if uninterested in what the fox was doing. It peeked, though, as the fox snuffled into the inches of straw and clacked its jaws. The fox had swallowed the winged tadpole in one bite.
“It's funny how the frogs let the fox and cat do that,” said Jake.
“Can they stop it?” Denario glanced up to the rafters, where a dozen frogs perched. One of them turned its lazy gaze downward.
“They can stop the cat, yes. Roughbottom doesn't want to incur the wrath of the adult frogs. But they don't seem to mind when he eats a fallen tadpole or even a frog that's been injured in a mating fight. If it's dropped to the floor, they don't care. But if Roughbottom climbs high enough to approach a bowl of eggs or tadpoles, the frogs dive at him. They've knocked him off the perches more than once. I think he's learned his lesson.”
“They wouldn't be so aggressive with the fox.”
“No, they wouldn't. Partly because the fox can eat them out of the air. He does it when they fly too low. But mostly because, unlike the cat, he can't climb.” Jake put a hand over his belly and laughed. Denario shook his head. He hadn't thought that out.
“In the wild areas, flying frogs can grow quite big.” Jake turned with his watering can raised high. He poured water into a bowl with elaborate care. “With enough magic, the frogs grow too large for even a fox to bother.”
They worked for over an hour on the frogs. Denario mostly held the ladder and fed the droopy-eared dog with a set of snacks that met Jake's approval. Denario also learned to use a pitchfork well enough to nearly kill a goat when he dropped it. After that incident, Jake stopped asking him to feed the big animals.
Finally, they finished. Jake climbed back down from the rafters. He wiped his hands on his trousers and picked up an armful of clover and hay that he had de-glued with his magic rock. He let the goats come over and eat some out of his hands.
“They certainly seem to like you,” Denario said, a bit jealous.
“They know me. And I suppose they sense that you aren't comfortable around them. That's why they shy away from you.”
“And because of the pitchfork.”
“And the pitchfork, yes. Again.” Jake rolled his eyes. Denario saw his point. The accountant had encountered problems with farm tools during the previous night’s work, too. “It's funny how awkward you are with long objects. I've seen how badly you practice with your spear. At first, I didn't quite believe you when you said you weren't a warrior. Not that I thought you were a liar exactly but I thought you might be unduly modest. Now I understand better.”
Denario sighed. There didn't seem to be much to say about the issue of combat. He knew he didn’t have much physical talent. However, there was no denying that he needed to hone his meager skills since he would depend on them to keep his body intact on the long walk to Oggli.
“You can't even string your bow,” Jake continued after a pause for thought. “It's good of you to keep trying, though.”
“I'm doing what the chief of the Mundredi told me to do. I'm not confident enough in my skills to disregard his advice.”
“You shouldn’t be. Now that I’ve met your acquaintance, I like you well enough to hope you never get into an actual fight. As you told us, you’re no warrior. But as a math tutor, you're wonderful. Hadewig and Tabitha are very happy with your lessons. That’s saying a lot for Tabitha. And you’ve done more, of course. I have to thank you for making that book of math practices for my daughter.”
“You're welcome.” Denario nodded in acknowledgment. “It's almost done.”
“You mean there's more to come? It looks quite good already. You're an amazing artist. The shapes you draw are perfect.”
“That's just a matter of having the right tools.” With a protractor, straightedge, and compass, Denario could draw almost anything. With a free hand, he was awful, not an artist at all, but that hardly mattered for a children's book.
“Your writing is incredibly precise. I've seen you making entries in your professional journals. You write in them quite a lot.”
“I've been concerned with odd questions about math. One of them, I'd have to say, I don't think anyone's formally ever asked before. It’s a question that came up when I was talking with the Mundredi war chief. Maybe there's no chance of me working out an answer but I feel bound to try. Math is my hobby as well as my profession.”
“Admirable.” Jake nodded. His agreement came as no surprise since he also was a man whose hobby was also his profession.
Denario subscribed to some interesting theories about geometry that he wanted to prove. The most important was that there was a way to approximate division by zero by using numbers that were really, really small. He drew thin rectangles inside of circles to do his basic calculations. He’d been working on that since he was nine. But the latest question to occupy his mind originated with the comment from Vir, “One apple isn't the same as another apple.” The chief hadn’t known it but he’d struck at the heart of mathematics. The axiom of x = x was a basic rule. And where did the basic rules come from? Was it possible to prove that any of them were correct? An Oggli native named Gauss had proved the assumption of primitive polynomial equations about a century ago. His name was famous in math although all he had done, in a formal way, was show that the integer coefficients of the product of two primitive equations would never be evenly divisible by any number greater than one.
It was the formal logic that had impressed everyone. Denario hoped he could use the same formal system to prove that numbers themselves were real. They had to be, he thought. But what was it about a number that gave it logical certitude? An apple wasn't the same as another apple but the number three was the same anywhere at any time. The quantity of something was a universal property. Denario had become determined to work out why.
“Might be worth taking to an engraver,” Jake muttered as he opened the barn door.
“What would?” Lost in thought, Denario hadn't followed whatever Jake kept on about. He numbly followed the animal doctor through the path in the dried glue.
In the house, Jake hugged his wife and endured squeals and squeezes from his daughter. She was always happy to see him even if he'd only been gone for a few minutes. He walked straight to the desk where Denario had made his math lessons pamphlet. Gingerly, he moved aside the protractor as if handling a rare and particularly fragile egg. Then he thumbed through the drawings beneath.
“Still a few blank pages left, I see,” he said at the end.
“Enough to do adding by tens, I think.” Denario glanced to Hadewig. “If that suits Tabitha. She seems to have the idea of adding the smaller numbers.”
“She's been at it all day,” said Hadewig with a mixture of exasperation and pride. “She won't leave me alone. It's a good thing Jake made her those wooden blocks this winter. She can count with them and let me bake.”
Although Denario slept in the barn as far away from the frogs as he could get, the Kaufmans fed him at their table. Their time alone must have seemed a bit too much because they sat on the edges of their seats for Denario's stories, even the ones about math.
“You know, a farmer on the next hill east sang me part of a story about an accountant,” Jake said as he wiped his hands on a napkin. “It sounds a bit like your adventures, only funnier.”
“It is me,” Denario sighed.
Jake howled. His wife joined in the laughter too, although she'd only heard the ballad third-hand from her husband. Their daughter laughed because everyone else was laughing. Together, they cajoled Denario into telling the details of his battles and of his accounting. They thought it was wonderful that the bards of Phart's Bad had written an saga for him. It made him 'official' in their eyes and also teased their vanity about Oggli being a city that produces great men from great guilds. Laceo had to admit he was grateful that the hill folk were paying any attention at all to math but he found it depressing that the story of his misadventures had already spread out ahead of him on his journey to Oggli.
“Don't worry, Denario.” Hadewig put a kind hand on his elbow. “You're the first accountant these folks have heard of. And you've represented your counting house very well, really.”
“If we get back to the city before you, we'll put in the good word.” Jake forgot the napkin and wiped his mouth on his sleeve. He ignored his wife's narrowed gaze.
“My father dotes on Tabitha,” she said after she sighed and visibly decided not to criticize her husband. “We'll show him your book for her.”
After Denario finished his second bowl of leek soup and devoured a loaf of Hadewig's fresh bread, he got up and grabbed the math tutorial booklet. Tabitha had announced that she was sleepy. She'd pulled on Denario's shoulder and said wanted a story. It was odd that a child would want him to read math lessons as if they were entertainment but Denario was willing. He sat next to her bedroll on the floor. He pointed to the drawings as he turned each page. Tabitha clapped when he reached the line about two ducks plus five ducks.
“Seven,” she said, eyes closed. Hadewig and Jake beamed. “Two big ducks. Five little ducks. Seven ducks.”
For a moment, Denario wondered what it might be like to be a parent. He dismissed the idea as outlandish. As it was, he had more than enough responsibility on his hands with five apprentices. He hoped Curo was reading math lessons to the youngest two, Guilder and Mark. If Curo didn't do it, he doubted Sheckel or Buck would take over the job. Maybe Kroner would. He could hope.
Finally Tabitha fell asleep. Denario managed to pry the corner of the booklet from her hand. He rose from the side of her bed and whispered, “I'll finish this tonight.”
“Just as well,” Jake said. He didn't bother to lower his voice much. “You'll be gone tomorrow.”
“You're kicking me out?”
“No. Can't you smell it above the scent of dried glue? It's fixing to storm tonight, a real rain. It's going to wash away the residual magic.”
Denario sniffed. His nose wasn't as trained as a Seven Valleys native, not even this rather non-native native. But Jake hadn't been wrong about weather or animals yet. So he checked his travel bags before he sat down to Hadewig's desk and lit a tall candle. Then he got back to his work.
Next: Chapter Thirteen, Scene Four
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