Chapter Baker's Dozen
Scene Six: Geometry by Ropes
After five days of travel in these hills, Denario wrote in his journal, I have seen several examples of tally sticks and find them remarkably consistent. I have drawn examples here of the symbols used.
The split-stick method cannot accommodate a complicated transaction. Yesterday I saw that that one farmer owes his neighbor thirty-two chicks while he is owed six pigs from that neighbor in turn. Even for this simple record with just two entries, the hill folk cut two sticks. What's more, the symbols for some types of livestock can be similar to others. There is a parallel problem with the symbols for clan houses. Despite what the hermit told me, I suspect that cheating goes on.
Tally sticks do not record anything more than debts. For farmlands and water rights, the hill folk rely on oral histories, totems carved in stone, and warfare. Thefts of sheep between clan herds and between houses are common. I have been fired upon twice, each time when I was mistaken for an enemy shepherd. Both arrows missed of course or I would not be writing this entry.
Everyone fights. Old women casually punch me on the shoulder to test my armor. Even the priests here fight over debts and territories. I was asked to adjudicate in the town of Double Bad on a border dispute between two churches. By popular reckoning, the local goddess is allowed two hundred ells distance from each spire of her church. (There are two, a north and a south spire.) The temple of the local god gets the pond, the well, and the lands west of the water. The border of the pond changed due to a snow melt years ago and the religious zealots on both sides have been fighting since.
I saw that the goddess's church spires formed formed the two foci for an ellipse that defines the border of her holy grounds. When I drew an example for the mayor, he was glad to pass the problem to me.
As Oggli accountants will know (most, anyway), the sum of the distances from any point on the ellipse to the two focal points is constant and it is known as the "major diameter." In this case, I knew it should be two hundred ells. The priests of both temples and many followers of both congregations came with me as I measured the bounds with ropes tied to the spires through the windows. They laid rocks and wooden totems along her holy border. There were surprises for everyone, as several private gardens turned out to be within the holy grounds and a church pagoda had to have a stone marker placed in it because it was not built precisely where it should have been.
Eventually, we came to the water. The priests were prepared for this. As it turns out, the goddess's lands now encompass about ten ells of marsh. That is not as far as the rival temple feared, so both holy men gave speeches, as planned, to say that the god had made a gift to the goddess. Then they laid down the largest marker stones in the watery border. A similar gift was made to the god from the goddess because lands once used by her congregation now clearly fell outside her border.
As all accountants will appreciate, I made sure to get paid in advance. In this case, the priests were not too unhappy so I had the luck of getting fed and paid again afterward. The mayor carved a note on a block of wood for me. He does not have parchment, much less paper. But he composed a letter of transit much like the one I received in Pharts Bad.
That letter has been useful, so I am glad to have another.
Denario used the last of his day's ink to darken the lines in his drawing of the churches. He carefully cross-hatched the pond between the structures until the pen gave out. Then put the quill away, shut his journal, and closed his eyes. He pulled up a blanket as he leaned against the trunk of a fir tree to sleep.
The heavens hadn't gone completely dark yet but there was no question of extending the day with a campfire. He'd tried to start a blaze from pyrites he'd been given in Double Bad. He'd failed despite having been shown how to use them. Fire remained a mystery. And without a fire, someone had whispered, the raccoons and wolves would 'get him for sure.' Denario worried about the possibility but there wasn't much he could do. He'd heard a wolf or two howling but none had come close, at least not while he was making so much noise. He'd been troubled by no other beasts.
His everyday challenges came from people, not wild animals. The local hill men shot at him, haggled terrible bargains for his food, cursed him for worshiping strange gods, accused him of acting like a child, demanded that he get their tribal tattoos, and seemed disinterested in math, which they considered the concern of "lowland folk."
Sometimes Denario wondered why Master Winkel had never warned him how many people would try to kill an ordinary accountant. But actually he had, hadn't he? Winkel had lent him the guild log books. Those were full of the deaths of book keepers and accountants. Details had not been spared. Clients became murderous when crossed. That was the underlying lesson that Denario had failed to understand in Ziegeburg, he supposed. And the carnage wasn't limited to small towns. Large cities could be worse. Hadn't Winkel said that the reason he didn't work directly for the Marquis de Oggli was that he'd seen the marquis chop off the heads from his treasure room staff? Witnessing that event had made an impression.
Denario tried to sleep but although his body was tired, his mind was not. He'd gotten himself wound up about geometry. Making his proof to the mayor about the properties of ellipses had gotten him interested in proofs in general. That line of thought led him back to how three must always equal three and how there had to be a formal way to prove it. He wished again for a campfire.
Pine needles, he thought.
Hadn't he heard something in one of the small towns about using pine needles to start fires? Denario had given it a try before but that had been when the needles were damp from an afternoon rain. The ones he was lying on right now were so dry they crackled beneath him like a fire already.
His eyes opened. He turned to his travel pack and dug down to the bottom of it, where the pyrites had come to rest.
In a minute or two, he was ready. He had his bags pushed to the side and his blankets folded neatly behind them. The sun had gone down behind the hills but there was still enough light to see his hands and the fire-starters in them. He took a swipe, pyrite against pyrite. Nothing. There was no spark. He tried it again. He tried it twelve times.
He flopped onto the mat of needles in defeat. Then he sat for a while, head in hands, and tried to think of what he must be doing wrong. The man he'd bartered with in town had bashed one of the pyrites with a piece of flint. The blow had produced plenty of sparks. Denario didn't have flint. There were rocks all around him, though. He'd felt them against his back as he lay on the bed of pine needles.
“What does flint look like?” Denario murmured to himself. It was a dark stone. That's all he knew. But maybe he could look for dark rocks and hope. He started shifting piles of needles with the toes of his boots.
The first big rock he found was nearly white. It was some kind of milky quartz. He found other pieces like it all over the ground. He covered them over and kept looking.
A minute later, he rushed back.
It's not a touch on good flint but quartz works just fine, the fellow had said. Denario hadn't paid much attention. He'd figured that the pyrites were the secret to fire. He hadn't wanted any yokel advice. But now he dug through the pine needles until he found a fist-sized lump. It was a solid chunk of quartz. Parts of the rock were the color curdled milk but other parts of it were rosy, he judged. It was getting hard to tell in the darkening sky.
He didn't have much time before true night hit. He reared back and took a big swing.
When the quartz met the pyrite, there was an explosion of sparks. Denario gaped. Hot bits of pyrite flew everywhere. One of them settled on the dry pine needles and began to smolder.
“Ha!” he shouted. He began to giggle like a maniac. Fortunately, there was no one to hear him. His feet did a little dance. He crouched down low to the pine needles and smashed the quartz into the flint again.
“Fire! Fire!” One of the sparks landed on a bed of dry needles. The needles immediately began to burn. Excited, Denario knelt down and blew on the embers as he'd seen many of the Mundredi do when they wanted to encourage a flame. “Hah! Finally.”
The diameter of charring pine needles grew. In a minute, with carefully applied long, slow breaths, the embers expanded into a broad flame. Denario began to relax. He'd done it. He'd made a fire. Tonight he would be warm. Even better, he'd be able to see. He could get back to mathematics. He could work all night if he wanted. That was the best part.
He spotted his accounting bag. It was a little too close to the burning pile of pine needles for his comfort. He plucked it by the strap. After watching the fire for a moment, he gathered his other bags. He grabbed his blankets, bow, buckler, and spear.
“Uh oh.” With all of his possessions in hand, he gazed upward at the dry pine tree. Flames from the pine needles on the floor beneath leaped to the lower branches. Denario suddenly realized that the needles, inches deep and dry almost all of the way to the ground, were too good a fuel. He saw no way to stop the burn.