A Time Traveller's Guide To Feudal Japan Chapter 105 - Mathematics

He broke the news to them, and set them about a new task. Using the dirt that had mounted up, they began building a wall of mud about the perimeter of the field, as they normally would. It had worked out well, because now they had the material required to do such a thing.
In fact, some of the workers suggested Gengyo had planned this from the start, but in truth, he had been so focused on the problem of getting water to the field, he had put the process of creating a wall out of his mind.
With them now focused on their new task, he began to focus on his. He needed the circle of the water wheel to be perfect, else there was no chance of the system working. There was a simple way to create a perfect circle, but he dared not do it on wood straight away.
He found a nearby stick, that looked rather sturdy, and hammered it into the ground using the end of his hoe. There was a bundle of twine left over by the food area, which had been brought along in case there were any problems with the oxen's harnesses.
He took that, and tied it around the end of the hammered stick, before tying the other piece of string to his hoe. He made sure both were not liable to slip, and began to trace out a circle, making sure that there was always maximum tension within the string.
It came out good and thick, as he used the blade of his hoe to draw it. It was about as thick as he would like the water wheel to be.
Tension had been kept throughout, and the circle came out perfectly, as he would want it.
He selected some men from the digging team, and set them the task of filling in the template of a circle that he had created using wood. They looked confused as to why they were being asked to do such a thing, but they left for home, and returned with an axe and some woodworking tools each.
He watched with approval as they went about their work, finding wood that was thick enough to form at least one-quarter of the circle after being trimmed down to size.
He had not expected them to be so proficient in their craft, as he had selected them at random, but he supposed when you had to build houses of your own, then you would develop some familiarity with the art of woodworking.
As they went about their work, some of the villagers peered over curiously, wondering what was going on.
Gengyo found a small stick, and began performing calculations in the sand, pondering how he might make sure each bucket was positioned an equal distance apart inside the circle.
If he had a protractor, his task would be easy, but alas such easy tools were not available to him, and he had to do things the proper way.
It was easy enough to place buckets – if you imagine the wheel as a clock – at 12 o'clock, 6 o'clock, 9 o'clock, and 3 o'clock. But he wished to have 8 buckets in total.
He pictured various sets of triangles within the circle, and realized something. In forming eight points, and connecting the eight points to each other via straight lines, you were effectively forming an octagon within the circle. Through this image, he realized that if he were to measure the direct distance between 12 o'clock and 3 o'clock, and half that, he would then be able to accurately find the point in the circle that existed exactly halfway between the two.
He would use a piece of wood equal to this half measurement, and place one end on 12 o'clock. And then, he would turn it, until it met the outer part edge of the circle. The point at which the end of the wood touched the circle would be the point exactly halfway between.
It was not extremely complicated mathematics, and he found that he was a little disappointed once he selected the obvious solution that was staring him in the face. But it was a change of pace, and he enjoyed it.
The men spent the entire rest of the day completing the circle, with four quarters matching up to each other perfectly. All they needed to do now was piece it together with a frame, and secure the buckets.
Elsewhere progress was steady. The villagers had run out of dirt after doing a single side of the field, so a team was assigned to dig, and another team was assigned to collect the dirt from them and apply it. They switched every hour, so the task would not grow too tiresome.
And with the ploughing team, they had managed to plough just over half the field. Everyone was in high spirits as they went home, including Gengyo. He was more than satisified with their progress, and had immensely enjoyed his return to a more engineering-based frame of mind.
…
…
The next day, work began early, and they were raring to go. The men who had been helping Gengyo in the construction of the wheel once more returned with their tools, whilst their master brought along 8 buckets that had been lined with tar, so that they were fully waterproof.
They began work in earnest, cutting down enough wood so that they could create the eight different beams required to hold the structure up. As it happened, the placement of the beams was identical to the placement of the buckets, so they were able to sort the specific measurements then and there.
Those were hammered into place quickly, as the men found themselves enjoying this less tedious bit of work. Yesterday they had been at their wit's end trying to shave the wood off perfectly so that it lined up with the circle. More than once they took too much off, and Gengyo had to insist that they start over, else it would not work as it should.
The buckets were attached so that they could swing freely. Because gravity was always pulling the buckets downwards, even if they were to turn around as they went around the wheel the water would not spill.
A series of paddles were attached, 18 in total. These were angled so that they would catch the flowing water of the river, and send the wheel spinning round. These were measured according to the same system Gengyo had devised before, but this time, instead of the length being half of the distance between 12 o'clock and 3 o'clock, it was a quarter.
The wheel was really shaping up, and the villagers could barely hold back their curiosity, as the four men carried it towards the river. It was 3 metres in diameter, so not extremely large for a water wheel, but it was certainly enough to catch the eye. Such systems were not unheard of in Japan, but they were far from being commonplace, and villagers like this had never seen such a thing in their lives.
Whilst the wheel had been complete, and looked sturdy enough, they still required a hub to attach it to. Like the buckets, the wheel needed to be able to spin freely, with as little resistance as possible. Too much resistance, and the force from the water pushing the wheel would be liable to tear the hub into pieces.
They also needed to construct a trough to catch the water in, and some pipes for the water to flow down into the field.
By the end of the second day, the ploughing was completed, and the dirt walls were about half a day away from completion. With that, they were able to begin casting seeds, so that the seedlings would have a chance to grow their roots before they were assaulted by the flooding water.
And on the third, the hub was complete. They mounted the wheel in place to test it. The villagers gathered round to watch.
It was lowered onto the hub, and within a split second it was already turning. They watched as the buckets were dunked underwater, travelling to the top, full of the liquid, before they hit a peg, which caused the bucket to tilt and dump the liquid back in the river. A trough was to be fixed in place at that point, but so far it was a resounding success.
The villagers cheered its successful first test. The calculations had been done to perfection, and there was not a single fault with it. Even with the river flowing as fast as it was, there was no sign of any strain upon the system. By now they were aware of the intentions of such a system, and understood the implications. If the wheel was complete, it meant less work for them.
"That's an impressive structure lad."
Jikouji commented, marvelling at how perfectly it functioned. He seemed to have warmed to their attempt at cultivating this land, now that he saw how high their chances of success were. He had not even asked about the other plan for a while.
"It's far from complete yet, old man. But once it's done, aye, it'll be impressive enough."
He agreed. In his mind, he did not take credit for its invention. He was merely a recreator of what had once been, and so, he had the utmost respect for the Romans, and the other ancients, that had been able to use water so perfectly.
They dismounted the wheel, so that it would not wear too much, and having it up there without a trough in place was pointless.
They had to create a ramp, all the way from the field edge, right to the top of where the trough was. It was made out of wood, though another material would have been preferable. The wood was just deep enough so that the water would not spill.
But nearing the end of the third day, this structure was far from completion. The rest of the villagers were at work planting the seeds, and would be finished by sundown. All the walls had been done, and their progress was satisfying to say the least.
As they were so absorbed in their work, a woman strode down the hill, towards the field. She had a great confidence about her, as though she was the wife of some Daimyo or other.
The villagers watched her come, curious as to who she was. But she, inturn, completely ignored them, and instead walked behind a man who was hard at work, cutting wood down to size to be used as scaffolding for the pipe structure.
"It's done."
"Hm, you took a little longer than expected, Isabella."
He commented, without turning around. It was the second time he had said that to her, and she had to resist the urge to slap him. She'd left straight away, and returned as soon as possible, and this was the greeting she had got.
"…"
She did not reply as she bit her lip.
"How did you get on, then?"
He turned around, wiping the sweat from his brow, as he dropped his axe to give her his full attention.
"I have the location."
"Good."
He nodded, she had done what was asked of her.
"I also know what room it's inside."
He raised his eyebrows in surprise. He had not expected her to go that far. But a smile soon broke out – the more information they had, the better their chances of success would be.
"Excellent. You will have your reward as soon as our task is complete. You have done a fine job, Isabella."
She wrinkled her nose against the smell of his sweat, but she could not help smiling beneath her veil at his praise.
'Of course I've done well, you foolish man. I never do anything less than brilliance.'
She thought internally, glancing around the field, noting all the work that was going on. She was about to ask about it, but he was already walking away, and he motioned for her to follow.