Mathematics and the Waterwheel
Free Ch. 131 of 655 9 min read
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“Other city-states believe that land is the greatest wealth, which is why they are unwilling to accept foreigners and freemen, unwilling to share the city’s benefits and fortunes with them. I, however, believe the opposite. I believe that people are the greatest wealth of a city-state! With people, you can cultivate grain, create works of art, and generate prosperity! With people, you can forge weapons, form armies, and defend the city! And with enough people, you can acquire far, far more land!” Davos declared, his voice ringing with unshakable conviction.

His words struck Kunogelata with a force he had never felt before. A man of his intelligence did not need to be told how one went about acquiring “far, far more land.” It was clear that Davos was implicitly revealing the source of the land grants for his new citizens. Kunogelata fell into a thoughtful silence.

It was Kunogelata’s eldest son, Sostratus, who broke the quiet. Seeing the partially completed warehouses and inns at the construction site, he asked, “I’ve heard you intend to attract merchants from Thurii and other foreign states to store their goods and lodge here. Are these the warehouses and inns you are building?”

“They are.”

“It looks like you are planning to build quite a few. Does the city-state intend to manage these properties itself?” Sostratus inquired.

Mersis understood the implication behind the question and immediately replied, “Of course not! The city-state doesn’t have that many public officials, and everyone has their own duties to attend to. We plan to hold a public auction and lease them out.”

“Public auction” was a novel term, but the innate commercial instincts of the Greeks allowed Sostratus to quickly grasp the concept. “Are we permitted to participate?” he asked.

“You would be most welcome!” Mersis answered enthusiastically.

***

Despite the exhaustion from a long day of moving and setting up their new home, Kunogelata awoke at dawn the next morning. Rising early was a habit ingrained over many years. After a slave helped him dress, he decided to take a short, supported walk before returning for breakfast.

Kunogelata’s new residence was situated at the foot of the mountain of Amendolara. After the distribution of housing to the new citizens, there were still numerous empty homes in this area. Only by being here, in the midst of it, could Kunogelata truly understand the depth of the original citizens’ animosity toward Thurii; the scars of their devastating losses were plain to see.

Most of these empty houses had been leased at low prices to freemen. As soon as Kunogelata stepped outside, he saw them in small groups, hurrying off to their day’s labor. Their faces shone with a vitality and passion he had never seen among the freemen of Thurii. Even upon seeing him, a complete stranger, they would offer a friendly greeting in their oddly accented Greek. Kunogelata knew the source of their zeal for life: they had been elevated from slaves to freemen, and their employers were paying the so-called “preparatory citizen tax” to the city-state on their behalf. As long as they abided by the law and worked hard, in two years they would become preparatory citizens of the city-state.

By now it was June, and the weather was growing hot. But here in Amendolara, built upon a mountain, it remained relatively cool, with sea breezes and mountain winds taking turns caressing the small city.

As Kunogelata made his way up the road, he frequently encountered citizens heading down the mountain to work their fields. Old citizens and new citizens alike greeted each other with jokes and laughter, their interactions remarkably harmonious. Even the slaves who followed them seemed to be in high spirits, undoubtedly filled with the hope inspired by the tales of the mercenaries who had earned their citizenship.

What truly astonished Kunogelata, however, was the presence of women among them. From their conversations, he gathered they were the wives of citizens, not freemen—whose wives often had to work to supplement the family’s income by running market stalls or waiting tables. These women were preparing to work in the fields alongside their husbands. He recalled his son mentioning some of the city’s strange customs the previous afternoon, and as he looked closer, he saw that the women were indeed Lucanian. A more traditional and conservative outsider would have been scandalized, but Kunogelata, a veteran of the political arena, immediately considered the significance of this phenomenon for the city-state. It was clear that Davos was not merely paying lip service to his ideas; he was actively using every means at his disposal to assimilate the Lucanians.

This young man possesses an uncommon ambition, Kunogelata thought. Under his leadership, the small city of Amendolara was so peculiar, so different from any other, yet in such a short time, it had been filled with a new and vibrant life. Kunogelata’s interest deepened. He noticed that while many people were heading down the mountain, a significant number were also heading up, among them many children. When he inquired, he was told that disciples of the Pythagorean school were giving free mathematics lectures in the agora.

The Pythagorean school is here? Kunogelata was shocked, but then he recalled the circumstances of his first meeting with Davos and had a flash of insight. Archytas must have been the one to forge this connection! The Pythagorean school was famously opposed to democracy and supportive of aristocratic rule. How could Davos tolerate their presence?

Then again, Kunogelata mused, as a archon of life, was Davos’s regime truly a democracy?

When he reached the agora, a small crowd had already gathered around a platform where a young man was teaching basic mathematics.

The young man was Mentoticeles. Sometime earlier, he and his colleague, Maticoris, had received Davos’s promise of support and had prepared to recruit a great number of apprentices. However, they had adhered strictly to the Pythagorean tradition, which required that all disciples possess a certain foundational knowledge of mathematics, to be verified by a test.

Brimming with enthusiasm, they went door to door, only to be met with rejection at every turn. The new citizens were largely uneducated and, having just received their land and homes, were too busy putting their lives in order to have time for tests and studies. The few who were interested in learning, hoping it might prepare them for future public office, were immediately discouraged by the entrance exam. The old citizens, though better educated, were preoccupied with mourning their dead. Furthermore, the Pythagorean school had a poor reputation in Magna Graecia; tales of it being a secret society, of its members worshipping a strange god—the Number—and of its opposition to the city-state had long since spread, making the two young disciples most unwelcome.

After days of fruitless effort, they had not managed to recruit a single student. Utterly dejected, they were wandering aimlessly through the city when they stumbled upon the “Hades’ Numbers”—in truth, Arabic numerals—that Davos had taught Mersis for easier bookkeeping. For Greek scholars of this era, mathematics presented a vexing problem: the Greek letters used to represent numbers were cumbersome to write, and representing any number over ten thousand was exceptionally difficult, creating a significant obstacle for complex calculations. Yet these Hades’ Numbers were simple, easy to learn, and easy to remember, elegantly solving one of the most persistent problems in their field. It was a discovery that would undoubtedly increase the efficiency of mathematical research and, most importantly, make mathematics far more accessible to the general populace. (Historically, it was Archimedes who solved this problem for the Greeks by inventing a method for writing large numbers.)

The two young men felt as if they had discovered a priceless treasure. However, when they asked Mersis to teach them the secret of these symbols, he refused. “This was taught to me by Lord Davos,” he said. “He claims it is a gift from the god Hades. You must have his permission to learn it.”

So, they were forced to seek out Davos once more, expecting a difficult negotiation. To their surprise, Davos agreed immediately. “Mathematics arises from daily life,” he told them. “Of course I hope that everyone can learn it, so they can apply it to their lives, solve the problems they encounter, and make their existence more convenient. That is the true purpose of acquiring knowledge!”

Davos’s words deeply moved the two young men.

Seizing the opportunity, Davos made a proposal: he asked them to hold free, public mathematics lectures in the agora, teaching the fundamentals to any citizen willing to learn. The Senate, he promised, would endorse their efforts and encourage the populace to attend. Over time, this would cultivate a large pool of mathematics enthusiasts for the Pythagorean school, allowing it to finally put down roots in Amendolara.

Unwilling to return to Tarentum as failures, the two ambitious young disciples deliberated and finally agreed to break with the hallowed tradition of the Pythagorean school and teach their art openly.

Davos was overjoyed. He immediately invited the two mathematicians to participate in the design and construction of the city’s waterwheels.

From his previous life in the water-rich regions of southern China, Davos remembered seeing waterwheels everywhere. They had seemed like simple contraptions. But when he had Alexis, the project’s supervising official, attempt to build one with the city’s carpenters, he discovered it was not easy at all. Prototypes collapsed because the support frames were unevenly stressed, or the main wheel wouldn’t turn smoothly because the axle was misaligned. The humble waterwheel, he realized, involved rather precise mathematical calculations and an application of mechanics. In this era, mathematicians were often also masters of physics, as the two fields were so closely intertwined. Thus, Davos had no choice but to seek help from the two young Pythagorean disciples.

When he showed them his simple diagrams of the waterwheel and explained its purpose, they were instantly captivated. After careful study, they concluded it was a perfect manifestation of the “God of Mathematics” helping humanity improve their lives (completely overlooking the fact that it was Davos’s idea). Furthermore, under the leadership of Archytas, the Pythagorean school in Tarentum had already been studying mathematical mechanics for several years. This was a perfect opportunity to put their knowledge into practice. The two men immediately pledged their full support to the project.

And so today, while Mentoticeles lectured on the platform, Maticoris was by the Sinni River, overseeing the final stages of the waterwheel’s construction. Though Mentoticeles was anxious to join him, he controlled his impatience, finished his lesson with care and quality, and diligently answered the flurry of questions from the eager crowd before hurrying down the mountain.

At a narrow section of the Sinni River, the rushing current pushed against the great wheel, causing it to turn in a slow, steady rotation. As each bucket, filled with river water, reached the apex of the wheel’s rotation, it tipped over, pouring its contents into a wooden trough. From there, the water flowed into a network of large and small, winding irrigation channels, carrying the cool, clear water of the Sinni to the fields on the higher ground.

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