Killing Pythagoras (Mediterranean Prize Winner 2015) (37 page)

CHAPTER 80

June 29
th
, 510 B.C.

Helicaon, secretary to the Croton Council, stumbled through the dark streets of the city. Every now and then he muttered a few words under his breath. He wasn’t used to getting drunk, and even less so to staying out late, but that day he had been drinking since the morning meeting in Cylon’s luxurious residence.

I haven’t even showed up at home yet
, he told himself guiltily. He was embarrassed to look his wife in the eye after having ratified that unjust decree of exile. He was still stunned that he had been persuaded to do it, but there had been something about the masked stranger’s voice that had invaded and become confused with his own thoughts. The bag of gold coins had also helped, of course, and he hoped his wife’s irascible temper would be subdued for a few days thanks to the gold. That afternoon, Helicaon had passed by a slave stall and bought a cook, something his wife was always asking for, but which until today they had never been able to afford. He had asked the slave trader to send her the following morning and have her prepare something special for his wife as a surprise. He hoped that would mitigate his wife’s anger over his late arrival and his drunkenness.

He looked up. Making an effort, he managed to merge the two images in front of him into one. He was relieved to find he was only fifty yards from his house.
Thank goodness I’m nearly there.
He shook his head, reproaching himself for his lack of prudence. He should have put the gold that remained after purchasing the slave in a safe place. It was still hidden in his tunic, and the night was disturbingly dark.

Suddenly, someone grasped the neck of his tunic and pulled him toward a narrow alleyway. He opened his mouth to shout, but a pair of strong hands quickly grabbed his head and smashed it against a wall.

Akenon contemplated the body lying unconscious at his feet. He knelt and searched the secretary’s tunic with expert hands.

Here it is!
he said to himself triumphantly.

He pulled out a bag of coins, hid it in his own tunic, and lost himself silently in the shadows of the night.

When he arrived at the community, everyone was asleep. Dawn wasn’t far away, so he decided to wait, sitting next to Pythagoras’ house. He was far too agitated to sleep.

The philosopher appeared a short while later.

“Let’s go to the Temple of the Muses,” Akenon suggested. He needed a brightly lit place far from prying eyes.

When he entered the temple, Akenon went to the light of the sacred fire. He took out the secretary’s bag and placed three gold coins in the palm of his hand. No sooner had Pythagoras seen them than he responded with certainty.

“They’re from Sybaris. There’s no doubt, though I had never seen them in gold.”

Akenon was already familiar with Sybarite coins. Following the prevailing custom in Magna Graecia, they were minted thin, with a decorative edge. In the case of Sybaris, they depicted a bull looking behind him. On the heads side of the coin the bull was in relief, whereas on the tails side it was concave. Akenon hadn’t seen them in gold either. In Magna Graecia coins were generally made of electrum—an alloy of gold and silver—or pure silver.

“Eritrius will be able to give you more information,” added Pythagoras. “He’s the foremost coin expert in Croton.”

Akenon was impatient to continue following that thread. He said goodbye to Pythagoras and rode back to Croton, arriving at the custodian’s establishment before he had opened his doors for business. When Eritrius appeared with his guards, he was surprised to see Akenon waiting for him, and began to stammer an apology.

“I’m sorry, Akenon. They had a sealed decree. I couldn’t say no. I just…”

“Don’t worry, Eritrius,” Akenon interrupted, “that’s not why I’m here. Pythagoras has assured me that at today’s Council session the decree will be rescinded. My fifteen thousand drachmas will return to your hands, and that’s where I want them to stay. Let’s go inside and I’ll tell you the reason I’ve come.”

In the privacy of Eritrius’ office, Akenon placed the coins on the table.

“I know they’re from Sybaris. Can you tell me anything else?”

Eritrius took one between his fingers and examined it with interest.

“I’ve never seen one of these.” He turned it over a few times in silence. “It’s the bull of Sybaris, you probably know that, and I’m even sure I recognize the stamp they’ve used. However…” He took the other coins and compared them. “There’s something odd here. To mint this kind of coin, the usual procedure would be to create a new stamp, but they’ve reused one that has already been used to mint silver coins. Also, it’s very rare to see a gold coin from Sybaris.” He continued examining them for a while, his gray eyebrows coming together in a thoughtful expression. Finally he held one close to Akenon’s eyes and pointed something out. “Do you see these letters?”

Akenon peered at the coin. There were two groups of lettering on the heads side, one above the bull and one below.

“What do they mean?”

“These letters here represent the name of the city, and these others,” he touched below the bull, “represent the aristocrat who ordered the gold to be minted. They’re only included when it’s a very important person, and even then only rarely.” He tapped the golden letters a few times. “In this case, the person who ordered them is Glaucus.”

Glaucus again!
Akenon sat on the edge of the table and stared at the coins.
Glaucus had this gold minted not long ago, and in such a hurry he didn’t have time to make a new stamp… A short while later, the gold appears in the bag carried by the secretary who signed my decree of exile…

He stood up determinedly and collected the coins.

“Thank you, Eritrius, you’ve been a great help.”

He rode back to the compound, spurring the horse so it flew along the road. Akenon felt full of energy. It had been agonizing groping around in the dark for so long while the murderer acted as he pleased. Now, at last, he was experiencing the familiar excitement of being involved in an investigation that was making progress.

Either Glaucus is the murderer, or he’s been in direct contact with him
.

Whatever the case, the Sybarite was the next step in the investigation.

Pythagoras’ theorem

…

Pythagoras’ theorem posits that, in a right triangle, the square of the hypotenuse (the longest side of the triangle) is equal to the sum of the squares of the two sides forming the right angle.

In the diagram: c²=a²+b²

For at least five thousand years, Ancient Egypt and other civilizations of Antiquity have known of groups of three numbers whose values correspond to the sides of right triangles. The simplest example is: 3, 4, 5 (5²=3²+4²). The right triangle whose sides have those values as their length is known as the
sacred Egyptian triangle
. It is believed to have been used in the design of multiple constructions, such as the Pyramid of Khafre (26
th
century B.C.).

During Pythagoras’ time, many peoples had already been using practical examples of right triangles for thousands of years. However, there is no evidence of anyone prior to Pythagoras theoretically demonstrating the relationship between its sides: i.e., that c²=a²+b² is always true, not just in specific cases such as the
sacred Egyptian triangle
.

Pythagoras’ theorem is another discovery of the Pythagorean School which shows, in a sublime and unequivocal way, the relationship between arithmetic and geometry: i.e., the link between numbers and physical space.

The theorem contributed significantly to the enormous confidence Pythagoreans had in their ideas, and to Pythagoras being definitively elevated to the status of one of history’s greatest geniuses.

…

Encyclopedia Mathematica.
Socram Ofisis. 1926.

CHAPTER 81

July 1
st
, 510 B.C.

Two days later, the heavy doors to Glaucus’ palace opened wide and the Sybarite passed through them to the sunshine outside, where he greeted his visitors with a placid, cordial smile.

“Akenon, Ariadne, it’s so lovely to see you again!” He looked at the rest of the group, widening his eyes as if finding a wonderful surprise. “By the earth and the sun’s resplendent light, if it’s not grand master Evander! What an honor to welcome such illustrious guests to my humble abode. But please, don’t stand out in the street. Come in, come in.”

Standing to one side of the entryway, he gestured toward the interior of the palace in an obsequious, fawning manner. He had regained his weight, his hair was coiffed, and he was wearing a clean, simple tunic. Akenon glanced at Ariadne from the corner of his eye and then at the interior of the palace, unconvinced.

The expedition had been organized to clarify Glaucus’ role in the murders, but also to find out how the problem of the circle had been solved using Pythagoras’ theorem. For this, the philosopher had decided that Evander should travel with Akenon to Sybaris. Aristomachus was a better mathematician, but his nervous disposition made him unsuited to the task.

Akenon was grateful for Evander’s presence on that expedition, since he was the most open and good-natured of the grand masters. He was also happy Ariadne had come. Although she still treated him with some reserve, something had changed in the past couple of days. The previous afternoon, as they were entering Sybaris, Akenon had caught her looking pensively at him. She had quickly diverted her gaze, as if she didn’t want him to read her thoughts. Could she be rethinking her decision? In any case, Ariadne was behaving more warmly toward him, and that was very welcome.

Pythagoras had given them several letters they could use should Glaucus refuse to see them. They were for other important members of the Sybaris government, Pythagorean initiates who had the ability to put pressure on Glaucus. The letters were still packed away because the unpredictable Sybarite had proven to be completely amenable from the first message they had sent him. In spite of that, and even with the ten élite soldiers that accompanied them, Akenon wasn’t so sure.

Glaucus put on a look of dismay.

“I see from your face that you doubt me, Akenon. It’s understandable, my behavior during your last visit left much to be desired. I wasn’t myself, but that’s all over now, believe me.”

Seeing Akenon still hesitant to enter the palace, the Sybarite spoke again.

“Besides, you don’t have to worry about Boreas anymore.”

What does he mean by that?
wondered Akenon in surprise. He turned to his right and looked at Evander and Ariadne. She shrugged and gestured with her head that they should enter. They had agreed they’d go into the palace if they were allowed to do so accompanied by their soldiers. Ariadne agreed that Glaucus’ radically changed attitude was unsettling, but he never seemed to be completely in his right mind, and his pleasant self would always be better than the violent side they’d seen the last time.

Evander also seemed in agreement, so they went through the entryway with Glaucus. Their ten soldiers followed close behind.

The enormous wooden circle was no longer in the inner courtyard. Akenon moved away from Glaucus and whispered to the commander of the hoplites that they should stay in the courtyard, beside the statue of Apollo, at the ready to come and help if he raised the alarm. He had already warned them about Boreas, and they were uneasy as they kept a lookout, prepared to use their weapons should a mountain of muscles come hurtling at them.

Akenon went back to Ariadne, Evander, and Glaucus, who led them in silence to the banquet hall.

The room had changed in appearance. The wall separating the hall from the kitchen and storerooms had been erected again, the silver panels had been neatly piled in a corner, and all the torches on the walls were lit, thanks to which they could see that the walls were still covered with inscriptions relating to the circle.

Glaucus turned to them.

“This is my new study hall.”

The Sybarite seemed a bit deranged, but there was genuine happiness in his face and a kind of gentle tranquility, as if he had achieved all his life’s goals. His movements were calm, lacking the explosive mania of a few weeks ago. He pointed to some tables covered with parchments in the middle of the hall.

“Here’s my treasure.” He smiled like someone introducing his own child to his guests. “It cost me a significant portion of my gold, but it’s worth it.” He inhaled and slowly released the air, then added to himself, “Well worth it.”

“Is it the method to find the approximation to the quotient?” asked Ariadne, approaching the table. She wanted to decipher whatever she could see in the parchments as quickly as possible, in case Glaucus hid them in an outburst of rage.

“That’s right,” answered the Sybarite solemnly. “The quotient between the circumference and the diameter of a circle, to four decimal places.” His face lit up, suddenly conscious that the people waiting for his explanations were grand master Evander and Pythagoras’ daughter. “As a sign of goodwill, so we can set aside any differences there may have been between us, I’m going to share something with you…” He shook his head slowly and opened his hands, implying it was something that couldn’t be expressed in words. “Something that up to now is known to only two people in the world: the person who discovered it, and me.”

“Who discovered it?” Akenon was quick to ask.

“We’ll talk about that later,” Ariadne immediately intervened. “Let’s see the method first.”

Ariadne threw a quick glance at Akenon. They had agreed that, if Glaucus cooperated, they would first try to get him to show them the method, and only afterwards ask him questions related to the crimes. If they tried the other way around, they might end up with nothing on either front.

Akenon clenched his jaw and looked away, acknowledging that Ariadne was right. It was what they had agreed to do, but he was finding it hard to wait without bringing up the subject. Glaucus could be the murderer and everything else could be nothing more than a charade. Besides, he was anxious. Even though the Sybarite had said they need not worry about Boreas, he couldn’t help turning every few minutes toward the door.

Glaucus passed his hand over the parchments, stroking them with delight, then turned to his guests.

“The method is based on a simple idea: the more sides a regular polygon has, the more its perimeter approximates the circumference of a circle. We can see clearly that an octagon is closer to a circle than a square. And a polygon with a thousand sides would, at a casual glance, be indistinguishable from a circle.”

Akenon nodded, as did Ariadne and Evander. He had understood Glaucus up to that point, but he suspected he’d soon be incapable of following the explanation.

“To calculate the quotient, we begin with a square inscribed in a circle. The diameter of this circle is one, which means that the length of its circumference is the quotient, which we knew was three and a bit.”
[5]

“The perimeter of the square inscribed in this circle will be equal to the number of its sides multiplied by the length of each side. Since it’s a square, it has four sides. And we can see that each side measures half the square root of two.”

As he spoke, he pointed out what he was explaining on a parchment with a circle lightly drawn on it and polygons inscribed inside it. In one of the quadrants of the circle there were many lines, which would prove to be the key to finding the highly coveted approximation to the quotient.

“Therefore,” continued Glaucus, “the first approximation we find to the quotient, starting from the square inscribed in it, is four times half the square root of two. In other words, 2.82.”

“It doesn’t seem like a very good approximation, when it’s already known to be about 3.1,” objected Akenon.

“No, of course it’s not,” replied Glaucus, amused, “but now we come to the magic behind the method. Starting from the square inscribed in the circle, we’re now going to find out how to double the number of sides. If we can do that, we’ll have a polygon with eight sides, whose perimeter will be much closer to that of the circle than the square. Then we’ll double the number of sides again, and the approximation will be even closer. And we’ll keep doubling the number of sides: 32, 64, 128…”

“It’s clear,” intervened Ariadne, “that with each doubling, the perimeter of that polygon will be a closer approximation to the quotient. We know the number of sides in every case. The key is knowing the length of each side. In the case of the square, it’s obvious it will be half the square root of two, but how can you tell for the next polygons?”

Glaucus turned toward her, suddenly excited. Akenon observed uneasily that his cheeks had become flushed and there was perspiration on his forehead.

“Exactly!” exclaimed the Sybarite, almost shouting. “That’s where your father’s theorem comes in. It shows us in an astonishingly simple and precise way the length of each side of the doubled polygon based on the length of the side of the polygon we began with. If we know the value of the side of the square, and we’ve seen that we do, with your father’s theorem we can obtain the value for the side of the octagon, and from that, for the side of the sixteen-sided polygon, and so forth.”

Akenon realized that Ariadne and Evander were becoming increasingly enthralled. They would experience few occasions like this in their lives, where they were on the verge of learning a geometric discovery which represented a huge leap compared to what was already known in the field.

“I didn’t know the proof of Pythagoras’ theorem,” continued Glaucus, “but the person who discovered this knew it perfectly, and showed me how to use the theorem to figure out the method for doubling the sides of a polygon.”

As an Egyptian versed in geometry, Akenon knew of some specific cases where the sides of a right triangle had precise values, but he was unfamiliar with Pythagoras’ theorem.

“Now you’ll be able to appreciate the genius of this method.” Glaucus was unaware that his hand was trembling over the parchments. He radiated so much enthusiasm he looked like a madman. “Pay attention, because this is the key to everything: the side of the duplicated polygon is the hypotenuse of a triangle whose long side is half the length of the side of the original polygon, and whose short side is the difference between the radius and the long side of another triangle whose short side is half the length of the side of the original polygon, and its hypotenuse is the radius.”

Complete silence descended on the banquet hall, as if these words had stopped time.

I got lost from the beginning
, thought Akenon, slightly embarrassed. From the corner of his eye, he observed his companions who were as still as statues, staring intently at the parchments. He concentrated on the drawings as well, trying to extract some conclusion.

Ariadne contemplated the geometric figures, trying to recreate Glaucus’ explanations. She concentrated on obtaining the eight-sided polygon from the four-sided one. After a while, she managed to do it, impressed with the process, and then immersed herself again in the diagram to see if the method could also be used to obtain the sixteen-sided polygon from the eight-sided one.

“It works,” she whispered in amazement after a while.

She turned to Evander, who had just finished checking it for himself. The grand master had the same astonished expression as Ariadne. If they hadn’t been standing next to their potential enemy, they would happily have stayed for days, exploring the geometric world they had just discovered.

Akenon continued looking at the diagram, frowning, and finally gave up. Maybe he would be able to understand it by analyzing it in his own time in a relaxed, safe environment, but he certainly couldn’t do it there. His priority was the investigation, and everything else was very much secondary. However, he respected the importance it had for the Pythagoreans, and could sense the enormous magnitude of that geometric discovery.

Glaucus stood there in expectant silence, giving them time to assimilate the method. When he saw that Ariadne and Evander had understood it, he was disappointed they didn’t get as carried away with euphoria as he had.

Evander spoke in an absent tone, without taking his eyes off the parchments.

“How many sides have been obtained by doubling?”

“Two hundred fifty-six,” replied Glaucus proudly, as if the merit were all his. “He’s done the doubling six times, starting from the square. The operations are cumbersome, but the discoverer has a superhuman ability to do numerical calculations. Besides, he uses numbers in a strange way, different than what I had seen up to now. As you can see from the parchments, it's an ingenious and very efficient method.”

Ariadne turned to Glaucus, almost afraid to ask.

“What approximation did he obtain?”

“3.1415. But don’t be in such a hurry, my dear brothers. I’m going to order a copy of all the parchments to be made so you can take it back to Croton with you.”

“You’re very kind, Glaucus.” Ariadne smiled in gratitude, but she feared that a new change of heart in the Sybarite would make him retract his offer. Because of that, she continued trying to memorize as much as she could of the parchments, as did Evander, while she continued asking questions. “How do you know the approximation is correct?”