Trisection

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Cannius the Younger thanked the boy who had come to fetch him, and moved to greet Duredda warmly. "How rare it is to receive a visit from a Brythonic chieftain," the ascetic said archly. "I had begun to worry that Britons' belief in the Glory of Christ had departed these lands along with the Romans."

"No such luck," Duredda grunted. He didn't have to reach too far back along his family tree to find barbaric relations who hated Rome and Christianity, but his great-grandfather had been hired as a mercenary by the legions, and now the Roman way of waging war and politics came as second nature to him. But he could take or leave the religion.

"Then I imagine you are not here to pray, or to seek absolution for your many sins," the priest said drily.

"No," admitted Duredda. "I seek a favor regarding ancient learnings. You're a well-read man, and I hope you can solve a problem for me."

"I am at your disposal," said Cannius, bowing his head.

"I think I recall from my studies, as a youth, a lesson regarding the Geometry of the Greeks," Duredda said. "To wit: if one has an angle, it cannot be easily trisected."

Cannius nodded. "I recall that fact, or something very like it," he said.

"Come with me to the garrison-house," Duredda asked. "Easier to show you than to tell you." The men walked down the hill, forded the stream, and crossed the fortified village to reach the reinforced stone building that had stood on its foundations since the time of Vespasian. Duredda led Cannius into the court where a gruesome scene was taking place. A man was stretched out on the ground with chains attached to his wrists and ankles. Three horses stood at the ready, and soldiers prepared to link the chains to their harness. Duredda stood over the captive, a wild-eyed and long-haired eastern barbarian, and waved his hands.

"You see my problem," he said. "Four limbs and chains; three horses; one Angle. What to do?"


Cannius the Younger's eyes narrowed. "Do you mean to say, sir," he asked mildly, "that you mean to rip this man into three equal pieces?"

"He's not a man, he's an Angle," Duredda corrected. "They've been boating over from their nasty marshes in Germanica forever now, killing Britons and pushing as far west as they dare. They're animals with a man-like appearance, that's all."

The Angle on the ground said something in his harsh language. Cannius couldn't understand a word of it, but nevertheless took it as read that the man had a number of negative opinions about Britons, as well.

"Yes, yes," said Cannius impatiently, "these are matters of nomenclature that no doubt would make for an amusing disputation. Of greater interest to me is this: why must you have him in thirds?"

"Oh!" Duredda stroked his mustaches. "Well, you see, this fellow's war-bands have been pushing even further west than the others. We actually caught him on this side of the Severn. It's important we send a message back to the Angles to have them stop this nonsense at once. As it happens, there are three kings that their kind recognizes: the kings of Northumbria, Mercia and East Anglia. I mean to divide this chap up into three pieces and send them back to each of 'em. And I want each piece to be similar in size, so they each know that we're watching 'em with equal vigor."

"Ah yes," said Cannius. "I believe I see the problem."

"Yes, but now I see what my teachers were going on about," said Duredda testily. "I could pull him in half easily enough, and quartering is certainly a procedure I'm familiar with, but divvying an Angle in equal thirds is an impossible geometric construction!"

"Or destruction, if you will," Cannius the Younger quipped.

"I come to you seeking help, and all you can do is make jokes," grumbled Duredda. "This is a serious matter, man!"

"There now, calm yourself," soothed Cannius. "Now, let us sort through one or two complications to the situation. First, are you aware that the Geometries of Pythagoras and the other Greeks do not apply to Jutlanders, Germanics or really any sort of creature at all?"

Duredda shifted his feet uncertainly. "I know I didn't pay very good attention during the classical bits of schooling," the chieftain admitted, "but I'm quite certain that dividing up Angles was mentioned."

"And right you are," said Cannius diplomatically. "But I'm afraid they were referring to a different sort of angle – a mathematical concept which describes the aperture between two lines which intersect at a point." He sketched briefly in the dirt with his toe. Duredda turned his head every which way, struggling to make sense of Cannius' drawings.

"Yes, but I don't care what sort of Angle they're talking about," Duredda said stubbornly. "The Greeks said trisecting an Angle is impossible, and by all the stupid saints, they were right!"

"Well, now, they didn't quite say it was impossible," said Cannius dogmatically. "The ancient Greek way of doing geometry involved using only a compass and a straightedge, undelineated by any sort of markings, to build their mathematical constructs. Pythagoras said that it was impossible to trisect an angle using only those two tools, and I believe he was right. Methods exist for trisecting an angle using other measuring devices, however."

"So, what I think you're saying is: I'm not using the right tools for the job?" asked Duredda. He snapped his fingers. "Bring me a saw!"

"Just a moment, just a moment," said Cannius hastily. "There might be an even more elegant solution. Tell me: is this unfortunate Angle the only one you have captured?"

"No, of course not," replied Duredda. "We have several hundred of them, including a few clan chiefs."

"I see," said Cannius. "Well, if you could procure a number of Angles that was itself divisible by three, then your problems would be solved trivially. You could simply send an equal division of captives to each king."

"And then just behead each of them," said Duredda, nodding. "An elegant solution, priest."

"Perhaps," said Cannius. "Myself, I'd consider sending them back alive."

"That would send the wrong message," Duredda argued. "We want the Angles to know we're a bloodthirsty, warlike sort. Showing mercy would appear as an act of weakness in their eyes."

"Then perhaps you might consider cutting off their left arms," suggested Cannius. "This would send a very clear message, regardless of the language of the speaker. The Angles will see: if they reach out upon the right side, all will be well, and no harm shall befall them. But should they reach out upon the left side…."

"… that limb shall be chopped off," finished Duredda. "Not bad, silly old priest. Not bad at all."

He walked Cannius the Younger back to his hermitage. "I fear it may be all in vain, though," said Duredda gloomily.

"Oh come now," said Cannius. "What's the matter? It cannot be as bad as all that."

"More and more Angles arrive by boat every day," said Duredda. "They're being pushed out of wherever they used to live; they have to come here. Their numbers are increasing; ours are dwindling, thanks to plague and emigrations and these never-ending turf wars. When seen with a military man's eye, our situation is a precarious one. It may be only a matter of a few generations before all Britons are pushed out by an unstoppable wave of Angles."

"Then you may need to stop looking at it with a military eye," Cannius suggested mildly. "If you cannot defeat them, consider joining with them. Better that than being swept away by some kind of Angular momentum."

"Conservation, and Angular momentum," said Duredda, nodding distractedly. "I'll think on it, priest."

"Then go with God," said Cannius the Younger, and disappeared within his humble cottage.

"Hm? Oh. Yes, whatever," said the chieftain, and stalked back down the hill again.