Hippasus of Metapontum

Robert Perron

The ship’s ladder swayed left and right, along with the contents of my stomach, forcing a frantic clamber from the center hold with no time to admire the hazy half-moon over the bow. I grasped the port-side rail as we surged uphill at the incline of a three-four-five triangle, teetered on wave’s crest, and plummeted at the same pitch. Except for last night’s supper, which rose and spewed over the rail into the white-capped brine.

“Aye,” said a voice beside me. “Let her all out. You’ll feel the better for it.”

Out it gushed, mutton and wine, followed by a dry retch. I pulled the sleeve of my tunic across my mouth and looked to the source of the nocturnal wisdom, a wide face of white stubble beneath a bald pate. His frame resembled a barrel set on two staves. He wore nothing but a sleeveless, white chiton [kiton] against the night draft, and stood upon the galloping deck as I would Mother Earth on a calm afternoon.

“My advice, sir,” he said, “take some air before retiring. It’ll settle ye stomach and steady ye legs.”

The ship fluttered on another crest, and my fingernails cleaved the rail as we dove for the wave’s trough. My companion smiled, planted to the deck as steady as the mast behind him.

“And I would welcome a chat,” he said, “to break the monotony of the watch.”

“If you please,” I said, “whom am I addressing?”

“Orestes the mariner, ye are, my young sir, sometimes called Orestes the ancient mariner.”

“What’s your age, then, ancient one?”

“Not a clue, sir, but I’ve been asea longer than anyone I know, including Kastor, the master of this boat.”

My stomach calmed, as did the ship, not soaring and diving as before, or perhaps my imagination and sickness had exaggerated the turbulence. I removed one hand from the rail and straightened my posture.

“Now, good sir,” said Orestes, “tell me, what is this distinguished group we convey to Miletus [myleetus]? Are ye mathematicians?”

“We’re mathematicians and philosophers,” I said. Then added sotto voce [sado voce], “Followers of Him.”

“Ah,” said the ancient mariner, “Pythagoreans.”

“Indeed,” I said, surprised that a simple sailor possessed knowledge beyond the myths of the sea. “We’re on our way to a conference.”

“And which of your greybeards is the illustrious Philolaus [Filolayus]? He must be the oldest and loudest, the one with the curly-haired lad in tow.”

Again, a surprise. That an ordinary seaman knew the name of a philosopher, albeit one of renown. However, he’d erred regarding identity.

“No,” I said, “that’s not Philolaus. The one with the lad is Hippasus.”

“Of course,” said Orestes. “I’ve heard of him, too. A contemporary of the great man, of Him himself, am I right?”

“You're correct.” What an extraordinary fellow, I thought—knowledgeable, inquisitive.

“And the lad?” said Orestes.

“The lad is Demetrios, assistant and ward to Hippasus.”

Orestes clapped his hands. “Aye, we should all have the assistance of such ruddy cheeks.” He extended an elbow as if to nudge me. “And I don’t mean the ones upon his face.” I indulged his coarse humor with a chuckle.

“If it’s Philolaus you’re looking for,” I said, “think of the most reserved, the sternest person aboard.”

“Ah, I know the one you mean. Who can miss him?”

“He represents the predominant view despite Hippasus’s seniority.”

“Do they get along,” said Orestes, “Hippasus and Philolaus?”

I moved my face close to the mariner’s toothless countenance. “Not at all,” I said. “Philolaus considers Hippasus a heretic.”

Orestes raised an eyebrow. “How so?”

“Hippasus doubts the harmonics of numbers. He says there exist numbers that are not true ratios.”

Orestes rubbed his chin as though considering the suggested abstraction. “Aye, but suppose there are such numbers?”

I took a half step back, releasing my grip on the rail and losing my balance. Orestes threw out a forearm which I clutched.

“No disrespect meant, sir. I’m just saying—is there any way to decide the issue?”

Again, I leaned into the sailor’s sagging face. “Tomorrow at noon Hippasus is making a presentation. We’re putting up a sandbox amidship, weather permitting.”

“It should be a fine day, sir. This bit of roughness will pass.”

***

Next day, on a calm sea, as Helios’s chariot approached its zenith, Demetrios, wearing a light blue citron [kiton] with gold trim, and no undergarments, raked the sandbox smooth. A semicircle of thirty Pythagoreans watched, Philolaus at one edge, arms crossed, Hippasus in the center. My perch lay midway in the second rank, where I craned my neck, edged out by the greybeards. I felt an elbow in my left rib cage and glanced to my side to see Orestes.

“You take an interest in mathematics?” I said.

“Aye, sir, and astronomy. It keeps us afloat.”

Hippasus stepped forward, turned, and faced us, tall body with an angular face and slanted smile, contemporary of the great man, Him himself. “We seek the truth,” he said. “Do we not?”

I looked to Philolaus for a hint of a reaction, but he was wont to give emotive displays.

“Yes, yes,” said the Pythagoreans, “get on with it.”

Hippasus turned to Demetrios and said, “Would you be so kind as to draw two lines, one perpendicular upon the other?”

The boy placed a stick with a cord in the sand, stretched the cord, and snapped it to create a line. He attached a second stick, drew arcs, aligned the cord on their vertices, and snapped a perpendicular.

“Well done,” said Hippasus. “Now mark off ten equal measures on each line.”

Demetrios shortened the length of cord between the two sticks and bent to the task.

“A fine lad,” whispered Orestes.

“Now,” said Hippasus, “connect the ends to form a triangle.”

I bent sideways toward Orestes. “Are you following this?”

“Aye, sir, he’s making a right isosceles triangle with the legs measuring ten units each.”

Hippasus raised a finger and his voice. “Do we agree that the area of each leg of this triangle comprises a hundred squares? And that the hypotenuse–” Hippasus pointed to the last line snapped by Demetrios. “And that therefore the hypotenuse is the root of two hundred squares.”

“Yes, yes,” said the greybeards.

“But what is the precise value?” said Hippasus. “Demetrios, mark off the hypotenuse using the same measure as for the legs.”

Swinging the two sticks, Demetrios marked fourteen measures along the hypotenuse, the last falling just short of the end. The boy swung again, and the fifteenth mark fell beyond the end of the hypotenuse. A murmur swept along the semicircle, more of dismay than surprise, for the problem was known, if not the solution.

“Clearly,” said Hippasus, “the hypotenuse cannot be apportioned in terms of the legs.”

“Come, come,” said a greybeard. “You just haven’t found the measure.”

“I’ve tried every number to ninety-six, with great care, on large plots of sand. Nothing comes out. As the measure grows smaller, the line approaches a harmonic. But it’s obvious that it will never get there.” Hippasus pointed into the sandbox. “The hypotenuse of this triangle is immeasurable.”

“Impossible,” said a greybeard.

“Absurd,” said another.

Philolaus unfolded his arms and stepped into the sandbox. He lacked the height of Hippasus and his sonorous delivery, but eyes of coal shriveled the strongest heart. He ground a sandal in the center of the hypotenuse, looked first at Hippasus, then the rest of us.

“This is not a proof,” he said.

“Of course it’s not a proof,” said Hippasus. “It’s a demonstration so even the densest among us can understand the issue. Even that bowlegged gaffer.”

For an instant, I thought Hippasus was pointing at me, my loins puckering until I realized he meant Orestes, who said, “Aye, sir, and I take your meaning.”

Laughter rippled from the Pythagoreans except for Philolaus.

“There is a proof,” said Hippasus. “A proof by contradiction which I intend to present at the conference.” He cast a finger at the trodden hypotenuse. “I will show beyond doubt that this line is immeasurable.”

A gasp arose, more for the tone of Hippasus’s remarks than their substance. Although the substance was, in fact, the problem.

“Your so-called proof is not on the syllabus,” said Philolaus.

“What are you afraid of?” said Hippasus.

***

Night passed without incident, neither the sea nor my stomach churning. I was making a breakfast of flatbread, goat cheese, and grapes when yells and thumps assailed the center hold from the adjacent greybeards’ cabin. A scream, the voice of Hippasus, then muffled echoes.

We looked around, us lesser philosophers, but made no move. It was not for us to dispute the actions of our begetters. We heard grunts rising toward the deck and took our own ladder into a post-dawn mist on an undulating deck. Eight greybeards stood about a squirming mass the length and breadth of Hippasus, wrapped in two cloaks, tied with cord top, middle, and bottom. There must have been a rag stuffed in his mouth for he emitted only inarticulate gobbles.

Two greybeards swayed across the deck and staggered back under the weight of a bag of sand gripped between them. They plopped it midway on the squirming sack of Hippasus and secured it with another cord.

I backed away from the proceedings and saw Orestes standing amidst his shipmates. He walked my way. I said, “They’re just trying to scare him.”

“I fear you’re wrong, sir. They mean to drown him.”

I edged closer to ensure our privacy. “But he had a mock court if any. These actions are akin to murder.”

“Not,” said Orestes, “if the master of the boat agrees.”

I looked to the helm where Kastor stood, legs apart, arms crossed, thick black hair rippling in the wind, eyes above the fray.

“How could he agree to such an extremity?” I said.

“Scuttlebutt has it he’s been promised the curly-haired lad for the duration of the voyage.”

I looked to Orestes for a sign of disgust or distress, but his demeanor appeared whimsical as he gazed west. I looked again to the master of the ship, then back to the commotion upon the deck. The greybeards, four to a side, were dragging Hippasus to the port-side rail.

I said to Orestes, “Did you understand the demonstration yesterday? The immeasurability of the hypotenuse?”

“Aye.”

“Doesn’t he make sense? Isn’t he on to the truth?”

At the port-side rail, sixteen hands grasped and lifted.

“Heave,” said Philolaus.

The wriggling cloaks, corded and sandbagged, held the sea’s surface for a pulse beat then sank. For a second pulse beat, the sack remained visible in the blue-green Aegean then faded beneath the ripples. Philolaus and the greybeards turned and marched away as having disposed of a dog gone mad.

“Aye, the truth,” said Orestes.

Of a sudden, the deck surged from port to starboard, and back. Whitecaps appeared on cresting waves.

“Poseidon is angry,” I said.

“Nonsense, sir,” said Orestes. “Ye don’t believe those fairy tales, do ye?” He pointed west. “This storm’s been in the making since afore dawn. But look at ye, sir.”

I stared into Orestes’s countenance to catch his meaning as wind and sea roiled the ship under scattered droplets. The deck had cleared except for Kastor and a handful of mariners trimming the sails.

“Why you’re rooted to the deck as if born asea, sir. You’re a natural ye are, once you got by the puking.”

I flushed with pride at my newfound mariner’s legs, but the next roll of the deck sent me skittering sideways.

Orestes gave a short laugh. “Nevertheless, sir, ye best get below.”

Aye, below.


About the Author

Robert Perron lives and writes in New Hampshire and New York City. Past life includes high-tech and military service. His stories have appeared in The Manchester Review, Sweet Tree Review, STORGY Magazine, Adelaide Literary Magazine, The Fictional Café, Front Porch Review, and other journals. Visit his website at https://robertperron.com.