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Footnotes on the Quadrature of the Circle

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Rare is the mortal who keeps their body after death, and rarer still is the bodied soul who turns up in the depths of Tartarus -- Zagreus can count on his fingers how many times that’s happened, Sisyphus and Bouldy, Tantalus eternally deprived, Ocnus with his rope, the other forty-nine Danaids and their broken sieves, Salmoneus cursed to eternally be the lesser. And fewer still those who end up in the House for good behaviour.

Thus there is already quite a stir when Zagreus emerges bloody and dark from the Styx, pointedly ignoring Hypnos’ nattering as soon as he sees the man talking to Achilles. And he is a man, or at least a dead man: pointed, grey at the temples, with a furrowed brow and a down-turned mouth as he seems to be arguing about something to an opponent unable to defend his position. Zagreus narrows his eyes and sees that he died of a broken heart -- and there are certainly worse ways to go.

“There you are, lad,” Achilles calls, very much sounding like a man eager for a way out of a debate he never asked to be a part of. “Your father has hired you a new tutor.”

“I was not consulted in this process,” the tutor says, which -- well, same. Zagreus hadn’t thought he’d needed another tutor, what with work going as well as it is; and besides, what else was there for him to learn? But time and misfortune have taught him that certain fights are not work picking as far as his father is concerned, so he strides forward and holds out his hand.

“Pleasure to meet you, sir,” Zagreus says.

The man looks him up and down, and perhaps deigns the wayward son of Hades to be worth his time -- so he takes the offered hand, and shakes it. “Pythagoras.”


He was a gemcutter’s son, born as Greek as the rest of the known world, called himself a lover of wisdom -- not to suggest that Athena ever shared his bed, but rather that he preferred to spend his days in deep thought regarding the true nature of things. In Miletus he had inherited an interest in demonstrative mathematics from Thales and Anaximander, alongside a certain heretical impulse to think of Zeus’ influence as not monarchic or paternal but rather geometric; later, he’d sojourned in a place called Egypt, where they worshipped different gods, and were taken to a different place after death.

He’d died, in his own words, not from a broken heart -- but from losing faith in humanity to solve its own problems with any elegance or tact. And now, as a final insult, he’d been taken to a place that made no godsdamned mathematical sense.

“If I might ask, what does mathematics have to do with it?” On the word it Zagreus motions outward, vaguely.

Pythagoras is leaning over a drafting table that Zagreus had, very thoughtfully, paid for in gemstones -- and therefore his own blood and sweat, thank you very much -- drawing shapes with a compass and straightedge. Mostly squares. “Mortals have two parts, the body and the soul. The soul is obviously immortal,” he drawls, as though talking to an idiot. In this regard, Zagreus has had much experience. “Upon dying the body becomes fertilizer to feed new bodies, and I had hypothesized that that the soul would take up a new body, to close the cycle. But now I’m told that we all end up here, forever?”

“Erm, yes, sir. That's how it has always been.”

“And what, pray tell, will happen when Hell is full?”

“Well, Tartarus will expand, I suppose.”

“Into what?”

“Into,” Zagreus wonders if this, somehow, is how Pythagoras intends to educate him. “Into Chaos?”

“Into Chaos, he says.” Pythagoras scoffs. “The world and the heavenly spheres, they are bound by mathematics, and move in time to a divine symphony. And we are beneath the world. If Hell expands, it must be -- it must be in tune with something equally contracting. But if this process sits in the centre of two eternities...” He regards his squares, and scratches his chin. “Leave me to my quadrature, boy.”

So Zagreus leaves him, and Pythagoras blows through a mountain of parchment drawing circles and squares and the occasional triangle, muttering to himself about music and vegetables. Zagreus spends the afternoon training in the spear with Achilles, and learns nothing of mathematics.


With twenty-five hundred years of hindsight, one might be forgiven for assuming that mathematics exists purely as the language of science, the tool by which one might calculate the velocity of a rocket or the processing speed of a computer; we can blame that mistake on anyone from Archimedes to Galileo to Lovelace and Turing.

But all of them were only some of descendants of Pythagoras, who was also forebear to Motzart, Descartes, and Thoreau; and Shakespeare remembered him primarily as a vegetarian; consider Dante, drawing Pythagoras’ numbers into his own spherical version of Hell, a different and yet not so different Hell than Zagreus’. Pythagoras is a philosopher and artist first and foremost, and his art is geometry, his problems that of the divine.

No writings of Pythagoras’ survived, and most records of his life were written after he died; the vast majority of those records came at least five hundred years after his passing, and are thus echoes of echoes, legends reconstructed from rumours. What separates the psuedohistorical from the mythological?

Nothing, if you ask Zagreus, who regards Pythagoras conversing with Orpheus, explaining how halving the string might change the octave. They are one and the same, though in his humble opinion the latter is significantly less of a drag than the former.


“Pythagoras, sir,” Zagreus says, after his umpteenth unsuccessful attempt at receiving a mathematics lesson from his tutor. “I’ve a gift for you, if you’ll accept it.”

Pythagoras, who has spent the last month (as far as Zagreus can tell time) contemplating a dodecahedron, gives him a theoretical side-eye. “Let’s see it, then.”

From the folds of his cloak Zagreus produces a bottle of Nectar, bright gold; Pythagoras regards it carefully before accepting the gift. Uncorks it, and gives it a sniff, and appears to approve.

“The smell isn’t unpleasant. What is it?”

“Nectar, sir. -- Erm, the food of the gods.”

“And where did this come from?”

“Well, you see.” Zagreus thinks back, dutifully. “In Asphodel, a boat took me to an island in the River Phlegethon, where I fought some Wave-Makers. And once they were defeated, I --”

“Zagreus.”

“-- Sir?”

Pythagoras takes a swig. His face relaxes the slightest bit. “I didn’t ask where you got it. I asked where it came from. Who forged the glass? Who spun the ribbon, dyed it violet? The drink itself, what are its components?”

“Pythagoras, sir, I.” He tries, valiantly, to come up with any answers; and having none, says instead: “I don’t know. To be honest, those never seemed like questions worth asking.”

“And yet, as immortals, they are the only questions worth asking. What manner of things can come up from nowhere, take up no space until they exist, and yet fill the universe?” Fishing about in the horrid mess that is his drafting-desk, Pythagoras procures a compass and straightedge, and hands them to Zagreus. “But before we unravel the mysteries of the world, let us begin by constructing a perpendicular bisector.”


Draw a line segment of any length smaller than infinity. Call the line AB -- or aleph-beta, whichever strikes your fancy. Lock your compass slightly over half the line’s distance, and draw an arc that intersects the line using point A as the circle’s centre, then again at point B. Call the points where the arcs intersect C and D -- or gamma and delta, though do be careful with those deltas, as they often represent change. Draw a line through C and D; this is the perpendicular bisector of AB.

To divide any line perfectly in half on a square angle is no small feat; and to do it with perfect mathematical precision was a tremendous demonstration of the beauty of the gods’ own creation. The Babylonians would have named the angle ninety, one-quarter of three-sixty, on account of its numerous divisors; the Greeks, orthogonal, al-Kashi, some two thousand years later, might have called it four diameter parts; Aristotle, of divergent interests to the many names of a right angle, took instead the idea of segment smaller than infinity to task, wondering boldly if infinity was just the name of a number, if number was the way by which the heavens spoke.

But let us not concern ourselves with Eudoxus of Cnidus, who passed the method of exhaustion to Archimedes, who used it to approximate the value of pi; who then passed it on to Cavalieri and Fermat, who might have called it a limit; who then passed it on to Newton and Liebniz, whose calculus confirmed that mathematics is indeed the language of motion, of thought, of the heavens and the earth. Let us concern ourselves instead with the beginning of the tale, Prince Zagreus, constructing angle bisectors on a morning or afternoon or evening, with the patience and eagerness and care of a child stringing beads onto the universe.


He takes the compass and straightedge with him on one jaunt upwards, for it is a keepsake, and keepsakes hold power.

He emerges from the River Styx and thinks to himself: oh, it’s a circle.

His work, that is, which always ends him exactly where he began.

Over a shared bottle of Nectar, Pythagoras introduces him to quadrature, the pinnacle of Greek mathematics and art. It is the method by which the area of a bound figure is geometrically determined, without need for anything as crass or rude as a number: using nothing but a few simple laws of congruency and symmetry, with one’s own compass and straightedge, constructing a square of equal size to any given shape.

“It is through this that the simple, the orderly, and the beautiful lead into the complex and intricate,” Pythagoras says, demonstrating the quadrature of a rectangle. “It shows to us that the world is governed by reason and order; we reduce the convoluted to the rational. And above all else, it reminds us that the gods may have given us a universe, but it it is the human spirit that will triumph over it.”

He then demonstrates the quadration of a triangle; and by extension, the general polygon; and since any rectilinear polygon can be divided into triangles, any straight-edged object is therefore quadrable. A beautiful and orderly universe, indeed.

“But what of --” Zagreus starts.

“-- Yes, the circle.” For Pythagoras died some sixty years before Hippocrates quadrated the right isosceles lune, and fifteen hundred years before Alhazen quadrated two more, and two thousand before Chebotaryov and Dorodnov determined the conditions that govern quadrable curvilinear shapes. “It stands to reason that, if the universe is as orderly as we assume, then the perfect shape should be perfectly quadrable. The circle must be squarable -- and now that I have apparently unending time…”

They spend the better part of a week on it, and get nowhere.


“The hexagon,” Zagreus starts.

“Say that word one more time and I will snap your neck,” Megara says.

He does, and she does, and you know how these things go.


The ratio of a circle’s circumference to its diameter is approximated as three to one in the Hebrew Bible; it was the square of four thirds to the Middle Kingdom Egyptians; it was twenty-two to seven to Archimedes. Ptolmey drew a polygon of three hundred sixty equal sides to approximate it to four decimal places; Zhu Chonzhi, a twenty-four thousand sided object to estimate it to six decimal places. Chasing Archimedes’ intuition and Zhu’s method, Francois Viète one thousand years hence drew a three hundred ninety-three thousand two-hundred sixteen sided object and found pi accurate to nine places. Ludolph van Ceulen, two hundred years later, roughhoused around with the newly invented decimal system and (one can assume) a very sharp pen to produce a polygon with roughly four quintillion sides, and found it to thirty-five places.

But one can only do so much with regular polygons and unmarked rulers. Newton and Liebniz and Sharp and Machin put the Archimedean method to rest with numerical approximations, wrestling the circumference-diameter ratio from a geometry question to an algebra problem.

Imagine number as a nesting doll. At its heart is the set of countable whole numbers; this was all that existed to the Sumerians. This nests inside the set of integers, whole numbers positive or negative; this was all that existed to the Qin Dynasty. This nests inside the set of rational numbers, numbers that can be expressed as fractions of whole number over whole number; this was all that existed to the Greeks, as shape without number can only exist in terms of ratios and equivalences, and Gods forbid you suggest that anything lies beyond.

“Hippasus of Metapontum.” Pythagoras draws a square. “Handsome fellow, too smart for his own good. He found the diagonal of a square of unit length --” He demonstrates this, to Zagreus and Thanatos -- “To be a number that was not a ratio of whole numbers. But all numbers must be ratios of whole numbers, and to suggest the converse is to suggest the Gods’ world is not orderly or beautiful at all…”

We now know, of course, that the square root of two is not the ratio of whole numbers; for the rational numbers nests inside an even greater doll of all real numbers, which includes the irrational numbers -- numbers expressed as decimals that go on and on and on, forever, neither terminating nor repeating; infinitely long strings of numerals whose true names can never be known or spoken.

Consider Hippasus, a fortnight dead, picked apart by the sea; and then ask yourself what the square root of negative one is.

“Hippasus drowned,” Thanatos says, quietly.

“We drowned him,” Pythagoras says. “Threw him off a boat in the dead of night. If such a secret got into the hearts of the people...”


Bertrand Russel, famously, chose mathematics over death. Srinivasa Ramanujan was not so lucky, as the choice was made for him.

Freud, on the other hand, gave our creative and destructive impulses another name: Eros and Thanatos, life and death.

In this aspect, Zagreus is reproducing the proof the sum of squares on a right triangle, and Thanatos is watching him, endlessly fascinated and deeply in love. Zagreus’ shapes are amateur and shaky, his intuition yet unformed, but a hundred million unspent lifetimes lie ahead of him, so there is time.

And that’s what separates the mythological from the historical, Hippocrates’ circles from Archimedes’ lines, Pythagoras’ transmigration of the soul from -- well, from whatever actually happens after death: we understand now that things are not circular but rather straight lines, and that all things have a beginning and an end, with us at the centre.


“A story,” he tells Pythagoras, who is busy at his drafting-table carving a circular cone into parabolas.

“A story?”

“Your question for me, sir. What comes from nowhere, takes up no space, and fills the universe?”

Euclid and Archimedes’ supposed works on conic sections, the fine art of slicing a cone on an angle, have never been published. We pretend its father is Appolonius, who came some two hundred years later -- but we forget that mathematics is not cumulative in nature. Rather, it is a series of quiet triumphs in dark rooms, bathtubs, laboratories, places of worship; and some things have been lost forever.

“I was -- well, I was running about in Elysium, and I got to thinking, sir. You told me that history is a straight line, and story is a circle --”

“So you think --” Pythagoras, furrowing his brow, slices his cone straight across, producing said circle. “You think we live not in a physical space, but a story?”

Ferdinand Lindemann, in 1882, proved that pi -- the ratio of a circle’s circumference to its diameter -- is a transcendental number; that is, an irrational number that cannot be derived algebraically. In doing so, he proved with finality that the circle cannot be squared. Five hundred years of Greek mathematics died to that fact, but out from the ashes rose David Hilbert, and a strange new dawn.

“I think so,” Zagreus says.

For even when a line is broken, there can always be a new beginning.

“So what,” Pythagoras says, “Happens when the story ends?”