Space, time and Shakespeare - Paul Glendinning

ELI5/TLDR

A mathematician decided the best way to talk about how ordinary people once thought about numbers, time, and space was to read Shakespeare. The plays are full of little maths gadgets — almanacs that predicted the moon, mirrors that had just become cheap, the new pocket watch — and the people in them fumble with these things the way we once fumbled with a first home computer. It’s not about Shakespeare being a secret scientist. It’s about seeing what “everyday knowledge” looked like 400 years ago, when that knowledge was changing fast.

The Full Story

The mathematical eye

Glendinning, an applied mathematician at Manchester, opens with a photo of a spider’s web after rain. Most people say “oh, pretty.” He can’t stop asking questions — why are the droplets evenly spaced, why don’t they slide together, why are they that size. He calls this the “mathematical eye,” and his real claim is that everyone has it. Mathematicians just have a vocabulary that makes the noticing sound impressive.

“You look at something and you see something through a mathematical lens… my contention would be everyone has that.”

He uses Shakespeare for two reasons. First, it’s a giant but finite world of words and characters — big enough to find interesting things in, small enough not to get lost. Second, and more cleverly, the historical distance does useful work. The plays are full of things that feel alien sitting right next to things that feel completely familiar. That friction — “why would somebody say that?” — is exactly what forces you to see how people used to think.

He’s explicit about what this is not: not a story of Shakespeare-the-scientist, and not a triumphant “look how clever we kept getting” march of progress. He just wants to watch people play with ideas.

Counting an army by multiplying

The first example is a scout in Henry IV Part 2 who reports an enemy force of “30,000.” How does he know? A modern mathematician would say: amount equals density times area. And that’s precisely what the scout does. He judges that the soldiers are packed at a standard spacing (“in goodly order” = uniform density), he eyeballs how much ground they cover, and he multiplies.

Here’s the lovely part — this trick is ancient and refuses to die.

“It’s so prevalent it’s got a name. It’s called Jacob’s technique.”

Julius Caesar used it in Germany, and even gamed it: he had his disciplined Roman troops pitch their tents unusually close together, so the enemy, eyeballing the small camp, badly underestimated his numbers. The same density-times-area method was later used to count American buffalo herds, and to argue about the crowd size at the 2017 inauguration. Today drones plus AI do the eyeballing. Same maths, 2,000 years running.

The almanac: a 17th-century smartphone

In A Midsummer Night’s Dream, the amateur actors need to know whether the moon will actually shine on the night of their play. Their solution: “Look in the almanac.”

Think of the almanac as the era’s smartphone — suddenly, printed information in your pocket. It told you the weather, the best days to sow and harvest, medical tips, tide tables, the phases of the moon, and “astronomical conjunctions with their astrological consequences.” By the mid-1600s, around 300,000 were printed every year — reprinted annually because the data changed.

Then Glendinning slows down to show the actual arithmetic a reader was expected to do, because the early almanacs (like Leonard Digges’) didn’t just hand you the answer — they handed you a recipe. The maths rests on one beautiful coincidence between the heavens:

The moon’s cycle (new → full → new) averages about 29.5 days.
The solar year is 365.24 days.
It happens that 235 lunar months ≈ 19 solar years — almost exactly.

So every 19 years the moon’s phases reset to where they started. Imagine two runners on circular tracks of different lengths; after 19 laps of the slow one, they line up at the start together again. That 19-year wheel is the engine. A number called the prime (or “golden number”) tells you where you are on the wheel. A second correction handles the fact that 12 lunar months fall about 11 days short of a solar year — so each year you slide 11 days, and a bit of “multiply by 11, divide by 30, keep the remainder” arithmetic (the epact) tells you the moon’s position on a given date.

A small detail he clearly relishes: the tables run 1 to 19 and quietly avoid zero. Where a modern mathematician would write 0, they wrote 19. Zero existed as a digit (1560 has one) but they wouldn’t use it to count with. He can’t resist a working-mathematician’s aside:

“I hate indexing… whether you go from 1 to n or naught to n minus one really matters when you’re trying to write something down clearly.”

Three mathematicians, and a manifesto

Quick portraits: John Dee — flamboyant “celebrity scientist,” dazzled Elizabeth I with light shows, kept an enormous library, possibly the model for Prospero, died penniless after annoying the wrong people. He wrote the preface to the first English translation of Euclid — which Glendinning calls a still-fantastic manifesto for applied mathematics. Dee mapped out fundamental maths (arithmetic, geometry, the two mixed) and then a long list of uses: perspective, astronomy, music, even measuring the human body. The highest art he called archimastry — linking theory to experiment. The pure-versus-applied divide, already being negotiated in 1570.

Leonard and Thomas Digges — father and son, possible inventors of the theodolite and an early telescope, early adopters of Copernicus. And crucially, they had real social ties to Shakespeare: Thomas’s widow remarried a man who oversaw Shakespeare’s will, and his poet son moved in Shakespeare’s circles. So these ideas weren’t distant — they were a few doors down in London.

Mirrors, or the first selfie

Cheap glass mirrors had just arrived from Venice. The trick: blow a small bottle, swill mercury and tin inside, cut it open — instant (convex) mirror. By the 16th century everyone had a hand mirror. Glendinning’s framing:

“I think of this as a selfie… suddenly you’re able to see yourself in real time.”

Before, you got a smear in burnished metal. Now you saw yourself sharply, anywhere — and could compare yourself to others. Cue moral panic: a 1575 French complaint moans about women bringing “scandalous mirrors hanging about their waists” to church.

Dee loved cylindrical mirrors that distort — useful for secret messages readable only through the right curve. And in the Euclid preface he describes visiting a friend who could rig mirrors so that you’d lunge with a sword and see a sword lunge back at your own face. That “seeing something that’s almost real” — one scholar argues it’s the kind of image in the air behind Macbeth’s hallucinated dagger (“Is this a dagger which I see before me?”). The underlying physics is Pepper’s Ghost: a sheet of glass, dark on one side, reflects an object so your brain places a ghost image behind the glass. Glendinning and his daughter reproduced it with an iPhone-lit knife and a dusk window onto the garden.

Sunrise, and the anxiety of two calendars

In Julius Caesar, Casca corrects someone pointing east at dawn: the sun, he says, rises further toward the south this time of year. To us this is gibberish. But anyone living without tall buildings knew the sunrise point wanders — northeast at midsummer, southeast at midwinter, due east only twice a year. So Casca is really saying: you’ve got the season wrong. (One scholar adds a poetic layer — sun equals Caesar, so the man who sees the sunrise first is the one who strikes Caesar first.)

But the timings don’t quite line up for the Ides of March — and that, two scholars argue, mirrors a real anxiety of the age: England was still on the old Julian calendar while Catholic Europe had switched to the Gregorian, so the two were days apart and even disagreed on when the year began (Julian new year: 25 March; Gregorian: 1 January). England didn’t fully switch until 1752. The play hums with unease about time itself.

Sonnet 77 and the nothing of Lear

Sonnet 77 gathers three new technologies in one breath: the glass (mirror) to see yourself age, the dial (watch/sundial) to watch minutes waste, and the blanks — the blank pages almanacs had started including so you could take notes (early users effectively kept diaries in them). Unusually for the sonnets, it’s optimistic: yes you age, yes you forget — but these tools help you manage the change and “much enrich thy book.” Glendinning calls it Shakespeare’s “white heat of technology” sonnet.

He runs out of time on King Lear, but leaves the seed: Lear is a play full of nothing, written just as zero was arriving and Roman numerals departing. Lear hands away his kingdom and becomes a placeholder — like the 0 that just marks an empty column — then madness drops him to literal nothing, and only death restores him to “something” as a number in history. He ends on a slide of dates with a grin: not a zero among them.

Key Takeaways

The “mathematical eye” — compulsively asking quantitative questions about ordinary things — is universal; mathematicians just have the vocabulary to dress it up.
Estimating a crowd or army as density × area (“Jacob’s technique”) is 2,000+ years old, was gamed by Julius Caesar, and is still used today via drones and AI.
Early almanacs were the era’s smartphone: ~300,000 printed yearly, packed with weather, farming, medical, tidal, lunar, and astrological data.
The lunar calendar works because 235 lunar months ≈ 19 solar years, so moon phases reset on a 19-year wheel (the “prime” / golden number tells you your position).
The lunar year falls ~11 days short of the solar year; the epact (roughly prime × 11 mod 30) corrects for this to locate the moon’s phase.
Early modern tables used zero as a digit but refused to count with it — they wrote 19 where a modern would write 0.
John Dee’s preface to the English Euclid is an early manifesto for applied mathematics, topped by archimastry — joining theory to experiment.
Cheap Venetian glass mirrors (blown, mercury-lined) made the “selfie” possible in the 1500s and triggered moral panic.
Macbeth’s floating dagger may echo mirror tricks (Pepper’s Ghost: glass reflects a ghost image your brain places behind it).
The sun’s rising point swings between southeast (winter) and northeast (summer); Casca uses this in Julius Caesar to flag a season — and the play registers England’s Julian-vs-Gregorian calendar anxiety.
Sonnet 77 bundles three new technologies — mirror, dial, and blank note-pages — into an unusually hopeful meditation on aging.
King Lear dramatizes the arrival of zero: king as placeholder → nothing → restored to a number through death.

Claude’s Take

This is a genuinely charming talk, and an honest one. Glendinning keeps telling you what he is not doing — not making Shakespeare a scientist, not selling a progress narrative — and he means it. The thesis is modest and true: looking at the maths-gadgets in the plays shows you what everyday knowledge felt like when it was in flux. The almanac-as-smartphone and mirror-as-selfie framings are exactly the right kind of bridge for a general audience.

Where it’s strongest is the texture — Caesar gaming the tent-count, the 19-year lunar wheel, the refusal to count with zero. Where it’s thinner is the literary payoff: he admits most of these are “occurrences” that don’t really illuminate the plays, and the two that promise to (Sonnet 77, Lear) get rushed or skipped because he ran out of time. So you leave with great anecdotes and only a partial argument. The Macbeth-mirror and Lear-zero readings are suggestive but, by his own admission, “what’s in the air” rather than demonstrated.

A 7: warm, well-told, full of durable facts, lightly under-delivered on the literary half it teases. Worth it for the history-of-everyday-mathematics alone.