Arctan of My Latitude Times a Week Without Sun

I didn’t know this until last week. I was adjusting the escapement on that twelve-dollar pocket watch — still haven’t fixed the kinked hairspring, still enjoying the disassembly more than the repair — and I got curious about how the twenty-four-hour day became standardized. Railway time, time zones, the International Meridian Conference of 1884. The usual Wikipedia spiral. But then I hit the entry for “equation of time” and something shifted.

A sundial doesn’t show clock time. It shows apparent solar time, which is what the sun actually does. And those two things — what the sun does and what the clock says — can differ by more than sixteen minutes depending on the season. Early November, a sundial runs fast. Mid-February, it runs slow. The discrepancy traces a figure-8 called an analemma, and if you’ve ever wondered what that weird shape printed in the Pacific Ocean on old classroom globes actually means, that’s it. It’s a correction chart. It’s the sun apologizing for having an elliptical orbit.

You’re probably thinking: okay, but who cares? We have phones now. And sure, yes, obviously. But I spent three hours last night working out the hour-line angles for my latitude and I haven’t been this engaged with trigonometry since university. For a horizontal sundial, the hour lines aren’t evenly spaced — that’s the first surprise. On an equatorial dial, where the face is tilted to match the celestial equator, hours are exactly 15° apart. Simple. But on a flat horizontal surface, the math gets interesting: each hour angle is arctan(sin(latitude) × tan(15° × hours from noon)). At 53.5°N, where I live, the 9 AM line and the 3 PM line are about 39° from the noon line, not 45° like you’d expect from naive division. Eyeballing it would give you a dial that looks right and reads wrong.

The gnomon has to point at the celestial pole. For me, that means angling it at 53.5° from horizontal — steep, nearly pointing at Polaris. I did something similar last winter when I was calibrating compass cards from star trails, finding true north by tracking the pole star’s tiny arc. Same destination, different approach. The sundial gnomon doesn’t care about my compass or the magnetic declination that drifts 14° east here in Alberta. It aligns with rotation, not magnetism.

I ordered a brass sheet and a protractor before realizing I’d committed to a hobby that requires going outside and waiting for the sun. The forecast calls for clouds through Wednesday.

The terminology is unexpectedly precise. The gnomon is the whole shadow-casting object, but the style is specifically the edge that indicates time. A wide triangular gnomon might cast a broad shadow, but only the style’s shadow edge matters for reading. The word gnomon comes from Greek — “one that knows or examines” — which is a good name for something that interrogates the sky’s geometry every clear day.

There’s a longitude correction too, because Edmonton sits at 113.5°W but Mountain Time is centred on 105°W. That’s 8.5° of offset, which translates to about 34 minutes. So: sixteen minutes for the equation of time (varies by season), plus thirty-four minutes for longitude (constant), plus the daylight saving hour (half the year). A sundial in my backyard could be nearly two hours off from my phone at certain times of year. Getting it to match requires either accepting a permanent correction table or getting clever with the dial design — double hour rings, figure-8 hour lines, all sorts of solutions people invented centuries before quartz oscillators made the problem irrelevant.

Irrelevant to everyone except the type of person who takes apart pocket watches to understand escapements instead of to fix them. That type of person — okay, it’s me — finds something satisfying in working out whether the sun or the clock is “right.” They’re both right, of course. They just measure different things. The clock measures a mathematical fiction called mean solar time, a perfectly averaged day that never actually happens. The sundial measures what’s actually overhead at this longitude, right now, today.

The brass arrived this afternoon. It’s thinner than I expected, about 0.5 mm, which will make cutting the gnomon easier but might not have enough presence for the finished dial. I might switch to something heavier for the final version. For now, I’m doing layout on paper, checking my angles, making sure I understand the geometry before I cut metal.

The pocket watch on my desk was engineered to measure time invisibly, nested inside a case, ticking in a pocket where no one would see it work. A sundial does the opposite. It sprawls in the garden, displays its mechanism openly, and stops functioning the moment the sun disappears. Both rely on geometry and precision. One hides its math; the other is nothing but math, visible, waiting for shadow.