Your latitude is
a morning's work.
Your longitude is
two centuries.

Part one

Fifteen degrees to the hour

Earth turns once in 24 hours, so it turns 15° every hour. That single fact is the whole method.

Local noon is easy to spot anywhere on Earth; the Sun is at its highest and a vertical stick's shadow is at its shortest. If, at that exact moment, you can read the time back at your reference meridian, the difference is your longitude.

Part two: the mode that matters

Wrong landfall

The theory above was understood by 1600. It was useless, because it needs a clock that keeps reference time through months of salt, damp, pitching decks and 30 °C temperature swings, and pendulum clocks do not work on a moving ship at all.

So the question was never how. It was how accurate is accurate enough. Set a clock, sail, and find out.

Part three

What this is really about

Time and position are the same problem. That is not a metaphor. It is why moonfield doctor checks your clock before it trusts any of your coordinates, and why GPS is, underneath, a constellation of atomic clocks broadcasting the time.

Small errors integrate. Four seconds a day sounds like nothing. Nobody could feel it. Over a two-month crossing it is a degree of longitude, and a degree of longitude is the difference between a harbour and a reef.

The hard part was not the physics. Everyone understood the method for a century and a half. The problem was building a clock that survived a ship, and that took John Harrison roughly forty years and four attempts.