Polaris Points at Your Latitude

"Polaris sits fixed at the centre of the sky's rotation and has never moved."

Free (or ~$50 for a sextant). On a clear night north of the equator, find Polaris (the last star in the Little Dipper's handle; locate it via the Big Dipper's pointer stars). Measure its elevation above the horizon - sextant, protractor on a string, or a phone angle app - and compare with your latitude from any map. Then repeat from a place a few hundred km south and compare again.

Polaris's elevation matches your latitude: ~51° in London, ~47° in New York, ~34° in Los Angeles - and it gets lower the further south you go (allow ~0.7°, since it isn't exactly on the pole). In the Southern Hemisphere the same trick works with the faint star Sigma Octantis, due south.

This is how sailors found their latitude for centuries. And the 'never moved' claim is false: the Nautical Almanacs list Polaris's declination as 88° 30' in 1850, 88° 46' in 1900 and 89° 20.3' in 2017 - it is 0.66° from the pole today, was ~3° away when Tycho Brahe catalogued it in 1572, and will be closest to the pole around 2100. The pole itself traces a 25,800-year circle (precession).

On a flat map with Polaris at the centre, every observer should see it at the same height. Instead its height changes by exactly the difference in latitude - which is what you'd expect standing on a sphere, looking up at the axis of rotation.