Recently my sister and mother moved into their new place in Malaga. It’s about 30 km from the sea, which you can see clearly on a fine day. The house is at about 900 m altitude. So how far away is the horizon, in fact?

Well let’s see: we need to construct the line tangent to the circle of the Earth (see diagram); cos a=R/(R+h) and D=R.a, so that makes D=R arccos R/(R+h).

R is 6400 km and h is 0.9 km, which gives D=107 km. That means we’re looking across 30 km of land and nearly 80 km of the Mediterranean sea, which is about half way to the African coast. To see the far coast, i.e. have the horizon about 190 km away, I’d need to get to about 2800 m altitude (in fact the Sierra Nevada gets to over 3400 m, and the Atlas mountains on the African side stick up and make the job of seeing Africa easier). If I were on the beach, with an altitude of h=2 m, the horizon would drop right down to 5 km, and to see twice as far, I need to be at an altitude of around 8 m, which is why the crow’s nest in a ship comes in handy.

Arago must known these numbers by heart from his time surveying the meridian.