Check the curve for yourself, what should the curve look like given the slope the earth would need if it really was 24,900 miles around.

Hmm, ok.

• distance to centre of earth, R, is 6360km, 6360E3 m, depending where on earth I am
• My viewing height is 2m, say
• d is the distance to the horizon, where my line of sight hits the horizon. Note that Flat Earth morons both disbelieve (paid to) and believe (every day observations) the existence of the horizon.

Question: does the calculated 'd' match the observed 'd'

Calculated d

The tangent (my view to the horizon) , which is d, is at right angles to the line of the earth's radius R. That is what a tangent is.

So there is a right-angled triangle:

• side length R to the centre of the spherical earth (side a)
• side length d my view to the horizon, length to be solved for (side b)
• hypotenuse R + my height. (side hyp)

a^2 + b^2 = hyp^2 : Pythagorean theorem

R^2 + d_to_horizon^2 = (R + my height)^2 : stick in our, my variables

R^2 + d_to_horizon^2 = R^2 + 2Rxh + h^2 : expand (a + b)^2 to a^2 +2ab + b^2

d_to_horizon^2 = 2Rxh + h^2 : remove R^2 from both sides

d_to_horizon = sqrt(2Rxh + h^2) : sqrt() of both sides

d_to_horizon = sqrt(2Rxh) : same

d_to_horizon = sqrt(2x 6360,000 x 2) or = sqrt(2 x 6360,000m x 2 + 4m ) : same

d_to_horizon = sqrt(2x6360,000x2)

d_to_horizon = 5044m = 5 km.

Observed d, d_to_horizon

Where I live, on the beach, there is an island about 5 km away. (thanks google maps). The horizon is slightly behind that, when viewed from the beach. I will be generous and say that the observed horizon is 6km (and also that the horizon actually exists).

Next time, maybe today (watch for an update) I will look for the horizon from the sea-line.

anyway, roughly correct:

• observed horizon 6km,
• calculated horizon 5km.

Of course.