The horizon on a spherical earth and at sea
Why a level line is really curved, and the sea horizon lies below the eye
Here Leonardo argues that a straight line of equal height is not truly level but curved, since it must follow the sphere of the earth. A first diagram sets two men a b with the horizon n at their height; a larger diagram of the earth's curvature g r h shows that for an eye raised above the surface the horizon falls below the feet by m f along the straight line a f k, and by b o along the vault of the earth. He then turns to the maritime horizon, stating that an observer standing at the water's edge sees the sea horizon lower than his own eye, by an amount tied to the height of his eye.
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A level line is in truth curved
A straight, level-lying line is not of equal height and therefore is not truly level-lying; a line called 'of equal height' can only be understood as curved. If a and b are two men, the horizon n comes level with their height.
On a spherical earth the horizon falls below the feet
If the earth is spherical, the horizon never reaches the height of an eye raised above the surface. With the eye's height n m, the wall b r, the horizon a, and the earth's curvature g r h, the horizon lies lower than the man's feet by the whole m f along the straight line a f k, and by the whole b o along the vault of the earth.
The sea horizon seen below the eye
An eye that views the maritime horizon while standing with its feet at the surface of the sea sees that horizon lower than itself. The sea horizon shows itself lower than the eye by as much as the height measured from the beholder's eye.