The Curvature of the Earth: How Far Two Men See the Horizon
With notes on portions of circles and the fading of shapes seen through fog
This leaf pairs short notes with a geometrical demonstration drawn over a semicircle at lower left. Leonardo argues that on the sphere of the world two men of equal height, standing on a calm 14-mile stretch of sea, each see the horizon at the other's feet, and at 7 miles at the height of the eyes. Above the diagram he notes that things seen through fog lose visibility toward their edges and the more remote they are, and at the top a terse remark on how the portions of circles vary against their figure. A set of pencil lines below the diagram, partly crossed out, restates the horizon argument.
On this page
Portions of circles varying against figure
A brief note at the top states that the portions of circles increase in proportion as they diminish in figure, and conversely diminish in number as they grow in figure. It is a compressed remark on the reciprocal relation between the size and count of circular parts.
Fading of shapes seen through fog
Written in another hand, this line observes that of things seen through fog the part nearer the edges is less visible, and the less so the more remote it is. It is an observation on how atmosphere degrades the visibility of contours with distance.
Two men and the horizon on the sphere of the world (b, c, o, a, m, n)
Over a semicircle Leonardo demonstrates that on a 14-mile stretch of calm sea two men of equal size each see the horizon at the other's feet, and at 7 miles at the height of the eyes. He labels the stretch n o m and the two men a n and c m, with the horizon point o where a man o b stands; the eye of man a sees the horizon o at the feet of man o b, and the eye of man c m does likewise. A label 'Dupla' (double) marks the paired figures.