Angle of Contingency; Why the Sea Seems Higher than Land
Geometry of the tangent to a circle, and the earth's surface set against the sphere of water
Two threads share the leaf. On the geometric side Leonardo defines the 'angle of contingency' — the vanishing contact where a tangent line b d meets a circle — deriving it from the rule that removing proportional parts from proportional wholes leaves the proportion unchanged, with a diagram lettered a, c, b, d, e, f. On the cosmographic side he explains why the sea appears higher than the land: because exposed earth is never lower than the sphere of water, a plain d b that carries a river to the sea is really a slope, so that extended straight it would run beneath the sea. Two pen diagrams at the right — a circle with a tangent triangle and an oval earth-and-water figure — illustrate the two arguments.
On this page
The angle of contingency at a tangent b d
Leonardo defines the angle of contingency as the contact a straight line makes with a circle. Working from the segments c f, a e, c d, a b, he argues that if the line b d is tangent to the circle, that contact is the said angle of contingency. Letter-labels: a c, b d, e f.
Proportional parts removed leave the proportion
If from proportionate things one takes away a part in the same proportion, the remainder does not change from its first proportion. Thus from c f (double of a e), removing the double parts c d and a b leaves d f and b e still double as before.
Why the sea appears higher than the exposed land
No part of the earth uncovered by water is ever lower than the surface of the water's sphere. The plain d b down which a river runs to the sea is really a slope — otherwise the river would not move — so extended straight to b a it would pass beneath the sea, which is why the sea a c d seems higher than the land. Letter-labels: a c b d.
The sea looks higher to simple observers
A short marginal note frames the theme: the sea, which to many simple folk appears higher than the land that forms its shore.