Cones and the surface of their base circle
Solid geometry: a cone's curved surface compared with the circle of its base
The page continues Leonardo's geometry of the cone, relating a cone's curved surface to the circle of its base according to the ratio between the base diameter and the slant side (the 'ipotenissa'). A bisected semicircle illustrates the axiom that if you remove half of a whole, the remainder equals what was taken. A circle inscribed in a semicircle (labelled n, a b c, p m o) is used to prove that a cone whose slant equals its base diameter has a surface double its base. A closing note treats nested cones of double surface and equal base as deriving from circles in quadruple proportion.
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A cone's surface set equal to the circle of its base
When a cone's base diameter is double its slant side, its curved surface equals the circle of the base. When instead the base diameter equals the slant, the surface is double the base.
Bisected semicircle and the half-equals-remainder axiom
A semicircle divided in half at a - b carries the axiom 'if you take away the half of a whole, the remainder is equal to what is taken away,' which Leonardo uses as the basis of the following demonstration.
Circle inscribed in a semicircle: cone with slant equal to base diameter
Using the labelled figure n, a b c, p m o, Leonardo proves that a cone whose slant equals its base diameter has a surface double that base. He appeals to the proposition that the greatest circle within a semicircle equals half the semicircle, and identifies the semidiameter n m as the cone's slant.
Nested cones from circles in quadruple proportion
Cones of double surface and equal base, and of double axis, are said to derive from circles standing in quadruple proportion one to another.