Why a Cone's Surface Cannot Equal Its Base
Studies on the surface, axis and height of the cone
This folio argues that the surface of any cone can never equal its base, since a circle only forms a conic surface when a sector is missing or an axis arises. Leonardo relates the cone's maximum width to the beginning of its axis, and its maximum height to the topmost axis, which equals its semidiameter. A second note observes that a cone with a lesser axis has the greater surface, concluding that the topmost height of the axis is generated from the destruction of the conic base. Marginal figures show a cone seen from above with a cross, a tall triangle, and a shallow cone on a wide base.
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The cone's surface cannot equal its base
It is impossible that any cone's surface equal its base, because the circle never makes a conic surface unless a sector is missing or its axis arises. The base's maximum width ends where the axis begins, for where the cone destroys its axis the conic surface reverts to the plane surface of the circle. The maximum height of the cone ends at its topmost axis, which is similar to its semidiameter.
Cone surface, axis and generated height
The cone with the lesser axis has the greater surface, and that with the greater axis the lesser surface. Its maximum height ends at the entire destruction of its surface, since that topmost height is similar to the semidiameter of its greatest base. Leonardo therefore says the topmost height of the axis is generated from the destruction of the conic base.