Curved pyramids and the rolling of circles
Quadrature of curvilinear figures inscribed in a large lettered circle
A large compass-drawn circle carrying the lettered points a b c d e f g h i K l fills the sheet, surrounded by studies of "curved pyramids" — sickle- and sector-shaped figures — and inscribed rectilinear figures. Leonardo argues that curved pyramids of equal base and equal curvature are equal to one another and comparable to straight-sided figures enclosing the same area. He grounds one equality in the motion of a smaller circle rolling through a larger one, citing the seventh proposition of motion, so that as much as it occupies in one place it vacates in the other.
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Curved pyramids of equal base and curvature
All curved pyramids that have equal base and sides of equal curvature will be equal to one another, and they are similar to the pyramids that can be constructed within a circle. The lettered points 1 K i h g f d and e a b c mark these figures on the great circle.
A straight-sided figure equal in area to the curvilinear one
For pyramids whose sides are of equal curvature and length, two straight lines drawn from the opposite angle to the base enclose a void of exactly the same capacity as the region formerly bounded by the curved lines. This equates a rectilinear figure to a curvilinear one of the same base.
The smaller circle rolling within the larger
Invoking the seventh proposition of motion, Leonardo notes that as the smaller circle moves through the larger, as much as it occupies in one place it vacates in the other. From this he deduces that the sickle-shaped pyramid b c e equals the pyramid l K h, and conversely that a b h equals K i d.