Squaring and dividing curved-sided triangles (falcate)
Cutting a chosen part from a two-curve figure; areas of nested triangles; light on curved and flat faces
A geometry sheet on the quadrature of the falcata, a triangle bounded by two equally curved sides. Leonardo shows how to remove any requested part of its height and give that part's quadrature by transposing a circular segment from one side to the opposite side, reducing a curvilinear triangle a n e p b to a rectilinear one d e c equal to a quarter of the great triangle f e g. Below, he divides such a curvilinear triangle into equal or unequal strips by dividing its base and drawing lines of the same curvature to the apex, while a cluster of small triangles carries the running areas 4, 16, 64 and 9, 36, 144 and the sum 72, 72, 144. A short note distinguishes the shadowed faces of the rectilinear pyramid from the bright faces of the curved one.
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Removing and squaring any part of a two-curve falcata
To remove from the equally two-sided curved figure whatever part of its height is asked, take half of triangle e f g at point c and draw from c the curve c d a. Where it cuts the falcata at a b draw the straight line, making the curvilinear triangle a n e p b equal to the rectilinear d e c. Moving the segment a n e to the opposite side e p b where it was lacking yields the rectilinear triangle a e b, equal to d e c, which was one quarter of the great triangle f e g.
Dividing a curvilinear triangle into equal or unequal parts
To divide such a curvilinear triangle from base to apex into equal or unequal parts, divide the base and draw lines of the same curvature from the base to the apex. Done this way the figure is exactly and correctly divided.
Areas of nested triangle subdivisions
The small triangles at the lower right are subdivided and tallied in running series: 4, 16, 64 within one, and 9, 36, 144 within another. A further note gives 72, 72, 144, the two halves summing to the whole.
Shadowed and lit faces of curved versus straight pyramids
The shadowed faces belong to the rectilinear pyramid, and the bright faces to the curved one. A companion note marks a form as twice curved, that is, toward the east and the south.