Making a curved figure equal to a straight-sided one
Equating curvilinear and rectilinear areas through motion
Leonardo works on transforming curved figures into straight-sided ones of equal area, treating a curvilinear 'parallelogram' abcd as equal to a rectilinear parallelogram arpq generated in the same time by an equal motion. He states a general rule that where every part of the curve and every part of the straight move alike, the whole curve equals the whole straight. The sheet is scattered with small segment, arch, curved-triangle (tent-like) and lens figures, plus a rectangle set over a circular sector, that carry the lettered demonstrations. One figure is struck through and marked as a good rule.
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A rectilinear surface equal to a curvilinear figure
Over a circular sector Leonardo writes that the rectilinear surface a b c is worth the curvilinear figure n m, because he removes two lesser portions and gives back m, a single portion worth those two. The two halves of the sheet each carry paired figures where a is worth b and c is worth d.
Good rule: the whole curve equals the whole straight
Beside a cancelled figure Leonardo marks a good rule: here every part of the curve and every part of the straight are of equal motion, whence it follows that the whole curve is equal to the whole straight.
Curvilinear parallelogram abcd equal to rectilinear arpq
A rectangle set above a circle-sector demonstrates that abcd is a true parallelogram because its sides are of equal curvature through the motion carrying curve bd into ac. The two curvatures enclose a surface equal to the triangle below; the joined straight line or, raised to pq, makes the parallelogram arpq equal to the curvilinear abcd, with side St equal to bd and the greater rectilinear side equal to ac.
Raising the curves to obtain the equal rectilinear space
The demonstration continues below: move the curve b d to the height e n, and the greater line a c rises into f m, and the straight S t rises to p q, so that the rectilinear parallel space equals the curvilinear parallel space. Because b d rises into a c without changing curvature, it is as if it stayed in its first place, and so the rectilinear parallelogram is found equal to the curvilinear one.
Segment, arch and curved-triangle figures
Across the sheet run small studies of circular segments, arch- and dome-like figures, curved 'tent' triangles, lens (bisangle) shapes, a square with an inscribed circle, and hatched sectors. These carry the lettered figures used to argue the equality of curved and straight areas.