Quadrature studies: doubling pyramids and equipartition of sectors
Comparing curved figures standing in double and quadruple ratio
The verso continues the recto's geometry of curved figures, covered with circular sectors, superimposed lunes and pyramidal cone-diagrams. Leonardo compares three figures whose parts overlap, then proposes to double three pyramids so their added quantities match, and insists one must give and take to reach an exact equipartition. A further note treats triangles standing in double and quadruple ratio, showing how removing half the largest leaves a curved-sided figure equal to the two triangles set upon it. Additional faint writing on the sheet is not part of the transcription.
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Three overlapping figures and the doubling of three pyramids
Three figures remain, of which the two o m a and n f d, joined together, equal the part m c f o n and overlap in the quantities o m and n f, leaving uncovered the part c, equal to the two parts a d. Leonardo then proposes to double the three pyramids a c d, adding equal quantity at c and at a d.
Giving and taking to make the equipartition
A short note beside four superimposed sectors (labelled m, n, e, d, b, a, c, f) states that here one must give and take and always adjust the parts in order to achieve the equipartition of the figure.
Triangles in double and quadruple ratio with curved portions
The triangle a b c p is twice the triangle e f g p and four times the triangle a n o S, their portions doing likewise. Removing e f g p (half the larger) leaves the curved-sided part a b c e f g, equal to the two triangles set upon it; its portion n o m is four times the portions n o and m, because born of a quadruple pyramid.