Quadrature of lunes: crescents equal to rectilinear areas
Proofs that curved-sided 'falcate' equal rectangles, with a circle of quadruple proportion
Folio 82r pursues the quadrature of lunes: crescent-shaped figures bounded by two arcs (Leonardo's 'falcate') are proved equal to straight-sided areas. Two lettered rectangles capped by arcs at top right support the claim that the lune q r m n S equals the lune q r n o t, while lower down a shaded circular sector headed 'Circle of quadruple proportion' underlies a delicate argument for finding by how much one squarable part exceeds another. The reasoning proceeds by adding and subtracting matching portions until a rectilinear remainder is left equal on both sides.
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Adding and removing matching portions to equate the figures
a b c d is set equal to a d e f, and equal pieces are stripped from each until a remainder of the one equals a remainder of the other. Removing the larger portion a c n d from both surfaces leaves a b d equal to d e c f a. The proof works purely by subtracting congruent parts.
Two crescents proved equal: q r m n S and q r n o t
The lune q r m n S is declared equal to the lune q r n o t. The two lettered rectangles capped by arcs at top right carry this equality of curved-sided figures.
Circle of quadruple proportion and a subtle quadrature
Under the heading 'Circle of quadruple proportion,' a d is taken equal to b c n while c n exceeds d. Subtracting d from c n and joining the remainder to the non-squarable b makes b equal to a, yielding a squared part that measures by how much a exceeds b — called a most subtle investigation.
Removing the excess to leave b c equal to a d
a is greater than b and c n is greater than d; the excess n that e has over d is removed, and m is taken from its sector. What remains, b c, is equal to a d.