Curved-sided triangles equal to a straight-sided figure
The rule for making two equal 'falcate' equal to a rectilinear surface
Folio 82v states the general rule behind the preceding quadrature studies: a 'triangle' bounded by two curved sides — a 'falcata' — can be made equal to a rectilinear surface provided its opposite curved sides are of equal length. A single rectangle at top right, crossed by diagonals and closed off by an arc, illustrates how such a crescent is set against straight-sided areas. Further faint diagrams show through from adjacent leaves.
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Rule for equating a two-curve triangle to a rectilinear surface
Triangles of two curved sides, also called 'falcate,' whose two curved sides are of equal length, can be made equal to a rectilinear surface when their opposite (curved) sides are equal. The statement generalizes the crescent-quadrature proofs of the facing page.
Rectangle crossed by diagonals and closed by an arc
At top right a rectangle is drawn with its diagonals and a curved arc springing across it, isolating a crescent against the straight-sided rectangle. The construction visualizes setting a 'falcata' equal to a rectilinear area.