Squaring a Curvilinear Figure Equal to a Rectilinear One
Quadrature with Euclid I.37: circular sectors and triangles between parallel lines
Folio 115r continues Leonardo's study of quadrature, the squaring of curved figures. Quarter-circle sectors, shaded curvilinear lobes and triangles inscribed between horizontal parallel lines accompany a demonstration that a surface a can be made equal to a square by removing equal parts from equal quantities. Leonardo invokes Euclid's Book I, Proposition 37 (triangles on the same base and between the same parallels are equal) to prove that the square o r p g equals two triangles and so squares the surface a.
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A curvilinear quadrilateral set equal to a rectilinear figure
Leonardo poses the problem: let there be given a curvilinear figure bounded by its four sides, equal to a rectilinear figure. This is the classic quadrature challenge of converting a curved-sided shape into a straight-edged one of equal area.
Squaring surface a by Euclid's Book I, Proposition 37
The surface a equals the quantity b c d, which is the square c e. The square o r p g, by the 37th of the first book, is double the triangle o p r because both lie between the parallel lines o r q g and on the same base o r; being double, it equals the two triangles t o r and q o r, and so the surface a is squared with the square o p r g.
Common notion: equals taken from equals leave equals
a b and b c are equal quantities; removing b from each leaves a equal to c. Leonardo cites the axiom that if equal parts are removed from two equal quantities the remainders are equal, concluding that c squared equals a.