Circles in quadruple proportion and curvilinear triangles
Squaring lunes: a quarter of the greater circle worth the whole lesser
This sheet develops Leonardo's geometry of curved figures, drawn as circles quartered into shaded lunes, curvilinear triangles and superimposed quadrants. He states the rule that among circles in quadruple proportion a quarter of the greater is worth the whole of the lesser, and argues from equals-minus-equals that a curvilinear triangle can equal a semicircle. A proof compares two surfaces equal in area but different in shape, showing that where they overlap they are equal and alike, while the projecting parts are equal in quantity yet unlike in figure. A faint pencil study of a sphere held in a curved cradle appears at the top, and further faint writing surrounds the figures.
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Circles in quadruple proportion: a quarter of the greater worth the whole lesser
For a circle and a joined quarter-circle (labelled c, b, a), the note gives the rule that among circles in quadruple proportion a quarter of the greater equals the whole of the smaller: a b (circle) is worth c b (a quarter-circle). Since taking equals from equals leaves equals, the curvilinear triangle c equals the semicircle a.
Equal figures minus equal parts leave equal remainders
If from two things equal in quantity but different in figure one removes an equal and similar part, the remainders will be equal and similar to each other. Hence a b are equal to c d, but not similar in shape.
Superimposed circle and quadrant: curvilinear triangles versus circle-portions
Taking two equal but differently shaped surfaces a m b and the m c d circle, the part m where they overlap is common to both, so they are equal and alike there. The non-touching parts a b and c d are equal in quantity but unlike, since a b are two curvilinear triangles and the others two portions of a circle.