Squaring the circle: lunes and doubled circles
Lunes proved equal to rectilinear figures by removing equal parts from circles double one another
A page headed 'On squaring [the circle]' filled with Leonardo's geometry of lunes. Three lettered figures, drawn at right and lower left, pair a larger and a smaller circle (or semicircle) whose areas he treats as double one another, and shade the crescent-shaped lunes formed where they overlap. The prose argues that if equal parts are subtracted from wholes that stand in a double, triple or unequal ratio, the equal or proportional remainders let a lune be shown equal to a triangle or to a smaller circular part. The reasoning leans on a stated proposition that two unequal circles meeting at the ends of a straight line, or touching the four sides of one square, are double one another.
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Two unequal circles that intersect are double one another
The largest circle here is worth twice the smaller. Leonardo removes half of the largest, s t x, by borrowing the portion a to complete the semicircle S t x, then subtracts from the greater lune the surplus it gained (the half-portion marked n). He asserts that two unequal circles intersecting at the ends of a straight line, like two touching the four sides of one square, are double one another.
Rule for subtracting equal parts from unequal or doubled wholes
If equal parts are taken from unequal things, the remainder stays unequal, but not in the first proportion; the excess of the greater quantity is increased. If equal parts are taken from double things, the remainder is equal plus half the greater; from triple things, plus two-thirds; and so on to infinity.
Lune b equal to the horn-lunes and to a rectilinear triangle
The lune b is declared worth c d, the little horn-lunes, and equal to the right-sided triangle. Since half of the greater of two doubled circles equals the whole smaller one, removing from each surface their common contact a and f g leaves the untouching remainders equal.
Sector proof: a quarter of the greater circle equals half the smaller
Of circles double one another, a quarter of the greater is worth half the smaller; removing an equal part n from each leaves the remainder o p equal. This holds because the semicircle is drawn on the side of the square as diameter while the other circle touches the corners of that same square.