Squaring Lunes: Circle Sectors, an Inscribed Square and Equal Areas
Doubling circles and reducing sectors and lunes to figures equal to a square
This geometrical page works toward the quadrature of lunes, reasoning about circles in a doubling ratio and the equal-area figures that result. Leonardo argues that of two circles where one is double the other, a quarter of the larger equals half of the smaller, then divides that quarter's periphery into four sectors and removes matching portions to leave a remainder equal to the semicircle, citing the rule that equal figures partly superimposed have equal touching and equal non-touching parts. In a circle with an inscribed square he shows the curvilinear triangles c and d to be double the lune a, which equals triangle b, so that c, d, a and b are of equal value, and concludes that the figure n is squared and its remainder equal to it in value. Shaded diagrams of a semicircular sector, a circle with an inscribed hatched square, and a small lune accompany the argument.
On this page
Doubling circles: a quarter of the greater equals half the lesser
Of two circles one double the other, the quarter of the greater is worth half the lesser. Dividing the periphery of that quarter into four sectors and removing the portions of the circle shown at their fronts, and removing as much on the straight side of the semicircle, the remainders of the quarter-circle are left equal to the remainder of the semicircle.
The lune a equal to the curvilinear triangles
With a square inscribed in the circle, a b are equal and b alone equals d (or c). Since the greater circle is double the lesser, removing equal parts leaves the remainder equal; removing b of the lesser and b p of the greater, which are equal and similar, leaves the curvilinear triangles c d double the lune a, which is worth the triangle b. Therefore c, d, a and b are of equal value.
Reducing the figure n to a squared value
The part n is worth half the triangle d and half the triangle b. Therefore the triangle n is squared, and its remainder, equal to it in value, is likewise squared. The reasoning treats the areas as proportional quantities in a doubling ratio to establish their equality.