Squaring curved figures: circles, lunes, sectors and annuli
Proving equal areas among concentric circles, inscribed squares and rings in ratio 1:4:16
This sheet is dense with Leonardo's geometry of curvilinear figures, arranged in two columns with many diagrams: concentric circles with inscribed squares, circular sectors, lunes and rings (annuli). He works to show that curved areas can be matched to equal figures, using circles set in the proportion 1:4:16 so that each is quadruple the next. The demonstrations pass portions between larger and smaller circles — a larger portion is worth four smaller ones, a sector equals a smaller circle, and a ring equals a middle circle — to prove surfaces equal in quantity though various in figure. Two small detached fragments mounted below the main sheet carry further tiny diagrams and a group of letters.
On this page
Concentric circles and inscribed squares
A figure of three concentric circles with three inscribed squares is analysed: a is worth b c, and b c is worth d e f g, so that a is worth the two middle portions and four of the smaller ones. The lettered parts h, d, g, b, e, f, a, c organise the comparison of areas.
Rings (annuli) equal to whole circles
Treating a sector f, Leonardo argues the larger portion is worth the four smaller portions, the sector f equals the smaller circle, and the larger ring equals the middle circle, so the ring is worth four times the sector f. He exchanges portions between the larger and middle rings because ring-to-ring keeps the same proportion as their whole circles.
Lunes and tangent circles in ratio 1:4:16
Three internally tangent circles in the proportion 1:4:16 give a b c equal to the lune a c f g, and the black sector f b a equal to the black circle L, since the largest circle is to the middle as the middle to the smallest, each quadruple the next. He removes quarters from sector and circle to establish both equalities.
Equal surfaces of different figure
Comparing rings, sectors and circles, he notes the ring m with its circle n is worth a quarter of the whole larger circle f c g and equals the side figure p o, so that p o and the middle r q are equal — an example of surfaces equal in quantity yet various in figure.
Circle and ring proportions
The analogous figure states that circle a is worth ring b, and circle-and-ring a b together are worth ring c, continuing the proportional chain among the concentric parts as each area is quadruple the next.