Geometry of Lunes: Semicircles, Sectors, and Triangles
Squaring three lunules by comparing sectors, portions, and triangles
Pen diagrams show a large semicircle carrying three lunules above a row of semicircles, with three sector-figures drawn as trapezoidal shapes and lettered a b c, f g h, and d l (one sector struck through). The note argues that the three sectors equal the four semicircles above, so that removing equal 'portions' from both leaves three lunules, and removing the portions from the sectors leaves three triangles larger than the lunules. Squaring the lunules and subtracting them from the triangles leaves three parallels equal to the fourth semicircle, labelled e. A caption also states that the three portions stand in sesquitertian (4:3) proportion to the larger.
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Three portions in sesquitertian proportion
A caption to the upper figure states that the three portions stand in sesquitertian proportion — a ratio of four to three — with the larger one. It fixes the numerical relation between the parts of the semicircle before the quadrature argument begins.
Lettered sector and semicircle figures
The sector-figures carry the letters a b c and f g h, with a further struck-through sector marked d l. These labels key the drawn shapes to the quadrature reasoning set out in the note below.
Squaring the three lunules
The three sectors are said to equal the four semicircles above; removing the three portions from both leaves the three lunules, while removing the portions from the sectors leaves three triangles larger than those lunules. Squaring the lunules and subtracting them from the triangles leaves three parallels equal to the fourth semicircle, labelled e.