Geometry: Squaring Lunes with Triangles, Circles and Ovals
Equal lunes, circles in double and quadruple ratio, and an oval that holds many circles
Folio 55v is headed 'Geometry' and pursues Leonardo's favourite problem of measuring curved figures by comparing lunes (crescent shapes) with straight-sided ones. A central diagram shows a circle with an inscribed square and shaded lunes lettered a, b, c, e, g, h, where he equates pairs of lunes with triangles; further sketches treat two internally tangent circles in quadruple ratio and an oval figure said to equal two circles, an oval lune, and a rectilinear triangle. The reasoning turns on removing equal 'portions' from figures standing in double or quadruple proportion so that the remainders can be set equal.
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Angles between chord and arc are all equal
The small figures of circle, semicircle and circular segment illustrate the claim that every kind of angle made between a chord and an arc is equal to every other of its kind. It is stated as a preliminary before the lune constructions.
Lunes equal to triangles in a circle with inscribed square
Taking two circles in double ratio, the smaller split into semicircles c d and e f, Leonardo removes the two portions d f from each. What remains, the lunes c e, equals the greater circle's remainder less its half a b. He concludes that the triangles g h are worth the two lunes c e.
Quadruple circles and the remaining lune
Of two circles in quadruple ratio, a quarter of the greater equals the whole of the smaller. Subtracting the smaller leaves a lune equal to that quarter, namely a b with two portions removed; two further equal portions n m are then taken from the lune.
An oval figure equal to circles, lune and triangle
The oval figure enclosed in the oval lune is said to be worth two circles f, the oval lune g, and the rectilinear triangle h. Leonardo adds that the oval f can be made to hold as many circles like the enclosed one as one pleases.