Squaring the Triangle of Three Curved Sides
Lunes and half-portions of hexagons in quadruple circles, divided to infinity
Folio 117r treats the squaring of a triangle with three curved sides (a lune-bounded figure), which Leonardo says he has proved by experiment. Lens-shaped and wedge figures accompany rules that in quadruple circles the larger half-portions of hexagons are quadruple the smaller, so four small half-portions equal one large. He shows a half-portion can be halved, quartered and subdivided to infinity, and closes by dividing the half-portion a b f into two equal parts using two equal triangles plus equal added portions.
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Squaring the curved triangle by moving portions
To square the triangle of 3 curved sides, the two portions a b are removed from their place and the half-portion c is restored to triangle d, restoring as much as was taken so the figure remains squared with parts d c. The proof: portion n c is 4 times portion a, since both are the sixth part of their circle and the larger's half equals two of the smaller.
Quadruple half-portions of hexagons
Found by experiment: in quadruple circles the larger half-portions of the hexagons are quadruple the smaller half-portions, so the four small half-portions together are worth the single larger half-portion.
Dividing a half-portion to infinity
v t is half of the whole half-portion x h l K, and a n r is a quarter of that half-portion; this rule divides it to infinity, generating ever-smaller equal subdivisions.
Halving the half-portion a b f with two equal triangles
To divide the half-portion a b f into two equal parts, the line a b is drawn to the angles of figure a m b f, forming two triangles c and d of equal base and height, hence equal. Adding equal half-portions e and n to c and d gives e c equal to n d.