Pyramids and Wedges Bisected by Semicircle Peripheries
Lunes and the proportional division of circular portions to infinity
On this leaf Leonardo argues that triangular 'pyramids' whose bases lie on a great circle and whose apex meets angle a are each cut exactly in half by the periphery of a semicircle half its size. Removing equal parts from two equal surfaces shows the parts removed were equal, so the wedges (coni) are always halved by the subduple circle and cut in thirds by the subtriple circle, and so on to infinity. Two radially divided semicircles and a large lune illustrate the proof, with parts numbered 3, 4, 5 and 6.
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Equal remainders prove equal parts removed
If two equal surfaces stay equal after diminution, the parts removed were equal; since portion a K o equals the lune K b o f a, removing parts 3 and 4 of the pyramid a b c leaves parts equal to lune i c o f a, proving 3 and 4 equal.
Straight-line pyramids halved by the subduple semicircle
All pyramids of straight lines rising on the semicircle's circumference and ending at one of its angles are cut into two equal parts by the periphery of a semicircle half the first, one of whose angles is joined to the angle of the greater semicircle.
Wedges cut in halves and thirds by nested circles
Because semicircle a b f is double the lesser, what the greater exceeds beyond the lesser equals the lesser; every wedge is thus always halved by the median line of the subduple circle and cut in thirds by the subtriple circle, and so on to infinity.
Double and triple proportion of the cutting line
If the greater circle is triple the lesser, the line of the lesser cuts the pyramids in triple proportion, just as the subduple circle cuts them in half.