Cutting a Third from a Semicircle Between Parallels
Dividing the arc into six curvilinear segments, and the limit at one third
To take a third of a semicircle between two parallel lines, Leonardo divides the periphery into 6 equal parts (six curvilinear 'pyramids') and draws chords to form the strip a f c b. As a general rule, four pyramids are always drawn off from the half-circle, counting the added pieces f d n c, and the remainder between the parallels is the part sought. He remarks that the construction cannot yield more than a third, though the semicircle can potentially be divided into infinitely many pyramids for finer fractions, and works a fraction routine reaching 16/3.
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Extracting a third by six curvilinear pyramids
Divide the periphery into 6 equal parts and draw the straight lines from the meta of the diameter, dividing the semicircle into six pyramids of curved base. Drawing the chord from base a to base b and another from base f to base c forms the space a f c b, equal to two pyramids, while the space a q b r occupies four of them once the added piece f d n c is counted.
The general rule and its limit at a third
By general rule one must always draw four pyramids from the half-circle, counting the added piece above; the remainder lies between equidistant lines and is the part sought. More than a third cannot be given, since the four pyramids exhaust the four, but requests from a third upward can be met infinitely.
Superposing the leftover pieces
a b is equal to d; c remains apart, because it was the piece of the contact. Now superpose b upon d, and whatever of b exceeds, together with the piece a, will be equal to the remainder of d.