The Rule of Subtracting Four to Divide a Semicircle
Taking 2/3, 3/4, 4/5 and 7/8 of a semicircle between two parallels
Continuing the previous demonstration, Leonardo concludes that removing 4 of the 6 sectors leaves the 2 sectors between the parallels a b and c d that form the promised third. He then generalises the construction into an arithmetic rule: whatever fraction is wanted, divide the semicircle into a suitable number of sectors and always remove 4 — so 12 sectors give 2/3, 16 give 3/4, 20 give 4/5, and 32 give 7/8. A small fan of radii forming a semicircle is drawn within the lower text.
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Removing four sectors leaves the promised third
Of the 6 sectors into which the semicircle was divided, removing 4 leaves 2 between the parallels a b, c d, which is the third of the whole that was promised. The sectors at the foot together with the supplement above always make up 4 sectors.
The general rule: always subtract four
For any fraction, divide the semicircle into a number of sectors and remove 4. Twelve sectors less 4 give 8 supplements for 2/3; 16 less 4 give 12 for 3/4; 20 less 4 give 16 for 4/5; and for 7/8 take a number receiving 4 eight times — 32 — and throw away 4.
Sector-fan of a semicircle drawn in the text
A small semicircle divided into radiating sectors is sketched among the lines, illustrating the fanned division that the arithmetic rule operates on.