Rule of proportion for equating parts of unequal circles
Fraction table: quarters, sixteenths and falcate pyramids
A table pairs each fraction of a quadruple circle with its counterpart in the subquadruple circle (1 against a quarter, a half against 1/8, a quarter against 1/16, and so on to 1/8 against 1/32). Leonardo gives the general rule: to find the part of a larger circle equal to a given part of a smaller one, multiply the given fraction by the ratio between the circles, so that if the larger is quadruple, 1/3 of the smaller equals 1/12 of the larger and 1/5 equals 1/20, continuing to infinity. He notes that 'falcate' pyramids of variously curved sides are made from curved lines of circles in various proportions. Lettered figures marked 8, 1/2 and 1/4 at the foot show that two smaller cones equal the larger, so the curved a equals the squared b c.
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Table matching fractions of quadruple and subquadruple circles
A column headed 'consequent part of the quadruple over the subquadruple' pairs 1 against a quarter, then a half with 1/8, 1/3 with 1/12, a quarter with 1/16, 1/5 with 1/20, 1/6 with 1/24, 1/7 with 1/28 and 1/8 with 1/32. Each fraction of the larger circle is a quarter of the matching fraction of the smaller.
Multiply the fraction by the circles' ratio
To equate a part of a larger circle to a given part of a smaller one, multiply the given fraction by the proportion the larger circle has over it. With a quadruple circle, 1/3 of the smaller equals 1/12 of the larger (3 times 4), and 1/5 of the smaller equals 1/20 of the larger (5 times 4), the rule following to infinity.
Two smaller cones equal the larger; a equals b c squared
The figures show an eighth and a quarter of circles, double one to the other. Therefore the two smaller cones are equal to the larger, and the curved a is equal to the b c squared.