Circles Quadruple to One Another and the Proportion of Lunes
Quarters of a quadruple circle superimposed, and the ratios of sickle-shaped segments
The recto continues Leonardo's study of circle proportions with a fan-shaped diagram of nested segments and lettered figures. He considers circles that are quadruple one to another, arguing that the four quarters of the smaller equal the quarter of the larger, so that two quarters of the smaller superimposed on the quarter of the larger leave an uncovered part equal to the covered part. A second thread analyses the proportion of a lune (falcata) to a portion m — whether one is double the other — and works out the fractions of the greater half that remain covered or uncovered. A final note relates the curvatures p n r, o b r and o f q as successively subduple circles.
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Quarters of a quadruple circle superimposed
When two circles are quadruple one to another, the four quarters of the smaller equal the quarter of the larger, so two quarters of the smaller equal half the quarter of the larger. Superimposing those two quarters entirely on the quarter of the larger, the part of the larger left uncovered equals the part covered. Leonardo letters the larger quarter a b c d e f n m and the two smaller quarters c d n m.
The proportion of a lune to portion m
Leonardo seeks the known proportion between the lune (falcata) and the portion m, testing whether one is double the other. Here the half-portion b m is double the whole m, so what is occupied by m equals m and the unoccupied part b also equals m. If the lesser portion is two-thirds of the greater half b m, the part of the greater left uncovered is one-third of its quantity.
Successively subduple curvatures
In the lower figure Leonardo relates lunes and arcs: if the lune c n equals m, and the lune a b equals c n, then b n and a c together equal m. He adds that the line p n r is the curvature of a circle subduple to the line o b r, which in turn is subduple to the line o f q.