Quadrature of lunes and proportions of circles
Squaring sickle-shapes and circle-sectors; circles in double, centuple and octuple ratio
A geometry sheet on the quadrature of lunes and the proportions of circles, its arguments numbered 'First', 'Second', 'Third' like propositions. Leonardo asserts that a lune, a curvilinear parallelogram and a sickle-shape are each squarable, that any part of a two-curved-sided figure with proportionally divided arcs is squarable, and works out equalities between semicircles and the sectors of double, quadruple and octuple circles, continuing 'to infinity'. He closes by noting that a quarter-circle cannot be reduced to a portion of equal arc, so it cannot be given.
On this page
First: parallelogram, lune and figure all squarable
The parallelogram a b is quadrable in itself, a is quadrable by itself, and the lune b is quadrable 'by Xenophon'. A drawing shows a rectangle with two curved sides and the included lunule labelled a-b.
Second: squaring the sickle-shape by transposing portions
Because lune, curvilinear parallelogram and whole are squarable, the sickle-shape c e is squarable. Leonardo adds the half-portion b o d and removes as much by taking away a b c, and since the triangle a lies outside b c o d e he restores it at o, concluding that the rectilinear d e equals the sickle c e.
General rule for two-curved-sided surfaces
Of every surface of two curved sides, all separated parts whose arcs are divided in the same proportion as the whole are each squarable, and so is the remainder. Two subdivided sickle-shapes illustrate the rule.
A sixth of a circle equal to a chord of a smaller circle
a b c is 1/6 of a circle, and o b r is a circle one-sixth of the whole, so the chord a c is worth o r. The figure sets a circle on a two-based circular segment.
Semicircle equal to sectors of larger circles, to infinity
The semicircle equals the sector holding 1/4 of a circle double its own, and equals the eighth of a circle quadruple the first, and the sixteenth of one octuple the first, and so on to infinity. The quarter equal to the half is a double proportion.
A quarter-circle cannot be given as an equal portion
Asked for a portion equal to a semicircle and then to a quarter of a circle, Leonardo objects that a quarter-circle reduced to a portion does not have a quarter of the arc but more, and therefore it cannot be given.