Geometry of Superimposed Circles and Sectors
Areas of overlaid semicircles and sectors in double and sesquialteral proportion; a definition of the circle
A dense geometrical sheet on the areas of circles, semicircles and sectors laid one over another. Leonardo shows that when a lesser figure is set on a greater in double proportion the exposed remainder equals the lesser, works the sesquialteral case (the exposed part is a third), proves two sectors equal as a quarter and an eighth of doubled circles, and defines the circle as a parallel space between circumference and center; he adds a theorem that uniformly-nonuniform pyramids on the same base are of equal capacity. The diagrams include two spirals, hatched triangles, semicircle pairs and rectangles ruled into little squares, and further untranscribed notes fill the sheet.
On this page
Lesser figure on greater: the exposed part equals the lesser
Given two surfaces in double proportion, a and b, with b twice a: setting a wholly upon b leaves an uncovered remainder of b exactly equal to a. Leonardo illustrates this with the labelled semicircles a, b and the remainder b - a.
The sesquialteral case: exposed part is one third
If one surface is in sesquialteral proportion to another and the lesser is set wholly on the greater, the part of the greater left uncovered is the third part of the lesser. The figures are marked with the numbers 3 1 and 1 2.
Definition of the circle
Leonardo writes a compact definition on the upper margin: the circle is a parallel space included between the circumference and the center of that circle.
Uniformly-nonuniform pyramids of equal capacity
Referring to the two circular figures in the central column, he states that all uniformly-nonuniform pyramids raised on the same base and ending at the center of such a circle are always of equal capacity, however infinitely their lengths vary.
Two equal sectors: quarter and eighth of doubled circles
The two sectors are equal because they are drawn from circles double one to the other, being the quarter of the smaller and the eighth of the greater set together at e f g. Removing b from d c leaves d equal to the crescent a; and since a e equals e f, removing the doubled e from each side leaves f equal to a and hence to d. Leonardo notes the second proof is shorter than the first.
Overlaid rectangles and the denomination of multiples
When surfaces partly overlap, the single uncovered part of the greater contains the whole lesser as many times as its multiple-denomination over the lesser, always less one, plus the single part of the lesser. For two surfaces a b in double proportion overlapping at d, the uncovered part e f g takes in the whole lesser c d at f g and still leaves e equal to c.