Quadrature of sectors, lunes and crowns; proportion of circles

Squaring a sector by adding its two lunes (S, a c, f)

Here the parallelogram b exceeds the sector S by its two lunes a c, which are squarable by rule. Leonardo therefore adds to the sector S a further squared surface equal to the two lunes.

Redistributing hatched triangles to form a parallelogram (m v, h i)

With the parallelogram a b c d taken equal to the sector e f, the hatched triangles a f are left over; he divides f and adds one part at o and the other at n, then joins everything into the parallelogram m v, lending it the known squared piece. Finally he divides parallelogram v m by the circumference, as parallelogram h i shows.

Equalizing an unequal quadrilateral (Euclid I.3)

If a rectilinear quadrilateral has unequal opposite sides and one wishes to take from it and render a portion, it must first be made equal with the help of the third proposition of Book I, without diminishing its original quantity. Otherwise the operation would be impossible.

Lending and rendering curvilinear pieces (Euclid I.4)

The square o is 'lent' and rendered with two opposite curved sides and two opposite straight sides while keeping the same quantity, by the fourth proposition of Book I. Any portion added in one figure is taken away below, as shown at n m, so that n joined with m remakes the exact quantity required by the construction.

Two concentric circles in the ratio 100 to 99

If two concentric circles stand in hundredfold proportion, the excess of the one equals the ninety-ninth part of the other, illustrated with smaller numbers (double, triple, 3 to 4). Thus for circles of 100 and 99 the greater exceeds the lesser by one ninety-ninth of itself, a single unit, or equally by one hundredth of itself.

Colonna destra. Settore di cerchio, con:

Parte di corona circolare, con:

Figure derivate dalla precedente, con:

Settore di cerchio con:

Settore di cerchio e due settori sovrapposti, con:

In alto. Quattro frammenti di corona circolare e grande settore di cerchio, con:

Piccola figura di triplice corona, con:

Sotto al grande settore, piccola figura di due porzioni, con:

Frammenti di corone circolari, con:

Colonna sinistra.

Girato il foglio sul lato sinistro, figura con:

Right-hand column. Sector of a circle, with:

Part of a circular crown, with:

Figures derived from the preceding one, with:

Sector of a circle, with:

Sector of a circle and two overlapping sectors, with:

At the top. Four fragments of a circular crown and a large sector of a circle, with:

Small figure of a triple crown, with:

Below the large sector, a small figure of two portions, with:

Fragments of circular crowns, with:

Left-hand column.

The sheet turned onto its left side, a figure with: