A Catalogue of Quadrable Curved Figures

Eight imperfect sectors and the squarable remainder

The eight imperfect sectors are worth the four largest portions of the greater circle, and the remainder is squarable. Leonardo reasons that four sectors equal four largest portions of the smaller circle plus four half-portions of the greater, so that the largest square drawn from the greater circle is worth all the 'white' of the circle above.

The white equals the largest square

In a circle with an inscribed square, all the 'white' area is worth the largest square of that circle, because the four sectors are worth the four largest portions of the enclosing circle. Several neighbouring variants restate the same equivalence between the hatched and blank regions.

Circles as the squares of their diameters

The proportion from circle to circle is such as that from square to square made by the multiplication of its diameter. This general ratio underlies the whole sheet of doubled and quadrupled circles.

The four lunules of Zenofonte

Between two concentric circles with an inscribed square, the four outer portions are worth the four half-largest inner portions and vice versa. This figure is worth the middle square and, Leonardo notes, is worth 'the four lunules of Zenofonte'.

Removing five lunules leaves a square

Within a rectangle holding circles and lunules lettered c a d b, if the five squarings of the five lunules are removed, the remainder of the enclosing quadrilateral stays squared. It is the closing demonstration of the sheet's quadrature programme.

Sixteen curvilinear triangles worth the four largest

Sixteen hatched surfaces bounded by three curved sides are together worth the four largest portions of the circle, and the remainder is squarable. A parallel figure of sixteen crescents makes the same claim.

Prima colonna.

Cerchio con settori imperfetti:

Quadrato iscritto nel cerchio:

Variante della figura precedente:

Figura analoga alla precedente:

Altra variante:

Quadrati, cerchio, ottagono iscritti:

Seconda colonna. Cerchio con bisangoli e settori imperfetti:

Cerchio e due quadrati iscritti:

Doppia figura:

Cerchi concentrici con ottagono iscritto:

Cerchi concentrici con quadrato iscritto:

Cerchi concentrici con quadrato e ottagono iscritti:

Margine superiore.

Terza colonna. Porzione divisa:

Quattro cerchi in un cerchio:

Cerchi concentrici con ottagoni iscritti:

Due quadrati e due cerchi in un cerchio:

Sedici quadrati curvilinei in un cerchio:

Quarta colonna. Quadrati e cerchi iscritti:

A sinistra della precedente:

Sotto la precedente:

Quadrato massimo e quadrati minori entro il cerchio:

Parte di corona circolare divisa in falcate con:

Cerchio con otto lunule:

Due mezze porzioni con:

Quinta colonna:

Due cerchi concentrici con quadrato iscritto:

Figura analoga alla precedente:

Cerchio con quadrato iscritto:

Cerchi e lunule entro un rettangolo, con:

First column.

Circle with imperfect sectors:

Square inscribed in the circle:

Variant of the preceding figure:

Figure analogous to the preceding one:

Another variant:

Squares, circle, octagon inscribed:

Second column. Circle with two-angled figures and imperfect sectors:

Circle and two inscribed squares:

Double figure:

Concentric circles with inscribed octagon:

Concentric circles with inscribed square:

Concentric circles with inscribed square and octagon:

Upper margin.

Third column. Divided portion:

Four circles within a circle:

Concentric circles with inscribed octagons:

Two squares and two circles within a circle:

Sixteen curvilinear squares within a circle:

Fourth column. Inscribed squares and circles:

To the left of the preceding:

Below the preceding:

Largest square and smaller squares within the circle:

Part of a circular ring divided into crescents, with:

Circle with eight lunules:

Two half-portions with:

Fifth column:

Two concentric circles with inscribed square: