Squaring the circle: dividing portions and 'quadrable' figures

Halving a circular portion by an equal-sided triangle (Euclid)

To halve the portion a f d, Leonardo invokes a proposition of Euclid: make the base angles at a b equal and draw the angle d over to c, converting the scalene triangle a b d into the equal-sided triangle a b c. The base of each triangle must always be the chord of the arc of the portion, and the equal partitions can then be repeated to infinity.

Curvilinear triangles that are 'in themselves quadrable'

The curvilateral triangles a b c d are quadrable because they lack the 'four maxima' — that is, sixteen 'minima' distributed as eight lunes and eight portions. This 'four maxima' bookkeeping recurs across the sheet as the test for when a curved figure equals a square.

Dividing a portion into similar sub-portions by proportion

A second method yields sub-portions geometrically similar to the whole. To take half of a given portion, Leonardo draws out a similar portion that is subduple to it — or subtriple, or in whatever proportion one wishes.

A cross built from sixteen equal curvilinear triangles

The cross inscribed in a double circle is composed of sixteen similar and equal triangles, each with two straight sides and one concave curve. Eight of them form the square that contains the cross and eight compose the cross itself; added to the circular 'parallel' it equals the maximum square of the maximum circle.

Prima colonna da destra:

Tre figure di settori di cerchio, una delle quali colle lettere:

Figura di lunula o falcata nel margine:

Figura colle lettere:

Figura colle lettere:

Colonna centrale.

Piccola figura di cerchio entro il testo:

Figura di cerchio contenente quattro cerchietti:

Figura analoga alla precedente:

Cerchio contenente sedici triangoli:

Come la precedente figura:

Cerchio contenente otto triangoli troncati:

Fino al termine della colonna figure e testi biffati. * Segmento circolare a due basi colle lettere:

*Figura analoga alla precedente colle lettere:

* Due cerchi con due diametri perpendicolari fra loro:

* Rettangolo con due lati curvi, contenente un bisangolo:

Altra semicolonna sotto la colonna centrale. Figura colle lettere:

Figura colla lettera:

Ultime due colonne a sinistra. Quadrato contenente un piccolo quadrato e quattro parti di cerchi:

Quadrato iscritto in un cerchio con altre figure:

Di fianco alle precedenti, figura, ripetuta due volte, di quadrato iscritto e circoscritto a due cerchi:

Croce iscritta in un doppio cerchio:

Figura affiancata alla precedente verso il margine:

Tre figure sulla stessa linea, una delle quali colle lettere:

Cancellato:

First column from the right

Three circle-sector figures, one of them lettered

Lune (falcata) figure in the margin

Lettered figure

Lettered figure

Central column

Small circle figure within the text

Circle containing four small circles

Figure analogous to the preceding

Circle containing sixteen triangles

As the preceding figure

Circle containing eight truncated triangles

To the end of the column, figures and crossed-out texts. * Circular segment with two bases, lettered

*Figure analogous to the preceding, lettered

*Two circles with two mutually perpendicular diameters

*Rectangle with two curved sides, containing a biangle

Another half-column below the central column. Lettered figure

Figure with the letter M

Last two columns on the left. Square containing a small square and four parts of circles

Square inscribed in a circle with other figures

Beside the preceding, a figure repeated twice: a square inscribed in and circumscribed to two circles

Cross inscribed in a double circle

Figure flanking the preceding toward the margin

Three figures on the same line, one lettered

Crossed out

On this page

Halving a circular portion by an equal-sided triangle (Euclid)

Curvilinear triangles that are 'in themselves quadrable'

Dividing a portion into similar sub-portions by proportion

A cross built from sixteen equal curvilinear triangles