Squaring the circle: lunes, rosettes and curvilinear octagons

Draw curved figures from a square, leaving a square

Leonardo sets the programme of the sheet: from a square let curved things be drawn out and let the remainder stay square, and within a circle let various curvilinear surfaces be drawn, mixed with rectilinear ones, so the remainder stays squarable. The columns that follow work this idea through many figures.

The four maxima and the greatest square

A recurring refrain across the diagrams states that where the four maxima are missing the remainder is worth the greatest square of the greatest circle. In one eight-pointed figure the labelled ab is squared in itself and cd is likewise squarable.

A great circle worth four lesser circles

In the cancelled figure the parallel made from the greater circle is worth a quarter of the whole circle, and that quarter is worth the lesser circle. Leonardo removes the middle circle, renders the same value with the greatest circular parallel outside, then removes the middle parallel worth the four maxima so the remainder stays square. Elsewhere the greatest circle is worth four least circles abcdefhnl, of which two are removed.

Removing two least circles and the half maxima

For the circle o - iah - mn - LbK - p the half is missing once the two least circles are removed, and it also lacks the half of the four maxima for want of op. Replacing them in the least circles nm and then removing the two maxima hiKL leaves the two triangles ab.

Prima colonna a destra:

Terza figura colle lettere:

In basso, cerchio con ottagono curvilineo:

Seconda colonna:

Cerchio colle lettere:

In basso, cerchio con otto punte triangolari:

Terza colonna. Cerchio con, a sinistra, nove figure distinte da:

Figura sottostante:

Sotto la precedente:

A sinistra delle due precedenti, figura biffata:

Sotto la precedente:

Angolo inferiore sinistro:

First column on the right:

Third figure with letters:

Below, circle with a curvilinear octagon:

Second column:

Circle with letters:

Below, circle with eight triangular points:

Third column. Circle with, on the left, nine figures distinguished by:

Figure below:

Below the preceding:

To the left of the two preceding, cancelled figure: