Areas of Circles, Lunes and Squares, with a Portal Study

Circles in double proportion

A heading rule states that among circles in double proportion, the quarter of the greater is worth the half of the lesser. This governs the many circle-area comparisons drawn below. Beneath a portal figure, a common notion is added: if from equals you take equals, the remainders are equal.

The value of a lune equals a quarter-circle

Working through a four-circle figure, Leonardo argues that the smaller lune is worth as much as the smaller circle, which is a quarter of the greatest circle, so the smaller lune equals a quarter of the greatest circle; together the lune and smallest circle make half the greatest circle. Removing the smallest lune leaves three quarters of the greatest circle, of which a third is the smallest circle and two quarters the greatest lune.

The largest circle worth four of the smallest

For the figure of four circles labelled n-c-b, f, e, a-d-m, a b equals c d and e f equals n m. Because the two smaller circles equal the middle circle and the middle equals the largest, the largest circle is worth four of the smallest and two of the middle ones. A column of numbers (54, 54, 216, 270, 2916) is set at the left.

Portal or aedicule with flanking columns

At the left of the sheet is a finished study of an architectural portal set within a framed aedicule, with a stepped base and paired supports; the transcription itself refers to "the figure of a portal." Slender column shafts and further architectural fragments run along the lower edge of the leaf.

Margine superiore.

A destra sotto la figura di un portale:

Figura composta di quattro cerchi:

Verso sinistra, figura con:

Margine destro, figura composta di quattro cerchi, con:

Margine sinistro, a metà. Tre cerchi internamente tangenti, con:

Tre cerchi, quadrato iscritto, con:

Figura analoga iscritta in un quadrato, con:

Quadrato entro quadrato, con:

Colonna sinistra.

Upper margin.

On the right, beneath the figure of a portal:

Figure composed of four circles:

Toward the left, figure with:

Right margin, figure composed of four circles, with:

Left margin, halfway down. Three internally tangent circles, with:

Three circles, inscribed square, with:

Similar figure inscribed in a square, with:

Square within a square, with:

Left column.

Areas of Circles, Lunes and Squares, with a Portal Study

On this page

Circles in double proportion

The value of a lune equals a quarter-circle

The largest circle worth four of the smallest

Portal or aedicule with flanking columns