Squaring lunes and portions of circles

Squaring the crescent lune (falcata)

To square the lune, Leonardo removes from beneath its curve a part such that the remainder becomes squarable. Working in four parts d; S; c; t, he takes S from the lune d and an equal arc t from c, so that each figure is left with arcs of equal value and can then be reduced to a square equal to the squared lune.

Circles on the sides of a right triangle

A right-angled triangle is drawn with three circles whose diameters equal its three sides. The note states that the circles made on the two sides are together worth the circle made on the hypotenuse, and that the same holds for portions proportional to those circles and for their lunes, both rational and irrational.

Equal areas by common subtraction

The figure a b c is worth a b d, because if two things are equal to a third they are equal to one another. From two figures that share a common part, casting away that shared contact leaves remainders of equal value — the axiom Leonardo relies on throughout to equate curved surfaces.

One portion equal to two unequal portions

A heading sets the task of giving two portions of various sizes equal to a single portion. The neighbouring sectors and triangles with hollowed-out portions work toward equating sums of circular segments to one another.

Metà destra. Colonna destra. Doppia figura colle lettere:

Le stesse parzialmente sovrapposte, con:

Le stesse diminuite e separate, con:

Le precedenti in quattro parti, con:

Settore di cerchio diminuito di porzioni:

Figura analoga, con:

A sinistra delle precedenti, settore di cerchio con porzioni vacue, e:

Triangolo e rettangolo con porzioni vacue, e

Colonna sinistra. Due settori con porzioni vacue, e:

Colonna sinistra. Due settori con porzioni vacue, e:

Tre piccole figure, con:

Figura analoga, con:

Sotto le precedenti, figura con:

In basso, sei figure con:

In basso, sei figure con:

Metà sinistra. Dall'alto.

Triangolo rettangolo e tre cerchi di diametro pari ai tre lati del triangolo, con:

Colonna destra.

Rettangolo curvilineo, con:

Presso il margine inferiore.

Porzione e falcata:

Settori sovrapposti, con:

Figura analoga, con:

Due rettangoli curvilinei, con:

Rettangolo curvilineo, con:

Più sotto, figurina:

Right half. Right column. Double figure with the letters:

The same figures partly overlaid, with:

The same reduced and separated, with:

The preceding figures in four parts, with:

Sector of a circle reduced by portions:

Analogous figure, with:

To the left of the preceding, a sector of a circle with empty portions, and:

Triangle and rectangle with empty portions, and

Left column. Two sectors with empty portions, and:

Left column. Two sectors with empty portions, and:

Three small figures, with:

Analogous figure, with:

Beneath the preceding, a figure with:

At the bottom, six figures with:

At the bottom, six figures with:

Left half. From the top.

Right triangle and three circles of diameter equal to the three sides of the triangle, with:

Right column.

Curvilinear rectangle, with:

Near the lower margin.

Portion and lune:

Overlaid sectors, with:

Analogous figure, with:

Two curvilinear rectangles, with:

Curvilinear rectangle, with:

Further below, a small figure:

On this page

Squaring the crescent lune (falcata)

Circles on the sides of a right triangle

Equal areas by common subtraction

One portion equal to two unequal portions