Squaring circles, lunes and curved figures

Reducing a rectilinear figure with a semicircle to a square

In the triangle figure Leonardo notes that n m equals o p, giving o p equal to a square. Drawing the diameter b c from angle S to angle S produces the semicircle b S c enclosed within the rectilinear surface abcd. Removing the square nm from that surface leaves a remainder that is itself a square with straight sides.

Two equal circles with equal portions removed

Two equal intersecting circles each lose an equal portion, each carrying a third of its circumference and a quarter of its diameter, because the circles touch one another at the centre with their circumference. Leonardo restates the Euclidean axiom that if equals are taken from equals the remainders are equal.

A sickle-shaped lune split into a squarable half

A lune of equal concave and convex curvature is described as being of double curvature in quantity. A related sickle-shaped (falcata) lune a b is divided into two equal parts, of which one is a square.

Pulley labelled b a

Among the geometrical figures Leonardo draws a pulley (carrucola) marked with the letters b a, and lower on the sheet refers to two pulleys beneath which a further circle is set out. The mechanism sits amid the quadrature diagrams rather than in a machine context.

Facciata destra. Grosso cerchio colle lettere:

Triangolo colle lettere:

Angolo inferiore destro, figura colle lettere:

Facciata sinistra, in alto, lunola:

Altra lunola con:

Sotto le precedenti, figura con:

Carrucola colle lettere:

Dall'alto lungo il margine sinistro, due cerchi secanti:

Figura colle lettere:

In basso, sotto le due carrucole, cerchio con:

A sinistra del precedente, settore con:

Sotto il precedente, lunula con:

A sinistra del precedente, settore con:

Right side. Large circle with letters:

Triangle with letters:

Lower right corner, figure with letters:

Left side, above, lune:

Another lune with:

Below the preceding, figure with:

Pulley with letters:

From the top along the left margin, two intersecting circles:

Figure with letters:

Below, under the two pulleys, circle with:

To the left of the preceding, sector with:

Below the preceding, lune with: