Studies in squaring lunes: crescents made rectilinear

Squaring a crescent into triangle and square (a, o, t)

The crescent a fills the portion o, leaving the triangle o t; the o then fills the concavity lying above it, so that n is left as a square. It is a step-by-step reduction of a curved lune to a rectilinear area by exchanging equal portions.

Reducing a crescent to a rectilinear figure (a, b, c)

Starting from the crescent b c with the portion a added, Leonardo forms the half-portion a b c, then removes the value of portion a by the quantity b. What remains, a c, is rectilinear and equal to the original crescent.

Two lunes from quarter-circles in double ratio (b, d, c, a)

The caption states that d b is worth a c, and b c, and a d, because the two portions a b c d and a c are born of the quarter of two circles, one double the other. An adjacent figure (a, n, m) argues a similar equality from a circle quadruple to another.

Filling portions to make two squares

Fill the two smaller whole portions with the larger whole portion, and an elongated tetragon (rectangle) is produced, from which two squares can be made. The instruction turns a set of lune-portions into a squarable rectilinear area.

Corner column of arithmetical calculations

The upper-left corner carries a separate block of figures worked in columns, including 12, 12, 1, 160 and 53 1/3, together with 24, 33, 3, 140, 144 and 16. It is unrelated numerical reckoning set apart from the geometric rows.

Dalla destra, seconda colonna, figura biffata con:

Due piccole figure a sinistra della precedente:

Terza colonna, figura biffata con:

Quarta colonna, figura con:

Colonna quinta, figura biffata con

Figura biffata con:

Figura biffata con

Figura biffata con

Colonna sesta, figura biffata con:

Figura biffata con:

Figura biffata con:

Colonna settima, figura biffata:

Colonna ottava, prima figura:

Figura biffata con:

Figura biffata con:

Colonna nona, figura di falcate con:

Figura analoga con:

Colonna decima, figura biffata con:

Figura biffata con:

Figura biffata con:

Colonna undecima, figura biffata con:

Figura biffata con:

Figura biffata con:

Figura biffata con:

Colonna dodicesima, figura biffata con:

Angolo superiore sinistro, operazioni aritmetiche:

Right side, second column, figure struck through, with:

Two small figures to the left of the preceding:

Third column, figure struck through, with:

Fourth column, figure with:

Fifth column, figure struck through, with:

Figure struck through, with:

Figure struck through, with:

Figure struck through, with:

Sixth column, figure struck through, with:

Figure struck through, with:

Figure struck through, with:

Seventh column, figure struck through:

Eighth column, first figure:

Figure struck through, with:

Figure struck through, with:

Ninth column, figure of crescents, with:

Analogous figure, with:

Tenth column, figure struck through, with:

Figure struck through, with:

Figure struck through, with:

Eleventh column, figure struck through, with:

Figure struck through, with:

Figure struck through, with:

Figure struck through, with:

Twelfth column, figure struck through, with:

Upper-left corner, arithmetical operations:

On this page

Squaring a crescent into triangle and square (a, o, t)

Reducing a crescent to a rectilinear figure (a, b, c)

Two lunes from quarter-circles in double ratio (b, d, c, a)

Filling portions to make two squares

Corner column of arithmetical calculations