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© Veneranda Biblioteca Ambrosiana, Milan

Lungo il margine superiore da destra a sinistra. Sei tipi di superfici:

Superfizie unicangolare1.
Superfizie triangolare curvilatera.
Bisangola.
Qu[a]drangola.
Lunare.
Circol paralello.

Figura con:

a -b

a b è la massima porzione del massimo circolo.

Porzione divisa in quattro parti, con :

b a - d e Divisa in qu[a]ttro parte equali: ab e d.

Entro le due parallele:

Tal proporzione è da circolo a circolo quale è da quadrato a quadrato fatto della multiplicazion del diamitro di tal circolo in se medesimo.

Due parallelogrammi, rettilineo e curvilineo, con:

d c; b a
a b vale c d, e le linie curve della superfizie a b sono d'equale curvità infra loro e son paralelle l'una all'altra, e per questo fanno il medesimo ufizio infra loro che fa la superfizie c d infra sé.

Due cerchi contenenti cerchi minori, con:

bdca; c-b fe a-d
n m vale ef.
ab lunole vagliano ed triangoli. Pruovasi così: li cerchi minori vagliano il cerchio mezzano. Leva le parte e f alli due cerchi minori e leva el simile al circolo mezzano. Li rimanenti restano infra loro di valuta equali. Addunque e d triangoli vagliano le due lunole a b, come è detto.

Figura analoga alla precedente, con:

Prima —c — ba-d
a b lunole vagliano c d triangoli e vagliano il massimo quadrato del cerchio mezzano, cioè a b c d. Ma a b lunole vagliano il massimo quadrato del minimo circolo e 'l simile vale li due triangoli c d.

Figura nel margine destro:

Questo è diviso in se[i] parte equali.

Cerchio contenente quattro cerchi e quattro bisangoli, con:

Seconda -b-ca-d
Tanto vagliano in questo secondo circolo li quattro bisangoli, quanto li 4 triangoli curvilateri.
ab c d vagliano le otto lunole della prima di sopra; e vagliano el massimo quadrato del massimo circolo. El rimanente vale le 4 massime del massimo circolo.

Sopra la precedente, porzione divisa, con:

b a — d c

Figura con:

a b a b è la massima porzione del massimo circolo.

Centro del foglio, cerchio diviso da quattro semicirconferenze, con:

Superfizie bisangola a.

Cerchio diviso da due semicirconferenze, con:

a b

Unangola ovvero solangola.

Figura ottagonale, con:

b a — c —Quarta
ab e vale un circolo del quale s'è remosso le 4 massime del massimo circolo, le quali 4 massime sono nel sito di e bisangolo.

Insieme di 18 semicirconferenze entro un cerchio:

Falsa

Gruppo di figure, con:

Seconda-c-ab; Prima - m n; Terza - c - f b d - g
La prima n m è un circolo diviso per equale. La seconda figura dimostra la levazione2 delle 4 massime del massimo circolo situate nel bisangolo c. Nella terza si dà la quadratura b dentro alle 4 porzioni d c f g. Nella quarta si dà la composizione della quadratura di 4 circoli e la quadratura del suo campo.

Angolo inferiore sinistro. Motivo di linee sinuose entro il cerchio:

Vale la metà del massimo quadrato del massimo circolo.

Idem colla lettera:

Vale il massimo quadrato del massimo circolo.

Along the upper margin, from right to left. Six types of surface:

One-angled surface. Curvilinear-triangular surface. Two-angled. Quadrangular. Lunar. Parallel circle.

Figure with:

a - b. a b is the greatest portion of the greatest circle.

Portion divided into four parts, with:

b a - d e. Divided into four equal parts: a b and d.

Between the two parallels:

Such proportion holds from circle to circle as from square to square made by the multiplication of the diameter of such circle by itself.

Two parallelograms, rectilinear and curvilinear, with:

d c; b a. a b is worth c d, and the curved lines of the surface a b are of equal curvature among themselves and are parallel to one another, and for this reason they perform the same office among themselves as the surface c d does within itself.

Two circles containing smaller circles, with:

b d c a; c - b f e a - d. n m is worth e f. The lunes a b are worth the triangles e d. It is proved thus: the smaller circles are worth the middle circle. Remove the parts e f from the two smaller circles and remove the like from the middle circle. The remainders stay equal in value among themselves. Therefore the triangles e d are worth the two lunes a b, as has been said.

Figure similar to the preceding, with:

First — c — b a - d. The lunes a b are worth the triangles c d and are worth the greatest square of the middle circle, that is a b c d. But the lunes a b are worth the greatest square of the smallest circle, and the like is worth the two triangles c d.

Figure in the right margin:

This is divided into six equal parts.

Circle containing four circles and four two-angled figures, with:

Second - b - c a - d. In this second circle the four two-angled figures are worth as much as the four curvilinear triangles. a b c d are worth the eight lunes of the first above; and they are worth the greatest square of the greatest circle. The remainder is worth the four greatest portions of the greatest circle.

Above the preceding, divided portion, with:

b a — d c

Figure with:

a b. a b is the greatest portion of the greatest circle.

Centre of the sheet, circle divided by four semicircumferences, with:

a. Two-angled surface a.

Circle divided by two semicircumferences, with:

a b. One-angled, that is, single-angled.

Octagonal figure, with:

b a — c — Fourth. a b is worth a circle from which the four greatest portions of the greatest circle have been removed, which four greatest portions are in the place of the two-angled figure e.

A set of eighteen semicircumferences within a circle:

False.

Group of figures, with:

Second - c - a b; First - m n; Third - c - f b d - g. The first, n m, is a circle divided equally. The second figure shows the removal of the four greatest portions of the greatest circle placed in the two-angled figure c. In the third is given the squaring b within the four portions d c f g. In the fourth is given the composition of the squaring of four circles and the squaring of its field.

Lower left corner. A motif of sinuous lines within the circle:

It is worth half the greatest square of the greatest circle.

The same, with the letter:

a. It is worth the greatest square of the greatest circle.

Squaring the circle: kinds of curved surface, lunes and triangles

A catalogue of curvilinear surfaces with figures showing lunes equal to triangles and squares

Geometry

This crowded sheet is a sustained attempt at squaring the circle. Across the top Leonardo names six kinds of surface — one-angled, curvilinear-triangular, two-angled, quadrangular, lunar and 'parallel circle' — and below fills the page with circles subdivided by semicircles into lunes, two-angled figures and rosette forms. He states that circle is to circle as the square of the diameter is to the square of the diameter, and repeatedly shows that lunes are worth triangles or squares of equal value. Some figures he marks 'first', 'second', 'third' and 'fourth', while one construction of eighteen semicircumferences he flatly labels 'false'. Further mirror-script notes are present on the sheet but not transcribed here.

A group of figures is read in sequence: the first n m is a circle divided equally; the second shows removal of the four greatest portions of the greatest circle set in the two-angled figure c; the third gives the squaring b within the portions d c f g; the fourth composes the squaring of four circles and of their field.

A curved figure worth half the greatest square

In the lower left corner a motif of sinuous lines within a circle is said to be worth half the greatest square of the greatest circle. A companion figure lettered a is instead worth the whole greatest square of the greatest circle.

Reader's Edition — described with AI from the e-Leo transcription.

Lungo il margine superiore da destra a sinistra. Sei tipi di superfici:

Figura con:

Porzione divisa in quattro parti, con :

Entro le due parallele:

Due parallelogrammi, rettilineo e curvilineo, con:

Due cerchi contenenti cerchi minori, con:

Figura analoga alla precedente, con:

Figura nel margine destro:

Cerchio contenente quattro cerchi e quattro bisangoli, con:

Sopra la precedente, porzione divisa, con:

Figura con:

Centro del foglio, cerchio diviso da quattro semicirconferenze, con:

Cerchio diviso da due semicirconferenze, con:

Figura ottagonale, con:

Insieme di 18 semicirconferenze entro un cerchio:

Gruppo di figure, con:

Angolo inferiore sinistro. Motivo di linee sinuose entro il cerchio:

Idem colla lettera:

Along the upper margin, from right to left. Six types of surface:

Figure with:

Portion divided into four parts, with:

Between the two parallels:

Two parallelograms, rectilinear and curvilinear, with:

Two circles containing smaller circles, with:

Figure similar to the preceding, with:

Figure in the right margin:

Circle containing four circles and four two-angled figures, with:

Above the preceding, divided portion, with:

Figure with:

Centre of the sheet, circle divided by four semicircumferences, with:

Circle divided by two semicircumferences, with:

Octagonal figure, with:

A set of eighteen semicircumferences within a circle:

Group of figures, with:

Lower left corner. A motif of sinuous lines within the circle:

The same, with the letter:

On this page

Six kinds of surface

Circle to circle as the square of the diameter

Lunes worth triangles (a b lunes and e d triangles)

A construction marked 'false'

Four figures: division, removal, quadrature, composition

A curved figure worth half the greatest square