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© Veneranda Biblioteca Ambrosiana, Milan

Sul margine superiore:

Regola del prestare e rendere.

Prima figura in alto a destra:

Primo. Falsa.

Andando verso sinistra. Figura colla lettera:

a co' sua aderenti¹ vale il massimo quadrato del massimo circolo.

Cerchio colle lunule:

a b

a b lunole vagliano il massimo quadrato del massimo circolo. Provasi, perché le due porzioni vagliano le 4 massime porzioni del massimo circolo.

A destra, figura con:

a b c d

a b c d vagliano il massimo quadrato del circolo. E perché d è quadrato in sé col riempiere la parte concava colla convessa che sta dalla sua parte opposita, ora tu trarrai d dalla compagnia del a b c, sua aderenti, e sarà la sua quadratura, come si dimostra qui da lato². Ogni quadratura fatta per questa regola ha due modi vari di quadrature almanco.

Verso sinistra, tre figure colle lettere:

A a b c –b d e f - c

a b c vale il quadrato d e f, perché li due vacui del a b c son della valuta che hanno la metà delli 4 biangoli³. Adunque riempi esso vacuo colla metà della larghezza che hanno essi biangoli e arai fatto il quadrato d e f. Le 3 superfizie a b c son d'egual quantità in varie figure. E quella di mezzo è la lor quadratura.

Figura colla lettera:

d vale 1/2 b.

Figura colla lettera:

e è la quadratura di d.

Figura colla lettera:

f vale 1/2 e.

Figura colle lettere:

n g

n è la quadratura dell'f

Figura sul margine sinistro:

Vale il massimo quadrato del massimo cerchio.

Tornando a destra, lungo il margine, figura con quadrato iscritto:

Qui è di tre sorte porzioni, delle quali la massima ha in corpo il diamitro del cerchio, la mezzana è dipennata, la minima è in corpo⁴ alla massima. Adunque la massima vale due mezzane e 4 minime; onde rendendo io due minime alla massima, essa resta di valuta d'una mezzana. Adunque essendo in due mezzane de la valuta di due bisangoli, resta le massime in valuta di 2 altre mezzane. Adunque a tal massima tranne 4 mezzane, e per questo el rimanente del circolo resta quadrato.

Figura a sinistra della precedente:

Se di quadrato trarrai quadrabile, el rimanente fia quadrabile.

Sotto la precedente:

Se di quadrabile trarrai quadrato, el rimanente è quadrabile.

Due figure legate da un tratto di penna:

ab -equali -ab
ab vale il massimo quadrato del massimo circolo.
ab vale a b di stanca⁵, e l'uno e l'altro per sé vale il massimo quadrato del massimo circolo.

In basso da destra a sinistra, figura con:

Vale il quadrato del massimo circolo.

Cerchio iscritto e circoscritto a quadrati:

Se 'l quadrato che tocca 'l circolo co' sua lati è duplo al quadrato che 'l tocca co' sua angoli.

Quadrato entro quadrato:

Quel quadrato che co' sua 4 lati tocca li 4 anguli dell'altro quadrato, sarà duplo uno dell'altro.

Triangolo entro triangolo:

Li triangoli, de' quali li 3 lati dell'uno son contin[gen]ti alli 3 lati dell'altro, saran quadrupli⁶ l'uno all'altro.

Along the upper margin:

Rule of lending and returning.

First figure, upper right:

First. False.

Moving to the left. Figure with the letter:

a — a with its adjoining parts is worth the greatest square of the greatest circle.

Circle with the lunules:

a b — the a b lunules are worth the greatest square of the greatest circle. It is proved, because the two portions are worth the four greatest portions of the greatest circle.

To the right, figure with:

a b c d — a b c d are worth the greatest square of the circle. And because d is squarable in itself by filling the concave part with the convex one on its opposite side, now you will take d away from the company of a b c, its adjoining parts, and it will be its quadrature, as is shown here to the side. Every quadrature made by this rule has at least two different modes of squaring.

To the left, three figures with the letters:

A: a b c – b d e f – c. a b c is worth the square d e f, because the two voids of a b c have the value of half of the four lens-shapes (biangles). Therefore fill that void with half the width those biangles have, and you will have made the square d e f. The three surfaces a b c are of equal quantity in various shapes. And the middle one is their quadrature.

Figure with the letter:

d — d is worth 1/2 b.

Figure with the letter:

e — e is the quadrature of d.

Figure with the letter:

f — f is worth 1/2 e.

Figure with the letters:

n g — n is the quadrature of f.

Figure on the left margin:

It is worth the greatest square of the greatest circle.

Returning to the right, along the margin, figure with an inscribed square:

Here there are three kinds of portions, of which the greatest contains the diameter of the circle, the middle one is hatched, and the smallest is contained within the greatest. Therefore the greatest is worth two middle ones and four small ones; whence, returning two small ones to the greatest, it remains worth one middle one. Therefore, since two middle ones have the value of two biangles, the greatest ones remain worth two other middle ones. Therefore from such a greatest take away four middle ones, and by this the remainder of the circle remains squared.

Figure to the left of the preceding:

If from a square you take away something squarable, the remainder will be squarable.

Below the preceding:

If from something squarable you take away a square, the remainder is squarable.

Two figures joined by a pen-stroke:

a b – equal – a b. a b is worth the greatest square of the greatest circle. a b is worth the a b on the left, and each one by itself is worth the greatest square of the greatest circle.

Below, from right to left, figure with:

It is worth the square of the greatest circle.

Circle inscribed in and circumscribed about squares:

The square that touches the circle with its sides is double the square that touches it with its corners.

Square within a square:

That square which with its four sides touches the four corners of the other square will be double, one of the other.

Triangle within a triangle:

The triangles of which the three sides of one are tangent to the three sides of the other will be quadruple, one to the other.

Squaring the circle: lunules and equivalent surfaces

The rule of 'lending and returning' areas; squares, circles and lunes compared

Geometry

A dense sheet of quadrature studies working out Leonardo's 'rule of lending and returning' — a method of transforming curved figures (lunules and circular portions) into equivalent squares. He repeatedly asserts that a given arrangement of lunules 'is worth the greatest square of the greatest circle,' proving each claim by showing that two curved portions equal four greatest portions of the circle. Further figures compare a square inscribed in a circle with one circumscribed about it (double in area) and nested triangles (quadruple), and he notes that every quadrature made by this rule admits at least two solutions.

The configuration a b c d is worth the greatest square of the circle. Because d is squarable in itself (filling its concave part with the convex from the opposite side), you take d away from its companions a b c to obtain the quadrature. Every quadrature made by this rule has at least two different solutions.

Lunules worth the greatest square of the circle

The lunules a b are worth the greatest square of the greatest circle. This is proved because the two portions are worth the four greatest portions of the greatest circle.

Three equal surfaces a b c and the square d e f

a b c is worth the square d e f, because the two voids of a b c equal half of the four lens-shapes (biangles); fill that void with half their width to obtain d e f. The three surfaces a b c are of equal quantity though in various shapes, and the middle one is their quadrature.

Square inscribed versus circumscribed on a circle

The square that touches the circle with its sides is double the square that touches it with its corners. It is stated as a rule that any square whose four sides touch the four corners of another square is double it.

Nested triangles are quadruple in area

The triangles of which the three sides of one are tangent to the three sides of the other will be quadruple, one to the other.

Adding and subtracting squarable areas

If from a square you take away a squarable part, the remainder will be squarable. If from a squarable area you take away a square, the remainder is squarable.

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

Sul margine superiore:

Prima figura in alto a destra:

Andando verso sinistra. Figura colla lettera:

Cerchio colle lunule:

A destra, figura con:

Verso sinistra, tre figure colle lettere:

Figura colla lettera:

Figura colla lettera:

Figura colla lettera:

Figura colle lettere:

Figura sul margine sinistro:

Tornando a destra, lungo il margine, figura con quadrato iscritto:

Figura a sinistra della precedente:

Sotto la precedente:

Due figure legate da un tratto di penna:

In basso da destra a sinistra, figura con:

Cerchio iscritto e circoscritto a quadrati:

Quadrato entro quadrato:

Triangolo entro triangolo:

Along the upper margin:

First figure, upper right:

Moving to the left. Figure with the letter:

Circle with the lunules:

To the right, figure with:

To the left, three figures with the letters:

Figure with the letter:

Figure with the letter:

Figure with the letter:

Figure with the letters:

Figure on the left margin:

Returning to the right, along the margin, figure with an inscribed square:

Figure to the left of the preceding:

Below the preceding:

Two figures joined by a pen-stroke:

Below, from right to left, figure with:

Circle inscribed in and circumscribed about squares:

Square within a square:

Triangle within a triangle:

On this page

The rule of 'lending and returning' to square a figure

Lunules worth the greatest square of the circle

Three equal surfaces a b c and the square d e f

Square inscribed versus circumscribed on a circle

Nested triangles are quadruple in area

Adding and subtracting squarable areas