100%

Reset zoom

© Veneranda Biblioteca Ambrosiana, Milan

Triangolo curvilineo, con:

Sia quadrato un triangolo curvilinio dalli sua 3 lati, del quale li 2 che contengano l'un delli angoli sieno concavi, e 'l terzo che guarda tale angolo, sia convesso.

Lo stesso fatto rettilineo:

Quadro rendendo quel ch'i tolsi, ovver l'equivalente.

Triangolo curvilineo, con:

* Sia quadrato il triangolo b con 3 lati curvi, de' quali li 2 lati convessi si giunghino nell'angolo opposito al terzo lato concavo.

Quadrilatero entro triangolo curvilineo, con:

a - bd-c

Il quadrato a b d c è equale al triangolo b.

Due figure con:

b; a

a è pari al b.

Riferiti alle figure sottostanti:

Porzioni di sesti [di] cerchi¹ quadrupli l'uno

all'altro.

Porzione di dodicesimi [di] cerchi quadrupli

l'uno all'altro.

Quadrante di cerchio sopra semicerchio, con:

a -b

a b sono equali.

Ottavo di cerchio su semicerchio, con:

a - b - d -

e c f

Leva a scempio e b disdoppia, che resta b s[c]empio.

Figura analoga, con:

c — d

c d son pari.

Sette cerchi internamente tangenti, con:

a b c d e f g

a b c d e f g sono equali.

Per dividere una falcata in parte equale.

Due figure di triangoli curvilinei, con:

a - b c - d; e - d - v - a - c b - n

* Se la mezza porzione c d b è doppia a tutta la porzione a b e quadrupla alla mezza a b, elli è segno che li cerchi loro interi son quadrupli l'uno all'altro. Seguita che, essendo ciascuna intera porzione il sesto del suo cerchio, che la metà della porzione del dodicesimo del cerchio maggiore è ancora lei dupla alla porzione del dodicesimo del cerchio minore. Seguita che la parte a b d c è quadrabile col torre e render parte equali.

Porzione divisa, con:

a — d — c b

* Io vo dividere la semiporzione a b c in 2 parti equali, e farò così. I' dividerò l'arco a d c in due parte equali e farò 2 equali porzioni, a d e d c, e poi restando rettilinio il rimanente, i' risolverò in triangoli e quadre[rò]², e farò della metà un paralellogramo, e giugnerollo dove mi accaderà, e per ora sia in n m o, e sarà equale alla porzione r o, e tirerò la linia n r, e col torre e rendere arò quadrato l'ango[lo] n r o.

Triangolo curvilineo, con:

r - n - o m

Sul margine sinistro:

* Col mezzo cerchio equipera un quarto d'un altro cerchio, il quale quarto sia equale per quantità a esso mezzo, e 'l simile fa con un ottavo e un sedicesimo e un trentaduesimo e sesto³ e quinto e ogni parte d'un altro cerchio.

Foglio capovolto. Settori di cerchio sovrapposti, con:

a – b - p m L d c - o

* I' ho qui 2 superfizie equali, cioè a b d L m o per una, e b d c L m p per l'altra; e ciascun per sé ha congiunto 2 superfizie curvilinie equali tutt'e 4 infra loro. Delle quali c b⁴ dell'una e a b dell'altra son 4, perché b v'è 2 volte. Leva a dell'una e c dell'altra, restano li rimanenti dell'una superfizie e dell'altra equali. Leva b 2 volte ch'è duppio e serve all'una e all'altra, resta ...

Figura analoga alla precedente, con:

a – b – e d - c

a b sono equali per esser b porzion di cerchio subduplo al cerchio di che a b è mezza porzione.

Curvilinear triangle, with:

Let a curvilinear triangle be squared by its 3 sides, of which the 2 that contain one of the angles be concave, and the third, which faces that angle, be convex.

The same made rectilinear:

I square by rendering back what I took, or the equivalent.

Curvilinear triangle, with:

Let the triangle b be squared with 3 curved sides, of which the 2 convex sides meet at the angle opposite the third, concave side.

Quadrilateral within a curvilinear triangle, with:

a - b d - c

The square a b d c is equal to the triangle b.

Two figures with:

b; a
a is equal to b.

Referring to the figures below:

Portions of sixths of circles fourfold one to the other.
Portion of twelfths of circles fourfold one to the other.

Quadrant of a circle over a semicircle, with:

a - b

a b are equal.

Eighth of a circle on a semicircle, with:

a - b - d -; e c f
Remove a single and un-double b, so that b single remains.

Analogous figure, with:

c - d

c d are equal.

Seven internally tangent circles, with:

a b c d e f g

a b c d e f g are equal.
To divide a crescent into equal parts.

Two figures of curvilinear triangles, with:

a - b c - d; e - d - v - a - c b - n
If the half-portion c d b is double the whole portion a b and quadruple the half a b, it is a sign that their whole circles are fourfold one to the other. It follows that, each whole portion being the sixth of its circle, the half of the portion of the twelfth of the larger circle is likewise double the portion of the twelfth of the smaller circle. It follows that the part a b d c is squarable by taking and rendering equal parts.

Divided portion, with:

a - d - c b

I want to divide the half-portion a b c into 2 equal parts, and I shall do thus. I shall divide the arc a d c into two equal parts and make 2 equal portions, a d and d c, and then, the remainder being rectilinear, I shall resolve it into triangles and square it, and I shall make of the half a parallelogram, and join it where it suits me, and for now let it be at n m o, and it will be equal to the portion r o, and I shall draw the line n r, and by taking and rendering I shall have squared the angle n r o.

Curvilinear triangle, with:

r - n - o m

In the left margin:

With the half circle, equate a quarter of another circle, which quarter is equal in quantity to that half, and do the same with an eighth and a sixteenth and a thirty-second and a sixth and a fifth and every part of another circle.

Sheet upside down. Overlapping circle sectors, with:

a - b - p m L d c - o

I have here 2 equal surfaces, that is a b d L m o for one, and b d c L m p for the other; and each by itself has joined to it 2 curvilinear surfaces, all 4 equal among themselves. Of which c b of the one and a b of the other are 4, because b is there 2 times. Remove a from the one and c from the other, and the remainders of the one surface and the other remain equal. Remove b 2 times, which is double and serves both the one and the other, there remains ...

Figure analogous to the preceding, with:

a - b - e d - c

a b are equal, because b is a portion of a circle half of the circle of which a b is a half-portion.

Squaring Curvilinear Triangles, Lunes and Circle Portions

A dense sheet of quadrature studies transforming curved figures into equal rectilinear squares.

Geometry

The leaf is packed with quadrature exercises turning curved figures into equal squares and triangles. Leonardo squares curvilinear triangles with concave and convex sides, equates portions of sixths and twelfths of circles that are fourfold one another, and shows that seven internally tangent circles yield equal figures. He gives constructions for dividing a crescent and a half-portion into equal parts and for equating a semicircle to a quarter, eighth, sixteenth or any part of another circle. The Ambrosiana catalogue additionally tags this leaf with mechanical devices (a crane, winch and screw-lift) and courtly notes, though the material transcribed here is entirely geometrical; further untranscribed text may be present on the sheet.

Leonardo proposes to square a curvilinear triangle a whose two sides containing one angle are concave and whose third side, facing that angle, is convex, and then to make it rectilinear by rendering back what was taken or its equivalent. A companion triangle b has two convex sides meeting at the angle opposite a concave third side.

Portions of fourfold circles proved equal

If the half-portion c d b is double the whole portion a b and quadruple the half a b, their whole circles are fourfold one another; each whole portion being the sixth of its circle, the half-twelfth of the larger circle is double the twelfth of the smaller. It follows that the part a b d c is squarable by taking and rendering equal parts.

Dividing a half-portion into two equal parts

To halve the half-portion a b c, Leonardo divides the arc a d c into two equal portions a d and d c, resolves the rectilinear remainder into triangles and squares it, forms a parallelogram n m o equal to portion r o, draws line n r and, by taking and rendering, squares the angle n r o.

Equating a semicircle to any fraction of another circle

In the margin Leonardo notes that a half circle can be equated to a quarter of another circle equal in quantity to that half, and likewise to an eighth, a sixteenth, a thirty-second, a sixth, a fifth and every part of another circle. Seven internally tangent circles are marked as equal figures to be used for dividing a crescent into equal parts.

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

Triangolo curvilineo, con:

Lo stesso fatto rettilineo:

Triangolo curvilineo, con:

Quadrilatero entro triangolo curvilineo, con:

Due figure con:

Riferiti alle figure sottostanti:

Quadrante di cerchio sopra semicerchio, con:

Ottavo di cerchio su semicerchio, con:

Figura analoga, con:

Sette cerchi internamente tangenti, con:

Due figure di triangoli curvilinei, con:

Porzione divisa, con:

Triangolo curvilineo, con:

Sul margine sinistro:

Foglio capovolto. Settori di cerchio sovrapposti, con:

Figura analoga alla precedente, con:

Curvilinear triangle, with:

The same made rectilinear:

Curvilinear triangle, with:

Quadrilateral within a curvilinear triangle, with:

Two figures with:

Referring to the figures below:

Quadrant of a circle over a semicircle, with:

Eighth of a circle on a semicircle, with:

Analogous figure, with:

Seven internally tangent circles, with:

Two figures of curvilinear triangles, with:

Divided portion, with:

Curvilinear triangle, with:

In the left margin:

Sheet upside down. Overlapping circle sectors, with:

Figure analogous to the preceding, with:

Squaring Curvilinear Triangles, Lunes and Circle Portions

On this page

Squaring a curvilinear triangle with concave and convex sides

Portions of fourfold circles proved equal

Dividing a half-portion into two equal parts

Equating a semicircle to any fraction of another circle