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© Veneranda Biblioteca Ambrosiana, Milan

Colonna destra. Costruzione del triangolo equilatero, con:

a b

Sia edificato un triangolo equilatero sopra la linia a b.

Costruzione di un quadrato, con:

f c - d - e -ba
Sopra la linia a b farai uno quadrangolo in questa forma. Poni ovvero ferma la punta de la sesta sopra il punto a e tira il circulo c b. Poi poni la punta del compasso nel punto b e fa la linia a d, e dividi la parte d b per metà nel punto e, e una d'esse metà poni in c d e arai la linia c a simile a ab, e poi fa la croce⁴ f.

Costruzione di un pentagono, con:

e f

Ancora farai sopra la linia e f un pentangolo equilatero e d'equali angoli.

Costruzione d'un esagono, con:

g h Farai un seangulo¹ sopra la linia g h.

Colonna sinistra. Triangolo circoscritto a pentagono, con:

r – d e - Sm -a -b -pKhq -cfn

Questa figura insegna fare il triangolo equilatero di fori del pentangolo, come il pentangolo dentro al triangolo. E se voli fare il triangolo di fori del pentangolo, piglia la linia e d e farai su il triangolo e r d. Di poi seguita la linia r d, tanto che pervenghi a c, e così fa che la linia r e capiti in n insino alla linia n c.

Per fare uno pentangolo in uno triangolo equilatero. Trova prima il mezzo, che fia dove s'intersegano le 2 linie, cioè m c, n S, le quali linie nascano nel mezzo delle facce e finiscano nelli angoli. Di poi dividi la linia a c in 6 parti equali. Di poi tira la linia f b al² fine de la prima parte b e dove termina in d, lì principia la grandezza delle facce del pentangolo e finisce in r. Adunq[ue] lascia indirieto il triangolo equilatero e r d e vieni innanzi col tuo pentangolo f q e d p, il quale farai in questa forma. Apri il sesto quanto è lo spazio e d e ferma l'una de le punte sopra d e fa nel punto p un poco di segno. Di poi ferma la punta del sesto in f e co' l'altra punta intersega un'altra volta in p, e in su la intersegazione de' 2 segni fia l'angolo p; e questo medesimo modo terrai in e q f; po' guarda se 'l circulo passa per tutti li angoli e fia giusto, se tutti li tocca. E sappi che h p è simile³ a e d e così K q è simile a d e e a d K.

Costruzione del triangolo massimo entro pentagono, con:

f-56-tS-m-ba
Se voli trovare il maggiore triangolo che sia in un pentangolo, dividi la linia ab in 2 parti equ[a]li e leva una delle parti in a m, e apri il sesto da f a, m, e fa la linia S t m. Po' poni il sesto in S e fa f t. Di poi poni il sesto in t e fa la linia f S.

Costruzione di un quadrato, con:

c d

Sia fatto sulla linia a d un quadrangolo che contenga in sé 4 d'esse linie.

Sul margine lacerato.

Se metterai i 5/6 dell... su per la linia c d ... [so]pra c d lo spazio ... del pentangolo ai s...⁵

Right column. Construction of the equilateral triangle, with:

a b. Let an equilateral triangle be built upon the line a b.

Construction of a square, with:

f c – d – e – b a. Upon the line a b you will make a square in this manner. Place, or rather fix, the point of the compass upon the point a and draw the circle c b. Then place the point of the compass on point b and make the line a d, and divide the part d b in half at the point e, and set one of those halves at c d, and you will have the line c a equal to a b; and then make the cross f.

Construction of a pentagon, with:

e f. Again you will make upon the line e f a pentagon that is equilateral and of equal angles.

Construction of a hexagon, with:

g h. You will make a hexagon upon the line g h.

Left column. Triangle circumscribed about a pentagon, with:

r – d e – S m – a – b – p K h q – c f n. This figure teaches how to make the equilateral triangle outside the pentagon, as [it does] the pentagon inside the triangle. And if you wish to make the triangle outside the pentagon, take the line e d and make upon it the triangle e r d. Then continue the line r d until it reaches c, and likewise make the line r e arrive at n as far as the line n c. To make a pentagon within an equilateral triangle: first find the middle, which will be where the two lines intersect, that is m c and n S, which lines spring from the middle of the faces and end at the angles. Then divide the line a c into 6 equal parts. Then draw the line f b to the end of the first part b, and where it ends at d, there begins the length of the faces of the pentagon and it finishes at r. Therefore leave behind the equilateral triangle e r d and come forward with your pentagon f q e d p, which you will make in this manner. Open the compass as much as the space e d and fix one of the points upon d and make a little mark at the point p. Then fix the compass point at f and with the other point intersect once more at p, and at the intersection of the two marks will be the angle p; and this same method you will keep at e q f; then see whether the circle passes through all the angles, and it will be correct if it touches them all. And know that h p is equal to e d, and likewise K q is equal to d e and to d K.

Construction of the largest triangle within a pentagon, with:

f – 5 6 – t S – m – b a. If you wish to find the largest triangle that may be in a pentagon, divide the line a b into 2 equal parts and carry one of the parts to a m, and open the compass from f to a and to m, and make the line S t m. Then place the compass at S and make f t. Then place the compass at t and make the line f S.

Construction of a square, with:

c d. Let there be made upon the line a d a square that contains within itself 4 of those lines.

On the torn margin.

If you set the 5/6 of the… up along the line c d … above c d the space … of the pentagon you have s…

Constructions of triangle, square, pentagon and hexagon

Regular polygons built on given lines, with a pentagon inscribed in and circumscribed by a triangle

Geometry

A dense geometry sheet in two columns. The right column steps through building an equilateral triangle, a square, a pentagon and a hexagon each on a given base line, while the right-hand drawings pair pointed-arch and polygon figures. The left column tackles the harder problems of circumscribing an equilateral triangle about a pentagon, inscribing a pentagon within a triangle, and finding the largest triangle that fits inside a pentagon, all worked with compass steps and lettered points. A torn left margin has carried away part of a further note involving the fraction 5/6 along line c d.

The long left-column note teaches both making an equilateral triangle e r d outside the pentagon and inscribing a pentagon inside a triangle by first finding the centre where lines m c and n S intersect, then dividing a c into 6 equal parts. Points p, q, K, h, f and the check that the circle passes through every angle are all worked out. It notes h p equals e d and K q equals d e.

Largest triangle contained in a pentagon

To find the greatest triangle that fits inside a pentagon, halve line a b, carry one part to a m, open the compass from f to a and m to draw line S t m, then draw f t from S and f S from t. Labels f, 5, 6, t, S, m, b, a locate the steps.

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

Colonna destra. Costruzione del triangolo equilatero, con:

Costruzione di un quadrato, con:

Costruzione di un pentagono, con:

Costruzione d'un esagono, con:

Colonna sinistra. Triangolo circoscritto a pentagono, con:

Costruzione del triangolo massimo entro pentagono, con:

Costruzione di un quadrato, con:

Sul margine lacerato.

Right column. Construction of the equilateral triangle, with:

Construction of a square, with:

Construction of a pentagon, with:

Construction of a hexagon, with:

Left column. Triangle circumscribed about a pentagon, with:

Construction of the largest triangle within a pentagon, with:

Construction of a square, with:

On the torn margin.

Constructions of triangle, square, pentagon and hexagon

On this page

Equilateral triangle on the line a b

Square raised on a b with the compass

Regular pentagon on line e f

Hexagon on line g h

Triangle circumscribed about a pentagon, pentagon within a triangle

Largest triangle contained in a pentagon