Triangles, circles and the rule of three for irrational lines

Transmuting a triangle's angle without changing its area

He wants to transmute the angle of a triangle to the thickness of a given angle in such a way that the triangle neither increases nor diminishes its quantity (area).

Thickening angle a b c and its diminishing axis

He thickens the angle a b c by the whole irrational quantity a d b, then seeks an irrational line that shows how much its axis diminishes.

Rule of three for irrational lines, with doubled circle-sectors

Beside portions of concentric circles labelled n, m, b, c he notes the 'rule of three' for irrational lines; elsewhere sectors a b c d of one circle are 'successively double one to another', and a equals the sector b c of a doubled circle.

Cutting a space in an angle equal to a given triangle

From an acute angle of indefinite sides he cuts a space equal to a given triangle of an obtuse or right angle; conversely, from a space enclosed in an obtuse angle of indefinite sides he draws a quantity equal to a given acute triangle.

Arithmetic working: 400 by 100 gives 40000

A block of figures sets out the multiplication of 400 by 100, carrying zeros through the columns to reach 40000.

Bird with outstretched wings amid flames

At the upper left is a finely drawn bird with spread wings rising from flames or foliage, an emblematic, phoenix-like creature; the transcription records only that a 'geometrical rule' is written above this figure of the bird.

Lungo il margine destro. Triangolo colle lettere:

Cinque porzioni di cerchi concentrici. Figura colle lettere:

Triangolo di proporzione, con:

A sinistra della figura precedente.

Figura con:

Margine superiore. Triangolo con:

Cerchio, tangenti e secanti, con :

Presso il centro, tra figure di angoli e triangoli:

In basso, presso la figura con :

Margine superiore a sinistra, figure con:

Sopra la figura dell'uccello.

Tre figure colle lettere:

Triangolo acutangolo e angolo ottuso, con:

Quattro settori di cerchio, con:

Operazioni aritmetiche.

Along the right margin. Triangle with letters:

Five portions of concentric circles. Figure with letters:

Triangle of proportion, with:

To the left of the preceding figure.

Figure with:

Upper margin. Triangle with:

Circle, tangents and secants, with:

Near the centre, among figures of angles and triangles:

Below, near the figure with:

Upper left margin, figures with:

Above the figure of the bird.

Three figures with the letters:

Acute-angled triangle and obtuse angle, with:

Four sectors of a circle, with:

Arithmetic operations.