Transmuting circle portions; the equilateral triangle

6 times 4 makes 24; 6 times 14 makes 84

Leonardo multiplies to count the portions of a circle: six times four gives 24 portions for the six greatest portions of the first figure, and six times fourteen gives 84. He thereby 'transmutes' the 6 portions of the first figure into 84 portions similar and equal to those of the fourth.

Transmutation of a number of portions into any other

By the second, third and fourth rules a given number of equal portions may be changed into any other number desired, and the same rule serves across the variety of circles and any other plane figure. He notes one can even change an even number into an odd number.

As parts grow in number they shrink in figure

Of a given continuous quantity divided into equal parts, the parts diminish as much as they grow in number. A given portion of a circle may be multiplied or diminished into any number one wishes without changing its figure.

Constructing the equilateral triangle e f m and its circle

Having drawn the parallel d c in the square a b c d, make an equilateral triangle e f m with sides equal to a side of the square; then, opening the compass to one side, set a point on the lower angle m and strike the circle e f S, which receives in its circumference the six greatest portions and holds exactly 84.

Definition of the equilateral triangle a b c

Each of the triangle's three sides is the semidiameter of one same circle and enters six times into that circle's circumference, as the triangle a b c demonstrates in itself.

Lungo il margine destro. Figura di cerchio con esagono iscritto:

Quadrato diviso in quattro rettangoli:

Quadrato diviso in quattordici rettangoli:

Trasformazione di un rettangolo in quadrato:

Unione delle figure seconda e terza:

Dal margine superiore:

Colonna centrale. Cerchio, quadrato della proporzionalità, con:

Trasformazione di rettangolo in quadrato, con:

Triangolo, quadrato, esagono entro un cerano, con

Colonna sinistra.

Figura per la costruzione del triangolo equilatero:

Along the right margin. Figure of a circle with an inscribed hexagon:

Square divided into four rectangles:

Square divided into fourteen rectangles:

Transformation of a rectangle into a square:

Union of the second and third figures:

From the upper margin:

Central column. Circle, square of proportionality, with:

Transformation of a rectangle into a square, with:

Triangle, square, hexagon within a circle, with:

Left column.

Figure for the construction of the equilateral triangle: