Squaring the circle: hexagons, bi-angles and proportional circles

Hexagon and circle in double proportion (2nd demonstration)

Beside a double-circle figure marked '2' Leonardo writes that it equals the greatest hexagon of the greatest circle, and that the two circles stand in double proportion.

Six bi-angles worth the six portions of the circle

A figure marked '3' repeats the hexagon-in-circle relation and states that its six bi-angles, set upon the circumference of the sub-double circle, are worth the six portions of the greatest circle.

Square on a chord: portions in quadruple proportion

A square built on the chord of a segment is divided into four squares and four portions: the greatest portion equals the four smallest portions, just as the great square equals its four smallest squares, and the circles drawn on the semidiameters of the chords are in quadruple proportion.

Reducing a square to an equal-sided octagon

For a double circle f inscribed in an octagon within a square, Leonardo directs: reduce this square into an octagon of equal sides, its four concavities a b c d e h g f filled with the value of the twelve bi-angles of the central star, that is, the twenty-four portions.

Multiplication 14 x 6

In a lower corner, after five small sketches (one crossed out), a short multiplication reads 14 x 6 = 84.

Perspective construction in the left margin

The left margin carries a one-point perspective construction in red chalk, its orthogonals converging to a single vanishing point.

Dall'alto in basso Figura di doppio cerchio contrassegnata con:

Figura di doppio cerchio contrassegnata con:

Figura di quadrato costruito sulla corda di una porzione, diviso in quattro quadrati e quattro porzioni:

Sopra la figura di doppio cerchio contrassegnata con:

Figura di doppio quadrato con due porzioni e sei rettangoli. Altra figura di doppio quadrato con porzione massima e divisione in sei rettangoli, uno dei quali trasformato in quadrato con porzione minima, e le lettere alfabetiche:

Figura di doppio cerchio simile alla figura f ma iscritto in un ottagono, iscritto a sua volta in un quadrato, colle lettere alfabetiche in senso orario:

A destra nell'angolo inferiore dopo i cinque piccoli disegni, uno dei quali cancellato, la moltiplicazione:

From top to bottom. Figure of a double circle marked with:

Figure of a double circle marked with:

Figure of a square constructed on the chord of a portion, divided into four squares and four portions:

Above the figure of a double circle marked with:

Figure of a double square with two portions and six rectangles. Another figure of a double square with the greatest portion and division into six rectangles, one of them transformed into a square with the smallest portion, and the alphabetical letters:

Figure of a double circle like figure f but inscribed in an octagon, itself inscribed in a square, with the alphabetical letters clockwise:

On the right in the lower corner, after the five small drawings (one of which is crossed out), the multiplication: