Squaring the circle through the hexagon and proportionality

Squaring the circle into six equilateral triangles

This circle is divided into six equal, like sectors; removing their six circle-portions leaves a figure squarable into six equilateral triangles. The six portions may be taken from inside or outside, in their own or other shapes, always keeping the same value; if you remove six the hexagon remains, and you may remove 12, 18, 24 or any multiple of six.

As the parts multiply, their size shrinks

The number of equal, like parts increases exactly as much as the figures of those parts decrease: made twelve from six, they grow by half in number and shrink by half in size; multiplied by three, they lose two thirds of their first size, and so on to infinity. Leonardo promises a general rule from this reciprocal law.

The square of proportionality

First compose the star of six bi-angles, and around it build whatever you please with a like number of equal bi-angles, keeping six always as the divisor; then apply the 'square of proportionality'. This square, with sides equal to the hexagon's and divided into parallels, is Leonardo's device for converting between numbers and sizes of portions.

Aliquot parts: four times six makes twenty-four

The 4 parallels of the first figure are an aliquot (whole-number) part of the 6 and the 24: four times six makes twenty-four, so the 24 smallest portions of the third circle equal in value the 6 greatest portions of the first. One portion of the first, at a b, is thus worth the four portions at d c in the same triangle.

Angolo superiore destro. Piccolo settore di cerchio, colla lettera:

Cerchio diviso in sei settori:

Da sinistra verso destra. Rettangolo trasformato in quadrato:

Figura analoga:

Esagono iscritto:

Cerchio con stella bisangolare:

Presso il margine sinistro:

Cerchio con:

Brano inquadralo con tratti di penna:

Da destra verso sinistra. Esagono iscritto nel cerchio:

Rettangolo trasformato in quadrato:

Settore di cerchio colle lettere

Doppio cerchio con stella:

A sinistra:

Upper right corner. Small sector of a circle, with the letter:

Circle divided into six sectors:

From left to right. Rectangle transformed into a square:

Analogous figure:

Inscribed hexagon:

Circle with a bi-angular star:

Near the left margin:

Circle with:

Passage framed with pen strokes:

From right to left. Hexagon inscribed in the circle:

Rectangle transformed into a square:

Sector of a circle with the letters:

Double circle with a star:

On the left:

Squaring the circle through the hexagon and proportionality

On this page

Squaring the circle into six equilateral triangles

As the parts multiply, their size shrinks

The square of proportionality

Aliquot parts: four times six makes twenty-four