Squaring circles by portions and curvilinear figures

Portions of circles in quadruple proportion

The 24 portions of the first figure are worth the 4 greatest portions of the second, because the smaller portions are one quarter of the larger and alike in shape, each made from a fourth of its circumference. Since the circles stand in quadruple proportion, so do their portions, and removing them leaves the square remaining in each figure equal.

Curvilinear triangles equal to two bi-angles

The curvilinear triangles a b are worth the two curvilinear bi-angles d e. Leonardo proves it: the curvilinear triangle a c b is worth a quarter of its whole greatest circle, and the circle c d e is worth the same; removing the curvilinear triangle c from each leaves a b equal to the two bi-angles d e.

Circles in double and sextuple proportion

Two brief rules govern nested circles: of circles in sextuple proportion, half of the smaller is worth a sixth of the larger; of circles in double proportion, a quarter of the larger is worth half of the smaller. They convert the concentric-circle diagrams into simple fractional statements.

Squaring by removing equal portions

At the lower right stands the common notion in bare form: if from equal things you take away equal parts, the remainders are equal. This is the principle that licenses every claim on the sheet that a hatched or 'squarable' region equals a given square once matching portions are struck out.

In alto da destra verso sinistra. Cerchio connesso con una piccola figura sottostante:

Cerchio con quadrato:

Due cerchi con:

A destra della precedente, figura con:

Sotto la precedente, due segmenti curvi:

Quattro piccoli cerchi in un cerchio, con:

Sotto, in basso, semicerchio con:

Angolo inferiore destro:

A metà della colonna sinistra, figura con:

Quadrato e quattro cerchi in un cerchio:

Sotto il precedente tre cerchi concentrici:

D'altra mano:

Top, from right to left. Circle connected with a small figure below:

Circle with a square:

Two circles with:

To the right of the preceding, figure with:

Below the preceding, two curved segments:

Four small circles within a circle, with:

Below, at the bottom, semicircle with:

Lower right corner:

Halfway down the left column, figure with:

Square and four circles within a circle:

Below the preceding, three concentric circles:

In another hand:

Squaring circles by portions and curvilinear figures

On this page

Portions of circles in quadruple proportion

Curvilinear triangles equal to two bi-angles

Circles in double and sextuple proportion

Squaring by removing equal portions