Squaring the Circle: Lunes and Inscribed Figures

Filling the four lunes to form an equal square

The four portions fill the eight portions of the four lunes and make an 'empty square' that is declared equal to the greatest square of the greatest circle. It is Leonardo's recurring device for reducing a circle to an equal rectilinear area by rearranging its curved pieces.

Circle-to-circle as square-to-square on diameters

With the numbers 2, 4 and the root of 20, the note states that the proportion from circle to circle equals that from square to square built on the circles' diameters. By this proposition of the Elements one may assign any proportion of a circle, whether rational or irrational.

Half-portion equals sector between parallels

The half-portion abcd equals the sector adbe, because triangles abc and abe stand between the parallel lines ab and dc and upon the same base ab. Leonardo cites the geometrical Elements for the equality of triangles on one base between parallels.

Squaring a rectangle with side e c

For a b times b d he reckons 2 times 6 makes 12; adding the square of b c makes 16, which is also the square of e c on itself. Therefore e c squares the parallelogram b d, of which the square on e c is one side.

A body falling on a slope makes no rebound

Along the lower margin a single maxim states that a thing falling upon a sloping (declining) surface takes no reflected motion. It sits apart from the geometry as a note on the mechanics of impact.

Figura con:

Figura di cerchio con 4 bisangoli:

Figura di cerchio con un bisangolo:

Figura con:

Figura colla lettera:

Figura sotto la precedente:

A destra, figura con:

Figura di cerchio con scudo:

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Figura con:

Figura sottostante le precedenti:

Figura:

Figura:

Figura colle lettere:

Lungo il margine inferiore:

Figure with:

Figure of a circle with 4 bi-angles (lenses):

Figure of a circle with one bi-angle:

Figure with:

Figure with the letter:

Figure below the preceding one:

On the right, figure with:

Figure of a circle with a shield:

Figure with:

Figure with:

Figure below the preceding ones:

Figure:

Figure:

Figure with the letters:

Along the lower margin:

Squaring the Circle: Lunes and Inscribed Figures

On this page

Filling the four lunes to form an equal square

Circle-to-circle as square-to-square on diameters

Half-portion equals sector between parallels

Squaring a rectangle with side e c

A body falling on a slope makes no rebound