Squaring the Circle: Lunes and 'Circles with a Foot'

Reducing lunes and 'circles with a foot' to squares

Taking n q r m as half the greatest circle, removing the two greatest portions n m leaves a squarable q r; adding the four triangles t, S, X, L completes the quadrilateral a b c d. Removing the squarable parts and the two curvilinear triangles leaves the two circles o p together with the four triangles, equal to four squares similar to the squares o p. It follows that the two circles below are worth four such squares, and that square f equals circle X with its foot t.

Two lunes equal to the greatest square of the greatest circle

Within circle a b c the two-angled lens surface c is removed, leaving the two squarable lunes whose square equals the greatest square of the greatest circle. This holds because the two-angled figure c is composed of two greatest portions of a circle double the circle a h e.

The two greatest inscribed circles equal half the circle

A marginal rule states that the two largest circles that can be made within a circle are worth half of that circle, and that these are the ones whose diameters together make up the whole diameter of the greatest circle.

Circle-to-circle as the squares of their diameters

The proportion from circle to circle is the same as from square to square built by the multiplication of their diameters; in a paired case a b is stated to be worth c d.

Balusters: field equal to solid

A final figure of balusters carries the rule that the empty field is worth as much as the solid body of the balusters, extending the area-equivalence game to a decorative profile.

Facciata destra. In alto da destra a sinistra. Cerchio con:

Figura: Seconda

Figura:

Due ' cerchi col piede ' e un quadrato, con:

Lungo il margine destro:

Due cerchi entro cerchio, con quadrati iscritti:

Due cerchi entro cerchio, con:

Cerchio in semicerchio, con:

Doppia figura, ripresa dalla figura fondamentale:

Facciata sinistra. Figura con:

Figura con:

Figura con:

Figura con:

Cerchi entro rettangolo. Cerchio iscritto in un quadrato. ' Cerchio col piede ' entro rettangolo, con:

Altro ' cerchio col piede ', con:

Due ' cerchi col piede ', con:

Figure analoghe, con:

Figura di ' balausti ':

Right-hand page. Top, right to left. Circle with:

Figure: Second

Figure:

Two 'circles with a foot' and a square, with:

Along the right margin:

Two circles within a circle, with inscribed squares:

Two circles within a circle, with:

Circle within a semicircle, with:

Double figure, taken up again from the fundamental figure:

Left-hand page. Figure with:

Figure with:

Figure with:

Figure with:

Circles within a rectangle. Circle inscribed in a square. 'Circle with a foot' within a rectangle, with:

Another 'circle with a foot', with:

Two 'circles with a foot', with:

Analogous figures, with:

Figure of 'balusters':

On this page

Reducing lunes and 'circles with a foot' to squares

Two lunes equal to the greatest square of the greatest circle

The two greatest inscribed circles equal half the circle

Circle-to-circle as the squares of their diameters

Balusters: field equal to solid