Definitions on circles in double proportion and squaring the circle

Circles in double proportion: quarter of the lesser equals eighth of the greater

First definition: of circles in double proportion, the quarter of the smaller equals the eighth of the greater, so the marked parts a and b are equal (two circles divided into four and eight parts).

Equal parts removed leave equal remainders

Second definition: if from equal things equal parts are taken away, the remainders are equal; removing part o from o n and o m leaves n and m equal. And of circles in double proportion, half the greater equals the whole of the smaller.

Moving a portion preserves the quantity

Taking the half-portion d from part c a and giving it to b leaves a c equal to d b; the half-portion d equals the whole portion c, being half of a portion double to c (curvilinear triangles d c - b a).

A construction toward squaring the circle

If a is made squarable and b is itself a square, the whole a b is a square; since it equals c, and c is the fourth part of the square from which the circle is removed, subtracting that square from the greater square leaves a remainder equal to the circle.

Ratios of circles in double proportion

The definitions rest on numerical ratios between concentric or paired circles standing in double proportion, matching quarters to eighths and halves to wholes across the figures.

Dalla destra in alto. Due cerchi divisi rispettivamente in quattro e otto parti:

Lungo il margine, figure colle lettere:

Cerchio e semicerchio sovrapposti colle lettere:

Settore e quadrante di cerchio sovrapposti:

Triangolo curvilineo colla lettera:

Lungo il margine superiore dal centro a sinistra. Triangolo con:

Triangolo curvilineo con:

Figura analoga con:

Ripetizione della figura dell'angolo superiore sinistro:

Tre figure colle lettere:

Upper right. Two circles divided respectively into four and eight parts:

Along the margin, figures with letters:

A circle and semicircle superimposed, with letters:

A sector and quadrant of a circle superimposed:

A curvilinear triangle with the letter:

Along the upper margin, from center to left. A triangle with:

A curvilinear triangle with:

An analogous figure with:

A repetition of the figure at the upper left corner:

Three figures with letters: