Geometry of circles, semicircles and their segments

A semicircle equal in quantity to a circle

A circle set upon a semicircle is labelled a - c b, with a worth b c, and Leonardo declares the semicircle equal in quantity to the circle. The same equality recurs where the shared part a b c is taken from both the semicircle and the circle, leaving equal remainders.

Equal figures minus equal hatched parts

A pair of joined figures labelled a and b are declared equal to one another; the crossed-out (hatched) portion taken from each is equal, so the remainder is equal. A companion pair repeats the rule: equals equal, and remove the remainder.

A greater semicircle double the lesser

A caption notes that below, the greater semicircle is double the smaller. A related circle figure (m a n - h - b) reasons that if the semicircle equals the circle, the circles are double one another and their portions are likewise double, since removing equal things from equal things leaves equal things.

Letter-labelled proportional equalities

Inscribed figures carry ratio statements: a b c d is worth e, and n b m c is worth c; K is worth g f, whence it follows that K is worth h and n m. The argument proceeds by taking equal quantities from equal quantities.

Hatched panel and spiral among the figures

At centre-right a rectangular panel filled with hatching and vertical bars sits beside a small tightly wound spiral, drawn among the circular figures. No transcription accompanies these two forms.

Colonna sinistra Figure di cerchio su semicerchio, con:

Semicerchio, con:

Semicerchio, con:

A sinistra della precedente, segmento circolare a due basi, con:

Cerchio su semicerchio, con:

Colonna sinistra, dall'alto. Cerchio su semicerchio, con:

Coppia di figure legate fra loro, con:.

Coppia di figure con:

Con riferimento alla figura successiva:

Semicerchio con mezzo quadrato e semicerchio iscritti, con:

Figura analoga, con riferimento alla figura successiva, con:

Figura analoga, con riferimento alla figura successiva, con:

Cerchio con:

Angolo inferiore destro. Coppia di figure con:

Left column. Figures of a circle on a semicircle, with:

Semicircle, with:

Semicircle, with:

To the left of the preceding, a circular segment with two bases, with:

Circle on a semicircle, with:

Left column, from the top. Circle on a semicircle, with:

Pair of figures joined together, with:

Pair of figures with:

With reference to the following figure:

Semicircle with a half-square and a semicircle inscribed, with:

Analogous figure, with reference to the following figure, with: