Squaring curved figures: lunes, circles and transformed spirals

Problems of squaring curved and square surfaces

Several statements set the program of the sheet: from a curved surface draw off a curved surface so the remainder is squarable, and likewise from a square surface. A related problem asks to draw a curvilinear surface out of a square one according to a given proportion, and to supply the quadrature of both the extracted surface and its remainder.

Three nested squares tangent to two circles

Three squares are drawn one within another, each double the next, proved so by being tangent to the two circles. The middle square equals the smaller square together with the 4 lunes; the 4 lunes equal the square they surround; and the excess of the greatest square equals the two smaller squares.

The greatest square equated to eight lunes

In the corner figure labelled a, d, c, b, the greatest square abcd equals the 4 lunes plus the middle square, hence equals 8 lunes; the remaining 4 greatest portions of the circle are each divided into 4 equal parts (16 in all). Removing the crossed-out portions, the uncrossed remainder equals the greatest square, or twice the middle square.

Transforming the spiral (elica) into rectilinear and curved figures

One rectangle divided into 8 squares carries a spiral, whose 'field' is stated as six-eighths of the sickle-shapes. Further notes ask that half the spiral be given for its thickness or length, and that the spiral be transformed into various rectilinear and curvilinear figures.

Portions, sickle-shapes and fields in proportion

The relations are stated as equalities of area: the 4 squares of the larger square are worth 4 times the smaller square, the 4 double sickle-shapes equal the 4 portions, and the field equals the solid, both worth two smaller squares. A companion note holds that like parts of circles stand in the same proportion as their wholes.

Marginal memo: a nude by Peruzzo

Among the geometric problems a short heading records 'Nude by Peruzzo', a memorandum referring to a figure study rather than to the constructions surrounding it. It stands with the opening quadrature problems at the top of the sheet.

Angolo superiore destro:

Rettangolo diviso in 8 quadrati, contenente un elica:

A destra, sopra disegni di eliche:

Sotto i disegni di eliche:

Più a sinistra, semicerchio con cerchio interno:

Tornando verso il margine:

Quadrati e cerchio entro un cerchio:

A sinistra del precedente, doppia figura con:

Sotto lo scritto precedente, figura con :

In basso dalla destra:

Due figure di cerchio iscritto in un quadrato, diviso in quattro parti:

Dall'alto a sinistra. Due cerchi biffati:

Ottagono curvilineo entro un cerchio:

A destra del precedente, figura colle lettere:

Figura sotto la precedente:

Tre figure colle lettere:

Quadrato fra cerchio iscritto e circoscritto:

Angolo inferiore, figura con:

Upper right corner

Rectangle divided into 8 squares, containing a spiral

At the right, above drawings of spirals

Beneath the drawings of spirals

Further left, semicircle with an inner circle

Returning toward the margin

Squares and a circle within a circle

To the left of the preceding, a double figure

Beneath the preceding, a figure

Below at the right

Two figures of a circle inscribed in a square, divided into four parts

From the upper left; two circles struck through

Curvilinear octagon within a circle

To the right of the preceding, a figure with letters

Figure beneath the preceding

Three figures with letters

Square between an inscribed and a circumscribed circle

Lower corner, figure with letters