Quadrature studies: transforming curved and straight surfaces

On transforming curvilinear into rectilinear surfaces

The heading sets the whole program of the sheet: the transformation of curvilinear surfaces into rectilinear ones, and of rectilinear back into curvilinear, that is, of surfaces conditioned by known proportions.

Octagon equal to the parallel circle and to triangle m

This octagon is worth the parallel (ring-shaped) circle and is worth the triangle m. The adjoining sector is the eighth part of a circle eightfold the smallest.

The eighth of an eightfold circle equals the whole other

Of two circles eightfold one to the other, the eighth part of the one is worth the whole of the other. The tailed-circle figures with added sectors make the equality visible.

Halving similar surfaces by proportion, to infinity

Of similar surfaces, half of the double is worth the whole sub-double, a third of the triple the whole sub-triple, a fourth of the quadruple the whole sub-quadruple, and so on to infinity. Rules follow on when removing a squarable or unsquarable part leaves a squarable or unsquarable remainder.

Removing the greatest bi-angle leaves the greatest square

If from an unsquarable figure you take an unsquarable part, the remainder is squarable. This is proved in circle a b c: taking away the greatest bi-angle b leaves a c equal to the greatest square that can fit in that circle, as the second figure shows.

The four greatest portions and the adversary's concession

Of the four greatest portions of a circle, four others equal to them can be found, as with portions a b in square c d e f. If an opponent concedes only portion a in the half-square e c d, he is forced to admit that if triangle e c d contains portion e a d, then triangle e d f contains portion e b d.

Facciata destra, margine superiore:

Colonna destra. Cerchio con ottagono iscritto e settore esterno, con:

Due figure colle lettere:

Due figure colle lettere:

Figura di cerchio entro un rettangolo, unita ad altra con tratto di penna:

Seconda figura, colle lettere:

Cerchi caudati coll'aggiunta di settori di cerchi:

Segmento circolare a doppia base colle lettere:

Colonna centrale. Cerchio con settore esterno e le lettere:

Quattro porzioni entro una corona circolare:

Doppia figura con:

Colonna sinistra. Quadrante di cerchio diviso con archi di cerchi proporzionali, con:

Quadrato iscritto tra due cerchi, con :

Un tratto di penna conduce a:

Due figure unite alla precedente con tratto di penna:

Tre figure accostate, colle lettere:

Figura stellare colle lettere in senso antiorario:

Facciata sinistra.

Cinque figure con:

Quadrato con cerchio iscritto e quattro bisangoli:

Bisangolo entro un cerchio, con:

La stessa figura con quadrato iscritto, con:

Figura ottogonale con croce e porzioni depennate:

Bisangolo in un cerchio iscritto in un quadrato, con:

A fianco del precedente paragrafo:

A sinistra del paragrafo precedente:

Numeri e operazione aritmetica:

Right side, top margin:

Right column. Circle with inscribed octagon and external sector, with:

Two figures with the letters:

Two figures with the letters:

Figure of a circle within a rectangle, joined to another by a pen-stroke:

Second figure, with the letters:

Tailed circles with added circular sectors:

Circular segment with double base, with the letters:

Central column. Circle with external sector and the letters:

Four portions within a circular ring:

Double figure with:

Left column. Quarter-circle divided with proportional circular arcs, with:

Square inscribed between two circles, with:

A pen-stroke leads to:

Two figures joined to the preceding by a pen-stroke:

Three figures set side by side, with the letters:

Star figure with the letters, counter-clockwise:

Left side:

Five figures with:

Square with inscribed circle and four bi-angles:

Bi-angle within a circle, with:

The same figure with an inscribed square, with:

Octagonal figure with a cross and hatched-out portions:

Bi-angle in a circle inscribed in a square, with:

Beside the preceding paragraph:

To the left of the preceding paragraph:

Numbers and arithmetical operation: