Curvilinear Geometry: Sectors, Lunes and the Limits of Squaring

Many sectors of equal value, from quarter to sixty-fourth

A row of five sector figures is labelled quarter, eighth, sixteenth, thirty-second and sixty-fourth, posing the problem of giving many various sectors of a circle all of the same value.

Dividing a curved-sided triangle into three equal parts

To divide a triangle of three curved sides into three equal parts, the triangle having two convex curved sides and a third concave curved side. Here all the curved lines of the triangle must be of equal curvature, and the dividing curve must be of like curvature.

Nested tangent circles doubling toward a point

Making a triangle tangent to two circles that are double one another and touch each other, all further circles that touch one another and the sides of the triangle toward the point, proceeding infinitely, are always double one another. The same holds toward the base if the triangle is extended, and for any figure wholly within these circles.

Which curvilinear portions can be squared

From three circles, three sectors of equal value are taken; removing their portions, one seeks the difference of the remainders. No difference can be known between the first and second, because the first has a curved side and the second all straight sides; but the rest, being rectilinear, remain squared, each equal to a curved portion.

On impossibility: crescents of unequal arcs

It is impossible to give a square equal to any crescent formed of arcs of one same circle whose arcs are of unequal length, because the value taken from one side cannot be returned on the other.

Squaring succeeds only for circles in rational proportion

Crescents of arcs of different circuits and lengths cannot be squared unless the circles stand in rational proportion — as arc a, of a circle double the circle of c, so that a quarter of one is worth an eighth of the other. Removing the whole portion c and returning the half-portion b squares the crescent c a, and so it follows for every proportion (triple, quadruple, quintuple) to infinity.

Figura colle lettere:

Figura colle lettere:

Cinque figure di settori di cerchio:

Cinque figure di settore con porzione:

Tre figure di parti di cerchio:

Due figure di doppio cerchio:

Figura di doppio cerchio con triangolo curvilineo:

Doppio cerchio con settore e le lettere:

Lungo il margine destro. Figura colle lettere:

Quattro settori di cerchio colle lettere:

Quattro cerchi concentrici con settore:

Tre cerchi di raggio crescente, divisi in settori e chiusi entro un triangolo aperto alla base, colle lettere:

A sinistra della predetta figura:

A sinistra dello scritto precedente, mezza porzione staccata da parte di un settore:

Angolo inferiore destro:

Facciata sinistra.

Falcata ampliata, colle lettere:

Figure with the letters:

Figure with the letters:

Five figures of circle sectors:

Five figures of a sector with a portion:

Three figures of parts of a circle:

Two figures of a double circle:

Figure of a double circle with a curvilinear triangle:

Double circle with a sector and the letters:

Along the right margin. Figure with the letters:

Four circle sectors with the letters:

Four concentric circles with a sector:

Three circles of increasing radius, divided into sectors and enclosed within a triangle open at the base, with the letters:

To the left of the aforesaid figure:

To the left of the preceding text, half a portion detached from part of a sector:

Lower right corner:

Left-hand side.

Enlarged crescent, with the letters: