Dividing lunes and squaring curved figures

Halving a lune with an inscribed circle

I want to divide the length of the lune a b c d e into 2 equal parts, and it will be divided by the circle d m r. And this rule will make such a lune divisible without limit along its length, with known parts.

A curved four-sided figure equal to a triangle

I want to make a figure of 2 opposite sides of various lengths and various curvatures, and of 2 other opposite sides of straight lines of various lengths, equal to a rectilinear triangle; and I will do it by the motion of a quarter-circle a b c moved from the place e b d.

Dividing the circle by the roots of 1, 2, 3, 4

I want to divide this circle into many equal parts using nothing but the circumferences of those parts; and I will do it with the roots of 1, of 2, of 3 and of 4, and then the lunes will likewise be divided into equal parts along their length.

Equals from equals leave equals

The thing that moves gains as much space as it loses. The surface a b is moved into a d e c and has left b equal to d e c, while a remains common and double; which is the proof that says: if from an equal thing an equal part is removed, the remainder is equal.

A worked calculation

A column of figures (1, 2, 13, 133, 200, 12, 200, 177...) is reduced to the result 11 13/17.

Dall'alto in basso. Figura colle lettere:

Figura colle lettere:

Figure di settori sovrapposti a segmenti circolari:

Figura colle lettere:

Figura colle lettere:

Due cerchi concentrici e triangolo curvilineo, colle lettere:

A destra. Figura colle lettere:

A sinistra, cerchi concentrici con triangolo curvilineo:

Operazione aritmetica:

From top to bottom. Figure with the letters:

Figure with the letters:

Figures of sectors laid over circular segments:

Figure with the letters:

Figure with the letters:

Two concentric circles and a curvilinear triangle, with the letters:

On the right. Figure with the letters:

On the left, concentric circles with a curvilinear triangle:

Arithmetical operation:

Dividing lunes and squaring curved figures

On this page

Halving a lune with an inscribed circle

A curved four-sided figure equal to a triangle

Dividing the circle by the roots of 1, 2, 3, 4

Equals from equals leave equals

A worked calculation