Lunes squared by proportion, and the percussion of balls

Eighth of the greater worth the half of the lesser

Of quadruple circles, the eighth of the greater equals the half of the lesser: the semicircle a b c equals an eighth of a circle. Removing equal sectors leaves the lune d equal to the portion b, and, by rearranging the squarable triangle f g, the rectilinear triangle c g is made equal to triangle a.

First, Second, Third, Fourth: rules by proportion

Four numbered constructions give the rule for each ratio: in double circles half the lesser equals a quarter of the greater (a m b); in quadruple circles a quarter of the greater equals the whole lesser (d n c); in double circles half the greater equals the whole lesser (f g e); the fourth combines these so lune q equals lune a of the first.

Percussion of balls: the striker stays, the struck flees

By the fourth proposition of percussion, a stone ball b attaches its power to ball a, and b will not leave its sheath. With copper balls in a blowpipe, the red pellet striking the black makes the black flee while the other stays; the gaming boards do the like.

Doubling and quadrupling as the key to each figure

Each construction turns on a stated ratio between circles — double or quadruple — from which the equality of a lune, sector or semicircle to a rectilinear part is deduced. A proportion is set between lune 7 and half-portion 8 as b is to d.

In alto. Quadrante di cerchio e cerchio sovrapposti, con:

Colonna destra. Quadrante di cerchio sovrapposto a semicerchio, con:

Quadrante di cerchio sovrapposto a cerchio, con:

Cerchio sovrapposto a semicerchio, con:

Cerchio sovrapposto a semicerchio, con:

Cerchio e semicerchio su semicerchio, con:

Due cerchi su semicerchio, con :

Colonna sinistra. Semicerchio su quadrante di cerchio, con:

Figura colle lettere:

Figura maggiore: due cerchi su semicerchio, con:

In basso, entro un cerchio, settore su semicerchio, con:

Above. Quarter-circle and circle superimposed, with:

Right column. Quarter-circle superimposed on a semicircle, with:

Quarter-circle superimposed on a circle, with:

Circle superimposed on a semicircle, with:

Circle superimposed on a semicircle, with:

Circle and semicircle on a semicircle, with:

Two circles on a semicircle, with:

Left column. Semicircle on a quarter-circle, with:

Figure with the letters:

Larger figure: two circles on a semicircle, with:

Below, within a circle, sector on a semicircle, with:

Lunes squared by proportion, and the percussion of balls

On this page

Eighth of the greater worth the half of the lesser

First, Second, Third, Fourth: rules by proportion

Percussion of balls: the striker stays, the struck flees

Doubling and quadrupling as the key to each figure