ſtruction:) hence by ſubſtraction BK = KG + BF, and by ſubtraction again

FK = KG.

Case
5th. Suppoſe the given circle A to include B, and it be required that

the circles to be deſcribed be touched outwardly by A and inwardly by B.

Then
let AB cut the circumferences in C and D, P and O: and biſecting

CO in I, and ſetting off from I towards P, IL = the difference of the ſemidia-

meters of the given circles, and with A and B foci and IL tranſverſe axis de-

ſcribing an ellipſe LKI, it will be the locus of the centers of the circles deſcribed,

and the demonſtration, mutatis mutandis, is the ſame as in the laſt caſe.

Cases
6th and 7th. Suppoſe the two given circles cut each other, and it be

required that the circles to be deſcribed either be touched and included in them

both, or be touched by them both and at the ſame time include them both.

These
two caſes are ſimilar to caſes 1ſt and 2d, and as there, ſo alſo here,

the tranſverſe axis of the two oppoſite hyperbolas, which are the loci required,

muſt be taken = the difference of the ſemidiameters of the given circles. The

demonſtration is ſo alike, it need not be repeated.

Having
a given point A, and a given right line BC, to determine the locus

of the centers of thoſe circles which ſhall paſs through A and touch BC.

From
A draw AG perpendicular to BC, then with focus A and directrix BC

let a parabola be deſcribed, and it will be the locus required; for by the propert

of the curve FA always equals FG drawn perpendicular to the directrix.

Having
a given point A, and a given circle whoſe center is B, to determine

the locus of the centers of all thoſe circles, which paſs through A, and at the

ſame time are touched by the given circle.

Cases
1ſt and 2d. Suppoſe the point A to lie out of the given circle, and

it be required that the circles to be deſcribed be either touched outwardly by the

given circle, or inwardly by it.

Let
AB be drawn, and let it cut the given circumference where it is convex

towards A in the point C, and where it is concave in the point O: then biſecting

AC in E, and ſetting off from E towards B, EH = BC the given radius, and

with A and B foci and EH tranſverſe Axis deſcribing two oppoſite Hyperbolas

KEK and LHL, it is evident that KEK will be the locus of the centers of thoſe

circles which paſs thrugh A and are touched outwardly by the given circle, and

LHL will be the locus of the centers of thoſe circles which paſs through A and

are touched inwardly by the given circle.