through the points H and G, and touches the plane ABC, touches likewiſe

the ſphere DFE.

##
50.
PROBLEM IX.

Let
there be given two ſpheres AB, DE, as alſo two points H and M; to find a ſphere which ſhall paſs through the two given points, and likewiſe

touch the two given ſpheres.

Let
the right line AF be drawn paſſing through the centers of the

ſpheres, and as the radius AB is to the radius DE, ſo make BF to EF, and

the point F will be given. Make the rectangle HFG = the rectangle NFA,

and the point G will be given. Now having given three points M, H, G,

as alſo a ſphere DE; find a ſphere by Problem III, which ſhall paſs through

the given points, and touch the given ſphere; and, by Lemma III, it will be

the ſphere here required.

##
51.
PROBLEM X.

Let
there be given two planes AB, BD, a point H, and a ſphere

EGF; to find a ſphere which ſhall paſs through the given point, and touch

the given ſphere, as alſo the two given planes.

Through
the center O of the given ſphere let a perpendicular to either of

the given planes CEOF be demitted, and make the rectangle HFI = the

rectangle CFE. Then having given the two points H and I, as alſo the

two planes AB, BD; find a ſphere, by Problem VII, which ſhall paſs

through the two given points, and likewiſe touch the two given planes; and,

by Lemma V, it will be the ſphere required.

##
52.
PROBLEM XI.

Let
there be given a point, a plane, and two ſpheres; to find a ſphere

which ſhall paſs through the point, touch the plane, and alſo the two

ſpheres.

This Problem, by a like method of reaſoning, is immediately reduced to

the VIIIth, where two points, a plane, and a ſphere are given, and that by

means of the Vth Lemma. But if you chuſe to uſe the IIId Lemma, it will

be reduced to the ſame Problem by a different method, and a different

conſtruction.