## Full text: Pergaeus, Apollonius: The two books of Apollonius Pergaeus, concerning tangencies, as they have been restored by Franciscus Vieta and Marinus Ghetaldus

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.

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