equal to UE: but RM is given in magnitude, being the radius of the given

ſphere, therefore UE is alſo given in magnitude. And ſince OE is perpen-

dicular to the plane DE, it will be alſo to the plane PU which is parallel

thereto. UE then being given in magnitude, and being the interval be-

tween two parallel planes DE, PU, whereof DE is given in poſition by hypo-

theſis, the other PU will alſo be given in poſition. In the ſame manner it

may be proved that the planes GH, IN, are given in poſition, and that the

lines OG, OI, are perpendicular thereto reſpectively, and each alſo equal to

OM. A ſphere therefore deſcribed with center O and OM radius will touch

the three planes PU, GH, IN, given in poſition: but the point M is given,

being the center of the given ſphere. The queſtion is then reduced to this,

Having three planes given PU, GH, IN, and a point M, to find the radius

of a ſphere which ſhall touch the given planes, and paſs through the given

point; which is the ſame as the preceeding Problem. [And this radius be-

ing increaſed or diminiſhed by MR, according as R is taken in the further or

nearer ſurface of the given ſphere, will give the radius of a ſphere which will

touch the three given planes DE, DB, BC, and likewiſe the given

ſphere. ]

By
a like method, when among the Data there are no points, but only

planes and ſpheres, we ſhall always be able to ſubſtitute a given point in the

place of a given ſphere.

##
43.
PROBLEM VII.

Having
two points H, M, as alſo two planes AB, BC, given, to find a

ſphere which ſhall paſs through the given points, and touch the given

planes.

Draw
HM and biſect it in I, the point I will be given, through the

point I let a plane be erected perpendicular to the right line HM, this plane

will be given in poſition, and the center of the ſphere required will be in this

plane. But becauſe it is alſo to touch the planes AB, BC, its center will be

alſo in another plane given in poſition (by what has been proved, Prob. IV.) and therefore in a right line which is their interſection, given in poſition,

which let be GE; to which line GE from one of the given points M demit-

ting a perpendicular MF, it will be given in magnitude and poſition, and

being continued to D ſo that FD equals MF, the point D will be given; and, from what has been proved before, will be in the ſpherical ſurface.