35(23)
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. ]
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.
planes and ſpheres, we ſhall always be able to ſubſtitute a given point in the
place of a given ſphere.
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.
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.
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.
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