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PROBLEM XII.
Let there be given a point and three ſpheres, to ſind a ſphere which ſhall
paſs through the point, and touch all the three ſpheres.
paſs through the point, and touch all the three ſpheres.
We aſſign no figure to this Problem alſo, becauſe, by help of Lemma III,
it may immediately be reduced to Problem IX, where two points and two
ſpheres are given.
it may immediately be reduced to Problem IX, where two points and two
ſpheres are given.
PROBLEM XIII.
Let there be two planes, and alſo two ſpheres given;
to find a ſphere
which ſhall touch the planes, as alſo the ſpheres.
which ſhall touch the planes, as alſo the ſpheres.
Suppoſe the thing done.
If therefore we imagine another ſpherical ſurface
parallel to that which is required, and which we now ſuppoſe found, and
whoſe radius is leſs than it's by the radius of the leſſer of the two given
ſpheres; this new ſpherical ſurface will touch two planes parallel to the two
given ones, and whoſe diſtance therefrom will be equal to the radius of the
leſſer of the given ſpheres; it will alſo touch a ſphere concentric to the
greater given one whoſe radius is leſs than it's by the radius of the leſſer given
one; and it will likewife paſs through the center of the leſſer given one.
The Queſtion is then reduced to Problem X, where a point, two planes and
a ſphere are given.
parallel to that which is required, and which we now ſuppoſe found, and
whoſe radius is leſs than it's by the radius of the leſſer of the two given
ſpheres; this new ſpherical ſurface will touch two planes parallel to the two
given ones, and whoſe diſtance therefrom will be equal to the radius of the
leſſer of the given ſpheres; it will alſo touch a ſphere concentric to the
greater given one whoſe radius is leſs than it's by the radius of the leſſer given
one; and it will likewife paſs through the center of the leſſer given one.
The Queſtion is then reduced to Problem X, where a point, two planes and
a ſphere are given.
PROBLEM XIV.
Having three ſpheres and a plane given;
to find a ſphere which ſhall
touch them all.
touch them all.
By a like method to what is uſed in the preceeding, and in the VIth Pro-
blem, this is reduced to Problem XI, where a point, a plane, and two
ſpheres are given.
blem, this is reduced to Problem XI, where a point, a plane, and two
ſpheres are given.
PROBLEM XV.
Having four ſpheres given;
to ſind a ſphere which ſhall touch them all.
Suppose the thing done.
As, in the treatiſe of Circular Tangencies, the
laſt Problem, where it is required, having three circles given, to find a fourth
which ſhall touch them all, is reduced to another, where a point and two
circles are given; ſo alſo this, by a like method, and ſimilar to what has been
uſed in the preceding Problems, is reduced to Problem XII, where three
ſpheres and a point are given.
laſt Problem, where it is required, having three circles given, to find a fourth
which ſhall touch them all, is reduced to another, where a point and two
circles are given; ſo alſo this, by a like method, and ſimilar to what has been
uſed in the preceding Problems, is reduced to Problem XII, where three
ſpheres and a point are given.
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