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the points I and H a ſphere be deſcribed which touches the plane AC, I ſay
it will alſo touch the ſphere EGF. From F draw FB to the point of contact
of the ſphere and plane, and make the rectangle BFN = the rectangle CFE,
and the point N will be in the ſurface of the ſphere EGF, by Lemma IV.
But the rectangle CFE, by conſtruction, = the rectangle IFH; therefore
IFH = BFN, and the point N will be alſo in the ſurſace of the ſphere IHB.
It remains then to be proved that theſe ſpheres touch in N, which is very eaſy
to be done. For from the point F through any point R in the ſpherical ſur-
face EGF let the line FR be drawn, which may cut the ſpherical ſurface
IBH in L and P, and the plane AC in K. The rectangle KFR = the rect-
angle CFE, by Lemma IV. = the rectangle IFH, by conſtruction, = the
rectangle PFL. Since then KFR = PFL, and KF is greater than PF, be-
cauſe the ſphere IHB touches the plane AC in B, therefore FR is leſs than
FL, and the point R is without the ſphere IHB, and the ſame may be ſhewn
of every other point in the ſpherical ſurface EGF, except the point N.
it will alſo touch the ſphere EGF. From F draw FB to the point of contact
of the ſphere and plane, and make the rectangle BFN = the rectangle CFE,
and the point N will be in the ſurface of the ſphere EGF, by Lemma IV.
But the rectangle CFE, by conſtruction, = the rectangle IFH; therefore
IFH = BFN, and the point N will be alſo in the ſurſace of the ſphere IHB.
It remains then to be proved that theſe ſpheres touch in N, which is very eaſy
to be done. For from the point F through any point R in the ſpherical ſur-
face EGF let the line FR be drawn, which may cut the ſpherical ſurface
IBH in L and P, and the plane AC in K. The rectangle KFR = the rect-
angle CFE, by Lemma IV. = the rectangle IFH, by conſtruction, = the
rectangle PFL. Since then KFR = PFL, and KF is greater than PF, be-
cauſe the ſphere IHB touches the plane AC in B, therefore FR is leſs than
FL, and the point R is without the ſphere IHB, and the ſame may be ſhewn
of every other point in the ſpherical ſurface EGF, except the point N.
Theſe Lemmas, though they be very eaſy, are very elegant and valuable,
eſpecially the IIId and Vth. In the IIId. though there be an inſinite num-
ber of ſpheres which, paſſing through the points T and S, may touch the
ſphere XM, yet they will all alſo touch the ſphere YN, by what is there
proved. In the Vth, though there be an infinite number of ſpheres which,
paſſing through the points I and H, may touch the plane AC, yet they will
all alſo touch the ſphere EGF, by what is there proved.
eſpecially the IIId and Vth. In the IIId. though there be an inſinite num-
ber of ſpheres which, paſſing through the points T and S, may touch the
ſphere XM, yet they will all alſo touch the ſphere YN, by what is there
proved. In the Vth, though there be an infinite number of ſpheres which,
paſſing through the points I and H, may touch the plane AC, yet they will
all alſo touch the ſphere EGF, by what is there proved.
We ſhall now be able to go through the remaining Problems with eaſe.
PROBLEM VIII.
Let there be given a plane ABC, and two points H and M, and alſo a
ſphere DFE; to find a ſphere which ſhall paſs through the given points, and
touch the given plane, and likewiſe the given ſphere.
ſphere DFE; to find a ſphere which ſhall paſs through the given points, and
touch the given plane, and likewiſe the given ſphere.
Through the center O of the given ſphere let EODB be demitted perpen-
dicular to the given plane ABC, and let HE be drawn, and make the rect-
angle HEG equal to the rectangle BED, and G will then be given. Find
then a ſphere, by Problem II. which ſhall paſs through the three points M,
H, G, and touch the plane ABC, and it will be the ſphere here required.
For it paſſes through the points M and H, and touches the plane ABC,
by conſtruction; it likewiſe touches the ſphere DFE, by Lemma V. For
ſince the rectangle HEG = the rectangle BED, every ſphere which
dicular to the given plane ABC, and let HE be drawn, and make the rect-
angle HEG equal to the rectangle BED, and G will then be given. Find
then a ſphere, by Problem II. which ſhall paſs through the three points M,
H, G, and touch the plane ABC, and it will be the ſphere here required.
For it paſſes through the points M and H, and touches the plane ABC,
by conſtruction; it likewiſe touches the ſphere DFE, by Lemma V. For
ſince the rectangle HEG = the rectangle BED, every ſphere which