Thercfore there are given three points H, M, D, as likewiſe a plane AB,

or AC, through which points the ſphere is to paſs, and alſo touch the given

plane. Hence it appears that this Problem is reduced to the IId of this

Supplement.

Before
we proceed, the following eaſy Lemmas muſt be premiſed.

##
44.
LEMMA I.

Let
there be a circle BCD, and a point E taken without it, and iſ from

E a line EDOB be drawn to paſs through the center, and another line ECA

to cut it any ways; we know from the Elements that the rectangle AEC is

equal to the rectangle BED. Let us now ſuppoſe a ſphere whoſe center is O,

and one of whoſe great circles is ACDB; if from the ſame point E a line

ECA be any-how drawn to meet the ſpherical ſurface in the points C and A,

I ſay the rectangle AEC will ſtill be equal to the rectangle BED. For if we

ſuppoſe the circle and right line ECA to revolve upon EDB as an immove-

able axis, the lines EC and EA will not be changed, becauſe the points C

and A deſcribe circles whoſe planes are perpendicular to that axis; and

therefore the rectangle AEC will in any plane be ſtill equal to the rectangle

BED.

##
45.
LEMMA II.

By
the ſame method of reaſoning, the Vth Lemma immediately preceed-

ing Problem XIII, in the Treatiſe of Circular Tangencies, may be extended

alſo to ſpheres, viz. that in any plane (ſee the Figures belonging to that

Lemma) MG X MB = MH X MA. And alſo that MF X MC = ME X MI.

##
46.
LEMMA III.

Let
there be two ſpheres YN, XM, through whoſe centers let the right

line RYNXMU paſs, and let it be as the radius YN to the radius XM, ſo

YU to XU; and from the point U let a line UTS be drawn in any plane,

and let the rectangle S U T be equal to the rectangle RUM; I fay that if

any ſphere OTS be deſcribed to paſs through the points T and S, and to

touch one of the given ſpheres XM as in O, it will alſo touch the other

given ſphere YN. For joining UO, and producing it to meet the ſurface of

the ſphere OTS in Q; the rectangle QUO = the rectangle SUT, by

Lemma I. but the rectangle SUT = the rectangle RUM. by conſtruction,