1the Right Line R Y in Y. We
72[Figure 72]
ſhall demonſtrate G H to be double
to H I, as it hathbeen demonſtra
ted, that O G is double to G X.
Then draw G ω touching the Section
A G Q L in G; and G C perpen di
cular to B D; and drawing a Line
from A to I, prolong it to que Now
A I ſhall be equall to I que and
A Q parallel to G ω. It is to be
demonſtrated, that the Portion being
demitted into the Liquid, and inclined, ſo, as that its Baſe touch
the Liquid, it ſhall ſtand ſo incli
73[Figure 73]
ned, as that its Axis ſhall make
an Angle with the Surface of the
Liquid leſſe than the Angle φ;
and its Baſe ſhall not in the leaſt
touch the Liquids Surface. For
let it be demitted into the Liquid,
and let it ſtand, ſo, as that its Baſe
do touch the Surface of the Liquid
in one Point only: and the Portion
being cut thorow the Axis by a
Plane erect unto the Surface of the Liquid, let the Section of
74[Figure 74]
the Portion be A N Z L, the Section
of a Rightangled Cone; that of
the Surface of the Liquid A Z; and
the Axis of the Portion and Dia
meter of the Section B D; and let
B D be cut in the Points K and R
as hath been ſaid above; and draw
N F parallel to A Z, and touching
the Section of the Cone in the Point
N; and N T parallel to B D; and
N S perpendicular to the ſame. Be
cauſe, now, that the Portion is in Gravity to the Liquid, as
the Square made of ψ is to the Square B D; and ſince that as the
Portion is to the Liquid in Gravity, ſo is the Square N T to the
Square B D, by the things that have been ſaid; it is plain, that
N T is equall to the Line ψ: And, therefore, alſo, the Portions
A N Z and A G Q are equall. And, ſeeing that in the Equall and
Like Portions A G Q L and A N Z L; there are drawn from the
Extremities of their Baſes, A Q and A Z which cut off equall Porti
ons: It is obvious, that with the Diameters of the Portions they
72[Figure 72]ſhall demonſtrate G H to be double
to H I, as it hathbeen demonſtra
ted, that O G is double to G X.
Then draw G ω touching the Section
A G Q L in G; and G C perpen di
cular to B D; and drawing a Line
from A to I, prolong it to que Now
A I ſhall be equall to I que and
A Q parallel to G ω. It is to be
demonſtrated, that the Portion being
demitted into the Liquid, and inclined, ſo, as that its Baſe touch
the Liquid, it ſhall ſtand ſo incli
73[Figure 73]ned, as that its Axis ſhall make
an Angle with the Surface of the
Liquid leſſe than the Angle φ;
and its Baſe ſhall not in the leaſt
touch the Liquids Surface. For
let it be demitted into the Liquid,
and let it ſtand, ſo, as that its Baſe
do touch the Surface of the Liquid
in one Point only: and the Portion
being cut thorow the Axis by a
Plane erect unto the Surface of the Liquid, let the Section of
74[Figure 74]the Portion be A N Z L, the Section
of a Rightangled Cone; that of
the Surface of the Liquid A Z; and
the Axis of the Portion and Dia
meter of the Section B D; and let
B D be cut in the Points K and R
as hath been ſaid above; and draw
N F parallel to A Z, and touching
the Section of the Cone in the Point
N; and N T parallel to B D; and
N S perpendicular to the ſame. Be
cauſe, now, that the Portion is in Gravity to the Liquid, as
the Square made of ψ is to the Square B D; and ſince that as the
Portion is to the Liquid in Gravity, ſo is the Square N T to the
Square B D, by the things that have been ſaid; it is plain, that
N T is equall to the Line ψ: And, therefore, alſo, the Portions
A N Z and A G Q are equall. And, ſeeing that in the Equall and
Like Portions A G Q L and A N Z L; there are drawn from the
Extremities of their Baſes, A Q and A Z which cut off equall Porti
ons: It is obvious, that with the Diameters of the Portions they