1
55[Figure 55]
And from the Point M draw M Y
touching the Section A M Q L in M;
and M C perpendicular to B D: and
laſtly, having drawn A N & prolong
ed it to Q, the Lines A N & N Q ſhall
be equall to each other. For in
regard that in the Like Portions
A M Q L and A X D the Lines A Q
and A N are drawn from the Baſes
unto the Portions, which Lines
contain equall Angles with the ſaid
Baſes, Q A ſhall have the ſame proportion to A M that L A hath
to A D: Therefore A N is equall to N Q, and A Q parallel to M Y.
It is to be demonſtrated that the Portion being demitted into the
Liquid, and ſo inclined as that its Baſe touch not the Liquid, it
ſhall continue inclined ſo as that its Baſe ſhall not in the leaſt touch
the Surface of the Liquid, and its Axis ſhall make an Angle with
the Liquids Surface greater than the Angle X. 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 let the Portion be cut
thorow the Axis by a Plane erect unto the Surface of the Liquid,
56[Figure 56]
and Let the Section of the Super
ficies of the Portion be A P O L,
the Section of a Rightangled Cone,
and let the Section of the Liquids
Surface be A O; And let the Axis
of the Portion and Diameter of the
Section be B D: and let B D be
cut in the Points K and R as hath
been ſaid; alſo draw P G Parallel to
A O and touching the Section
A P O L in P; and from that Point
draw P T Parallel to B D, and P S perpendicular to the ſame B D.
Now, foraſmuch as the Portion is unto the Liquid in Gravity, as
the Square made of the Line ψ is to the Square B D; and ſince that
as the portion is unto the Liquid in Gravitie, ſo is the part thereof
ſubmerged unto the whole Portion; and that as the part ſubmerged
is to the whole, ſo is the Square T P to the Square B D; It follow
eth that the Line ψ ſhall be equall to T P: And therefore the Lines
M N and P T, as alſo the Portions A M Q and A P O ſhall like
wiſe be equall to each other. And ſeeing that in the Equall and
Like Portions A P O L and A M Q L the Lines A O and A Q
are drawn from the extremites of their Baſes, ſo, as that the Portions
cut off do make Equall Angles with their Diameters; as alſo the
55[Figure 55]And from the Point M draw M Y
touching the Section A M Q L in M;
and M C perpendicular to B D: and
laſtly, having drawn A N & prolong
ed it to Q, the Lines A N & N Q ſhall
be equall to each other. For in
regard that in the Like Portions
A M Q L and A X D the Lines A Q
and A N are drawn from the Baſes
unto the Portions, which Lines
contain equall Angles with the ſaid
Baſes, Q A ſhall have the ſame proportion to A M that L A hath
to A D: Therefore A N is equall to N Q, and A Q parallel to M Y.
It is to be demonſtrated that the Portion being demitted into the
Liquid, and ſo inclined as that its Baſe touch not the Liquid, it
ſhall continue inclined ſo as that its Baſe ſhall not in the leaſt touch
the Surface of the Liquid, and its Axis ſhall make an Angle with
the Liquids Surface greater than the Angle X. 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 let the Portion be cut
thorow the Axis by a Plane erect unto the Surface of the Liquid,
56[Figure 56]and Let the Section of the Super
ficies of the Portion be A P O L,
the Section of a Rightangled Cone,
and let the Section of the Liquids
Surface be A O; And let the Axis
of the Portion and Diameter of the
Section be B D: and let B D be
cut in the Points K and R as hath
been ſaid; alſo draw P G Parallel to
A O and touching the Section
A P O L in P; and from that Point
draw P T Parallel to B D, and P S perpendicular to the ſame B D.
Now, foraſmuch as the Portion is unto the Liquid in Gravity, as
the Square made of the Line ψ is to the Square B D; and ſince that
as the portion is unto the Liquid in Gravitie, ſo is the part thereof
ſubmerged unto the whole Portion; and that as the part ſubmerged
is to the whole, ſo is the Square T P to the Square B D; It follow
eth that the Line ψ ſhall be equall to T P: And therefore the Lines
M N and P T, as alſo the Portions A M Q and A P O ſhall like
wiſe be equall to each other. And ſeeing that in the Equall and
Like Portions A P O L and A M Q L the Lines A O and A Q
are drawn from the extremites of their Baſes, ſo, as that the Portions
cut off do make Equall Angles with their Diameters; as alſo the