1and F leſs than B R. Let R Ψ be equall to F; and draw Ψ E
perpendicular to B D; which let be in power the half of that
which the Lines K R and Ψ B containeth; and draw a Line from
B to E: I ſay that the Portion demitted into the Liquid, ſo as that
its Baſe be wholly within the Liquid, ſhall ſo ſtand, as that its Axis
do make an Angle with the Liquids Surface, equall to the Angle B.
For let the Portion be demitted into the Liquid, as hath been ſaid;
and let the Axis not make an Angle with the Liquids Surface, equall
to B, but firſt a greater: and the ſame being cut thorow the Axis
by a Plane erect unto the Surface of the Liquid, let the Section of
the Portion be A P O L, the Section of a Rightangled Cone; the
Section of the Surface of the Liquid Γ I; and the Axis of the
Portion and Diameter of the Section N O; which let be cut in
the Points ω and T, as before: and draw Y P, parallelto Γ I, and
touching the Section in P, and MP parallel to N O, and P S perpen
dicular to the Axis. And becauſe now that the Axis of the Portion
maketh an Angle with the Liquids Surface greater than the Angle
B, the Angle S Y P ſhall alſo be greater than the Angle B: And,
therefore, the Square P S hath greater proportion to the Square
S Y, than the Square Ψ E hath to the Square Ψ B: And, for that
cauſe, K R hath greater proportion to S Y, than the half of K R
hath to Ψ B: Therefore, S Y is leſs than the double of Ψ B; and
S O leſs than ψ B: And, therefore, S ω is greater than R ψ; and
P H greater than F. And, becauſe that the Portion hath the
ſame proportion in Gravity unto the Liquid, that the Exceſs by
which the Square B D, is greater than the Square F Q, hath unto
the Square B D; and that as the Portion is in proportion to the
Liquid in Gravity, ſo is the part thereof ſubmerged unto the whole
Portion; It followeth that the part ſubmerged, hath the ſame
proportion to the whole Portion, that the Exceſs by which the
Square B D is greater than the Square F Q hath unto the Square
B D: And, therefore, the whole Portion ſhall have the ſame propor
tion to that part which is above the
45[Figure 45]
Liquid, that the Square B D hath to
the Square F Q: But as the whole
Portion is to that part which is above
the Liquid, ſo is the Square N O unto
the Square P M: Therefore, P M
ſhall be equall to F Q: But it
hath been demonſtrated, that P H is
greater than F. And, therefore,
MH ſhall be leſs than que and P H
greater than double of H M. Let
therefore, P Z be double to Z M:
perpendicular to B D; which let be in power the half of that
which the Lines K R and Ψ B containeth; and draw a Line from
B to E: I ſay that the Portion demitted into the Liquid, ſo as that
its Baſe be wholly within the Liquid, ſhall ſo ſtand, as that its Axis
do make an Angle with the Liquids Surface, equall to the Angle B.
For let the Portion be demitted into the Liquid, as hath been ſaid;
and let the Axis not make an Angle with the Liquids Surface, equall
to B, but firſt a greater: and the ſame being cut thorow the Axis
by a Plane erect unto the Surface of the Liquid, let the Section of
the Portion be A P O L, the Section of a Rightangled Cone; the
Section of the Surface of the Liquid Γ I; and the Axis of the
Portion and Diameter of the Section N O; which let be cut in
the Points ω and T, as before: and draw Y P, parallelto Γ I, and
touching the Section in P, and MP parallel to N O, and P S perpen
dicular to the Axis. And becauſe now that the Axis of the Portion
maketh an Angle with the Liquids Surface greater than the Angle
B, the Angle S Y P ſhall alſo be greater than the Angle B: And,
therefore, the Square P S hath greater proportion to the Square
S Y, than the Square Ψ E hath to the Square Ψ B: And, for that
cauſe, K R hath greater proportion to S Y, than the half of K R
hath to Ψ B: Therefore, S Y is leſs than the double of Ψ B; and
S O leſs than ψ B: And, therefore, S ω is greater than R ψ; and
P H greater than F. And, becauſe that the Portion hath the
ſame proportion in Gravity unto the Liquid, that the Exceſs by
which the Square B D, is greater than the Square F Q, hath unto
the Square B D; and that as the Portion is in proportion to the
Liquid in Gravity, ſo is the part thereof ſubmerged unto the whole
Portion; It followeth that the part ſubmerged, hath the ſame
proportion to the whole Portion, that the Exceſs by which the
Square B D is greater than the Square F Q hath unto the Square
B D: And, therefore, the whole Portion ſhall have the ſame propor
tion to that part which is above the
45[Figure 45]Liquid, that the Square B D hath to
the Square F Q: But as the whole
Portion is to that part which is above
the Liquid, ſo is the Square N O unto
the Square P M: Therefore, P M
ſhall be equall to F Q: But it
hath been demonſtrated, that P H is
greater than F. And, therefore,
MH ſhall be leſs than que and P H
greater than double of H M. Let
therefore, P Z be double to Z M: