1Portion demitted into the Liquid, like as hath been ſaid, ſhall ſtand
enclined ſo as that its Axis do make an Angle with the Surface of
the Liquid equall unto the Angle E B Ψ. For demit any Portion
into the Liquid ſo as that its Baſe
39[Figure 39]
touch not the Liquids Surface;
and, if it can be done, let the
Axis not make an Angle with the
Liquids Surface equall to the
Angle E B Ψ; but firſt, let it be
greater: and the Portion being
cut thorow the Axis by a Plane e
rect unto [or upon] the Surface of
the Liquid, let the Section be A P
O L the Section of a Rightangled
Cone; the Section of the Surface of the Liquid X S; and let the
Axis of the Portion and Diameter of the Section be N O: and
draw P Y parallel to X S, and touching the Section A P O L in P;
and P M parallel to N O; and P I perpendicular to N O: and
moreover, let B R be equall to O ω, and R K to T ω; and let ω H
be perpendicular to the Axis. Now becauſe it hath been ſuppoſed
that the Axis of the Portion doth make an Angle with the Surface
of the Liquid greater than the Angle B, the Angle P Y I ſhall be
greater than the Angle B: Therefore the Square P I hath greater
proportion to the Square Y I, than the Square E Ψ hath to the
Square Ψ B: But as the Square P I is to the Square Y I, ſo is the
Line K R unto the Line I Y; and as the Square E Ψ is to the Square
Ψ B, ſo is half of the Line K R unto the Line Ψ B: Wherefore
(a) K R hath greater proportion to I Y, than the half of K R hath
to Ψ B: And, conſequently, I Y isleſſe than the double of Ψ B,
and is the double of O I: Therefore O I is leſſe than Ψ B; and I ω
greater than Ψ R: but Ψ R is equall to F: Therefore I ω is greater
than F. And becauſe that the Portion is ſuppoſed to be in Gra
vity unto the Liquid, as the Square F Q is to the Square B D; and
ſince that as the Portion is to the Liquid in Gravity, ſo is the part
thereof ſubmerged unto the whole Portion; and in regard that as
the part thereof ſubmerged is to the whole, ſo is the Square P M to
the Square O N; It followeth, that the Square P M is to the Square
N O, as the Square F Q is to the Square B D: And therefore F
Q is equall to P M: But it hath been demonſtrated that P H is
greater than F: It is manifeſt, therefore, that P M is leſſe than
ſeſquialter of P H: And conſequently that P H is greater than
the double of H M. Let P Z be double to Z M: T ſhall be the Cen
tre of Gravity of the whole Solid; the Centre of that part of it
which is within the Liquid, the Point Z; and of the remaining
part the Centre ſhall be in the Line Z T prolonged unto G. In
enclined ſo as that its Axis do make an Angle with the Surface of
the Liquid equall unto the Angle E B Ψ. For demit any Portion
into the Liquid ſo as that its Baſe
39[Figure 39]touch not the Liquids Surface;
and, if it can be done, let the
Axis not make an Angle with the
Liquids Surface equall to the
Angle E B Ψ; but firſt, let it be
greater: and the Portion being
cut thorow the Axis by a Plane e
rect unto [or upon] the Surface of
the Liquid, let the Section be A P
O L the Section of a Rightangled
Cone; the Section of the Surface of the Liquid X S; and let the
Axis of the Portion and Diameter of the Section be N O: and
draw P Y parallel to X S, and touching the Section A P O L in P;
and P M parallel to N O; and P I perpendicular to N O: and
moreover, let B R be equall to O ω, and R K to T ω; and let ω H
be perpendicular to the Axis. Now becauſe it hath been ſuppoſed
that the Axis of the Portion doth make an Angle with the Surface
of the Liquid greater than the Angle B, the Angle P Y I ſhall be
greater than the Angle B: Therefore the Square P I hath greater
proportion to the Square Y I, than the Square E Ψ hath to the
Square Ψ B: But as the Square P I is to the Square Y I, ſo is the
Line K R unto the Line I Y; and as the Square E Ψ is to the Square
Ψ B, ſo is half of the Line K R unto the Line Ψ B: Wherefore
(a) K R hath greater proportion to I Y, than the half of K R hath
to Ψ B: And, conſequently, I Y isleſſe than the double of Ψ B,
and is the double of O I: Therefore O I is leſſe than Ψ B; and I ω
greater than Ψ R: but Ψ R is equall to F: Therefore I ω is greater
than F. And becauſe that the Portion is ſuppoſed to be in Gra
vity unto the Liquid, as the Square F Q is to the Square B D; and
ſince that as the Portion is to the Liquid in Gravity, ſo is the part
thereof ſubmerged unto the whole Portion; and in regard that as
the part thereof ſubmerged is to the whole, ſo is the Square P M to
the Square O N; It followeth, that the Square P M is to the Square
N O, as the Square F Q is to the Square B D: And therefore F
Q is equall to P M: But it hath been demonſtrated that P H is
greater than F: It is manifeſt, therefore, that P M is leſſe than
ſeſquialter of P H: And conſequently that P H is greater than
the double of H M. Let P Z be double to Z M: T ſhall be the Cen
tre of Gravity of the whole Solid; the Centre of that part of it
which is within the Liquid, the Point Z; and of the remaining
part the Centre ſhall be in the Line Z T prolonged unto G. In