1and drawing a Line from Z to T pro
272[Figure 272]
long it unto G. The Centre of
Gravity of the whole Portion ſhall
be T; of that part which is above
the Liquid Z; and of the Remain
der which is within the Liquid, the
Centre ſhall be in the Line Z T pro
longed; let it be in G: It ſhall be
demonſtrated, as before, that T H
is perpendicular to the Surface of
the Liquid, and that the Lines
drawn thorow Z and G parallel to the ſaid T H, are alſo perpen
diculars unto the ſame: Therefore, the Part which is above the
Liquid ſhall move downwards, along that which paſseth thorow Z;
and that which is within it, ſhall move upwards, along that which
paſseth thorow G: And, therefore, the Portion ſhall not remain
ſo inclined, nor ſhall ſo turn about, as that its Axis be perpendicular
unto the Surface of the Liquid; for the parts towards L ſhall move
downwards, and thoſe towards A upwards; as may appear by
the things already demonſtrated. And, if the Axis ſhould make
an Angle with the Surface of the Liquid, leſs than the Angle B;
it ſhall in like manner be demonſtrated, that the Portion will not
reſt, but incline untill that its Axis do make an Angle with the
Surface of the Liquid, equall to the Angle B.
272[Figure 272]long it unto G. The Centre of
Gravity of the whole Portion ſhall
be T; of that part which is above
the Liquid Z; and of the Remain
der which is within the Liquid, the
Centre ſhall be in the Line Z T pro
longed; let it be in G: It ſhall be
demonſtrated, as before, that T H
is perpendicular to the Surface of
the Liquid, and that the Lines
drawn thorow Z and G parallel to the ſaid T H, are alſo perpen
diculars unto the ſame: Therefore, the Part which is above the
Liquid ſhall move downwards, along that which paſseth thorow Z;
and that which is within it, ſhall move upwards, along that which
paſseth thorow G: And, therefore, the Portion ſhall not remain
ſo inclined, nor ſhall ſo turn about, as that its Axis be perpendicular
unto the Surface of the Liquid; for the parts towards L ſhall move
downwards, and thoſe towards A upwards; as may appear by
the things already demonſtrated. And, if the Axis ſhould make
an Angle with the Surface of the Liquid, leſs than the Angle B;
it ſhall in like manner be demonſtrated, that the Portion will not
reſt, but incline untill that its Axis do make an Angle with the
Surface of the Liquid, equall to the Angle B.
A
B
C
D
E
F
G
COMMANDINE.
And, therefore, the Square B D doth more exceed the Square
F Q, than doth the Square B C: And, conſequently, the Line
F Q, is leſs than B C; and F leſs than B R.] Becauſe the Exceſs by
which the Square B D exceedeth the Square B C; having leſs proportion unto the Square B D,
than the Exceſs by which the Square B D exceedeth the Square F Q, hath to the ſaid Square;
(a) the Exceſs by which the Square B D exceedeth the Square B C ſhall be leſs than the Exceſs
by which it exceedeth the Square F Q: Therefore, the Square F Q is leſs than the Square B C:
and, conſquently, the Line F Q leſs than the Line BC: But F Q hath the ſameproportion
to F, that B C hath to B R; for the Antecedents are each Seſquialter of their conſequents:
And (b) F Q being leſs than B C, F ſhall alſo be leſs than B R.
F Q, than doth the Square B C: And, conſequently, the Line
F Q, is leſs than B C; and F leſs than B R.] Becauſe the Exceſs by
which the Square B D exceedeth the Square B C; having leſs proportion unto the Square B D,
than the Exceſs by which the Square B D exceedeth the Square F Q, hath to the ſaid Square;
(a) the Exceſs by which the Square B D exceedeth the Square B C ſhall be leſs than the Exceſs
by which it exceedeth the Square F Q: Therefore, the Square F Q is leſs than the Square B C:
and, conſquently, the Line F Q leſs than the Line BC: But F Q hath the ſameproportion
to F, that B C hath to B R; for the Antecedents are each Seſquialter of their conſequents:
And (b) F Q being leſs than B C, F ſhall alſo be leſs than B R.
A
(a) By 8. of the
fifth.
fifth.
(b) By 14. of the
fifth.
fifth.
And, for that cauſe, K R hath greater proportion to S Y, than
the half of K R hath to ψ B.] For K R is to S Y, as the Square P S is to the Square
the half of K R hath to ψ B.] For K R is to S Y, as the Square P S is to the Square
S Y: and the half of the Line K R is to the Line ψ B, as the Square E ψ is to the Square ψ B.
B
And S O leſs than ψ B.] For S Y is double of S O.
C
And P H greater than F.] For P H is equall to S ω, and R ψ equall to F.
D
And, therefore, the whole Portion ſhall have the ſame propor
tion to that part which is above the Liquid, that the Square B D
hath to the Square F Q] Becauſe that the part ſubmerged, being to the whole Portion
as the Exceſs by which the Square B D is greater than the Square F Q, is to the Square B D;
the whole Portion, Converting, ſhall be to the part thereof ſubmerged, as the Square B D is to