1the ſame manner we might demon
266[Figure 266]
ſtrate the Line T H to be perpendi
cular unto the Surface of the Liquid:
and that the Portion demerged with
in the Liquid moveth or aſcend
eth out of the Liquid according to
the Perpendicular that ſhall be
drawn thorow Z unto the Surface
of the Liquid; and that the part
that is above the Liquid deſcendeth
into the Liquid according to that
drawn thorow G: therefore the Portion will not continue ſo inclined
as was ſuppoſed: But neither ſhall it return to Rectitude or Per
pendicularity; For that of the Perpendiculars drawn thorow Z and
G, that paſſing thorow Z doth fall on thoſe parts which are to
wards L; and that that paſſeth thorow G on thoſe towards A:
Wherefore it followeth that the Centre Z do move upwards,
and G downwards: Therefore the parts of the whole Solid which
are towards A ſhall move downwards, and thoſe towards L up
wards. Again let the Propoſition run in other termes; and let
the Axis of the Portion make an Angle with the Surface of the
Liquid leſſe than that which is at B. Therefore the Square P I
hath leſſer Proportion unto the Square
267[Figure 267]
I Y, than the Square E Ψ hath to the
Square Ψ B: Wherefore K R hath
leſſer proportion to I Y, than the half
of K R hath to Ψ B: And, for the
ſame reaſon, I Y is greater than dou
ble of Ψ B: but it is double of O I:
Therefore O I ſhall be greater than
Ψ B: But the Totall O ω is equall
to R B, and the Remainder ω I leſſe
than ψ R: Wherefore P H ſhall alſo
be leſſe than F. And, in regard that
M P is equall to F Q, it is manifeſt that P M is greater than ſeſqui
alter of P H; and that P H is leſſe than double of H M. Let P Z
be double to Z M. The Centre of Gravity of the whole Solid ſhall
again be T; that of the part which is within the Liquid Z; and
drawing a Line from Z to T, the Centre of Gravity of that which
is above the Liquid ſhall be found in that Line portracted, that is
in G: Therefore, Perpendiculars being drawn thorow Z and G
unto the Surface of the Liquid that are parallel to T H, it followeth
that the ſaid Portion ſhall not ſtay, but ſhall turn about till
that its Axis do make an Angle with the Waters Surface greater than
that which it now maketh. And becauſe that when before we
266[Figure 266]ſtrate the Line T H to be perpendi
cular unto the Surface of the Liquid:
and that the Portion demerged with
in the Liquid moveth or aſcend
eth out of the Liquid according to
the Perpendicular that ſhall be
drawn thorow Z unto the Surface
of the Liquid; and that the part
that is above the Liquid deſcendeth
into the Liquid according to that
drawn thorow G: therefore the Portion will not continue ſo inclined
as was ſuppoſed: But neither ſhall it return to Rectitude or Per
pendicularity; For that of the Perpendiculars drawn thorow Z and
G, that paſſing thorow Z doth fall on thoſe parts which are to
wards L; and that that paſſeth thorow G on thoſe towards A:
Wherefore it followeth that the Centre Z do move upwards,
and G downwards: Therefore the parts of the whole Solid which
are towards A ſhall move downwards, and thoſe towards L up
wards. Again let the Propoſition run in other termes; and let
the Axis of the Portion make an Angle with the Surface of the
Liquid leſſe than that which is at B. Therefore the Square P I
hath leſſer Proportion unto the Square
267[Figure 267]I Y, than the Square E Ψ hath to the
Square Ψ B: Wherefore K R hath
leſſer proportion to I Y, than the half
of K R hath to Ψ B: And, for the
ſame reaſon, I Y is greater than dou
ble of Ψ B: but it is double of O I:
Therefore O I ſhall be greater than
Ψ B: But the Totall O ω is equall
to R B, and the Remainder ω I leſſe
than ψ R: Wherefore P H ſhall alſo
be leſſe than F. And, in regard that
M P is equall to F Q, it is manifeſt that P M is greater than ſeſqui
alter of P H; and that P H is leſſe than double of H M. Let P Z
be double to Z M. The Centre of Gravity of the whole Solid ſhall
again be T; that of the part which is within the Liquid Z; and
drawing a Line from Z to T, the Centre of Gravity of that which
is above the Liquid ſhall be found in that Line portracted, that is
in G: Therefore, Perpendiculars being drawn thorow Z and G
unto the Surface of the Liquid that are parallel to T H, it followeth
that the ſaid Portion ſhall not ſtay, but ſhall turn about till
that its Axis do make an Angle with the Waters Surface greater than
that which it now maketh. And becauſe that when before we