1ſaid K ω in H, and A S is parallel unto the Line that toucheth in
P; It is neceſſary that P I hath unto P H either the ſame proportion
that N ω hath to ω O, or greater; for this hath already been de
monſtrated: But N ω is ſeſquialter of ω O; and P I, therefore, is
either Seſquialter of H P, or more than ſeſquialter: Wherefore
P H is to H I either double, or leſſe than double. Let P T be
double to T I: the Centre of Gravity of the part which is within
the Liquid ſhall be the Point T. Therefore draw a Line from T
to F prolonging it; and let the Centre of
256[Figure 256]
Gravity of the part which is above the Liquid
be G: and from the Point B at Right Angles
unto N O draw B R. And ſeeing that P I is
parallel unto the Diameter N O, and B R
perpendicular unto the ſaid Diameter, and F
B equall to the Semi-parameter; It is mani
feſt that the Line drawn thorow the Points
F and R being prolonged, maketh equall
Angles with that which toucheth the Section
A P O L in the Point P: and therefore doth alſo make Right An
gles with A S, and with the Surface of the Liquid: and the Lines
drawn thorow T and G parallel unto F R ſhall be alſo perpendicu
lar to the Surface of the Liquid: and of the Solid Magnitude A P
O L, the part which is within the Liquid moveth upwards according
to the Perpendicular drawn thorow T; and the part which is above
the Liquid moveth downwards according to that drawn thorow G:
The Solid A P O L, therefore, ſhall turn about, and its Baſe ſhall
not in the leaſt touch the Surface of the Liquid, And if P I do not
cut the Line K ω, as in the ſecond Figure, it is manifeſt that the
Point T, which is the Centre of Gravity of the ſubmerged Portion,
falleth betwixt P and I: And for the other particulars remaining,
they are demonſtrated like as before.
P; It is neceſſary that P I hath unto P H either the ſame proportion
that N ω hath to ω O, or greater; for this hath already been de
monſtrated: But N ω is ſeſquialter of ω O; and P I, therefore, is
either Seſquialter of H P, or more than ſeſquialter: Wherefore
P H is to H I either double, or leſſe than double. Let P T be
double to T I: the Centre of Gravity of the part which is within
the Liquid ſhall be the Point T. Therefore draw a Line from T
to F prolonging it; and let the Centre of
256[Figure 256]Gravity of the part which is above the Liquid
be G: and from the Point B at Right Angles
unto N O draw B R. And ſeeing that P I is
parallel unto the Diameter N O, and B R
perpendicular unto the ſaid Diameter, and F
B equall to the Semi-parameter; It is mani
feſt that the Line drawn thorow the Points
F and R being prolonged, maketh equall
Angles with that which toucheth the Section
A P O L in the Point P: and therefore doth alſo make Right An
gles with A S, and with the Surface of the Liquid: and the Lines
drawn thorow T and G parallel unto F R ſhall be alſo perpendicu
lar to the Surface of the Liquid: and of the Solid Magnitude A P
O L, the part which is within the Liquid moveth upwards according
to the Perpendicular drawn thorow T; and the part which is above
the Liquid moveth downwards according to that drawn thorow G:
The Solid A P O L, therefore, ſhall turn about, and its Baſe ſhall
not in the leaſt touch the Surface of the Liquid, And if P I do not
cut the Line K ω, as in the ſecond Figure, it is manifeſt that the
Point T, which is the Centre of Gravity of the ſubmerged Portion,
falleth betwixt P and I: And for the other particulars remaining,
they are demonſtrated like as before.
A
B
C
D
E
COMMANDINE.
A
It is to be demonſtrated that the ſaid Portion ſhall not continue
ſo, but ſhall turn about in ſuch manner as that its Baſe do in no wiſe
touch the Surface of the Liquid.] Theſe words are added by us, as having been
omitted by Tartaglia.
ſo, but ſhall turn about in ſuch manner as that its Baſe do in no wiſe
touch the Surface of the Liquid.] Theſe words are added by us, as having been
omitted by Tartaglia.
Now becauſe N O hath greater proportion to F ω than unto
the Semi parameter.] For the Diameter of the Portion N O hath unto F ω the
ſame proportion as fifteen to fower: But it was ſuppoſed to have leſſe proportion unto the
Semi-parameter than fifteen to fower: Wherefore N O hath greater proportion unto F ω
than unto the Semi-parameter: And therefore (a) the Semi-parameter ſhall be greater
than the ſaid F ω.
B
(a) By 10. of the
fifth.
fifth.
Foraſmuch, therefore, as in the Portion A P O L, contained, be
twixt the Right Line and the Section of the Rightangled Cone K
ω is parallel to A L, and P I parallel unto the Diameter, and cut by
twixt the Right Line and the Section of the Rightangled Cone K
ω is parallel to A L, and P I parallel unto the Diameter, and cut by