PROP. V. THEOR. V.
The Right Portion of a Right-Angled Conoid lighter
than the Liquid, when it ſhall have its Axis great
er than Seſquialter of the Semi-parameter, if it have
not greater proportion in Gravity to the Liquid [of
equal Maſs] than the Exceſſe by which the Square
made of the Axis is greater than the Square made
of the Exceſſe by which the Axis is greater than
ſeſquialter of the Semi-Parameter hath to the
Square made of the Axis being demitted into the Li
quid, ſo as that its Baſe be wholly within the Liquid,
and being ſet inclining, it ſhall not remain ſo inclined,
but ſhall turn about till that its Axis ſhall be accor
ding to the Perpendicular.
than the Liquid, when it ſhall have its Axis great
er than Seſquialter of the Semi-parameter, if it have
not greater proportion in Gravity to the Liquid [of
equal Maſs] than the Exceſſe by which the Square
made of the Axis is greater than the Square made
of the Exceſſe by which the Axis is greater than
ſeſquialter of the Semi-Parameter hath to the
Square made of the Axis being demitted into the Li
quid, ſo as that its Baſe be wholly within the Liquid,
and being ſet inclining, it ſhall not remain ſo inclined,
but ſhall turn about till that its Axis ſhall be accor
ding to the Perpendicular.
For let any Portion be demitted into the Liquid, as hath been
ſaid; and let its Baſe be wholly within the Liquid, And being
cut thorow its Axis by a Plain erect upon the Surface of the
Liquid; its Section ſhall be the Section
254[Figure 254]
of a Rightangled Cone: Let it be
A P O L, and let the Axis of the Por
tion and Diameter of the Section be
N O; and the Section of the Surface of
the Liquid I S. And becauſe the Axis
is not according to the Perpendicu
lar, N O will not be at equall angles
with I S. Draw K ω touching the Se
ction A P O L in P, and parallel unto
I S: and thorow P, draw P F parallel unto N O: and take the
Centres of Gravity; and of the Solid A P O L let the Centre be
R; and of that which lyeth above the Liquid let the Centre be B;
and draw a Line from B to R, prolonging it to G; which let be the
Centre of Gravity of the Solid demerged within the Liquid: and
moreover, take R H equall to the Semi-parameter, and let O H be
double to H M; and do in the reſt as hath been ſaid (a) above.
Now foraſmuch as it was ſuppoſed that the Portion hath not greater
proportion in Gravity to the Liquid, than the Exceſſe by which
the Square N O is greater than the Square M O, hath to the ſaid
Square N O: And in regard that whatever proportion in Gravity
ſaid; and let its Baſe be wholly within the Liquid, And being
cut thorow its Axis by a Plain erect upon the Surface of the
Liquid; its Section ſhall be the Section
254[Figure 254]of a Rightangled Cone: Let it be
A P O L, and let the Axis of the Por
tion and Diameter of the Section be
N O; and the Section of the Surface of
the Liquid I S. And becauſe the Axis
is not according to the Perpendicu
lar, N O will not be at equall angles
with I S. Draw K ω touching the Se
ction A P O L in P, and parallel unto
I S: and thorow P, draw P F parallel unto N O: and take the
Centres of Gravity; and of the Solid A P O L let the Centre be
R; and of that which lyeth above the Liquid let the Centre be B;
and draw a Line from B to R, prolonging it to G; which let be the
Centre of Gravity of the Solid demerged within the Liquid: and
moreover, take R H equall to the Semi-parameter, and let O H be
double to H M; and do in the reſt as hath been ſaid (a) above.
Now foraſmuch as it was ſuppoſed that the Portion hath not greater
proportion in Gravity to the Liquid, than the Exceſſe by which
the Square N O is greater than the Square M O, hath to the ſaid
Square N O: And in regard that whatever proportion in Gravity