Therefore, A Q and A M do make equall Acute Angles with
the Diameters of the Portions.] We demonſtrate this as in the Commentaries
upon the ſecond Concluſion.
the Diameters of the Portions.] We demonſtrate this as in the Commentaries
upon the ſecond Concluſion.
It is to be demonſtrated in the ſame manner, that the Portion
that hath the ſame proportion in Gravity to the Liquid, that the
Square P F hath to the Square B D,
being demitted into the Liquid, ſo,
65[Figure 65]
as that its Baſe touch not the Li
quid, it ſhall ſtand inclined, ſo, as
that its Baſe touch the Surface of the
Liquid in one point only; and its Axis
ſhall make therewith an angle equall
to the Angle φ.] Let the Portion be to the
Liquid in Gravity, as the Square P F to the
Square B D: and being demitted into the
Liquid, ſo inclined, as that its Baſe touch not
the Liquid, let it be cut thorow the Axis by a
Plane erect to the Surface of the Liquid, that
that the Section may be A M O L, the Section
of a Rightangled Cone; and, let the Section of the Liquids Surface be I O; and the Axit
of the Portion and Diameter of the Section B D; which let be cut into the ſame parts as
we ſaid before, and draw M N parallel to I O, that it may touch the Section in the Point
M; and M T parallel to B D, and P M S perpe ndicular to the ſame. It is to be demon
strated, that the Portion ſhall not reſt, but ſhall incline, ſo, as that it touch the Liquids
Surface, in one Point of its Baſe only. For,
66[Figure 66]
draw P C perpendicular to B D; and drawing
a Line from A to F, prolong it till it meet with
the Section in que and thorow P draw P φ pa
rallel to A Q: Now, by the things allready de
monſtrated by us, A F and F Q ſhall be equall
to one another. And being that the Portion hath
the ſame proportion in Gravity unto the Liquid,
that the Square P F hath to the Square B D; and
ſeeing that the part ſubmerged, hath the ſame pro-
partion to the whole Portion; that is, the Squàre
M T to the Square B D; (g) the Square M T
ſhall be equall to the Square P F; and, by the
ſame reaſon, the Line M T equall to the Line
P F. So that there being drawn in the equall & like
portions A P Q Land A M O L, the Lines A Q and I O which cut off equall Portions, the
firſt from the Extreme term of the Baſe, the laſt not from the Extremity; it followeth, that
A Q drawn from the Extremity, containeth a leſſer Acute Angle with the Diameter of the
Portion, than I O: But the Line P φ is parallel to the Line A Q, and M N to I O: There
fore, the Angle at φ ſhall be leſſer than the Angle at N; but the Line B C greater than B S;
and S R, that is, M X, greater than C R, that is, than P Y: and, by the ſame reaſon, X T
leſſer than Y F. And, ſince P Y is double to Y F, M X ſhall be greater than double to
Y F, and much greater than double of X T. Let M H be double to H T, and draw a
Line from H to K, prolonging it. Now, the Centre of Gravity of the whole Portion
ſhall be the Point K; of the part within the Liquid H; and of the Remaining part above
the Liquid in the Line H K produced, as ſuppoſe in ω It ſhall be demonſtrated in the ſame
manner, as before, that both the Line K H and thoſe that are drawn thorow the Points H
and ω parallel to the ſaid K H, are perpendicular to the Surface of the Liquid: The
Portion therefore, ſhall not reſt; but when it ſhall be enclined ſo far as to touch the Sur
face of the Liquid in one Point and no more, then it ſhall ſtay. For the Angle at N
that hath the ſame proportion in Gravity to the Liquid, that the
Square P F hath to the Square B D,
being demitted into the Liquid, ſo,
65[Figure 65]as that its Baſe touch not the Li
quid, it ſhall ſtand inclined, ſo, as
that its Baſe touch the Surface of the
Liquid in one point only; and its Axis
ſhall make therewith an angle equall
to the Angle φ.] Let the Portion be to the
Liquid in Gravity, as the Square P F to the
Square B D: and being demitted into the
Liquid, ſo inclined, as that its Baſe touch not
the Liquid, let it be cut thorow the Axis by a
Plane erect to the Surface of the Liquid, that
that the Section may be A M O L, the Section
of a Rightangled Cone; and, let the Section of the Liquids Surface be I O; and the Axit
of the Portion and Diameter of the Section B D; which let be cut into the ſame parts as
we ſaid before, and draw M N parallel to I O, that it may touch the Section in the Point
M; and M T parallel to B D, and P M S perpe ndicular to the ſame. It is to be demon
strated, that the Portion ſhall not reſt, but ſhall incline, ſo, as that it touch the Liquids
Surface, in one Point of its Baſe only. For,
66[Figure 66]draw P C perpendicular to B D; and drawing
a Line from A to F, prolong it till it meet with
the Section in que and thorow P draw P φ pa
rallel to A Q: Now, by the things allready de
monſtrated by us, A F and F Q ſhall be equall
to one another. And being that the Portion hath
the ſame proportion in Gravity unto the Liquid,
that the Square P F hath to the Square B D; and
ſeeing that the part ſubmerged, hath the ſame pro-
partion to the whole Portion; that is, the Squàre
M T to the Square B D; (g) the Square M T
ſhall be equall to the Square P F; and, by the
ſame reaſon, the Line M T equall to the Line
P F. So that there being drawn in the equall & like
portions A P Q Land A M O L, the Lines A Q and I O which cut off equall Portions, the
firſt from the Extreme term of the Baſe, the laſt not from the Extremity; it followeth, that
A Q drawn from the Extremity, containeth a leſſer Acute Angle with the Diameter of the
Portion, than I O: But the Line P φ is parallel to the Line A Q, and M N to I O: There
fore, the Angle at φ ſhall be leſſer than the Angle at N; but the Line B C greater than B S;
and S R, that is, M X, greater than C R, that is, than P Y: and, by the ſame reaſon, X T
leſſer than Y F. And, ſince P Y is double to Y F, M X ſhall be greater than double to
Y F, and much greater than double of X T. Let M H be double to H T, and draw a
Line from H to K, prolonging it. Now, the Centre of Gravity of the whole Portion
ſhall be the Point K; of the part within the Liquid H; and of the Remaining part above
the Liquid in the Line H K produced, as ſuppoſe in ω It ſhall be demonſtrated in the ſame
manner, as before, that both the Line K H and thoſe that are drawn thorow the Points H
and ω parallel to the ſaid K H, are perpendicular to the Surface of the Liquid: The
Portion therefore, ſhall not reſt; but when it ſhall be enclined ſo far as to touch the Sur
face of the Liquid in one Point and no more, then it ſhall ſtay. For the Angle at N