1Section of the Portion be A P O L, the Section of a Rightangled
Cone; and let the Axis of the Portion and Diameter of the Section
be N O, and the Section of the Surface of the Liquid I S. If now
the Portion be not erect, then N O ſhall not be at equall Angles with
I S. Draw R ω touching the Section of the Rightangled Conoid
in P, and parallel to I S: and from the Point P and parall to O N
draw P F: and take the Centers of Gravity; and of the Solid A
P O L let the Centre be R; and of that which lyeth within the
Liquid let the Centre be B; and draw a Line from B to R pro
longing it to G, that G may be the Centre of Gravity of the Solid
that is above the Liquid. And becauſe N O is ſeſquialter of R
O, and is greater than ſeſquialter of the Semi-Parameter; it is ma
nifeſt that (a) R O is greater than the
252[Figure 252]
Semi-parameter. ^{*}Let therefore R
H be equall to the Semi-Parameter,
^{*} and O H double to H M. Foraſ
much therefore as N O is ſeſquialter
of R O, and M O of O H, (b) the
Remainder N M ſhall be ſeſquialter
of the Remainder R H: Therefore
the Axis is greater than ſeſquialter
of the Semi parameter by the quan
tity of the Line M O. And let it be
ſuppoſed that the Portion hath not leſſe proportion in Gravity unto
the Liquid of equall Maſſe, than the Square that is made of the
Exceſſe by which the Axis is greater than ſeſquialter of the Semi
parameter hath to the Square made of the Axis: It is therefore ma
nifeſt that the Portion hath not leſſe proportion in Gravity to the
Liquid than the Square of the Line M O hath to the Square of N
O: But look what proportion the Portion hath to the Liquid in
Gravity, the ſame hath the Portion ſubmerged to the whole Solid:
for this hath been demonſtrated (c) above: ^{*}And look what pro
portion the ſubmerged Portion hath to the whole Portion, the
ſame hath the Square of P F unto the Square of N O: For it hath
been demonſtrated in (d) Lib. de Conoidibus, that if from a Right
angled Conoid two Portions be cut by Planes in any faſhion pro
duced, theſe Portions ſhall have the ſame Proportion to each
other as the Squares of their Axes: The Square of P F, therefore,
hath not leſſe proportion to the Square of N O than the Square of
M O hath to the Square of N O: ^{*}Wherefore P F is not leſſe than
M O, ^{*}nor B P than H O. ^{*}If therefore, a Right Line be drawn
from H at Right Angles unto N O, it ſhall meet with B P, and ſhall
fall betwixt B and P; let it fall in T: (e) And becauſe P F is
parallel to the Diameter, and H T is perpendicular unto the ſame
Diameter, and R H equall to the Semi-parameter; a Line drawn
from R to T and prolonged, maketh Right Angles with the Line
Cone; and let the Axis of the Portion and Diameter of the Section
be N O, and the Section of the Surface of the Liquid I S. If now
the Portion be not erect, then N O ſhall not be at equall Angles with
I S. Draw R ω touching the Section of the Rightangled Conoid
in P, and parallel to I S: and from the Point P and parall to O N
draw P F: and take the Centers of Gravity; and of the Solid A
P O L let the Centre be R; and of that which lyeth within the
Liquid let the Centre be B; and draw a Line from B to R pro
longing it to G, that G may be the Centre of Gravity of the Solid
that is above the Liquid. And becauſe N O is ſeſquialter of R
O, and is greater than ſeſquialter of the Semi-Parameter; it is ma
nifeſt that (a) R O is greater than the
252[Figure 252]Semi-parameter. ^{*}Let therefore R
H be equall to the Semi-Parameter,
^{*} and O H double to H M. Foraſ
much therefore as N O is ſeſquialter
of R O, and M O of O H, (b) the
Remainder N M ſhall be ſeſquialter
of the Remainder R H: Therefore
the Axis is greater than ſeſquialter
of the Semi parameter by the quan
tity of the Line M O. And let it be
ſuppoſed that the Portion hath not leſſe proportion in Gravity unto
the Liquid of equall Maſſe, than the Square that is made of the
Exceſſe by which the Axis is greater than ſeſquialter of the Semi
parameter hath to the Square made of the Axis: It is therefore ma
nifeſt that the Portion hath not leſſe proportion in Gravity to the
Liquid than the Square of the Line M O hath to the Square of N
O: But look what proportion the Portion hath to the Liquid in
Gravity, the ſame hath the Portion ſubmerged to the whole Solid:
for this hath been demonſtrated (c) above: ^{*}And look what pro
portion the ſubmerged Portion hath to the whole Portion, the
ſame hath the Square of P F unto the Square of N O: For it hath
been demonſtrated in (d) Lib. de Conoidibus, that if from a Right
angled Conoid two Portions be cut by Planes in any faſhion pro
duced, theſe Portions ſhall have the ſame Proportion to each
other as the Squares of their Axes: The Square of P F, therefore,
hath not leſſe proportion to the Square of N O than the Square of
M O hath to the Square of N O: ^{*}Wherefore P F is not leſſe than
M O, ^{*}nor B P than H O. ^{*}If therefore, a Right Line be drawn
from H at Right Angles unto N O, it ſhall meet with B P, and ſhall
fall betwixt B and P; let it fall in T: (e) And becauſe P F is
parallel to the Diameter, and H T is perpendicular unto the ſame
Diameter, and R H equall to the Semi-parameter; a Line drawn
from R to T and prolonged, maketh Right Angles with the Line