PROP. IX. THE OR. IX.
The Right Portion of a Rightangled Conoid, when it
ſhall have its Axis greater than Seſquialter of the
Semi-parameter, but leſſer than to be unto the ſaid
Semi-parameter in proportion as fifteen to four, and
hath greater proportion in Gravity to the Liquid, than
the exceſs by which the Square made of the Axis is
greater than the Square made of the Exceſs, by which
the Axis is greater than Seſquialter of the Semi
parameter, hath to the Square made of the Axis,
being demitted into the Liquid, ſo as that its Baſe
be wholly within the Liquid, and being ſet inclining
it ſhall neither turn about, ſo as that its Axis ſtand
according to the Perpendicular, nor remain inclined,
ſave only when the Axis makes an Angle with
the Surface of the Liquid, equall to that aßigned
as before.
ſhall have its Axis greater than Seſquialter of the
Semi-parameter, but leſſer than to be unto the ſaid
Semi-parameter in proportion as fifteen to four, and
hath greater proportion in Gravity to the Liquid, than
the exceſs by which the Square made of the Axis is
greater than the Square made of the Exceſs, by which
the Axis is greater than Seſquialter of the Semi
parameter, hath to the Square made of the Axis,
being demitted into the Liquid, ſo as that its Baſe
be wholly within the Liquid, and being ſet inclining
it ſhall neither turn about, ſo as that its Axis ſtand
according to the Perpendicular, nor remain inclined,
ſave only when the Axis makes an Angle with
the Surface of the Liquid, equall to that aßigned
as before.
Let there be a Portion as was ſaid; and ſuppoſe D B equall to
the Axis of the Portion: and let B K be double to K D; and
K R equall to the Semi-parameter: and C B Seſquialter of
B R. And as the Portion is to the Liquid in Gravity, ſo let the Ex
ceſſe by which the Square B D exceeds the Square F Q be to the
Square B D: and let F be double to Q: It is manifeſt, therefore,
that the Exceſſe by which the
270[Figure 270]
Square B D is greater than the
Square B C hath leſser proportion
to the Square B D, than the Exceſs
by which the Square B D is greater
than the Square F Q hath to the
Square B D; for B C is the Exceſs
by which the Axis of the Portion is
greater than Seſquialter of the
Semi-parameter: And, therefore,
the Axis of the Portion: and let B K be double to K D; and
K R equall to the Semi-parameter: and C B Seſquialter of
B R. And as the Portion is to the Liquid in Gravity, ſo let the Ex
ceſſe by which the Square B D exceeds the Square F Q be to the
Square B D: and let F be double to Q: It is manifeſt, therefore,
that the Exceſſe by which the
270[Figure 270]Square B D is greater than the
Square B C hath leſser proportion
to the Square B D, than the Exceſs
by which the Square B D is greater
than the Square F Q hath to the
Square B D; for B C is the Exceſs
by which the Axis of the Portion is
greater than Seſquialter of the
Semi-parameter: And, therefore,
the Square B D doth more exceed
the Square F Q, than doth the
Square B C: And, conſequently, the Line F Q is leſs than B C;