1parameter be equall to K R: and
47[Figure 47]
let D S be Seſquialter of K R: but
S B is alſo Seſquialter of B R:
Therefore, draw a Line from A to
B; and thorow C draw C E Per
pendicular to B D, cutting the Line
A B in the Point E; and thorow E
draw E Z parallel unto B D. Again,
A B being divided into two equall
parts in T, draw T H parallel to the
ſame B D: and let Sections of
Rightangled Cones be deſcribed, A E I about the Diameter E Z;
and A T D about the Diameter T H; and let them be like to the
Portion A B L: Now the Section of the Cone A E I, ſhall paſs
thorow K; and the Line drawn from R perpendicular unto B D,
ſhall cut the ſaid A E I; let it cut it in the Points Y G: and
thorow Y and G draw P Y Q and O G N parallels unto B D, and
cutting A T D in the Points F and X: laſtly, draw P Φ and O X
touching the Section A P O L in the Points P and O. In regard,
therefore, that the three Portions A P O L, A E I, and A T D are
contained betwixt Right Lines, and the Sections of Rightangled
Cones, and are right alike and unequall, touching one another, upon
one and the ſame Baſe; and N X G O being drawn from the
Point N upwards, and Q F Y P from Q: O G ſhall have to G X
a proportion compounded of the proportion, that I L hath to L A,
and of the proportion that A D hath to DI: But I L is to L A,
as two to five: And C B is to B D, as ſix to fifteen; that is, as two
to five: And as C B is to B D, ſo is E B to B A; and D Z to
D A: And of D Z and D A, L I and L A are double: and A D
is to D I, as five to one: But the proportion compounded of the
proportion of two to five, and of the proportion of five to one, is
the ſame with that of two to one: and two is to one, in double
proportion: Therefore, O G is double of GX: and, in the ſame
manner is P Y proved to be double of Y F: Therefore, ſince that
D S is Seſquialter of K R; B S ſhall be the Exceſs by which the
Axis is greater than Seſquialter of the Semi-parameter. If there
fore, the Portion have the ſame proportion in Gravity unto the
Liquid, as the Square made of the Line B S, hath to the Square
made of B D, or greater, being demitted into the Liquid, ſo as hat
its Baſe touch not the Liquid, it ſhall ſtand erect, or perpendicular:
For it hath been demonſtrated above, that the Portion whoſe
Axis is greater than Seſquialter of the Semi-parameter, if it have
not leſser proportion in Gravity unto the Liquid, than the Square
47[Figure 47]let D S be Seſquialter of K R: but
S B is alſo Seſquialter of B R:
Therefore, draw a Line from A to
B; and thorow C draw C E Per
pendicular to B D, cutting the Line
A B in the Point E; and thorow E
draw E Z parallel unto B D. Again,
A B being divided into two equall
parts in T, draw T H parallel to the
ſame B D: and let Sections of
Rightangled Cones be deſcribed, A E I about the Diameter E Z;
and A T D about the Diameter T H; and let them be like to the
Portion A B L: Now the Section of the Cone A E I, ſhall paſs
thorow K; and the Line drawn from R perpendicular unto B D,
ſhall cut the ſaid A E I; let it cut it in the Points Y G: and
thorow Y and G draw P Y Q and O G N parallels unto B D, and
cutting A T D in the Points F and X: laſtly, draw P Φ and O X
touching the Section A P O L in the Points P and O. In regard,
therefore, that the three Portions A P O L, A E I, and A T D are
contained betwixt Right Lines, and the Sections of Rightangled
Cones, and are right alike and unequall, touching one another, upon
one and the ſame Baſe; and N X G O being drawn from the
Point N upwards, and Q F Y P from Q: O G ſhall have to G X
a proportion compounded of the proportion, that I L hath to L A,
and of the proportion that A D hath to DI: But I L is to L A,
as two to five: And C B is to B D, as ſix to fifteen; that is, as two
to five: And as C B is to B D, ſo is E B to B A; and D Z to
D A: And of D Z and D A, L I and L A are double: and A D
is to D I, as five to one: But the proportion compounded of the
proportion of two to five, and of the proportion of five to one, is
the ſame with that of two to one: and two is to one, in double
proportion: Therefore, O G is double of GX: and, in the ſame
manner is P Y proved to be double of Y F: Therefore, ſince that
D S is Seſquialter of K R; B S ſhall be the Exceſs by which the
Axis is greater than Seſquialter of the Semi-parameter. If there
fore, the Portion have the ſame proportion in Gravity unto the
Liquid, as the Square made of the Line B S, hath to the Square
made of B D, or greater, being demitted into the Liquid, ſo as hat
its Baſe touch not the Liquid, it ſhall ſtand erect, or perpendicular:
For it hath been demonſtrated above, that the Portion whoſe
Axis is greater than Seſquialter of the Semi-parameter, if it have
not leſser proportion in Gravity unto the Liquid, than the Square