1
71[Figure 71]
ſtanding as before, we will again
demonſtrate, that N T is equall to
V I; and that the Portions A V Q
and A N Z are equall to each other.
Therefore, in regard, that in the
Equall and Like Portions A V Q L
and A N Z G, there are drawn
A Q and A Z cutting off equall Por
tions, they ſhall with the Diameters
of the Portions, contain equall
Angles. Therefore, in the Triangles
N L S and V ω C, the Angles at
the Points L and ω are equall; and the Right Line B S equall to
B C; S R to C R; N X to V H; and X T to H I: And, ſince
V Y is double to Y I, N X ſhall be greater than double of X T.
Let therefore, N M be double to M T. It is hence again manifeſt,
that the Portion will not remain, but ſhall incline on the part
towards A: But it was ſuppoſed, that the ſaid Portion did
touch the Surface of the Liquid in one ſole Point: Therefore,
its Baſe muſt of neceſſity ſubmerge farther into the Liquid.
71[Figure 71]ſtanding as before, we will again
demonſtrate, that N T is equall to
V I; and that the Portions A V Q
and A N Z are equall to each other.
Therefore, in regard, that in the
Equall and Like Portions A V Q L
and A N Z G, there are drawn
A Q and A Z cutting off equall Por
tions, they ſhall with the Diameters
of the Portions, contain equall
Angles. Therefore, in the Triangles
N L S and V ω C, the Angles at
the Points L and ω are equall; and the Right Line B S equall to
B C; S R to C R; N X to V H; and X T to H I: And, ſince
V Y is double to Y I, N X ſhall be greater than double of X T.
Let therefore, N M be double to M T. It is hence again manifeſt,
that the Portion will not remain, but ſhall incline on the part
towards A: But it was ſuppoſed, that the ſaid Portion did
touch the Surface of the Liquid in one ſole Point: Therefore,
its Baſe muſt of neceſſity ſubmerge farther into the Liquid.
CONCLVSION V.
If the Portion have leſſer proportion in Gravity to
the Liquid, than the Square F P to the Square
B D, being demitted into the Liquid, and in
clined, ſo, as that its Baſe touch not the Liquid,
it ſhall ſtand ſo inclined, as that its Axis ſhall
make an Angle with the Surface of the Liquid,
leſſe than the Angle ψ; And its Baſe ſhall
not in the leaſt touch the Liquids Surface.
the Liquid, than the Square F P to the Square
B D, being demitted into the Liquid, and in
clined, ſo, as that its Baſe touch not the Liquid,
it ſhall ſtand ſo inclined, as that its Axis ſhall
make an Angle with the Surface of the Liquid,
leſſe than the Angle ψ; And its Baſe ſhall
not in the leaſt touch the Liquids Surface.
Finally, let the Portion have leſſer proportion to the Liquid
in Gravity, than the Square F P hath to the Square B D; and
as the Portion is in Gravity to the Liquid, ſo let the
Square made of the Line ψ be to the Square B D. ψ ſhall be
leſſer than P F. Again, apply any Right Line as G I, falling
betwixt the Sections A G Q L and A X D, and parallel to B D;
and let it cut the Middle Conick Section in the Point H, and
in Gravity, than the Square F P hath to the Square B D; and
as the Portion is in Gravity to the Liquid, ſo let the
Square made of the Line ψ be to the Square B D. ψ ſhall be
leſſer than P F. Again, apply any Right Line as G I, falling
betwixt the Sections A G Q L and A X D, and parallel to B D;
and let it cut the Middle Conick Section in the Point H, and