1and touching the Section in P, and T P parallel to B D; and P S perpen
dicular unto B D. It is to be demonſtrated that the Portion ſhall
61[Figure 61]
not ſtand ſo, but ſhall encline until
that the Baſe touch the Surface of
the Liquid, in one Point only, for let
the ſuperior figure ſtand as it was,
and draw O C, Perpendicular to B D;
and drawing a Line from A to X,
prolong it to Q: A X ſhalbe equall
to X que Then draw O X parallel
to A que And becauſe the Portion
is ſuppoſed to have the ſame pro
portion in Gravity to the Liquid
that the ſquare X O hath to the
Square B D; the part thereof ſubmerged ſhall alſo have the ſame
proportion to the whole; that is, the Square T P to the Square
B D; and ſo T P ſhall be equal to X O: And ſince that of the Portions
I P M and A O Q the Diameters are equall, the portions ſhall alſo be
equall. Again, becauſe that in the Equall and Like Portions A O Q L
and AP ML the Lines A Q and I M, which cut off equall Por
tions, are drawn, that, from the Extremity of the Baſe, and this
not from the Extremity; it appeareth that that which is drawn from
the end or Extremity of the Baſe, ſhall make the Acute Angle with
the Diameter of the whole Portion leſset. And the Angle at X
being leſſe than the Angle at N, B C ſhall be greater than B S; and
C R leſſer than S R: And, therfore O G ſhall be leſſer than P Z;
and G X greater than Z T: Therfore P Z is greater than double of
Z T; being that O G is double of G X. Let P H be double to H T;
and drawing a Line from H to K, prolong it to ω. The Center of
Gravity of the whole Portion ſhall be K; the Center of the part
which is within the Liquid H, and that of the part which is above
the Liquid in the Line K ω; which ſuppoſed to be ω. Therefore it
ſhall be demonſtrated, both, that K H is perpendicular to the Surface
of the Liquid, and thoſe Lines alſo that are drawn thorow the Points
Hand ω parallel to K H: And therfore the Portion ſhall not reſt, but
ſhall encline untill that its Baſe do touch the Surface of the Liquid
in one Point; and ſo it ſhall continue. For in the Equall Portions
A O Q L and A P M L, the
62[Figure 62]
Lines A Q and A M, that cut off
equall Portions, ſhall be dawn
from the Ends or Terms of the Baſes;
and A O Q and A P M ſhall be
demonſtrated, as in the former, to
be equall: Therfore A Q and A M,
do make equall Acute Angles with
the Diameters of the Portions; and
dicular unto B D. It is to be demonſtrated that the Portion ſhall
61[Figure 61]not ſtand ſo, but ſhall encline until
that the Baſe touch the Surface of
the Liquid, in one Point only, for let
the ſuperior figure ſtand as it was,
and draw O C, Perpendicular to B D;
and drawing a Line from A to X,
prolong it to Q: A X ſhalbe equall
to X que Then draw O X parallel
to A que And becauſe the Portion
is ſuppoſed to have the ſame pro
portion in Gravity to the Liquid
that the ſquare X O hath to the
Square B D; the part thereof ſubmerged ſhall alſo have the ſame
proportion to the whole; that is, the Square T P to the Square
B D; and ſo T P ſhall be equal to X O: And ſince that of the Portions
I P M and A O Q the Diameters are equall, the portions ſhall alſo be
equall. Again, becauſe that in the Equall and Like Portions A O Q L
and AP ML the Lines A Q and I M, which cut off equall Por
tions, are drawn, that, from the Extremity of the Baſe, and this
not from the Extremity; it appeareth that that which is drawn from
the end or Extremity of the Baſe, ſhall make the Acute Angle with
the Diameter of the whole Portion leſset. And the Angle at X
being leſſe than the Angle at N, B C ſhall be greater than B S; and
C R leſſer than S R: And, therfore O G ſhall be leſſer than P Z;
and G X greater than Z T: Therfore P Z is greater than double of
Z T; being that O G is double of G X. Let P H be double to H T;
and drawing a Line from H to K, prolong it to ω. The Center of
Gravity of the whole Portion ſhall be K; the Center of the part
which is within the Liquid H, and that of the part which is above
the Liquid in the Line K ω; which ſuppoſed to be ω. Therefore it
ſhall be demonſtrated, both, that K H is perpendicular to the Surface
of the Liquid, and thoſe Lines alſo that are drawn thorow the Points
Hand ω parallel to K H: And therfore the Portion ſhall not reſt, but
ſhall encline untill that its Baſe do touch the Surface of the Liquid
in one Point; and ſo it ſhall continue. For in the Equall Portions
A O Q L and A P M L, the
62[Figure 62]Lines A Q and A M, that cut off
equall Portions, ſhall be dawn
from the Ends or Terms of the Baſes;
and A O Q and A P M ſhall be
demonſtrated, as in the former, to
be equall: Therfore A Q and A M,
do make equall Acute Angles with
the Diameters of the Portions; and