1did ſuppoſe that it made an Angle greater than the Angle B, the
Poriton did not reſt then neither; It is manifeſt that it ſhall ſtay
or reſt when it ſhall make an Angle eqnall to B. For ſo ſhall I O
be equall to Ψ B; and ω I equall to
268[Figure 268]
Ψ R; and P H equall to F: There
fore M P ſhall be ſeſquialter of P H,
and P H double of H M: And there
fore ſince H is the Centre of Gravity
of that part of it which is within the
Liquid, it ſhall move upwards along
the ſame Perpendicular according to
which the whole Portion moveth;
and along the ſame alſo ſhall the part
which is above move downwards:
The Portion therefore ſhall reſt; for
aſmuch as the parts are not repulſed by each other.
Poriton did not reſt then neither; It is manifeſt that it ſhall ſtay
or reſt when it ſhall make an Angle eqnall to B. For ſo ſhall I O
be equall to Ψ B; and ω I equall to
268[Figure 268]Ψ R; and P H equall to F: There
fore M P ſhall be ſeſquialter of P H,
and P H double of H M: And there
fore ſince H is the Centre of Gravity
of that part of it which is within the
Liquid, it ſhall move upwards along
the ſame Perpendicular according to
which the whole Portion moveth;
and along the ſame alſo ſhall the part
which is above move downwards:
The Portion therefore ſhall reſt; for
aſmuch as the parts are not repulſed by each other.
A
B
C
D
E
F
G
(a) By 13. of the
fifth.
fifth.
H
K
L
M
N
O
P
Q
COMMANDINE.
And let C B be ſeſquialter of B R: C D ſhall alſo be ſeſquialter
of K R.] In the Tranſlation it is read thus: Sit autem & CB quidem hemeolia
ipſius B R: C D autem ipſius K R. But we at the reading of this paſſage have thought
fit thus to correctit; for it is not ſuppoſed ſo to be, but from the things ſuppoſed is proved to
be ſo. For if B ψ be double of ψ D, D B ſhall be ſeſquialter of B ψ. And becauſe E B is
ſeſquialter of B R, it followeth that the (a) Remainder C D is ſeſquialter of ψ R; that is, of
the Semi-parameter: Wherefore B C ſhall be the Exceſſe by which the Axis is greater than
ſeſquialter of the Semi-parameter.
of K R.] In the Tranſlation it is read thus: Sit autem & CB quidem hemeolia
ipſius B R: C D autem ipſius K R. But we at the reading of this paſſage have thought
fit thus to correctit; for it is not ſuppoſed ſo to be, but from the things ſuppoſed is proved to
be ſo. For if B ψ be double of ψ D, D B ſhall be ſeſquialter of B ψ. And becauſe E B is
ſeſquialter of B R, it followeth that the (a) Remainder C D is ſeſquialter of ψ R; that is, of
the Semi-parameter: Wherefore B C ſhall be the Exceſſe by which the Axis is greater than
ſeſquialter of the Semi-parameter.
A
(a) By 19. of the
fifth.
fifth.
And therefore F Q is leſſe than B C.] For in regard that the Portion hath
the ſame proportion in Gravity unto the Liquid, as the Square F Q hath to the Square D B;
and hath leſſer proportion than the Square made of the Exceſſe by which the Axis
is greater than Seſquialter of the Semi parameter, hath to the Square made of the Axis; that
is, leßer than the Square C B hath to the Square B D; for the Line B D was ſuppoſed to be
equall unto the Axis: Therefore the Square F Q ſhall have to the Square D B leſſer proporti
on than the Sqnare C B to the ſame Square B D: And therefore the Square (b) F Q ſhall be
leße than the Square C B: And, for that reaſon, the Line F Q ſhall be leße than B C.
the ſame proportion in Gravity unto the Liquid, as the Square F Q hath to the Square D B;
and hath leſſer proportion than the Square made of the Exceſſe by which the Axis
is greater than Seſquialter of the Semi parameter, hath to the Square made of the Axis; that
is, leßer than the Square C B hath to the Square B D; for the Line B D was ſuppoſed to be
equall unto the Axis: Therefore the Square F Q ſhall have to the Square D B leſſer proporti
on than the Sqnare C B to the ſame Square B D: And therefore the Square (b) F Q ſhall be
leße than the Square C B: And, for that reaſon, the Line F Q ſhall be leße than B C.
B
(b) By 8 of the
fifth.
fifth.
And, for the ſame reaſon, F is leſſe than B R.] For C B being ſeſqui-
alter of B R, and F Q ſeſquialter of F: (c) F Q ſhall be likewiſe leſſe than B C; and F
leße than B R.
alter of B R, and F Q ſeſquialter of F: (c) F Q ſhall be likewiſe leſſe than B C; and F
leße than B R.
C
(c) By 14 of the
fifth.
fifth.
Now becauſe it hath been ſuppoſed that the Axis of the Portion
doth make an Angle with the Surface of the Liquid greater than
the Angle B, the Angle P Y I ſhall be greater than the Angle B.]
For the Line P Y being parallel to the Surface of the Liquid, that is, to XS; (d) the Angle
P Y I ſhall be equall to the Angle contained betwixt the Diameter of the Portion N O, and the
Line X S: And therefore ſhall be greater than the Angle B.
doth make an Angle with the Surface of the Liquid greater than
the Angle B, the Angle P Y I ſhall be greater than the Angle B.]
For the Line P Y being parallel to the Surface of the Liquid, that is, to XS; (d) the Angle
P Y I ſhall be equall to the Angle contained betwixt the Diameter of the Portion N O, and the
Line X S: And therefore ſhall be greater than the Angle B.
D
(d) By 29 of the
firſt.
firſt.
Therefore the Square P I hath greater proportion to the Square
Y I, than the Square E Ψ hath to the Square Ψ B] Let the Triangles P I Y
and E ψ B, be deſcribed apart: And ſeeing that the Angle P Y I is greater
than the Angle E B ψ, unto the Line I Y, and at the Point Y aſſigned in
269[Figure 269]
the ſame, make the Angle V Y I equall to the Angle E B ψ; But
the Right Angle at I, is equall unto the Right Angle at ψ; therefore the
Y I, than the Square E Ψ hath to the Square Ψ B] Let the Triangles P I Y
and E ψ B, be deſcribed apart: And ſeeing that the Angle P Y I is greater
than the Angle E B ψ, unto the Line I Y, and at the Point Y aſſigned in
269[Figure 269]the ſame, make the Angle V Y I equall to the Angle E B ψ; But
the Right Angle at I, is equall unto the Right Angle at ψ; therefore the