1Planotum, be in the Line V Q prolonged: But that is impoſſible; for it is in the Axis
G: It followeth, therefore, that the Center of Gravity of the Portion demerged in
Liquid be in the Line N K: which we propounded to be proved.
G: It followeth, therefore, that the Center of Gravity of the Portion demerged in
Liquid be in the Line N K: which we propounded to be proved.
C
(a) By 29. of the
firſt of Encl.
firſt of Encl.
(b) By 3. of the
third.
third.
But the Centre of Gravity of the whole Portion is in the Line
T, betwixt the Point R and the Point F; let us ſuppoſe it to be
the Point X.] Let the Sphære becompleated, ſo as that there be added of that Portion
the Axis T Y, and the Center of Gravity Z. And becauſe that from the whole Sphære,
whoſe Centre of Gravity is K, as we have alſo demonſtrated in the (c) Book before named, the
is cut off the Portion E Y H, having the Centre of Gravity Z; the Centre of the remaind
of the Portion E F H ſhall be in the Line Z K prolonged: And therefore it muſt of neceſſity
fall betwixt K and F.
T, betwixt the Point R and the Point F; let us ſuppoſe it to be
the Point X.] Let the Sphære becompleated, ſo as that there be added of that Portion
the Axis T Y, and the Center of Gravity Z. And becauſe that from the whole Sphære,
whoſe Centre of Gravity is K, as we have alſo demonſtrated in the (c) Book before named, the
is cut off the Portion E Y H, having the Centre of Gravity Z; the Centre of the remaind
of the Portion E F H ſhall be in the Line Z K prolonged: And therefore it muſt of neceſſity
fall betwixt K and F.
D
(c) By 8 of the
firſt of Archimedes.
firſt of Archimedes.
E
The remainder, therefore, of the Figure, elevated above the Sur
face of the Liquid, hath its Center of Gravity in the Line R X
prolonged.] By the ſame 8 of the firſt Book of Archimedes, de Centro Gravita
tis Planorum.
face of the Liquid, hath its Center of Gravity in the Line R X
prolonged.] By the ſame 8 of the firſt Book of Archimedes, de Centro Gravita
tis Planorum.
Now the Gravity of the Figure that is above the Liquid ſhall
preſs from above downwards according to S L; and the Gravit
of the Portion that is ſubmerged in the Liquid ſhall preſs from be
low upwards, according to the Perpendicular R L.] By the ſecond Sup
poſition of this. For the Magnitude that is demerged in the Liquid is moved upwards with as
much Force along R L, as that which is above the Liquid is moved downwards along S L; as
may be ſhewn by Propoſition 6. of this. And becauſe they are moved along ſeverall other Lines,
neither cauſeth the others being leſs moved; the which it continually doth when the Portion
is ſet according to the Perpendicular: For then the Centers of Gravity of both the Magnitudes
do concur in one and the ſame Perpendicular, namely, in the Axis of the Portion: and look
with what force or Impetus that which is in the Lipuid tendeth upwards, and with the like
doth that which is above or without the Liquid tend downwards along the ſame Line: And
therefore, in regard that the one doth not ^{*} exceed the other, the Portion ſhall no longer move
but ſhall ſtay and reſt allwayes in one and the ſame Poſition, unleſs ſome extrinſick Cauſe
chance to intervene.
preſs from above downwards according to S L; and the Gravit
of the Portion that is ſubmerged in the Liquid ſhall preſs from be
low upwards, according to the Perpendicular R L.] By the ſecond Sup
poſition of this. For the Magnitude that is demerged in the Liquid is moved upwards with as
much Force along R L, as that which is above the Liquid is moved downwards along S L; as
may be ſhewn by Propoſition 6. of this. And becauſe they are moved along ſeverall other Lines,
neither cauſeth the others being leſs moved; the which it continually doth when the Portion
is ſet according to the Perpendicular: For then the Centers of Gravity of both the Magnitudes
do concur in one and the ſame Perpendicular, namely, in the Axis of the Portion: and look
with what force or Impetus that which is in the Lipuid tendeth upwards, and with the like
doth that which is above or without the Liquid tend downwards along the ſame Line: And
therefore, in regard that the one doth not ^{*} exceed the other, the Portion ſhall no longer move
but ſhall ſtay and reſt allwayes in one and the ſame Poſition, unleſs ſome extrinſick Cauſe
chance to intervene.
F
* Or overcome.
PROP. IX. THEOR. IX.
* In ſome Greek
Coppies this is no
diſtinct Propoſi
tion, but all
Commentators,
do divide it
from the Prece
dent, as having a
diſtinct demon
ſtration in the
Originall.
Coppies this is no
diſtinct Propoſi
tion, but all
Commentators,
do divide it
from the Prece
dent, as having a
diſtinct demon
ſtration in the
Originall.
^{*} But if the Figure, lighter than the Liquid, be demit
ted into the Liquid, ſo, as that its Baſe be wholly
within the ſaid Liquid, it ſhall continue in ſuch
manner erect, as that its Axis ſhall ſtand according
to the Perpendicular.
ted into the Liquid, ſo, as that its Baſe be wholly
within the ſaid Liquid, it ſhall continue in ſuch
manner erect, as that its Axis ſhall ſtand according
to the Perpendicular.
For ſuppoſe, ſuch a Magnitude as that aforenamed to be de
mitted into the Liquid; and imagine a Plane to be produced
thorow the Axis of the Portion, and thorow the Center of the
Earth: And let the Section of the Surface of the Liquid, be the Cir
cumference A B C D, and of the Figure the Circumference E F H
And let E H be a Right Line, and F T the Axis of the Portion. If
now it were poſſible, for ſatisfaction of the Adverſary, let it be
ſuppoſed that the ſaid Axis were not according to the Perpendicu
lar: we are now to demonſtrate that the Figure will not ſo
mitted into the Liquid; and imagine a Plane to be produced
thorow the Axis of the Portion, and thorow the Center of the
Earth: And let the Section of the Surface of the Liquid, be the Cir
cumference A B C D, and of the Figure the Circumference E F H
And let E H be a Right Line, and F T the Axis of the Portion. If
now it were poſſible, for ſatisfaction of the Adverſary, let it be
ſuppoſed that the ſaid Axis were not according to the Perpendicu
lar: we are now to demonſtrate that the Figure will not ſo