1is the ſixth part of E C: Which was the Propoſition.
This being pre-demonſtrated, we will prove that
PROPOSITION.
The Center of Gravity of the Parabolick
Conoid doth ſo divide the Axis, as that the
part towards the Vertex is double to the re
maining part towards the Baſe.
Conoid doth ſo divide the Axis, as that the
part towards the Vertex is double to the re
maining part towards the Baſe.
Let there be a Parabolick Conoid whoſe Axis let be A B divided in
N ſo as that A N be double to N B. It is to be proved that the Cen
ter of Gravity of the Conoid is the point N. For if it be not N, it
ſhall be either above or below it. Firſt let it be below; and let it be X:
And ſet off upon ſome place by it ſelf the Line L O equal to N X; and let
L O be divided at pleaſure in S: and look what proportion B X and
O S both together have to O S, and the ſame ſhall the Conoid have to
the Solid R. And in the Conoid let Figures be deſcribed by Cylinders
having equal Altitudes, ſo, as that that which lyeth between the Center
of Gravity and the point N be leſs than L S: and let the exceſs of the
Conoid above it be leſs than the Solid R: and that this may be done is
clear. Take therefore the inſcribed, whoſe Center of Gravity let be I:
now I X ſhall be greater than S O: And becauſe that as X B with S O
is to S O, ſo is the Conoid to the Solid R: (and R is greater than the
exceſs by which the Conoid exceeds the inſcribed Figure:) the proporti
on of the Conoid to the ſaid exceſs ſhall be greater than both B X and
O S unto S O: And, by Diviſion, the inſcribed Figure ſhall have grea
ter proportion to the ſaid exceſs than B X to S O: But B X hath to
X I a proportion yet leſs than to S O: Therefore the inſcribed Figure
ſhall have much greater proportion to the reſt of the proportions than
B X to X I: Therefore what proportion the inſcribed Figure hath to
thereſt of the portions, the ſame ſhall a certain other Line have to X I:
which ſhall neceſſarily be greater than B X: Let it, therefore, be M X.
We have therefore the Center of Gravity of the Conoid X: But the
Center of Gravity of the Figure inſcribed in it is I: of the reſt of the
portions by which the Conoid exceeds the inſcribed Figure the Center of
Gravity ſhall be in the Line X M, and in it that point in which it ſhall
be ſo terminated, that look what proportion the inſcribed Figure hath
to the exceſs by which the Conoid exceeds it, the ſame it ſhall have to
X I: But it hath been proved, that this proportion is that which M X
hath to X I: Therefore M ſhall be the Center of Gravity of thoſe pro
portions by which the Conoid exceeds the inſcribed Figure: Which
certainly cannot be. For if along by M a Plane be drawn equidiſtant to
the Baſe of the Conoid, all thoſe proportions ſhall be towards one and
N ſo as that A N be double to N B. It is to be proved that the Cen
ter of Gravity of the Conoid is the point N. For if it be not N, it
ſhall be either above or below it. Firſt let it be below; and let it be X:
And ſet off upon ſome place by it ſelf the Line L O equal to N X; and let
L O be divided at pleaſure in S: and look what proportion B X and
O S both together have to O S, and the ſame ſhall the Conoid have to
the Solid R. And in the Conoid let Figures be deſcribed by Cylinders
having equal Altitudes, ſo, as that that which lyeth between the Center
of Gravity and the point N be leſs than L S: and let the exceſs of the
Conoid above it be leſs than the Solid R: and that this may be done is
clear. Take therefore the inſcribed, whoſe Center of Gravity let be I:
now I X ſhall be greater than S O: And becauſe that as X B with S O
is to S O, ſo is the Conoid to the Solid R: (and R is greater than the
exceſs by which the Conoid exceeds the inſcribed Figure:) the proporti
on of the Conoid to the ſaid exceſs ſhall be greater than both B X and
O S unto S O: And, by Diviſion, the inſcribed Figure ſhall have grea
ter proportion to the ſaid exceſs than B X to S O: But B X hath to
X I a proportion yet leſs than to S O: Therefore the inſcribed Figure
ſhall have much greater proportion to the reſt of the proportions than
B X to X I: Therefore what proportion the inſcribed Figure hath to
thereſt of the portions, the ſame ſhall a certain other Line have to X I:
which ſhall neceſſarily be greater than B X: Let it, therefore, be M X.
We have therefore the Center of Gravity of the Conoid X: But the
Center of Gravity of the Figure inſcribed in it is I: of the reſt of the
portions by which the Conoid exceeds the inſcribed Figure the Center of
Gravity ſhall be in the Line X M, and in it that point in which it ſhall
be ſo terminated, that look what proportion the inſcribed Figure hath
to the exceſs by which the Conoid exceeds it, the ſame it ſhall have to
X I: But it hath been proved, that this proportion is that which M X
hath to X I: Therefore M ſhall be the Center of Gravity of thoſe pro
portions by which the Conoid exceeds the inſcribed Figure: Which
certainly cannot be. For if along by M a Plane be drawn equidiſtant to
the Baſe of the Conoid, all thoſe proportions ſhall be towards one and