1leaſt which is T hangeth at A; and the reſt are ordinately diſpoſed.
And again there is another Ballance A B in which other Magnitudes
equal in number and Magnitude to the former are diſpoſed in the ſame
order. Wherefore the Ballances A B and A D are divided by the Cen
ter of all the Magnitudes according to the ſame proportion: But the
Center of Gravity of the aforeſaid Magnitudes is X: Wherefore X
divideth the Ballances B A and A D according to the ſame proportion;
ſo that as B X is to X A, ſo is X A to X D: Wherefore B X is double
to X A, by the Lemma aforegoing: Which was to be proved.
And again there is another Ballance A B in which other Magnitudes
equal in number and Magnitude to the former are diſpoſed in the ſame
order. Wherefore the Ballances A B and A D are divided by the Cen
ter of all the Magnitudes according to the ſame proportion: But the
Center of Gravity of the aforeſaid Magnitudes is X: Wherefore X
divideth the Ballances B A and A D according to the ſame proportion;
ſo that as B X is to X A, ſo is X A to X D: Wherefore B X is double
to X A, by the Lemma aforegoing: Which was to be proved.
PROPOSITION.
If in a Parabolical Conoid Figure be deſcribed,
and another circumſcribed by Cylinders of
equal Altitude; and the Axis of the ſaid Co
noid be divided in ſuch proportion that the
part towards the Vertex be double to that to
wards the Baſe; the Center of Gravity of the
inſcribed Figure of the Baſe portion ſhall be
neareſt to the ſaid point of diviſion; and the
Center of Gravity of the circumſcribed from
the Baſe of the Conoid ſhall be more remote:
and the diſtance of either of thoſe Centers
from that ſame point ſhall be equal to the Line
that is the ſixth part of the Altitude of one of
the Cylinders of which the Figures are com
poſed.
and another circumſcribed by Cylinders of
equal Altitude; and the Axis of the ſaid Co
noid be divided in ſuch proportion that the
part towards the Vertex be double to that to
wards the Baſe; the Center of Gravity of the
inſcribed Figure of the Baſe portion ſhall be
neareſt to the ſaid point of diviſion; and the
Center of Gravity of the circumſcribed from
the Baſe of the Conoid ſhall be more remote:
and the diſtance of either of thoſe Centers
from that ſame point ſhall be equal to the Line
that is the ſixth part of the Altitude of one of
the Cylinders of which the Figures are com
poſed.
Take therefore a Parabolical Conoid, and the Figures that have
been mentioned: let one of them be inſcribed, the other circum
ſcribed; and let the Axis of the Conoid, which let be A E, be di
vided in N, in ſuch proportion as that A N be double to N E. It is to
be proved that the Center of Gravity of the inſcribed Figure is in the
Line N E, but the Center of the circumſcribed in the Line A N. Let
the Plane of the Figures ſo diſpoſed be cut through the Axis, and let
the Section be that of the Parabola B A C: and let the Section of the
cutting Plane, and of the Baſe of the Conoid be the Line B C; and
let the Sections of the Cylinders be the Rectangular Figures; as ap
peareth in the deſcription. Firſt, therefore, the Cylinder of the inſcri
bed whoſe Axis is D E, hath the ſame proportion to the Cylinder whoſe
Axis is D Y, as the Quadrate I D hath to the Quadrate S Y; that is,
as D A hath to A Y: and the Cylinder whoſe Axis is D Y is potentia
been mentioned: let one of them be inſcribed, the other circum
ſcribed; and let the Axis of the Conoid, which let be A E, be di
vided in N, in ſuch proportion as that A N be double to N E. It is to
be proved that the Center of Gravity of the inſcribed Figure is in the
Line N E, but the Center of the circumſcribed in the Line A N. Let
the Plane of the Figures ſo diſpoſed be cut through the Axis, and let
the Section be that of the Parabola B A C: and let the Section of the
cutting Plane, and of the Baſe of the Conoid be the Line B C; and
let the Sections of the Cylinders be the Rectangular Figures; as ap
peareth in the deſcription. Firſt, therefore, the Cylinder of the inſcri
bed whoſe Axis is D E, hath the ſame proportion to the Cylinder whoſe
Axis is D Y, as the Quadrate I D hath to the Quadrate S Y; that is,
as D A hath to A Y: and the Cylinder whoſe Axis is D Y is potentia
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