COROLLARY.
Hence it is manifeſt, that a Conoid may be inſcribed in a Para
bolical Figure, and another circumſcribed, ſo, as that the
Centers of their Gravities may be diſtant from the point N
leſs than any Line given.
bolical Figure, and another circumſcribed, ſo, as that the
Centers of their Gravities may be diſtant from the point N
leſs than any Line given.
For if we aſſume a Line ſexcuple of the propoſed Line, and make the
Axis of the Cylinders, of which the Figures are compounded given
leſſer than this aſſumed Line, there ſhall fall Lines between the Centers
of Gravities of theſe Figures and the mark N that are leſs than the
Line propoſed.
Axis of the Cylinders, of which the Figures are compounded given
leſſer than this aſſumed Line, there ſhall fall Lines between the Centers
of Gravities of theſe Figures and the mark N that are leſs than the
Line propoſed.
The former Propoſition another way.
Let the Axis of the Conoid (which let be C D) be divided in
O, ſo, as that C O be double to O D. It is to be proved that the
Center of Gravity of the inſcribed Figure is in the Line O D;
and the Center of the circumſcribed in C O. Let the Plane of the Fi
gures be cut through the Axis and C, as hath been ſaid. Becauſe there
fore the Cylinders S N, T M, V I,
168[Figure 168]
X E are to one another as the Squares
of the Lines S D, T N, V M, X I;
and theſe are to one another as the
Lines N C, C M, C I, C E: but
theſe do exceed one another equally;
and the exceſs is equal to the leaſt, to
wit, C E: And the Cylinder T M is
equal to the Cylinder Q N; and the
Cylinder V I equal to P N; and X E
is equal to L N: Therefore the Cylin
ders S N, Q N, P N, and L N do
equally exceed one another, and the
exceſs is equal to the leaſt of them,
namely, to the Cylinder L N. But
the exceſs of the Cylinder S N, above
the Cylinder Q N is a Ring whoſe
height is Q T; that is, N D; and
its breadth S que And the exceſs of the Cylinder Q N above P N, is a
Ring, whoſe breadth is Q P. And the exceſs of the Cylinder P N above
L N is a Ring, whoſe breadth is P L. Wherefore the ſaid Rings S Q,
Q P, P L, are equal to another, and to the Cylinder L N. Therefore the
Ring S T equalleth the Cylinder X E: the Ring Q V, which is double
to S T, equalleth the Cylinder V I; which likewiſe is double to the
O, ſo, as that C O be double to O D. It is to be proved that the
Center of Gravity of the inſcribed Figure is in the Line O D;
and the Center of the circumſcribed in C O. Let the Plane of the Fi
gures be cut through the Axis and C, as hath been ſaid. Becauſe there
fore the Cylinders S N, T M, V I,
168[Figure 168]X E are to one another as the Squares
of the Lines S D, T N, V M, X I;
and theſe are to one another as the
Lines N C, C M, C I, C E: but
theſe do exceed one another equally;
and the exceſs is equal to the leaſt, to
wit, C E: And the Cylinder T M is
equal to the Cylinder Q N; and the
Cylinder V I equal to P N; and X E
is equal to L N: Therefore the Cylin
ders S N, Q N, P N, and L N do
equally exceed one another, and the
exceſs is equal to the leaſt of them,
namely, to the Cylinder L N. But
the exceſs of the Cylinder S N, above
the Cylinder Q N is a Ring whoſe
height is Q T; that is, N D; and
its breadth S que And the exceſs of the Cylinder Q N above P N, is a
Ring, whoſe breadth is Q P. And the exceſs of the Cylinder P N above
L N is a Ring, whoſe breadth is P L. Wherefore the ſaid Rings S Q,
Q P, P L, are equal to another, and to the Cylinder L N. Therefore the
Ring S T equalleth the Cylinder X E: the Ring Q V, which is double
to S T, equalleth the Cylinder V I; which likewiſe is double to the
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