1I ſay now, that the Line E S is leſſer than K. For if not, then let C A
be ſuppoſed equal to E O. Becauſe therefore O E hath to K the ſame
proportion that L hath to X; and the inſcribed Figure is not leſs than
the Cylinder L; and the exceſs with which the ſaid Figure is exceeded
by the circumſcribed is leſs than the Solid X: therefore the inſcribed
Figure ſhall have to the ſaid exceſs
176[Figure 176]
greater proportion than O E hath to
K: But the proportion of O E to K is
not leſs than that which O E hath to
E S with E S. Let it not be leſs than
K. Therefore the inſcribed Figure
hath to the exceſs of the circumſcri
bed Figure above it greater propor
tion than O E hath to E S. Therefore
as the inſcribed is to the ſaid exceſs,
ſo ſhall it be to the Line E S. Let E R
be a Line greater than E O; and the
Center of Gravity of the inſcribed
Figure is S; and the Center of the cir
cumſcribed is E. It is manifeſt there
fore, that the Center of Gravity of
the remaining proportions by which
the circumſcribed exceedeth the in
ſcribed is in the Line R E, and in that point by which it is ſo termina
ted, that as the inſcribed Figure is to the ſaid proportions, ſo is the Line
included betwixt E and that point to the Line E S. And this propor
tion hath R E to E S. Therefore the Center of Gravity of the remain
ing proportions with which the circumſcribed Figure exceeds the in
ſcribed ſhall be R, which is impoſſible. For the Plane drawn thorow
R equidiſtant to the Baſe of the Cone doth not cut thoſe proportions. It
is therefore falſe that the Line E S is not leſſer than K. It ſhall therefore
be leſs. The ſame alſo may be done in a manner not unlike this in Pyra
mides, as ne could demonſtrate.
be ſuppoſed equal to E O. Becauſe therefore O E hath to K the ſame
proportion that L hath to X; and the inſcribed Figure is not leſs than
the Cylinder L; and the exceſs with which the ſaid Figure is exceeded
by the circumſcribed is leſs than the Solid X: therefore the inſcribed
Figure ſhall have to the ſaid exceſs
176[Figure 176]greater proportion than O E hath to
K: But the proportion of O E to K is
not leſs than that which O E hath to
E S with E S. Let it not be leſs than
K. Therefore the inſcribed Figure
hath to the exceſs of the circumſcri
bed Figure above it greater propor
tion than O E hath to E S. Therefore
as the inſcribed is to the ſaid exceſs,
ſo ſhall it be to the Line E S. Let E R
be a Line greater than E O; and the
Center of Gravity of the inſcribed
Figure is S; and the Center of the cir
cumſcribed is E. It is manifeſt there
fore, that the Center of Gravity of
the remaining proportions by which
the circumſcribed exceedeth the in
ſcribed is in the Line R E, and in that point by which it is ſo termina
ted, that as the inſcribed Figure is to the ſaid proportions, ſo is the Line
included betwixt E and that point to the Line E S. And this propor
tion hath R E to E S. Therefore the Center of Gravity of the remain
ing proportions with which the circumſcribed Figure exceeds the in
ſcribed ſhall be R, which is impoſſible. For the Plane drawn thorow
R equidiſtant to the Baſe of the Cone doth not cut thoſe proportions. It
is therefore falſe that the Line E S is not leſſer than K. It ſhall therefore
be leſs. The ſame alſo may be done in a manner not unlike this in Pyra
mides, as ne could demonſtrate.
Hence it is manifeſt, that a given Cone may circumſcribe one
Figure and inſcribe another conſiſting of Cylinders of equal
Altitudes ſo, as that the Lines which are intercepted betwixt
their Centers of Gravity and the point which ſo divides the
Axis of the Cone, as that the part towards the Vertex is tri
ple to the leſt, are leſs than any given Line.
Figure and inſcribe another conſiſting of Cylinders of equal
Altitudes ſo, as that the Lines which are intercepted betwixt
their Centers of Gravity and the point which ſo divides the
Axis of the Cone, as that the part towards the Vertex is tri
ple to the leſt, are leſs than any given Line.