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.

COROLLARY.

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.

For, ſince it hath been demonſtrated, that the ſaid point dividing the

Axis, as was ſaid, is alwaies found betwixt the Centers of Gravity

Axis, as was ſaid, is alwaies found betwixt the Centers of Gravity