Let there be a Conoid whoſe Axis is A B, and the Center of the
circumſcribed Figure C, and the Center of the inſcribed O. I ſay
the Center of the Conoid is betwixt the points C and O. For if
not, it ſhall be either above them, or below them, or in one of them. Let
it be below, as in R. And becauſe R is the Center of Gravity of the
whole Conoid; and the Center of Gravity of the inſcribed Figure is O:
Therefore of the remaining proportions by which the Conoid exceeds
the inſcribed Figure the Center of Gravity ſhall be in the Line O R ex
tended towards R, and in that point in which it is ſo determined, that,
what proportion the ſaid proportions have to the inſcribed Figure, the
ſame ſhall O R have to the Line falling betwixt R and that falling point.
Let this proportion be that of O R to R X. Therefore X falleth either
without the Conoid or within, or in its
170[Figure 170]
Baſe. That it falleth without, or in its
Baſe it is already manifeſt to be an abſur
dity. Let it fall within: and becauſe X R
is to R O, as the inſcribed Figure is to
the exceſs by which the Conoid exceeds
it; the ſame proportion that B R hath to
R O, the ſame let the inſcribed Figure
have to the Solid K: Which neceſſarily
ſhall be leſſer than the ſaid exceſs. And let
another Figure be inſcribed which may be
exceeded by the Conoid a leſs quantity
than is K, whoſe Center of Gravity falleth betwixt O and C. Let it
be V. And, becauſe the firſt Figure is to K as B R to R O, and the ſe
cond Figure, whoſe Center V is greater than the firſt, and exceeded
by the Conoid a leſs quantity than is K; what proportion the ſecond
Figure hath to the exceſs by which the Conoid exceeds it, the ſame
ſhall a Line greater than B R have to R V. But R is the Center of Gra
vity of the Conoid; and the Center of the ſecond inſcribed Figure V:
The Center therefore of the remaining proportions ſhall be without
the Conoid beneath B: Which is impoſſible. And by the ſame means
we might demonſtrate the Center of Gravity of the ſaid Conoid not to
be in the Line C A. And that it is none of the points betwixt C and
O is manifeſt. For ſay, that there other Figures deſcribed, greater
ſomething than the inſcribed Figure whoſe Center is O, and leſs than
that circumſcribed Figure whoſe Center is C, the Center of the Conoid
would fall without the Center of theſe Figures: Which but now was
concluded to be impoſſible: It reſts therefore that it be betwixt the Cen
ter of the circumſcribed and inſcribed Figure. And if ſo, it ſhall ne
ceſſarily be in that point which divideth the Axis, ſo as that the part
towards the Vertex is double to the remainder; ſince N may circum
ſcribe and inſcribe Figures, ſo, that thoſe Lines which fall between
circumſcribed Figure C, and the Center of the inſcribed O. I ſay
the Center of the Conoid is betwixt the points C and O. For if
not, it ſhall be either above them, or below them, or in one of them. Let
it be below, as in R. And becauſe R is the Center of Gravity of the
whole Conoid; and the Center of Gravity of the inſcribed Figure is O:
Therefore of the remaining proportions by which the Conoid exceeds
the inſcribed Figure the Center of Gravity ſhall be in the Line O R ex
tended towards R, and in that point in which it is ſo determined, that,
what proportion the ſaid proportions have to the inſcribed Figure, the
ſame ſhall O R have to the Line falling betwixt R and that falling point.
Let this proportion be that of O R to R X. Therefore X falleth either
without the Conoid or within, or in its
170[Figure 170]Baſe. That it falleth without, or in its
Baſe it is already manifeſt to be an abſur
dity. Let it fall within: and becauſe X R
is to R O, as the inſcribed Figure is to
the exceſs by which the Conoid exceeds
it; the ſame proportion that B R hath to
R O, the ſame let the inſcribed Figure
have to the Solid K: Which neceſſarily
ſhall be leſſer than the ſaid exceſs. And let
another Figure be inſcribed which may be
exceeded by the Conoid a leſs quantity
than is K, whoſe Center of Gravity falleth betwixt O and C. Let it
be V. And, becauſe the firſt Figure is to K as B R to R O, and the ſe
cond Figure, whoſe Center V is greater than the firſt, and exceeded
by the Conoid a leſs quantity than is K; what proportion the ſecond
Figure hath to the exceſs by which the Conoid exceeds it, the ſame
ſhall a Line greater than B R have to R V. But R is the Center of Gra
vity of the Conoid; and the Center of the ſecond inſcribed Figure V:
The Center therefore of the remaining proportions ſhall be without
the Conoid beneath B: Which is impoſſible. And by the ſame means
we might demonſtrate the Center of Gravity of the ſaid Conoid not to
be in the Line C A. And that it is none of the points betwixt C and
O is manifeſt. For ſay, that there other Figures deſcribed, greater
ſomething than the inſcribed Figure whoſe Center is O, and leſs than
that circumſcribed Figure whoſe Center is C, the Center of the Conoid
would fall without the Center of theſe Figures: Which but now was
concluded to be impoſſible: It reſts therefore that it be betwixt the Cen
ter of the circumſcribed and inſcribed Figure. And if ſo, it ſhall ne
ceſſarily be in that point which divideth the Axis, ſo as that the part
towards the Vertex is double to the remainder; ſince N may circum
ſcribe and inſcribe Figures, ſo, that thoſe Lines which fall between