1the ſame part, and not by it divided. Therefore the Center of Gravity
of the ſaid Conoid is not below the point N: Neither is it above. For,
if it may, let it be H: and again, as before, ſet the Line L O by it ſelf
equalto the ſaid H N, and divided at pleaſure in S: and the ſame pro
portion that B N and S O both together have to S L, let the Conoid
have to R: and about the Conoid let a Figure be circumſcribed conſi
ſting of Cylinders, as hath been ſaid: by which let it be exceeded a leſs
quantity than that of the Solid R: and let the Line betwixt the Center
of Gravity of the circumſcribed Figure and the point N be leſſer than
S O: the remainder V H ſhall be greater than S L. And becauſe that as
both B N and O S is to SL, ſo is the
169[Figure 169]
Conoid to R: (and R is greater
than the exceſs by which the circum
ſcribed Figure exceeds the Conoid:)
Therefore B N and S O hath leſs pro
portion to S L than the Conoid to the
ſaid exceſs. And B V is leſſer than
both B N and S O; and V H is grea
ter than S L: much greater proporti
on, therefore, hath the Conoid to the
ſaid proportions, than B V hath to
V H. Therefore whatever proporti
on the Conoid hath to the ſaid pro
portions, the ſame ſhall a Line greater
than B V have to V H. Let the ſame be M V: And becauſe the Center
of Gravity of the circumſcribed Figure is V, and the Center of the
Conoid is H. and ſince that as the Conoid to the reſt of the proportions,
ſois M V to V H, M ſhall be the Center of Gravity of the remaining
proportions: which likewiſe is impoſſible: Therefore the Center of
Gravity of the Conoid is not above the point N: But it hath been de
monſtrated that neither is it beneath: It remains, therefore, that it ne
ceſſarily be in the point N it ſelf. And the ſame might be demonſtrated
of Conoidal Plane cut upon an Axis not erect. The ſame in other terms,
as appears by what followeth:
of the ſaid Conoid is not below the point N: Neither is it above. For,
if it may, let it be H: and again, as before, ſet the Line L O by it ſelf
equalto the ſaid H N, and divided at pleaſure in S: and the ſame pro
portion that B N and S O both together have to S L, let the Conoid
have to R: and about the Conoid let a Figure be circumſcribed conſi
ſting of Cylinders, as hath been ſaid: by which let it be exceeded a leſs
quantity than that of the Solid R: and let the Line betwixt the Center
of Gravity of the circumſcribed Figure and the point N be leſſer than
S O: the remainder V H ſhall be greater than S L. And becauſe that as
both B N and O S is to SL, ſo is the
169[Figure 169]Conoid to R: (and R is greater
than the exceſs by which the circum
ſcribed Figure exceeds the Conoid:)
Therefore B N and S O hath leſs pro
portion to S L than the Conoid to the
ſaid exceſs. And B V is leſſer than
both B N and S O; and V H is grea
ter than S L: much greater proporti
on, therefore, hath the Conoid to the
ſaid proportions, than B V hath to
V H. Therefore whatever proporti
on the Conoid hath to the ſaid pro
portions, the ſame ſhall a Line greater
than B V have to V H. Let the ſame be M V: And becauſe the Center
of Gravity of the circumſcribed Figure is V, and the Center of the
Conoid is H. and ſince that as the Conoid to the reſt of the proportions,
ſois M V to V H, M ſhall be the Center of Gravity of the remaining
proportions: which likewiſe is impoſſible: Therefore the Center of
Gravity of the Conoid is not above the point N: But it hath been de
monſtrated that neither is it beneath: It remains, therefore, that it ne
ceſſarily be in the point N it ſelf. And the ſame might be demonſtrated
of Conoidal Plane cut upon an Axis not erect. The ſame in other terms,
as appears by what followeth:
PROPOSITION.
The Center of Gravity of the Parabolick Co
noid falleth betwixt the Center of the cir
cumſcribed Figure and the Center of the in
ſcribed.
noid falleth betwixt the Center of the cir
cumſcribed Figure and the Center of the in
ſcribed.