1L M, V C: they ſhall be ordinately applyed. Divide therefore E B in
to three equal parts, of which let the middlemoſt be Q Y: and divide
this ſo in the point I that Q I may have the ſame proportion to I Y, as
the Baſe whoſe Diameter is V C hath to the Baſe whoſe Diameter is
L M; that is, that the Square V C hath to Square L M. It is to be de
monſtrated that I is the Center of Gravity of the Fruſtrum L M C.
Draw the Line N S, by the by, equall to B R: and let S X be equal to
E R: and unto N S and S X aſſume a third proportional S G: and as
N G is to G S, ſo let B Q be to I O. And it nothing matters whether
the point O fall above or below L M. And becauſe in the Section V R C
the Lines L M and V C are ordinately
172[Figure 172]
applyed, it ſhall be that as the Square
V C is to the Square L M, ſo is the Line
B R to R E: And as the Square V C is
to the Square L M, ſo is Q I to I Y: and
as B R is to R E, ſo is N S to S X: There
fore Q I is to I Y, as R S is to S X. Where
fore as G Y is to Y I, ſo ſhall both N S and
S X be to S X: and as E B is to Y I, ſo
ſhall the compound of triple N S and tri
ple S X be to S X: But as E B is to B Y,
ſo is the compound of triple N S and S X
both together to the compound of N S and S X: Therefore, as E B is to
B I, ſo is the compound of triple N S and triple S X to the compound of
N S and double S X. Therefore N S, S X, and S G are three proporti
onal Lines: And as S G is to G N, ſo is the aſſumed O I to two thirds
of E B; that is, to N X: And as the compound of N S and double
S X is to the compound of triple N S and triple S X, ſo is another aſſu
med Line I B to B E; that is, to N X. By what therefore hath been
above demonſtrated, thoſe Lines taken together are a third part of N S;
that is, of R B: Therefore R B is triple to B O: Wherefore O ſhall
be the Center of Gravity of the Conoid v R C. And let it be the Cen
ter of Gravity of the Fruſtrum L R M of the Conoid: Therefore the
Center of Gravity of V L M C is in the Line O B, and in that point
which ſo terminates it, that as V L M C of the Fruſtrum is to the
proportion L R M, ſo is the Line A O to that which intervenes betwixt
O and the ſaid point. And becauſe R O is two thirds of R B; and
R A two thirds of R E; the remaining part A O ſhall be two thirds
of the remaining part E B. And becauſe that as the Fruſtum V L M C
is to the proportion L R M, ſo is N G to G S: and as N G to G S, ſo is
two thirds of E B to O I: and two thirds of E B is equal to the Line
A O: it ſhall be that as the Fruſtum V L M O is to the proportion
L R M, ſo is A O to O I. It is manifeſt therefore that of the Fruſtum
V L M C the Center of Gravity is the point I, and ſo divideth the Axis,
[as?] that the part towards the leſſer Baſe is to the part towards the grea-
to three equal parts, of which let the middlemoſt be Q Y: and divide
this ſo in the point I that Q I may have the ſame proportion to I Y, as
the Baſe whoſe Diameter is V C hath to the Baſe whoſe Diameter is
L M; that is, that the Square V C hath to Square L M. It is to be de
monſtrated that I is the Center of Gravity of the Fruſtrum L M C.
Draw the Line N S, by the by, equall to B R: and let S X be equal to
E R: and unto N S and S X aſſume a third proportional S G: and as
N G is to G S, ſo let B Q be to I O. And it nothing matters whether
the point O fall above or below L M. And becauſe in the Section V R C
the Lines L M and V C are ordinately
172[Figure 172]applyed, it ſhall be that as the Square
V C is to the Square L M, ſo is the Line
B R to R E: And as the Square V C is
to the Square L M, ſo is Q I to I Y: and
as B R is to R E, ſo is N S to S X: There
fore Q I is to I Y, as R S is to S X. Where
fore as G Y is to Y I, ſo ſhall both N S and
S X be to S X: and as E B is to Y I, ſo
ſhall the compound of triple N S and tri
ple S X be to S X: But as E B is to B Y,
ſo is the compound of triple N S and S X
both together to the compound of N S and S X: Therefore, as E B is to
B I, ſo is the compound of triple N S and triple S X to the compound of
N S and double S X. Therefore N S, S X, and S G are three proporti
onal Lines: And as S G is to G N, ſo is the aſſumed O I to two thirds
of E B; that is, to N X: And as the compound of N S and double
S X is to the compound of triple N S and triple S X, ſo is another aſſu
med Line I B to B E; that is, to N X. By what therefore hath been
above demonſtrated, thoſe Lines taken together are a third part of N S;
that is, of R B: Therefore R B is triple to B O: Wherefore O ſhall
be the Center of Gravity of the Conoid v R C. And let it be the Cen
ter of Gravity of the Fruſtrum L R M of the Conoid: Therefore the
Center of Gravity of V L M C is in the Line O B, and in that point
which ſo terminates it, that as V L M C of the Fruſtrum is to the
proportion L R M, ſo is the Line A O to that which intervenes betwixt
O and the ſaid point. And becauſe R O is two thirds of R B; and
R A two thirds of R E; the remaining part A O ſhall be two thirds
of the remaining part E B. And becauſe that as the Fruſtum V L M C
is to the proportion L R M, ſo is N G to G S: and as N G to G S, ſo is
two thirds of E B to O I: and two thirds of E B is equal to the Line
A O: it ſhall be that as the Fruſtum V L M O is to the proportion
L R M, ſo is A O to O I. It is manifeſt therefore that of the Fruſtum
V L M C the Center of Gravity is the point I, and ſo divideth the Axis,
[as?] that the part towards the leſſer Baſe is to the part towards the grea-
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