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-