1double B C: But triple A B with ſexcuple B C, are triple to A B with
double B C. Therefore A O is triple to S N.
double B C. Therefore A O is triple to S N.
Again, becauſe O C is to C A as triple C B is to triple A B with tri
ple C B: and becauſe as C A is to A F, ſo is triple A B to triple B C:
Therefore, ex equali, by perturbed proportion, as O C is to C F, ſo ſhall
triple A B be to triple A B with treble B C: And, by Converſion of
proportion, as O F is to F C, ſo is triple B C to triple A B with triple
B C: And as C F is to F B, ſo is A C to C B, and triple A C to triple
C B: Therefore, ex equali, by Perturbation of proportion, as O F is
to F B, ſo is triple A C to the triple of both A B and A C together.
And becauſe F C and C A are in the ſame proportion as C B and B A;
it ſhall be that as F C is to C A, ſo ſhall B C be to B A. And, by Com
poſition, as F A is to A C, ſo are both B A and B C to B A: and ſo the
triple to the triple: Therefore as F A is to A C, ſo the compound of tri
ple B A and triple B C is to triple A B. Wherefore, as F A is to two
thirds of A C, ſo is the compound of triple B A and triple B C to two
thirds of triple B A; that is, to double B A: But as F A is to two thirds
of A C, ſo is F B to M S: Therefore, as F B is to M S, ſo is the compound
of triple B A and triple B C to double B A: But as O B is to F B, ſo
was Sexcuple A B to triple of both A B and B C: Therefore, ex equa
li, O B ſhall have to M S the ſame proportion as Sexcuple A B hath to
double B A. Wherefore M S ſhall be the third part of O B: And it
hath been demonſtrated, that S N is the third part of A O: It is mani
feſt therefore, that MN is a third part likewiſe of A B: And this is
that which was to be demonſtrated.
ple C B: and becauſe as C A is to A F, ſo is triple A B to triple B C:
Therefore, ex equali, by perturbed proportion, as O C is to C F, ſo ſhall
triple A B be to triple A B with treble B C: And, by Converſion of
proportion, as O F is to F C, ſo is triple B C to triple A B with triple
B C: And as C F is to F B, ſo is A C to C B, and triple A C to triple
C B: Therefore, ex equali, by Perturbation of proportion, as O F is
to F B, ſo is triple A C to the triple of both A B and A C together.
And becauſe F C and C A are in the ſame proportion as C B and B A;
it ſhall be that as F C is to C A, ſo ſhall B C be to B A. And, by Com
poſition, as F A is to A C, ſo are both B A and B C to B A: and ſo the
triple to the triple: Therefore as F A is to A C, ſo the compound of tri
ple B A and triple B C is to triple A B. Wherefore, as F A is to two
thirds of A C, ſo is the compound of triple B A and triple B C to two
thirds of triple B A; that is, to double B A: But as F A is to two thirds
of A C, ſo is F B to M S: Therefore, as F B is to M S, ſo is the compound
of triple B A and triple B C to double B A: But as O B is to F B, ſo
was Sexcuple A B to triple of both A B and B C: Therefore, ex equa
li, O B ſhall have to M S the ſame proportion as Sexcuple A B hath to
double B A. Wherefore M S ſhall be the third part of O B: And it
hath been demonſtrated, that S N is the third part of A O: It is mani
feſt therefore, that MN is a third part likewiſe of A B: And this is
that which was to be demonſtrated.
PROPOSITION.
Of any Fruſtum or Segment cut off from a Para
bolick Conoid the Center of Gravity is in the
Right Line that is Axis of the Fruſtum; which
being divided into three equal parts the Cen
ter of Gravity is in the middlemoſt and ſo di
vides it, as that the part towards the leſſer Baſe
hath to the part towards the greater Baſe, the
ſame proportion that the greater Baſe hath to
the leſſer.
bolick Conoid the Center of Gravity is in the
Right Line that is Axis of the Fruſtum; which
being divided into three equal parts the Cen
ter of Gravity is in the middlemoſt and ſo di
vides it, as that the part towards the leſſer Baſe
hath to the part towards the greater Baſe, the
ſame proportion that the greater Baſe hath to
the leſſer.
From the Conoid whoſe Axis is R B let there be cut off the Solid
whoſe Axis is B E; and let the cutting Plane be equidiſtaut to
the Baſe: and let it be cut in another Plane along the Axis erect
upon the Baſe, and let it be the Section of the Parabola V R C: R B
ſhall be the Diameter of the proportion, or the equidiſtant Diameter
whoſe Axis is B E; and let the cutting Plane be equidiſtaut to
the Baſe: and let it be cut in another Plane along the Axis erect
upon the Baſe, and let it be the Section of the Parabola V R C: R B
ſhall be the Diameter of the proportion, or the equidiſtant Diameter