For let A B, B C, B D, and B E be four proportional Lines. And

as B E is to E A, ſo let F G be to 3/4 of A C. And as the Line equal

to A B and to double B C and to triple B D is to the Line equal

to the quadruples of A B, B C, and B D, ſo let H G be to A C. It is

to be proved, that H F is a fourth part of A B. Foraſmuch therefore

as A B, B C, B D, and B E

178[Figure 178]

are proportionals, A C,

C D, and D E ſhall be in

the ſame proportion: And

as the quadruple of the ſaid

A B, B C, and B D is to

A B with the double of B C and triple of B D, ſo is the quadruple of

A C, C D, and D E; that is, the quadruple of A E; to A C with the

double of C D, and triple of D E. And ſo is A C to H G. Therefore

as the triple of A E is to A C, with the double of C D and triple of

D E, ſo is 3/4 of A C to H G. And as the triple of A E is to the triple of

E B, ſo is 3/4 A C to G F: Therefore, by the Converſe of the twenty

fourth of the fifth, As triple A E is to A C with double C D and tri

ple D B, ſo is 3/4 of A C to H F: And as the quadruple of A E is to A C

with the double of C D and triple of D B; that is, to A B with C B and

B D, ſo is A C to H F. And, by Permutation, as the quadruple of A E

is to A C, ſo is A B with C B and B D to H F. And as A C is to A E, ſo

is A B to A B with C B and B D. Therefore, ex æquali, by Perturbed

proportion, as quadruple A E is to A E, ſo is A B to H F. Wherefore it

is manifeſt that H F is the fourth part of A B.

as B E is to E A, ſo let F G be to 3/4 of A C. And as the Line equal

to A B and to double B C and to triple B D is to the Line equal

to the quadruples of A B, B C, and B D, ſo let H G be to A C. It is

to be proved, that H F is a fourth part of A B. Foraſmuch therefore

as A B, B C, B D, and B E

178[Figure 178]

are proportionals, A C,

C D, and D E ſhall be in

the ſame proportion: And

as the quadruple of the ſaid

A B, B C, and B D is to

A B with the double of B C and triple of B D, ſo is the quadruple of

A C, C D, and D E; that is, the quadruple of A E; to A C with the

double of C D, and triple of D E. And ſo is A C to H G. Therefore

as the triple of A E is to A C, with the double of C D and triple of

D E, ſo is 3/4 of A C to H G. And as the triple of A E is to the triple of

E B, ſo is 3/4 A C to G F: Therefore, by the Converſe of the twenty

fourth of the fifth, As triple A E is to A C with double C D and tri

ple D B, ſo is 3/4 of A C to H F: And as the quadruple of A E is to A C

with the double of C D and triple of D B; that is, to A B with C B and

B D, ſo is A C to H F. And, by Permutation, as the quadruple of A E

is to A C, ſo is A B with C B and B D to H F. And as A C is to A E, ſo

is A B to A B with C B and B D. Therefore, ex æquali, by Perturbed

proportion, as quadruple A E is to A E, ſo is A B to H F. Wherefore it

is manifeſt that H F is the fourth part of A B.

PROPOSITION.

The Center of Gravity of the Fruſtum of any Py

ramid or Cone, cut equidiſtant to the Plane

of the Baſe, is in the Axis, and doth ſo divide

the ſame, that the part towards the leſſer Baſe

is to the remainder, as the triple of the greater

Baſe, with the double of the mean Space be

twixt the greater and leſſer Baſe, together

with the leſſer Baſe is to the triple of the leſſer

Baſe, together with the ſame double of the

mean Space, as alſo of the greater Baſe.

ramid or Cone, cut equidiſtant to the Plane

of the Baſe, is in the Axis, and doth ſo divide

the ſame, that the part towards the leſſer Baſe

is to the remainder, as the triple of the greater

Baſe, with the double of the mean Space be

twixt the greater and leſſer Baſe, together

with the leſſer Baſe is to the triple of the leſſer

Baſe, together with the ſame double of the

mean Space, as alſo of the greater Baſe.