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.
zoom in
zoom out
zoom area
full page
page width
set mark
remove mark
get reference
digilib