1their Centers and the ſaid points, may be leſſer than any other Lines.

To expreſs the ſame in other terms, we have reduced it to an impoſſibi

lity, that the Center of the Conoid ſhould not fall betwixt the Centers of

the inſcribed and circumſcribed Figures.

To expreſs the ſame in other terms, we have reduced it to an impoſſibi

lity, that the Center of the Conoid ſhould not fall betwixt the Centers of

the inſcribed and circumſcribed Figures.

PROPOSITION.

Suppoſing three proportional Lines, and that

what proportion the leaſt hath to the exceſs

by which the greateſt exceeds the leaſt, the

ſame ſhould a Line given have to two thirds of

the exceſs by which the greateſt exceeds the

middlemoſt: and moreover, that what pro

portion that compounded of the greateſt, and

of double the middlemoſt, hath unto that com

pounded of the triple of the greateſt and mid

dlemoſt, the ſame hath another Line given, to

the exceſs by which the greateſt exceeds the

middle one; both the given Lines taken toge

ther ſhall be a third part of the greateſt of the

proportional Lines.

what proportion the leaſt hath to the exceſs

by which the greateſt exceeds the leaſt, the

ſame ſhould a Line given have to two thirds of

the exceſs by which the greateſt exceeds the

middlemoſt: and moreover, that what pro

portion that compounded of the greateſt, and

of double the middlemoſt, hath unto that com

pounded of the triple of the greateſt and mid

dlemoſt, the ſame hath another Line given, to

the exceſs by which the greateſt exceeds the

middle one; both the given Lines taken toge

ther ſhall be a third part of the greateſt of the

proportional Lines.

Let A B, B C, and B F, be three proportional Lines; and what

proportion B F hath to F A, the ſame let M S have to two thirds

of C A. And what proportion that compounded of A B and the

double of B C hath to that compounded of the triple of both A B and

B C, the ſame let another, to wit S N, have to A C. Becauſe therefore

that A B, B C, and C F,

171[Figure 171]

are proportionals, A G

and C F ſhall, for the ſame

reaſon, be likewiſe ſo.

Therefore, as A B is to

B C, ſo is A C to C F:

and as the triple of A B is to the triple of B C, ſo is A C to C F:

Therefore, what proportion the triple of A B with the triple of B C

hath to the triple of C B, the ſame ſhall A C have to a Line leſs than

C F. Let it be C O. Wherefore by Compoſition and by Converſion of

proportion, O A ſhall have to A C, the ſame proportion, as triple A B

with Sextuple B C, hath to triple A B with triple B C. But A C hath

to S N the ſame proportion, that triple A B with triple B C hath to A B

with double B C: Therefore, ex equali, O A to NS ſhall have the

ſame proportion, as triple A B with Sexcuple B C hath to A B with

proportion B F hath to F A, the ſame let M S have to two thirds

of C A. And what proportion that compounded of A B and the

double of B C hath to that compounded of the triple of both A B and

B C, the ſame let another, to wit S N, have to A C. Becauſe therefore

that A B, B C, and C F,

171[Figure 171]

are proportionals, A G

and C F ſhall, for the ſame

reaſon, be likewiſe ſo.

Therefore, as A B is to

B C, ſo is A C to C F:

and as the triple of A B is to the triple of B C, ſo is A C to C F:

Therefore, what proportion the triple of A B with the triple of B C

hath to the triple of C B, the ſame ſhall A C have to a Line leſs than

C F. Let it be C O. Wherefore by Compoſition and by Converſion of

proportion, O A ſhall have to A C, the ſame proportion, as triple A B

with Sextuple B C, hath to triple A B with triple B C. But A C hath

to S N the ſame proportion, that triple A B with triple B C hath to A B

with double B C: Therefore, ex equali, O A to NS ſhall have the

ſame proportion, as triple A B with Sexcuple B C hath to A B with