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