Again, let there be two like Portions, contained betwixt Right
Lines and the Sections of Right-angled Cones, as in the fore
going figure, A B C, whoſe Diameter is B D; and E F C,
whoſe Diameter is F G; and from the Point E, draw the
Line E H parallel to the Diameters B D and F G; and let it
cut the Section A B C in K: and from the Point C draw C H
touching the Section A B C in C, and meeting with the Line
E H in H; which alſo toucheth the Section E F C in the ſame
Point C, as ſhall be demonſtrated: I ſay that the Line drawn
from C H unto the Section E F C ſo as that it be parallel to
the Line E H, ſhall be divided in the ſame proportion by the
Section A B C, in which the Line C A is divided by the Section
E F C; and the part of the Line C A which is betwixt the
two Sections, ſhall anſwer in proportion to the part of the Line
drawn, which alſo falleth betwixt the ſame Sections: that is,
as in the foregoing Figure, if D B be produced untill it meet
with C H in L, that it may interſect the Section E F C in the
Point M, the Line L B ſhall have to B M the ſame proportion
that C E hath to E A.
Lines and the Sections of Right-angled Cones, as in the fore
going figure, A B C, whoſe Diameter is B D; and E F C,
whoſe Diameter is F G; and from the Point E, draw the
Line E H parallel to the Diameters B D and F G; and let it
cut the Section A B C in K: and from the Point C draw C H
touching the Section A B C in C, and meeting with the Line
E H in H; which alſo toucheth the Section E F C in the ſame
Point C, as ſhall be demonſtrated: I ſay that the Line drawn
from C H unto the Section E F C ſo as that it be parallel to
the Line E H, ſhall be divided in the ſame proportion by the
Section A B C, in which the Line C A is divided by the Section
E F C; and the part of the Line C A which is betwixt the
two Sections, ſhall anſwer in proportion to the part of the Line
drawn, which alſo falleth betwixt the ſame Sections: that is,
as in the foregoing Figure, if D B be produced untill it meet
with C H in L, that it may interſect the Section E F C in the
Point M, the Line L B ſhall have to B M the ſame proportion
that C E hath to E A.
For let G F be prolonged untill it meet the ſame Line C H in N, cutting the Section A B C
in O; and drawing a Line from B to C, which ſhall paſſe by F, as hath been ſhewn, the
52[Figure 52]
Triangles C G F and C D B ſhall be alike; as
alſo the Triangles C F N and C B L: Wherefore
(a) as G F is to D B, ſo ſhall C F b to C B:
And as (b) C F is to C B, ſo ſhall F N be
to B L: Therefore G F ſhall be to D B, as F N
to B L: And, Permutando, G F ſhall be to
F N, as D B to B L: But D B is equall to
B L, by 35 of our Firſt Book of Conicks:
Therefore (c) G F alſo ſhall be equall to F N:
And by 33 of the ſame, the Line C H touch
eth the Section E F C in the ſame Point. There
fore, drawing a Line from C to M, prolong it
untill it meet with the Section A B C in P; and
from P unto A C draw P Q parallel to B D.
Becauſe, now, that the Line C H toucheth the
Section E F C in the Point C; L M ſhall have
the ſame proportion to M D that C D hath to D E,
by the Fifth Propoſition of Archimedes in his
Book De Quadratura Patabolæ: And by
reaſon of the Similitude of the Triangles C M D
and C P Q, as C M is to C D, ſo ſhall C P
be to C Q: And, Permutando, as C M is to
C P, ſo ſhall C D be to C Q: But as C M is to C P, ſo is C E to C A,; as we have but
even now demonſtrated: And therefore, as C E is to C A, ſo is C D to C que that is as the
whole is to the whole, ſo is the part to the part: The remainder, therefore, D E is to the
Remainder Q A, as C E is to C A; that is, as C D is to C Q: And, Permutando, C D
is to D E, as C Q is to Q A: And L M is alſo to M D, as C D to D E: Therefore L M is
in O; and drawing a Line from B to C, which ſhall paſſe by F, as hath been ſhewn, the
52[Figure 52]Triangles C G F and C D B ſhall be alike; as
alſo the Triangles C F N and C B L: Wherefore
(a) as G F is to D B, ſo ſhall C F b to C B:
And as (b) C F is to C B, ſo ſhall F N be
to B L: Therefore G F ſhall be to D B, as F N
to B L: And, Permutando, G F ſhall be to
F N, as D B to B L: But D B is equall to
B L, by 35 of our Firſt Book of Conicks:
Therefore (c) G F alſo ſhall be equall to F N:
And by 33 of the ſame, the Line C H touch
eth the Section E F C in the ſame Point. There
fore, drawing a Line from C to M, prolong it
untill it meet with the Section A B C in P; and
from P unto A C draw P Q parallel to B D.
Becauſe, now, that the Line C H toucheth the
Section E F C in the Point C; L M ſhall have
the ſame proportion to M D that C D hath to D E,
by the Fifth Propoſition of Archimedes in his
Book De Quadratura Patabolæ: And by
reaſon of the Similitude of the Triangles C M D
and C P Q, as C M is to C D, ſo ſhall C P
be to C Q: And, Permutando, as C M is to
C P, ſo ſhall C D be to C Q: But as C M is to C P, ſo is C E to C A,; as we have but
even now demonſtrated: And therefore, as C E is to C A, ſo is C D to C que that is as the
whole is to the whole, ſo is the part to the part: The remainder, therefore, D E is to the
Remainder Q A, as C E is to C A; that is, as C D is to C Q: And, Permutando, C D
is to D E, as C Q is to Q A: And L M is alſo to M D, as C D to D E: Therefore L M is