1
49[Figure 49]
from K to C, cutting the Diameter F G in L:
and, thorow L, unto the Section E F. G, on the
part E, draw the Line L M, parallel unto the
ſame Baſe A C. And, of the Section A B C,
let the Line B N be the Parameter; and, of the
Section E F C, let F O be the Parameter. And,
becauſe the Triangles C B D and C F G are alike;
(b) therefore, as B C is to C F, ſo ſhall D C be
to C G, and B D to F G. Again, becauſe the
Triangles C K B and C L F, are alſo alike to
one another; therefore, as B C is to C F, that is,
as B D is to F G, ſo ſhall K C be to C L, and B K to F L: Wherefore, K C to C L, and,
B K to F L, are as D C to C G; that is, (c) as their duplicates A C and C E: But as
B D is to F G, ſo is D C to C G; that is, A D to E G: And, Permutando, as B D is to
A D, ſo is F G to E G: But the Square A D, is equall to the Rectangle D B N, by the 11
of our firſt of Conicks: Therefore, the (d) three Lines B D, A D and B N are
Proportionalls. By the ſame reaſon, likewiſe, the Square E G being equall to the Rectangle
G F O, the three other Lines F G, E G and F O, ſhall be alſo Proportionals: And, as B D is
to A D, ſo is F G to E G: And, therefore, as A D is to B N, ſo is E G to F O: Ex equali,
therefore, as D B is to B N, ſo is G F to F O: And, Permutando, as D B is to G F, ſo is
B N to F O: But as D B is to G F, ſo is B K to F L: Therefore, B K is to F L, as
B N is to F O: And, Permutando, as B K is to B N, ſo is F L to F O. Again,
becauſe the (e) Square H K is equall to the Rectangle B N; and the Square M L, equall
to the Rectangle L F O, therefore, the three Lines B K, K H and B N ſhall be Proportionals:
and F L, L M, and F O ſhall alſo be Proportionals: And, therefore, (f) as the Line
B K is to the Line B N, ſo ſhall the Square B K, be to the Square H K: And, as the
Line F L is to the Line F O, ſo ſhall the Square F L be to the Square L M:
Therefore, becauſe that as B K is to B N, ſo is F L to F O; as the Square
B K is to the Square K H, ſo ſhall the Square F L be to the Square L M: Therefore,
(g) as the Line B K is to the Line K H, ſo is the Line F L to L M: And, Permutando,
as B K is to F L, ſo is K H to L M: But B K was to F L, as K C to C L: Therefore,
K H is to L M, as K C to C L: And, therefore, by the preceding Lemma, it is manifeſt that
the Line H C alſo ſhall paſs thorow the Point M: As K C, therefore, is to C L, that is,
as A C to C E, ſo is H C to C M; that is, to the ſame part of it ſelf, that lyeth betwixt C and
the Section E F C. And, in like manner might we demonſtrate, that the ſame happeneth
in other Lines, that are produced from the Point C, and the Sections E B C. And, that
B C hath the ſame proportion to C F, plainly appeareth; for B C is to C F, as D C to C G;
that is, as their Duplicates A C to C E.
49[Figure 49]from K to C, cutting the Diameter F G in L:
and, thorow L, unto the Section E F. G, on the
part E, draw the Line L M, parallel unto the
ſame Baſe A C. And, of the Section A B C,
let the Line B N be the Parameter; and, of the
Section E F C, let F O be the Parameter. And,
becauſe the Triangles C B D and C F G are alike;
(b) therefore, as B C is to C F, ſo ſhall D C be
to C G, and B D to F G. Again, becauſe the
Triangles C K B and C L F, are alſo alike to
one another; therefore, as B C is to C F, that is,
as B D is to F G, ſo ſhall K C be to C L, and B K to F L: Wherefore, K C to C L, and,
B K to F L, are as D C to C G; that is, (c) as their duplicates A C and C E: But as
B D is to F G, ſo is D C to C G; that is, A D to E G: And, Permutando, as B D is to
A D, ſo is F G to E G: But the Square A D, is equall to the Rectangle D B N, by the 11
of our firſt of Conicks: Therefore, the (d) three Lines B D, A D and B N are
Proportionalls. By the ſame reaſon, likewiſe, the Square E G being equall to the Rectangle
G F O, the three other Lines F G, E G and F O, ſhall be alſo Proportionals: And, as B D is
to A D, ſo is F G to E G: And, therefore, as A D is to B N, ſo is E G to F O: Ex equali,
therefore, as D B is to B N, ſo is G F to F O: And, Permutando, as D B is to G F, ſo is
B N to F O: But as D B is to G F, ſo is B K to F L: Therefore, B K is to F L, as
B N is to F O: And, Permutando, as B K is to B N, ſo is F L to F O. Again,
becauſe the (e) Square H K is equall to the Rectangle B N; and the Square M L, equall
to the Rectangle L F O, therefore, the three Lines B K, K H and B N ſhall be Proportionals:
and F L, L M, and F O ſhall alſo be Proportionals: And, therefore, (f) as the Line
B K is to the Line B N, ſo ſhall the Square B K, be to the Square H K: And, as the
Line F L is to the Line F O, ſo ſhall the Square F L be to the Square L M:
Therefore, becauſe that as B K is to B N, ſo is F L to F O; as the Square
B K is to the Square K H, ſo ſhall the Square F L be to the Square L M: Therefore,
(g) as the Line B K is to the Line K H, ſo is the Line F L to L M: And, Permutando,
as B K is to F L, ſo is K H to L M: But B K was to F L, as K C to C L: Therefore,
K H is to L M, as K C to C L: And, therefore, by the preceding Lemma, it is manifeſt that
the Line H C alſo ſhall paſs thorow the Point M: As K C, therefore, is to C L, that is,
as A C to C E, ſo is H C to C M; that is, to the ſame part of it ſelf, that lyeth betwixt C and
the Section E F C. And, in like manner might we demonſtrate, that the ſame happeneth
in other Lines, that are produced from the Point C, and the Sections E B C. And, that
B C hath the ſame proportion to C F, plainly appeareth; for B C is to C F, as D C to C G;
that is, as their Duplicates A C to C E.
From whence it is manifeſt, that all Lines ſo drawn, ſhall be cut by the
ſaid Section in the ſame proportion. For, by Diviſion and Converſion,
C M is to M H, and C F to F B, as C E to E A.
ſaid Section in the ſame proportion. For, by Diviſion and Converſion,
C M is to M H, and C F to F B, as C E to E A.
LEMMA. III.
And, hence it may alſo be proved, that the Lines which are
drawn in like Portions, ſo, as that with the Baſes, they con
tain equall Angles, ſhall alſo cut off like Portions; that is,
as in the foregoing Figure, the Portions H B C and M F C,
which the Lines C H and C M do cut off, are alſo alike to
each other.
drawn in like Portions, ſo, as that with the Baſes, they con
tain equall Angles, ſhall alſo cut off like Portions; that is,
as in the foregoing Figure, the Portions H B C and M F C,
which the Lines C H and C M do cut off, are alſo alike to
each other.