(a) By 4. of the
ſixth.
ſixth.
* Or permitting.
(b) By 22. of the
ſixth.
ſixth.
(c) By Cor. of 20.
of the ſixth.
of the ſixth.
LEMMA IV.
The ſame things aſſumed again, and M Q being drawn from the
Point M unto the Diameter, let it touch the Section in the
Point M. I ſay that H Q hath to Q O, the ſame proportion
that G H hath to C N.
Point M unto the Diameter, let it touch the Section in the
Point M. I ſay that H Q hath to Q O, the ſame proportion
that G H hath to C N.
For make H R equall to G F; and ſeeing that
260[Figure 260]
the Triangles A F C and O P N are alike, and
P N equall to F C, we might in like manner de
monſtrate P O and F A to be equall to each other:
Wherefore P O ſhall be double to F B: But H O
is double to G B: Therefore the Remainder P H
is alſo double to the Remainder F G; that is, to
R H: And therefore is followeth that P R, R H
and F G are equall to one another; as alſo that
R G and P F are equall: For P G is common to
both R P and G F. Since therefore, that H B is
to B G, as G B is to B F, by Converſion of Pro
portion, B H ſhall be to H G, as B G is to G F:
But Q H is to H B, as H O to B G. For by 35
of our firſt Book of Conicks, in regard that Q
M toucheth the Section in the Point M, H B and
B Q ſhall be equall, and Q H double to H B:
Therefore, ex æquali, Q H ſhall be to H G, as
H O to G F; that is, to H R: and, Permu
tando, Q H ſhall be to H O, as H G to H R: again, by Converſion, H Q ſhall be to Q
O, as H G to G R; that is, to P F; and, by the ſame reaſon, to C N: Whichwas to be de
monſtrated.
260[Figure 260]the Triangles A F C and O P N are alike, and
P N equall to F C, we might in like manner de
monſtrate P O and F A to be equall to each other:
Wherefore P O ſhall be double to F B: But H O
is double to G B: Therefore the Remainder P H
is alſo double to the Remainder F G; that is, to
R H: And therefore is followeth that P R, R H
and F G are equall to one another; as alſo that
R G and P F are equall: For P G is common to
both R P and G F. Since therefore, that H B is
to B G, as G B is to B F, by Converſion of Pro
portion, B H ſhall be to H G, as B G is to G F:
But Q H is to H B, as H O to B G. For by 35
of our firſt Book of Conicks, in regard that Q
M toucheth the Section in the Point M, H B and
B Q ſhall be equall, and Q H double to H B:
Therefore, ex æquali, Q H ſhall be to H G, as
H O to G F; that is, to H R: and, Permu
tando, Q H ſhall be to H O, as H G to H R: again, by Converſion, H Q ſhall be to Q
O, as H G to G R; that is, to P F; and, by the ſame reaſon, to C N: Whichwas to be de
monſtrated.
Theſe things therefore being explained, we come now to that
which was propounded. I ſay, therefore, firſt that N C hath
to C K the ſame proportion that H G hath to G B.
which was propounded. I ſay, therefore, firſt that N C hath
to C K the ſame proportion that H G hath to G B.
For ſince that H Q is to Q O, as H G to C N;
261[Figure 261]
that is, to A O, equall to the ſaid C N: The Re
mainder G Q ſhall be to the Remainder Q A, as
H Q to Q O: and, for the ſame cauſe, the Lines
A C and G L prolonged, by the things that wee
have above demonstrated, ſhall interſect or meet
in the Line Q M. Again, G Q is to Q A,
as H Q to Q O: that is, as H G to F P; as
(a) was bnt now demonstrated, But unto (b) G
Q two Lines taken together, Q B that is H B, and
B G are equall: and to Q A H F is equall; for
if from the equall Magnitudes H B and B Q there
be taken the equall Magnitudes F B and B A, the
Re mainder ſhall be equall; Therefore taking H
G from the two Lines H B and B G, there ſhall re
main a Magnitude double to B G; that is, O H:
and P F taken from F H, the Remainder is H P:
Wherefore (c) O H is to H P, as G Q to Q A:
But as G Q is to Q A, ſo is H Q to Q O;
261[Figure 261]that is, to A O, equall to the ſaid C N: The Re
mainder G Q ſhall be to the Remainder Q A, as
H Q to Q O: and, for the ſame cauſe, the Lines
A C and G L prolonged, by the things that wee
have above demonstrated, ſhall interſect or meet
in the Line Q M. Again, G Q is to Q A,
as H Q to Q O: that is, as H G to F P; as
(a) was bnt now demonstrated, But unto (b) G
Q two Lines taken together, Q B that is H B, and
B G are equall: and to Q A H F is equall; for
if from the equall Magnitudes H B and B Q there
be taken the equall Magnitudes F B and B A, the
Re mainder ſhall be equall; Therefore taking H
G from the two Lines H B and B G, there ſhall re
main a Magnitude double to B G; that is, O H:
and P F taken from F H, the Remainder is H P:
Wherefore (c) O H is to H P, as G Q to Q A:
But as G Q is to Q A, ſo is H Q to Q O;