CONIC SECTIONS. 
PROP. XVI. 
Fiff. §8. A right line, drawn from a 
vertex of a diameter of an ellipse, or a pa- 
rabola, or from the vertex of a transverse 
diameter of a hyperbola, so as to be parallel 
to a line ordinately applied to that diame- 
ter, is a tangent of the curve. 
Fig. 28. Let F H be a diameter of an 
ellipse or a parabola, or a transverse diame- 
ter of a hyperbola, and R S T, a line ordi- 
nately applied to tliat diameter ; then F M, 
drawn from a vertex of the diameter, so as 
to be parallel to R T, is a tangent of the 
curve. For, if F M be not a tangent, it will 
cut the section again in another point (Cor. 2. 
Def. 8,) let it cut the section again in K, and 
bisect F K in I. Then, if a diameter of the 
section be drawn through I, that diame- 
ter would bisect R T pardlel to F K, Pr. 
15. Therefore R T would be bisected by 
two different diameters ; viz. by the diame- 
ter F H, and by that drawn through I. But, 
in the ellipse and hyperbola, all the diame- 
ters pass through the centre ; and, in the 
parabola, they are all parallel to one ano- 
ther ; therefore two diameters of a conic 
section will cut every straight line (which 
does not pass through the centre of the el- 
lipse and hyperbola) in two different points. 
Therefore RT cannot be bisected by two 
different diameters. Therefore F M, pa- 
rallel to R T, does not cut the curve again ; 
that is F M is a tangent of tire conic sec- 
tion. 
Cor. 1. If R T be ordinately applied to 
the diameter F H, it is parallel to a tan- 
gent, F M, at a vertex of that diameter. 
For there cannot be two tangents of a 
conic section at the same point of the 
cune. 
Cor. 2. All right lines ordinately applied 
to the same diameter of a conic section are 
parallel to one another. 
For they are all parallel to a tangent at a 
vertex of that diameter. 
PROP. xvir. 
Fig. 29. A right line D E terminated 
both ways by the curve of a conic section, 
and parallel to a tangent F H, is ordinately 
applied to the diameter B C drawn througli 
the point of contact B. 
Take B F and B H, in the tangent on 
opposite sides of the point of contact, equal 
to one another, and of such a magnitude that 
lines drawn through F and H parallel to 
the, diameter B C may cut the curve 
in K and L : join K L. It is plain 
that K L is bisected by B C : therefore 
K L is parallel to the tangent FH (Cor, 
1, 16.) ; and consequently it is also pa- 
rallel to D E (30. 1. E.) ; therefore D E is 
bisected by the same diameter which bi- 
sects KL (Cor. 14.) 
Def. 14. Two diameters of an ellipse 
or of opposite hyperbolas, that are mutually 
parallel to one another’s ordinate, are called 
conjugate diameters. 
Cor. It is plain that two conjugate 
diameters of opposite hyperbolas cannot be 
both transverse, nor both second diameters. 
PROP. XVIII. 
Fig. SO and 31. If adiameterof an ellipse, 
or of opposite hyperbolas, be parallel to the 
ordinates of another diameter, these two are 
conjugate diameters. 
Let the diameter El) be parallel to 
P Q S an ordinate of the diameter F H ; 
draw the diameter PR and join SR cutting 
E D in T. Because P Q = Q S, and 
P G =: G R ; therefore S R is parallel to 
F H. And because E D is parallel to P Q S, 
and P G = G R ; therefore R T =; T S. 
Therefore R S is an ordinate of the diame- 
ter E D, and it is parallel to F H ; there- 
fore E D and F H are conjugate diameters, 
Def. 14. 
Cor. If a diameter of an ellipse, as E D, 
be parallel to F O, a tangent at a vertex of 
another diameter F H ; then F H is paral- 
lel to D I, a tangent at a vertex of E D. 
For a tangent at a vertex of a diameter 
is parallel to the ordinates of that diame- 
ter. 
PROP.' XIX. 
If a point be assumed without or within 
an ellipse, and two right lines, parallel to 
two diameters, be drawn from it to cut or 
touch the ellipse ; then, as the rectangle 
under the segments of the secant, or the 
square of the tangent, parallel to one of the 
diameters, is to the rectangle under the 
segments of the secant, or tlie square of the 
tangent, parallel to the other diameter, so 
is the square of the first diameter to the 
square of the second diameter. And the 
same thing is true of two transverse diame- 
ters of opposite hyperbolas, and any two 
lines, parallel to these, drawn through a 
point to cut the two curves. 
For diameters of an ellipse, and of oppo- 
site hyperbolas, are secants that intersect in 
the centre : and, because they are bi- 
sected there, this proposition is manifest 
from Pr. 10. 
Drf. 15. Fig. 32. Let a point, as O, be 
