CONIC SECTIONS. 
therefore H L = K N ; and it is plain that 
HA passes tlirough K, and that HK is 
bisected in the centre A. 
Cor. It follows from this proposition, that 
a right line drawn through the centre of 
two opposite hyperbolas from a point H 
in one of them will meet the other. 
PROP. xni. 
Ftg. 23. An ellipse, or opposite hyper- 
bolas, have only one centre. 
If there were two centres of an ellipse, 
tlien the right line drawn through them, and 
terminated by the periphery, would be bi- 
sected in two different points (12), wliich is 
absurd. 
If it be possible, let A and D be both 
centres of two opposite hyperbolas, and 
from C, a point in one of the hyperbolas, 
draw CAB and CDF through A and D 
to meet the opposite hyperbola : also from 
B and F draw B D E and F A G to meet 
the first hyperbola, and join D A, G C, and 
C E. Because A and D are both centres, 
therefore B A = A C, and B D = D E, and 
C E is parallel to D A. In like manner, 
because F D =: D C, and F A := A G, there- 
fore CG is parallel to DA. Therefore 
GC and CE, drawn through the same 
point and parallel to the same line, make 
only one right line that meets a conic sec- 
tion in three points, which is absurd. 
Cor. All the diameters of an ellipse, or 
opposite hyperbolas, intersect in the cen- 
tre, and mutually bisect one another. 
For if not, tlien there would be more 
than one centre. 
PROP. XIV. 
Fig. 24, 25, 26. Every right line drawn 
through the centre of an ellipse is a diame- 
ter ; and every right line drawn through the 
centre of two opposite hyperbolas so as to 
be terminated by the opposite hyperbolas, 
or so as to be parallel to a right line termi- 
nated by one of tire hyperbolas, is a dia- 
meter. 
Wlien a line drawn through the centre A 
of two opposite hyperbolas is parallel to 
HR (fig. 23), a line terminated in one hy- 
perbola, draw the diameters H A G, F A K, 
and join FH and GK; and when a line 
drawn through the centre is terminated by 
an ellipse (fig. 24, 25), or opposite hyper- 
bolas, draw H K parallel to it, and make 
the same construction as before. Because 
HA=::AG, and KA=AF (Def. 10.) 
the two triangles F A H and GAR are 
equal in all respects, and it is manifest that 
F H and G R are parallel, and are bisected 
by the line through the centre parallel to 
HR: therefore that line is a diameter. 
(Def. 9.) 
Cor. A right line drawn through the 
centre of an ellipse, or opposite hyperbolas 
which bisect one right line not passing 
tlirough the centre, and terminated by the 
ellipse, or one of the hyperbolas, or both, 
will bisect all right lines terminated in the 
like manner, and parallel to the former 
line. 
For the right line which bisects all the 
parallels passes ^through the centre : and 
therefore it must coincide with the line that 
bisects one of the parallels, and is drawn 
through the centre. 
PROP. XV. 
Fig. 27. All the diameters of a parabola 
are parallel to one another. 
Let B C be a diameter of a parabola 
bisecting the lines D E and F G ; take any 
point L within the parabola, and draw M N 
through it parallel to D E or F G, and ter- 
minated by the curve: tlien BC will bisect 
M N ; and as tliis is true however remote 
from the lines D E and F G the line M N is 
drawn, it follows that the diameter B C 
cannot meet the curve in more than one 
point : and the same tiling may be shewn of 
every other diameter as P Q. But all 
those right lines are parallel to one another 
which cut a parabola in one point only. 
(Cor. 2. Def. 6.) 
Cor. A right line, parallel to a diameter 
of a parabola, which bisects one right line, 
terminated by the parabola, will bisect all 
other right lines parallel to the former and 
terminated by the parabola. 
Def. 11. A diameter of two opposite hy- 
perbolas, which is terminated by the two 
curve.s, is called a transverse diameter : and 
a diameter which meets neither of the 
curves is called a second diameter. 
Def. 12. A vertex of a diameter is a 
point where the diameter meets the conic 
section. 
The magnitude of a diameter, that meets 
a conic section or opposite sections in two 
points, is the line between the two vertices. 
Def. 13. A right line, not passing 
through the centre, terminated by a conic 
section, or opposite sections, and bisected 
by a diameter, is said to be ordinately ap- 
plied to that diameter : or, it is called a 
double oidinate, and the half it, an oydi- 
nate. 
