CONIC SECTIONS. 
gemi-axis D C is less than any other seiiii- 
tliameter C H. 
Fig. 42. In the hyperbola, a tangent of 
.the curve drawn from the extremity of the 
axis C A, as AT, falls between the centre 
and the curve ; and because C A, the semi- 
axis, is less than any other line drawn from 
C to A T, much more is it less than a semi- 
diameter C H drawn from C to tlie curve 
on the other side of A’T. 
Cor. Hence it is plain, that an ellipse, or 
opposite hyperbolas, have only two axes. 
Def. 17. The greater axis of an ellipse is 
called the transverse axis ; and the less, the 
conjugate axis ; and, in the hyperbola, that 
one is the transverse axis which is a trans- 
verse diameter, and tlie other is the conju- 
gate axis. 
PROP. xxvn. 
Fig. 41 and 42. A diameter of an ellipse 
nearer the transverse axis is greater than one 
more remote ; and a transverse diameter 
of the hyperbola nearer the transverse axis 
is less than one more remote. 
Let C K and C H (fig. 41.) be two semi- 
diameters of an ellipse ; join HK, and draw 
A G parallel to H K ; join C G and draw 
CL to bisect HK. Because CL bisects 
H K, it will likewise bisect A G. Cor. 14. 
And because A M = M G, and A C is 
greater than C G, therefore the angle A M C 
is greater than the angle GM C, (25. 1. E) ; 
that is, the angle K L C is greater than the 
angle H L C. And because H L = L K, 
therefore KC, nearer to CA, is greater 
than HC more remote from CA, 24. 
1. E. 
In the hyperbola, the same construction 
being made, because A C is less than C G, 
therefore the angle A M C, or K L C, is less 
than the angle GMC, or HLC. There- 
fore C K is less than C H. 
PROP. XXVIII. 
Fig. 43. A parabola has only one axis. 
Let OS, terminated by the curve, be 
perpendicular to any diameter, and draw 
(he diameter P Q to bisect O S , and, be- 
cause all the diameters of the curve are pa- 
rallel, therefore PQ is perpendicular to 
Os, and an axis of the curve, Def. 17. 
And because • O S can be an ordinate of 
only one diameter, therefore there is only 
one axis. 
Def. 19. Fig. 44, 45, and 46. Let AB 
(fig. 44 and 46.) be the transverse axis, 
D E the conjugate axis, and C the centre 
of an ellipse, or hypei bola, or opposite hy- 
perbolas : and let C F and Cf be taken in 
the transverse axis, such that C F^ and C/^ 
are each equal to CA^ — CIP in the el- 
lipse, and to C A^ -j- C in the hyperbola ; 
then the two points F and / are called the 
foci of the ellipse, hyperbola, or opposite 
hyperbolas. 
But the focus of a parabola (fig. 45.) is a 
point F in the axis within the curve, and 
distant from the vertex by a line equal to 
one fourth part of the parameter of the 
axis. 
Cor. The distance of each foci of an el- 
lipse from either extremity of the conjugate 
axis is equal to half the transverse axis ; and 
the distance of either of the foci of a hyper- 
bola from the centre is equal to the distance 
between the extremities of the ti’ansverse 
and conjugated axes. 
DfSO. If F (fig. 44 and 46) be a focus of 
an ellipse, or hyperbola, or opposite hyper- 
bolas, and A G be taken in the transverse axis 
(on the opposite side of tlie vertex to the 
focus F), such, that A F is to A G as CF is 
to C A; then a line, as HK, drawn through 
G peipendicular to the transverse axis, is 
called a directrix of the ellipse, or hyper- 
bola, or opposite hyperbolas. 
Fig. 45. But the directi-ix of a parabola 
is a line, as H K, perpendicular to the axis, 
drawn through a point G as far distant from 
the vertex of the axis on the one side as tlie 
focus is on the other side. 
Cor. An ellipse, hyperbola, or opposite 
hyperbolas, have two directrices ; one cor- 
responding to each focus. For the same 
construction that is made for one focus may 
be made for the other focus. 
PROP, XXIX. 
Fig. 44 and 46. Let A B be the trans- 
verse, and DE the conjugate axis of an 
ellipse, or hyperbola, or opposite hyperbo- 
las ; from any point in the curve, or oppo- 
site curves, as M, let M C be drawn to the 
centre, and M P perpendicular to the trans- 
verse axis, and take C O in the same axis, 
such that C may be equal to M — 
C in the ellipse, and to M C^ -(- C in 
the hyperbola ; then as A C is to C F so is 
PC to CO. 
For, because A B and D E are conjugate 
diameters, therefore, 
AC^iCD^ ::AP X PB;MP^ (Cor. 3. 
Def. 15.) therefore, A C^ ; A C^ ^ C D^ :: 
AP X PB:AP X PBqpMP^ Butin 
theellipseAC"— CD= = CF2j and AP x 
