CONIC SECTIONS. 
HEK are parallel to one another: there- 
fore by this proposition, 
R D X D G> or R ; H E X E K or 
Cor. 2. If the square of one ordinate, of 
the diameter of a parabola, as RD, be 
made equal to a rectangle contained by the 
corresponding abscissa B D and the line P : 
then, it is manifest from the last corollary, 
that the square of any other ordinate of the 
same diameter, as H E, will be equal to a 
rectangle under the corresponding abscissa 
B E, and the same line P. 
The line P is called the parameter of the 
diameter to which the ordinates are drawn. 
Fig. 35. Def. 16. If two right lines, as 
G C S and F C T, be drawn through the 
centre of two opposite hyperbolas, so as to 
be parallel to the two lines in the conic sur- 
face, which are the intersections of that 
surface, and a plane drawn through the ver- 
tex of the cone, parallel to the plane of the 
hyperbolas, (viz. to the lines V m and V n, 
in fig. 14) : these two lines G S and F T 
are called the asymptotes of the hyperbolas. 
Cor. 1. Every line drawn through the 
centre, within the angles of the asymptotes 
that are turned to the hyperbolas, is a trans- 
verse diameter: and every line drawn 
through the centre within the adjacent 
angle is a second diameter. 
For the former lines are parallel to lines 
(such as V O in fig. 14.) drawn within the 
cone in the angle contained by the two 
lines (mV and mV, fig. ,14.) in the conic 
surface, that are parallel to the asymptotes ; 
and the latter lines are parallel to lines 
(such as R VS, fig. 14.) without the cone : 
whence the truth of the corollary is mani- 
fest by Cor. 2, Def. 7. and Prop. 14. 
PROP. XXII. 
The asymptotes do not meet either of 
the opposite hyperbolas. 
For if an asymptote be supposed to meet 
one of the hyperbolas, being drawm through 
the centre, it will likewise meet the other 
hyperbola (Cor. 12) : and thus a line, drawn 
parallel to a line contained in the surfece 
of a cone, would meet both the opposite 
conic surfaces, which is impossible (Pr. 2.) 
PROP. XXIII. 
Fig. 35 and 36. If a point be assumed 
without a hyperbola, but within the 
asymptotes, and a right line be drawn 
from it to touch or cut the hyperbola, or 
opposite hyperbolas : then the square of the 
tangent, or the rectangle under the seg- 
ments of the secant, is less than the square 
of the semi-diameter parallel to the tan- 
gent or secant; but if the point be assumed 
without both the hyperbola and the asymp- 
totes, the square of the tangent, or the rec- 
tangle under the segments of the secant, 
is greater than the square of the semi-dia- 
meter parallel to the tangent or secant. 
First, let the point P he without the hy- 
pei'bola, and within the asymptotes, and 
let PH, (fig. 35.) parallel to the semi-dia- 
meter C D touch the hyperbola ; because 
P is a point within the asymptotes, the line 
drawn from it through the centre will he 
a transverse diameter : thus, 
CE^:CD^;:EP X PK:Pff (19 and 
Cor. 1. Def. 15.) 
But C£^ is greater than EP X PK; 
therefore C D^ is greater than P H*. And 
in like manner may the proposition be de- 
monstrated when the line drawn from P 
does not touch, but cuts the hyperbola, or 
opposite hyperbolas. 
Next, let P be without the asymptotes : 
draw RS (fig. 36), terminated by one of 
the hyperbolas parallel to CP, the line 
drawn from P to the centre : draw the dia- 
meter C E to bisect R S and M N tlirough 
P parallel to C E. Because the diameter 
C E bisects R S parallel to C P, therefore 
M N, parallel to C E, is ordinately applied 
to the second diameter C P (18). Let C Q 
be the magnitude of this seraidiameter, 
then 
CQ* : CE^ :: CQ^-j-CP^ : P (20) 
And CE2 : CD" :: M P X P N, or PN» 
: PH" 
Ex equo, C Q" : C D" :: C Q" -f C P : PH". 
But CQ"-|-CP" is greater than CQ", 
therefore P H" is greater C D". And in 
like manner may the proposition be proved, 
when the line drawn through P does not 
touch, but cuts a hyperbola, or opposite hy- 
perbolas. 
PROP. XXIV. 
Fig. 37, 38. If from a point (PorQ) in an 
asymptote of a right line be drawn to touch 
or cut the hyperbola, or opposite hyperbo- 
las (P H or Q R S) : the square of tiie tan- 
gent, or the rectangle under the segments of 
the secant (P H" or Q R x Q S) is equal to 
the square of the semidiameter (C D") pa- 
rallel to the tangent or secant. 
For if not, make H O" and R 0‘ x O' S. 
equal to C D" : then O and O' are without 
the hyperbola, and they must be eithei;, 
within the asymptotes or without them. In 
the former case H O" and R O' x O' S 
would be less than CD" (23.); an,d in the 
latter case H O" and R O' X O' S would be 
