CONIC SECTIONS. 
assumed in the plane of two opposite hyper- 
bolas, and let the secant O H li be drawn 
through it parallel to a transverse diameter 
B A ; and the secants R O S, G O L, &c. 
parallel to any second diameters M N, 
P Q, &c. : in these diameters take the seg- 
ments M N, P Q, &c. all bisected in the 
centre, such that the squares of M N, P Q, 
&c. may severally be to the square of the 
transverse diameters A B, as the rectangles 
R O X O S, G O X O H, &c. contained 
by the segments of the secants parallel 
to tlie second diameters are to K O X 
O H, the-rectangle under the segments of 
the secant parallel to the transverse dia- 
meter: then the magnitudes of the se- 
cond diameters are the segments MN, 
P Q &c. 
Because the ratios of the rectangles 
KO X OH,SO X O R,GO x OH,&c. 
are invariably the same wherever the point 
O is assumed, (10,) it is plain that the mag- 
nitudes of the second diameters M N,PQ, 
&c. are also invariably the same wherever 
the point O is assumed. 
And because the ratio of the rectangles 
K O X O H to the square of the transverse 
diameter A B is the same as the ratio of the 
rectangle contained by the segments of any 
secant drawn through O parallel to a trans- 
verse diameter, to the square of the trans- 
verse diameter to wHiich it is parallel, (19), 
it is also manifest that the magnitudes of the 
second diameters are the same from what- 
ever transverse diameter they are de- 
duced. 
Cor. 1. And hence, taking the magni- 
tudes of the transverse diameters as here 
defined, Prop. 19, may be enunciated for 
the hyperbola as generally as it is for the 
ellipse : that is, the rectangle under the 
segments of a secant, or the square of a 
tangent parallel to one diameter (whether 
a transverse or a second diameter) of oppo- 
site hyperbolas, is to the rectangle under 
the segments of a secant, or tlie square of a 
tangent, parallel to another diameter, as 
the square of the first diameter is to the 
square of the second diameter. 
Cor. 2. If two tangents be drawn to an 
ellipse, or a hyperbola, or opposite hyper- 
bolas, from the same point, then these tan- 
gents are proportional to the diameters, or 
semi-diameters, drawn parallel to the tan- 
gents. 
For the squares of the tangents are pro- 
portional to the squares of the diameters. 
Cor. 3. If a right line be ordinately ap- 
plied to a diameter of an ellipse, or to a 
transverse diameter of a hyperbola ; then as 
the square of the diameter is to the square 
of the cojpugate diameter, so is the rectan- 
gle contained by the abscisses of the diame- 
ter, between the vertices and ordinate, to 
the square of the ordinate. 
For the double-ordinate is bisected by 
the diameter, and it is parallel to the conju- 
gate diameter. 
PROP. XX. 
Fig. 33. If an ordinate be drawn to a se- 
cond diameter of opposite hyperbolas ; the 
square of this second diameter is to the squai e 
of the conjugate diameter, as the sum of 
the squares of half tlie second diameter, 
and the paid of it between the ordinate and 
the centre, is to the square of the ordinate. 
LetAB and MN be conjugate diame- 
ters of opposite hyperbolas, HK an oidi- 
nate to the second diameter MN, and 
draw K D S parallel to M N ; then K. D S 
is ordinately applied to A B (18.) ; there- 
fore 
MC^:CB^;:KR%orCU:AD xDB, 
orCD" — CB' (Cor. 3. Def. 15.) 
therefore, M : C B* :: MC^-^- 
CL':CD% orKL^ 
\ 
PROP. XXI. 
Fig. 34. If two parallel lines be drawn 
from two points in the diameter of a paia- 
bola to cut or touch the curve : then as the 
rectangle under the segments of the secant, 
or the square of the tangent, drawn from 
one point, is to the rectangle under the seg- 
ment of the secant, or the square ot the tan- 
gent drawn from the other point, so is the 
abscissa of the diameter between the first 
point and the curve to the abscissa between 
the second point and the curve. 
Let the parallel secants MN and PQ 
meet the diameter of a parabola in D and 
E : it has been shewn (Prop. 15.) that the 
diameters of a parabola meet the curve 
only in one point; and therefore (Cor. 1st. 
Def. 7.) they ai'e all parallel to a line in the 
surface of the cone by the section of which 
the parabola is produced {viz. to the line 
VB (fig. 13.) in which th-^ touclung plane, 
parallel to the plane of the parabola, meets 
the conic surface) : therefore Prop. 8. 
MDxGN: PExEQ;:BD:BE. 
Cor. 1. The squares of the ordinates 
drawn to a diameter of a parabola are pro- 
portional to the abscissas of the diameter 
between the ordinates and the vertex. 
For the double ordinates RDG and 
