HYPERBOLA. 
L M double G E or A H, E M shall also be 
double C H ; but because of the circle, E L 
or Aa: E F : : E K : E M ; and taking their 
halves, it will be as C A : C F : : C D : 
C H, 
Prop. III. (fig. 7and 8.) the same things be- 
ing supposed, if from A the extremity of the 
transverse axis nearest to the point Gy there 
is set off a right line A H on the axis pro- 
duced, equal to the distance of the point G 
from the focus F, nearest to the said extre- 
mity ; the square of the perpendicular G D 
shall be equal to the excess of the rectangle 
EHF, contained under the segments be- 
tween H (the extremity of the right line 
A H) and the foci, above the rectangle 
AD a contained under the segments cut off 
between the perpendicular and the extre- 
mities of the axis. 
For since the right line CH is any how 
cut in A, the squares of CA and C H 
together will be equal to twice the rect- 
angle A C H, and the square of A H, 
(7. 2.) i. e. because C A, C F, CD, CH 
are proportionals, to twice the rectangle 
F C D, and to the square of AH or G F ; 
that is, to twice the rectangle F C D and 
the squares of F D and D G, that is, to the 
squares of F C, C D, and D G, (7. 2.) 
wherefore the two squares of C A and C H 
are equal to the three squares of F C, C D, 
and DG; and taking away the squares of 
C A and C F from both sides, the remaining 
rectangle EHF, will be equal to the re- 
maining rectangle A D a, and to the square 
of D G (6. 2.) 
Prop. IV. (fig. 7 and 8. ) If from any point 
G of the hyperbola, there is drawn a right 
line parallel to the second axis B b, meeting 
the transverse axis A a in D ; the square of 
the transverse axis shall be to the square 
of the second axis, as the rectangle con- 
tained under the segments of the tranverse- 
axis between the parallel and its extremes, 
to the square of the parallel. 
Prop; V. (fig. 8.) If from any point 
G of the hyperbola there is drawn a right 
line parallel to the transverse axis A a, 
meeting the second axis in N ; the square 
of the second axis shall be, to the square 
of the transverse, as the sum of the squares 
of the half of the second axis and its seg- 
ment, between the centre and the right line, 
to the square of the line itself ; that is, 
C B 2 : C A 2 . : C B 2 -f G D 2 : C A 2 + the 
rectangle A D a ; that is, as C B 2 C N 2 
is to CD 2 or G N 2 . 
Prop. VI. (fig. 9.) It is another pro- 
perty of the hyperbola, that the asymptotes, 
D d, E e, do never absolutely meet with the 
curve. See Asymptote. 
Prop. VII. If through any point F 
(fig. 9.) of the hyperbola, there is drawn a 
right line I F L parallel to the second axis, 
and meeting the asymptotes in I and L ; the 
rectangle contained under the right lines 
which are intercepted between the asymp- 
totes and the hyperbola, is equal to the 
square of the half of the se'cond axis, that is, 
CB 2 =IFL= IHL. 
Prop. VIII. (fig. 10.) If from any point 
F of the hyperbola, there is drawn to the 
transverse diameter, A B, a right line ordi- 
nately applied to it F G ; and from the ex- 
tremity of the diameter there is drawn A H 
perpendicular to it, and equal to the latus 
rectum ; the square of the ordinate shall be 
eqyal to the rectangle applied to the latus 
rectum, being of the breadth of the abscissa 
between the ordinate and the vertex, and 
which exceeds it by a figure like and alike 
situated to that which is contained under 
the diameter and the latus rectum. 
For join B H, and from the point G let 
there be drawn G M parallel to A H, and 
meeting B H in M, and through M let 
there be drawn M N parallel to A B 
meeting A II in N, and let the rectangles 
M NHO, B A H P, be completed. Then 
since the rectangle A G B, is to the square 
of G F, as A B is to A H, i. e. as G B is to 
G M, i. e. as the rectangle A G B is to the 
rectangle ACM; AG B shall be to the 
square of G F, as the same A G B to the 
rectangle A G M : wherefore the square of 
G F is equal to the rectangle A G M, which 
is applied to the latus rectum A H, having 
the breadth AG, aqd exceeds the rectangle 
H A G O, by the rectangle M N H O, like 
to B A H P ; from which excess the name 
of hyperbola was given to this curve by 
Apollonius. 
Prob. 1. An easy method to describe the 
hyperbola, fig. it. having the transverse dia- 
meter, D E, and the foci N n given. From 
N, at any distance,, as N F, strike an arch ; 
and w ith the same opening of the compasses 
with one foot in E, the vertex, set off E G 
equal to N F in the axis continued; then 
with the distance G D, and one foot in n, 
the other focus, cross the former arch in F. 
So F is a point in the hyperbola : and by this 
method repeated may be found any other 
point, /, further on, and as many more as 
you please. 
An asymptote being taken for a diameter; 
divided into equal parts, and through all the 
divisions, which form so many abscisses 
