HYPERBOLA. 
fixed in the point E, is fixed in the point F, 
and the end of the thread is fixed in the 
point E, and the same things performed as 
before, there will be described another line 
opposite to the former, which is likewise 
called an hyperbola ; and both together are 
called opposite hyperbolas. These lines may 
be extended to any greater distance from 
the points EE, ®iz. if a thread is taken of a 
length greater than that distance. 2. The 
points E and F are called the foci. 3. And 
the point C, which bisects the right line 
between the two focus’s, is called the centre 
of the hyperbola, or of the opposite hy- 
perbolas. 4. Any right line passing through 
the centre, and meeting the hyperbolas, is 
called a transverse diameter ; and the 
points in which it meets them, their ver- 
tices : but the right line, which passes 
through the centre, and bisects any right 
line terminated by the opposite hyperbolas’s, 
but not passing through the centre, is called 
a right diameter. 5. The diameter which 
passes through the foci, is called the trans- 
verse axis. 6 If from A or a, the extremities 
of the transverse axis, there is put a right 
line A D equal to the distance of the centre 
C from either focus, and with A, as a cen- 
tre, and the distance A D, there is a circle 
described, meeting the right line, which is 
drawn through the centre of the hyperbola 
at right angles to the transverse axis, in 
B b ; the line B b, is called the second axis. 
7. Two diameters, either of which bisects 
all the right lines parallel to the other, and 
which are terminated both ways by the hy- 
perbola, or opposite hyperbolas, are called 
conjugate diameters. 8. Any right line not 
passing through the centre, but terminated 
both ways by the hyperbola, or opposite hy- 
perbolas, and bisected by a diameter, is 
called an ordinate applied, or simply an or- 
, dinate to that diameter : the diameter like- 
wise, which is parallel to that other right 
line ordinately applied to the other diame- 
ter, is said to be ordinately applied to it. 
9. The right line which meets the hyperbola 
in one point only, but produced both ways 
falls without the opposite' hyperbolas, is said 
to touch it in that point, or is a tangent to 
it. 10. If through the vertex of the trans- 
verse axis a right line is drawn equal and 
parallel to the second axis, and is bisected 
by the transverse axis, the right lines drawn 
through the centre and the extremities of 
the parallel line are called asymptotes. It. 
The right line drawn through the centre of 
the hyperbola, parallel to the tangent, and 
equal to the segment of the tangent be- 
tween the asymptotes, and which is bisected 
in the centre, is called the second diameter 
of that which is drawn through the point of 
contact. 12. A third proportional to two 
diameters, one of which is the transverse, 
the other second to it, is called the lotus 
rectum, or parameter of that diameter, which 
is the first of the three proportionals. And, 
13. Lastly, fig. 9. If upon two right lines 
A a, B b, mutually bisecting each other at 
right angles, the opposite hyperbolas A G, 
a g, are described ; and if upon the same 
right lines there are described two other 
opposite hyperbolas, B K, b k, of which 
the transverse axis, B b, is the. second axis 
of the two first ; and the second axis of 
the two last, A a, is the transverse axis of 
the two first ; these four are called conju- 
gated hyperbolas, and their asymptotes 
shall be common. 
Prop. I. (fig. 6.) The square of the 
half of the second axis is equal to the rect- 
angle contained by the right lines between 
the foci and the vertexes of the transverse 
axis. 
Let A « be the tranverse axis, C the 
centre, E and F the foci, and B b the se- 
cond axis, which is evidently bisected in 
the centre C, from the definition ; let A B be 
joined : then since (by def. 6) A B and 
C F are equal ; the squares of A C and 
C B together, will be equal to the square of 
C F, that is, (6. 2.) to the square of A C and 
the rectangle A F a together ; wherefore 
taking away the square of A C which is 
common, the square of C B will be equal to 
the rectangle A F a. 
Prop, II. If from any point G (fig. 7 and 
8. ) of the hyperbola, a right line G D is drawn 
at right angles to the transverse axis A «, 
and if from the same point there is drawn 
the right line G F to the focus nearest to 
that point ; the half of the transverse axig 
C A will be to the distance of the focus from 
the centre, viz. C F, as the distance of the 
perpendicular C D, is to the sum of the half 
of the transverse axis, and the right line 
drawn to the focus. 
Let G E be drawn to the other focus, 
and on the axis a A produced, let there be 
set off A H equal G F ; then with the cen- 
tre G, and the distance G F, describe a 
circle cutting the axis a A in K and F, and 
the right line EG in the points, L and M : 
then since E F is double C F, and F K 
double FD, EK shall be also double C D ; 
and since EL or A a, is double C A, and 
