PARABOLA. 
a square rule, D E F, so <1101 tlie side D E 
may be applied to the right line A B, and 
tlie other E F turned to the side on which 
the point C is situated. This done, and 
the thread F G C, exactly of the length of 
the side of the rule, E F, being fixed at one 
end to the extremity of the rule F, and at 
the other to the point C, if you slide the 
side of the rule, D E, along the right line 
A B, and by means of a pin, G, continually 
apply the thread to the side of the rule, 
E F, so as to keep it always stretched as 
the rule is moved along, the point of this 
pin, will describe a parabola GH O. 
Definitions. 1. The right line A B is called 
the directrix. 2. The point C is the focus 
of the parabola. 3. All perpendiculars to 
the directrix, as LK, M O, Sec. are called 
diameters ; the points, where these cut the 
parabola, are called its vertices ; the dia- 
meter B I, which passes through the focus 
C, is called the axis of the parabola; and 
its vertex, H, the principal vertex. 4. A 
right line, terminated on each side by the 
parabola, and bisected by a diameter, is 
called the ordinate applicate, of simply the 
ordinate, to that diameter. 5. A line equal 
to four times the segment of any diameter ; 
intercepted between the directrix and the 
vertex where it cuts the parabola, is call- 
ed the latus rectum, or parameter of that 
diameter. 6. A right line which touches 
the parabola only in one point, and being 
produced on each side falls without it, is a 
tangent to it in that point. 
Prop. 1. Any right line, as G E, drawn 
from any point of the parabola, G, per- 
pendicular to A B, is equal to a line, G C, 
drawn from the same point to the focus. 
This is evident from the description ; for 
the length of the thread, F G C, being equal 
to the side of the rule E F, if the part, F G, 
common to both, be taken away, there 
remains E G = G C. Q. E. D. 
The reverse of this proposition is equally 
evident, viz. that if the distance of any 
point from the focus of a parabola, be equal 
to the perpendicular drawn from it to the 
directrix, then shall that point fall in the 
curve of the parabola. 
Prop. 2. If from a point of the parabola, 
D, (fig. 2) a right line be drawn to the 
focus, C ; and another, D A, perpendicular 
to the directrix ; then shall the right line, 
D E, which bisects the angle, ADC, con- 
tained between them, be a tangent to the 
parabola in the point D : a line also, as 
H K, drawn through the vertex of the axis, 
and perpendicular to it, is a tangent to the 
parabola in that point. 
1. Let any point, F, be taken in the line 
D E, and let FA, FC, and AC be joined; 
also let F G be drawn perpendicular to the 
directrix. Then, because (by Prop, l), 
DA =-pC, DF common to both, and 
the angle FDA = FDC, FC will be 
equal to FA; but FA greater than FG 
therefore F C greater than F G, and con- 
sequently the point, F, falls without the 
parabola ; and as the same can be demon- 
strated of every other point of D E, except 
D, it follows that D E is a tangent to the 
parabola in D. Q. E. D. 
2. If every/ point of H K, except H, falls 
without the parabola, then is H K a tan- 
gent in H. To demonstrate this, from any 
point K draw I\ L perpendicular to A B, 
and join Iv C ; then because K C is greater 
4:han C H = H B = K L, it follows that 
K C is greater than K L, and consequently 
that the point K. falls without the parabola : 
and as this holds of every other point, ex- 
cept H, it follows that K H is a tangent to 
the parabola in H. Q. E. D, 
Prop. 3. Every right line, parallel to a 
tangent, and terminated on each side by 
the parabola, is bisected by the diameter 
passing through the point of conlact ; that 
is, it will be an ordinate to that diameter. 
For let Ee (fig. 3 and 4) terminating in 
the parabola in the points E e, be parallel 
to the tangent D K ; and let AD be a dia- 
meter passing tlirough the point of contact 
D, and meeting E e in L; then shall E L = 
Ee. 
Let AD meet the directrix in A, and 
from the points E e, let perpendicular* 
E F, ef, be drawn to the directrix ; let C A 
be drawn, meeting E e in G ; and on the 
centre E, with the distance EC, let a 
circle be described, meeting A C again in 
H, and touching the directrix in F; and 
let D C be joined. Then because T) A — 
D C, and tlie angle A D K = the angle 
C D K, it follows (4. 1.) that 1) K per- 
pendicular to AC; wherefore Ee perpen- 
dicular to AC, and CG = GH (3. 3); 
so that e C = e H (4. 1), and a circle des- 
cribed upon the centre c, with the radius 
e C, must pass through H ; and because e C 
= ef,it must likewise pass through f. Now 
because F/ is a tangent to both these 
circles, and AHC cuts them, the square 
A F = the rectangle C A H (36. 3) = the 
square Af; therefore A F = A/, and F E, 
A L, and f e are parallel; and consequently 
LE = Le. Q. E. D. 
Prop. 4. If from any point of a parabola, 
D, (fig. 5) a perpendicular, D H, be drawn 
to a diameter B H, so as to be an ordinate 
