Dr Taylor , Geometrical Notes on Theorems , etc. 
153 
Geometrical Notes on Theorems of Halley and Fregier. By 
C. Taylor. D.D., Master of St John’s College. 
The two subjects of these notes are Halley’s construction 
for the normals or normal from a given point to a parabola by 
means of a circle, and Fregier ’s theorem that a chord of a conic 
which subtends a right angle at a fixed point of the curve passes 
through a fixed point on the normal thereat. 
A. 
Halley. 
1. Apollonius shewed how to draw the normals to a conic 
from a given point by means of a rectangular hyperbola. 
This cuts the conic in four or fewer real points, each lying on 
a normal to the conic from the given point. 
In the course of his investigation Apollonius found the 
coordinates of what we call the centre of curvature, and thus 
virtually the equation of the evolute of a conic. 
From the given point H in certain positions normals can be 
drawn to a central conic at four points P, p, Q, q. 
When the conic is a parabola one of the four points, as q, is at 
infinity, and the remaining three, P, p, Q lie on a circle through 
the vertex A. Halley shewed how to draw the normals HP, 
Hp , HQ by means of this circle. 
2. Draw the ordinates PN, pn, QM, supposing Q to be either 
of the two points on the parabola having the same abscissa AM 
and such that 
QM = PN±pn. 
The chords AQ, pP make equal angles with the axis, and in 
this section they may be parallel. 
Taking the upper sign, draw Ppl to the axis, and let 4 a be 
the latus rectum. 
Then AM = IN + In = AN + An + 2AI. 
Writing (PN +pn) 2 /4<a for AM and subtracting AN An 
from both sides, we get 
PN.pn = 4 a.AI. 
