in the higher Geometry. 38 1 
intersections of the ellipse with all the parabolas of orders not 
given, but which may be found, lying between the lines drawn 
at right angles to each other, the lines so drawn from the point 
found, shall be normals to the parabolas at their intersections 
with the ellipse. 
PROP. IV. PORISM. 
A conic hyperbola being given, if through any point thereof 
a straight line be drawn parallel to the transverse axis, (and 
cutting the opposite hyperbolas,) a point may be found, such, 
that if from it there be drawn straight lines, to the innumerable 
intersections of the given curve with all the hyperbolas of 
orders to be found, lying between straight lines which may be 
found, the straight lines so drawn shall be normals to the hy- 
perbolas at the points of section. 
Scholium. The two last propositions give an instance of the 
many curious and elegant analogies between the hyperbola and 
ellipse ; failing, however, when by equating the axes we change 
the ellipse into a circle. 
PROP. V. LOCAL THEOREM. Fig. 2 . 
If, from a given point A, a straight line DE moves parallel 
to itself, and another CS, from a given point C, moves along 
with it round C ; and a point I moves along AB, from H, the 
middle point of AB, with a velocity equal to half the velocity 
of DE; then, if AP be always taken a third proportional to 
AS and BC, and through P, with asymptotes D'E' and AB, a 
conic hyperbola be described ; also, focus I and axis AB, a 
conic parabola be described, the radius vector from C to M, the 
