A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

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A SEQUEL TO EUCLID..'. the As AOE, A'O'E are equiangular;.'. the L OAE
= A'EO';.'. if the points A, A' change position, the
lines EA, EA' will revolve in the same direction, and
through equal angles. Hence the Z AEA' is constant.
In the same manner, another point F can be found on
the other side of 00' such that the Z AFA' is constant.
Cor. 1.-Since the line AA' subtends a constant angle
at E, the locus of the pole of AA' with respect to a circle
whose centre is E is a circle. Hence the properties of
lines joining corresponding points on two lines divided
homographically may be inferred from the properties of
a system of points on a circle.
Cor. 2.-Since when A'goes to infinity A coincides
with 0, then OA is one of the lines joining corresponding points. And so in like manner is O'A', and the poles
of these lines will be points on the circle which is the
locus of the pole of AA'.
Cor. 3. —The locus of the foot of the perpendicular
from E on the line AA' is a circle, namely, the inverse
of the circle which is the locus of the pole of AA'.
Cor. 4.-If two lines be divided homographically,
any four lines joining corresponding points are divided
equianharmonically by all the remaining lines joining
corresponding points. This follows from the fact that
any four points on a circle are subtended equianharmonically by all the remaining points of the circle.
Prop. 4.-If any figure A be given, by tacing the pole
of every line, and the polar of every point in it with
respect to any arbitrary circle X, we can construct a new
figure B, which is called the reciprocal of A with respect
to X. Thus we see that to any system of collinear points
or concurrent lines of A there will correspond a system of
concurrent lines or collinear points of B; and to anypair
of lines divided homographically in A there will correspond
in B two homographic pencils of lines. Lastly, the angle
which any two points of A subtend at the centre of the
reciprocating circle is equal to the angle made by their
polars in B.