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.

IAljA2 is isosceles. Hence A1A2 is bisected in C1, and
OA1 = OA2. Similarly OAG = OA3, &c. Hence the
points A,, A2... are concyclic; and since the polygon
formed by them has a symmedian point K, it is a harmonic polygon.
Prop. 9.-If the Brocard angles o, o' of a Tarry's
line be unequal, it cannot be a closed polygon.
Dem.-If o' be > o, AIC1 is > C0A2. Hence OA1
> OA2. Similarly OA2 > OA3. Hence the points Al, A2... are continually approaching O. Hence the proposition is evident.
Cou.- When Tarry's line has equal Brocard angles,
its symmedian point is diametrically opposite to 0 on
the circle of similitude.
For the parallels to A1,A, A2A3, &c., through the invariable points, meet in K. Now, since IK, IO are
respectively parallel and perpendicular to A1A2, the
angle OI1K is right. Hence OK is the diameter of Z.
Exercises.
1. If an associated system of figures have a common centre of
similitude, the figures formed by inverting them from that point
form an associated system.
2. In the same case, the figures formed by reciprocating them
from the centre of similitude form an associated system.
3. If a series of directly similar triangles be inscribed in a
given triangle, they have a common centre of similitude.
4. If a series of directly similar triangles be circumscribed to
a given triangle, they have a common centre of similitude.
5. In an associated system of four directly similar figures
there exists one system of four homologous points which are
collinear.
6. In the same case the four director points of the four triads
which are obtained from them are collinear.