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.

(6). If A and B be the points of contact, upon two
circles X and Y, of tangents drawn from any point of
their circle of similitude, then the tangent from A to Y
is equal to the tangent from B to X.
Prop. 2.-Two circles being given, it is required to
describe a system of circles coaxal with them.
Con.-If the circles have real points of intersection,
the problem is solved by describing circles through
these points and any third point taken arbitrarily.
If the given circles have not real points of intersection, we proceed as follows:
Let X and Y be the given Os, P and Q their
centres: draw AB, the radical axis of X and Y, intersecting PQ in O: from 0 draw two tangents 00, OD
A
B
to X and Y; then C0 = OD, and the ( described with
0 as centre and OD as radius will cut the two Os X
and Y orthogonally. Now take any point E in this
orthogonal 0, and draw the tangent ER meeting the
line PQ in R: from R as centre, and RE as radius,
describe a ( Z; then Z will be coaxal with X and Y.
For the line ER being a tangent to the 0 CDE, the
Z OER is right,.*. OE is a tangent to Z; and since
OD = OE, the tangents from 0 to the Os Y and Z are