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.

136 A SEQUEL TO EUCLID.
to the anharmonic ratio of their four corresponding points
on the other, are said to be homographic, and the lines are
said to be homographically divided. The points 0, 0'
are called the centres of the systems.
Cor. 1.-The point O on OX is the point corresponding to infinity on O'X'; and the point O' on O'X'
corresponds to infinity on OX.
DEF.-If the line O'X' be superimposed on OX, but so
that the point O' will not coincide with 0, the two systems
of points on OX divide it homographically, and the points
of one system which coincide with their homologous points
of the other are called the double points of the homographic system.
Prop. 10.- Given three pairs of corresponding points
of a line divided homographically, to find the double points.
Let A,A'; B, o A B C A' B' ' X
B'; C, C', be the ' ' ' '
three pairs of corresponding points, and 0 one of the
required double points; then the conditions of the
question give us the anharmonic ratio
(OABC) = (O A'B'C');
(OA. C OA'. B'C'
therefore OB. AC 0 B1'. A'C'
OA. OB' B'C'. AC
OA'.OB BC.A'C'
equal constant, say.2.
Now OA. OB', OA'. OB are the squares of tangents
drawn from 0 to the circles described on the lines AB'
and A'B as diameters; hence the ratio of these tangents is given; but if the ratio of tangents from a
variable point to two fixed circles be given, the locus
of the point is a circle coaxal with the given circles.
Hence the point 0 is given as one of the points of
intersection of a fixed circle with OX, and these intersections are the two double points of the homographic
system.