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.

Cor. 4. —If a circle be described about the triangle
CEF, its envelope will be a circle concentric with the
circle whose centre is O;
that is, with the circle D
whose chord is CF. c
(When a line or circle
moves according to any given / \
law, the curve which it
touches in all its positions 1
is called its envelope.)
Produce EK till it meets
the circumference in D; \ \ y
then because the Z CEF
is bisected by ED, the arc
CDF is bisected in D; hence the line 00, which joins
the centres of the circles, passes through D and is I to
CF;.-. O'K is 1I to OD;.-. O'K: OD::EO': EO;
hence the ratio of O'K: OD is given; but O'K is given;
therefore OD is given, and the ( whose centre is 0
and radius OD is given in position, and the ( CEF
touches it in D; hence the Proposition is proved.
Prop. 9.-If a system of coaxal circles have two real
points of intersection, all
lines drawn through either A
point are divided proportionally by the circles.
Let A, B be the points of
intersection of the coaxal
system: through A draw
two lines intersecting the
circles again in the two
systems of points C, D, E; B
C', D', E'; then
CD: DE:: C'D': D'E'.
Dem.-Join the points C, D, E, C', D', E' to B; then
the Ls BCD, BC'D' are evidently equiangular, as are