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.

inversion, and the inverse of the common centre of
the original system.
Cor. 2.-If a variable circle touch two concentric
circles, it will cut any other circle concentric with them
at a constant angle. Hence, by inversion, if a variable
circle touch two circles of a coaxal system, it will cut
any other circle of the system at a constant angle.
Cor. 3.-If a variable circle touch two fixed circles,
its radius has a constant ratio to the perpendicular from
its centre on the radical axis of the two circles, for it
cuts the radical axis at a constant angle.
Cor. 4.-The inverse of a system of concurrent lines
is a system of coaxal (Os intersecting in two real points.
Cor. 5.-If a system of coaxal circles having real
limiting points be inverted from either limiting point,
they will invert into a concentric system of circles.
Cor. 6.-If a coaxal system of either species be inverted from any arbitrary point, it inverts into another
system of the same species.
Prop. 7.-If 'a variable circle touch two fixed circles,
its radius has a constant ratio to the perpendicular from
its centre on the radical axis.
Dem,-This is Cor. 3 of the last Proposition; but it
is true universally, and
not only as proved there
for the case where the ( /
cuts the radical axis. On / c
account of its importance
we give an independent D 0 0
proof here. Letthe centres of the fixed ( s be O,
0', and that of the variable
( 0". Join 00', and produce it to meet the fixed
Os in the points C, C': - A B
upon CC' describe a 0:
let 0"' be its centre: let fall the is O"A, O"'B on the
radical axis: let D be the point of contact of 0" with
0; then the lines CD and 0''0" will meet in the centre