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.

BOOK V1.

145

Hence it is evident that, from theorems which hold
for A, we can get other theorems which are true for B.
This method, which is called reciprocation, is due to
Poncelet, and is one of the most important known to
Geometers.
We give a few Theorems proved by this method:(1). Any two fixed tangents to a circle are cut homographically by any variable T
tangent.
Dem.-Let AT, BT be the
A
two fixed tangents touching B
the circle at the fixed points A
and B, and CD a variable tan- c
gent touching at P. Join AP, /
BP. Now AP is the polar of
C, and BP the polar of D; and if the point P take four
different positions, the point C will take four different
positions, and so will the point D; and the anharmonic
ratio of the four positions of C equal the anharmonic ratio
of the pencil from A to the four positions of P (Prop. 1).
Similarly the anharmonic ratio of the four positions of
D equal the anharmonic ratio of the pencil from B to the
four positions of P; but the pencil from A equal the
pencil from B; therefore the anharmonic ratio of the
four positions of C equal the anharmonic ratio of the
four positions of D.
(2). Any four fixed tangents to a circle are cut by any
fifth variable tangent in four points whose anharmonic
ratio is constant.
Dem. —The lines joining the point of contact of the
variable tangent to the points of contact of the fixed
tangents are the polars of the points of intersection of
the variable tangent with the fixed tangents; but the
anharmonic ratio of the pencil of four lines from a
variable point to four fixed points on a circle is constant; hence the anharmonic ratio of their four polesthat is, of the four points in which the variable tangent
cuts the fixed tangent-is constant.

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