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.

CONTENTS.

xic

SECTION III.
PAGE
Theory of Harmonic Section,. 87
Harmonic Section defined-Harmonic conjugates,. 87
Circles described on harmonic segments intersect orthogonally,... 87
Relation between Arithmetic, Geometric, and Harmonic means,...... 88
Harmonic properties of a line cutting a circle,.. 89
Harmonic properties of a line cutting sides of a triangle and a parallel to the base through the
vertex,.... 90
Harmonic pencils,....... 90
Harmonic properties of a complete quadrilateral,. 91
The polars of four harmonic points form a harmonic
pencil,.... 91
Harmonic system of points on a circle,... 92
Harmonic properties of a quadrilateral inscribed in a
circle,........ 93
Harmonic triangle of a quadrilateral,... 94
SECTION IV.
Theory of Inversion,....... 95
Inverse figures defined,..... 95
Angle of intersection of two curves remains unaltered
by inversion,... 96
Two circles can be inverted into themselves,.. 97
Two circles inverted into two equal circles,.. 98
Property of perpendiculars on sides of an inscribed
polygon,... 99
Invariant property of four collinear points,.. 100
Square of common tangent of two circles divided by
the rectangle of their diameters is an invariant,. 101