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.

THEORY OF HARMONIC POLYGONS.

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COR. 5.-A harmonic quadrilateral is the figure whose
vertices are four harmonic points on a circle [Book VI.,
Sect. ii., Prop. 9, Cor. 2]. Hence, the rectangle contained
by one pair of opposite sides is equal to the rectangle contained by the other pair.]
COR. 6. If 1, 2, 3... 2n be the vertices of a harmonic polygon of an even number of sides, the polygon
formed by the alternate vertices 1, 3, 5... 2n - 1 is a
harmonic polygon, and so is the polygon formed by the
vertices 2, 4, 6,... 2n, and these three polygons have a
common symmedian point.
Prop 2.-To invert a harmonic polygon into a regular
polygon.
Sol.-Let AB be a side of the harmonic polygon, Z
its circumcircle, 0 the circumcentre, and K the symP
median point. Upon OK as diameter describe a circle
OKX; and let S, S' be the limiting points of Z and
OKX; join SA, SB, and produce, if necessary, to meet Z
again in A', B'. Then A'B'is the side of a regular polygon.