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.

11. If A, B, C be any three consecutive vertices of a harmonic
polygon, whose symmedian point is K, prove, if K' be the symmedian point of the triangle ABC, and B' the point where KK'
intersects AC, thatthe anharmonic ratio (KB'K'B) is constant.
12. If through the vertices A, B, C, &c., of a harmonic polygon F, be drawn lines making the same angle ( with the sides
AB, BC, &c., and in the same direction of rotation, prove that
the polygon F2 formed by these lines is a harmonic polygon, and
similar to the original.
13. If a polygon F3 be formed by lines which are the isogonal
conjugates of F2, with respect to the angles of F1, prove that F3
is equal to F2 in every respect.
14. If F1, F2, F3 be considered as three directly similar figures,
prove that their symmedian points are the invariable points, and
that the double points are the circumcentre of F, and its Brocard
points.
15. The symmedian point of a harmonic polygon is the mean
centre of the feet of isogonal lines drawn from it to the sides of
the polygon.
This follows from the fact that the isogonal lines make
equal angles with, and are proportional to, the sides of a closed
polygon.
16. The square of any side of a harmonic polygon is proportional to the rectangle contained by the perpendiculars from its
extremities on the harmonic polar.
17. If from the angular points of a harmonic polygon tangents
tl, t2,.. t,, be drawn to its Brocard circle, prove that
V(l/t2) = a/ (R2 - 2).
18 If ABCD, &c., be a harmonic polygon, R one of its
Brocard points, prove that the lines AR, Bas, &c., meet the circumcircle again in points which form the vertices of a harmonic
polygon equal in every respect, and that n will be one of its
Brocard points.
The extension of recent Geometry to a harmonic quadrilateral
was made by Mr. Tucker in a Paper read before the Mathematical Society of London, February 12, 1885. His researches were
continued by Neuberg in Mathesis, vol. v., Sept., Oct., Nov.,
Dec., 1885. The next generalization was made by me in a Paper
read before the Royal Irish Academy, January 26, 1886, " On
the Harmonic Hexagon of a Triangle." Both extensions are
special cases of the theory contained in this section, the whole
of which I discovered since the date of the latter Paper, and
which M. Brocard remarks, "parait etre le couronnement de ces