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 VI.

135

(2). A right line, cutting a circle and the sides of an
inscribed quadrilateral, is cut L
in involution.
Dem.-Join AR, AR',
CR, CR'; then the anhar- M
monic ratio of the pencil \
(A. LRMR')= (A. DRBR')
= (C. DRBR')=(C. 'RL'R')
=(C.LLR'}VIR).
Cor.-The points N, N'
belong to the involution.
(3). If three chords of a circle be concurrent, their six
points of intersection with the
circle are in involution.
Let AA', BE', CC' be the A
three chords intersecting in
the point 0. Join AC, AC', o
AB', CB'; then the anhar- c
monic ratio (A. CA'B'C')
(B'. CBAC') = (B'. C'ABC).
Cor.-The pencil formed by any six lines from the
pairs of homologous points A, A'; B, B; C, C, to any
seventh point in the circumference is in involution.
Prop. 9.-If 0, O' be two fixed points on two given
lines OX, O'X', and if on OX we take any system of
points A, B, C, &c., and on O'X' a corresponding system
A', B', C', &c., such that the rectangles OA. O'A'
= OB. O'B' = OC. O'C', &c., equal constant, say k2;
then the anharmonic ratio of any four points on OX equal
the anharmonic ratio of theirfour corresponding points on
O'X'.
This is evident by superposition of O'X' on OX, so that
the point O' will coincide with O (see Prop. 7); then
the two ranges on OX will form a system in involution.
DEF.-Two systems of points on two lines, such that
the anharmonic ratio of any four points on one line is equal