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.

BOO3K VI.

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two triads of points be A, E, C, D, 1B, F, and the proof
of the Proposition can be applied, word for word, except
that the pencil (A. FBDE) is equal to the pencil
(C. FBDE), for a different reason, viz., they have a
common transversal.
Prop. 5.-If two equal pencils have a common ray, the
intersections of the remaining three homologous pairs of
rays are collinear.
Let the pencils be (. O'ABC), (0'. OABC), having
the common ray 00'; then, if possible, let the line
joining the points A and C intersect the rays OB, O'B
in different points B', B"; then, c
since the pencils are equal,
the anharmonic ratio of the
points D, A, B', C equal the
anharmonic ratio of the points
D, A, B", C, which is impos-
sible. Hence the points A, B, C
must be collinear. o D '
Cor. l. —If A, B, C; A', B', C' be two triads of
points on two lines intersecting in 0, and if the anharmonic ratio (OABC) = (OA'B'C'), the three lines
AA', BB', CC' are concurrent. For, let AA', BB',
intersect in D; join CD, intersecting OA' in E; then
the anharmonic ratio (OA'B'E)= (OABC) = (OA'B'C')
by hypothesis; therefore
the point E coincides
with C'. Hence the
Proposition is proved.
Cor. 2. —If two As
ABC, A'B'C' have lines
joining corresponding
vertices concurrent, the
intersections of corre- 0 A
sponding sides must be
collinear. For, join P, B i
the point of intersection
of the sides BC, B'C',