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.

127

segments in the three couples are respectively AB, BC,
CA, exactly corresponding to the sides of a triangle ABC,
taken in order. Now, if we take the rectangles formed by
these three pairs of segments, the six quotients obtained
by dividing each rectangle by the two remaining ones are
called the six anharmonic ratios of the four points A, B,
C, D. Thus these six functions are
AB.CD BC. AD CA. BD
BC.AD' CA.BD' AB.CD;
and their reciprocals
BC. AD CA. BD AB. CD
AB. CD' BC.AD' CA.BD'
It is usual to call any one of these six functions the anharmonic ratio of the four points A, B, C, D.
Prop. 1.-If (0. ABCD) be a pencil of four rays
passing through the four points
A, B, C, D; and if through
any of these points B we draw
a line parallel to a ray passing M
through any of the other points, \M
and cutting the two remaining
rays in the points M, N, the
six anharmonic ratios of A, B, A \ C
C, D can be expressed in terms
of the ratios of the segments
MB B, NM, M.
Dem.-From similartriangles, N
MB AB
OD AD'
OD CD
andBN B
]Bx = ~C'