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.

128

A SEQUEL TO EUCLID.

Hence, MB AB. CD
BN BC. AD'
therefore MB: BN:: AB. CD: BC. AD.
ComponendoMN: B:: AB. CD+BC. AD:BC. AD;.. N: B::AC. BD: BC. AD;.MB: BN: NM::AB. CD: BC. AD: CA.BD.
Prop. 2.-If a pencil of four rays be cut by two transversals ABCD, A'B'C'D', then (see last fig.) any of the
anharmonic ratios of the points A, B, C, D is equal to the
corresponding ratio for the points A', B', C', D'.
Dem.-Through the points B, B' draw MN, M'N'
parallel to OD; then (Section I., Prop. 3) we have
IMB: BN: M1'B': B'N';
AB. CD A'B'. C'D'
BC. AD B'C'. A'D
Cor. 1.-We may suppose the rays of the pencil
produced through the vertex, and the transversal to
cut any of the rays produced without altering the anharmonic ratio.
DEF.-The anharmonic ratio of the four points on any
transversal cutting a pencil being constant, it is called the
anharmonic ratio of the pencil.
Cor. 2.-If two pencils have equal anharmonic ratios
and a common vertex; and if three rays of one pencil
be the production of three rays of the other, then the
fourth ray of one is the production of the fourth ray of
the other.
Cor. 3.-If two pencils have a common transversal,
they are equal; that is, they have equal anharmonic
ratios,