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.

102

A SEQUEL TO EUCLID.

Dem.-Let X, Y be the original Gs, X', Y' their
inverse Gs, ABCD the line through the centres of X
and Y, and let the inverse of the line ABCD be the (
A'B'C'D'; then, since the line ABCD cuts orthogonally
the (s X, Y, its inverse, the 0 A'B'C'D', cuts orthogonally the Os X', Y'. Let abcd be the line through the
A \D
cens of th s, Y; then a cuts the s
centies of the Os X', Y'; then abcd cuts the Os X',
Y' orthogonally; hence the 0 A'B'C'D' is the inverse
of the line abed with respect to a ( of inversion, which
inverts the Gs X', Y' into themselves (see Prop. 4,
Cor. 3). Hence, by Prop. 8, each of the ratios
AC.BD ac. bd
AB. CD' ab. cd
is equal to the ratio
A'C'. B'D'
A'B'. C'D"
AC. 1BD a c. bd
therefore AC. CD =ab. cd
AB. CD ab. cd'

The numerators of these fractions are equal respectively