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.

103

to the squares of the common tangents of the pairs of
circles X, Y; X', Y' (see Prop. 8, Section I., Book III).
Hence the Proposition
is proved.
Cor. 1.-If 0Q, C2, 3 c
C3, &c., be a series of
circles, touching two
parallel lines, and also
touching each other;
then it is evident, by making the diagram, that the
square of the direct common tangent of any two of
these circles, such as C,,, C,,+, which are separated by
(n - 1) circles, is = n2 times the rectangle contained by
their diameters. Hence, by inversion and by the theorem of this Article, we have the following theorem:If A and B be any two semicircles in contact with each
other, and also in contact with another semicircle, on
whose diameter they are described; and if circles CQ, C2, C3
be described, touching them as in the diagram, the._
from the centre of C, on the line AB = n times the
diameter of C,,, where n denotes any of the natural numbers 1, 2, 3, Ic.
This theorem will immediately follow by completing
the semicircles, and describing another system of circles
on the other side equal to the system CQ, C2, Ca, &c., and
similarly placed."
Prop. 10.-If four circles be all touched by the same
circle; then, denoting by 12, the common tangent of the
1st and 2nd, sc.,
12. 34 + 14. 23 = 13. 24.
Dem.-Let A, B, C, D be four points taken in order
on a right line; then, by Prop. 7, Section I., Book II.,
we have
AB. CD + BC. AD = AC. BD.
Now, let four arbitrary circles touch the line at the

* The theorem of this Cor. is due to Pappus. See Steiner's
Gesanmmelte ierice, Band I., Seite 47.