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 III.

37

Prop. 15.-Through one of the points of intersection of
two given circles to draw a line, the sum of whose segments
intercepted by the circles
shall be a maximum.
Analysis.-Let thes /
intersect in the points P,
R, and let APB be any o
line through P. From
0, 0', the centres of the
Os, let fall the Is 00, E
O'D, and draw O'E 11 to A c / B
AB. Now, it is evident
that AB = 2CD = 20'E; and that the semicircle de.
scribed on 00' as diameter will pass through E. Hence
it follows that if AB is a maximum, the chord O'E
will coincide with 00'. Therefore AB must be II to
the line joining the centres of the Os.
Cor. 1.-If it were required to draw through P a
line such that the sum of the segments AP, PB may be
equal to a given line, we have only to describe a 0
from 0' as centre, with a line equal half the given line
as radius; and the place where this 0 intersects the (
on 00' as diameter will determine the point E; and
then through P draw a II to O'E.
DEF. —A triangle is said to be given in species when
its angles are given.
Prop. 16.-To describe a triangle of given species
whose sides shall pass through three given points, and
whose area shall be a maximum.
Analysis.-Let A, B, C be the given points, DEF
the required A; then, since the triangle DEF is given
in species, the Zs D, E, F are given, and the lines AB,
BC, CA are given by hypothesis;.'. the Os about the
As ABF, BCD, CAE are given. These three Os will
intersect in a common point. For, let the two first intersect in 0. Join AO, BO, CO; then ZAFB + AOB =
tworightZ s; andBDC +BOC =tworightZ s;.-.the s
AFB, BDC, AOB, COB = four right Ls, and the Zs