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.

39

in species. Now, since the A ABC is a minimum, the
side AB is a minimum;.~. OA is a minimum; and
since 0 is a given point, OA must be 1 to EF. Hence
the method of inscribing the minimum L has been
found.
Cor.-From the foregoing analysis the method is
obvious of inscribing in a given A another A whose
sides shall be respectively equal to three given right
lines.
Prop. 18. —If ABC be a triangle, and CD a perpendicular to AB; then if AE = DB, it is required to prove
that AB is the minimum c
line that can be drawn\
through E, meeting the two
ixed lines AC, BC.
Dem.-Describe a 0
about the A ABC; produce CD to meet it in, A\L
and erect EKL to AB. / B
Join AK, BK. Through '/
E draw any other line FG; -
draw KO I to E'G, and
produce it to meet AB in H; through H draw JI Il to
FG. Join JK, iK, K, KL. Now, since AE = DB, it
is evident that EK = DL. Hence KL is II to AB;.'. the Z KLC = ADC, and is consequently a right Z;.-. KC is the diameter of the 0;.-. the Z KBC is
right, and the Z KHII is right;.'. KHIB is a quadrilateral inscribed in a circle;.'. the Z KIl = KBA.
In like manner, the Z KJHI = KAB;.-. the As IJK
and 3BAK are equiangular; and since IK is greater
than KB (the Z IBIK being right), it follows that IJ is
greater than AB; but FG is evidently greater than IJ;.. much more is FG greater than AB. Hence AB is
the minimum line that can be drawn through E.

If in the foregoing Jig. the line M3A receive an infinitely small
change of position, namely, 13 along BC, and A along AC; then