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.

4

A SEQUEL TO EUCLID.

figure EFHG is evidently a =, since the opposite sides
EF, GlH are each 11 to AB (3), and = AB (Prop. 2, Cor.).
Again, let the lines EG, FHB meet AB in the points
M, N; then = EFNM = i- ABC (Prop. 3, Cor. 2),
and = GHINM = I A ABD. Hence = EFHG =
i (ABC + ABD).
Dem. 2. —When ABC, ABD are on the same side
of AB, we have evidently r= EFGIH = EFNM -
GHNM - I(ABC - ABD).
Observation.-The second case of this proposition may be
inferred from the first if we make the convention of regarding the
sign of the area of the A ABD to change from positive to negative, when the A goes to the other side of the base. This affords
a simple instance of a convention universally adopted by modern
geometers, namely-when a geometrical magnitude of any kind,
which varies continuously according to any law, passes through a
zero value to give it the algebraic signs, plus and minus, on different sides of the zero-in other words, to suppose it to change
sign in passing through zero, unless zero is a maximum or minimum.
Prop. 6.-If two equal triangles ABC, ABD be on the
same base AB, but on opposite sides, the line joining the
vertices C, D is bisected by AB.
Dem.-Through A and B
draw AE, BE 11 respectively
to BD, AD; join EC. Now,
since AEBD is a,, the
A AEB = ADB (xxxiv.); but A /M —
ADB = ACB (hyp.);.'. AEB
= ACB;.-. CE is 1[ to AB
(xxxix.). Let CD, ED meet
AB in the points M, N, respec- X
tively. Now, since AEBD is
a =n, ED is bisected in N (1); and since NM is I] to
EC, CD is bisected in M (2).
Cor.-If the line joining the vertices of two As on
the same base, but on opposite sides, be bisected by the
base, the As are equal.

Prop. 7.-If the opposite sides AB, CD of a guadrilateral