A history of Greek mathematics - Wilbourhall.org

264 TRIGONOMETRY
Eucl. I. 16, 32 are not true of spherical triangles, and
Menelaus has therefore the corresponding but different propositions.
I. 10 proves that, with the usual notation a, b, c,
A, B, 0, for the sides and opposite angles of a spherical
triangle, the exterior angle at C, or 180° — G, < = or >A
according as c + a> = or < 180°, and vice versa. The proof
of this and the next proposition shall be given as specimens.
In the triangle ABC suppose that c + a > = or < 180°
;
let
D be the pole opposite to A.
Then, according as c + a > = or < 180°, BC > = or < BD
(since AD = 180°),
and therefore ID > = or < I BCD (= 180°-C), [I. 9]
i.e. (since ID = LA) 180°- G < = or > A.
Menelaus takes the converse for granted.
As a consequence of this, I. 1 1 proves that A + B + C> 180°.
Take the same triangle ABG, with the pole D opposite
to A, and from B draw the great circle BE such that
Z.DBE = IBDE.
Then GE+EB = CD < 180°, so that, by the preceding
proposition, the exterior angle AGB to the triangle BGE is
greater than LGBEy
i.e.
C>ACBE.
Add A dv D (= IEBD) to the unequals;
therefore
G + A > L GBD,
whence A + B + C > IGBD + B or 180°.
After two lemmas I. 21, 22 we have some propositions introducing
M, N, P the middle points of a, 6, c respectively. I. 23
proves, e.g., that the arc MN of a great circle >-|c, and I. 20
that AM < = or > \a according as A > = or < (B + C). The
last group of propositions, 26-35, relate to the figure formed