20 MATHEMATICS. 
Obs. If the side a, opposite the given angle, A, be less than the side 3, 
there will be two solutions possible, since for B, we may take the 
acute angle answering to sin. B in the tables, and likewise its obtuse 
supplemental angle, whence there will also be two values forC and c. 
3. Given two sides, a, b, and the included angle C ; then tang. } A—B 
a—b)tan A+B 
Cs one HO), Ai(A4 B) +4(A—B); BG om 
—+3(A—B); ¢ remains as before. 
4. Given the three sides, a, b,c. Then indicating by s, the half sum 
of the sides, a eg: = 5s; we have tang.4;A — RA (SS sae , 
(s—a) s 
we aA we tom 
2. SPHERICAL TRIGONOMETRY. 
Spherical Trigonometry teaches the calculation of spherical triangles ; 
that is, of such triangles as are formed on the surface of a sphere, by ares 
of great circles. In such a triangle there are also six parts, of which three 
must be given to determine the rest. 
Every spherical triangle answers to a three-sided solid angle, from whose 
vertex, with any radius, circles are described. Consequently the three 
sides of the spherical triangle on the surface of the sphere, measure the 
plane angles at the centre forming the solid angle, and its angles, the 
inclination of their planes. Hence spherical trigonometry serves for 
calculating solid angles, and may thus be called solid trigonometry. 
On account of what is to follow, some of the most important properties of 
spherical triangles may here be introduced, although they belong properly 
to Stereometry. Every two sides of a spherical triangle are together 
greater than a third (pl. 3, fig. 111). If through the centre of the sphere, 
and the sides of the spherical triangle abc, we pass three planes, these latter 
will form a solid angle, whose three plane angles are measured by the arcs, 
ab, ac, be. Since any one of these three plane angles is less than the sum 
of the other two, the same must be true with respect to the three arcs or 
sides of the spherical triangle. 
The sum of the three angles, aob, aoc, boc, is less than four right angles ; 
likewise the sum of the three sides is less than the entire circumference 
or 360°. 
The area of a spherical triangle is proportional to the excess of the sum 
of its angles over two right angles (called the spherical excess). A spherical 
triangle, def, is called the polar or supplemental triangle of another, abe 
(pl. 3, fig. 112), where the vertices of the angles of this second triangle are 
respectively poles of the sides of the first. If def be the polar triangle of 
abc, the latter will be, on the other hand, the polar triangle of the former. 
Every angle of the polar triangle is measured by a semi-circumference 
minus the side lying opposite to it in the other triangle, whence the name 
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