Page:Spherical Trigonometry (1914).djvu/33

30. The sum of the three sides of a spherical triangle is less than the circumference of a great circle. (See the figure of Art. 18.)

For the sum of the three plane angles which form the solid angle at $O$ is less than four right angles (Euclid, xi. 21);

therefore ${\dfrac {AB}{OA}}+{\dfrac {BC}{OA}}+{\dfrac {CA}{OA}}$ is less than $2\pi$,

therefore, $AB+BC+CD$ is less than $2\pi \times OA$;

that is, the sum of the arcs is less than the circumference of a great circle.

31. The propositions contained in the preceding two Articles may be extended. Thus, if there be any polygon which has each of its angles less than two right angles, any one side is less than the sum of all the others. This may be proved by repeated use of Art. 29. Suppose, for example, that the figure has four sides, and let the angular points be denoted by $A$, $B$, $C$, $D$.

Then $AD+BC$ is greater than $AC$;

therefore, $AB+BC+CD$ is greater than $AC+CD$,

and à fortiori greater than $AD$.

Again, if there be any polygon which has each of its angles less than two right angles, the sum of its sides will be less than the circumference of a great circle. This follows from Euclid, XI, 21, in the manner shewn in Art. 30.

32. The three angles of a spherical triangle are together greater than two right angles and less than six right angles.

Let $A$, $B$, $C$ be the angles of a spherical triangle; let $a'$, $b'$, $c'$ be the sides of the polar triangle. Then by Art. 30, $$a'+b'+c'\mathrm {\ is\ less\ than\ } 2\pi ,$$ that is, $\pi -A+\pi -B+\pi -C$ is less than $2\pi$;

therefore, $A+B+C$ is greater than $\pi$.

And since each of the angles $A$, $B$, $C$ is less than $\pi$, the sum $A+B+C$ is less than $3\pi$.