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§24] semicircle; this is of course a convention, and it is adopted, for the present, partly because it is traditional and partly because it simplifies the study of Spherical Trigonometry for the beginner.

Thus, in the figure, the arc $$ADEB$$ is greater than a semi-circumference, and we might, if we pleased, consider $$ADEB$$, $$AC$$, and $$BC$$ as forming a triangle, having its angular points at $$A$$, $$B$$, and $$C$$. But we agree to exclude such triangles from our consideration; and the triangle having its angular points at $$A$$, $$B$$, and $$C$$, will be understood to be that formed by $$AFB$$, $$BC$$, and $$CA$$.

23. From the restriction of the preceding Article it will follow that any angle of a spherical triangle is less than two right angles.

For suppose a triangle formed by $$BC$$, $$CA$$, and $$BEDA$$, having the angle $$BCA$$ greater than two right angles. Then the parts of the are $$BEDA$$, which are in the immediate neighbourhoods of $$B$$ and of $$A$$ respectively, clearly lie on opposite sides of the plane of the great circle $$BC$$. Hence the are must cut this plane; let it do so in a point $$D$$; then $$D$$ lies on the arc $$BC$$ produced. By Art. 10, $$BED$$ is a semicircle, and therefore $$BEA$$ is greater than a semicircle ; thus the proposed triangle is not one of those which we consider.

24. The relations between the sides and angles of a Spherical Triangle, which are investigated in treatises on Spherical Trigonometry, are chiefly such as involve the Trigonometrical Functions of the sides and angles. Before proceeding to these, however, we shall consider some theorems which involve the sides and angles themselves, and not their trigonometrical ratios.

Definitions. The following definitions are important.

A lune is that portion of the surface of a sphere which is comprised between two great semicircles.