...\a sin ¿6 sin \c sin ^d hence the diagonal б is given in terms of а, Ъ, с, and d. Again, since the area of a spherical triangle is proportional to the excess of the sum of its three angles above two right angles, technically termed the spherical excess, which (spherical excess)...

...180° : A+B + C— 180° :: wr" : area of ABC; and it appears, therefore, that (in the same sphere) the area of a spherical triangle is proportional to the excess of the sum of its angles above two right angles, or to what is called its SPHERICAL EXCESS.* It is also plain, that, when the...

...next Prob.). Cor. 2. The measure of a spherical triangle is the arc of a great circle subtending half the excess of the sum of its angles over two right angles, multiplied by the diameter of the sphere. This depends on the above and Prop. XVII., cor. 1, Spher....

...Hence, if r = 1, the area = A + В + С — тг ; .-. area a A" + В" + С° — 180°. COR. 2. Hence the area of a spherical triangle is proportional to the excess of the sum of its angles above two right angles. This is commonly called the spherical excess. Ex. Let A = 90°, В = 80°,...

...respect to the three arcs or sides of the spherical triangle. The sum of the three angles, aob, aoc, hoc, is less than four right angles ; likewise the sum...polar or supplemental triangle of another, abc (pi. 3, fig. 112), where the vertices of the angles of this second triangle are respectively poles of the sides...

...Stereometry. Every two sides of a spherical triangle are together greater than a third (pi. 3, fig. 111). If through the centre of the sphere, and the...polar or supplemental triangle of another, abc (pi. 3, fig. 112), where the vertices of the angles of this second triangle are respectively poles of the sides...

...will be equal to the sum of the angles of the polygon. Now, the area of each triangle is measured by the excess of the sum of its angles over two right angles, multiplied by the tri-rectangular triangle. Hence the sum of the areas of all the triangles, or the...

...as a unit, by A, B) and (7, we shall have, 2. The spherical excess of any spherical polygon is equal to the excess of the sum of its angles over two right angles taken as many times as the polygon has sides, less two. If we denote the spherical excess by E, the...

...134. The area of a spherical triangle is known to be pro portional to the " spherical excess," ie, the excess of the sum of its angles over two right angles, or the excess of four right Area of angles over the sum of its exterior angles. The area of a spherical...

...angle, as a unit, by A, B, and C, we shall have, The spherical excess of any spherical polygon is equal to the excess of the sum of its angles over two right angles taken as many times as the polygon has sides, less two. If we denote the spherical excess by E, the...