The following propositions and remarks concerning spherical triangles, will render their calculation perspicuous and free from ambiguity. 1st. A spherical trian gle is equilateral, isucelar, or scalene, ac cording ark has its three angles all equal, or two of them equal, or all three unequal. 2d. The greatest side is always opposite the greatest angle, and the smallest side opposite the smallest angle. 3d. Any two sides, taken together, are greater than the third. 4,11. If the three angles are all a cute, or all right, or all obtuse, the three sides will be, accordingly, all less than or 90°, or greater than 90°. 5th. If from the three angles A, B, C, of a trian gle A B C (fig. 23), as poles, there be described on the surface of the sphere, three arches of a great circle DE, DF, FE forming by their intersections a new spherical triangle DM: ; each side of the new triangle will be the supplement of the angle at its pole; and each angle of the same triangle will be the supplement of the side opposite to it in the triangle A IIC. 61.11, In any triangle A B C (fig. 29), or A
b C, right-angled in A : lst, The angles at the hypothenuse. are always of the same kind as their opposite sides. 2dly. The hypothenuse is greater or lesser than a quadrant, according as the sides, includ ing the right angle, are of the same, or different kinds ; that is to say, according as the same sides, are either both acute, Or both obtuse: or, as one is acute, and the other obtuse. And vice versa: 1st. The sides including the right angles, are always of the same kind as their opposite angles. 2dly, The sides, including the right angles, will be of the same, or different kinds, according as the hypothenuse is less, or more, than 90° ; but, one at least of them will be of 90°, ;Ellie hypothenuse is so.
Considering it impossible to give a po pular idea of this highly important branch of mathematics, in any brief form, we must refer those readers, who wish to be come proficients therein, to the various excellent treatises published on that sub ject ; particularly those by Simpson, Bo nycastle, Payne, &c.