Spherical Trigonometry. D E FIN IT ION S. I. Great-Circle of a Sphere is a Sečtion of the Sphere by a Plane passing thro’ the Center thereof. 2: The Axis of a Great-Circle is a Right-line passing thro’ the Center, perpendicular to the Plane of the Circle: And the two Points, where the Axis intersects the Surface of the Sphere, are call'd the Poles of the Circle. 3. A spherical Angle is the Inclination of two Great-Circles. 4. A spherical Triangle is a Part of the Surface of the Sphere included by the Arches of three Great-Circles: Which Arches are called the Sides of the Triangle. 5. If thro’ the Poles A and F of two GreatCircles D F and DA, standing atRight-angles, two other Great-Circles . ACE and FCB be conceived to pass,and thereby form two spherical Triangles A B C. and FCE, the latter of the Triangles so formed is said to be the Complement of the former; and vice versa. C 3 CoRo B F. its Pole; and that the .4 ** Measure of a spherical Angle CAD “ is an Arch of a Great-Circle intercepted by the two Circles ACB, ADB forming that Angle, and whose Pole is the angular Point A. For let the Diameter AB be the Interse&tion of the Great Circles ADB and ACB (see Corol. 1.) and let the Plane, or Great-Circle, DEC be conceived perpendicular to that Diameter, interseóting the Surface of the Sphere in the Arch CD; then it is manifest that AD = BD = 90°, and AC = BC = 90° (Cor. 1.) and that CD is the Measure of the Angle DEC (or CAD) the Inclination of the two proposed Circles. * Note, Altho' a spherical Angle is, properly, the Inclination of two Great-Circles, yet it is commonly expressed by the Inclination of their Peripheries at the Point where they inters & each other. 4. Hence 4. Hence it is also manifest, that the Angles B and E, of the ComplementalTriangles ABC and FCE, are both Right-angles; and that CE is the Complement | of AC, CF of BC, B D . (or the Angle F) of AB, B and EF of ED (or the Angle A). making an Angle DOE, measured by the Arch Hence it follows, that the Sines of the Angles of any oblique spherical Triangle A DC are to one another, directly, as the Sines of the opposite Sides, |