SPHERICAL
, something relating to the sphere. As,
Spherical Angle, is the angle formed on the surface of a Sphere or globe by the circumferences of | two great circles. This angle, formed by the circumferences, is equal to that formed by the planes of the same circles, or equal to the inclination of those two planes; or equal to the angle made by their tangents at the angular point. Thus, the inclination of the two planes CAF, CEF, forms the Spherical Angle ACE, equal to the tangential angle PCQ.
The measure of a Spherical Angle, ACE, is an arc of a great circle AE, described from the vertex C, as from a pole, and intercepted between the legs CA and CE.
Hence, 1st, Since the inclination of the plane CEF to the plane CAF, is every where the same, the angles in the opposite intersections, C and F, are equal.—2d, Hence the measure of a Spherical Angle ACE, is an arc described at the interval of a quadrant CA or CE, from the vertex C between the legs CA, CE.— 3d, If a circle of the sphere CEFG cut another AEBG, the adjacent angles AEC and BEC are together equal to two right angles; and the vertical angles AEC, BEF are equal to one another. Also all the angles formed at the same point, on the same side of a circle, are equal to two right angles, and all those quite around any point equal to four right angles.
Spherical Triangle, is a triangle formed upon the surface of a sphere, by the intersecting arcs of three great circles; as the triangle ACE.
Spherical Triangles are either right-angled, oblique, equilateral, isosceles, or scalene, in the same manner as plane triangles. They are also said to be quadrantal, when they have one side a quadrant. Two sides or two angles are said to be of the same affection, when they are at the same time either both greater, or both less than a quadrant or a right angle or 90°; and of different offections, when one is greater and the other less than 90 degrees.
1. Spherical Triangles have many properties in common with plane ones: Such as, That, in a triangle, equal sides subtend equal angles, and equal angles are subtended by equal sides: That the greater angles are subtended by the greater sides, and the less angles by the less sides.
2. In every Spherical Triangle, each side is less than a semicircle: any two sides taken together are greater than the third side: and all the three sides taken together are less than the whole circumference of a circle.
3. In every Spherical Triangle, any angle is less than 2 right angles; and the sum of all the three angles taken together, is greater than 2, but less than 6, right angles.
4. In an oblique Spherical Triangle, if the angles at the base be of the same affection, the perpendicular from the other angle falls within the triangle; but if they be of different affections, the perpendicular falls without the triangle.
Dr. Maskelyne's remarks on the properties of Spherical Triangles, are as follow: (See the Introd. to my Logs. pa. 160, 2d edition.)
5. “A Spherical Triangle is equilateral, isoscelar, or scalene, according as it has its three angles all equal, or two of them equal, or all three unequal; and vice versa.
6. The greatest side is always opposite the greatest angle, and the smallest side opposite the smallest angle.
7. Any two sides taken together are greater than the third.
8. If the three angles are all acute, or all right, or all obtuse; the three sides will be, accordingly, all less than 90°, or equal to 90°, or greater than 90°; and vice versa.
9. If from the three angles A, B, C, of a triangle ABC, as poles, there be described, upon the surface of the sphere, three arches of a great circle DE, DF, FE, forming by their intersections a new Spherical Triangle DEF; 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 ABC.
10. In any triangle GHI or GhI, right angled in G, 1st, The angles at the hypotenuse are always of the same kind as their opposite sides; 2dly, The hypotenuse is less or greater than a quadrant, according as the sides including the right angle, are of the same or different kinds; that is to say, according as these 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 angle, are always of the same kind as their opposite angles; 2dly, The sides including the right angle will be of the same or different kinds, according as the hypotenuse is less or more than 90°; but one at least of them will be of 90°, if the hypotenuse is so.”
Of the Area of a Spherical Triangle. The mensuration of Spherical Triangles and polygons was first found out by Albert Girard, about the year 1600, and is given at large in his Invention Nouvelle en l'Algebre, pa. 50, &c; 4to, Amst. 1629. In any Spherical Triangle, the area, or surface inclosed by its three sides upon the surface of the globe, will be found by this proportion: As 8 right angles or 720°, Is to the whole surface of the sphere; Or, as 2 right angles or 180°, To one great circle of the sphere; So is the excess of the 3 angles above 2 right angles, To the area of the Spherical Triangle.
Hence, if a denote .7854, d = diam. of the globe, and s = sum of the 3 angles of the triangle; | then add X (s - 180)/180 = area of the Spherical Triangle.
Hence also, if r denote the radius of the sphere, and c its circumference; then the area of the triangle will thus be variously expressed; viz, , in square degrees, when the radius r is estimated in degrees; for then the circumference c is = 360°.
Farther, because the radius r, of any circle, when estimated in degrees, is, =180/(3.14159 &c.) = 57.2957795, the last rule r X ―(s - 180), for expressing the area A of the Spherical Triangle, in square degrees, will be barely very nearly.
Hence may be found the sums of the three angles in any Spherical Triangle, having its area A known; for the last equation gives the sum .
So that, for a Triangle on the surface of the earth, whose three sides are known; if it be but small, as of a few miles extent, its area may be found from the known lengths of its sides, considering it as a plane Triangle, which gives the value of the quantity A; and then the last rule above will give the value of s, the sum of the three angles; which will serve to prove whether those angles are nearly exact, that have been taken with a very nice instrument, as in large and extensive measurements on the surface of the earth.
Resolution of Spherical Triangles. See Triangle, and Trigonometry.
Spherical Polygon, is a figure of more than three sides, formed on the surface of a globe by the intersecting arcs of great circles.
The area of any Spherical Polygon will be found by the following proportion; viz, As 8 right-angles or 720°, To the whole surface of the sphere; Or, as 2 right angles or 180°, To a great circle of the sphere; So is the excess of all the angles above the product of 180 and 2 less than the number of angles, To the area of the spherical polygon.
| That is, putting n | = the number of angles, |
| s | = sum of all the angles, |
| d | = diam. of the sphere, |
| a | = .78539 &c; |
Hence other rules might be found, similar to those for the area of the Spherical Triangle.
Hence also, the sum s of all the angles of any Spherical Polygon, is always less than 180n, but greater than 180 (n - 2), that is less than n times 2 right angles, but greater than n - 2 times 2 right angles.
Spherical Astronomy, that part of astronomy which considers the universe such as it appears to the eye. See Astronomy.
Under Spherical Astronomy, then, come all the phenomena and appearances of the heavens and heavenly bodies, such as we perceive them, without any enquiry into the reason, the theory, or truth of them. By which it is distinguished from theorical astronomy, which considers the real structure of the universe, and the causes of those phenomena.
In the Spherical Astronomy, the world is conceived to be a concave Spherical surface, in whose centre is the earth, or rather the eye, about which the visible frame revolves, with stars and planets fixed in the circumference of it. And on this supposition all the other phenomena are determined.
The theorical astronomy teaches us, from the laws of optics, &c, to correct this Scheme and reduce the whole to a juster system.
Spherical Compasses. See Compasses.
Spherical Geometry, the doctrine of the sphere; particularly of the circles described on its surface, with the method of projecting the same on a plane; and measuring their arches and angles when projected.
Spherical Numbers. See Circular Numbers.
Spherical Trigonometry. See Spherical TRIGONOMETRY.
