SPHERE, DOCTRINE OF TILE. This phrase is generally used to signify the application of the simple geometrical notions in the article 6PUERE to geography and astronomy. It comes between spherical trigonometry and those two sciences, being merely the explanation of the circumstances under which the former is to be applied to the latter, and the nomenclature which is employed to faci litate explanation.
In geography the end is almost gained when a distinct notion is acquired of the meaning of the terms terrestrial latitude and terrestial longitude, generally abbreviated into latitude and longitude. These are only names given to a pair of spherical co-ordinates as described in SPUME, the axis of rotation of the earth furnishing the means of pre scribing the necessary data. The earth revolves round an axis, say r q (see the diagram in SPUERE), and the great circle perpendicular to that axis is the equator (c u A D). An arbitrary point u is chosen as an origin ; and r being the pole which is called north, v A is the east direction and u c the west. The English choose the point u in such a way that the secondary r v passes through the Observatory at Green wich : the French pay the same compliment to their Observatory at Paris, and so on. The co-ordinate v A (or its angle) is called longitude, east or west according as it falls; and the co-ordinate A P (or its angle) is called latitude, north or south according to the pole towards which it is directed. Thus the place F (r v passing through Greenwich) would be described as in longitude u A east of Greenwich, and F A of north latitude; but if the fundamental secondary, P LT, be moved any number of degrees to the east, every east longitude must be diminished and every west longitude increased as much ; and all places which the secondary passes over in the transfer, must have the names of the directions of their longitudes changed, and take for their new longitudes the excesses of the angle of transfer over their former longitudes. Again, longitude might be measured all the way round in one direction : thus D, instead of being described as in u c of west longitude, might be considered as in 360°— u c of east longitude.
There are few problems of much interest connected with geography merely; and it must be remembered that the common terrestrial globe, with its brazen secondary to the equator (called a meridian, very incorrectly, except as meaning that it may be made a meridian to any place), its ecliptic, and figured horizon, is almost as much a represen tative of the sphere of the heavens as of the earth; and the most useful problems are those in which the sphere is used conjointly in these capacities. But, merely to show what we asserted at first, that the
description and nomenclature which arc called the doctrine of the sphere are nothing but the connecting link of geography, &e., and spherical trigonometry, let us ask the following question :—Given a table of latitudes and longitudes, required the distance between two places mentioned I Let n and at be the places (see diagram in SPIIEUE), then Fri is the co-latitude of n, or 90°—lat. of n, and eat (on account of it's south latitude) is 90° + lat. of it ; while the spherical angle, a r at (which is the angle of the are A o), is, on account of the longitudes being of different names, the sum of the longitudes of D and at. Hence, if n and is be joined by the arc of a great circle, we have given (from the tables) two sides and the angle included, in the spherical triangle n rat. From these data the third side, n m, can be found, in degrees, &c. : convert this into miles, at the rate of 69 miles to a degree (which is accurate enough for the purpose), and the result will be the distance required.
We now make the passage from the terrestrial to the celestial sphere. The latter is a fiction, derived from the impossibility of dis tinguishing the distances of the heavenly bodies, on which account they all seem at the same distances, on a sphere so great that the earth, its centre, is but a point in comparison. But it must be remembered that the appearances of the heavenly bodies conform themselves to this fiction, so that the development of the consequences of the latter amounts to an explanation of the phenomena of the heavens. And first, the rotation of the earth from west to cast gives to the sphere of the heavens an apparent motion from east to west, round an axis which is obtained, by lengthening the axis of the earth. The point of the heavens which answers, for the moment, to the spectator's position on the earth, is that point which is directly over his head, or his zenith. And since the spectator is not exactly at the centre of the celestial sphere, we give the following diagram, illustra tive of the manner in which the effect of this misplacement is destroyed by the largeness of the sphere.