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sum of the three observed angles of any of the great triangles in geodesical operations is always found to be rather more than 180°: were the earth's surface a plane, it ought to be exactly 180°; and this excess, which is called the spherical excess, is so far from being a proof of incorrectness in the work, that it is essential to its accuracy, and offers at the same time another palpable proof of the earth's sphericity.

(229.) The true way, then, of conceiving the subject of a trigonometrical survey, when the spherical form of the earth is taken into consideration, is to regard the network of triangles with which the country is covered, as the bases of an assemblage of pyramids converging to the center of the earth. The theodolite gives us the true measures of the angles included by the planes of these pyramids; and the surface of an imaginary sphere on the level of the sea intersects them in an assemblage of spherical triangles, above whose angles, in the radii prolonged, the real stations of observation are raised, by the superficial inequalities of mountain and valley. The operose calculations of spherical trigonometry which this consideration would seem to render necessary for the reductions of a survey, are dispensed with in practice by a very simple and easy rule, called the rule for the spherical excess, which is to be found in most works on trigonometry.* If we would take into account the ellipticity of the earth, it may also be done by appropriate processes of calculation, which, however, are too abstruse to dwell upon in a work like the present.

(230.) Whatever process of calculation we adopt, the result will be a reduction to the level of the sea, of all the triangles, and the consequent determination of the geographical latitude and longitude of every station observed. Thus we are at length enabled to construct maps of countries; to lay down the outlines of continents and islands; the courses of rivers; the direction of mountain ridges, and the places of their principal

Lardner's Trigonometry, prop. 94. Woodhouse's ditto, p. 148. 1st

edition.

summits; and all those details which, as they belong to physical and statistical, rather than to astronomical geography, we need not here dilate on. A few words, however, will be necessary respecting maps, which are used as well in astronomy as in geography.

(231.) A map is nothing more than a representation, upon a plane, of some portion of the surface of a sphere, on which are traced the particulars intended to be expressed, whether they be continuous outlines or points. Now, as a spherical surface * can by no contrivance be extended or projected into a plane, without undue enlargement or contraction of some parts in proportion to others; and as the system adopted in so extending or projecting it will decide what parts shall be enlarged or relatively contracted, and in what proportions; it follows, that when large portions of the sphere are to be mapped down, a great difference in their representations may subsist, according to the system of projection adopted.

(232.) The projections chiefly used in maps, are the orthographic, stereographic, and Mercator's. In the orthographic projection, every point of the hemisphere is referred to its diametral plane or base, by a perpendicular let fall on it, so that the representation of the hemi

sphere thus mapped on its base, is such as it would actually appear to an eye placed at an infinite distance from it. It is obvious, from the annexed figure, that in this projection only the central por

tions are represented of their true forms, while all the exterior is more and more distorted and crowded toge ther as we approach the edges of the map. Owing to this cause, the orthographic projection, though very good for small portions of the globe, is of little service for large ones.

(233.) The stereographic projection is in great mea

We here neglect the ellipticity of the earth, which, for such a purpose as map-making, is too trifling to have any material influence.

sure free from this defect. To understand this projection, we must conceive an eye to be placed at E, one extremity of a diameter, E C B, of the sphere, and to view the concave surface of the sphere, every point of which, as P, is referred to the diametral plane A D F,

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perpendicular to EB by the visual line P ME. The stereographic projection of a sphere, then, is a true perspective representation of its concavity on a diametral plane; and, as such, it possesses some singularly elegant geometrical properties, of which we shall state one or two of the principal.

(234.) And first, then, all circles on the sphere are represented by circles in the projection. Thus the circle X is projected into x. Only great circles passing through the vertex B are projected into straight lines traversing the center C: thus, BPA is projected into C A.

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2dly. Every very small triangle, G H K, on the sphere, is represented by a similar triangle, g hk, in the projection. This is a very valuable property, as it insures a general similarity of appearance in the map to the reality in all its parts, and enables us to project at least a hemisphere in a single map, without any violent distortion of the configurations on the surface from their real forms. As in the orthographic projection, the bor

ders of the hemisphere are unduly crowded together; in the stereographic, their projected dimensions are, on the contrary, somewhat enlarged in receding from the center.

(235.) Both these projections may be considered natural ones, inasmuch as they are really perspective representations of the surface on a plane. Mercator's is entirely an artificial one, representing the sphere as it cannot be seen from any one point, but as it might be seen by an eye carried successively over every part

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of it. In it, the degrees of longitude, and those of latitude, bear always to each other their due proportion : the equator is conceived to be extended out into a straight line, and the meridians are straight lines at right angles to it, as in the figure. Altogether, the general character of maps on this projection is not very dissimilar to what would be produced by referring every point in the globe to a circumscribing cylinder, by lines drawn from the center, and then unrolling the cylinder into a plane. Like the stereographic projection, it gives a true representation, as to form, of every particular small part, but varies greatly in point of scale in its different regions; the polar portions in particular being extravagantly enlarged; and the whole map, even of a single hemisphere, not being comprizable within any finite limits.

(236.) We shall not, of course, enter here into any geographical details; but one result of maritime

discovery on the great scale is, so to speak, massive enough to call for mention as an astronomical feature. When the continents and seas are laid down on a globe (and since the discovery of Australia we are sure that no very extensive tracts of land remain unknown, except perhaps at the south pole), we find that it is possible so to divide the globe into two hemispheres, that one shall contain nearly all the land; the other being almost entirely sea. It is a fact, not a little interesting to Englishmen, and, combined with our insular station in that great highway of nations, the Atlantic, not a little explanatory of our commercial eminence, that London occupies nearly the center of the terrestrial hemisphere. Astronomically speaking, the fact of this divisibility of the globe into an oceanic and a terrestrial hemisphere is important, as demonstrative of a want of absolute equality in the density of the solid material of the two hemispheres. Considering the whole mass of land and water as in a state of equilibrium, it is evident that the half which protrudes must of necessity be buoyant; not, of course, that we mean to assert it to be lighter than water, but, as compared with the whole globe, in a less degree heavier than that fluid. We leave to geologists to draw from these premises their own conclusions (and we think them obvious enough) as to the internal constitution of the globe, and the immediate nature of the forces which sustain its continents at their actual elevation; but in any future investigations which may have for their object to explain the local deviations of the intensity of gravity, from what the hypothesis of an exact elliptic figure would require, this, as a general fact, ought not to be lost sight of.

(237.) Our knowledge of the surface of our globe is incomplete, unless it include the heights above the sea level of every part of the land, and the depression of the bed of the ocean below the surface over all its extent. The latter object is attainable (with whatever difficulty and however slowly) by direct sounding; the former by two distinct methods: the one consisting in

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