servation, and their sides AC and B C, we can thence calculate the lengths AG, CG, and BF, CF. Again, CG and C F being known, and the included angle G C F, G F may be calculated, and so on. Thus may all the stations be accurately determined and laid down, and as this process may be carried on to any extent, a map of the whole county may be thus constructed, and filled in to any degree of detail we please.

(227.) Now, on this process there are two important remarks to be made. The first is, that it is necessary to be careful in the selection of stations, so as to form triangles free from any very great inequality in their angles. For instance, the triangle K B F would be a very improper one to determine the situation of F from observations at B and K, because the angle F being very acute, a small error in the angle K would produce a great one in the place of F upon the line B F. Such ill-conditioned triangles, therefore, must be avoided. But if this be attended to, the accuracy of the determination of the calculated sides will not be much short of that which would be obtained by actual measurement (were it practicable); and, therefore, as we recede from the base on all sides as a center, it will speedily become practicable to use as bases, the sides of much larger triangles, such as G F, G H, H K, &c.; by which means the next step of the operation will come to be carried on on a much larger scale, and embrace far greater intervals, than it would have been safe to do (for the above reason) in the immediate neighbourhood of the base. Thus it becomes easy to divide the whole face of a country into great triangles of from 30 to 100 miles in their sides (according to the nature of the ground), which, being once well determined, may be afterwards, by a second series of subordinate operations, broken up into smaller ones, and these again into others of a still minuter order, till the final filling in is brought within the limits of personal survey and draftsmanship, and till a map is constructed, with any required degree of detail.

CORRECTION FOR THE EARTH'S SPHERICITY. 149

(228.) The next remark we have to make is, that all the triangles in question are not, rigorously speaking, plane, but spherical-existing on the surface of a sphere, or rather, to speak correctly, of an ellipsoid. In very small triangles, of six or seven miles in the side, this may be neglected, as the difference is imperceptible; but in the larger ones it must be taken into consideration.

M

It is evident that, as every object used for pointing the telescope of a theodolite has some certain elevation, not only above the soil, but above the level of the sea, and as, moreover, these elevations differ in every instance, a reduction to the horizon of all the measured angles would appear to be required. But, in fact, by the construction of the theodolite (art. 155.) which is nothing more than an altitude and azimuth instrument, this reduction is made in the very act of reading off the horizontal angles. Let E be the center of the earth; A, B, C, the places on its spherical surface, to which three stations, A, P, Q, in a country are referred by radii E A, EBP, ECQ. If a theodolite be stationed at A, the axis of its horizontal circle will point to E when truly adjusted, and its plane will be a tangent to the sphere at A, intersecting the radii E B P, EC Q, at M and N, above the spherical surface. The

telescope of the theodolite, it is true, is pointed in succession to P, and Q; but the readings off of its azimuth circle give-not the angle PA Q between the directions of the telescope, or between the objects P, Q, as seen from A; but the azimuthal angle M A N, which is the measure of the angle A of the spherical triangle BA C. Hence arises this remarkable circumstance,-that the

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.

CHAP. III.

PROJECTIONS OF THE SPHERE.

151

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 an nexed 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 together 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,

perpendicular to EB by the visual line P M E. 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 CA.

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