does so move among the stars, while the latter hold constantly, with respect to each other, the same relative position, the notice of a few nights, or even hours, will satisfy the commencing student, and this is all that at present we require.

(223.) There is only one circumstance wanting to make our analogy complete. Suppose the hands of our clock, instead of moving quite close to the dial-plate, were considerably elevated above, or distant in front of it. Unless, then, in viewing it, we kept our eye just in the line of their center, we should not see them exactly thrown or projected upon their proper places on the dial. And if we were either unaware of this cause of optical change of place, this parallax or negligent in not taking it into account we might make great mistakes in reading the time, by referring the hand to the wrong mark, or incorrectly appreciating its distance from the right. On the other hand, if we took care to note, in every case, when we had occasion to observe the time, the exact position of the eye, there would be no difficulty in ascertaining and allowing for the precise influence of this cause of apparent displacement. Now, this is just what obtains with the apparent motion of the moon among the stars. The former (as will appear) is comparatively near to the earth the latter immensely distant; and in consequence of our not occupying the center of the earth, but being carried about on its surface, and constantly changing place, there arises a parallax, which displaces the moon apparently among the stars, and must be allowed for before we can tell the true place she would occupy if seen from the center.

(224.) Such a clock as we have described might, no doubt, be considered a very bad one; but if it were our only one, and if incalculable interests were at stake on a perfect knowledge of time, we should justly regard it as most precious, and think no pains ill bestowed in studying the laws of its movements, or in facilitating the means of reading it correctly. Such, in the parallel we are drawing, is the lunar theory, whose object is to

L

reduce to regularity, the indications of this strangely irregular-going clock, to enable us to predict, long beforehand, and with absolute certainty, whereabouts among the stars, at every hour, minute, and second, in every day of every year, in Greenwich local time, the moon would be seen from the earth's center, and will be seen from every accessible point of its surface; and such is the lunar method of longitudes. The moon's apparent angular distances from all those principal and conspicuous stars which lie in its course, as seen from the earth's center, are computed and tabulated with the utmost care and precision in almanacks published under national control. No sooner does an observer, in any part of the globe, at sea or on land, measure its actual distance from any one of those standard stars (whose places in the heavens have been ascertained for the purpose with the most anxious sollicitude), than he has, in fact, performed that comparison of his local time with the local times of every observatory in the world, which enables him to ascertain his difference of longitude from one or all of them.

(225.) The latitudes and longitudes of any number of points on the earth's surface may be ascertained by the methods above described; and by thus laying down a sufficient number of principal points, and filling in the intermediate spaces by local surveys, might maps of counties be constructed, the outlines of continents and islands ascertained, the courses of rivers and mountain chains traced, and cities and towns referred to their proper localities. In practice, however, it is found simpler and easier to divide each particular nation into a series of great triangles, the angles of which are stations conspicuously visible from each other. Of these triangles, the angles only are measured by means of the theodolite, with the exception of one side only of one triangle, which is called a base, and which is measured with every refinement which ingenuity can devise or expense command. This base is of moderate extent, rarely surpassing six or seven miles, and purposely selected in a perfectly horizontal plane, otherwise

Its

conveniently adapted for purposes of measurement. length between its two extreme points (which are dots on plates of gold or platina let into massive blocks of stone, and which are, or at least ought to be, in all cases preserved with almost religious care, as monumental records of the highest importance), is then measured, with every precaution to ensure precision*, and its position with respect to the meridian, as well as the geographical positions of its extremities, carefully ascertained.

(226.) The annexed figure represents such a chain of triangles. A B is the base, O, C, stations visible from

both its extremities (one of which, O, we will suppose to be a national observatory, with which it is a principal object that the base should be as closely and immediately connected as possible); and D, E, F, G, H, K, other stations, remarkable points in the county, by whose connection its whole surface may be covered, as it were, with a network of triangles. Now, it is evident that the angles of the triangle A, B, C being observed, and one of its sides, A B, measured, the other two sides, A C, B C, may be calculated by the rules of trigonometry; and thus each of the sides A C and BC becomes in its turn a base capable of being employed as known sides of other triangles. For instance, the angles of the triangles ACG and BCF being known by ob

The greatest possible error in the Irish base of between seven and eight miles, near Londonderry, is supposed not to exceed two inches.

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.

(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

B

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 P AQ 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