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At 9h. 54m. 26-8s. A.M.} By chronometer.
311° 47' 20" morning observation.
359 46 35 reading of approximate
meridian. The sun's change of declination in one hour of mean time on May 12 appears, by the Nautical Almanac, = 37":53, therefore for 2h. 5.6m., the half interval, it is = 78":5.
The magnetic bearing of the pole star, or of any circumpolar star at its upper or lower culmination, gives at once the variation of the compass; a meridian may likewise be traced by observing the azimuths of a star at its greatest elongations, and taking the
If only one elongation is observed, the sine of the angular
sin polar distance of star which added to, or subdistance =
cosine latitude tracted from, the observed azimuth, gives the direction of the meridian.
The time at which any star is at its greatest elongation is thus found. The cosine of the hour angle in space = tan polar dist. x tan lat. This hour angle divided by 15 gives the interval in sidereal time.
The other methods of finding the variation of the compass by the amplitude of the sun at sunrise or sunset, and by his azimuth at any period of the day, requiring more calculation, will be found among the Astronomical Problems.
A meridian line can be marked on the ground, without the aid of any instrument, with sufficient accuracy to obtain the variation of the needle for common purposes, by driving a picket vertically into the ground on a perfectly level surface. At three or four hours before noon, measure the length of its shadow on the ground, and from the bottom of the picket, as a centre, describe an arc with this distance as radius. Observe, when the shadow intersects this arc about the same time in the afternoon; and the middle point between these, and the picket, gives the line of the meridian. It is of course better to have three or four observations at different periods before and after noon; and these several middle points afford means of laying out the line more correctly.
The method hitherto described of laying down stations by triangulation, or by means of distances calculated from astronomical observation, is, however, only applicable within certain limits; as, on account of the spherical figure of the earth, the relative positions of places on the globe cannot be represented by any projection in geographical maps embracing very large portions of its surface, except by altering more or less their real distances, the content of
various tracts of territory, and in fact, distorting the whole appearance, when compared with the different portions of the same country represented as plane surfaces.
Either a true projection or some arbitrary arrangement of the meridians and parallels is therefore necessarily adopted, and each place is marked on this skeleton according to its relative latitude and longitude. Those projections should be preferred in which the geographical lines are most easily traced, and whose arrangement distorts as little as possible the linear and superficial dimensions.
Descriptions of various projections will be found in the works of Puissant, Francæur, and other authors on the subject; and some very useful explanations of the projections of the sphere, in a treatise on “ Practical Geometry and Projection,” published by the Society of Useful Knowledge.
The following short but clear definition of the three species of projection commonly used in maps, viz., the orthographic, stereographic, and Mercator's, is taken from Sir J. F. Herschel's “ Astronomy :"
“ 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 its representation, 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 that in this projection only the central portions are represented in their true forms, while the exterior is more and more distorted and crowded together as it approaches 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.
“ The stereographic projection is in a great measure free from this defect. To understand this method, we must conceive an eye to be placed at E, one extremity of a diameter ECB of the sphere, and to view the concave surface of the sphere, every point of which, as P, is referred to the diametral plane ADF perpendicular to E B 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 singular geometrical properties, of which the following are two of the principal :-first, 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 centre C; thus B P A is projected into CA.
Secondly, every very small triangle G H K on the sphere is represented by a similar triangle ghk in the projection. This valuable property ensures 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 borders of the hemisphere are unduly crowded together; in the stereographic, their projected dimensions are, on the contrary, somewhat enlarged in receding from the centre.”
Both these projections may be considered natural ones, inasmuch as they are really perspective representations of the surface on a plane ; but Mercator's projection 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 of it. The degrees of longitude are assumed equal, and of the value of those at the equator. The degrees of latitude are extended each way from the equator, retaining always their proper proportion to those of longitude; consequently the intervals between the parallels of latitude increase from the equator to the poles. 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 centre, 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 regions, in particular, being extravagantly enlarged; and the whole map, even of a single hemisphere, not being comprisable within any finite limits. .
The following simple directions are given by Mr. Arrowsmith for a projection, adapted to a map to comprehend only a limited portion of the globe; for instance, that between the parallels of 44°
and 48° 30' north latitude, and longitudes 9° and 18° east of Greenwich. Draw a line A B for a central meridian; divide it into the