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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, Francœur, and other authors on the subject; and some very useful explanations of the projections of the sphere, in a treatise on 66 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.

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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 : only great circles passing through the vertex B are projected into straight lines traversing the centre C; thus BPA 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

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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°

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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


required number of degrees of latitude (42); through one of these points of division (say 46°) draw CD intersecting the meridian at right angles, and likewise draw lines through the other points parallel to CD.

Take the breadth in minutes of a degree of longitude in lat. 46° = 41.63; from M towards C and D, set off each way one-half of this, 20-84, (ME. MG). Again, from N lay off on each side one-half of the length of a degree in lat. 47° 40′92 — N F, N H. Measure the diagonals G H, E F, and putting one point of the compasses successively on F, G, H, and E, describe the arcs, x x x x.

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Take 41.68, the whole measurement of a longitudinal degree in lat. 46°, and lay off the distance, GO, EO, intersecting the arcs xxx at 00. Again, take the value of a degree in latitude 47° 40.92, and lay off the distances EP, HP.

This process continued until the parallels of 46° and 47° are completed, the whole projection may be carried on in the same manner, the two parallels first drawn furnishing the respective points of each meridian.

It would occupy too much space to pursue the subject further; explanations of all the most useful projections will be found in the sixth chapter of Francœur's "Géodesie," and in other works of the same character.




BEFORE proceeding to the solution of the few simple problems by which the latitude, longitude, and time can be determined under different circumstances, it is considered advisable to explain the meaning of such terms as are most constantly met with in practical astronomy, and the corrections necessary to be applied to all observations.

The Sextant; Reflecting Circle, or Dollond's Repeating Circle; with the Artificial Horizon and Chronometer; are the description of portable instruments generally used in taking astronomical observations. In an observatory, or for any extensive geodesical operation, instruments are required of firmer construction, and admitting from their size of more minute graduation; but these are mostly confined to permanent establishments.

In all reflecting instruments the angle formed by the planes of the two mirrors is only half the observed angle, but the arc or circle is graduated to meet this effect of the principle of their construction; thus an angle of 60° is marked on the limb of the sextant 120°; and the entire circle reads 720°.

Descriptions of the methods of using and adjusting the sextant and reflecting circle are given in Mr. J. Simms' "Treatise on Mathematical Instruments," which little work is, or should be, in the hands of every observer; but as no allusion is there made to the repeating circle*, which is, at all events in theory, the most

* The repeating circle here spoken of, is a reflecting circle, having the power of repetition. For the determination of latitudes and longitudes on surveys of the magnitude of the Ordnance Survey of Great Britain, or for very important and delicate geodesical operations, the Zenith Sector, Altitude and Azimuth Instrument, and Portable Transit are employed. This latter, though properly an observatory instrument, can be used upon a stand formed by the stump of a large tree, or by three or four strong posts driven into the ground, supporting a top, on which the transit is placed. A rough pedestal of masonry or brick-work of course answers the same purpose, great care being taken to secure its steadiness, and prevent its

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