Theoretical Since the object of maps and charts is to accurately represent by suitable symbols on a plane surface the relative position of objects, etc., on the spherical sur face of the earth, the problem requiring solution is one that is geometrically impossible, and therefore any map, no matter how accurately constructed, must necessarily be only an ap proximation to the truth. Yet by the assump tion of certain lines of reference which corre spond to the actual latitudes and longitudes on the surface of the earth according to certain assumed or fixed geometric laws, the approxi mation to the truth may be so nearly attained as to be quite as useful as the truth itself for 'stereographic' with the eye supposed upon the surface of the sphere and occupying the pole of a great circle, the plane of which is the plane of projection, and (3) •the ((gnomonic° or 'centrals projection, with the eye assumed at the centre of the sphere and the plane of projec tion tangent to its surface. To this class be longs also the globular or equidistant projec tion designed to avoid to a certain extent the contraction of the orthographic and the exag geration of the stereographic projections near their respective outer edges. In this projection the eye is supposed at a distance of 4, times the radius above the surface of the sphere. The perspective projections, in their astronom ical and geographical application, are generally used for representing large areas and hemi spheres and are seldom used in the construction of maps of small extent on large scales, in which case they are too laborious of construe tion and cannot be made to satisfy any special conditions not included within the scope of their inherent characteristics. The three principal perspective projections were known to the an cients, the orthographic and the stereographic being accredited to Hipparchus and the gno monic to Thales. The globular projection was devised by Nicolisi. They served to introduce projection methods, but with the exception of the gnomonic, which is still used for the con struction of star charts and some classes of nautical charts, they have been superseded in the construction of modern maps by the de veloped projections.
Developed Projections are obtained by the substitution of a cylindrical or a conical sur face for the ordinary plane of projection, and that surface, being subsequently developed or rolled out in a plane, gives the projection. The application of this results in two general classes the great obliquity of the intersections of the meridians and the parallels in the higher lati tudes. This projection on account of the close similarity of the figures on the surface of the earth to the corresponding figures on the pro jection, and the uniformity of scale over the whole map is now extensively used for mapping not only areas of comparatively limited extent, but also for the representation of continental areas of Marge extent.
For the mathematical operations involved in the computation of the elements of these pro jections and for illustrations showing how they are plotted in the actual construction of maps, see article under the title PaojEcrioN in this Encyclopedia.
Practical Methods of Construction.—As a rule, and with but a few exceptions, the draw ing of an accurate map is necessarily preceded by an accurate survey of the things to be of developed projections— those employing a cylinder tangent to the sphere, usually at the equator, and those employing a cone tangent, usually at the middle parallel of latitude of the arta mapped. Of these, the two most com manly used at the present time are the Mercator and the polyconic projections, the former ern-, ploying a tangent cylinder and the latter an infinite number of tangent cones. The Mercator was purposely designed for the use of nautical maps and is principally employed for that pur pose at the present time. On a map based on this projection the loxodromic curve, that is, the course of a vessel which intersects the suc cessive meridians at a constant angle, is repre sented on the map as a straight line, thus facili tating the plotting of that course from day to day during the progress of the voyage, and ma terially assisting in determining the true dis tance and bearing of the objective point. The polyconic projection appears to have been de vised for the purpose of providing a suitable base for the mapping of the Atlantic coast line of the United States by the United States Coast and Geodetic Survey. The direction of this coast line being nearly diagonal to the merid ians and parallels, and having a great north and south amplitude, required for its representation a projection which avoided the inherent dis tortions of the various conic projections due to mapped, and the character of those things to gether with the purposes for which the survey is made, not only define the methods of survey ing to be employed, but also the character of the map and its relative scale. The various methods of surveying are described under the title SURVEYING in this Encyclopedia. The various kinds of maps may be grouped into the follow ing named general classes: eGeographical,' ((physiographical," ((topographicaP and "geo logical" maps. The representations of astro nomical, hydrographical, nautical and statistical data belong more properly to the category of charts. The various forms of railroad maps are merely diagrams only useful for consulta tion with the accompanying railroad schedules. The amount of distortion required to fit them for the purposes for which they are used makes them entirely unreliable for any other purpose. The military maps are of various types and differ greatly according to the purposes for which they are used. Staff maps usually show every feature, natural and artificial, on the sur face of the country mapped in the minutest de tail and with the greatest accuracy. The differ ences of elevation of the various points are a matter of the greatest moment, and, therefore, are usually treated in the most complete manner both as to graphic effect and accuracy of in formation. The sketch maps are simpler in character and represent in the most elemen tary way the most important features which have a direct bearing on the particular kind of military operations involved.