PROJECTION is the representation on any surface of objects or figures as they appear to the eye of an observer. It thus includes perspective (q.v.), and is most simply illus trated by time shadow of an object thrown by a candle on a wall; the shadow being the projection, and the place of the light the position of the eye. The theory of projec tions is of great importance, both in mathematics and geography, being, in the former case, perfectly general in its application; while in the tatter only the projection of .the sphere is required. Projections of the sphere are of various kinds, depending upon the position and distance of the eye from the sphere, and the form of the surface on which the projection is thrown; thus we have the orthographic, stereographic, globular, conical, and cylindrical or. Mercator's projections, all of which are treated of under the article MAP. Another projection frequently employed is the gnomonic. In the gnomonic pro jection, the eye is supposed to be situated at the center of the sphere, and the surface on the projection is thrown is a plane surface which touches the sphere at any one point (called the principal point). It is evident that a map constructed on the gnomonic projection is sensibly correct only for a circular area whose circumference is at a small angular distance from the principal point. From the position of the eye in the gnomonic projection, it follows that all great circles, or portions of great circles, of the sphere are represented by straight lines, for their planes pass through the eye. The distance of two 'points on the sphere, when measured along the surface, is least if they are measured along a great circle; and as the distance of the projections of these points on the plane is represented by a straight line, which is the shortest distance between two points on a plane, this projection, if employed in the construction of mariners' charts, would at once show the shortest course. Maps of the earth's surface have been projected by the gno
monic method, the surface of projection being the interior surface of a cube circum scribing the sphere, and the complete series consequently amounting to six maps; but it is not fitted for the construction of maps of large portions of the earth's surface. The gnomonic projection derives its name from its connection with the mode of'describing a gnomon or dial (q.v.). The orthographic and stereographic projections were employed by the Greek astronomers for the construction of maps of the heavens; the former, or analemma, being the best-known and most used. The stereographic, called planisphere by the Greeks, is said to have been invented by Hipparchus, and the gnomonic is described by Ptolemy. The others are of modern invention.
In mathematics, the theory of projections is general in its application, and has been employed within the last few years to generalize the ancient geometry, and as a power ful aid to algebra. Its basis is the investigation and determinatiomof those properties which, being true of a figure, are also true of its projections, such properties being neces sarily dependent, not on the "magnitude," but on the "position" of the lines and angles belonging to the figure. These properties are generally denominated projective properties. For instance, the three conic sections, the parabola, ellipse, and hyperbola, are merely various projections of a circle on a plane, and all "positional" properties of the circle are at once, by this theory, connected with similar properties of the three conic sections. The theory is also largely employed in demonstrative mechanics.—See, for further information, Mulcahy's Modern 'Geometry, Salmon's Conic Sections, Monge's Geoin ark Descriptive, Poncelet's Proprietis des Figures Projectives, and Poisson's Traite de ffecanique.