_Central or Gnomonic (Perspective) Projection._--In this projection the eye is imagined to be at the centre of the sphere. It is evident that, since the planes of all great circles of the sphere pass through the centre, the representations of all great circles on this projection will be straight lines, and this is the special property of the _central projection_, that any great circle (i.e. shortest line on the spherical surface) is represented by a straight line. The plane of projection may be either parallel to the plane of the equator, in which case the parallels are represented by concentric circles and the meridians by straight lines radiating from the common centre; or the plane of projection may be parallel to the plane of some meridian, in which case the meridians are parallel straight lines and the parallels are hyperbolas; or the plane of projection may be inclined to the axis of the sphere at any angle [lambda]. In the latter case, which is the most general, if [theta] is the angle any meridian makes (on paper) with the central meridian, [alpha] the longitude of any point P with reference to the central meridian, l the latitude of P, then it is clear that the central meridian is a straight line at right angles to the equator, which is also a straight line, also tan [theta] = sin [lambda] tan [alpha], and the distance of p, the projection of P, from the equator along its meridian is (on paper) m sec [alpha] sin l / sin (l + x), where tan x = cot [lambda] cos [alpha], and m is a constant which defines the scale. The three varieties of the central projection are, as is the case with other perspective projections, known as polar, meridian or horizontal, according to the inclination of the plane of projection. [Illustration: (From _Text Book of Topographical Surveying_, by permission of the Controller of H. M. Stationery Office.) FIG. 14.--Part of the Atlantic Ocean on a Meridian Central Projection. The shortest path between any two points is shown on this projection by a straight line.] Fig. 14 is an example of a _meridian central projection_ of part of the Atlantic Ocean. The term "gnomonic" was applied to this projection because the projection of the meridians is a similar problem to that of the graduation of a sun-dial. It is, however, better to use the term "central," which explains itself. The central projection is useful for the study of direct routes by sea and land. The United States Hydrographic Department has published some charts on this projection. False notions of the direction of shortest lines, which are engendered by a study of maps on Mercator's projection, may be corrected by an inspection of maps drawn on the central projection. There is no projection which accurately possesses the property of showing shortest paths by straight lines when applied to the spheroid; one which very nearly does so is that which results from the intersection of terrestrial normals with a plane.