When the eye is supposed to be infinitely distant, so that the lines of vision are parallel to one another and perpendicular to the primitive, the projection is called orthographic. When the primitive is a tangent plane to the sphere, and the eye is supposed to be at the center of the sphere, the projection is called gnomonic. When the eye is supposed to be at the surface of the sphere, and the primitive to pass through the center, so as to have the eye in its pole, the projection is called stereographic. The projection is further termed equatorial, meridional, or horizontal, according as the primitive coincides with, or is parallel to, the equator, or the meridian or horizon of any place. 30 60 60 30 To delineate the Orthographic Projection of the Circles of the Terrestrial Sphere upon the Plane of the Meridian of any place. With a radius according to the contemplated scale of the projection, describe the circle w NES for the circumference of the primitive, and draw the vertical and horizontal diameters N s and w E, which will be the projections of a meridian perpendicular to the primitive, and of the equator, respectively. Take out from the line of sines the sines of the latitudes through which the parallels are to Wo 300 30 60 90 120 150 E 30 30 GO be drawn, and, reducing these sines to the radius of the pri -mitive*, set off these reduced distances both ways from the center upon the line N s; and also both ways from the center upon the line w E, for the sines of the angles which the meridians, to be drawn at the same intervals as the parallels, make with the meridian N s. Through the divisions thus set off upon the line N s draw straight lines parallel to w E, and such straight lines will be the projections of the several paral lels of latitude, which are to be numbered 0 to 90, from the equator to either pole for the latitudes. With distances from the center to the divisions set off upon w E as semi-minor axes, and the distance from c to N or s, equal to radius of primitive, *If the proportional compasses be set in the proportion of the sine 90° on the line of sines to the radius of the primitive, one pair of points will give, reduced to this radius, the sines taken off by the other pair of points. The manner of taking from the sector a sine to any radius will be hereafter pointed out. as a common major axis, describe semi-ellipses *, and they will be the projections of the several meridians, which are to be numbered either way from the first meridian for the longitudes. In the figure the primitive coincides with the plane of the meridian of a place in 30° west longitude, or 150° east longitude, the sum of these two being 180°, as must always be the case. To delineate the Gnomonic Projection of the Circles of the Terrestrial Sphere upon a Plane parallel to the Equator.-Ir this case the meridians 30 60 90 45 120 60 60 150 180 120 150 will all be projected into straight lines, making the same angles one with another that their originals do on the surface of the sphere; the projection of the pole will be the center of the primitive, and the projections of the parallels of latitude will be circles described from the projection of the pole, as center, with distances equal to the tangents of the respective colatitudes reduced to the radius of the primitive. The parallel of 45° will, therefore, coincide with the circumference of the primitive; the parallels of latitudes greater than 45° will lie within the primitive; and for latitudes less than 45° the parallels will fall without the primitive, the radii of their projections increasing as the latitude decreases, until the radius for projecting the equator becomes infinite 45 GO 90 30 Describe, then, a N circle for the primitive; draw straight lines radiating from its center, and equally inclined one to another for the projections of equidistant meridians; and number them 0 to 180 both ways from the first meridian for the longitudes. With the tangents of the colatitudes, taken at intervals equal to the angle between two successive meridians, and reduced to the radius of the primitive, as distances, describe from the center of the primitive concentric circles; and number them 90 to 45 from the pole to the primitive for the latitudes, continuing the graduation beyond for the lower latitudes. The gnomonic projection affords a good representation of the polar regions, but all places in latitudes lower than 60° appear greatly distorted. The gnomonic projection enlarges the representations of places at a distance from the center of projection beyond their proportionate true dimensions; and the orthographic, on the contrary, unduly contracts them; while both are adapted for representing best the countries at only a moderate distance from the center of projection. To delineate the Stereographic Projection of the Circles of the Terrestrial Sphere upon the Horizon of any place.—With radius determined upon describe a circle for the primitive, and draw its vertical and horizontal diameters, N s and w E, which will be the projections of the meridian of the place and of the prime vertical respectively. From the center c set off upon the radius c s, produced, if necessary, the distance c 4, equal to the tangent of the latitude of the place reduced to the radius of the primitive; and with center A and distance a w or A E describe the circle w N E, which will be the projection of the meridian at right angles to N s, the meridian of the place; and, consequently, N will be the projection of the pole. Through 4 draw the right line A в at right angles to Ac, and another line A D making any convenient angle with ▲ B, and, setting off 4 B equal to the radius of the primitive, and 4 D equal to the sine of the colatitude, taken from the line of sines, join B D. Now take from the line of tangents the angles which the other meridians to be drawn are to make with the meridian w N E, or the complements of the angles which they are to make with N s, and set them off both ways from a upon the line A D; through each of the divisions L, thus found, draw Lo, parallels to B D, and we have at o the centers of the circles for describing the meridians* With centers o and distances o N, describe the The distance 40= r cot. L cos. where represents the radius of the pri meridians, and number them 0 to 180, both ways, from the first meridian, for the longitudes. For a parallel through any given latitude, take the difference of the complement of the given latitude and of the colatitude of the place from the line of semitangents, and, having reduced it to the radius of the primitive, set it off at r from c towards N for latitudes greater than the latitude of the place, and from c towards s for latitudes less than the latitude of the place: - again, take the sum of the complement of the given latitude and of the colatitude of the place from the line of semitangents, and set it off at s from c upon c N produced: then the circle described upon r s* as diameter will be the parallel required. Draw these parallels for intervals of latitude equal to the angles made by two successive meridians, and number them 90 to 0 from the pole N for the north latitudes, and again in creasing from 0 on the other side of the equator for the south mitive, the latitude of the place, and L the angle at which the meridian is inclined to the meridian of the place. (c+8) where c * Diameter of parallel =rtan (cd) + rtan. =colatitude of place, and d colatitude of parallel. = latitudes, if the place be in north latitude-or the converse, if the place be in south latitude. The practical application of the preceding methods of projection is usually confined to the representation of an entire hemisphere, or at least of a considerable portion of a sphere; but for laying down smaller portions of the sphere the method of development may be advantageously adopted. In this method the portion of the sphere to be represented is considered as coincident with a portion of a cone, touching the sphere in a circle which is the middle parallel of latitude of the country to be represented, and this portion of the cone when developed forms a portion of a sector of a circle. To lay down the meridians and parallels of latitude for this development. 1. Take a straight line, B C A, for the middle meridian of the intended map, and divide it into equal parts, to represent degrees and minutes of latitude according to the scale determined for the map. upon 2. From one of these divisions, a, which is conveniently situated to form the center of the map, set off from A to c the cotangent of the middle latitude, reduced to a radius equal to 57.3 of the divisions previously marked off as degrees, or to 3438 of those marked off as minutes. 3. With c as a center and radius c A, describe the arc D A E for the middle parallel of latitude, and divide it into equal parts to represent degrees and minutes of longitude, the lengths of these parts having, to the lengths. of the parts previously set the meridian off on degrees and latitude, the for minutes of ratio cosine of middle latitude : radius. 4. With c as center, describe concentric arcs, through the divisions on c E, for the parallels of latitude; and draw straight lines, radiating from c, through the divisions on D A E for the meridians. In our figure the middle latitude is 55°; A B is equal to the length of 57.3°, the radius of the sphere, or D. B |