PROJECTION. Projection of the Sphere in Astronomy and Geography signifies in general a perspective representation of the surface of the sphere on a plane; and it is of various kinds according to the different positions of the eye and the plane of projection.

As the principles of the different projections commonly employed in the construction of maps and charts, and the method of applying them, have already been explained in the article Geography, (chap, iv.,) we shall merely give a general view of the subject in this place, referring the reader to the article just cited,for thedetailsand demonstrations.

The simplest of all the projections is the Orthographic. Considered as a perspective delineation of the surface of the sphere, this supposes the eye to be placed at an infinite distance, so that all the rays coming from the different parts of the surface are parallel; and the plane of projection, or plane upon which the representation is made, is assumed to be perpendicular to the direction of the visual rays, and to pass through the centre of the sphere. The orthographic projection of any point is therefore the intersection of a perpendicular from that point with the plane of projection. When the object is to represent only a small portion of the surface of the sphere, situated near the pole of projection, or point in which the straight line from the eye to the centre of the sphere intersects the surface, this projection answers every purpose; but it is not so well adapted to the delineation of a whole hemisphere, or even a considerable portion of it; for the distances from the pole being represented by the sines of the arcs by which they are measured on the sphere, and as the sines of large arcs increase in a much smaller ratio than the arcs themselves, the map is necessarily crowded and distorted towards the extremities, and gives a very imperfect representation of the objects delineated.

The principal properties of the orthographic projection are, 1st, That all the great circles of the sphere, the planes of which pass through the eye, are represented by straight lines: 2d, That all circles, (they can only be small circles) the planes of which are perpendicular to the line of sight, are represented by circles: 3d. That all other circles, whether large or small, being seen obliquely, are represented by ellipses. This method of projection was known to the ancient astronomers, and employed by them in the construction of the Analemma, a sort of instrument for the graphical solution of various problems of the sphere, on which Ptolemy wrote a treatise, which has been preserved through the medium of an Arabic version, and of which a Latin translation was published by Commandine at Rome in 1562. The orthographic projection is conveniently used for various astronomical purposes, such as the geometrical delineation of eclipses, transits, and occultations; and it has numerous applications in the arts, the plans and sections by which artificers execute their constructions, being merely orthographical projections of the things to be constructed.

In the Stereographic projection the eye is supposed to be situated at the surface of the sphere, and the plane of projection is the plane of the great circle, at the pole of which the eye is placed. The hemisphere which is projected, is that which is opposite to the eye, and consequently the sphere must be supposed transparent, in order that the different points of its surface may be visible. According to the testimony of two ancient authors, Synesius and Proclus, this projection was invented by Hipparchus. It was called the Planisphere, on which also there is a treatise by Ptolemy, preserved through the Arabic. The term Stereo-graphia (derived from στepeos∙, solid, because it results from the intersection of two solids, the sphere and the cone) was, according to Delambre, first applied to it by the Jesuit Aguilon in a Treatise on Optics, published at Antwerp in 1613. The astrolabe, an instrument formerly much in use for the graphical solution of astronomical problems connected with the diurnal motion, was constituted on this projection.

The two principal properties of the stereographic projection are, 1st, That all circles on the sphere which do not pass through the eye, whether great or small, are projected into circles; and, 2d, That the circles of projection intersect each other under the same angles as their corresponding circles on the sphere. The first of these properties is of great importance, by reason of the facility which it affords for the construction of maps. It is demonstrated in the planisphere of Ptolemy for all the particular cases that occur in the projections there described; and although the general proposition is not formally mentioned, it can scarcely be supposed to have been unknown to him, as his demonstrations are easily rendered applicable to every case. It is, indeed, an almost obvious consequence of a property demonstrated by Apollonius relative to the subcontrary sections of the oblique cone. The demonstration of Ptolemy for the particular cases is, however, entirely independent of the theorem of Apollonius. Delambre (Astronomie Ancienne, tom. ii. p. 456) states, that the oldest work in which he has met with the enunciation of the general property is the Planisphere of Jordanus, published with some other astronomical tracts, at Toulouse in 1544. It may be remarked, that Jordanus, instead of projecting the hemisphere on the plane of the equator, chooses for the plane of projection,. the plane touching the sphere at the north pole. The property of the equality of angles, which is also very curious and important, appears to be of modern discovert, [7:18:657]but the author of it is not certainly known. Delambre (Ms- tronomie, tom. iii p. 674) says, he has searched for it in vain in the large treatise of Clavius, in that of Stoflerinus, and in Bion; and (Histoire de P Astronomie au Dix-Huitième Siècle, p. 88,) that the oldest demonstration of it known to him is in Leadbetteds Astronomy, London, 1728. It had, however, been previously demonstrated by Dr. Halley in 1696, in No. 219 of the Philosophical Transactions, in which he attributes the first discovery of the proposition to Demoivre or Hook.

As the surface of a sphere is equal to four times the area of one of its great circles, it follows that in the stereographic projection on the plane of a great circle, the surface of the hemisphere is reduced to one-half of its original dimensions. This w ould be no defect, if the reduction were uniform; which, however, is far from being the case; for at the centre, the linear dimensions are reduced to one-half, and consequently the superficial to one-fourth; and the reduction becomes less and less towards the extremities of the map. Any portion of the upper hemisphere projected in this way would be enlarged. Hence there is little resemblance between a given portion of the spherical surface and its representation, which is the great defect of the projection. In maps of the stars, for which it is most frequently employed, there is also an inconveniency in the difficulty of finding the places of particular stars, for three or more objects, which in the apparent heavens are situated in the same straight line, are placed in the map on the circumference of a circle which is not represented. Nevertheless, the projection answers conveniently enough all the purposes, for which a map is required by the astronomer.

In the Gnomonic or Central projection, the eye is supposed to be placed at the centre of the sphere, and the plane of projection is a plane touching the sphere at any point assumed at pleasure. The point of contact is called the principal point ; the projection of all other points are at the extremities of the tangents of the arcs intercepted between them and the principal point, and the tangents make with each other angles equal to those in which their arcs intersect at the pole. This projection is also described by Ptolemy in the Analemma, (Delambre, Astr. Ancienne, tom. ii. p. 486); but as the Greeks had no idea of the trigonometrical tangents, the polar distances arc in that work expressed by the ratios of the sines to the cosines of the arcs. As the tangents of arcs exceeding 45° increase very rapidly, and become infinite for arcs of 90°, the projection cannot be employed for a whole hemisphere. The celestial maps published by the Society for the Diffusion of Useful Knowledge, are projected in this manner, and the surface embraced is one-sixth of the whole sphere, or the sphere is projected on the six sides of its circumscribing cube. For a map of the stars this projection possesses some advantage over the stereographic. All stars which appear in the same straight line in the heavens, are projected in the same straight line, so that their places can be readily found on the map; and from the position of the eye at the centre of the sphere, the appearance of the heavens is better preserved. The map is necessarily distorted in some degree near the extremities and corners, but not to so great an extent as to render it a bad representation. Besides, the distortion is equal for lines equally distant from the centre, and there is no angular distortion with respect to lines which pass through the centre of the map. This projection is particularly applicable to the construction of sun-dials and gnomons, whence its name.

The three kinds of projection now described, are chiefly employed in the construction of celestial maps; but for the purpose of representing portions of the earth’s surface, the Globular projection is frequently had recourse to. This may be considered as a modification of the stereographic, or in a manner intermediate between the stereographic and orthographic. In the orthographic, the proportion of the parts of the projection to the parts on the sphere which they represent, decreases from the centre to the circumference of the projection; in the stereographic the projection increases; and as in the first case the eye is supposed at an infinite distance, and in the second, on the surface of the sphere, there must be some position at a finite distance from the surface, in which if the eye is placed, the proportion between the parts of the surface of the sphere and their representations, will be the same or nearly so, whether those parts be taken at the centre or circumference of the map. It is easy to shew that the height above the sphere at which this takes place is equal to the sine of 45 degrees, and therefore very nearly 71-hundredths of the radius. This projection was first proposed by La Hire, in the Memoirs of the Academy of Sciences of Paris for 1701. It is employed in many cases, on account of the small change of configuration; but by the removal of the eye from the surface the characteristic advantages of the stereographic projection arc lost.

Projections of the circles of the sphere on a plane, are easily found by the methods of the descriptive Geometry, by spherical Trigonometry, or common Algebra. Whatever be the situation of the eye, every circle on the sphere, unless its plane pass through the eye, forms the base of a cone, of which the eye is the apex; and the intersection of this cone with the plane of projection, gives the curve into which the circle is projected. Hence the curve is necessarily a conic section and its equation cannot exceed the second degree. On forming the general equation of the curve, and assigning a determinate position to the eye and the plane of projection, all the properties of the different kinds of projection, which are merely particular cases of the same general problem, are readily deduced.

Although it is convenient to employ the principles of perspective in representing the lines and circles of the sphere on a plane surface, there is nothing in the nature of the thing itself, (unless the map is to be regarded as a picture) which renders such a mode of proceeding necessary. The meridians and parallels may be represented on the map by any lines whatever, and the map will still be accurate, if the different points are laid down so as to have respectively the same relative positions with respect to those lines as the points represented have to the corresponding circles of latitude and longitude; for in thjs case, any point on the map can be referred to its proper place on the sphere, which is the essential condition of a correct map. In order, however, to render the problem determinate, the lines chosen to represent the meridians and parallels of the sphere must be such as satisfy certain conditions capable of being expressed by an algebraic equation. But these conditions may have no reference to perspective. Mercator’s Chart is an example. In this chart the condition which determines the relative situations of the meridians and parallels is, that the rhumbs shall be straight lines, and make with each other the same angles as on the sphere. This condition determines the well known property of the chart, namely, that the degrees of longitude are all equal, while the degrees of latitude vary in the inverse ratio of the sines of their polar distances.

The theory of the construction of maps on this general principle, was first considered by Lambert, who, in the third volume of his Beitrage zum Gebrauche der Mathematik, &c., published at Berlin in 1772, undertook the solution of the problem to determine the nature and situation of the lines on the map representing the meridians and parallels, where the condition to be fulfilled is, that the angles formed by lines on the map shall be equal to the corresponding angles on the sphere. This condition is important; for it follows from the equality of the angles that any indefinitely small portion of the surface of the sphere, and its projection[7:18:658] on the map, are similar figures; and consequently that the representations of such parts differ from the original only in respect of magnitude. The same problem was considered by Euler in the Petersburg Commentaries for 1777, but neither of these illustrious geometers prosecuted the subject farther than to shew that the known properties of the stereographic projection, and Mercator’s Chart, are comprehended in their general solution.

In the volume of the Berlin Memoirs for 1779, there are two memoirs by Lagrange on the same subject. In the first he gave a solution of the problem of Lambert, of which he shewed the stereographic projection to be only a particular case; and in the second, he undertook the more difficult task of determining the form which the arbitrary functions that enter into the general solution must have in order that the lines representing the meridians and parallels may be of a given nature.

The general problem of projection has been more recently considered by the celebrated Professor Gauss of Göttingen, in a memoir written in answer to a prize question proposed by the Academy of Sciences of Copenhagen. In this memoir the author, instead of confining himself to the particular case of the representation of a spherical surface on a plane, undertook the solution of the general problem; namely, to represent the parts of any given surface on any other given surface, subject to the condition that the differential elements of the first surface shall be similar to their representations on the second. Having determined the differential expressions of the co-ordinates of any point on the map, in terms of the co-ordinates of the corresponding point of the primitive surface, he applies the general solution to the particular cases of a plane surface projected on another plane, the surface of a cone on a plane, of a sphere and spheroid on a plane, and lastly, of an ellipsoid of revolution on the surface of a sphere. Gauss’s memoir is published in No. 3 of Schumacher’s Abhandlungen, and a translation of it in the Philosophical Magazine for August and September 1828. Considered in this general view, the theory of projection becomes an interesting and very difficult mathematical speculation; but it is one of which, it must be confessed, the results are of no great practical use.