Page:Encyclopædia Britannica, Ninth Edition, v. 19.djvu/830

806 P R P R O perspective of a circle may be any conic, not necessarily an ellipse. Similarly the perspective of the shadow of a circle on a plane is some conic. 57. A few words must l>e said about the determination of shadows in perspective. The theory of their construction is very simple. We have given, say, a figure and a point L as source of light. "We join the point L to any point of which we want to find the shadow and produce this line till it cuts the surface on which the shadow falls. These constructions must in many cases first be performed in plan and elevation, and then the point in the shadow has to be found in perspective. The constructions are different according as we take as the source of light a finite point (say, the flame of a lamp), or the sun, which we may suppose to be at an infinite distance. If, for instance, in fig. 23, A is a source of light, EIIGF a vertical wall, and C a point whose shadow has to be determined, then the shadow must lie on the line joining A to C. To see where this ray meets the tloor we draw through the source of light and the point C a vertical plane. This will cut the floor in a line which contains the feet A lf C, of the perpendiculars drawn from the points A, C to the floor, or the plans of these points. At C , where the line AjC, cuts AC, will be the shadow of C on the floor. If the wall EHGF prevents the shadow from falling on the floor, we determine the intersection K of the line AjCj with the base EF of the wall and draw a vertical through it, this gives the intersection of the wall with the vertical plane through A and C. Where it cuts AC is the shadow C" of C on the wall. If the shadow of a screen CDD^ has to be found we find the shadow D of D which falls on the floor ; then DjD is the shadow of DjD and D C is the shadow on the floor of the line DC, The shadow of D X D, however, is intercepted by the wall at L. Here then the wall takes up the shadow, which must extend to D" as the shadow of a line on a plane is a line. Thus the shadow of the screen is found in the shaded part in the figure. 58. If the shadows are due to the sun, we have to find first the perspective of the sun, that is, the vanishing point of its rays. This will always be a point in the picture plane ; but we have to distin guish between the cases where the sun is in the front of the picture, and so behind the spectator, or behind the picture plane, and so in front of the spectator. In the second case only does the vanishing point of the rays of the sun actually represent the sun itself. It will be a point above the horizon. In the other case the vanishing point of the rays will lie below the horizon. It is the point where a ray of the sun through the centre of sight S cuts the picture plane, or it will be the shadow of the eye on the picture. In either case the ray of the sun through any point is the line joining the per spective of that point to the vanishing point of the sun s rays. But in the one case the shadow falls away from the vanishing point, in the other it falls towards it. The direction of the sun s rays may be given by the plan and elevation of one ray. For the construction of the shadow of points it is convenient first to draw a perpendicular from the point to the ground and to find its shadow on the ground. But the shadows of verticals from a point at infinity will be parallel ; hence they have in perspective a vanishing point Lj in the horizon. To find this point, we draw that vertical plane through the eye which contains a ray of the sun. This cuts the horizon in the required point Lj and the picture plane in a vertical line which contains the vanishing point of the sun s rays themselves. Let then (fig. 24) L be the vanishing Fig. 24. point of the sun s rays, Lj be that of their projection in a horizon tal plane, and let it be required to find the shadow of the vertical column AH. We draw ALi and EL ; they meet at E , which is the shadow of E. Similarly we find the shadows of F, G, H. Then E F G H will be the shadow of the quadrilateral EFGH. For the shadow of the column itself we join E to A, &c., but only piitrk the outlines ; F B, the shadow of BF, does not appear as such in the figure. If the shadow of E has to be found when falling on any other surface we use the vertical plane through E, determine its inter section with the surface, and find the point where this intersection is cut by the line EL. This will be the required shadow of E. 59. If the picture is not to be drawn on a vertical but on an other plane say, the ceiling of a room the rules given have to bo slightly modified. The general principles will remain true. But if the picture is to be on a curved surface the constructions become somewhat more complicated. In the most general case conceivable it would be necessary to have a representation in plan and eleva tion of the figure required and of the surface on which the pro jection has to be made. A number of points might also be found by calculation, using coordinate geometry. But into this we do not enter. As an example we take the case of a panorama, where the surface is a vertical cylinder of revolution, the eye being in the axis. The ray projecting a point A cuts the cylinder in two points on opposite sides of the eye, hence geometrically speaking every point has two projections ; of these only the one lying on the half ray from the eye to the point can be used in the picture. But the other has sometimes to be used in constructions, as the projection of a line has to pass through both. Parallel lines have two vanish ing points which are found by drawing a line of the given direction through the eye ; it cuts the cylinder in the vanishing points required. This operation may be performed by drawing on the ground the plan of the ray through the foot of the axis, and through the point where it cuts the cylinder a vertical, on which the point required must lie. Its height above is easily found by making a drawing of a vertical section on a reduced scale. Parallel planes have in the same manner a vanishing curve. This will be for horizontal planes a horizontal circle of the height of the eye above the ground. For vertical planes it will be a pair of generators of the cylinder. For other planes the vanishing curves will be ellipses having their centre at the eye. The projections of vertical lines will be vertical lines on the cylinder. Of all other lines they will be ellipses with the centre at the eye. If the cylinder be developed into a plane, then these ellipses will be changed into curves of sines. Parallel lines are thus represented by curves of sines which have two points in common. There is no difficulty in making all the constructions on a small scale on the drawing board and then transferring them to the cylinder. 60. A variety of instruments have been proposed to facilitate perspective drawings. If the problem is to make a drawing from nature then a camera obscura or, better, Wollaston s camera lucida may be used. Other instruments are made for the construction of perspective drawings. It will often happen that the vanishing point of some direction which would be very useful in the construc tion falls at a great distance off the paper, and various methods have been proposed of drawing lines through such a point. For some of these see Stanley s Descriptive Treatise on Mathematical Drawing Instruments. Literature. Descriptive geometry dates from Monge, whose Geometrie Descrip tive appeared in 1800. Before his time plans and elevations, especially of build ings, had been in use, and rules had been developed to determine by construction from drawings the shapes of the stones required in buildings, especially in vaults and arches. These rules were reduced to a consistent method by Monge. Perspective was investigated much earlier, as painters felt the need of it. It beginnings date from the time of the Greek mathematicians, but its modern development from the time of the Renaissance, when the first books on the subject appeared in Italy. Albrecht Purer also published a treatise on it and constructed a machine for making perspective drawings of objects. Of later writers we men tion in the 17th century Desargues, and in the 18th Dr Brook Taylor, whose Linear Perspective appeared first in 1715 and New Principles of Linear Perspectice in 1719. At present perspective is generally treated as a special case of projec tion, and included in books on descriptive geometry. For the literature of projection in general, we refer to the list of books given under GEOMETRY, vol. x. p. 407. For descriptive geometry and peispective, see Monge, Gi omftrie Descriptive ; Leroy, Traite de Ge omArie Descriptive ; Fiedler, Darstellende Geometrie; Goumerle, Traite de Perspective; Mannheim, Ge ometrie Descriptive (1880) and Elements de la Geometrie Descriptive (188 J); J. Woolley, Descriptive Geometry (185C), which is based on Leroy s work, and is the only scientific publication on the subject in England. A number of other publications with titles such as " Practical Geometry" and "Geometrical Drawing" contain more or less full explanations of the methods of descriptive geometry. These arc based generally on Euclidian as opposed to projective geometry, and are there fore in their theoretical part more or less unsatisfactory. We may mention Angel, Practical Plane Geometry and Projection. (0. H.) PROJECTION OF THE SPHERE. See GEOGRAPHY. PROME, a district in Pegu division, British Burmah, India, between 18 30 and 19 15 N. lat., and 94 40 and 96 E. long., containing an area of 2887 square miles. It occupies the whole breadth of the valley of the Irawadi, between Thayet district on the north and Hen- zada and Tharawadi districts on the south, and originally extended as far as the frontier of the province of Burmah, but in 1870 Thayet was formed into an independent juris diction. There are two mountain ranges in Prome, which form respectively the eastern and western boundaries. The Arakan Yoma extends along the whole of the western side, and that portion of the district lying on the right

bank of the Irawadi is broken up by thickly wooded spurs