the line h'p' will be the perspective width of the pier ; draw k' D; and from k", k" D; from g" the intersection of the curve of the arch by the perpendicular to g', draw g" D, the intersection with " D' will be one point in the curve of the arch on the opposite side of the bridge: in the same way, from any point in the nearer arc draw lines to D, and the intersection with lines in the same planes on the opposite side of the bridge, will furnish points for the further arch: all below the first only will be visible to the spectator.

To Draw in Parallel Perspective the Interior of a Room.-Plate V. We propose to construct this by scale without laying down the plan. Draw the horizon line D V D', and the base M M', making D and D' the point of distance. Let the room be 20 feet wide, 14 feet high, and 12 feet deep; on the base M M', lay off the rectangle of the section in our figure on a scale of 8 feet to the inch, 20 feet x 14 feet. From the four corners draw lines to the centre of view V; from S' lay off to the right or left on M M' 12 feet, and through this point draw lines to D'or D as the case may be: through the point of intersection a' of this line with S' V draw a line parallel to M M'; at the intersections of this line with M V and M' V erect a perpendicular, cutting the vanishing lines of the upper angle of the room at d and e; connect d e and the perspective of the room is complete. To draw the aperture for a door or window on the side, measure off from S' the distance of the near side from the plane of the picture, and in addition thereto the width of the aperture; from these two points draw lines to the proper point of distance, and at their intersection with S' V, draw parallels to M M', cutting the lower angles of the room, and erect perpendiculars, the height of which will be determined by a line drawn from ƒ, the height of the window above the floor measured on MD. Should the window be recessed, the farther jamb will be visible; extend the farther parallel to M M', and cut it by a line g V. Mg being the depth of the recess, the rest of the construction may be easily understood by inspection of the figure. At the extremity of the apartment a door is represented half open, hence as the plane of the door is at right angles to the plane of the picture, the top and bottom lines will meet in the point of view; if the door were open at an angle of 45°, these lines would meet in the points of distance; if at any other angle, the vanishing points would have to be determined by constructing a plan, drawing a line parallel to the side of the door through the station point, and projecting it upon the horizon line. The chair in the middle of the room is placed diagonally, and the table parallel to the plane of the picture; their projection is simple.

To Draw in Perspective a Flight of Stairs.-Plate VI. Lay off the

base line, horizon, centre of view, and point of distance of the picture, construct the solid a b c d, ef gh, containing the stairs, and in the required position in the plane of the picture, divide the rise a c into equal parts according to the number of stairs, four for instance; divide perspectively the line ab into the same number of parts; at the points of division of this latter erect perpendiculars, and through the former draw lines to the centre of view; one will form the rise and the other the tread of the steps. From the top of the first step to the top of the upper continue a line a d, till it meets the perpendicular S′ V prolonged in v; this line will be the inclination or pitch of the stair; if through the top of the step at the other extremity a similar line be drawn, it will meet the central perpendicular at the same point v, and will define the length of the lines of nosing of the steps, and the other lines may be completed. As the pitch lines of both sides of the stairs meet the central vertical in the same point, in like mannerv will be the vanishing point of all lines having a similar inclination to the plane of the picture. The projection of the other flight of stairs will be easily understood from the lines of construction perpendicular to the base line or parallel thereto, lying in planes.

To Find the Reflection of Objects in the Water.-Plate VI. Let B be a cube suspended above the water; we find the reflection of the point a, but letting fall a perpendicular from it, and setting off the distance a' w below the plane of the water equal to the line a w above this line; the line wf' will also be equal to the line wf; find in the same way the points b' and e', through these points construct perspectively a cube in this lower plane, and we have the reflection of the cube above.

To find the reflection of the square pillar D removed from the shore : suppose the plane of the water extended beneath the pillar, and proceed as in the previous example.

It will be observed that those lines of an object which meet in the centre of view V, in the original; their corresponding reflected lines will converge to the same point. If the originals converge to the points of distance, the reflected ones will do the same. To find the reflection of any inclined line, find the reflection of the rectangle of which it is the diagonal, if the plane of the rectangle is perpendicular to the plane of the picture; if the line is inclined in both directions enclose it in a parallelopided and project the reflection of the solid.

To find the Perspective Projection of Shadows.-Plate VII. Let the construction points and lines of the picture be plotted. Let A be the perspective projection of a cube placed against another block, of which the face is parallel to the plane of the picture: to find the shadow upon the