to it. Lay off, at pleasure, CD for the distance of the picture, if C be the centre of the picture. Draw a line from D, touching the end B of the line to be divided: draw DBE, cutting the groundline in E. Then AE represents the actual dimensions of the line AB, which is seen in perspective. (Here it may be observed, that this gives a rule also for finding the real length of any line which tends to a vanishing-point.) Divide AE into the same number of equal parts into which you proposed to divide the given line AB; as A1, 12, 23, &c. Then from these different divisions draw lines to D, cutting the line AB in a, b, c, d, &c., which will represent the required number of equal parts, but diminishing in size as they are farther removed from the eye. If it be wished to divide the line AB into any number of unequal parts, or to lay off doors, windows, &c. upon it, the line AE, found as before, must be divided in the required proportion; and lines drawn from those to D will give the required divisions on AB, from which perpendiculars may be drawn for the doors, windows, &c.

To draw a circle in perspective.

The perspective representation of every circle is a regular ellipsis, when the eye is without the circle, which may be demonstrated, by considering that the rays from the circumference of the circle to the eye, form an oblique cone. But it is well known to those who are acquainted with conic sections, that every section of a cone, whether right or oblique is a true ellipsis, except in one case only, which is, when the section is taken sub-contrary to its base, a situation which happens so rarely in. drawings, that it may be disregarded altogether, and the section of a cone, or the perspective of a circle, in all cases considered as a perfect ellipsis.

The most correct and easy method of drawing an ellipsis is to find the transverse and conjugate axes, and then to complete the curve by a trammel, or by hand. But as it is very difficult to find the transverse and conjugate axes of the ellipses which are the perspective representations of circles, recourse is generally had to another method of obtaining the curve. The circle is circumscribed by a square, as K L M N, in Fig. 3., and the diagonals and the lines across the centre, and parallel to the sides, are drawn; also the lines, al, cd, are drawn parallel to the sides, through the points where the circle is cut by the diagonals. This square, with all these lines drawn across it, is now put in perspective as follows: Draw A B for the horizontal line, and fix B for the centre of the picture, and A B for the distance of the picture. Make DC equal to the width of the square, and draw CB, DB; draw C A to the distance-point A, cutting off D G equal to the depth of the square; then draw GF, parallel to DC, which completes the perspective of the square; also draw the diagonal DF. Take now the distances Ma, c N; and transfer them to Da, o C; from these points and o draw lines to the vanishing point B, cutting the diagonals of the square. The points in this reticulated square in perspective, which correspond to those in the square KL MN, where the circle passes through, must now be observed, and a curve traced through them with a steady hand: it will be the perspective required. Even in this process, it is of considerable use to know that the curve you are tracing is a regular ellipsis; for though you cannot easily ascertain the axes exactly, yet you may very nearly; and the eye very soon discovers whether the curve which has been drawn, be that of a regular ellipsis or not.

Upon the same principle the row of arches (Fig. 4.) is drawn. The width of the arches and piers is obtained in the same manner as was shown in Fig. 2.; viz. by laying their dimensions upon the ground-line AB, and drawing lines to the distance-point. The curves of the arches are then found, by drawing the lines which correspond to those in half the square, Fig. 3., in the same manner as described above for the circle.

Fig.5. shows the appearance of circles drawn upon a cylinder, when HI is the horizontal line. The circle drawn on the cylinder at that place is seen exactly edgeways, and appears only as a straight line; that next above it is seen a little underneath; the next still more; and so on, as they rise higher, appearing like so many ellipses of the same transverse diameter, but whose conjugate diameters continually increase in length as they rise above the horizontal line. On the contrary, you see the under sides of the circles drawn below the horizontal lines; but they observe the same law, being so many ellipses whose conjugate diameters vary in the same proportion. A little reflection on this simple example will enable those who draw to avoid many ridiculous mistakes which are sometimes committed, such as showing the two ends of a cask, or the top and bottom of a cylinder, at the same time.

Pl. 7. Fig. 1. shows the method of drawing a building, or other object, in oblique perspective. A B is the horizontal line, and CD the ground-line parallel to it as before. Here neither of the sides of the house is parallel to the picture, but each goes to its respective vanishing point. Having fixed on the nearest corner E, draw EB, at pleasure, for one side, and choose any point F for the

centre of the picture; then, to find the other side, lay off F G equal to the distance of the picture, which, as before, depends upon taste only; draw BG and GA perpendicular to BG, cutting the horizontal line in A, the other vanishing point. Draw now E A for the other side. To cut off the several widths of the two sides of the house, which as yet are only drawn to an indefinite extent, two distance-points must be laid down, viz. one for each vanishing point. To do this, extend the compass from B to G, and lay the distance taken in it from B to H, which will give H for the distance-point of B, and which is to cut off all the divisions on the side E B. Also extend the compasses from AG, and lay down AI. I is the distance-point of A, and is used for transferring all divisions upon the side E A, from the ground-line CE. These points and lines being adjusted, the process is not much different from parallel perspective; only here, equal divisions on each side of the building, as doors, windows, diminish as they recede, in the same way as on the side BE FC, Pl. 6. Fig. 1. Take the real length of the side E L, from the same scale used for laying down the horizontal line, and lay it down on the ground-line from E to C, and draw C I, cutting off E L for the perspective length of the building. For the other side of the house, lay its width down in the same manner, from E to D, and draw DH, cutting off E N for the perspective width. Raise the perpendiculars EM, L K, and NO, for the three angles of the house. Lay the height of the building upon the corner that comes to the groundline, as EM, and draw MK and MO to their several vanishing points. Also lay all the heights of the doors and windows, and other divisons, upon EM, and draw them to the vanishing points A and B.

To lay down the widths of the doors and windows, put their actual widths upon CE, and draw from them to the distance-point I, which cuts off all divisions upon the side L E, and then raise the perpendiculars. The gable-end is found exactly in the same manner as was described in Pl. 6. only taking care to use the proper distance-point H. The manner of finding the width of the chimney is different. Lay off b a for the height of the chimney above the top of the gable, and draw a c parallel to the horizontal line; then put a c equal to the actual thickness of the chimney, and draw ad to the vanishing point A; draw also cd to the distancepoint I, cutting off a d in d: then having drawn eƒ from the nearest corner of the chimney, which was found as in Pl. 6. Fig. 1. Draw df to the vanishing point B, cutting off ef for the exact perspective width.

Fig. 2. represents the method of finding the perspective of a circle in oblique perspective. AB is the horizontal line, C the centre of the picture, and D, E, the distance-points. The process is exactly the same as that just described, the several divisions of the reticulated square in Pl. 6. Fig. 3. being laid upon the ground-line FG, and from these lines are drawn to the distance-points. The perspective of the square is then drawn with all the lines across it, and the curve traced through the different points.

By drawing these examples frequently over, to large scale, and reflecting upon them with attention, the student will become familiar with their use; and as they include the cases which most frequently occur, he will find great benefit from the knowledge of them.