Let a b, Fig-. 28., be the curve. Enclose it in a rectangle, c d e f. Fix the position of the point c or d, and draw the rectangle. (Prob. VIII. Cor. I.)1

Fig. 28.

Let c D E F, Fig. 29., be the rectangle so drawn.

If an extremity of the curve, as A, is in a side of the rectangle, divide the side c E, Fig. 29., so that A c shall be (in perspective ratio) to A e as a c is to a e in Fig. 28. (Prob. V. Cor. II.)

Similarly determine the points of contact of the curve and rectangle e, f, g.

1 Or if the curve is in a vertical plane, Coroll. to Problem IX. As a rectangle may be drawn in any position round any given curve, its position with respect to the curve will in either case be regulated by convenience. See the Exercises on this Prob em, in the Appendix, p. 289.

Problem XI - Corollary

If an extremity of the curve, as B, is not in a side of the rectangle, let fall the perpendiculars B a, b b on the rectangle sides. Determine the correspondent points a and b in Fig. 29., as you have already determined a, B, e, and f.

From b, Fig. 29., draw b B paralled to C D1, and from a draw a b to the vanishing-point of D F, cutting each other in B. Then B is the extremity of the curve.

Determine any other important point in the curve, as p, in the same way, by letting fall p q and p r on the rectangle sides.

Any number of points in the curve may be thus determined, and the curve drawn through the series; in most cases, three or four will be enough. Practically, complicated curves may be better drawn in perspective by an experienced eye than by rule, as the fixing of the various points in haste involves too many chances of error; but it is well to draw a good many by rule first, in order to give the eye its experience.2

## Corollary

If the curve required be a circle, Fig. 30., the rectangle which encloses it will become a square, and the curve will have four points of contact, A B c d, in the middle of the sides of the square.

Draw the square, and as a square may be drawn

Fig. 29.

1 Or to its vanishing-point, if c D has one.

2 Of course, by dividing the original rectangle into any number of equal rectangles, and dividing the perspective rectangle similarly, the curve may be approximately drawn without any trouble; but, when accuracy is required, the points should be fixed, as in the problem.About a circle in any position, draw it with its nearest side, E G, parallel to the sight-line.

Let e f, Fig. 31., be the square so drawn.

Draw its diagonals E F, G h; and through the centre of the square (determined by their intersection) draw A B to the vanishing-point of G F, and c D parallel to e g. Then the points a b c d are the four points of the circle's contact.

On e g describe a half square, e l; draw the semicircle k A l; and from its centre, r, the diagonals r e, r g, cutting the circle in x, y.

Fig. 30.

Fig. 31.

From the points x, y, where the circle cuts the diagonals, raise perpendiculars, p x, q y, to e g.

From p and Q draw p p', Q Q', to the vanishing-point of G F, cutting the diagonals in m, n, and 0, p.

Then m, n, o, p are four other points in the circle.

Through these eight points the circle may be drawn by the hand accurately enough for general purpos s; but any number of points required may, of course, be determined, as in Problem XI.

The distance e p is approximately one seventh of E g, and may be assumed to be so in quick practice, as the error involved is not greater than would be incurred in the hasty operation of drawing the circle and diagonals.

It may frequently happen that, in consequence of associated constructions, it may be inconvenient to draw E G parallel to the sight-line, the square being perhaps first constructed in some oblique direction. In such cases, Q G and E P must be determined in perspective ratio by the dividing-point, the line E G being used as a measuring-line.

[0bs. In drawing Fig. 31. the station-point has been taken much nearer the paper than is usually advisable, in order to show the character of the curve in a very distinct form.

If the student turns the book so that E G may be vertical, Fig. 31. will represent the construction for drawing a circle in a vertical plane, the sight-line being then of course parallel to G L; and the semicircles A D B, A c B, on each side of the diameter A B, will represent ordinary semicircular arches seen in perspective. In that case, if the book be held so that the line E H is the top of the square, the upper semicircle will represent a semicircular arch, above the eye, drawn in perspective. But if the book be held so hat the line G F is the top of the square, the upper se. icircle will represent a semicircular arch, below the eye, drawn in per pective.

If the book be turned upside down, the figure will represent a circle drawn on the ceiling, or any other horizontal plane above the eye; and the construction is, of course, accurate in every case.]

## Problem XI

It is seldom that any complicated curve, except occasionally a spiral, needs to be drawn in perspective; but the student will do well to practise for some time any fantastic

Fig. 63.

shapes which he can find drawn on flat surfaces as on wall-papers, carpets, c, in order to accustom himself to the strange and great changes which perspective causes in them.

The curves most required in architectural drawing, after the circle, are those of pointed arches; in which, however, all that will be generally needed is to fix the apex, and two points in the sides. Thus if we have to draw a range of pointed arches, such as A P B, Fig. 63., draw the measured arch to its sight-magnitude first neatly in a rectangle, A B C D; then draw the diagonals A D and B C; where they cut the curve draw a horizontal line (as at the level E in the figure), and carry it along the range to the vanishing-point, fixing the points where the arches cut their diagonals all along. If the arch is cusped, a line should be drawn at F to mark the height of the cusps, and verticals raised at G and H, to determine the interval between them.

Fig. 64.

Any other points may be similarly determined, but these will usually be enough. Figure 63. shows the perspective construction of a square niche of good Veronese Gothic, with anuncusped arch of similar size and curve beyond.

In Fig. 64. the more distant arch only is lettered, as the construction of the nearer explains itself more clearly to the eye without letters. The more distant arch shows the general construction for all arches seen underneath, as of bridges, cathedral aisles, c. The rectangle A B C D is first drawn to contain the outside arch; then the depth of the arch, A a, is determined by the measuring-line, and the rectangle, a b c d, drawn for the inner arch.

a a, Bb, c, go to one vanishing-point; A B, a b, c, to the opposite one.

In the nearer arch another narrow rectangle is drawn to determine the cusp. The parts which would actually come into sight are slightly shaded.