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be rigidly adhered to, the relation will be exhibited by the following construction ; and if varied, the effect of the variation can be readily traced.

If the distance OE be given, and HL be a horizontal line through the centre of the picture; take any equal distances Oh, 01, on each side of o, and describe the equilateral triangle ehl; draw Oe, and make OE in it equal to the given distance of the picture; and draw EH, EL parallel to eh, el. Then H, L are the lateral boundaries of the picture.

(6.) The height of the eye above the horizon is usually between five and six feet, and when the supposed observer stands on the same horizontal plane with the building to be put in perspective, this height becomes a fixed standard for forming all the scales of the picture. This in fact it is which forms the fundamental scale ; the distance of the eye, and consequently the breadth of the picture, being regulated by it.

The height of the top of the picture above the horizontal line is a question less easy of decision. Of course it may include an angle of 302 on the ground of our distinct vision being limited by a right cone of rays, the axis of which cone is the perpendicular from the eye to the picture. But in fact, from our being so habituated to elevate the optic axis itself that we do it unconsciously to some extent, we can take in without pain a more extended space vertically than laterally.

(c.) Having fixed upon the boundaries and marked the centre of the picture, draw a line through that centre horizontally. This is called the horizontal line.

(d.) To form a scale of heights. Let HR be taken of the height of the eye, and divide the entire line

PR in conformity with it; draw lines to any point G in HQL: then the lines ab, cd, ef, etc., are the perspective heights of the line PR when removed backwards from the picture, so that the perspective of R shall be distant from the picture so far as to be represented on the horizontal lines bb', cc', dd', etc.

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* If the horizontal breadth be taken much more than this, it cannot be seen in all its parts without moving the axis of the eye so as to see one part separately from another (which would be tantamount to looking at the perspective of a perspective), or else distressing the eye to attempt to take it all in at one view. If, on the contrary, it be much less, the picture has precisely the same effect as an agreeable picture would be made to have by cutting off a portion at cach end-a mutilated appearance.

(e.) To form a scale of distances. Let the figure be supposed to be a section of the entire system by a

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plane perpendicular to the ground plane, passing through the eye. It may be considered, then, as a subsidiary figure in a plane at right angles to the picture and to the horizon.

Set off upon the line RQ in which this plane intersects the horizontal plane the several distances Q1, Q2, Q3, etc. Make QO (from the same scale) equal to the height of the eye; and OE parallel to QR equal to the distance of the picture. Then El, E2, E3, etc., will cut fronı QO segments Q1, Q2, Q3, etc., which represent on the picture the corresponding distances on the horizontal plane.

The proof is obvious. (f.) To form a scale of breadths. Divide the ground line RQ per scale; draw lines RO, a0, 60, co,

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etc., to the centre 0 (or any point in HL): then if 1, 2, 3, etc., be the divisions for distances, and parallels be drawn to RQ, as in the figure, a, b, or b, c, etc., will be the perspective breadth at the distance 1. Similarly a, b, or be Ce, etc., will be that at the distance 2; and so on.

These scales, though often used by practical draughtsmen, are not essential. Though given here in accordance with such frequent usage, they are not recommended to the draughtsman, who is also a geometer. When used, it is better to keep them as subsidiary figures than to encumber the actual drawing with them.

PROPOSITION II.

To put a plane figure in perspective. As a general statement of the rule, it may be directed to put each entire component line of a rectilnear figure in perspective. These perspectives, by their mutual intersections, give the perspective required.

In the case of curves, set a series of points sufficiently near for the purpose, and having put these in perspective, trace a curve through them by hand or by instruments, as the case may admit.

In practical perspective, however, we have seldom to operate upon figures which do not possess some kind of symmetry, which greatly simplifies the entire work by diminishing the number of independent steps. Thus the ground-plan of a building is mostlycomposed of rectangles, or of figures somehow related to rectangles ; the walls are invariably rectangular, and the tops and bottoms of a range of windows are each in line; the roofs are rectangles, or occasionally isosceles triangles ; and so on. It is the same in machinery, in a great degree, for even wheels do not often occur except in horizontal or vertical positions. On this account directions somewhat more special may be given for operating.

(a.) Let the plane of the figure be parallel to the picture.

This, on the general principles of perspective, is an isolated or exceptive case; since the vanishing line of the plane itself being infinitely distant, cannot be brought into use. It is usually waved in practice by making it dependent on the construction of another case; viz., hy finding the perspectives of some of the points by feigning them to be the intersections of the trace of the plane on the ground-line and some other lines. The following construction is more simple in principle as well as in practice.

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ANALYSIS.—Let PQ be the picture-plane, RS a plane parallel to it, containing the figure ABC to be put in perspective: draw lines, AE, BE, CE to the eye E, meeting the picture in abc. Then the figure abc will be the perspective of ABC, by the definition.

Draw perpendiculars AA', BB', CC' to the picture, and likewise the perpendicular EON to the picture, meeting the picture at 0, and the parallel plane at N. Draw also the planes ANE, BNE, CNE, cutting PQ in A'O, B'O, C'O.

Then the lines AN, A'O, AA', NOE, AaE are in one plane; and hence since this plane cuts the parallel planes, each line in one is equal and parallel to the corresponding line in the other, namely, A'B' to AB, A'O to AN, etc. Also since AE, A'O are in one plane, the point a, which is the perspective of A, is in the line A'O; and similarly b, c, etc, are in B'O, C'O, etc.

Again, since all'is parallel to AN, the triangles EOа, ENA are similar, and hence

AN:a0 :: EN: EO, a given ratio. Now O is the projection of the eye upon the picture and A'B'C' is VOL II.

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the projection on the picture of the figure to be put in perspective. Wherefore, if we take O the centre of the picture, and describe the given figure on the picture-plane in its appropriate position, we shall get the following construction :

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Let ABCDEF be any figure, regular or irregular, to be put in perspective, and the centre of the picture, PQ. Draw AO, BO, .. FO; make OG (in any direction, but generally in the horizontal line as most convenient) equal to the distance of the picture, and OK that of the plane of the given figure from the eye. Draw KA, and Ga parallel to it meeting AO in a: then a is the perspective of A.

The concluding part may either be effected similarly (as is done in the figure for b): or still more simply by drawing ab parallel to AB, bc to BC, etc.

For the figures ABCDEF and abcdef are evidently similar, and have their homologous sides parallel. Also their sides are to one another as OG to OK, that is, as EO to ON of the former figure. Wherefore all the conditions of the entire figure, being the perspective of the former, are fulfilled.

When the given figure is in any respect regular, the process is or

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Let ABCD be, for instance, a parallelogram ; PQ the picture-plane, and the other parts as before. Draw EV parallel to BC or AD, and EW to AB or DC. Then, completing the figure, abcd is the perspective of ABCD.

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Let, now, the planes HT, RQ simultaneously revolve about HL, KQ, respectively, so as always to be parallel to one another. If the direction of their revolution be such as to bring them finally into the extension of PQ, the parts bearing the several letters quoted in these figures will take the positions in Fig. 1. If, on the contrary, they revolve in the opposite directions, they will ultimately coincide with the picture plane as in Fig. 2.

Now the line FG or trace of the ground-plane, the horizontal line HL and centre o of the picture, and the distance EO of the eye from the picture are all given. The figure ABCD is also given, and hence the traces F, M, K, G, of its sides are also given. But EV is drawn through a given point E parallel to a given line AD or BC, and meets the given line HL in V. Whence V is a given point. Similarly W is a given point. Whence the entire perspectives of the sides of the figure ABCD are given ; viz., KV, MV, those of BC, AD, and FW, GW those of AB, CD. Wherefore, their intersection gives the perspective sought. Whence, the rule.

Place the given figure in its appropriate position on the groundplane; from 0 the centre of the picture draw OE at right angles to the horizontal line and equal to the given distance of the picture. Produce the sides of the figure to meet the trace of the ground-line, and parallel to them draw lines to meet the horizontal line. They will intersect it in the vanishing points of those lines. Having now the traces and vanishing points of the lines, draw their entire perspectives ; and these, by their intersections, give the perspective sought.

The two figures above give the same result. The first interferes less with the part on which the drawing is to be made, as well as affords more distinct work : but it on the other hand requires more space above and below the picture for tracing the component parts. When possible to be used, it is always the preferable. The process is the same too for any horizontal plane. The only

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