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compelled to take in a wider range of scene on one side than the other. It is otherwise with respect to the height, inasmuch as we habitually refer all altitudes and depressions to our own horizon; and a scene which supposes the eye to be at the middle of the vertical height of the picture would be less satisfactory to the observer than one where it was far out of the middle of the horizontal breadth.

8. By the distance of the picture is meant the distance of the eye from it, or the length of the perpendicular drawn to it from the eye. This is sometimes marked on the horizontal line through the centre, each way from that centre; sometimes a circle with this distance as a radius is described on the picture-plane about that centre, and sometimes it is merely reserved without any separate exhibition.

Some degree of confusion is created by most writers neglecting to explain what is really meant by “ the distance of the picture.” It is not that 18 or 20 inches at which we hold the paper from the eye when making a sketch, or the distance at which we stand from the picture when viewing it from a proper position.

Let abcd represent the frame of the picture interiorly (or the visible

boundary); E the eye ; PQ the plane of the scene (ordinarily horizontal); EH the height of the eye above it; Eabcd the pyramid of rays having the eye for vertex and the frame for base; ČD the line in which cEd cuts the plane PQ; and ABCD the section of this pyramid by a plane through CD parallel to abcd. This plane ABCD is that upon which the perspective drawing is supposed to be made ; whilst the picture (the actual paper or canvas) which is presented to our view is the “ reduced picture,” or picture of the real perspective, abcd.

Draw EG perpendicular to ABCD cutting abcd in g. Then gE is the distance at which we stand to view the actual picture; whilst EG is the real distance of the picture upon which the perspective is drawn. These may be called the inspecting and constructive distances respectively.

The pictures on ABCD and abcd are, however, similar, and all the parts of the one proportional to the corresponding parts of the other; as are, likewise, all lines connected perspectively with them. Draw the plane GEH cutting ABCD in GK, abcd in gk, and PQ in KH. Also draw kh parallel to KH. Then it follows from the similar figures, that if all the parts of the scene were diminished in the ratio of GE: gE, we might actually construct on the inspection plane abcd, In direct perspective, however, this is never done; but the drawing on a reduced scale is made upon abcd from that actually constructed on ABCD.

9. Perspective drawings are rarely made to any special scale (as an inch to a yard or to a furlong), and general landscape drawings never. The practical difficulty is not in these cases felt so strongly-often not felt at all. But the confusion of mind about the relation between ABCD and abcd is strongly felt by the reflecting student. Cases, however, do occur where it creates practical difficulty as well as theoretical.

In the inverse perspective it is found in a more troublesome form than in the direct; but of this hereafter.

10. A plane to be put in perspective is given by means of its trace on the picture-plane actually exhibited, and its inclination to that plane not exhibited but reserved. We are required, then, to connect the data with its perspective, thus giving rise to the following problem :Given the trace and inclination of a plane in respect to the picture

plane, together with the place of the eye : to find the vanishing line of that plane.

Let PQ be the picture ; O its centre; HL the horizontal line (where LQ denotes the height of the eye above the horizontal plane in or- al dinary. drawings); KM the distance of the picture; MKN the complement of the inclination of the given plane to the picture; and AB its trace upon the picture.

Draw MN perpendicular to MK; with centre 0 and radius MN describe a circle on the picture-plane; draw tangents to it, viz., CGD and C'G'D', parallel to AB.

These are the vanishing lines of the plane according as the given inclination is towards o or the contrary, as is too obvious to need a formal statement of the proof here.

Cor. 1. If either of the tangents CD, C'D' should coincide with AB, it indicates that the plane in question passes through the eye.

Cor. 2. When the plane to be put in perspective is parallel to the picture-plane, this mode of complete representation is inapplicable, inasmuch as such plane can have no trace (and consequently no vanishing line) situated on the picture. We can hence only exhibit the perspective of a limited portion of the plane; and this itself will require a modified process, which will be duly explained further on.

11. A line is most simply given for the purposes of perspective, when it is taken as the intersection of two given planes. Its entire perspective is then the intersection of the perspectives of the two planes : that is, its trace is the intersection of the traces of the planes, and its vanishing point the intersection of the vanishing lines of the planes, as has been shown.

12. A second manner in which a line is often given is, as having a given position in a given plane.

With a view to the analysis of the geometry of the problem, it will be advisable to view the planes in a general manner, apart from the perspective operations.

Let PQ be the given plane and RS the picture plane; AB the given line in the plane PQ, and RQ the given trace of the plane, and B that of the line.

Take any point A in AB; draw AC perpendicular to the plane RS, and AR in the plane PQ perpendicular to RQ; and join RC, BC.

Now the plane ARC passing through the perpendicular AC is perpendicular to the plane RS; and hence ACR and ACB are right angles. Also, since AC is perpendicular to the plane RS, and AR perpendicular to a line RQ in it, the line CR is also perpendicular to RQ (Pls. 11. 13). Wherefore ARC is the profile of the given dihedral angle of the object-plane and picture-plane, and is hence given.

Again, the point A in the given line being given (or assumed as given), together with the angle ARC, and ACR being a right angle, the lines AC, CR are given. The line AB being also given as well as AC, and the angle ACB being a right angle; the line BC and the angle ABC are also given. That is, the angle made by the given line with its projection, together with the projection itself, on the pictureplane are given.

This reduces the problem to another form which will be presently investigated ; and hence we shall here only give the construction up to the stage already analysed. It is more conveniently performed as byework apart from the drawing.

In reference to the preceding figure, the small letters here will represent the capitals in the analysis.

Take ab equal to any segment of the given line, estimated from its trace b, as a working datum. Draw the perpendicular from a to the trace of the plane, and place ar equal to it in the semicircle on ab. Make the angle arc equal to the given profile angle of the plane and picture; draw ac perpendicular to rc, and set off in the semicircle ad equal to ac. Then abd is the inclination of the given line to its projection.

This construction is, obviously, only a performance upon one plane of operations indicated by the analysis as belonging to several ; which process, where magnitudes only and not positions are concerned, is always to be adopted in practice.

13. A third method of defining a line for perspective purposes is :Given its plane, and its projection on the picture-plane, to find its per

spective.

[Sometimes the angle which the line makes with its projection is substituted for the last datum.]

In this case we have virtually a second plane through the line given, as well as that which contains the line. This plane, moreover, being perpendicular to the picture-plane, its vanishing line passes through the centre of the picture, and is parallel to the trace, that is, to the given projection. It is therefore given in position. Whence the following construction:

Let PQ be the picture-plane, AB the trace, and CD the vanishing line of the plane which contains the given line ; let GF be the projection of the line on the plane of the picture intersecting AB in K; and let O be the centre of the picture.

Draw OV parallel to FG meeting CD in V: then V is the vanishing point, and KV the perspective of the line.

14. Finally, a line is given by its containing plane, its trace, and its inclination to the picture-plane.

A line drawn through the eye parallel to the given line will make an angle with the picture-plane equal to the given one; which angle will therefore be given. But the whole of the lines through the eye which do this, make equal angles with the perpendicular from the eye to the picture-plane; and hence they are situated on a right cone which has that perpendicular for its axis, and a circle on the pictureplane for its base. The centre of this circle is the centre of the picture; and the axis of the cone being given in magnitude, the radius of the circle is given in magnitude; and the circle itself is given in magnitude and position.

Whence the following construction :

Let AB be the given trace of the containing plane; K the trace of the line; CD the vanishing line of the containing plane; and ( the centre of the picture. Make km the distance of the eye from the picture; mkn the complement of the given inclination of the line to the picture; and kmn a right angle. With centre O and radius OG equal lo mn describe a circle cutting CD in GG'.

Then KG or KG' will be the perspective of the entire line, according as the given inclination is towards the centre of the picture or in the contrary direction.

The unfinished part of the construction of the second case of the data is identical with this : for the inclination of a line to a plane is the same thing as its inclination to its own projection on that plane.

The perspective of a point is always viewed practically (that is, the operations for finding it always imply its being so) as the intersection of the entire perspectives of two lines which pass through it. This of course renders further remark unnecessary, as it leads simply to direct constructions of those two perspectives.

SECTION II. PRACTICAL PERSPECTIVE. In general the application of perspective is to objects having certain features of regularity and symmetry which enable us greatly to abridge the actual work that would be required to put each angular point (as an isolated point) in perspective. For instance, nearly all the planes in which lines or points are actually given are either parallel or perpendicular to the horizon; or their relations to such planes are easily determined both in fact and in perspective. Again, most of the plane figures to be put in perspective are rectangles, or have easily assignable relations to rectangles or to parallelograms; and most of the solid angles that occur are trihedral right angles, having one of the faces horizontal. Even where these conditions are not fulfilled, the data can almost invariably be connected by subsidiary processes with data that do fulfil them; and in any particular case where even this may fail, the general methods already given will be adequate to a full and perfect construction of the perspective.

In the case of curves, as it is rarely that any one but the circle occurs in the practice of the draughtsman or engineer, it may be sufficient to state that it is deemed in practice sufficient to find the perspectives of eight points equidistant from each other (the angular points of an inscribed regular octagon), and then carefully tracing by hand an ellipse through them.

A few incidental notes, which though not geometrical perhaps as to strict form, are yet founded on geonietrical and physical principles, may be given in initio. For the sake of reference, they are designated as Propositions.

PROPOSITION I.

To prepare the drawing and its scales. (a.) To eyes of ordinary conformation, horizontal vision is painful and indistinct when the bounding line of light makes an angle of more than 30' with the optic axis. When, therefore, the breadth of the picture is fixed upon, the distance of the eye from it, to form an agreeable perspective, becomes also determined—at least within very narrow limits of variation; and when the distance of the eye is first fixed upon, the breadth of the picture becomes, conversely, dependent. If this rule

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