Thus, let PQ be the picture, BH the plane to be put in perspective (supposed to be infinitely extended on the side of PQ opposite to the eye E) and AB its trace on the picture plane. Then if a plane CL be drawn through E parallel to HB, the trace of which on the picture is CD; the band of the picture plane, infinitely produced both ways, is the perspective of the infinitely extended plane of which HABG is a part.

For, as before, the perspective of every point in HABG is situated somewhere in that band; and there is no point in that band which is not the perspective of some point in HABG.

The line CD is called the vanishing line of the plane HABG, from the perspective of the plane terminating or vanishing in that line.

It follows, that all parallel planes have the same vanishing line. Moreover, all lines situated in the same plane, as MN, etc., will have their vanishing points, as V, etc., in the vanishing line of that plane.

For HB, CL being parallel planes, and EV, MN parallel lines, one of which MN is (Hypoth.) in one of the planes HB, the other must be in the other (Pls. 1. 4). Whence since EV is in the plane CL, its trace V must be in the trace CD of the plane CL. That is, the vanishing point of MN in the plane HB is situated in the vanishing line CD of the plane HB.

And again, all lines anyhow situated in any number of parallel planes, have their vanishing points in the common vanishing line of those planes.

These simple theorems will find important applications presently.

6. It only remains, in this part of the subject, to justify the remarks made at the opening of the section, by showing:

(1.) The intersection of the perspectives of two lines is the perspective of the intersection of the two lines themselves.

Let AC, BC be two lines which intersect in C, and have A, B for their traces on the picture plane; and let V, W be their vanishing points respectively. Then AV, BW will be their indefinite perspectives, intersecting in c; and it is alleged that c is the perspective of C, or, in geometrical language, that C, c, E are in one straight line.

This has been already proved at p. 157.

(2.) The intersection of the perspectives of two planes is the perspective of the line in which those planes intersect.

Let MA, NA be two planes which intersect in CA, and their traces on the picture-plane be GH, KL; and let VS, VR be planes through the eye E parallel to them, having EV for their intersection, and BC, DF for their traces on the picture-plane. Then the bands BCHG, DFLK are the perspectives of those planes, having the line AV for their intersection.

Now, since the planes MA, AN which intersect in AC are respectively parallel to the planes SV, VR which intersect in EV, the line EV is parallel to AC (Pls. 1. 16). Whence V is the vanishing point of AC, and AV is the perspective of the line AC. Wherefore, the intersection of the perspectives of the two planes MA, AN is the perspective of their intersection AC, as stated in the enunciation.

We have next to show the method of defining the data of a perspective problem, considering all the lines employed to be traced upon the picture-plane of the paper. We cannot, indeed, exhibit the immediate data in the direct manner that we are able to do in the commencement of a problem in the Descriptive Geometry; but with very little subsidiary construction we are able to render the exhibition of a perspective problem in a light quite as simple and intelligible.*

7. In all direct problems in perspective, the position of the eye with respect to the picture forms part of the data. A perpendicular from the eye to the picture meets it in a specific point, called the centre of the picture; and a line through it parallel to the horizon (or the top and bottom of the picture itself) is called the horizontal line. When this point is not taken at, or very near to, the middle of the horizontal line, the effect of the picture is that of a painful distortion on account of the eye being

*In fact, all things considered, the Descriptive Geometry itself involves subsidiary considerations and constructions not less operose than those required by Perspective. Indeed, Descriptive Geometry is mainly conversant with inverse operations operations upon the projections of the direct figures, so as to get the projections of the quæsita of the problem, and thence the quæsita themselves. English writers have confined themselves almost entirely to the projections of the figures which constitute the data of a problem: the French to the processes which are requisite for the determination of the quæsita. The French begin where the English writers leave off; the one is the complement of the other: neither is perfect and complete in itself; together they form a complete and perfect system. What the English “Orthographic Projection" is to the Descriptive Geometry, this determination of the working data from the absolute conditions of the problem is to the practical operations of perspective solution.

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

B

E

C

boundary); E the eye; PQ the plane of the scene (ordinarily horizontal); EH the height of the eye above it; Eabed the pyramid of rays having the eye for vertex and the frame for base; CD 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 pictureplane, together with the place of the eye: to find the vanishing line of *that plane.

Let PQ be the picture; O its P centre; HL the horizontal line (where LQ denotes the height of the eye above the horizontal plane in ordinary 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 O and radius MN describe a circle on the picture-plane ; draw tangents to it, viz., CGD and C'G'D', parallel to AB.

M

N

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. II. 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.

a

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 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.

to ac.