or PROSOPOPOEIA, is, in rhetoric and composition, a species of metaphor, in which the highest degree of energy is produced by representing inanimate objects as endowed with life and action. Perspective.
Projection Perspective, or Perspective Drawing, is the art of representing solid bodies by means of pictures traced upon a surface. This surface is usually, though not necessarily, flat, and is most commonly placed upright.
The principle of the art is very simple, being founded on the fact that light moves in straight lines. If a transparent plate were placed between the eye and the object to be represented, and if the various points at which the light passes through were noticed, these would indicate the form of the picture, and if proper colours were laid on these, would produce upon the eye nearly the same effect which the object did, and would be a perspective picture of that object.
Such a picture, however, though accurately drawn, and though coloured quite in accordance with nature, can never produce on the eye exactly the same effect which the object itself does. The adjustment of the eye to distinct vision is not the same in both; and, in truth, the light which comes from any point in the object is not a mere line, but a conical pencil, having the aperture of the pupil for its base. Hence, however skilfully finished, a picture is always recognised as such.
The various contrivances which were used for enabling the painter to mark off with precision the outlines of his subject, are almost entirely superseded by the new processes of photography. The well-known instrument, misnamed the camera lucida, was one of the most ingenious of these; and the most simple of them was to set up an open frame, across which a network of threads was stretched; by placing the eye behind this frame and observing through which mesh each part of the object is seen, an expert draughtsman could readily transfer the outlines to a sheet of paper, on which a similar network had been traced.
The art of drawing in perspective is useful to architects and engineers, by enabling them to make views of intended buildings or machines. The ordinary plans, sections, and elevations, which are needed for carrying out a piece of work, hardly enable us to realise the appearance which the structure will have when completed, and hence the importance, particularly in matters of taste, of having perspective representations.
Projection of Points.
Let PQRS (fig. 1), be the ground plan of an intended building, and suppose that we wish to have its projection upon a vertical plane standing on the line AB, the point of sight being at E. We have only to join E with each point of the structure, and ascertain where the lines cut the picture plane. Thus having joined EP, the point p at which this line cuts the plane on AB, is the representative of P.
When the point of sight can be marked on the same sheet of paper with the plan, the simple joining of P, Q, R, S, &c., with it gives the representative points; but this rarely happens, and some contrivance has to be made for obtaining these points without actually drawing the lines.
Having let fall EC, a normal to the picture plane from the point of sight, C is called the centre of the picture, and EC the distance of sight. Produce EC to meet PT drawn parallel to the line BA, then we have the proportion ET : EC :: PT : pC; so that if ET and TP were known, the distance Cp could be very easily found. Now ET and TP are the co-ordinates of the point P referred to axes passing through the point E; hence a very convenient arrangement. Measure off along ET and CB a series of equal parts, and from the points of section draw lines parallel to AB and ET and we have a network, by help of which the co-ordinates of the various points on the plan can be found.
This network may be drawn in pencil upon the working plans without injuring them; but, for those who are in the habit of making perspectives, it is convenient to have lines etched upon the surface of a plate of glass which can be laid down upon the plans in any required position.
In fig. 1, the distance of sight EC is ten of the actual divisions; calling it 100 and referring to the figure, we find ET = 187, TP = 104, whence 187 : 100 :: 104 : 55·6; so that Cp = 55·6.
Again, referring to fig. 2, which is the elevation of our supposed building, we find the points P to be below the eye-level 16 and 24, whence the distances of their representatives below the eye-level on the picture must be 8·6 and 12·8, as found by the proportions 187 : 100 :: 16 : 8·6 and 187 : 100 :: 24 : 12·8.
Let fig. 3 be the actual picture, C being the centre, and Projection CU the eye level; then the representation of the corner of Straight of the base at P is obtained by laying off CU = 55°6' and Lines. Up, 8°6' and 12°8' respectively.
These calculations may be readily made by help of a sliding rule or of a circular logarithmic scale. The number 187 on the one slide being brought opposite to 100 on the other, we find opposite to the three numbers 104, 16, and 24, the three results 55°6', 8°6', and 12°8'.
Denoting the co-ordinates of the various points of the structure by X, Y, Z; X being in the direction ET, Y in the direction TP, and Z in the vertical direction; and at the same time denoting by y and z the co-ordinates of the projections on the picture plane, reckoning from the point C, we have
\[ y = EC \frac{Y}{X}; \quad z = EC \frac{Z}{X}. \]
Similar operations give the projections of the other points of the structure, and the whole details may be recorded in a tabular form as under:
| Point | X | Y | Z | y | z | |-------|-----|-----|-----|-----|-----| | At P Base | 137 + 104 | -16 + 55°6' | -8°6' | | Wall | 188 + 96 | -16 + 51°1' | -8°5' | | At Q Base | 117 - 6 | -16 - 51°1' | -8°5' | | Wall | 125 - 5 | -16 - 4°0' | -12°8' | | At R Base | 183 - 48 | -16 - 2°2' | -8°7' | | Wall | 182 - 49 | -16 - 2°2' | -8°7' | | At S Base | 233 + 62 | -16 + 24°3' | -6°3' | | Vanishing point | B A | -157°1' | 0°0' |
The positions of the points on the picture are most readily marked by help of a graduated T square and drawing-board; the scale for the y being marked off along the edge AB (fig. 4) of the drawing-board, and that for the z along the edge CD of the square; and pieces with verniers may be fitted to slide on the blade of the square so as to give a great degree of precision. It is to be observed, that the scale used for the drawing need not be the same as that used for the plans and elevation, but may be made larger or smaller, according to the nature of the case—the distance of sight being increased or diminished in the same ratio.
**PROJECTION OF STRAIGHT LINES**
The projection of a straight line is also straight; for all the lines drawn from the eye to points in a straight line lie in one plane, and the intersection of this plane with the picture is a straight line; therefore, if the projections of two points in a straight line be found, the projection of that line is known.
If we draw through the point of sight E (fig. 1) a line EB parallel to PQ, and meeting the picture plane in B, this line is in the plane EPQ, and therefore B is in the projection of QP upon the picture. Hence the projection of the line RS, and, in general of any line parallel to QP, must also pass through B. This point B is called the vanishing point for lines parallel to QP. Similarly, the vanishing-point for lines parallel to QR is found by drawing EA of Straight parallel to QR. Having then measured off on the actual picture (fig. 3), CB and CA equal to CB, CA, of fig. 1, we perceive that the face and end lines of the structure tend respectively to B and A.
The positions of the vanishing-points can be easily computed, since we have
\[ EC : CB :: X_P - X_Q : Y_P - Y_Q \] \[ EC : CA :: X_R - X_Q : Y_R - Y_Q. \]
or in our example
\[ 70 : 110 :: 100 : 157°1' = CB, \] \[ 65 : -42 :: 100 : -63°6' = CA, \]
and when the angle PQR is right it is clear that the distance of sight EC is a mean proportional between the distances CB and CA of the two vanishing-points.
The determination of the points A and B facilitates greatly the delineation of the picture, since all the lines of the cornices, doors, and windows tend toward them. In no properly arranged drawing can both of these points be found within the limits of the paper, and therefore we must either support a pin at the proper place by a cumbrous erection, or make use of some instrument for drawing lines to a distant point. The centrolineal, contrived by the late Mr David Dick is, perhaps, the most convenient for the perspective draughtsman: it consists of a parallel ruler of the ordinary rhomboidal form, on the cross-ties, AB and CD (fig. 5) of which two pieces, BE and DF, are made to slide; these are retained in their places by pinching-screws at G and H. From the ends of the sliding-pieces two thin angular bits of steel project downwards to the paper, and against these a tracing-rule KL is laid. The edges of the steel at E and F must be accurately in the straight line passing through the pins of the parallel ruler. In order to facilitate the application of this instrument, notches are cut in the principal bar at M and N, exactly in the line of the joints. If, now, MN be applied to one of two converging lines, and the bar MN be secured against moving, by means of weights; and if, while the edge of KL is applied to the second line, the sliding-pieces at E and F be adjusted to bear upon it, the instrument is ready to trace lines tending to the same point.
The vanishing-points for all horizontal lines are found in the eye-level; but for lines which are inclined to the horizon, the vanishing-points are above or below that level. If, for example, there were a pediment on the face PQ of our supposed building, the vanishing-points for the cornices of that pediment would be found in a vertical line drawn through the point B of fig. 3, and at distances equal to the height of a pediment of which EB of fig. 1 is the half-breadth. Thus in Greek buildings the rise is about one-fourth part of the horizontal distance, in which case the positions of the vanishing-points will be found by laying one-fourth part of EB, fig. 1, above and below B of fig. 3. When a row of equidistant vertical lines is put in perspective, the distances and the lengths of the projections are in harmonic progression; and if any two of these projections be given, all the rest can readily be found.
Let, for example, \( Pp \) and \( Qq \) (fig. 6) be the perspective representations of two equal vertical lines, and let it be proposed to continue the series either way.
Join \( PQ \), \( pq \), and produce the lines till they meet in \( B \); \( B \) is the vanishing-point for horizontal lines in the plane of the series, and a horizontal line drawn through \( B \) must be the eye-level. Join now \( pQ \), and produce it to meet a vertical line drawn through \( B \); the point of intersection \( b \) must be the vanishing-point for all the lines connecting the top of one upright with the bottom of the next; so that the intersection \( R \) of \( qB \) with \( BP \) marks the position of the perspective projection of the next upright, and so that the intersection \( o \) of \( bP \) with \( Bp \) gives the position of the next on the other side.
Again \( Oo \) and \( Ss \) being the projections of two uprights in a series, let it be proposed to divide the interval between them as into four equal parts. Having found the vanishing-point \( B \) as before, draw the vertical line through \( B \), produce \( Os \) to meet it in \( C \), and measure off \( Bb \) equal to four times \( BC \); then \( b \) is the vanishing-point for diagonals of single intervals. The rest of the construction is obvious.
**PROJECTION OF CURVED LINES.**
Generally speaking, the most convenient method of drawing the perspective representation of a curved line is, to assume a number of points in the curve, to project these separately, and then to trace, by hand, a curve through the several projections. This process is applicable, whether the line to be projected be a fanciful curve sketched merely to please the eye, or a curve of a definite character. In the latter case, however, it is more satisfactory to determine strictly the nature of the projection.
The lines which join the point of sight with the various points in the given curve, trace out a conical or conoidal surface; and the intersection of the picture plane with that surface is the projection sought for; so that this branch of perspective drawing belongs essentially to the doctrine of conic sections.
When the proposed line is a circle, or any other line of the second order, the conoidal surface belongs to a cone proper, right, or oblique, as the case may be. Now the section of such a cone by a plane is also a line of the second order; wherefore the perspective projections of *circles*, *ellipses*, *parabolas*, and *hyperbolas* belong to the same class.
As the circle occurs in all machinery, and very frequently in buildings, the projection of the circle is a matter of special interest, and a ready means of obtaining it is a desideratum.
When five points in a line of the second order are known, the whole line is determined, and can be easily traced out by the method of radiants. If, then, we project five points in a circle, we can obtain its perspective representation. But as this process resolves itself into finding a number of points in the curve, and then tracing, by hand, a line through them, it is not preferable in practice to the simple process of assuming a number of points in the circle, and projecting these directly.
When the circle to be projected lies entirely on one side of a plane passing through the point of sight and parallel to the picture plane, its projection is a closed curve, generally elliptical, and sometimes, though very rarely, circular. Now, we can trace ellipses by help of mechanical contrivances, called *elliptographs*. These instruments can be readily applied when the major and minor axes of the curve are known; and therefore the practical problem becomes this, "To find the major and minor axes of the ellipse."
If the plane of the circle be parallel to the picture plane the projection is circular, and the centre of the projection is the projection of the centre. Otherwise, let the plane of the circle be produced to intersect the picture plane along the line \( PQ \) (fig. 7); this line may happen to be vertical, horizontal, or inclined.
From \( O \), the centre of the circle, let fall \( OR \) perpendicular to \( PQ \), and circumscribe the circle by the square \( KEMN \), touching at \( S \), \( U \), \( T \), and \( V \).
From the centre of the picture also draw \( CQ \) perpendicular to \( PQ \), and suppose that the inclination of the two planes is represented by \( i \).
Since the two lines \( LK \) and \( MN \) are parallel to the picture plane, their projections must be parallel to each other, and to the line \( PQ \); let \( Ik \), \( mK \) (fig. 8) be these projections. The vanishing-point for the parallels \( LM \), \( RSO \), and \( KN \) must be in the continuation of \( QC \) as at \( B \). Since \( O \) is the intersection of the lines \( LN \) and \( MK \), its projection must be at the meeting of the two lines \( Ik \) and \( mK \) on the picture.
Having drawn through \( B \) a line parallel to \( PQ \), produce \( ln \) to meet it in \( b \); \( b \) is the vanishing-point for all lines parallel to \( LN \). Now we have, in general, \( NK : KL :: EB : Bb \), wherefore, in this case, \( Bb \) must be equal to \( EB \); but \( EB \) is parallel to the plane of the circle, so that \( EB : BC :: R : \cos i \), or \( Bb = BC \cdot \sec i \). But \( BC = EC \cdot \cot i \); wherefore, since \( \cot i \cdot \sec i = \csc i \), \( Bb = EC \cdot \csc i \).
The vanishing-points for lines parallel to \( KM \) is at \( b' \), as far on the other side of \( B \).
If we join \( V \) with \( F \), the middle of the arc \( VS \), the angle \( KVF \) is \( 22^\circ \), and the vanishing-point for lines parallel to VF, as SI, GT, UH, is found by making Bd equal to the tangent of \(22^\circ\) for Bb as a radius; while, by laying off an equal distance Ba', we obtain the vanishing-point for the lines VI, TF, SH, UG. By help of one of these we readily find the points f, g, h, i, and, by help of the other, we can check the accuracy of the work. (The letters f, g, h, i, are, for the sake of clearness, omitted in the figure.) Observing that the tangents at f and h tend to b, those at g and i to b', we obtain an octagon circumscribing the desired ellipse, after which an ordinarily expert draughtsman can trace it out.
All this supposes that the vanishing-points are within the range of the paper, or implies the use and frequent re-adjustment of the centrolinead. Those who possess an ellipograph would naturally prefer to determine the major and minor axes of the ellipse, so as to be able to apply the instrument. These axes can be obtained by means of the doctrine of conic sections, but the processes, whether geometrical or algebraic, are so tedious that the method of points is to be preferred in practice.
When the circumference of the circle is graduated, as in the case of arches or of toothed wheels, the projection by successive points is unavoidable.
Retaining the ordinates X in their former position, let us, for the sake of convenience, place the Y in the direction QC, the Z in the direction QP, the origin being still at the point of sight. Having assumed any point W in the circumference of the circle, let us denote the angle UOW by \(a\), the radius of the circle being \(r\), then the co-ordinates of the point W are,
\[ \begin{align*} X_w &= X_0 + r \sin i \cdot \sin a \\ Y_w &= Y_0 + r \cos i \cdot \sin a \\ Z_w &= Z_0 + r \cos a, \end{align*} \]
so that the co-ordinates of its projection on the picture plane are,
\[ \begin{align*} y_w &= EC \cdot \frac{Y_0 + r \cos i \cdot \sin a}{X_0 + r \sin i \cdot \sin a} \\ z_w &= EC \cdot \frac{Z_0 + r \cos a}{X_0 + r \sin i \cdot \sin a}. \end{align*} \]
By giving to \(a\) the values corresponding to the various divisions of the circle, we can compute, by means of these formulas, their several projections.
When the perspective of any other curve is wanted, the best method is to assume points in the proposed curve, and to find the perspective representations of these separately.
**Curved Surfaces.**
The representation of a curved surface in perspective is a matter of considerable difficulty. When the surface is so placed that lines can be drawn from the point of sight to touch it, these lines form the surface of an enveloping conoid, and the intersection of the picture plane with this conoidal surface is the boundary of the perspective representation; but it is also the representation of the line in which the conoid touches the proposed surface, and, without the help of shading or colouring, the eye cannot distinguish whether the trace be meant to represent the line of contact, or the curved surface.
For example, lines drawn from the point of sight to touch a sphere lie in the surface of a right cone touching the sphere along a circle. The intersection of this cone with the picture plane is an ellipse, which is, indifferently, the projection of the sphere, or of the circle of contact. Nay, it might have been intended to represent any other curved surface enveloped by the same cone. The mere line then cannot convey the desired impression to the eye.
If the equation of the curved surface be represented by the symbol \(U = 0\), \(U\) being a function of the three ordinates \(X, Y\) and \(Z\), we must also have \(\delta U = 0\), or
\[ \left( \frac{\delta U}{\delta X} \right) \delta X + \left( \frac{\delta U}{\delta Y} \right) \delta Y + \left( \frac{\delta U}{\delta Z} \right) \delta Z = 0; \]
so that the equation of a plane touching the surface at the point of which the ordinates are \(X', Y', Z'\), must be
\[ \left( \frac{\delta U}{\delta X} \right) (X - X') + \left( \frac{\delta U}{\delta Y} \right) (Y - Y') + \left( \frac{\delta U}{\delta Z} \right) (Z - Z') = 0. \]
If this plane pass through the point of sight, the point at which it touches the curved surface is in the line of contact, so that the character of this line is determined by the two equations
\[ U = 0, \quad \text{and} \quad \left( \frac{\delta U}{\delta X} \right) X + \left( \frac{\delta U}{\delta Y} \right) Y + \left( \frac{\delta U}{\delta Z} \right) Z = 0. \]
If now \(x\) be the distance of sight, \(y\) and \(z\) the co-ordinates of the perspective limit, we have
\[ xy = XY; \quad xz = XZ, \]
and on eliminating \(X, Y\) and \(Z\) from these four equations, we obtain an equation in \(y\) and \(z\), which determines the form of the perspective outline.
The equation of a sphere, of which the centre is O and the radius \(r\) is
\[ (X - X_o)^2 + (Y - Y_o)^2 + (Z - Z_o)^2 - r^2 = 0, \quad (1) \]
which corresponds to the general equation \(U = 0\). Treating this by the method of partial differentials, we obtain, on halving,
\[ (X - X_o)X + (Y - Y_o)Y + (Z - Z_o)Z = 0, \quad (2) \]
and this subtracted from the preceding equation, leaves
\[ XX_o + YY_o + ZZ_o = X_o^2 + Y_o^2 + Z_o^2 - r^2, \quad (3) \]
which is the equation of a plane perpendicular to the line joining E with O; the intersection of this plane with the surface of the sphere gives the line of contact. By adding the double of equation (3.) to (1.), we obtain
\[ X^2 + Y^2 + Z^2 = X_o^2 + Y_o^2 + Z_o^2 - r^2, \quad (4.) \]
which belongs to a sphere having its centre at the origin, and its radius the side of a right-angled triangle, having the distance EO for its hypotenuse, and \(r\) for its other side. The equation of the ellipse may be now found by eliminating from (3.) and (4.), the quantities \(X, Y, Z\), by means of the equations
\[ xy = XY; \quad xz = XZ, \]
the result being
\[ (x^2 + y^2 + z^2)(X_o^2 + Y_o^2 + Z_o^2 - r^2) = (X_o x + Y_o y + Z_o z)^2, \]
which indicates a line of the second order, the major axis of which passes through the centre of the picture.
From this example of the application of the general method to a very simple case, it will be seen that the strict analytic process is far too laborious to be used in projecting the curved surfaces which are found on buildings or in machinery; a process readier, though perhaps not so satisfactory, is needed by the practical draughtsman.
It is of no use to assume, at random, points upon the proposed surface, since the projections of these would not give the limiting line of the picture. We must endeavour to select points on the line of contact.
A convenient though, in some cases, rather a tedious process, is to make a series of sections of the proposed surface, to project these, and then to draw a line touching and inclosing the projections. In the case of a cylinder or of a truncated cone, it is enough to project the two ends and to draw straight lines touching the two ellipses, while for a pointed cone, tangents drawn from the projection of the apex to the ellipse which represents the base, are the limits of the picture.
For an annular surface, as that of the torus of a column, it is convenient to take sections made by planes passing along the axis of the solid, as these give equal curves; and it may be remarked in general, that only a small part of each curve need be projected.
**Inverse Problem.**
When a drawing, purporting to be the perspective representation of an object is presented to us, we endeavour to realize the form of that object. This we can only do strictly, by resolving the problem "of what object can such a drawing be the true picture?" Shading and colouring aside, the lines of the picture may be the projections of an infinity of diverse solids, the corners and edges of which are in the straight lines joining the eye with the various traces; and these again change with every alteration in the position of the spectator. Hence the inverse problem is incapable of a strict solution.
A correct drawing of any familiar object, though in outline only, at once suggests the idea of the object which it was intended to represent; and though it cannot be demonstrated strictly to be the picture of such and such a thing, the mind, through the force of association, is satisfied of the fact. Adding to the actual delineation the knowledge that the picture is meant to represent, say a house, we may, to a certain degree, thence determine the proportions of the structure.
Let us suppose, for example, that fig. 9 is placed before us. We recognize that it is intended to portray the block of a building, and infer that the building is rectangular. Query, with this inference can we deduce the actual proportions of the structure from its picture in perspective?
The first thing to be discovered is the position of the point of sight. Having produced QP, qP to meet in B, QR, qR to meet in A, we obtain the two principal vanishing points, and having joined AB we have the level of the eye. If the picture plane have been placed upright, and if the drawing have been accurately made, the lines Rr, Qq, Pp, are perpendicular to AB. The point of sight E must be in a plane perpendicular to the plane of the paper, and meeting it along the line AB. The line EA is parallel to the lines represented by QR, qR; EB to those represented by QP, qP; wherefore the angle AEB must be right, and the point of sight must be found somewhere in the circumference of a semicircle described on AB as a diameter; but we have no positive means of determining at what point in that semi-circumference the eye ought to be placed. However, it is customary to place ourselves right in front of a picture, so that if we take C, the middle of that portion of the eye-level which is included within the limits of the paper, we may assume it as the middle of the Binocular picture. Erecting then at C a normal to the picture-plane, equal to a mean proportional between CA and CB, we obtain the position of the point of sight, and thus we can construct the part ABCE of fig. 1.
In order to discover the proportions of the parts of the building, draw the diagonals Qp, Qr on the picture, and produce them to meet vertical lines drawn through the vanishing points B and A; then, since the line joining the eye with b is parallel to that represented by Qp, we have the proportion EB : Bb :: length represented by QP : length represented by Pp, and thus we obtain the ratio of the length to the height of the building. Similarly EA : Aa :: breadth : height.
The ground-plan of the building may be reproduced by measuring from C, in fig. 1, a horizontal distance CP, taken from the picture, joining Ep, and producing it to any supposed distance pP. A line drawn through P, parallel to BE, is the line of the front, and the length of it may be found by measuring off Cq equal to the horizontal distance on the picture, by joining Eq, and producing it to meet the line of the front.
From these examples it is easy to see how the whole details of the structure could be scrutinized, if it ever should happen that the actual dimensions had to be obtained from a perspective drawing. It is evident that these operations cannot give the actual dimensions: some object of a known magnitude must form part of the picture, so as to be a standard of comparison.
Unless a picture be viewed from its proper point of sight, the impression produced must be different from that intended by the artist. In landscape and figure paintings the effect of a misplacement of the eye is not very conspicuous, but in drawings of buildings and machinery it is painfully felt. When the picture of a house is viewed from beyond the point of sight, that corner which is towards the eye appears acute; it seems, in fact, equal to the angle of two straight lines joining the eye with the vanishing-points A and B; and conversely, if the spectator come too close to the picture, the corner appears to be obtuse. By changing slowly one's distance from a perspective drawing, and studying the effect of the change, the proper distance may be passably well ascertained.
On moving the eye across the picture, the faces of the building, seeming always to be parallel to the lines joining the eye with the vanishing-points, appear to change their directions, as if the whole structure turned to follow the observer. This effect is well known in the case of portraits in which the model has been looking at the painter; the eyes of such portraits seem to follow the spectator wherever he may go.
In preparing vignettes and frontispieces, it is very common for engravers to reduce some well executed drawing; the effect of this is to reduce also the distance of sight, and to bring it far within the limits of distinct vision. The eye cannot be placed nearer the paper than twelve or ten inches, while distance of sight for vignettes is reduced often to three or four inches, sometimes even to less. Hence a painful and even ludicrous distortion: if there be buildings, they look as if sharpened at the corners to angles of 50, 40, or 20 degrees; and if it be a landscape, the retirement of the distant objects is out of all proportion to the breadth of the foreground.
**Binocular Perspective.**
We cannot realize the meaning of a perspective drawing without the aid of the association of ideas; but if we had two drawings of the same object as seen from two known separate points of view, we could thence deduce strictly the dimensions and form of the object represented by them. The distance between the two points of sight would form the base of a triangulation, and the problem would become a case in trigonometry.
Actually the appearances presented to the two eyes are two such pictures, and the convergence which we find it necessary to give to the two optical axes in order to bring the images of a given point to coincide, enables us to estimate the nearness of that point. Of this we have a beautiful example in the habits of the chameleon. The eyes of that lizard are set upon two projecting balls which turn freely in their sockets, and their motions are quite independent of each other, so that while the one eye may be intently watching some insect, the other may be moving about in search of a more convenient prey. The field of view is very small, and hence an incessant motion of exploration. As soon as the animal has perceived a fly tolerably within range, he brings both eyes to bear upon it, and advances cautiously till it be within the reach of his long tongue; he never mistakes the distance; single vision does not enable him to estimate it with sufficient nicety, but the convergence of the two eyes does so.
Professor Wheatstone showed, many years ago, that if two properly projected drawings be presented, one to each eye, the effect of the double vision is to give the appearance of solidity. The only thing which hindered Wheatstone's stereoscope from at once becoming a valuable addition to our stock of illustrative apparatus was the extreme difficulty of making the drawings sufficiently exact. Only simple geometrical forms could be attempted, and the invention had almost been forgotten. But the recent discovery of photography, by enabling us to obtain minutely exact views of actual objects, has invested the doctrine of binocular perspective with new importance.
The projection of twin drawings for the stereoscope differs in no respect from the processes of ordinary perspective; the only thing to be attended to is the choice of the points of sight. If we were to make two pictures of a large building as seen from points only $2\frac{1}{2}$ inches apart, the differences between them would be almost imperceptible; in the language of the surveyor, the base would be far too short. Hence we are obliged to separate considerably the points of sight; but when we come to unite the two pictures, we see them from the eyes, which are $2\frac{1}{2}$ inches separate, and hence the impression is not that of a large building but of a small model. In other words, the stereoscope is applicable to objects at hand, and can only give diminutive views of remote objects.
The accompanying figures (fig. 10) are the projections of a rhomboidal solid, with one of its diagonals drawn. When viewed through any of the ordinary stereoscopes, the illusion is complete; but if the wood engraver had attempted to draw the other three diagonals, they would have been seen, in all probability, to have passed each other at the middle, as any one may convince himself by drawing them in pencil; the reason is, that the slightest deviation from the proper place occasions a great change in the apparent position of the line; and hence the great difficulty of making stereoscopic engravings, or even of touching up by hand photographic pictures.
It must be observed, that a pair of stereoscopic views can never produce on the eyes exactly the same effect which the sight of the object would; for when we look from one part of an object to another, the adjustment of the focal length, as well as the convergence of the axes, is changed, and these two, from habit, are inseparably connected. Now, it is only the convergence of the eyes which is accommodated in the arrangement of binocular drawings, so that the use of the stereoscope must always do violence to the natural association of the muscular efforts.