Chapter V: Book IV (1)

CXLVII. The Perspective of Cast Shadows 229
CXLVIII. The Two Kinds of Shadows 230
CXLIX. Shadows Cast by the Sun 232
CL. The Sun in the Same Plane as the Picture 233
CLI. The Sun Behind the Picture 234
CLII. Sun Behind the Picture, Shadows Thrown on a Wall 238
CLIII. Sun Behind the Picture Throwing Shadow on
an Inclined Plane 240
CLIV. The Sun in Front of the Picture 241
CLV. The Shadow of an Inclined Plane 244
CLVI. Shadow on a Roof or Inclined Plane 245
CLVII. To Find the Shadow of a Projection or Balcony
on a Wall 246
CLVIII. Shadow on a Retreating Wall, Sun in Front 247
CLIX. Shadow of an Arch, Sun in Front 249
CLX. Shadow in a Niche or Recess 250
CLXI. Shadow in an Arched Doorway 251
CLXII. Shadows Produced by Artificial Light 252
CLXIII. Some Observations on Real Light and Shade 253
CLXIV. Reflection 257
CLXV. Angles of Reflection 259
CLXVI. Reflections of Objects at Different Distances 260
CLXVII. Reflection in a Looking-glass 262
CLXVIII. The Mirror at an Angle 264
CLXIX. The Upright Mirror at an Angle of 45 deg to
the Wall 266
CLXX. Mental Perspective 269

BOOK FIRST

THE NECESSITY OF THE STUDY OF PERSPECTIVE TO PAINTERS, SCULPTORS, AND ARCHITECTS

Leonardo da Vinci tells us in his celebrated _Treatise on Painting_ that the young artist should first of all learn perspective, that is to say, he should first of all learn that he has to depict on a flat surface objects which are in relief or distant one from the other; for this is the simple art of painting. Objects appear smaller at a distance than near to us, so by drawing them thus we give depth to our canvas. The outline of a ball is a mere flat circle, but with proper shading we make it appear round, and this is the perspective of light and shade.

'The next thing to be considered is the effect of the atmosphere and light. If two figures are in the same coloured dress, and are standing one behind the other, then they should be of slightly different tone, so as to separate them. And in like manner, according to the distance of the mountains in a landscape and the greater or less density of the air, so do we depict space between them, not only making them smaller in outline, but less distinct.'[1]

[Footnote 1: Leonardo da Vinci's _Treatise on Painting_.]

Sir Edwin Landseer used to say that in looking at a figure in a picture he liked to feel that he could walk round it, and this exactly expresses the impression that the true art of painting should make upon the spectator.

There is another observation of Leonardo's that it is well I should here transcribe; he says: 'Many are desirous of learning to draw, and are very fond of it, who are notwithstanding void of a proper disposition for it. This may be known by their want of perseverance; like boys who draw everything in a hurry, never finishing or shadowing.' This shows they do not care for their work, and all instruction is thrown away upon them. At the present time there is too much of this 'everything in a hurry', and beginning in this way leads only to failure and disappointment. These observations apply equally to perspective as to drawing and painting.

Unfortunately, this study is too often neglected by our painters, some of them even complacently confessing their ignorance of it; while the ordinary student either turns from it with distaste, or only endures going through it with a view to passing an examination, little thinking of what value it will be to him in working out his pictures. Whether the manner of teaching perspective is the cause of this dislike for it, I cannot say; but certainly most of our English books on the subject are anything but attractive.

All the great masters of painting have also been masters of perspective, for they knew that without it, it would be impossible to carry out their grand compositions. In many cases they were even inspired by it in choosing their subjects. When one looks at those sunny interiors, those corridors and courtyards by De Hooghe, with their figures far off and near, one feels that their charm consists greatly in their perspective, as well as in their light and tone and colour. Or if we study those Venetian masterpieces by Paul Veronese, Titian, Tintoretto, and others, we become convinced that it was through their knowledge of perspective that they gave such space and grandeur to their canvases.

I need not name all the great artists who have shown their interest and delight in this study, both by writing about it and practising it, such as Albert Duerer and others, but I cannot leave out our own Turner, who was one of the greatest masters in this respect that ever lived; though in his case we can only judge of the results of his knowledge as shown in his pictures, for although he was Professor of Perspective at the Royal Academy in 1807--over a hundred years ago--and took great pains with the diagrams he prepared to illustrate his lectures, they seemed to the students to be full of confusion and obscurity; nor am I aware that any record of them remains, although they must have contained some valuable teaching, had their author possessed the art of conveying it.

However, we are here chiefly concerned with the necessity of this study, and of the necessity of starting our work with it.

Before undertaking a large composition of figures, such as the 'Wedding-feast at Cana', by Paul Veronese, or 'The School of Athens', by Raphael, the artist should set out his floors, his walls, his colonnades, his balconies, his steps, &c., so that he may know where to place his personages, and to measure their different sizes according to their distances; indeed, he must make his stage and his scenery before he introduces his actors. He can then proceed with his composition, arrange his groups and the accessories with ease, and above all with correctness. But I have noticed that some of our cleverest painters will arrange their figures to please the eye, and when fairly advanced with their work will call in an expert, to (as they call it) put in their perspective for them, but as it does not form part of their original composition, it involves all sorts of difficulties and vexatious alterings and rubbings out, and even then is not always satisfactory. For the expert may not be an artist, nor in sympathy with the picture, hence there will be a want of unity in it; whereas the whole thing, to be in harmony, should be the conception of one mind, and the perspective as much a part of the composition as the figures.

If a ceiling has to be painted with figures floating or flying in the air, or sitting high above us, then our perspective must take a different form, and the point of sight will be above our heads instead of on the horizon; nor can these difficulties be overcome without an adequate knowledge of the science, which will enable us to work out for ourselves any new problems of this kind that we may have to solve.

Then again, with a view to giving different effects or impressions in this decorative work, we must know where to place the horizon and the points of sight, for several of the latter are sometimes required when dealing with large surfaces such as the painting of walls, or stage scenery, or panoramas depicted on a cylindrical canvas and viewed from the centre thereof, where a fresh point of sight is required at every twelve or sixteen feet.

Without a true knowledge of perspective, none of these things can be done. The artist should study them in the great compositions of the masters, by analysing their pictures and seeing how and for what reasons they applied their knowledge. Rubens put low horizons to most of his large figure-subjects, as in 'The Descent from the Cross', which not only gave grandeur to his designs, but, seeing they were to be placed above the eye, gave a more natural appearance to his figures. The Venetians often put the horizon almost on a level with the base of the picture or edge of the frame, and sometimes even below it; as in 'The Family of Darius at the Feet of Alexander', by Paul Veronese, and 'The Origin of the "Via Lactea"', by Tintoretto, both in our National Gallery. But in order to do all these things, the artist in designing his work must have the knowledge of perspective at his fingers' ends, and only the details, which are often tedious, should he leave to an assistant to work out for him.

We must remember that the line of the horizon should be as nearly as possible on a level with the eye, as it is in nature; and yet one of the commonest mistakes in our exhibitions is the bad placing of this line. We see dozens of examples of it, where in full-length portraits and other large pictures intended to be seen from below, the horizon is placed high up in the canvas instead of low down; the consequence is that compositions so treated not only lose in grandeur and truth, but appear to be toppling over, or give the impression of smallness rather than bigness. Indeed, they look like small pictures enlarged, which is a very different thing from a large design. So that, in order to see them properly, we should mount a ladder to get upon a level with their horizon line (see Fig. 66, double-page illustration).

We have here spoken in a general way of the importance of this study to painters, but we shall see that it is of almost equal importance to the sculptor and the architect.

A sculptor student at the Academy, who was making his drawings rather carelessly, asked me of what use perspective was to a sculptor. 'In the first place,' I said, 'to reason out apparently difficult problems, and to find how easy they become, will improve your mind; and in the second, if you have to do monumental work, it will teach you the exact size to make your figures according to the height they are to be placed, and also the boldness with which they should be treated to give them their full effect.' He at once acknowledged that I was right, proved himself an efficient pupil, and took much interest in his work.

I cannot help thinking that the reason our public monuments so often fail to impress us with any sense of grandeur is in a great measure owing to the neglect of the scientific study of perspective. As an illustration of what I mean, let the student look at a good engraving or photograph of the Arch of Constantine at Rome, or the Tombs of the Medici, by Michelangelo, in the sacristy of San Lorenzo at Florence. And then, for an example of a mistake in the placing of a colossal figure, let him turn to the Tomb of Julius II in San Pietro in Vinculis, Rome, and he will see that the figure of Moses, so grand in itself, not only loses much of its dignity by being placed on the ground instead of in the niche above it, but throws all the other figures out of proportion or harmony, and was quite contrary to Michelangelo's intention. Indeed, this tomb, which was to have been the finest thing of its kind ever done, was really the tragedy of the great sculptor's life.

The same remarks apply in a great measure to the architect as to the sculptor. The old builders knew the value of a knowledge of perspective, and, as in the case of Serlio, Vignola, and others, prefaced their treatises on architecture with chapters on geometry and perspective. For it showed them how to give proper proportions to their buildings and the details thereof; how to give height and importance both to the interior and exterior; also to give the right sizes of windows, doorways, columns, vaults, and other parts, and the various heights they should make their towers, walls, arches, roofs, and so forth. One of the most beautiful examples of the application of this knowledge to architecture is the Campanile of the Cathedral, at Florence, built by Giotto and Taddeo Gaddi, who were painters as well as architects. Here it will be seen that the height of the windows is increased as they are placed higher up in the building, and the top windows or openings into the belfry are about six times the size of those in the lower story.

WHAT IS PERSPECTIVE?

Perspective is a subtle form of geometry; it represents figures and objects not as they are but as we see them in space, whereas geometry represents figures not as we see them but as they are. When we have a front view of a figure such as a square, its perspective and geometrical appearance is the same, and we see it as it really is, that is, with all its sides equal and all its angles right angles, the perspective only varying in size according to the distance we are from it; but if we place that square flat on the table and look at it sideways or at an angle, then we become conscious of certain changes in its form--the side farthest from us appears shorter than that near to us, and all the angles are different. Thus A (Fig. 2) is a geometrical square and B is the same square seen in perspective.

The science of perspective gives the dimensions of objects seen in space as they appear to the eye of the spectator, just as a perfect tracing of those objects on a sheet of glass placed vertically between him and them would do; indeed its very name is derived from _perspicere_, to see through. But as no tracing done by hand could possibly be mathematically correct, the mathematician teaches us how by certain points and measurements we may yet give a perfect image of them. These images are called projections, but the artist calls them pictures. In this sketch _K_ is the vertical transparent plane or picture, _O_ is a cube placed on one side of it. The young student is the spectator on the other side of it, the dotted lines drawn from the corners of the cube to the eye of the spectator are the visual rays, and the points on the transparent picture plane where these visual rays pass through it indicate the perspective position of those points on the picture. To find these points is the main object or duty of linear perspective.

Perspective up to a certain point is a pure science, not depending upon the accidents of vision, but upon the exact laws of reasoning. Nor is it to be considered as only pertaining to the craft of the painter and draughtsman. It has an intimate connexion with our mental perceptions and with the ideas that are impressed upon the brain by the appearance of all that surrounds us. If we saw everything as depicted by plane geometry, that is, as a map, we should have no difference of view, no variety of ideas, and we should live in a world of unbearable monotony; but as we see everything in perspective, which is infinite in its variety of aspect, our minds are subjected to countless phases of thought, making the world around us constantly interesting, so it is devised that we shall see the infinite wherever we turn, and marvel at it, and delight in it, although perhaps in many cases unconsciously.

In perspective, as in geometry, we deal with parallels, squares, triangles, cubes, circles, &c.; but in perspective the same figure takes an endless variety of forms, whereas in geometry it has but one. Here are three equal geometrical squares: they are all alike. Here are three equal perspective squares, but all varied in form; and the same figure changes in aspect as often as we view it from a different position. A walk round the dining-room table will exemplify this.

It is in proving that, notwithstanding this difference of appearance, the figures do represent the same form, that much of our work consists; and for those who care to exercise their reasoning powers it becomes not only a sure means of knowledge, but a study of the greatest interest.

Perspective is said to have been formed into a science about the fifteenth century. Among the names mentioned by the unknown but pleasant author of _The Practice of Perspective_, written by a Jesuit of Paris in the eighteenth century, we find Albert Duerer, who has left us some rules and principles in the fourth book of his _Geometry_; Jean Cousin, who has an express treatise on the art wherein are many valuable things; also Vignola, who altered the plans of St. Peter's left by Michelangelo; Serlio, whose treatise is one of the best I have seen of these early writers; Du Cerceau, Serigati, Solomon de Cause, Marolois, Vredemont; Guidus Ubaldus, who first introduced foreshortening; the Sieur de Vaulizard, the Sieur Dufarges, Joshua Kirby, for whose _Method of Perspective made Easy_ (?) Hogarth drew the well-known frontispiece; and lastly, the above-named _Practice of Perspective_ by a Jesuit of Paris, which is very clear and excellent as far as it goes, and was the book used by Sir Joshua Reynolds.[2] But nearly all these authors treat chiefly of parallel perspective, which they do with clearness and simplicity, and also mathematically, as shown in the short treatise in Latin by Christian Wolff, but they scarcely touch upon the more difficult problems of angular and oblique perspective. Of modern books, those to which I am most indebted are the _Traite' Pratique de Perspective_ of M. A. Cassagne (Paris, 1873), which is thoroughly artistic, and full of pictorial examples admirably done; and to M. Henriet's _Cours Rational de Dessin_. There are many other foreign books of excellence, notably M. Thibault's _Perspective_, and some German and Swiss books, and yet, notwithstanding this imposing array of authors, I venture to say that many new features and original problems are presented in this book, whilst the old ones are not neglected. As, for instance, How to draw figures at an angle without vanishing points (see p. 141, Fig. 162, &c.), a new method of angular perspective which dispenses with the cumbersome setting out usually adopted, and enables us to draw figures at any angle without vanishing lines, &c., and is almost, if not quite, as simple as parallel perspective (see p. 133, Fig. 150, &c.). How to measure distances by the square and diagonal, and to draw interiors thereby (p. 128, Fig. 144). How to explain the theory of perspective by ocular demonstration, using a vertical sheet of glass with strings, placed on a drawing-board, which I have found of the greatest use (see p. 29, Fig. 29). Then again, I show how all our perspective can be done inside the picture; that we can measure any distance into the picture from a foot to a mile or twenty miles (see p. 86, Fig. 94); how we can draw the Great Pyramid, which stands on thirteen acres of ground, by putting it 1,600 feet off (Fig. 224), &c., &c. And while preserving the mathematical science, so that all our operations can be proved to be correct, my chief aim has been to make it easy of application to our work and consequently useful to the artist.

[Footnote 2: There is another book called _The Jesuit's Perspective_
which I have not yet seen, but which I hear is a fine work.]

The Egyptians do not appear to have made any use of linear perspective. Perhaps it was considered out of character with their particular kind of decoration, which is to be looked upon as picture writing rather than pictorial art; a table, for instance, would be represented like a ground-plan and the objects upon it in elevation or standing up. A row of chariots with their horses and drivers side by side were placed one over the other, and although the Egyptians had no doubt a reason for this kind of representation, for they were grand artists, it seems to us very primitive; and indeed quite young beginners who have never drawn from real objects have a tendency to do very much the same thing as this ancient people did, or even to emulate the mathematician and represent things not as they appear but as they are, and will make the top of a table an almost upright square and the objects upon it as if they would fall off.

No doubt the Greeks had correct notions of perspective, for the paintings on vases, and at Pompeii and Herculaneum, which were either by Greek artists or copied from Greek pictures, show some knowledge, though not complete knowledge, of this science. Indeed, it is difficult to conceive of any great artist making his perspective very wrong, for if he can draw the human figure as the Greeks did, surely he can draw an angle.

The Japanese, who are great observers of nature, seem to have got at their perspective by copying what they saw, and, although they are not quite correct in a few things, they convey the idea of distance and make their horizontal planes look level, which are two important things in perspective. Some of their landscapes are beautiful; their trees, flowers, and foliage exquisitely drawn and arranged with the greatest taste; whilst there is a character and go about their figures and birds, &c., that can hardly be surpassed. All their pictures are lively and intelligent and appear to be executed with ease, which shows their authors to be complete masters of their craft.

The same may be said of the Chinese, although their perspective is more decorative than true, and whilst their taste is exquisite their whole art is much more conventional and traditional, and does not remind us of nature like that of the Japanese.

We may see defects in the perspective of the ancients, in the mediaeval painters, in the Japanese and Chinese, but are we always right ourselves? Even in celebrated pictures by old and modern masters there are occasionally errors that might easily have been avoided, if a ready means of settling the difficulty were at hand. We should endeavour then to make this study as simple, as easy, and as complete as possible, to show clear evidence of its correctness (according to its conditions), and at the same time to serve as a guide on any and all occasions that we may require it.

To illustrate what is perspective, and as an experiment that any one can make, whether artist or not, let us stand at a window that looks out on to a courtyard or a street or a garden, &c., and trace with a paint-brush charged with Indian ink or water-colour the outline of whatever view there happens to be outside, being careful to keep the eye always in the same place by means of a rest; when this is dry, place a piece of drawing-paper over it and trace through with a pencil. Now we will rub out the tracing on the glass, which is sure to be rather clumsy, and, fixing our paper down on a board, proceed to draw the scene before us, using the main lines of our tracing as our guiding lines.

If we take pains over our work, we shall find that, without troubling ourselves much about rules, we have produced a perfect perspective of perhaps a very difficult subject. After practising for some little time in this way we shall get accustomed to what are called perspective deformations, and soon be able to dispense with the glass and the tracing altogether and to sketch straight from nature, taking little note of perspective beyond fixing the point of sight and the horizontal-line; in fact, doing what every artist does when he goes out sketching.

THE THEORY OF PERSPECTIVE

DEFINITIONS

I

Fig. 7. In this figure, _AKB_ represents the picture or transparent vertical plane through which the objects to be represented can be seen, or on which they can be traced, such as the cube _C_.

The line _HD_ is the +Horizontal-line+ or +Horizon+, the chief line in perspective, as upon it are placed the principal points to which our perspective lines are drawn. First, the +Point of Sight+ and next _D_, the +Point of Distance+. The chief vanishing points and measuring points are also placed on this line.

Another important line is _AB_, the +Base+ or +Ground line+, as it is on this that we measure the width of any object to be represented, such as _ef_, the base of the square _efgh_, on which the cube _C_ is raised. _E_ is the position of the eye of the spectator, being drawn in perspective, and is called the +Station-point+.

Note that the perspective of the board, and the line _SE_, is not the same as that of the cube in the picture _AKB_, and also that so much of the board which is behind the picture plane partially represents the +Perspective-plane+, supposed to be perfectly level and to extend from the base line to the horizon. Of this we shall speak further on. In nature it is not really level, but partakes in extended views of the rotundity of the earth, though in small areas such as ponds the roundness is infinitesimal.

Fig. 8. This is a side view of the previous figure, the picture plane _K_ being represented edgeways, and the line _SE_ its full length. It also shows the position of the eye in front of the point of sight _S_. The horizontal-line _HD_ and the base or ground-line _AB_ are represented as receding from us, and in that case are called vanishing lines, a not quite satisfactory term.

It is to be noted that the cube _C_ is placed close to the transparent picture plane, indeed touches it, and that the square _fj_ faces the spectator _E_, and although here drawn in perspective it appears to him as in the other figure. Also, it is at the same time a perspective and a geometrical figure, and can therefore be measured with the compasses. Or in other words, we can touch the square _fj_, because it is on the surface of the picture, but we cannot touch the square _ghmb_ at the other end of the cube and can only measure it by the rules of perspective.

II

THE POINT OF SIGHT, THE HORIZON, AND THE POINT OF DISTANCE

There are three things to be considered and understood before we can begin a perspective drawing. First, the position of the eye in front of the picture, which is called the +Station-point+, and of course is not in the picture itself, but its position is indicated by a point on the picture which is exactly opposite the eye of the spectator, and is called the +Point of Sight+, or +Principal Point+, or +Centre of Vision+, but we will keep to the first of these.

If our picture plane is a sheet of glass, and is so placed that we can see the landscape behind it or a sea-view, we shall find that the distant line of the horizon passes through that point of sight, and we therefore draw a line on our picture which exactly corresponds with it, and which we call the +Horizontal-line+ or +Horizon+.[3] The height of the horizon then depends entirely upon the position of the eye of the spectator: if he rises, so does the horizon; if he stoops or descends to lower ground, so does the horizon follow his movements. You may sit in a boat on a calm sea, and the horizon will be as low down as you are, or you may go to the top of a high cliff, and still the horizon will be on the same level as your eye.

[Footnote 3: In a sea-view, owing to the rotundity of the earth, the
real horizontal line is slightly below the sea line, which is noted
in Chapter I.]

This is an important line for the draughtsman to consider, for the effect of his picture greatly depends upon the position of the horizon. If you wish to give height and dignity to a mountain or a building, the horizon should be low down, so that these things may appear to tower above you. If you wish to show a wide expanse of landscape, then you must survey it from a height. In a composition of figures, you select your horizon according to the subject, and with a view to help the grouping. Again, in portraits and decorative work to be placed high up, a low horizon is desirable, but I have already spoken of this subject in the chapter on the necessity of the study of perspective.

III

POINT OF DISTANCE

Fig. 11. The distance of the spectator from the picture is of great importance; as the distortions and disproportions arising from too near a view are to be avoided, the object of drawing being to make things look natural; thus, the floor should look level, and not as if it were running up hill--the top of a table flat, and not on a slant, as if cups and what not, placed upon it, would fall off.

In this figure we have a geometrical or ground plan of two squares at different distances from the picture, which is represented by the line _KK_. The spectator is first at _A_, the corner of the near square _Acd_. If from _A_ we draw a diagonal of that square and produce it to the line _KK_ (which may represent the horizontal-line in the picture), where it intersects that line at _A'_ marks the distance that the spectator is from the point of sight _S_. For it will be seen that line _SA_ equals line _SA'_. In like manner, if the spectator is at _B_, his distance from the point _S_ is also found on the horizon by means of the diagonal _BB"_, so that all lines or diagonals at 45 deg are drawn to the point of distance (see Rule 6).

Figs. 12 and 13. In these two figures the difference is shown between the effect of the short-distance point _A'_ and the long-distance point _B'_; the first, _Acd_, does not appear to lie so flat on the ground as the second square, _Bef_.

From this it will be seen how important it is to choose the right point of distance: if we take it too near the point of sight, as in Fig. 12, the square looks unnatural and distorted. This, I may note, is a common fault with photographs taken with a wide-angle lens, which throws everything out of proportion, and will make the east end of a church or a cathedral appear higher than the steeple or tower; but as soon as we make our line of distance sufficiently long, as at Fig. 13, objects take their right proportions and no distortion is noticeable.

In some books on perspective we are told to make the angle of vision 60 deg, so that the distance _SD_ (Fig. 14) is to be rather less than the length or height of the picture, as at _A_. The French recommend an angle of 28 deg, and to make the distance about double the length of the picture, as at _B_ (Fig. 15), which is far more agreeable. For we must remember that the distance-point is not only the point from which we are supposed to make our tracing on the vertical transparent plane, or a point transferred to the horizon to make our measurements by, but it is also the point in front of the canvas that we view the picture from, called the station-point. It is ridiculous, then, to have it so close that we must almost touch the canvas with our noses before we can see its perspective properly.

Now a picture should look right from whatever distance we view it, even across the room or gallery, and of course in decorative work and in scene-painting a long distance is necessary.

We need not, however, tie ourselves down to any hard and fast rule, but should choose our distance according to the impression of space we wish to convey: if we have to represent a domestic scene in a small room, as in many Dutch pictures, we must not make our distance-point too far off, as it would exaggerate the size of the room.

The height of the horizon is also an important consideration in the composition of a picture, and so also is the position of the point of sight, as we shall see farther on.

In landscape and cattle pictures a low horizon often gives space and air, as in this sketch from a picture by Paul Potter--where the horizontal-line is placed at one quarter the height of the canvas. Indeed, a judicious use of the laws of perspective is a great aid to composition, and no picture ever looks right unless these laws are attended to. At the present time too little attention is paid to them; the consequence is that much of the art of the day reflects in a great measure the monotony of the snap-shot camera, with its everyday and wearisome commonplace.

IV

PERSPECTIVE OF A POINT, VISUAL RAYS, &C.

We perceive objects by means of the visual rays, which are imaginary straight lines drawn from the eye to the various points of the thing we are looking at. As those rays proceed from the pupil of the eye, which is a circular opening, they form themselves into a cone called the +Optic Cone+, the base of which increases in proportion to its distance from the eye, so that the larger the view which we wish to take in, the farther must we be removed from it. The diameter of the base of this cone, with the visual rays drawn from each of its extremities to the eye, form the angle of vision, which is wider or narrower according to the distance of this diameter.

Now let us suppose a visual ray _EA_ to be directed to some small object on the floor, say the head of a nail, _A_ (Fig. 17). If we interpose between this nail and our eye a sheet of glass, _K_, placed vertically on the floor, we continue to see the nail through the glass, and it is easily understood that its perspective appearance thereon is the point _a_, where the visual ray passes through it. If now we trace on the floor a line _AB_ from the nail to the spot _B_, just under the eye, and from the point _o_, where this line passes through or under the glass, we raise a perpendicular _oS_, that perpendicular passes through the precise point that the visual ray passes through. The line _AB_ traced on the floor is the horizontal trace of the visual ray, and it will be seen that the point _a_ is situated on the vertical raised from this horizontal trace.

V

TRACE AND PROJECTION

If from any line _A_ or _B_ or _C_ (Fig. 18), &c., we drop perpendiculars from different points of those lines on to a horizontal plane, the intersections of those verticals with the plane will be on a line called the horizontal trace or projection of the original line. We may liken these projections to sun-shadows when the sun is in the meridian, for it will be remarked that the trace does not represent the length of the original line, but only so much of it as would be embraced by the verticals dropped from each end of it, and although line _A_ is the same length as line _B_ its horizontal trace is longer than that of the other; that the projection of a curve (_C_) in this upright position is a straight line, that of a horizontal line (_D_) is equal to it, and the projection of a perpendicular or vertical (_E_) is a point only. The projections of lines or points can likewise be shown on a vertical plane, but in that case we draw lines parallel to the horizontal plane, and by this means we can get the position of a point in space; and by the assistance of perspective, as will be shown farther on, we can carry out the most difficult propositions of descriptive geometry and of the geometry of planes and solids.

The position of a point in space is given by its projection on a vertical and a horizontal plane--

Thus _e'_ is the projection of _E_ on the vertical plane _K_, and _e''_ is the projection of _E_ on the horizontal plane; _fe''_ is the horizontal trace of the plane _fE_, and _e'f_ is the trace of the same plane on the vertical plane _K_.

VI

SCIENTIFIC DEFINITION OF PERSPECTIVE

The projections of the extremities of a right line which passes through a vertical plane being given, one on either side of it, to find the intersection of that line with the vertical plane. _AE_ (Fig. 20) is the right line. The projection of its extremity _A_ on the vertical plane is _a'_, the projection of _E_, the other extremity, is _e'_. _AS_ is the horizontal trace of _AE_, and _a'e'_ is its trace on the vertical plane. At point _f_, where the horizontal trace intersects the base _Bc_ of the vertical plane, raise perpendicular _fP_ till it cuts _a'e'_ at point _P_, which is the point required. For it is at the same time on the given line _AE_ and the vertical plane _K_.

This figure is similar to the previous one, except that the extremity _A_ of the given line is raised from the ground, but the same demonstration applies to it.

And now let us suppose the vertical plane _K_ to be a sheet of glass, and the given line _AE_ to be the visual ray passing from the eye to the object _A_ on the other side of the glass. Then if _E_ is the eye of the spectator, its projection on the picture is _S_, the point of sight.

If I draw a dotted line from _E_ to little _a_, this represents another visual ray, and _o_, the point where it passes through the picture, is the perspective of little _a_. I now draw another line from _g_ to _S_, and thus form the shaded figure _ga'Po_, which is the perspective of _aAa'g_.

Let it be remarked that in the shaded perspective figure the lines _a'P_ and _go_ are both drawn towards _S_, the point of sight, and that they represent parallel lines _Aa'_ and _ag_, which are at right angles to the picture plane. This is the most important fact in perspective, and will be more fully explained farther on, when we speak of retreating or so-called vanishing lines.

RULES

VII

THE RULES AND CONDITIONS OF PERSPECTIVE

The conditions of linear perspective are somewhat rigid. In the first place, we are supposed to look at objects with one eye only; that is, the visual rays are drawn from a single point, and not from two. Of this we shall speak later on. Then again, the eye must be placed in a certain position, as at _E_ (Fig. 22), at a given height from the ground, _S'E_, and at a given distance from the picture, as _SE_. In the next place, the picture or picture plane itself must be vertical and perpendicular to the ground or horizontal plane, which plane is supposed to be as level as a billiard-table, and to extend from the base line, _ef_, of the picture to the horizon, that is, to infinity, for it does not partake of the rotundity of the earth.

We can only work out our propositions and figures in space with mathematical precision by adopting such conditions as the above. But afterwards the artist or draughtsman may modify and suit them to a more elastic view of things; that is, he can make his figures separate from one another, instead of their outlines coming close together as they do when we look at them with only one eye. Also he will allow for the unevenness of the ground and the roundness of our globe; he may even move his head and his eyes, and use both of them, and in fact make himself quite at his ease when he is out sketching, for Nature does all his perspective for him. At the same time, a knowledge of this rigid perspective is the sure and unerring basis of his freehand drawing.

RULE 1

All straight lines remain straight in their perspective appearance.[4]

[Footnote 4: Some will tell us that Nature abhors a straight line,
that all long straight lines in space appear curved, &c., owing to
certain optical conditions; but this is not apparent in short straight
lines, so if our drawing is small it would be wrong to curve them; if
it is large, like a scene or diorama, the same optical condition which
applies to the line in space would also apply to the line in the
picture.]

RULE 2

Vertical lines remain vertical in perspective, and are divided in the same proportion as _AB_ (Fig. 24), the original line, and _a'b'_, the perspective line, and if the one is divided at _O_ the other is divided at _o'_ in the same way.

It is not an uncommon error to suppose that the vertical lines of a high building should converge towards the top; so they would if we stood at the foot of that building and looked up, for then we should alter the conditions of our perspective, and our point of sight, instead of being on the horizon, would be up in the sky. But if we stood sufficiently far away, so as to bring the whole of the building within our angle of vision, and the point of sight down to the horizon, then these same lines would appear perfectly parallel, and the different stories in their true proportion.

RULE 3

Horizontals parallel to the base of the picture are also parallel to that base in the picture. Thus _a'b'_ (Fig. 25) is parallel to _AB_, and to _GL_, the base of the picture. Indeed, the same argument may be used with regard to horizontal lines as with verticals. If we look at a straight wall in front of us, its top and its rows of bricks, &c., are parallel and horizontal; but if we look along it sideways, then we alter the conditions, and the parallel lines converge to whichever point we direct the eye.

This rule is important, as we shall see when we come to the consideration of the perspective vanishing scale. Its use may be illustrated by this sketch, where the houses, walls, &c., are parallel to the base of the picture. When that is the case, then objects exactly facing us, such as windows, doors, rows of boards, or of bricks or palings, &c., are drawn with their horizontal lines parallel to the base; hence it is called parallel perspective.

RULE 4

All lines situated in a plane that is parallel to the picture plane diminish in proportion as they become more distant, but do not undergo any perspective deformation; and remain in the same relation and proportion each to each as the original lines. This is called the front view.

RULE 5

All horizontals which are at right angles to the picture plane are drawn to the point of sight.

Thus the lines _AB_ and _CD_ (Fig. 28) are horizontal or parallel to the ground plane, and are also at right angles to the picture plane _K_. It will be seen that the perspective lines _Ba'_, _Dc'_, must, according to the laws of projection, be drawn to the point of sight.

This is the most important rule in perspective (see Fig. 7 at beginning of Definitions).

An arrangement such as there indicated is the best means of illustrating this rule. But instead of tracing the outline of the square or cube on the glass, as there shown, I have a hole drilled through at the point _S_ (Fig. 29), which I select for the point of sight, and through which I pass two loose strings _A_ and _B_, fixing their ends at _S_.

As _SD_ represents the distance the spectator is from the glass or picture, I make string _SA_ equal in length to _SD_. Now if the pupil takes this string in one hand and holds it at right angles to the glass, that is, exactly in front of _S_, and then places one eye at the end _A_ (of course with the string extended), he will be at the proper distance from the picture. Let him then take the other string, _SB_, in the other hand, and apply it to point _b"_ where the square touches the glass, and he will find that it exactly tallies with the side _b"f_ of the square _a'b"fe_. If he applies the same string to _a'_, the other corner of the square, his string will exactly tally or cover the side _a'e_, and he will thus have ocular demonstration of this important rule.

In this little picture (Fig. 30) in parallel perspective it will be seen that the lines which retreat from us at right angles to the picture plane are directed to the point of sight _S_.

RULE 6

All horizontals which are at 45 deg, or half a right angle to the picture plane, are drawn to the point of distance.

We have already seen that the diagonal of the perspective square, if produced to meet the horizon on the picture, will mark on that horizon the distance that the spectator is from the point of sight (see definition, p. 16). This point of distance becomes then the measuring point for all horizontals at right angles to the picture plane.

Thus in Fig. 31 lines _AS_ and _BS_ are drawn to the point of sight _S_, and are therefore at right angles to the base _AB_. _AD_ being drawn to _D_ (the distance-point), is at an angle of 45 deg to the base _AB_, and _AC_ is therefore the diagonal of a square. The line 1C is made parallel to _AB_, consequently A1CB is a square in perspective. The line _BC_, therefore, being one side of that square, is equal to _AB_, another side of it. So that to measure a length on a line drawn to the point of sight, such as _BS_, we set out the length required, say _BA_, on the base-line, then from _A_ draw a line to the point of distance, and where it cuts _BS_ at _C_ is the length required. This can be repeated any number of times, say five, so that in this figure _BE_ is five times the length of _AB_.

RULE 7

All horizontals forming any other angles but the above are drawn to some other points on the horizontal line. If the angle is greater than half a right angle (Fig. 32), as _EBG_, the point is within the point of distance, as at _V"_. If it is less, as _ABV""_, then it is beyond the point of distance, and consequently farther from the point of sight.

In Fig. 32, the dotted line _BD_, drawn to the point of distance _D_, is at an angle of 45 deg to the base _AG_. It will be seen that the line _BV"_ is at a greater angle to the base than _BD_; it is therefore drawn to a point _V"_, within the point of distance and nearer to the point of sight _S_. On the other hand, the line _BV""_ is at a more acute angle, and is therefore drawn to a point some way beyond the other distance point.

_Note._--When this vanishing point is a long way outside the picture, the architects make use of a centrolinead, and the painters fix a long string at the required point, and get their perspective lines by that means, which is very inconvenient. But I will show you later on how you can dispense with this trouble by a very simple means, with equally correct results.

RULE 8

Lines which incline upwards have their vanishing points above the horizontal line, and those which incline downwards, below it. In both cases they are on the vertical which passes through the vanishing point (_S_) of their horizontal projections.

This rule is useful in drawing steps, or roads going uphill and downhill.

RULE 9

The farther a point is removed from the picture plane the nearer does its perspective appearance approach the horizontal line so long as it is viewed from the same position. On the contrary, if the spectator retreats from the picture plane _K_ (which we suppose to be transparent), the point remaining at the same place, the perspective appearance of this point will approach the ground-line in proportion to the distance of the spectator.

Therefore the position of a given point in perspective above the ground-line or below the horizon is in proportion to the distance of the spectator from the picture, or the picture from the point.

Figures 38 and 39 are two views of the same gallery from different distances. In Fig. 38, where the distance is too short, there is a want of proportion between the near and far objects, which is corrected in Fig. 39 by taking a much longer distance.

RULE 10

Horizontals in the same plane which are drawn to the same point on the horizon are parallel to each other.

This is a very important rule, for all our perspective drawing depends upon it. When we say that parallels are drawn to the same point on the horizon it does not imply that they meet at that point, which would be a contradiction; perspective parallels never reach that point, although they appear to do so. Fig. 40 will explain this.

Suppose _S_ to be the spectator, _AB_ a transparent vertical plane which represents the picture seen edgeways, and _HS_ and _DC_ two parallel lines, mark off spaces between these parallels equal to _SC_, the height of the eye of the spectator, and raise verticals 2, 3, 4, 5, &c., forming so many squares. Vertical line 2 viewed from _S_ will appear on _AB_ but half its length, vertical 3 will be only a third, vertical 4 a fourth, and so on, and if we multiplied these spaces _ad infinitum_ we must keep on dividing the line _AB_ by the same number. So if we suppose _AB_ to be a yard high and the distance from one vertical to another to be also a yard, then if one of these were a thousand yards away its representation at _AB_ would be the thousandth part of a yard, or ten thousand yards away, its representation at _AB_ would be the ten-thousandth part, and whatever the distance it must always be something; and therefore _HS_ and _DC_, however far they may be produced and however close they may appear to get, can never meet.

Fig. 41 is a perspective view of the same figure--but more extended. It will be seen that a line drawn from the tenth upright _K_ to _S_ cuts off a tenth of _AB_. We look then upon these two lines _SP_, _OP_, as the sides of a long parallelogram of which _SK_ is the diagonal, as _cefd_, the figure on the ground, is also a parallelogram.

The student can obtain for himself a further illustration of this rule by placing a looking-glass on one of the walls of his studio and then sketching himself and his surroundings as seen therein. He will find that all the horizontals at right angles to the glass will converge to his own eye. This rule applies equally to lines which are at an angle to the picture plane as to those that are at right angles or perpendicular to it, as in Rule 7. It also applies to those on an inclined plane, as in Rule 8.

With the above rules and a clear notion of the definitions and conditions of perspective, we should be able to work out any proposition or any new figure that may present itself. At any rate, a thorough understanding of these few pages will make the labour now before us simple and easy. I hope, too, it may be found interesting. There is always a certain pleasure in deceiving and being deceived by the senses, and in optical and other illusions, such as making things appear far off that are quite near, in making a picture of an object on a flat surface to look as if it stood out and in relief by a kind of magic. But there is, I think, a still greater pleasure than this, namely, in invention and in overcoming difficulties--in finding out how to do things for ourselves by our reasoning faculties, in originating or being original, as it were. Let us now see how far we can go in this respect.

VIII

A TABLE OR INDEX OF THE RULES OF PERSPECTIVE

The rules here set down have been fully explained in the previous pages, and this table is simply for the student's ready reference.

RULE 1

All straight lines remain straight in their perspective appearance.

RULE 2

Vertical lines remain vertical in perspective.

RULE 3