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Henry Frowde, M.A. Publisher to the University of Oxford London, Edinburgh, New York Toronto and Melbourne

THE THEORY AND PRACTICE OF PERSPECTIVE

by

G. A. STOREY, A.R.A.

Teacher of Perspective at the Royal Academy



Oxford At the Clarendon Press 1910

Oxford Printed at the Clarendon Press by Horace Hart, M.A. Printer to the University



DEDICATED to

SIR EDWARD J. POYNTER Baronet

President of the Royal Academy

in Token of Friendship and Regard



PREFACE

It is much easier to understand and remember a thing when a reason is given for it, than when we are merely shown how to do it without being told why it is so done; for in the latter case, instead of being assisted by reason, our real help in all study, we have to rely upon memory or our power of imitation, and to do simply as we are told without thinking about it. The consequence is that at the very first difficulty we are left to flounder about in the dark, or to remain inactive till the master comes to our assistance.

Now in this book it is proposed to enlist the reasoning faculty from the very first: to let one problem grow out of another and to be dependent on the foregoing, as in geometry, and so to explain each thing we do that there shall be no doubt in the mind as to the correctness of the proceeding. The student will thus gain the power of finding out any new problem for himself, and will therefore acquire a true knowledge of perspective.



CONTENTS

BOOK I Page THE NECESSITY OF THE STUDY OF PERSPECTIVE TO PAINTERS, SCULPTORS, AND ARCHITECTS 1 WHAT IS PERSPECTIVE? 6 THE THEORY OF PERSPECTIVE: I. Definitions 13 II. The Point of Sight, the Horizon, and the Point of Distance. 15 III. Point of Distance 16 IV. Perspective of a Point, Visual Rays, &c. 20 V. Trace and Projection 21 VI. Scientific Definition of Perspective 22 RULES: VII. The Rules and Conditions of Perspective 24 VIII. A Table or Index of the Rules of Perspective 40

BOOK II

THE PRACTICE OF PERSPECTIVE: IX. The Square in Parallel Perspective 42 X. The Diagonal 43 XI. The Square 43 XII. Geometrical and Perspective Figures Contrasted 46 XIII. Of Certain Terms made use of in Perspective 48 XIV. How to Measure Vanishing or Receding Lines 49 XV. How to Place Squares in Given Positions 50 XVI. How to Draw Pavements, &c. 51 XVII. Of Squares placed Vertically and at Different Heights, or the Cube in Parallel Perspective 53 XVIII. The Transposed Distance 53 XIX. The Front View of the Square and of the Proportions of Figures at Different Heights 54 XX. Of Pictures that are Painted according to the Position they are to Occupy 59 XXI. Interiors 62 XXII. The Square at an Angle of 45 deg 64 XXIII. The Cube at an Angle of 45 deg 65 XXIV. Pavements Drawn by Means of Squares at 45 deg 66 XXV. The Perspective Vanishing Scale 68 XXVI. The Vanishing Scale can be Drawn to any Point on the Horizon 69 XXVII. Application of Vanishing Scales to Drawing Figures 71 XXVIII. How to Determine the Heights of Figures on a Level Plane 71 XXIX. The Horizon above the Figures 72 XXX. Landscape Perspective 74 XXXI. Figures of Different Heights. The Chessboard 74 XXXII. Application of the Vanishing Scale to Drawing Figures at an Angle when their Vanishing Points are Inaccessible or Outside the Picture 77 XXXIII. The Reduced Distance. How to Proceed when the Point of Distance is Inaccessible 77 XXXIV. How to Draw a Long Passage or Cloister by Means of the Reduced Distance 78 XXXV. How to Form a Vanishing Scale that shall give the Height, Depth, and Distance of any Object in the Picture 79 XXXVI. Measuring Scale on Ground 81 XXXVII. Application of the Reduced Distance and the Vanishing Scale to Drawing a Lighthouse, &c. 84 XXXVIII. How to Measure Long Distances such as a Mile or Upwards 85 XXXIX. Further Illustration of Long Distances and Extended Views. 87 XL. How to Ascertain the Relative Heights of Figures on an Inclined Plane 88 XLI. How to Find the Distance of a Given Figure or Point from the Base Line 89 XLII. How to Measure the Height of Figures on Uneven Ground 90 XLIII. Further Illustration of the Size of Figures at Different Distances and on Uneven Ground 91 XLIV. Figures on a Descending Plane 92 XLV. Further Illustration of the Descending Plane 95 XLVI. Further Illustration of Uneven Ground 95 XLVII. The Picture Standing on the Ground 96 XLVIII. The Picture on a Height 97

BOOK III

XLIX. Angular Perspective 98 L. How to put a Given Point into Perspective 99 LI. A Perspective Point being given, Find its Position on the Geometrical Plane 100 LII. How to put a Given Line into Perspective 101 LIII. To Find the Length of a Given Perspective Line 102 LIV. To Find these Points when the Distance-Point is Inaccessible 103 LV. How to put a Given Triangle or other Rectilineal Figure into Perspective 104 LVI. How to put a Given Square into Angular Perspective 105 LVII. Of Measuring Points 106 LVIII. How to Divide any Given Straight Line into Equal or Proportionate Parts 107 LIX. How to Divide a Diagonal Vanishing Line into any Number of Equal or Proportional Parts 107 LX. Further Use of the Measuring Point O 110 LXI. Further Use of the Measuring Point O 110 LXII. Another Method of Angular Perspective, being that Adopted in our Art Schools 112 LXIII. Two Methods of Angular Perspective in one Figure 115 LXIV. To Draw a Cube, the Points being Given 115 LXV. Amplification of the Cube Applied to Drawing a Cottage 116 LXVI. How to Draw an Interior at an Angle 117 LXVII. How to Correct Distorted Perspective by Doubling the Line of Distance 118 LXVIII. How to Draw a Cube on a Given Square, using only One Vanishing Point 119 LXIX. A Courtyard or Cloister Drawn with One Vanishing Point 120 LXX. How to Draw Lines which shall Meet at a Distant Point, by Means of Diagonals 121 LXXI. How to Divide a Square Placed at an Angle into a Given Number of Small Squares 122 LXXII. Further Example of how to Divide a Given Oblique Square into a Given Number of Equal Squares, say Twenty-five 122 LXXIII. Of Parallels and Diagonals 124 LXXIV. The Square, the Oblong, and their Diagonals 125 LXXV. Showing the Use of the Square and Diagonals in Drawing Doorways, Windows, and other Architectural Features 126 LXXVI. How to Measure Depths by Diagonals 127 LXXVII. How to Measure Distances by the Square and Diagonal 128 LXXVIII. How by Means of the Square and Diagonal we can Determine the Position of Points in Space 129 LXXIX. Perspective of a Point Placed in any Position within the Square 131 LXXX. Perspective of a Square Placed at an Angle. New Method 133 LXXXI. On a Given Line Placed at an Angle to the Base Draw a Square in Angular Perspective, the Point of Sight, and Distance, being given 134 LXXXII. How to Draw Solid Figures at any Angle by the New Method 135 LXXXIII. Points in Space 137 LXXXIV. The Square and Diagonal Applied to Cubes and Solids Drawn Therein 138 LXXXV. To Draw an Oblique Square in Another Oblique Square without Using Vanishing-points 139 LXXXVI. Showing how a Pedestal can be Drawn by the New Method 141 LXXXVII. Scale on Each Side of the Picture 143 LXXXVIII. The Circle 145 LXXXIX. The Circle in Perspective a True Ellipse 145 XC. Further Illustration of the Ellipse 146 XCI. How to Draw a Circle in Perspective Without a Geometrical Plan 148 XCII. How to Draw a Circle in Angular Perspective 151 XCIII. How to Draw a Circle in Perspective more Correctly, by Using Sixteen Guiding Points 152 XCIV. How to Divide a Perspective Circle into any Number of Equal Parts 153 XCV. How to Draw Concentric Circles 154 XCVI. The Angle of the Diameter of the Circle in Angular and Parallel Perspective 156 XCVII. How to Correct Disproportion in the Width of Columns 157 XCVIII. How to Draw a Circle over a Circle or a Cylinder 158 XCIX. To Draw a Circle Below a Given Circle 159 C. Application of Previous Problem 160 CI. Doric Columns 161 CII. To Draw Semicircles Standing upon a Circle at any Angle 162 CIII. A Dome Standing on a Cylinder 163 CIV. Section of a Dome or Niche 164 CV. A Dome 167 CVI. How to Draw Columns Standing in a Circle 169 CVII. Columns and Capitals 170 CVIII. Method of Perspective Employed by Architects 170 CIX. The Octagon 172 CX. How to Draw the Octagon in Angular Perspective 173 CXI. How to Draw an Octagonal Figure in Angular Perspective 174 CXII. How to Draw Concentric Octagons, with Illustration of a Well 174 CXIII. A Pavement Composed of Octagons and Small Squares 176 CXIV. The Hexagon 177 CXV. A Pavement Composed of Hexagonal Tiles 178 CXVI. A Pavement of Hexagonal Tiles in Angular Perspective 181 CXVII. Further Illustration of the Hexagon 182 CXVIII. Another View of the Hexagon in Angular Perspective 183 CXIX. Application of the Hexagon to Drawing a Kiosk 185 CXX. The Pentagon 186 CXXI. The Pyramid 189 CXXII. The Great Pyramid 191 CXXIII. The Pyramid in Angular Perspective 193 CXXIV. To Divide the Sides of the Pyramid Horizontally 193 CXXV. Of Roofs 195 CXXVI. Of Arches, Arcades, Bridges, &c. 198 CXXVII. Outline of an Arcade with Semicircular Arches 200 CXXVIII. Semicircular Arches on a Retreating Plane 201 CXXIX. An Arcade in Angular Perspective 202 CXXX. A Vaulted Ceiling 203 CXXXI. A Cloister, from a Photograph 206 CXXXII. The Low or Elliptical Arch 207 CXXXIII. Opening or Arched Window in a Vault 208 CXXXIV. Stairs, Steps, &c. 209 CXXXV. Steps, Front View 210 CXXXVI. Square Steps 211 CXXXVII. To Divide an Inclined Plane into Equal Parts—such as a Ladder Placed against a Wall 212 CXXXVIII. Steps and the Inclined Plane 213 CXXXIX. Steps in Angular Perspective 214 CXL. A Step Ladder at an Angle 216 CXLI. Square Steps Placed over each other 217 CXLII. Steps and a Double Cross Drawn by Means of Diagonals and one Vanishing Point 218 CXLIII. A Staircase Leading to a Gallery 221 CXLIV. Winding Stairs in a Square Shaft 222 CXLV. Winding Stairs in a Cylindrical Shaft 225 CXLVI. Of the Cylindrical Picture or Diorama 227

BOOK IV

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

Horizontals parallel to the base of the picture are also parallel to that base in the picture.

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. This is called the front view.

RULE 5

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

RULE 6

All horizontals which are at 45 deg to the picture plane are drawn to the point of distance.

RULE 7

All horizontals forming any other angles but the above are drawn to some other points on the horizontal line.

RULE 8

Lines which incline upwards have their vanishing points above the horizon, and those which incline downwards, below it. In both cases they are on the vertical which passes through the vanishing point of their ground-plan or horizontal projections.

RULE 9

The farther a point is removed from the picture plane the nearer does it appear to approach the horizon, so long as it is viewed from the same position.

RULE 10

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



BOOK SECOND

THE PRACTICE OF PERSPECTIVE

In the foregoing book we have explained the theory or science of perspective; we now have to make use of our knowledge and to apply it to the drawing of figures and the various objects that we wish to depict.

The first of these will be a square with two of its sides parallel to the picture plane and the other two at right angles to it, and which we call

IX

THE SQUARE IN PARALLEL PERSPECTIVE

From a given point on the base line of the picture draw a line at right angles to that base. Let P be the given point on the base line AB, and S the point of sight. We simply draw a line along the ground to the point of sight S, and this line will be at right angles to the base, as explained in Rule 5, and consequently angle APS will be equal to angle SPB, although it does not look so here. This is our first difficulty, but one that we shall soon get over.



In like manner we can draw any number of lines at right angles to the base, or we may suppose the point P to be placed at so many different positions, our only difficulty being to conceive these lines to be parallel to each other. See Rule 10.



X

THE DIAGONAL

From a given point on the base line draw a line at 45 deg, or half a right angle, to that base. Let P be the given point. Draw a line from P to the point of distance D and this line PD will be at an angle of 45 deg, or at the same angle as the diagonal of a square. See definitions.



XI

THE SQUARE

Draw a square in parallel perspective on a given length on the base line. Let ab be the given length. From its two extremities a and b draw aS and bS to the point of sight S. These two lines will be at right angles to the base (see Fig. 43). From a draw diagonal aD to point of distance D; this line will be 45 deg to base. At point c, where it cuts bS, draw dc parallel to ab and abcd is the square required.



We have here proceeded in much the same way as in drawing a geometrical square (Fig. 47), by drawing two lines AE and BC at right angles to a given line, AB, and from A, drawing the diagonal AC at 45 deg till it cuts BC at C, and then through C drawing EC parallel to AB. Let it be remarked that because the two perspective lines (Fig. 48) AS and BS are at right angles to the base, they must consequently be parallel to each other, and therefore are perspectively equidistant, so that all lines parallel to AB and lying between them, such as ad, cf, &c., must be equal.



So likewise all diagonals drawn to the point of distance, which are contained between these parallels, such as Ad, af, &c., must be equal. For all straight lines which meet at any point on the horizon are perspectively parallel to each other, just as two geometrical parallels crossing two others at any angle, as at Fig. 49. Note also (Fig. 48) that all squares formed between the two vanishing lines AS, BS, and by the aid of these diagonals, are also equal, and further, that any number of squares such as are shown in this figure (Fig. 50), formed in the same way and having equal bases, are also equal; and the nine squares contained in the square abcd being equal, they divide each side of the larger square into three equal parts.





From this we learn how we can measure any number of given lengths, either equal or unequal, on a vanishing or retreating line which is at right angles to the base; and also how we can measure any width or number of widths on a line such as dc, that is, parallel to the base of the picture, however remote it may be from that base.





XII

GEOMETRICAL AND PERSPECTIVE FIGURES CONTRASTED

As at first there may be a little difficulty in realizing the resemblance between geometrical and perspective figures, and also about certain expressions we make use of, such as horizontals, perpendiculars, parallels, &c., which look quite different in perspective, I will here make a note of them and also place side by side the two views of the same figures.



























XIII

OF CERTAIN TERMS MADE USE OF IN PERSPECTIVE

Of course when we speak of Perpendiculars we do not mean verticals only, but straight lines at right angles to other lines in any position. Also in speaking of lines a right or straight line is to be understood; or when we speak of horizontals we mean all straight lines that are parallel to the perspective plane, such as those on Fig. 52, no matter what direction they take so long as they are level. They are not to be confused with the horizon or horizontal-line.



There are one or two other terms used in perspective which are not satisfactory because they are confusing, such as vanishing lines and vanishing points. The French term, fuyante or lignes fuyantes, or going-away lines, is more expressive; and point de fuite, instead of vanishing point, is much better. I have occasionally called the former retreating lines, but the simple meaning is, lines that are not parallel to the picture plane; but a vanishing line implies a line that disappears, and a vanishing point implies a point that gradually goes out of sight. Still, it is difficult to alter terms that custom has endorsed. All we can do is to use as few of them as possible.



XIV

HOW TO MEASURE VANISHING OR RECEDING LINES

Divide a vanishing line which is at right angles to the picture plane into any number of given measurements. Let SA be the given line. From A measure off on the base line the divisions required, say five of 1 foot each; from each division draw diagonals to point of distance D, and where these intersect the line AC the corresponding divisions will be found. Note that as lines AB and AC are two sides of the same square they are necessarily equal, and so also are the divisions on AC equal to those on AB.



The line AB being the base of the picture, it is at the same time a perspective line and a geometrical one, so that we can use it as a scale for measuring given lengths thereon, but should there not be enough room on it to measure the required number we draw a second line, DC, which we divide in the same proportion and proceed to divide cf. This geometrical figure gives, as it were, a bird's-eye view or ground-plan of the above.





XV

HOW TO PLACE SQUARES IN GIVEN POSITIONS

Draw squares of given dimensions at given distances from the base line to the right or left of the vertical line, which passes through the point of sight.



Let ab (Fig. 55) represent the base line of the picture divided into a certain number of feet; HD the horizon, VO the vertical. It is required to draw a square 3 feet wide, 2 feet to the right of the vertical, and 1 foot from the base.

First measure from V, 2 feet to e, which gives the distance from the vertical. Second, from e measure 3 feet to b, which gives the width of the square; from e and b draw eS, bS, to point of sight. From either e or b measure 1 foot to the left, to f or f'. Draw fD to point of distance, which intersects eS at P, and gives the required distance from base. Draw Pg and B parallel to the base, and we have the required square.

Square A to the left of the vertical is 2-1/2 feet wide, 1 foot from the vertical and 2 feet from the base, and is worked out in the same way.

Note.—It is necessary to know how to work to scale, especially in architectural drawing, where it is indispensable, but in working out our propositions and figures it is not always desirable. A given length indicated by a line is generally sufficient for our requirements. To work out every problem to scale is not only tedious and mechanical, but wastes time, and also takes the mind of the student away from the reasoning out of the subject.



XVI

HOW TO DRAW PAVEMENTS, &C.

Divide a vanishing line into parts varying in length. Let BS' be the vanishing line: divide it into 4 long and 3 short spaces; then proceed as in the previous figure. If we draw horizontals through the points thus obtained and from these raise verticals, we form, as it were, the interior of a building in which we can place pillars and other objects.



Or we can simply draw the plan of the pavement as in this figure.





And then put it into perspective.



XVII

OF SQUARES PLACED VERTICALLY AND AT DIFFERENT HEIGHTS, OR THE CUBE IN PARALLEL PERSPECTIVE

On a given square raise a cube.



ABCD is the given square; from A and B raise verticals AE, BF, equal to AB; join EF. Draw ES, FS, to point of sight; from C and D raise verticals CG, DH, till they meet vanishing lines ES, FS, in G and H, and the cube is complete.



XVIII

THE TRANSPOSED DISTANCE

The transposed distance is a point D' on the vertical VD', at exactly the same distance from the point of sight as is the point of distance on the horizontal line.

It will be seen by examining this figure that the diagonals of the squares in a vertical position are drawn to this vertical distance-point, thus saving the necessity of taking the measurements first on the base line, as at CB, which in the case of distant objects, such as the farthest window, would be very inconvenient. Note that the windows at K are twice as high as they are wide. Of course these or any other objects could be made of any proportion.





XIX

THE FRONT VIEW OF THE SQUARE AND OF THE PROPORTIONS OF FIGURES AT DIFFERENT HEIGHTS

According to Rule 4, all lines situated in a plane parallel to the picture plane diminish in length as they become more distant, but remain in the same proportions each to each as the original lines; as squares or any other figures retain the same form. Take the two squares ABCD, abcd (Fig. 61), one inside the other; although moved back from square EFGH they retain the same form. So in dealing with figures of different heights, such as statuary or ornament in a building, if actually equal in size, so must we represent them.





In this square K, with the checker pattern, we should not think of making the top squares smaller than the bottom ones; so it is with figures.

This subject requires careful study, for, as pointed out in our opening chapter, there are certain conditions under which we have to modify and greatly alter this rule in large decorative work.



In Fig. 63 the two statues A and B are the same size. So if traced through a vertical sheet of glass, K, as at c and d, they would also be equal; but as the angle b at which the upper one is seen is smaller than angle a, at which the lower figure or statue is seen, it will appear smaller to the spectator (S) both in reality and in the picture.



But if we wish them to appear the same size to the spectator who is viewing them from below, we must make the angles a and b (Fig. 64), at which they are viewed, both equal. Then draw lines through equal arcs, as at c and d, till they cut the vertical NO (representing the side of the building where the figures are to be placed). We shall then obtain the exact size of the figure at that height, which will make it look the same size as the lower one, N. The same rule applies to the picture K, when it is of large proportions. As an example in painting, take Michelangelo's large altar-piece in the Sistine Chapel, 'The Last Judgement'; here the figures forming the upper group, with our Lord in judgement surrounded by saints, are about four times the size, that is, about twice the height, of those at the lower part of the fresco. The figures on the ceiling of the same chapel are studied not only according to their height from the pavement, which is 60 ft., but to suit the arched form of it. For instance, the head of the figure of Jonah at the end over the altar is thrown back in the design, but owing to the curvature in the architecture is actually more forward than the feet. Then again, the prophets and sybils seated round the ceiling, which are perhaps the grandest figures in the whole range of art, would be 18 ft. high if they stood up; these, too, are not on a flat surface, so that it required great knowledge to give them their right effect.



Of course, much depends upon the distance we view these statues or paintings from. In interiors, such as churches, halls, galleries, &c., we can make a fair calculation, such as the length of the nave, if the picture is an altar-piece—or say, half the length; so also with statuary in niches, friezes, and other architectural ornaments. The nearer we are to them, and the more we have to look up, the larger will the upper figures have to be; but if these are on the outside of a building that can be looked at from a long distance, then it is better not to have too great a difference.







These remarks apply also to architecture in a great measure. Buildings that can only be seen from the street below, as pictures in a narrow gallery, require a different treatment from those out in the open, that are to be looked at from a distance. In the former case the same treatment as the Campanile at Florence is in some cases desirable, but all must depend upon the taste and judgement of the architect in such matters. All I venture to do here is to call attention to the subject, which seems as a rule to be ignored, or not to be considered of importance. Hence the many mistakes in our buildings, and the unsatisfactory and mean look of some of our public monuments.



XX

OF PICTURES THAT ARE PAINTED ACCORDING TO THE POSITION THEY ARE TO OCCUPY

In this double-page illustration of the wall of a picture-gallery, I have, as it were, hung the pictures in accordance with the style in which they are painted and the perspective adopted by their painters. It will be seen that those placed on the line level with the eye have their horizon lines fairly high up, and are not suited to be placed any higher. The Giorgione in the centre, the Monna Lisa to the right, and the Velasquez and Watteau to the left, are all pictures that fit that position; whereas the grander compositions above them are so designed, and are so large in conception, that we gain in looking up to them.

Note how grandly the young prince on his pony, by Velasquez, tells out against the sky, with its low horizon and strong contrast of light and dark; nor does it lose a bit by being placed where it is, over the smaller pictures.

The Rembrandt, on the opposite side, with its burgomasters in black hats and coats and white collars, is evidently intended and painted for a raised position, and to be looked up to, which is evident from the perspective of the table. The grand Titian in the centre, an altar-piece in one of the churches in Venice (here reversed), is also painted to suit its elevated position, with low horizon and figures telling boldly against the sky. Those placed low down are modern French pictures, with the horizon high up and almost above their frames, but placed on the ground they fit into the general harmony of the arrangement.

It seems to me it is well, both for those who paint and for those who hang pictures, that this subject should be taken into consideration. For it must be seen by this illustration that a bigger style is adopted by the artists who paint for high places in palaces or churches than by those who produce smaller easel-pictures intended to be seen close. Unfortunately, at our picture exhibitions, we see too often that nearly all the works, whether on large or small canvases, are painted for the line, and that those which happen to get high up look as if they were toppling over, because they have such a high horizontal line; and instead of the figures telling against the sky, as in this picture of the 'Infant' by Velasquez, the Reynolds, and the fat man treading on a flag, we have fields or sea or distant landscape almost to the top of the frame, and all, so methinks, because the perspective is not sufficiently considered.

Note.—Whilst on this subject, I may note that the painter in his large decorative work often had difficulties to contend with, which arose from the form of the building or the shape of the wall on which he had to place his frescoes. Painting on the ceiling was no easy task, and Michelangelo, in a humorous sonnet addressed to Giovanni da Pistoya, gives a burlesque portrait of himself while he was painting the Sistine Chapel:—

"I'ho gia' fatto un gozzo in questo stento."

Now have I such a goitre 'neath my chin That I am like to some Lombardic cat, My beard is in the air, my head i' my back, My chest like any harpy's, and my face Patched like a carpet by my dripping brush. Nor can I see, nor can I budge a step; My skin though loose in front is tight behind, And I am even as a Syrian bow. Alas! methinks a bent tube shoots not well; So give me now thine aid, my Giovanni.

At present that difficulty is got over by using large strong canvas, on which the picture can be painted in the studio and afterwards placed on the wall.

However, the other difficulty of form has to be got over also. A great portion of the ceiling of the Sistine Chapel, and notably the prophets and sibyls, are painted on a curved surface, in which case a similar method to that explained by Leonardo da Vinci has to be adopted.

In Chapter CCCI he shows us how to draw a figure twenty-four braccia high upon a wall twelve braccia high. (The braccia is 1 ft. 10-7/8 in.). He first draws the figure upright, then from the various points draws lines to a point F on the floor of the building, marking their intersections on the profile of the wall somewhat in the manner we have indicated, which serve as guides in making the outline to be traced.





XXI

INTERIORS



To draw the interior of a cube we must suppose the side facing us to be removed or transparent. Indeed, in all our figures which represent solids we suppose that we can see through them, and in most cases we mark the hidden portions with dotted lines. So also with all those imaginary lines which conduct the eye to the various vanishing points, and which the old writers called 'occult'.



When the cube is placed below the horizon (as in Fig. 59), we see the top of it; when on the horizon, as in the above (Fig. 69), if the side facing us is removed we see both top and bottom of it, or if a room, we see floor and ceiling, but otherwise we should see but one side (that facing us), or at most two sides. When the cube is above the horizon we see underneath it.

We shall find this simple cube of great use to us in architectural subjects, such as towers, houses, roofs, interiors of rooms, &c.

In this little picture by de Hoogh we have the application of the perspective of the cube and other foregoing problems.



XXII

THE SQUARE AT AN ANGLE OF 45 DEG.

When the square is at an angle of 45 deg to the base line, then its sides are drawn respectively to the points of distance, DD, and one of its diagonals which is at right angles to the base is drawn to the point of sight S, and the other ab, is parallel to that base or ground line.



To draw a pavement with its squares at this angle is but an amplification of the above figure. Mark off on base equal distances, 1, 2, 3, &c., representing the diagonals of required squares, and from each of these points draw lines to points of distance DD". These lines will intersect each other, and so form the squares of the pavement; to ensure correctness, lines should also be drawn from these points 1, 2, 3, to the point of sight S, and also horizontals parallel to the base, as ab.





XXIII

THE CUBE AT AN ANGLE OF 45 DEG.

Having drawn the square at an angle of 45 deg, as shown in the previous figure, we find the length of one of its sides, dh, by drawing a line, SK, through h, one of its extremities, till it cuts the base line at K. Then, with the other extremity d for centre and dK for radius, describe a quarter of a circle Km; the chord thereof mK will be the geometrical length of dh. At d raise vertical dC equal to mK, which gives us the height of the cube, then raise verticals at a, h, &c., their height being found by drawing CD and CD" to the two points of distance, and so completing the figure.





XXIV

PAVEMENTS DRAWN BY MEANS OF SQUARES AT 45 DEG.





The square at 45 deg will be found of great use in drawing pavements, roofs, ceilings, &c. In Figs. 73, 74 it is shown how having set out one square it can be divided into four or more equal squares, and any figure or tile drawn therein. Begin by making a geometrical or ground plan of the required design, as at Figs. 73 and 74, where we have bricks placed at right angles to each other in rows, a common arrangement in brick floors, or tiles of an octagonal form as at Fig. 75.





XXV

THE PERSPECTIVE VANISHING SCALE

The vanishing scale, which we shall find of infinite use in our perspective, is founded on the facts explained in Rule 10. We there find that all horizontals in the same plane, which are drawn to the same point on the horizon, are perspectively parallel to each other, so that if we measure a certain height or width on the picture plane, and then from each extremity draw lines to any convenient point on the horizon, then all the perpendiculars drawn between these lines will be perspectively equal, however much they may appear to vary in length.



Let us suppose that in this figure (76) AB and A'B' each represent 5 feet. Then in the first case all the verticals, as e, f, g, h, drawn between AO and BO represent 5 feet, and in the second case all the horizontals e, f, g, h, drawn between A'O and B'O also represent 5 feet each. So that by the aid of this scale we can give the exact perspective height and width of any object in the picture, however far it may be from the base line, for of course we can increase or diminish our measurements at AB and A'B' to whatever length we require.

As it may not be quite evident at first that the points O may be taken at random, the following figure will prove it.



XXVI

THE VANISHING SCALE CAN BE DRAWN TO ANY POINT ON THE HORIZON

From AB (Fig. 77) draw AO, BO, thus forming the scale, raise vertical C. Now form a second scale from AB by drawing AO' BO', and therein raise vertical D at an equal distance from the base. First, then, vertical C equals AB, and secondly vertical D equals AB, therefore C equals D, so that either of these scales will measure a given height at a given distance.



(See axioms of geometry.)







XXVII

APPLICATION OF VANISHING SCALES TO DRAWING FIGURES

In this figure we have marked off on a level plain three or four points a, b, c, d, to indicate the places where we wish to stand our figures. AB represents their average height, so we have made our scale AO, BO, accordingly. From each point marked we draw a line parallel to the base till it reaches the scale. From the point where it touches the line AO, raise perpendicular as a, which gives the height required at that distance, and must be referred back to the figure itself.





XXVIII

HOW TO DETERMINE THE HEIGHTS OF FIGURES ON A LEVEL PLANE

First Case.

This is but a repetition of the previous figure, excepting that we have substituted these schoolgirls for the vertical lines. If we wish to make some taller than the others, and some shorter, we can easily do so, as must be evident (see Fig. 79).

Note that in this first case the scale is below the horizon, so that we see over the heads of the figures, those nearest to us being the lowest down. That is to say, we are looking on this scene from a slightly raised platform.

Second Case.

To draw figures at different distances when their heads are above the horizon, or as they would appear to a person sitting on a low seat. The height of the heads varies according to the distance of the figures (Fig. 80).

Third Case.

How to draw figures when their heads are about the height of the horizon, or as they appear to a person standing on the same level or walking among them.

In this case the heads or the eyes are on a level with the horizon, and we have little necessity for a scale at the side unless it is for the purpose of ascertaining or marking their distances from the base line, and their respective heights, which of course vary; so in all cases allowance must be made for some being taller and some shorter than the scale measurement.





XXIX

THE HORIZON ABOVE THE FIGURES

In this example from De Hoogh the doorway to the left is higher up than the figure of the lady, and the effect seems to me more pleasing and natural for this kind of domestic subject. This delightful painter was not only a master of colour, of sunlight effect, and perfect composition, but also of perspective, and thoroughly understood the charm it gives to a picture, when cunningly introduced, for he makes the spectator feel that he can walk along his passages and courtyards. Note that he frequently puts the point of sight quite at the side of his canvas, as at S, which gives almost the effect of angular perspective whilst it preserves the flatness and simplicity of parallel or horizontal perspective.





XXX

LANDSCAPE PERSPECTIVE

In an extended view or landscape seen from a height, we have to consider the perspective plane as in a great measure lying above it, reaching from the base of the picture to the horizon; but of course pierced here and there by trees, mountains, buildings, &c. As a rule in such cases, we copy our perspective from nature, and do not trouble ourselves much about mathematical rules. It is as well, however, to know them, so that we may feel sure we are right, as this gives certainty to our touch and enables us to work with freedom. Nor must we, when painting from nature, forget to take into account the effects of atmosphere and the various tones of the different planes of distance, for this makes much of the difference between a good picture and a bad one; being a more subtle quality, it requires a keener artistic sense to discover and depict it. (See Figs. 95 and 103.)

If the landscape painter wishes to test his knowledge of perspective, let him dissect and work out one of Turner's pictures, or better still, put his own sketch from nature to the same test.



XXXI

FIGURES OF DIFFERENT HEIGHTS

THE CHESSBOARD

In this figure the same principle is applied as in the previous one, but the chessmen being of different heights we have to arrange the scale accordingly. First ascertain the exact height of each piece, as Q, K, B, which represent the queen, king, bishop, &c. Refer these dimensions to the scale, as shown at QKB, which will give us the perspective measurement of each piece according to the square on which it is placed.



This is shown in the above drawing (Fig. 83) in the case of the white queen and the black queen, &c. The castle, the knight, and the pawn being about the same height are measured from the fourth line of the scale marked C.





XXXII

APPLICATION OF THE VANISHING SCALE TO DRAWING FIGURES AT AN ANGLE WHEN THEIR VANISHING POINTS ARE INACCESSIBLE OR OUTSIDE THE PICTURE

This is exemplified in the drawing of a fence (Fig. 84). Form scale aS, bS, in accordance with the height of the fence or wall to be depicted. Let ao represent the direction or angle at which it is placed, draw od to meet the scale at d, at d raise vertical dc, which gives the height of the fence at oo'. Draw lines bo', eo, ao, &c., and it will be found that all these lines if produced will meet at the same point on the horizon. To divide the fence into spaces, divide base line af as required and proceed as already shown.