Transcriber's Note:

 The html version (see above) is strongly recommended to the reader because of its explanatory illustrations. In chapters LXII and later, the numerals in V1, V2, M1, M2 were printed as superscripts. Other letter-number pairs represent lines. 

 Points and lines were printed either as lower-case italicized letters, or as small uppercase letters. Most will be shown here with _lines_ representing italics. 

 Words and phrases in bold face have been enclosed between + signs (+this is bold face+)

Henry Frowde, M. A. Publisher to the University of OxfordLondon, Edinburgh, New YorkToronto and Melbourne

THE THEORY AND PRACTICE OF PERSPECTIVE

by

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

Teacher of Perspective at the Royal Academy

[Illustration: 'QUÎ FIT?']

OxfordAt the Clarendon Press1910

OxfordPrinted at the Clarendon Pressby 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 isgiven for it, than when we are merely shown how to do it without beingtold why it is so done; for in the latter case, instead of beingassisted by reason, our real help in all study, we have to rely uponmemory or our power of imitation, and to do simply as we are toldwithout thinking about it. The consequence is that at the very firstdifficulty we are left to flounder about in the dark, or to remaininactive till the master comes to our assistance. 

Now in this book it is proposed to enlist the reasoning faculty from thevery first: to let one problem grow out of another and to be dependenton the foregoing, as in geometry, and so to explain each thing we dothat there shall be no doubt in the mind as to the correctness of theproceeding. The student will thus gain the power of finding out any newproblem for himself, and will therefore acquire a true knowledge ofperspective. 

CONTENTS

BOOK I PageTHE NECESSITY OF THE STUDY OF PERSPECTIVE TO PAINTERS, SCULPTORS, AND ARCHITECTS 1WHAT IS PERSPECTIVE? 6THE 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 22RULES: 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° 64 XXIII. The Cube at an Angle of 45° 65 XXIV. Pavements Drawn by Means of Squares at 45° 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 143LXXXVIII. 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 212CXXXVIII. 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° to the Wall 266 CLXX. Mental Perspective 269

BOOK FIRST

THE NECESSITY OF THE STUDY OF PERSPECTIVETO PAINTERS, SCULPTORS, AND ARCHITECTS

Leonardo da Vinci tells us in his celebrated _Treatise on Painting_ thatthe 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 surfaceobjects which are in relief or distant one from the other; for this isthe simple art of painting. Objects appear smaller at a distance thannear to us, so by drawing them thus we give depth to our canvas. Theoutline of a ball is a mere flat circle, but with proper shading we makeit appear round, and this is the perspective of light and shade. 

'The next thing to be considered is the effect of the atmosphere andlight. If two figures are in the same coloured dress, and are standingone behind the other, then they should be of slightly different tone, so as to separate them. And in like manner, according to the distance ofthe 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 inoutline, 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 picturehe liked to feel that he could walk round it, and this exactly expressesthe impression that the true art of painting should make upon thespectator. 

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

Unfortunately, this study is too often neglected by our painters, someof them even complacently confessing their ignorance of it; while theordinary student either turns from it with distaste, or only enduresgoing through it with a view to passing an examination, little thinkingof what value it will be to him in working out his pictures. Whether themanner of teaching perspective is the cause of this dislike for it, I cannot say; but certainly most of our English books on the subject areanything 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 theirgrand compositions. In many cases they were even inspired by it inchoosing their subjects. When one looks at those sunny interiors, thosecorridors and courtyards by De Hooghe, with their figures far off andnear, one feels that their charm consists greatly in their perspective, as well as in their light and tone and colour. Or if we study thoseVenetian masterpieces by Paul Veronese, Titian, Tintoretto, and others, we become convinced that it was through their knowledge of perspectivethat they gave such space and grandeur to their canvases. 

I need not name all the great artists who have shown their interest anddelight in this study, both by writing about it and practising it, suchas Albert Dürer and others, but I cannot leave out our own Turner, whowas one of the greatest masters in this respect that ever lived; thoughin his case we can only judge of the results of his knowledge as shownin his pictures, for although he was Professor of Perspective at theRoyal Academy in 1807--over a hundred years ago--and took great painswith the diagrams he prepared to illustrate his lectures, they seemed tothe students to be full of confusion and obscurity; nor am I aware thatany record of them remains, although they must have contained somevaluable 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, hiscolonnades, his balconies, his steps, &c. , so that he may know where toplace his personages, and to measure their different sizes according totheir distances; indeed, he must make his stage and his scenery beforehe introduces his actors. He can then proceed with his composition, arrange his groups and the accessories with ease, and above all withcorrectness. But I have noticed that some of our cleverest painters willarrange their figures to please the eye, and when fairly advanced withtheir work will call in an expert, to (as they call it) put in theirperspective for them, but as it does not form part of their originalcomposition, it involves all sorts of difficulties and vexatiousalterings 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, tobe in harmony, should be the conception of one mind, and the perspectiveas much a part of the composition as the figures. 

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

Then again, with a view to giving different effects or impressions inthis decorative work, we must know where to place the horizon and thepoints of sight, for several of the latter are sometimes required whendealing with large surfaces such as the painting of walls, or stagescenery, or panoramas depicted on a cylindrical canvas and viewed fromthe centre thereof, where a fresh point of sight is required at everytwelve or sixteen feet. 

Without a true knowledge of perspective, none of these things can bedone. The artist should study them in the great compositions of themasters, by analysing their pictures and seeing how and for what reasonsthey applied their knowledge. Rubens put low horizons to most of hislarge figure-subjects, as in 'The Descent from the Cross', which notonly gave grandeur to his designs, but, seeing they were to be placedabove the eye, gave a more natural appearance to his figures. TheVenetians often put the horizon almost on a level with the base of thepicture or edge of the frame, and sometimes even below it; as in 'TheFamily of Darius at the Feet of Alexander', by Paul Veronese, and 'TheOrigin of the "Via Lactea"', by Tintoretto, both in our NationalGallery. But in order to do all these things, the artist in designinghis work must have the knowledge of perspective at his fingers' ends, and only the details, which are often tedious, should he leave to anassistant to work out for him. 

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

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

A sculptor student at the Academy, who was making his drawings rathercarelessly, asked me of what use perspective was to a sculptor. 'In thefirst place, ' I said, 'to reason out apparently difficult problems, andto 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 tomake your figures according to the height they are to be placed, andalso the boldness with which they should be treated to give them theirfull effect. ' He at once acknowledged that I was right, proved himselfan efficient pupil, and took much interest in his work. 

I cannot help thinking that the reason our public monuments so oftenfail to impress us with any sense of grandeur is in a great measureowing to the neglect of the scientific study of perspective. As anillustration of what I mean, let the student look at a good engraving orphotograph of the Arch of Constantine at Rome, or the Tombs of theMedici, by Michelangelo, in the sacristy of San Lorenzo at Florence. Andthen, 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 onlyloses much of its dignity by being placed on the ground instead of inthe niche above it, but throws all the other figures out of proportionor harmony, and was quite contrary to Michelangelo's intention. Indeed, this tomb, which was to have been the finest thing of its kind everdone, was really the tragedy of the great sculptor's life. 

The same remarks apply in a great measure to the architect as to thesculptor. The old builders knew the value of a knowledge of perspective, and, as in the case of Serlio, Vignola, and others, prefaced theirtreatises on architecture with chapters on geometry and perspective. Forit showed them how to give proper proportions to their buildings and thedetails thereof; how to give height and importance both to the interiorand exterior; also to give the right sizes of windows, doorways, columns, vaults, and other parts, and the various heights they shouldmake their towers, walls, arches, roofs, and so forth. One of the mostbeautiful examples of the application of this knowledge to architectureis the Campanile of the Cathedral, at Florence, built by Giotto andTaddeo Gaddi, who were painters as well as architects. Here it will beseen that the height of the windows is increased as they are placedhigher up in the building, and the top windows or openings into thebelfry are about six times the size of those in the lower story. 

WHAT IS PERSPECTIVE?

 [Illustration: Fig. 1. ]

Perspective is a subtle form of geometry; it represents figures andobjects not as they are but as we see them in space, whereas geometryrepresents figures not as we see them but as they are. When we have afront view of a figure such as a square, its perspective and geometricalappearance is the same, and we see it as it really is, that is, with allits sides equal and all its angles right angles, the perspective onlyvarying in size according to the distance we are from it; but if weplace that square flat on the table and look at it sideways or at anangle, then we become conscious of certain changes in its form--the sidefarthest from us appears shorter than that near to us, and all theangles are different. Thus A (Fig. 2) is a geometrical square and B isthe same square seen in perspective. 

 [Illustration: Fig. 2. ]

 [Illustration: Fig. 3. ]

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

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

 [Illustration: Fig. 4. ]

 [Illustration: Fig. 5. ]

In perspective, as in geometry, we deal with parallels, squares, triangles, cubes, circles, &c. ; but in perspective the same figure takesan endless variety of forms, whereas in geometry it has but one. Hereare three equal geometrical squares: they are all alike. Here are threeequal perspective squares, but all varied in form; and the same figurechanges 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 notonly a sure means of knowledge, but a study of the greatest interest. 

Perspective is said to have been formed into a science about thefifteenth century. Among the names mentioned by the unknown but pleasantauthor of _The Practice of Perspective_, written by a Jesuit of Paris inthe eighteenth century, we find Albert Dürer, who has left us some rulesand principles in the fourth book of his _Geometry_; Jean Cousin, whohas 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 earlywriters; Du Cerceau, Serigati, Solomon de Cause, Marolois, Vredemont;Guidus Ubaldus, who first introduced foreshortening; the Sieur deVaulizard, the Sieur Dufarges, Joshua Kirby, for whose _Method ofPerspective made Easy_ (?) Hogarth drew the well-known frontispiece; andlastly, 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 bookused by Sir Joshua Reynolds. [2] But nearly all these authors treatchiefly of parallel perspective, which they do with clearness andsimplicity, and also mathematically, as shown in the short treatisein Latin by Christian Wolff, but they scarcely touch upon the moredifficult problems of angular and oblique perspective. Of modernbooks, those to which I am most indebted are the _Traité Pratiquede Perspective_ of M. A. Cassagne (Paris, 1873), which is thoroughlyartistic, and full of pictorial examples admirably done; and toM. Henriet's _Cours Rational de Dessin_. There are many other foreignbooks of excellence, notably M. Thibault's _Perspective_, and someGerman and Swiss books, and yet, notwithstanding this imposing array ofauthors, I venture to say that many new features and original problemsare 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 whichdispenses with the cumbersome setting out usually adopted, and enablesus to draw figures at any angle without vanishing lines, &c. , and isalmost, if not quite, as simple as parallel perspective (see p. 133, Fig. 150, &c. ). How to measure distances by the square and diagonal, andto draw interiors thereby (p. 128, Fig. 144). How to explain the theoryof perspective by ocular demonstration, using a vertical sheet of glasswith strings, placed on a drawing-board, which I have found of thegreatest use (see p. 29, Fig. 29). Then again, I show how all ourperspective can be done inside the picture; that we can measure anydistance 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 onthirteen acres of ground, by putting it 1, 600 feet off (Fig. 224), &c. , &c. And while preserving the mathematical science, so that all ouroperations can be proved to be correct, my chief aim has been to make iteasy 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 ofdecoration, which is to be looked upon as picture writing rather thanpictorial art; a table, for instance, would be represented like aground-plan and the objects upon it in elevation or standing up. A rowof chariots with their horses and drivers side by side were placed oneover the other, and although the Egyptians had no doubt a reason forthis kind of representation, for they were grand artists, it seems to usvery primitive; and indeed quite young beginners who have never drawnfrom real objects have a tendency to do very much the same thing as thisancient people did, or even to emulate the mathematician and representthings not as they appear but as they are, and will make the top of atable an almost upright square and the objects upon it as if they wouldfall off. 

No doubt the Greeks had correct notions of perspective, for thepaintings on vases, and at Pompeii and Herculaneum, which were either byGreek artists or copied from Greek pictures, show some knowledge, thoughnot complete knowledge, of this science. Indeed, it is difficult toconceive of any great artist making his perspective very wrong, for ifhe can draw the human figure as the Greeks did, surely he can draw anangle. 

The Japanese, who are great observers of nature, seem to have got attheir perspective by copying what they saw, and, although they are notquite correct in a few things, they convey the idea of distance and maketheir horizontal planes look level, which are two important things inperspective. Some of their landscapes are beautiful; their trees, flowers, and foliage exquisitely drawn and arranged with the greatesttaste; whilst there is a character and go about their figures and birds, &c. , that can hardly be surpassed. All their pictures are lively andintelligent and appear to be executed with ease, which shows theirauthors to be complete masters of their craft. 

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

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

To illustrate what is perspective, and as an experiment that any one canmake, whether artist or not, let us stand at a window that looks out onto a courtyard or a street or a garden, &c. , and trace with apaint-brush charged with Indian ink or water-colour the outline ofwhatever view there happens to be outside, being careful to keep the eyealways in the same place by means of a rest; when this is dry, place apiece of drawing-paper over it and trace through with a pencil. Now wewill rub out the tracing on the glass, which is sure to be ratherclumsy, and, fixing our paper down on a board, proceed to draw the scenebefore us, using the main lines of our tracing as our guiding lines. 

If we take pains over our work, we shall find that, without troublingourselves much about rules, we have produced a perfect perspective ofperhaps a very difficult subject. After practising for some little timein this way we shall get accustomed to what are called perspectivedeformations, and soon be able to dispense with the glass and thetracing altogether and to sketch straight from nature, taking littlenote of perspective beyond fixing the point of sight and thehorizontal-line; in fact, doing what every artist does when he goes outsketching. 

 [Illustration: Fig. 6. This is a much reduced reproduction of a drawing made on my studio window in this way some twenty years ago, when the builder started covering the fields at the back with rows and rows of houses. ]

THE THEORY OF PERSPECTIVE

DEFINITIONS

I

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

 [Illustration: Fig. 7. ]

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

Another important line is _AB_, the +Base+ or +Ground line+, as it is onthis 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 inperspective, and is called the +Station-point+. 

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

 [Illustration: Fig. 8. ]

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_ arerepresented as receding from us, and in that case are called vanishinglines, a not quite satisfactory term. 

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

II

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

There are three things to be considered and understood before we canbegin a perspective drawing. First, the position of the eye in front ofthe picture, which is called the +Station-point+, and of course is notin the picture itself, but its position is indicated by a point on thepicture which is exactly opposite the eye of the spectator, and iscalled the +Point of Sight+, or +Principal Point+, or +Centre ofVision+, but we will keep to the first of these. 

 [Illustration: Fig. 9. ]

 [Illustration: Fig. 10. ]

If our picture plane is a sheet of glass, and is so placed that we cansee the landscape behind it or a sea-view, we shall find that thedistant line of the horizon passes through that point of sight, and wetherefore draw a line on our picture which exactly corresponds with it, and which we call the +Horizontal-line+ or +Horizon+. [3] The height ofthe horizon then depends entirely upon the position of the eye of thespectator: if he rises, so does the horizon; if he stoops or descends tolower ground, so does the horizon follow his movements. You may sit in aboat on a calm sea, and the horizon will be as low down as you are, oryou may go to the top of a high cliff, and still the horizon will be onthe 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 theeffect 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, thehorizon should be low down, so that these things may appear to towerabove you. If you wish to show a wide expanse of landscape, then youmust survey it from a height. In a composition of figures, you selectyour horizon according to the subject, and with a view to help thegrouping. Again, in portraits and decorative work to be placed high up, a low horizon is desirable, but I have already spoken of this subject inthe 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 greatimportance; as the distortions and disproportions arising from too neara view are to be avoided, the object of drawing being to make thingslook natural; thus, the floor should look level, and not as if it wererunning up hill--the top of a table flat, and not on a slant, as if cupsand what not, placed upon it, would fall off. 

In this figure we have a geometrical or ground plan of two squares atdifferent 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 tothe line _KK_ (which may represent the horizontal-line in the picture), where it intersects that line at _A·_ marks the distance that thespectator 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_, hisdistance from the point _S_ is also found on the horizon by means of thediagonal _BB´_, so that all lines or diagonals at 45° are drawn to thepoint of distance (see Rule 6). 

Figs. 12 and 13. In these two figures the difference is shown betweenthe 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 asthe second square, _Bef_. 

From this it will be seen how important it is to choose the right pointof 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 commonfault with photographs taken with a wide-angle lens, which throwseverything out of proportion, and will make the east end of a church ora cathedral appear higher than the steeple or tower; but as soon as wemake our line of distance sufficiently long, as at Fig. 13, objects taketheir right proportions and no distortion is noticeable. 

 [Illustration: Fig. 11. ]

 [Illustration: Fig. 12. ]

 [Illustration: Fig. 13. ]

In some books on perspective we are told to make the angle of vision60°, so that the distance _SD_ (Fig. 14) is to be rather less than thelength or height of the picture, as at _A_. The French recommend anangle of 28°, and to make the distance about double the length of thepicture, as at _B_ (Fig. 15), which is far more agreeable. For we mustremember that the distance-point is not only the point from which we aresupposed to make our tracing on the vertical transparent plane, or apoint transferred to the horizon to make our measurements by, but it isalso 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 closethat we must almost touch the canvas with our noses before we can seeits perspective properly. 

 [Illustration: Fig. 14. ]

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

 [Illustration: Fig. 15. ]

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

 [Illustration: Fig. 16. Cattle. By Paul Potter. ]

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

In landscape and cattle pictures a low horizon often gives space andair, as in this sketch from a picture by Paul Potter--where thehorizontal-line is placed at one quarter the height of the canvas. Indeed, a judicious use of the laws of perspective is a great aid tocomposition, and no picture ever looks right unless these laws areattended 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 greatmeasure the monotony of the snap-shot camera, with its everyday andwearisome commonplace. 

IV

PERSPECTIVE OF A POINT, VISUAL RAYS, &C. 

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

Now let us suppose a visual ray _EA_ to be directed to some small objecton the floor, say the head of a nail, _A_ (Fig. 17). If we interposebetween this nail and our eye a sheet of glass, _K_, placed verticallyon the floor, we continue to see the nail through the glass, and it iseasily understood that its perspective appearance thereon is the point_a_, where the visual ray passes through it. If now we trace on thefloor a line _AB_ from the nail to the spot _B_, just under the eye, andfrom the point _o_, where this line passes through or under the glass, we raise a perpendicular _oS_, that perpendicular passes through theprecise point that the visual ray passes through. The line _AB_ tracedon the floor is the horizontal trace of the visual ray, and it will beseen that the point _a_ is situated on the vertical raised from thishorizontal trace. 

 [Illustration: Fig. 17. ]

V

TRACE AND PROJECTION

If from any line _A_ or _B_ or _C_ (Fig. 18), &c. , we dropperpendiculars from different points of those lines on to a horizontalplane, the intersections of those verticals with the plane will be ona line called the horizontal trace or projection of the original line. We may liken these projections to sun-shadows when the sun is in themeridian, for it will be remarked that the trace does not represent thelength of the original line, but only so much of it as would be embracedby the verticals dropped from each end of it, and although line _A_ isthe same length as line _B_ its horizontal trace is longer than that ofthe other; that the projection of a curve (_C_) in this upright positionis a straight line, that of a horizontal line (_D_) is equal to it, andthe projection of a perpendicular or vertical (_E_) is a point only. The projections of lines or points can likewise be shown on a verticalplane, 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 bythe assistance of perspective, as will be shown farther on, we can carryout the most difficult propositions of descriptive geometry and of thegeometry of planes and solids. 

 [Illustration: Fig. 18. ]

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

 [Illustration: Fig. 19. ]

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 thehorizontal trace of the plane _fE_, and _e·f_ is the trace of the sameplane on the vertical plane _K_. 

VI

SCIENTIFIC DEFINITION OF PERSPECTIVE

The projections of the extremities of a right line which passes througha vertical plane being given, one on either side of it, to find theintersection of that line with the vertical plane. _AE_ (Fig. 20) is theright line. The projection of its extremity _A_ on the vertical plane is_a·_, the projection of _E_, the other extremity, is _e·_. _AS_ is thehorizontal 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 thevertical 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 thegiven line _AE_ and the vertical plane _K_. 

 [Illustration: Fig. 20. ]

This figure is similar to the previous one, except that the extremity_A_ of the given line is raised from the ground, but the samedemonstration applies to it. 

 [Illustration: Fig. 21. ]

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 theobject _A_ on the other side of the glass. Then if _E_ is the eye of thespectator, its projection on the picture is _S_, the point of sight. 

If I draw a dotted line from _E_ to little _a_, this represents anothervisual ray, and _o_, the point where it passes through the picture, isthe 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 theyrepresent parallel lines _Aa·_ and _ag_, which are at right angles tothe picture plane. This is the most important fact in perspective, andwill be more fully explained farther on, when we speak of retreating orso-called vanishing lines. 

RULES

VII

THE RULES AND CONDITIONS OF PERSPECTIVE

The conditions of linear perspective are somewhat rigid. In the firstplace, 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 thiswe shall speak later on. Then again, the eye must be placed in a certainposition, 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 perpendicularto the ground or horizontal plane, which plane is supposed to be aslevel 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 notpartake of the rotundity of the earth. 

We can only work out our propositions and figures in space withmathematical precision by adopting such conditions as the above. Butafterwards the artist or draughtsman may modify and suit them to a moreelastic view of things; that is, he can make his figures separate fromone another, instead of their outlines coming close together as they dowhen we look at them with only one eye. Also he will allow for theunevenness of the ground and the roundness of our globe; he may evenmove his head and his eyes, and use both of them, and in fact makehimself quite at his ease when he is out sketching, for Nature does allhis perspective for him. At the same time, a knowledge of this rigidperspective is the sure and unerring basis of his freehand drawing. 

 [Illustration: Fig. 22. ]

 [Illustration: Fig. 23. Front view of above figure. ]

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 thesame proportion as _AB_ (Fig. 24), the original line, and _a·b·_, theperspective line, and if the one is divided at _O_ the other is dividedat _o·_ in the same way. 

 [Illustration: Fig. 24. ]

It is not an uncommon error to suppose that the vertical lines of a highbuilding should converge towards the top; so they would if we stood atthe foot of that building and looked up, for then we should alter theconditions of our perspective, and our point of sight, instead of beingon the horizon, would be up in the sky. But if we stood sufficiently faraway, so as to bring the whole of the building within our angle ofvision, and the point of sight down to the horizon, then these samelines would appear perfectly parallel, and the different stories intheir true proportion. 

RULE 3

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

 [Illustration: Fig. 25. ]

 [Illustration: Fig. 26. ]

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

RULE 4

All lines situated in a plane that is parallel to the picture planediminish in proportion as they become more distant, but do not undergoany perspective deformation; and remain in the same relation andproportion each to each as the original lines. This is called the frontview. 

 [Illustration: Fig. 27. ]

RULE 5

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

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

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

An arrangement such as there indicated is the best means of illustratingthis rule. But instead of tracing the outline of the square or cube onthe 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 whichI pass two loose strings _A_ and _B_, fixing their ends at _S_. 

 [Illustration: Fig. 28. ]

 [Illustration: Fig. 29. ]

As _SD_ represents the distance the spectator is from the glass orpicture, I make string _SA_ equal in length to _SD_. Now if the pupiltakes 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 distancefrom the picture. Let him then take the other string, _SB_, in the otherhand, and apply it to point _b´_ where the square touches the glass, andhe 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 thesquare, his string will exactly tally or cover the side _a·e_, and hewill thus have ocular demonstration of this important rule. 

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

 [Illustration: Fig. 30. ]

RULE 6

All horizontals which are at 45°, or half a right angle to the pictureplane, are drawn to the point of distance. 

We have already seen that the diagonal of the perspective square, ifproduced to meet the horizon on the picture, will mark on that horizonthe distance that the spectator is from the point of sight (seedefinition, p. 16). This point of distance becomes then the measuringpoint 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° to the base _AB_, and_AC_ is therefore the diagonal of a square. The line 1C is madeparallel to _AB_, consequently A1CB is a square in perspective. Theline _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 thepoint 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 berepeated any number of times, say five, so that in this figure _BE_is five times the length of _AB_. 

 [Illustration: Fig. 31. ]

RULE 7

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

 [Illustration: Fig. 32. ]

In Fig. 32, the dotted line _BD_, drawn to the point of distance _D_, isat an angle of 45° 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 apoint _V´_, within the point of distance and nearer to the point ofsight _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 distancepoint. 

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

RULE 8

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

 [Illustration: Fig. 33. ]

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

 [Illustration: Fig. 34. ]

RULE 9

The farther a point is removed from the picture plane the nearer doesits perspective appearance approach the horizontal line so long as it isviewed from the same position. On the contrary, if the spectatorretreats from the picture plane _K_ (which we suppose to betransparent), the point remaining at the same place, the perspectiveappearance of this point will approach the ground-line in proportion tothe distance of the spectator. 

 [Illustrations: Fig. 35. Fig. 36. The spectator at two different distances from the picture. ]

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

 [Illustration: Fig. 37. ]

 [Illustrations: The picture at two different distances from the point. Fig. 38. Fig. 39. ]

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

RULE 10

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

 [Illustration: Fig. 40. ]

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

Suppose _S_ to be the spectator, _AB_ a transparent vertical plane whichrepresents the picture seen edgeways, and _HS_ and _DC_ two parallellines, mark off spaces between these parallels equal to _SC_, the heightof 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 afourth, and so on, and if we multiplied these spaces _ad infinitum_ wemust 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 tobe also a yard, then if one of these were a thousand yards away itsrepresentation at _AB_ would be the thousandth part of a yard, or tenthousand yards away, its representation at _AB_ would be theten-thousandth part, and whatever the distance it must always besomething; and therefore _HS_ and _DC_, however far they may be producedand however close they may appear to get, can never meet. 

 [Illustration: Fig. 41. ]

Fig. 41 is a perspective view of the same figure--but more extended. Itwill be seen that a line drawn from the tenth upright _K_ to _S_ cutsoff a tenth of _AB_. We look then upon these two lines _SP_, _OP_, asthe 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 ruleby placing a looking-glass on one of the walls of his studio and thensketching himself and his surroundings as seen therein. He will findthat all the horizontals at right angles to the glass will converge tohis own eye. This rule applies equally to lines which are at an angle tothe picture plane as to those that are at right angles or perpendicularto it, as in Rule 7. It also applies to those on an inclined plane, asin Rule 8. 

 [Illustration: Fig. 42. Sketch of artist in studio. ]

With the above rules and a clear notion of the definitions andconditions of perspective, we should be able to work out any propositionor any new figure that may present itself. At any rate, a thoroughunderstanding of these few pages will make the labour now before ussimple and easy. I hope, too, it may be found interesting. There isalways a certain pleasure in deceiving and being deceived by the senses, and in optical and other illusions, such as making things appear far offthat are quite near, in making a picture of an object on a flat surfaceto look as if it stood out and in relief by a kind of magic. But thereis, I think, a still greater pleasure than this, namely, in inventionand in overcoming difficulties--in finding out how to do things forourselves 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 tothat base in the picture. 

RULE 4

All lines situated in a plane that is parallel to the picture planediminish in proportion as they become more distant, but do not undergoany perspective deformation. This is called the front view. 

RULE 5

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

RULE 6

All horizontals which are at 45° to the picture plane are drawn to thepoint of distance. 

RULE 7

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

RULE 8

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

RULE 9

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

RULE 10

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

BOOK SECOND

THE PRACTICE OF PERSPECTIVE

In the foregoing book we have explained the theory or science ofperspective; we now have to make use of our knowledge and to apply it tothe 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 tothe picture plane and the other two at right angles to it, and which wecall

IX

THE SQUARE IN PARALLEL PERSPECTIVE

From a given point on the base line of the picture draw a line at rightangles 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 tothe point of sight _S_, and this line will be at right angles to thebase, as explained in Rule 5, and consequently angle _APS_ will be equalto angle _SPB_, although it does not look so here. This is our firstdifficulty, but one that we shall soon get over. 

 [Illustration: Fig. 43. ]

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

 [Illustration: Fig. 44. ]

X

THE DIAGONAL

From a given point on the base line draw a line at 45°, or half a rightangle, to that base. Let _P_ be the given point. Draw a line from _P_ tothe point of distance _D_ and this line _PD_ will be at an angle of 45°, or at the same angle as the diagonal of a square. See definitions. 

 [Illustration: Fig. 45. ]

XI

THE SQUARE

Draw a square in parallel perspective on a given length on the baseline. 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 atright angles to the base (see Fig. 43). From _a_ draw diagonal _aD_ topoint of distance _D_; this line will be 45° to base. At point _c_, where it cuts _bS_, draw _dc_ parallel to _ab_ and _abcd_ is the squarerequired. 

 [Illustration: Fig. 46. ]

We have here proceeded in much the same way as in drawing a geometricalsquare (Fig. 47), by drawing two lines _AE_ and _BC_ at right angles toa given line, _AB_, and from _A_, drawing the diagonal _AC_ at 45° tillit 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 beparallel to each other, and therefore are perspectively equidistant, sothat all lines parallel to _AB_ and lying between them, such as _ad_, _cf_, &c. , must be equal. 

 [Illustration: Fig. 47. ]

So likewise all diagonals drawn to the point of distance, which arecontained between these parallels, such as _Ad_, _af_, &c. , must beequal. For all straight lines which meet at any point on the horizon areperspectively parallel to each other, just as two geometrical parallelscrossing two others at any angle, as at Fig. 49. Note also (Fig. 48)that all squares formed between the two vanishing lines _AS_, _BS_, andby the aid of these diagonals, are also equal, and further, that anynumber of squares such as are shown in this figure (Fig. 50), formed inthe same way and having equal bases, are also equal; and the ninesquares contained in the square _abcd_ being equal, they divide eachside of the larger square into three equal parts. 

 [Illustration: Fig. 48. ]

 [Illustration: Fig. 49. ]

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 atright angles to the base; and also how we can measure any width ornumber of widths on a line such as _dc_, that is, parallel to the baseof the picture, however remote it may be from that base. 

 [Illustration: Fig. 50. ]

XII

GEOMETRICAL AND PERSPECTIVE FIGURES CONTRASTED

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

 [Illustration: Fig. 51 A. The geometrical view. ]

 [Illustration: Fig. 51 B. The perspective view. ]

 [Illustration: Fig. 51 C. A geometrical square. ]

 [Illustration: Fig. 51 D. A perspective square. ]

 [Illustration: Fig. 51 E. Geometrical parallels. ]

 [Illustration: Fig. 51 F. Perspective parallels. ]

 [Illustration: Fig. 51 G. Geometrical perpendicular. ]

 [Illustration: Fig. 51 H. Perspective perpendicular. ]

 [Illustration: Fig. 51 I. Geometrical equal lines. ]

 [Illustration: Fig. 51 J. Perspective equal lines. ]

 [Illustration: Fig. 51 K. A geometrical circle. ]

 [Illustration: Fig. 51 L. A perspective circle. ]

XIII

OF CERTAIN TERMS MADE USE OF IN PERSPECTIVE

Of course when we speak of +Perpendiculars+ we do not mean verticalsonly, but straight lines at right angles to other lines in any position. Also in speaking of +lines+ a right or +straight line+ is to beunderstood; or when we speak of +horizontals+ we mean all straight linesthat are parallel to the perspective plane, such as those on Fig. 52, nomatter what direction they take so long as they are level. They are notto be confused with the horizon or horizontal-line. 

 [Illustration: Fig. 52. Horizontals. ]

There are one or two other terms used in perspective which are notsatisfactory because they are confusing, such as vanishing lines andvanishing points. The French term, _fuyante_ or _lignes fuyantes_, orgoing-away lines, is more expressive; and _point de fuite_, instead ofvanishing point, is much better. I have occasionally called the formerretreating lines, but the simple meaning is, lines that are not parallelto the picture plane; but a vanishing line implies a line thatdisappears, and a vanishing point implies a point that gradually goesout of sight. Still, it is difficult to alter terms that custom hasendorsed. 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 planeinto any number of given measurements. Let _SA_ be the given line. From_A_ measure off on the base line the divisions required, say five of1 foot each; from each division draw diagonals to point of distance _D_, and where these intersect the line _AC_ the corresponding divisions willbe found. Note that as lines _AB_ and _AC_ are two sides of the samesquare they are necessarily equal, and so also are the divisions on _AC_equal to those on _AB_. 

 [Illustration: Fig. 53. ]

The line _AB_ being the base of the picture, it is at the same time aperspective line and a geometrical one, so that we can use it as a scalefor measuring given lengths thereon, but should there not be enough roomon it to measure the required number we draw a second line, _DC_, whichwe divide in the same proportion and proceed to divide _cf_. Thisgeometrical figure gives, as it were, a bird's-eye view or ground-planof the above. 

 [Illustration: Fig. 54. ]

XV

HOW TO PLACE SQUARES IN GIVEN POSITIONS

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

 [Illustration: Fig. 55. ]

Let _ab_ (Fig. 55) represent the base line of the picture divided into acertain number of feet; _HD_ the horizon, _VO_ the vertical. It isrequired to draw a square 3 feet wide, 2 feet to the right of thevertical, and 1 foot from the base. 

First measure from _V_, 2 feet to _e_, which gives the distance from thevertical. Second, from _e_ measure 3 feet to _b_, which gives the widthof the square; from _e_ and _b_ draw _eS_, _bS_, to point of sight. Fromeither _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 therequired distance from base. Draw _Pg_ and _B_ parallel to the base, andwe have the required square. 

Square _A_ to the left of the vertical is 2½ feet wide, 1 foot from thevertical 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 inarchitectural drawing, where it is indispensable, but in working out ourpropositions and figures it is not always desirable. A given lengthindicated by a line is generally sufficient for our requirements. Towork out every problem to scale is not only tedious and mechanical, butwastes time, and also takes the mind of the student away from thereasoning out of the subject. 

XVI

HOW TO DRAW PAVEMENTS, &C. 

Divide a vanishing line into parts varying in length. Let _BS·_ be thevanishing line: divide it into 4 long and 3 short spaces; then proceedas in the previous figure. If we draw horizontals through the pointsthus obtained and from these raise verticals, we form, as it were, theinterior of a building in which we can place pillars and other objects. 

 [Illustration: Fig. 56. ]

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

 [Illustration: Fig. 57. ]

 [Illustration: Fig. 58. ]

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. 

 [Illustration: Fig. 59. ]

_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·_, atexactly the same distance from the point of sight as is the point ofdistance on the horizontal line. 

It will be seen by examining this figure that the diagonals of thesquares in a vertical position are drawn to this verticaldistance-point, thus saving the necessity of taking the measurementsfirst on the base line, as at _CB_, which in the case of distantobjects, such as the farthest window, would be very inconvenient. Notethat the windows at _K_ are twice as high as they are wide. Of coursethese or any other objects could be made of any proportion. 

 [Illustration: Fig. 60. ]

XIX

THE FRONT VIEW OF THE SQUARE AND OF THE PROPORTIONS OF FIGURESAT DIFFERENT HEIGHTS

According to Rule 4, all lines situated in a plane parallel to thepicture plane diminish in length as they become more distant, but remainin the same proportions each to each as the original lines; as squaresor 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 ofdifferent heights, such as statuary or ornament in a building, ifactually equal in size, so must we represent them. 

 [Illustration: Fig. 61. ]

 [Illustration: Fig. 62. ]

In this square _K_, with the checker pattern, we should not think ofmaking the top squares smaller than the bottom ones; so it is withfigures. 

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

 [Illustration: Fig. 63. ]

In Fig. 63 the two statues _A_ and _B_ are the same size. So if tracedthrough a vertical sheet of glass, _K_, as at _c_ and _d_, they wouldalso be equal; but as the angle _b_ at which the upper one is seen issmaller than angle _a_, at which the lower figure or statue is seen, itwill appear smaller to the spectator (_S_) both in reality and in thepicture. 

 [Illustration: Fig. 64. ]

But if we wish them to appear the same size to the spectator who isviewing them from below, we must make the angles _a_ and _b_ (Fig. 64), at which they are viewed, both equal. Then draw lines through equalarcs, as at _c_ and _d_, till they cut the vertical _NO_ (representingthe side of the building where the figures are to be placed). We shallthen obtain the exact size of the figure at that height, which will makeit look the same size as the lower one, _N_. The same rule applies tothe picture _K_, when it is of large proportions. As an example inpainting, take Michelangelo's large altar-piece in the Sistine Chapel, 'The Last Judgement'; here the figures forming the upper group, with ourLord in judgement surrounded by saints, are about four times the size, that is, about twice the height, of those at the lower part of thefresco. The figures on the ceiling of the same chapel are studied notonly according to their height from the pavement, which is 60 ft. , butto suit the arched form of it. For instance, the head of the figure ofJonah at the end over the altar is thrown back in the design, but owingto the curvature in the architecture is actually more forward than thefeet. Then again, the prophets and sybils seated round the ceiling, which are perhaps the grandest figures in the whole range of art, wouldbe 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. 

 [Illustration: Fig. 65. ]

Of course, much depends upon the distance we view these statues orpaintings from. In interiors, such as churches, halls, galleries, &c. , we can make a fair calculation, such as the length of the nave, if thepicture is an altar-piece--or say, half the length; so also withstatuary in niches, friezes, and other architectural ornaments. Thenearer we are to them, and the more we have to look up, the larger willthe upper figures have to be; but if these are on the outside of abuilding that can be looked at from a long distance, then it is betternot to have too great a difference. 

 [Illustration: Fig. 66. 1909. ]

These remarks apply also to architecture in a great measure. Buildingsthat can only be seen from the street below, as pictures in a narrowgallery, require a different treatment from those out in the open, thatare to be looked at from a distance. In the former case the sametreatment as the Campanile at Florence is in some cases desirable, butall must depend upon the taste and judgement of the architect in suchmatters. 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 ofimportance. Hence the many mistakes in our buildings, and theunsatisfactory and mean look of some of our public monuments. 

XX

OF PICTURES THAT ARE PAINTED ACCORDING TO THE POSITIONTHEY 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 inwhich they are painted and the perspective adopted by their painters. Itwill be seen that those placed on the line level with the eye have theirhorizon lines fairly high up, and are not suited to be placed anyhigher. The Giorgione in the centre, the Monna Lisa to the right, andthe Velasquez and Watteau to the left, are all pictures that fit thatposition; 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 outagainst the sky, with its low horizon and strong contrast of light anddark; nor does it lose a bit by being placed where it is, over thesmaller pictures. 

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

It seems to me it is well, both for those who paint and for those whohang pictures, that this subject should be taken into consideration. Forit must be seen by this illustration that a bigger style is adopted bythe artists who paint for high places in palaces or churches than bythose who produce smaller easel-pictures intended to be seen close. Unfortunately, at our picture exhibitions, we see too often that nearlyall the works, whether on large or small canvases, are painted for theline, and that those which happen to get high up look as if they weretoppling over, because they have such a high horizontal line; andinstead of the figures telling against the sky, as in this picture ofthe 'Infant' by Velasquez, the Reynolds, and the fat man treading on aflag, we have fields or sea or distant landscape almost to the top ofthe frame, and all, so methinks, because the perspective is notsufficiently considered. 

_Note. _--Whilst on this subject, I may note that the painter in hislarge decorative work often had difficulties to contend with, whicharose from the form of the building or the shape of the wall on which hehad to place his frescoes. Painting on the ceiling was no easy task, andMichelangelo, in a humorous sonnet addressed to Giovanni da Pistoya, gives a burlesque portrait of himself while he was painting the SistineChapel:--

 _"I'ho già 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, onwhich the picture can be painted in the studio and afterwards placed onthe wall. 

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

In Chapter CCCI he shows us how to draw a figure twenty-four bracciahigh 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 drawslines to a point _F_ on the floor of the building, marking theirintersections on the profile of the wall somewhat in the manner we haveindicated, which serve as guides in making the outline to be traced. 

 [Illustration: Fig. 67. 

'Draw upon part of wall _MN_ half the figure you mean to represent, andthe other half upon the cove above (_MR_). ' Leonardo da Vinci's_Treatise on Painting_. ]

XXI

INTERIORS

 [Illustration: Fig. 68. Interior by de Hoogh. ]

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

 [Illustration: Fig. 69. ]

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

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

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

XXII

THE SQUARE AT AN ANGLE OF 45°

When the square is at an angle of 45° to the base line, then its sidesare drawn respectively to the points of distance, _DD_, and one of itsdiagonals which is at right angles to the base is drawn to the point ofsight _S_, and the other _ab_, is parallel to that base or ground line. 

 [Illustration: Fig. 70. ]

To draw a pavement with its squares at this angle is but anamplification of the above figure. Mark off on base equal distances, 1, 2, 3, &c. , representing the diagonals of required squares, and from eachof these points draw lines to points of distance _DD´_. These lines willintersect each other, and so form the squares of the pavement; to ensurecorrectness, lines should also be drawn from these points 1, 2, 3, tothe point of sight _S_, and also horizontals parallel to the base, as_ab_. 

 [Illustration: Fig. 71. ]

XXIII

THE CUBE AT AN ANGLE OF 45°

Having drawn the square at an angle of 45°, as shown in the previousfigure, 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 thegeometrical 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 twopoints of distance, and so completing the figure. 

 [Illustration: Fig. 72. ]

XXIV

PAVEMENTS DRAWN BY MEANS OF SQUARES AT 45°

 [Illustration: Fig. 73. ]

 [Illustration: Fig. 74. ]

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

 [Illustration: Fig. 75. ]

XXV

THE PERSPECTIVE VANISHING SCALE

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

 [Illustration: Fig. 76. ]

Let us suppose that in this figure (76) _AB_ and _A·B·_ each represent5 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 allthe horizontals _e_, _f_, _g_, _h_, drawn between _A·O_ and _B·O_ alsorepresent 5 feet each. So that by the aid of this scale we can give theexact perspective height and width of any object in the picture, howeverfar it may be from the base line, for of course we can increase ordiminish our measurements at _AB_ and _A·B·_ to whatever length werequire. 

As it may not be quite evident at first that the points _O_ may be takenat 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, raisevertical _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 willmeasure a given height at a given distance. 

 [Illustration: Fig. 77. ]

(See axioms of geometry. )

 [Illustration: Fig. 79. Schoolgirls. ]

 [Illustration: Fig. 80. Cavaliers. ]

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 ourfigures. _AB_ represents their average height, so we have made our scale_AO_, _BO_, accordingly. From each point marked we draw a line parallelto the base till it reaches the scale. From the point where it touchesthe line _AO_, raise perpendicular as _a_, which gives the heightrequired at that distance, and must be referred back to the figureitself. 

 [Illustration: Fig. 78. ]

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 havesubstituted these schoolgirls for the vertical lines. If we wish to makesome taller than the others, and some shorter, we can easily do so, asmust be evident (see Fig. 79). 

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

_Second Case. _

To draw figures at different distances when their heads are above thehorizon, or as they would appear to a person sitting on a low seat. Theheight 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 thehorizon, or as they appear to a person standing on the same level orwalking among them. 

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

 [Illustration: Fig. 81. ]

XXIX

THE HORIZON ABOVE THE FIGURES

In this example from De Hoogh the doorway to the left is higher up thanthe figure of the lady, and the effect seems to me more pleasing andnatural for this kind of domestic subject. This delightful painter wasnot only a master of colour, of sunlight effect, and perfectcomposition, but also of perspective, and thoroughly understood thecharm it gives to a picture, when cunningly introduced, for he makes thespectator feel that he can walk along his passages and courtyards. Notethat he frequently puts the point of sight quite at the side of hiscanvas, as at _S_, which gives almost the effect of angular perspectivewhilst it preserves the flatness and simplicity of parallel orhorizontal perspective. 

 [Illustration: Fig. 82. Courtyard by De Hoogh. ]

XXX

LANDSCAPE PERSPECTIVE

In an extended view or landscape seen from a height, we have to considerthe perspective plane as in a great measure lying above it, reachingfrom the base of the picture to the horizon; but of course pierced hereand there by trees, mountains, buildings, &c. As a rule in such cases, we copy our perspective from nature, and do not trouble ourselves muchabout mathematical rules. It is as well, however, to know them, so thatwe may feel sure we are right, as this gives certainty to our touch andenables us to work with freedom. Nor must we, when painting from nature, forget to take into account the effects of atmosphere and the varioustones of the different planes of distance, for this makes much of thedifference between a good picture and a bad one; being a more subtlequality, 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, butthe chessmen being of different heights we have to arrange the scaleaccordingly. First ascertain the exact height of each piece, as _Q_, _K_, _B_, which represent the queen, king, bishop, &c. Refer thesedimensions to the scale, as shown at _QKB_, which will give us theperspective measurement of each piece according to the square on whichit is placed. 

 [Illustration: Fig. 83. Chessboard and Men. ]

This is shown in the above drawing (Fig. 83) in the case of the whitequeen and the black queen, &c. The castle, the knight, and the pawnbeing about the same height are measured from the fourth line of thescale marked _C_. 

 [Illustration: Fig. 84. ]

XXXII

APPLICATION OF THE VANISHING SCALE TO DRAWING FIGURES AT AN ANGLEWHEN 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 bedepicted. Let _ao_ represent the direction or angle at which it isplaced, 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 willmeet at the same point on the horizon. To divide the fence into spaces, divide base line _af_ as required and proceed as already shown. 

XXXIII

THE REDUCED DISTANCE. HOW TO PROCEED WHEN THE POINT OF DISTANCEIS INACCESSIBLE

It has already been shown that too near a point of distance isobjectionable on account of the distortion and disproportion resultingfrom it. At the same time, the long distance-point must be some way outof the picture and therefore inconvenient. The object of the reduceddistance is to bring that point within the picture. 

 [Illustration: Fig. 85. ]

In Fig. 85 we have made the distance nearly twice the length of the baseof the picture, and consequently a long way out of it. Draw _Sa_, _Sb_, and from _a_ draw _aD_ to point of distance, which cuts _Sb_ at _o_, anddetermines the depth of the square _acob_. But we can find that samepoint if we take half the base and draw a line from ½ base to ½distance. But even this ½ distance-point does not come inside thepicture, so we take a fourth of the base and a fourth of the distanceand draw a line from ¼ base to ¼ distance. We shall find that it passesprecisely through the same point _o_ as the other lines _aD_, &c. Weare thus able to find the required point _o_ without going outside thepicture. 

Of course we could in the same way take an 8th or even a 16th distance, but the great use of this reduced distance, in addition to the above, is that it enables us to measure any depth into the picture with thegreatest ease. 

It will be seen in the next figure that without having to extend thebase, as is usually done, we can multiply that base to any amount bymaking use of these reduced distances on the horizontal line. This isquite a new method of proceeding, and it will be seen is mathematicallycorrect. 

XXXIV

HOW TO DRAW A LONG PASSAGE OR CLOISTER BY MEANS OF THE REDUCED DISTANCE

 [Illustration: Fig. 86. ]

In Fig. 86 we have divided the base of the first square into four equalparts, which may represent so many feet, so that A4 and _Bd_ being theretreating sides of the square each represents 4 feet. But we foundpoint ¼D by drawing 3D from ¼ base to ¼ distance, and by proceedingin the same way from each division, _A_, 1, 2, 3, we mark off on _SB_four spaces each equal to 4 feet, in all 16 feet, so that by taking thewhole base and the ¼ distance we find point _O_, which is distant fourtimes the length of the base _AB_. We can multiply this distance to anyamount by drawing other diagonals to 8th distance, &c. The same ruleapplies to this corridor (Fig. 87 and Fig. 88). 

 [Illustration: Fig. 87. ]

 [Illustration: Fig. 88. ]

XXXV

HOW TO FORM A VANISHING SCALE THAT SHALL GIVE THE HEIGHT, DEPTH, AND DISTANCE OF ANY OBJECT IN THE PICTURE

If we make our scale to vanish to the point of sight, as in Fig. 89, wecan make _SB_, the lower line thereof, a measuring line for distances. Let us first of all divide the base _AB_ into eight parts, each partrepresenting 5 feet. From each division draw lines to 8th distance; bytheir intersections with _SB_ we obtain measurements of 40, 80, 120, 160, &c. , feet. Now divide the side of the picture _BE_ in the samemanner as the base, which gives us the height of 40 feet. From theside _BE_ draw lines 5S, 15S, &c. , to point of sight, and from eachdivision on the base line also draw lines 5S, 10S, 15S, &c. , topoint of sight, and from each division on _SB_, such as 40, 80, &c. , draw horizontals parallel to base. We thus obtain squares 40 feet wide, beginning at base _AB_ and reaching as far as required. Note how theheight of the flagstaff, which is 140 feet high and 280 feet distant, isobtained. So also any buildings or other objects can be measured, suchas those shown on the left of the picture. 

 [Illustration: Fig. 89. ]

XXXVI

MEASURING SCALE ON GROUND

A simple and very old method of drawing buildings, &c. , and giving themtheir right width and height is by means of squares of a given size, drawn on the ground. 

 [Illustration: Fig. 90. ]

In the above sketch (Fig. 90) the squares on the ground represent 3 feeteach way, or one square yard. Taking this as our standard measure, wefind the door on the left is 10 feet high, that the archway at the endis 21 feet high and 12 feet wide, and so on. 

 [Illustration: Fig. 91. Natural Perspective. ]

 [Illustration: Fig. 92. Honfleur. ]

Fig. 91 is a sketch made at Sandwich, Kent, and shows a somewhat similarsubject to Fig. 84, but the irregularity and freedom of the perspectivegives it a charm far beyond the rigid precision of the other, while itconforms to its main laws. This sketch, however, is the real artist'sperspective, or what we might term natural perspective. 

XXXVII

APPLICATION OF THE REDUCED DISTANCE AND THE VANISHING SCALETO DRAWING A LIGHTHOUSE, &C. 

[Above illustration:Perspective of a lighthouse 135 feet high at 800 feet distance. ]

 [Illustration: Fig. 93. Key to Fig. 92, Honfleur. ]

In the drawing of Honfleur (Fig. 92) we divide the base _AB_ as in theprevious figure, but the spaces measure 5 feet instead of 3 feet: sothat taking the 8th distance, the divisions on the vanishing line _BS_measure 40 feet each, and at point _O_ we have 400 feet of distance, butwe require 800. So we again reduce the distance to a 16th. We thusmultiply the base by 16. Now let us take a base of 50 feet at _f_ anddraw line _fD_ to 16th distance; if we multiply 50 feet by 16 we obtainthe 800 feet required. 

The height of the lighthouse is found by means of the vanishing scale, which is 15 feet below and 15 feet above the horizon, or 30 feet fromthe sea-level. At _L_ we raise a vertical _LM_, which shows the positionof the lighthouse. Then on that vertical measure the height required asshown in the figure. 

The 800 feet could be obtained at once by drawing line _fD_, or 50 feet, to 16th distance. The other measurements obtained by 8th distance servefor nearer buildings. 

XXXVIII

HOW TO MEASURE LONG DISTANCES SUCH AS A MILE OR UPWARDS

The wonderful effect of distance in Turner's pictures is not to beachieved by mere measurement, and indeed can only be properly done bystudying Nature and drawing her perspective as she presents it to us. Atthe same time it is useful to be able to test and to set out distancesin arranging a composition. This latter, if neglected, often leads togreat difficulties and sometimes to repainting. 

To show the method of measuring very long distances we have to work witha very small scale to the foot, and in Fig. 94 I have divided the base_AB_ into eleven parts, each part representing 10 feet. First draw _AS_and _BS_ to point of sight. From _A_ draw _AD_ to ¼ distance, and weobtain at 440 on line _BS_ four times the length of _AB_, or 110 feet× 4 = 440 feet. Again, taking the whole base and drawing a line from _S_to 8th distance we obtain eight times 110 feet or 880 feet. If now weuse the 16th distance we get sixteen times 110 feet, or 1, 760 feet, one-third of a mile; by repeating this process, but by using the base at1, 760, which is the same length in perspective as _AB_, we obtain 3, 520feet, and then again using the base at 3, 520 and proceeding in the sameway we obtain 5, 280 feet, or one mile to the archway. The flags showtheir heights at their respective distances from the base. By the scaleat the side of the picture, _BO_, we can measure any height above or anydepth below the perspective plane. 

 [Illustration: Fig. 94. ]

_Note_. --This figure (here much reduced) should be drawn large by thestudent, so that the numbering, &c. , may be made more distinct. Indeed, many of the other figures should be copied large, and worked out withcare, as lessons in perspective. 

XXXIX

FURTHER ILLUSTRATION OF LONG DISTANCES AND EXTENDED VIEWS

An extended view is generally taken from an elevated position, so thatthe principal part of the landscape lies beneath the perspective plane, as already noted, and we shall presently treat of objects and figures onuneven ground. In the previous figure is shown how we can measureheights and depths to any extent. But when we turn to a drawing byTurner, such as the 'View from Richmond Hill', we feel that the only wayto accomplish such perspective as this, is to go and draw it fromnature, and even then to use our judgement, as he did, as to how much wemay emphasize or even exaggerate certain features. 

 [Illustration: Fig. 95. Turner's View from Richmond Hill. ]

Note in this view the foreground on which the principal figures stand ison a level with the perspective plane, while the river and surroundingpark and woods are hundreds of feet below us and stretch away for milesinto the distance. The contrasts obtained by this arrangement increasethe illusion of space, and the figures in the foreground give as it werea standard of measurement, and by their contrast to the size of thetrees show us how far away those trees are. 

XL

HOW TO ASCERTAIN THE RELATIVE HEIGHTS OF FIGURES ON AN INCLINED PLANE

 [Illustration: Fig. 96. ]

The three figures to the right marked _f_, _g_, _b_ (Fig. 96) are onlevel ground, and we measure them by the vanishing scale _aS_, _bS_. Those to the left, which are repetitions of them, are on an inclinedplane, the vanishing point of which is _S·_; by the side of this planewe have placed another vanishing scale _a·S·_, _b·S·_, by which wemeasure the figures on that incline in the same way as on the levelplane. It will be seen that if a horizontal line is drawn from the footof one of these figures, say _G_, to point _O_ on the edge of theincline, then dropped vertically to _o·_, then again carried on to _o··_where the other figure _g_ is, we find it is the same height and alsothat the other vanishing scale is the same width at that distance, sothat we can work from either one or the other. In the event of therising ground being uneven we can make use of the scale on the levelplane. 

XLI

HOW TO FIND THE DISTANCE OF A GIVEN FIGURE OR POINT FROM THE BASE LINE

 [Illustration: Fig. 97. ]

Let _P_ be the given figure. Form scale _ACS_, _S_ being the point ofsight and _D_ the distance. Draw horizontal _do_ through _P_. From _A_draw diagonal _AD_ to distance point, cutting _do_ in _o_, through _o_draw _SB_ to base, and we now have a square _AdoB_ on the perspectiveplane; and as figure _P_ is standing on the far side of that square itmust be the distance _AB_, which is one side of it, from the baseline--or picture plane. For figures very far away it might be necessaryto make use of half-distance. 

XLII

HOW TO MEASURE THE HEIGHT OF FIGURES ON UNEVEN GROUND

In previous problems we have drawn figures on level planes, which iseasy enough. We have now to represent some above and some below theperspective plane. 

 [Illustration: Fig. 98. ]

Form scale _bS_, _cS_; mark off distances 20 feet, 40 feet, &c. Supposefigure _K_ to be 60 feet off. From point at his feet draw horizontal tomeet vertical _On_, which is 60 feet distant. At the point _m_ wherethis line meets the vertical, measure height _mn_ equal to width ofscale at that distance, transfer this to _K_, and you have the requiredheight of the figure in black. 

For the figures under the cliff 20 feet below the perspective plane, form scale _FS_, _GS_, making it the same width as the other, namely5 feet, and proceed in the usual way to find the height of the figureson the sands, which are here supposed to be nearly on a level with thesea, of course making allowance for different heights and various otherthings. 

XLIII

FURTHER ILLUSTRATION OF THE SIZE OF FIGURES AT DIFFERENT DISTANCESAND ON UNEVEN GROUND

 [Illustration: Fig. 99. ]

Let _ab_ be the height of a figure, say 6 feet. First form scale _aS_, _bS_, the lower line of which, _aS_, is on a level with the base or onthe perspective plane. The figure marked _C_ is close to base, the groupof three is farther off (24 feet), and 6 feet higher up, so we measurethe height on the vanishing scale and also above it. The two girlscarrying fish are still farther off, and about 12 feet below. To tellhow far a figure is away, refer its measurements to the vanishing scale(see Fig. 96). 

XLIV

FIGURES ON A DESCENDING PLANE

In this case (Fig. 100) the same rule applies as in the previousproblem, but as the road on the left is going down hill, the vanishingpoint of the inclined plane is below the horizon at point _S·_; _AS_, _BS_ is the vanishing scale on the level plane; and _A·S·_, _B·S·_, thaton the incline. 

Fig. 101. This is an outline of above figure to show the working moreplainly. 

Note the wall to the left marked _W_ and the manner in which it appearsto drop at certain intervals, its base corresponding with the inclinedplane, but the upper lines of each division being made level are drawnto the point of sight, or to their vanishing point on the horizon; it isimportant to observe this, as it aids greatly in drawing a road goingdown hill. 

 [Illustration: Fig. 100. ]

 [Illustration: Fig. 101. ]

 [Illustration: Fig. 102. ]

XLV

FURTHER ILLUSTRATION OF THE DESCENDING PLANE

In the centre of this picture (Fig. 102) we suppose the road to bedescending till it reaches a tunnel which goes under a road or leads toa river (like one leading out of the Strand near Somerset House). It isdrawn on the same principle as the foregoing figure. Of course to seethe road the spectator must get pretty near to it, otherwise it will beout of sight. Also a level plane must be shown, as by its contrast tothe other we perceive that the latter is going down hill. 

XLVI

FURTHER ILLUSTRATION OF UNEVEN GROUND

An extended view drawn from a height of about 30 feet from a road thatdescends about 45 feet. 

 [Illustration: Fig. 103. Farningham. ]

In drawing a landscape such as Fig. 103 we have to bear in mind theheight of the horizon, which being exactly opposite the eye, shows us atonce which objects are below and which are above us, and to draw themaccordingly, especially roofs, buildings, walls, hedges, &c. ; also itis well to sketch in the different fields figures of men and cattle, as from the size of these we can judge of the rest. 

XLVII

THE PICTURE STANDING ON THE GROUND

Let _K_ represent a frame placed vertically and at a given distance infront of us. If stood on the ground our foreground will touch the baseline of the picture, and we can fix up a standard of measurement both onthe base and on the side as in this sketch, taking 6 feet as about theheight of the figures. 

 [Illustration: Fig. 104. Toledo. ]

XLVIII

THE PICTURE ON A HEIGHT

If we are looking at a scene from a height, that is from a terrace, or awindow, or a cliff, then the near foreground, unless it be the terrace, window-sill, &c. , would not come into the picture, and we could not seethe near figures at _A_, and the nearest to come into view would bethose at _B_, so that a view from a window, &c. , would be as it werewithout a foreground. Note that the figures at _B_ would be (accordingto this sketch) 30 feet from the picture plane and about 18 feet belowthe base line. 

 [Illustration: Fig. 105. ]

BOOK THIRD

XLIX

ANGULAR PERSPECTIVE

Hitherto we have spoken only of parallel perspective, which iscomparatively easy, and in our first figure we placed the cube withone of its sides either touching or parallel to the transparent plane. We now place it so that one angle only (_ab_), touches the picture. 

 [Illustration: Fig. 106. ]

Its sides are no longer drawn to the point of sight as in Fig. 7, norits diagonal to the point of distance, but to some other points on thehorizon, although the same rule holds good as regards their parallelism;as for instance, in the case of _bc_ and _ad_, which, if produced, wouldmeet at _V_, a point on the horizon called a vanishing point. In thisfigure only one vanishing point is seen, which is to the right of thepoint of sight _S_, whilst the other is some distance to the left, andoutside the picture. If the cube is correctly drawn, it will be foundthat the lines _ae_, _bg_, &c. , if produced, will meet on the horizon atthis other vanishing point. This far-away vanishing point is one of theinconveniences of oblique or angular perspective, and therefore it willbe a considerable gain to the draughtsman if we can dispense with it. This can be easily done, as in the above figure, and here our geometrywill come to our assistance, as I shall show presently. 

L

HOW TO PUT A GIVEN POINT INTO PERSPECTIVE

Let us place the given point _P_ on a geometrical plane, to show how farit is from the base line, and indeed in the exact position we wish it tobe in the picture. The geometrical plane is supposed to face us, to hangdown, as it were, from the base line _AB_, like the side of a table, thetop of which represents the perspective plane. It is to that perspectiveplane that we now have to transfer the point _P_. 

 [Illustration: Fig. 107. ]

From _P_ raise perpendicular _Pm_ till it touches the base line at _m_. With centre _m_ and radius _mP_ describe arc _Pn_ so that _mn_ is nowthe same length as _mP_. As point _P_ is opposite point _m_, so must itbe in the perspective, therefore we draw a line at right angles to thebase, that is to the point of sight, and somewhere on this line will befound the required point _P·_. We now have to find how far from _m_ mustthat point be. It must be the length of _mn_, which is the same as _mP_. We therefore from _n_ draw _nD_ to the point of distance, which being atan angle of 45°, or half a right angle, makes _mP_· the perspectivelength of _mn_ by its intersection with _mS_, and thus gives us thepoint _P·_, which is the perspective of the original point. 

LI

A PERSPECTIVE POINT BEING GIVEN, FIND ITS POSITIONON THE GEOMETRICAL PLANE

To do this we simply reverse the foregoing problem. Thus let _P_ be thegiven perspective point. From point of sight _S_ draw a line through _P_till it cuts _AB_ at _m_. From distance _D_ draw another line through_P_ till it cuts the base at _n_. From _m_ drop perpendicular, and thenwith centre _m_ and radius _mn_ describe arc, and where it cuts thatperpendicular is the required point _P·_. We often have to make use ofthis problem. 

 [Illustration: Fig. 108. ]

LII

HOW TO PUT A GIVEN LINE INTO PERSPECTIVE

This is simply a question of putting two points into perspective, instead of one, or like doing the previous problem twice over, for thetwo points represent the two extremities of the line. Thus we have tofind the perspective of _A_ and _B_, namely _a·b·_. Join those points, and we have the line required. 

 [Illustration: Fig. 109. ]

 [Illustration: Fig. 110. ]

If one end touches the base, as at _A_ (Fig. 110), then we have but tofind one point, namely _b_. We also find the perspective of the angle_mAB_, namely the shaded triangle mAb. Note also that the perspectivetriangle equals the geometrical triangle. 

 [Illustration: Fig. 111. ]

When the line required is parallel to the base line of the picture, thenthe perspective of it is also parallel to that base (see Rule 3). 

LIII

TO FIND THE LENGTH OF A GIVEN PERSPECTIVE LINE

A perspective line _AB_ being given, find its actual length and theangle at which it is placed. 

This is simply the reverse of the previous problem. Let _AB_ be thegiven line. From distance _D_ through _A_ draw _DC_, and from _S_, pointof sight, through _A_ draw _SO_. Drop _OP_ at right angles to base, making it equal to _OC_. Join _PB_, and line _PB_ is the actual lengthof _AB_. 

This problem is useful in finding the position of any given line orpoint on the perspective plane. 

 [Illustration: Fig. 112. ]

LIV

TO FIND THESE POINTS WHEN THE DISTANCE-POINT IS INACCESSIBLE

 [Illustration: Fig. 113. ]

If the distance-point is a long way out of the picture, then the sameresult can be obtained by using the half distance and half base, asalready shown. 

From _a_, half of _mP_·, draw quadrant _ab_, from _b_ (half base), drawline from _b_ to half Dist. , which intersects _Sm_ at _P_, precisely thesame point as would be obtained by using the whole distance. 

LV

HOW TO PUT A GIVEN TRIANGLE OR OTHER RECTILINEAL FIGURE INTO PERSPECTIVE

Here we simply put three points into perspective to obtain the giventriangle _A_, or five points to obtain the five-sided figure at _B_. So can we deal with any number of figures placed at any angle. 

 [Illustration: Fig. 114. ]

Both the above figures are placed in the same diagram, showing how anynumber can be drawn by means of the same point of sight and the samepoint of distance, which makes them belong to the same picture. 

It is to be noted that the figures appear reversed in the perspective. That is, in the geometrical triangle the base at _ab_ is uppermost, whereas in the perspective _ab_ is lowermost, yet both are nearest tothe ground line. 

LVI

HOW TO PUT A GIVEN SQUARE INTO ANGULAR PERSPECTIVE

Let _ABCD_ (Fig. 115) be the given square on the geometrical plane, where we can place it as near or as far from the base and at any anglethat we wish. We then proceed to find its perspective on the picture byfinding the perspective of the four points _ABCD_ as already shown. Notethat the two sides of the perspective square _dc_ and _ab_ beingproduced, meet at point _V_ on the horizon, which is their vanishingpoint, but to find the point on the horizon where sides _bc_ and _ad_meet, we should have to go a long way to the left of the figure, whichby this method is not necessary. 

 [Illustration: Fig. 115. ]

LVII

OF MEASURING POINTS

We now have to find certain points by which to measure those vanishingor retreating lines which are no longer at right angles to the pictureplane, as in parallel perspective, and have to be measured in adifferent way, and here geometry comes to our assistance. 

 [Illustration: Fig. 116. ]

Note that the perspective square _P_ equals the geometrical square _K_, so that side _AB_ of the one equals side _ab_ of the other. With centre_A_ and radius _AB_ describe arc _Bm·_ till it cuts the base line at_m·_. Now _AB_ = _Am·_, and if we join _bm·_ then triangle _BAm·_ is anisosceles triangle. So likewise if we join _m·b_ in the perspectivefigure will m·Ab be the same isosceles triangle in perspective. Continueline _m·b_ till it cuts the horizon in _m_, which point will be themeasuring point for the vanishing line _AbV_. For if in an isoscelestriangle we draw lines across it, parallel to its base from one side tothe other, we divide both sides in exactly the same quantities andproportions, so that if we measure on the base line of the picture thespaces we require, such as 1, 2, 3, on the length _Am·_, and thenfrom these divisions draw lines to the measuring point, these lineswill intersect the vanishing line _AbV_ in the lengths and proportionsrequired. To find a measuring point for the lines that go to the othervanishing point, we proceed in the same way. Of course great accuracyis necessary. 

Note that the dotted lines 1, 1, 2, 2, &c. , are parallel in theperspective, as in the geometrical figure. In the former the lines aredrawn to the same point _m_ on the horizon. 

LVIII

HOW TO DIVIDE ANY GIVEN STRAIGHT LINE INTO EQUAL OR PROPORTIONATE PARTS

 [Illustration: Fig. 117. ]

Let _AB_ (Fig. 117) be the given straight line that we wish to divideinto five equal parts. Draw _AC_ at any convenient angle, and measureoff five equal parts with the compasses thereon, as 1, 2, 3, 4, 5. From5C draw line to 5B. Now from each division on _AC_ draw lines 4, 4, 3, 3, &c. , parallel to 5, 5. Then _AB_ will be divided into the required numberof equal parts. 

LIX

HOW TO DIVIDE A DIAGONAL VANISHING LINE INTO ANY NUMBEROF EQUAL OR PROPORTIONAL PARTS

In a previous figure (Fig. 116) we have shown how to find a measuringpoint when the exact measure of a vanishing line is required, but if itsuffices merely to divide a line into a given number of equal parts, then the following simple method can be adopted. 

We wish to divide _ab_ into five equal parts. From _a_, measure off onthe ground line the five equal spaces required. From 5, the point towhich these measures extend (as they are taken at random), draw a linethrough _b_ till it cuts the horizon at _O_. Then proceed to draw linesfrom each division on the base to point _O_, and they will intersect anddivide _ab_ into the required number of equal parts. 

 [Illustration: Fig. 118. ]

 [Illustration: Fig. 119. ]

The same method applies to a given line to be divided into variousproportions, as shown in this lower figure. 

 [Illustration: Fig. 120. ]

 [Illustration: Fig. 121. ]

LX

FURTHER USE OF THE MEASURING POINT O

One square in oblique or angular perspective being given, draw anynumber of other squares equal to it by means of this point _O_ and thediagonals. 

Let _ABCD_ (Fig. 120) be the given square; produce its sides _AB_, _DC_till they meet at point _V_. From _D_ measure off on base any number ofequal spaces of any convenient length, as 1, 2, 3, &c. ; from 1, throughcorner of square _C_, draw a line to meet the horizon at _O_, and from_O_ draw lines to the several divisions on base line. These lines willdivide the vanishing line _DV_ into the required number of parts equalto _DC_, the side of the square. Produce the diagonal of the square _DB_till it cuts the horizon at _G_. From the divisions on line _DV_ drawdiagonals to point _G_: their intersections with the other vanishingline _AV_ will determine the direction of the cross-lines which form thebases of other squares without the necessity of drawing them to theother vanishing point, which in this case is some distance to the leftof the picture. If we produce these cross-lines to the horizon we shallfind that they all meet at the other vanishing point, to which of courseit is easy to draw them when that point is accessible, as in Fig. 121;but if it is too far out of the picture, then this method enables us todo without it. 

Figure 121 corroborates the above by showing the two vanishing pointsand additional squares. Note the working of the diagonals drawn to point_G_, in both figures. 

LXI

FURTHER USE OF THE MEASURING POINT O

Suppose we wish to divide the side of a building, as in Fig. 123, or todraw a balcony, a series of windows, or columns, or what not, or, inother words, any line above the horizon, as _AB_. Then from _A_ we draw_AC_ parallel to the horizon, and mark thereon the required divisions 5, 10, 15, &c. : in this case twenty-five (Fig. 122). From _C_ draw a linethrough _B_ till it cuts the horizon at _O_. Then proceed to draw theother lines from each division to _O_, and thus divide the vanishingline _AB_ as required. 

 [Illustration: Fig. 122 is a front view of the portico, Fig. 123. ]

 [Illustration: Fig. 123. ]

In this portico there are thirteen triglyphs with twelve spaces betweenthem, making twenty-five divisions. The required number of parts to drawthe columns can be obtained in the same way. 

LXII

ANOTHER METHOD OF ANGULAR PERSPECTIVE, BEING THAT ADOPTEDIN OUR ART SCHOOLS

In the previous method we have drawn our squares by means of ageometrical plan, putting each point into perspective as required, andthen by means of the perspective drawing thus obtained, finding ourvanishing and measuring points. In this method we proceed in exactly theopposite way, setting out our points first, and drawing the square (orother figure) afterwards. 

 [Illustration: Fig. 124. ]

Having drawn the horizontal and base lines, and fixed upon the positionof the point of sight, we next mark the position of the spectator bydropping a perpendicular, _S ST_, from that point of sight, making itthe same length as the distance we suppose the spectator to be from thepicture, and thus we make _ST_ the station-point. 

To understand this figure we must first look upon it as a ground-plan orbird's-eye view, the line V2V1 or horizon line representing the pictureseen edgeways, because of course the station-point cannot be in thepicture itself, but a certain distance in front of it. The angle at_ST_, that is the angle which decides the positions of the two vanishingpoints V1, V2, is always a right angle, and the two remaining angleson that side of the line, called the directing line, are together equalto a right angle or 90°. So that in fixing upon the angle at which thesquare or other figure is to be placed, we say 'let it be 60° and 30°, or 70° and 20°', &c. Having decided upon the station-point and the angleat which the square is to be placed, draw TV1 and TV2, till they cutthe horizon at V1 and V2. These are the two vanishing points towhich the sides of the figure are respectively drawn. But we still wantthe measuring points for these two vanishing lines. We therefore takefirst, V1 as centre and V1T as radius, and describe arc of circle tillit cuts the horizon in M1, which is the measuring point for all linesdrawn to V1. Then with radius V2T describe arc from centre V2 tillit cuts the horizon in M2, which is the measuring point for allvanishing lines drawn to V2. We have now set out our points. Let usproceed to draw the square _Abcd_. From _A_, the nearest angle (in thisinstance touching the base line), measure on each side of it the equallengths _AB_ and _AE_, which represent the width or side of the square. Draw EM2 and BM1 from the two measuring points, which give us, bytheir intersections with the vanishing lines AV1 and AV2, theperspective lengths of the sides of the square _Abcd_. Join _b_ and V1and dV2, which intersect each other at _C_, then _Adcb_ is the squarerequired. 

This method, which is easy when you know it, has certain drawbacks, thechief one being that if we require a long-distance point, and a smallangle, such as 10° on one side, and 80° on the other, then the size ofthe diagram becomes so large that it has to be carried out on the floorof the studio with long strings, &c. , which is a very clumsy andunscientific way of setting to work. The architects in such cases makeuse of the centrolinead, a clever mechanical contrivance for gettingover the difficulty of the far-off vanishing point, but by the method Ihave shown you, and shall further illustrate, you will find that you candispense with all this trouble, and do all your perspective eitherinside the picture or on a very small margin outside it. 

Perhaps another drawback to this method is that it is not self-evident, as in the former one, and being rather difficult to explain, the studentis apt to take it on trust, and not to trouble about the reasons for itsconstruction: but to show that it is equally correct, I will draw thetwo methods in one figure. 

LXIII

TWO METHODS OF ANGULAR PERSPECTIVE IN ONE FIGURE

 [Illustration: Fig. 125. ]

It matters little whether the station-point is placed above or below thehorizon, as the result is the same. In Fig. 125 it is placed above, asthe lower part of the figure is occupied with the geometrical plan ofthe other method. 

In each case we make the square _K_ the same size and at the same angle, its near corner being at _A_. It must be seen that by whichever methodwe work out this perspective, the result is the same, so that both arecorrect: the great advantage of the first or geometrical system being, that we can place the square at any angle, as it is drawn withoutreference to vanishing points. 

We will, however, work out a few figures by the second method. 

LXIV

TO DRAW A CUBE, THE POINTS BEING GIVEN

As in a previous figure (124) we found the various working points ofangular perspective, we need now merely transfer them to the horizontalline in this figure, as in this case they will answer our purposeperfectly well. 

 [Illustration: Fig. 126. ]

Let _A_ be the nearest angle touching the base. Draw AV1, AV2. From_A_, raise vertical _Ae_, the height of the cube. From _e_ draw eV1, eV2, from the other angles raise verticals _bf_, _dh_, _cg_, to meeteV1, eV2, fV2, &c. , and the cube is complete. 

LXV

AMPLIFICATION OF THE CUBE APPLIED TO DRAWING A COTTAGE

 [Illustration: Fig. 127. ]

Note that we have started this figure with the cube _Adhefb_. We havetaken three times _AB_, its width, for the front of our house, and twice_AB_ for the side, and have made it two cubes high, not counting theroof. Note also the use of the measuring-points in connexion with themeasurements on the base line, and the upper measuring line _TPK_. 

LXVI

HOW TO DRAW AN INTERIOR AT AN ANGLE

Here we make use of the same points as in a previous figure, with theaddition of the point _G_, which is the vanishing point of the diagonalsof the squares on the floor. 

 [Illustration: Fig. 128. ]

From _A_ draw square _Abcd_, and produce its sides in all directions;again from _A_, through the opposite angle of the square _C_, draw adiagonal till it cuts the horizon at _G_. From _G_ draw diagonalsthrough _b_ and _d_, cutting the base at _o_, _o_, make spaces _o_, _o_, equal to _Ao_ all along the base, and from them draw diagonals to _G_;through the points where these diagonals intersect the vanishing linesdrawn in the direction of _Ab_, _dc_ and _Ad_, _bc_, draw lines to theother vanishing point V1, thus completing the squares, and so coverthe floor with them; they will then serve to measure width of door, windows, &c. Of course horizontal lines on wall 1 are drawn to V1, andthose on wall 2 to V2. 

In order to see this drawing properly, the eye should be placed about3 inches from it, and opposite the point of sight; it will then standout like a stereoscopic picture, and appear as actual space, butotherwise the perspective seems deformed, and the angles exaggerated. To make this drawing look right from a reasonable distance, the point ofdistance should be at least twice as far off as it is here, and thiswould mean altering all the other points and sending them a long way outof the picture; this is why artists use those long strings referred toabove. I would however, advise them to make their perspective drawing ona small scale, and then square it up to the size of the canvas. 

LXVII

HOW TO CORRECT DISTORTED PERSPECTIVE BY DOUBLING THE LINE OF DISTANCE

Here we have the same interior as the foregoing, but drawn with doublethe distance, so that the perspective is not so violent and the objectsare truer in proportion to each other. 

 [Illustration: Fig. 129. ]

To redraw the whole figure double the size, including the station-point, would require a very large diagram, that we could not get into this bookwithout a folding plate, but it comes to the same thing if we double thedistances between the various points. Thus, if from _S_ to _G_ in thesmall diagram is 1 inch, in the larger one make it 2 inches. If from _S_to M2 is 2 inches, in the larger make it 4, and so on. 

Or this form may be used: make _AB_ twice the length of _AC_ (Fig. 130), or in any other proportion required. On _AC_ mark the points as in thedrawing you wish to enlarge. Make _AB_ the length that you wish toenlarge to, draw _CB_, and then from each division on _AC_ draw linesparallel to _CB_, and _AB_ will be divided in the same proportions, as Ihave already shown (Fig. 117). 

There is no doubt that it is easier to work direct from the vanishingpoints themselves, especially in complicated architectural work, but atthe same time I will now show you how we can dispense with, at allevents, one of them, and that the farthest away. 

 [Illustration: Fig. 130. ]

LXVIII

HOW TO DRAW A CUBE ON A GIVEN SQUARE, USING ONLY ONE VANISHING POINT

_ABCD_ is the given square (Fig. 131). At _A_ raise vertical _Aa_ equalto side of square _AB·_, from _a_ draw _ab_ to the vanishing point. Raise _Bb_. Produce _VD_ to _E_ to touch the base line. From _E_ raisevertical _EF_, making it equal to _Aa_. From _F_ draw _FV_. Raise _Dd_and _Cc_, their heights being determined by the line _FV_. Join _da_ andthe cube is complete. It will be seen that the verticals raised at eachcorner of the square are equal perspectively, as they are drawn betweenparallels which start from equal heights, namely, from _EF_ and _Aa_ tothe same point _V_, the vanishing point. Any other line, such as _OO·_, can be directed to the inaccessible vanishing point in the same way as_ad_, &c. 

_Note. _ This is only one of many original figures and problems in thisbook which have been called up by the wish to facilitate the work of theartist, and as it were by necessity. 

 [Illustration: Fig. 131. ]

LXIX

A COURTYARD OR CLOISTER DRAWN WITH ONE VANISHING POINT

 [Illustration: Fig. 132. ]

In this figure I have first drawn the pavement by means of the diagonals_GA_, _Go_, _Go_, &c. , and the vanishing point _V_, the square at _A_being given. From _A_ draw diagonal through opposite corner till it cutsthe horizon at _G_. From this same point _G_ draw lines through theother corners of the square till they cut the ground line at _o_, _o_. Take this measurement _Ao_ and mark it along the base right and left of_A_, and the lines drawn from these points _o_ to point _G_ will givethe diagonals of all the squares on the pavement. Produce sides ofsquare _A_, and where these lines are intersected by the diagonals _Go_draw lines from the vanishing point _V_ to base. These will give us theoutlines of the squares lying between them and also guiding points thatwill enable us to draw as many more as we please. These again will giveus our measurements for the widths of the arches, &c. , or between thecolumns. Having fixed the height of wall or dado, we make use of _V_point to draw the sides of the building, and by means of proportionatemeasurement complete the rest, as in Fig. 128. 

LXX

HOW TO DRAW LINES WHICH SHALL MEET AT A DISTANT POINT, BY MEANS OF DIAGONALS

This is in a great measure a repetition of the foregoing figure, andtherefore needs no further explanation. 

 [Illustration: Fig. 133. ]

I must, however, point out the importance of the point _G_. In angularperspective it in a measure takes the place of the point of distance inparallel perspective, since it is the vanishing point of diagonals at45° drawn between parallels such as _AV_, _DV_, drawn to a vanishingpoint _V_. The method of dividing line _AV_ into a number of parts equalto _AB_, the side of the square, is also shown in a previous figure(Fig. 120). 

LXXI

HOW TO DIVIDE A SQUARE PLACED AT AN ANGLE INTO A GIVEN NUMBEROF SMALL SQUARES

_ABCD_ is the given square, and only one vanishing point is accessible. Let us divide it into sixteen small squares. Produce side _CD_ to baseat _E_. Divide _EA_ into four equal parts. From each division draw linesto vanishing point _V_. Draw diagonals _BD_ and _AC_, and produce thelatter till it cuts the horizon in _G_. Draw the three cross-linesthrough the intersections made by the diagonals and the lines drawn to_V_, and thus divide the square into sixteen. 

 [Illustration: Fig. 134. ]

This is to some extent the reverse of the previous problem. It alsoshows how the long vanishing point can be dispensed with, and theperspective drawing brought within the picture. 

LXXII

FURTHER EXAMPLE OF HOW TO DIVIDE A GIVEN OBLIQUE SQUARE INTOA GIVEN NUMBER OF EQUAL SQUARES, SAY TWENTY-FIVE

Having drawn the square _ABCD_, which is enclosed, as will be seen, in adotted square in parallel perspective, I divide the line _EA_ into fiveequal parts instead of four (Fig. 135), and have made use of the devicefor that purpose by measuring off the required number on line _EF_, &c. Fig. 136 is introduced here simply to show that the square can bedivided into any number of smaller squares. Nor need the figure benecessarily a square; it is just as easy to make it an oblong, as _ABEF_(Fig. 136); for although we begin with a square we can extend it in anydirection we please, as here shown. 

 [Illustration: Fig. 135. ]

 [Illustration: Fig. 136. ]

LXXIII

OF PARALLELS AND DIAGONALS

 [Illustration: Fig. 137 A. ]

 [Illustration: Fig. 137 B. ]

 [Illustration: Fig. 137 C. ]

To find the centre of a square or other rectangular figure we have butto draw its two diagonals, and their intersection will give us thecentre of the figure (see 137 A). We do the same with perspectivefigures, as at B. In Fig. C is shown how a diagonal, drawn from oneangle of a square _B_ through the centre _O_ of the opposite side of thesquare, will enable us to find a second square lying between the sameparallels, then a third, a fourth, and so on. At figure _K_ lying onthe ground, I have divided the farther side of the square _mn_ into ¼, 1/3, ½. If I draw a diagonal from _G_ (at the base) through the halfof this line I cut off on _FS_ the lengths or sides of two squares;if through the quarter I cut off the length of four squares on thevanishing line _FS_, and so on. In Fig. 137 D is shown how easily anynumber of objects at any equal distances apart, such as posts, trees, columns, &c. , can be drawn by means of diagonals between parallels, guided by a central line _GS_. 

 [Illustration: Fig. 137 D. ]

LXXIV

THE SQUARE, THE OBLONG, AND THEIR DIAGONALS

 [Illustration: Fig. 138. ]

 [Illustration: Fig. 139. ]

Having found the centre of a square or oblong, such as Figs. 138 and139, if we draw a third line through that centre at a given angle andthen at each of its extremities draw perpendiculars _AB_, _DC_, wedivide that square or oblong into three parts, the two outer portionsbeing equal to each other, and the centre one either larger or smalleras desired; as, for instance, in the triumphal arch we make the centreportion larger than the two outer sides. When certain architecturaldetails and spaces are to be put into perspective, a scale such as thatin Fig. 123 will be found of great convenience; but if only a readydivision of the principal proportions is required, then these diagonalswill be found of the greatest use. 

LXXV

SHOWING THE USE OF THE SQUARE AND DIAGONALS IN DRAWING DOORWAYS, WINDOWS, AND OTHER ARCHITECTURAL FEATURES

This example is from Serlio's _Architecture_ (1663), showing whatexcellent proportion can be obtained by the square and diagonals. Thewidth of the door is one-third of the base of square, the heighttwo-thirds. As a further illustration we have drawn the same figure inperspective. 

 [Illustration: Fig. 140. ]

 [Illustration: Fig. 141. ]

LXXVI

HOW TO MEASURE DEPTHS BY DIAGONALS

If we take any length on the base of a square, say from _A_ to _g_, andfrom _g_ raise a perpendicular till it cuts the diagonal _AB_ in _O_, then from _O_ draw horizontal _Og·_, we form a square AgOg·, and thusmeasure on one side of the square the distance or depth _Ag·_. So can wemeasure any other length, such as _fg_, in like manner. 

 [Illustration: Fig. 142. ]

 [Illustration: Fig. 143. ]

To do this in perspective we pursue precisely the same method, as shownin this figure (143). 

To measure a length _Ag_ on the side of square _AC_, we draw a line from_g_ to the point of sight _S_, and where it crosses diagonal _AB_ at _O_we draw horizontal _Og_, and thus find the required depth _Ag_ in thepicture. 

LXXVII

HOW TO MEASURE DISTANCES BY THE SQUARE AND DIAGONAL

It may sometimes be convenient to have a ready method by which tomeasure the width and length of objects standing against the wall of agallery, without referring to distance-points, &c. 

 [Illustration: Fig. 144. ]

In Fig. 144 the floor is divided into two large squares with theirdiagonals. Suppose we wish to draw a fireplace or a piece of furniture_K_, we measure its base _ef_ on _AB_, as far from _B_ as we wish it tobe in the picture; draw _eo_ and _fo_ to point of sight, and proceed asin the previous figure by drawing parallels from _Oo_, &c. 

Let it be observed that the great advantage of this method is, that wecan use it to measure such distant objects as _XY_ just as easily asthose near to us. 

There is, however, a still further advantage arising from it, and thatis that it introduces us to a new and simpler method of perspective, towhich I have already referred, and it will, I hope, be found of infiniteuse to the artist. 

_Note. _--As we have founded many of these figures on a given square inangular perspective, it is as well to have a ready and certain means ofdrawing that square without the elaborate setting out of a geometricalplan, as in the first method, or the more cumbersome and extended systemof the second method. I shall therefore show you another method equallycorrect, but much simpler than either, which I have invented for ouruse, and which indeed forms one of the chief features of this book. 

LXXVIII

HOW BY MEANS OF THE SQUARE AND DIAGONAL WE CAN DETERMINETHE POSITION OF POINTS IN SPACE

Apart from the aid that perspective affords the draughtsman, there is afurther value in it, in that it teaches us almost a new science, whichwe might call the mystery of aspect, and how it is that the objectsaround us take so many different forms, or rather appearances, althoughthey themselves remain the same. And also that it enables us, with, I think, great pleasure to ourselves, to fathom space, to work outdifficult problems by simple reasoning, and to exercise those inventiveand critical faculties which give strength and enjoyment to mental life. 

And now, after this brief excursion into philosophy, let us come down tothe simple question of the perspective of a point. 

 [Illustration: Fig. 145. ]

 [Illustration: Fig. 146. ]

Here, for instance, are two aspects of the same thing: the geometricalsquare _A_, which is facing us, and the perspective square _B_, which wesuppose to lie flat on the table, or rather on the perspective plane. Line _A·C·_ is the perspective of line _AC_. On the geometrical squarewe can make what measurements we please with the compasses, but on theperspective square _B·_ the only line we can actually measure is thebase line. In both figures this base line is the same length. Suppose wewant to find the perspective of point _P_ (Fig. 146), we make use of thediagonal _CA_. From _P_ in the geometrical square draw _PO_ to meet thediagonal in _O_; through _O_ draw perpendicular _fe_; transfer length_fB_, so found, to the base of the perspective square; from _f_ draw_fS_ to point of sight; where it cuts the diagonal in _O_, drawhorizontal _OP·_, which gives us the point required. In the same way wecan find the perspective of any number of points on any side of thesquare. 

LXXIX

PERSPECTIVE OF A POINT PLACED IN ANY POSITION WITHIN THE SQUARE

Let the point _P_ be the one we wish to put into perspective. We havebut to repeat the process of the previous problem, making use of ourmeasurements on the base, the diagonals, &c. 

 [Illustration: Fig. 147. ]

Indeed these figures are so plain and evident that further descriptionof them is hardly necessary, so I will here give two drawings oftriangles which explain themselves. To put a triangle into perspectivewe have but to find three points, such as _fEP_, Fig. 148 A, and thentransfer these points to the perspective square 148 B, as there shown, and form the perspective triangle; but these figures explain themselves. Any other triangle or rectilineal figure can be worked out in the sameway, which is not only the simplest method, but it carries itsmathematical proof with it. 

 [Illustration: Fig. 148 A. ]

 [Illustration: Fig. 148 B. ]

 [Illustration: Fig. 149 A. ]

 [Illustration: Fig. 149 B. ]

LXXX

PERSPECTIVE OF A SQUARE PLACED AT AN ANGLE NEW METHOD

As we have drawn a triangle in a square so can we draw an oblique squarein a parallel square. In Figure 150 A we have drawn the oblique square_GEPn_. We find the points on the base _Am_, as in the previous figures, which enable us to construct the oblique perspective square _n·G·E·P·_in the parallel perspective square Fig. 150 B. But it is not necessaryto construct the geometrical figure, as I will show presently. It ishere introduced to explain the method. 

 [Illustration: Fig. 150 A. ]

 [Illustration: Fig. 150 B. ]

Fig. 150 B. To test the accuracy of the above, produce sides _G·E·_ and_n·P·_ of perspective square till they touch the horizon, where theywill meet at _V_, their vanishing point, and again produce the othersides _n·G·_ and _P·E·_ till they meet on the horizon at the othervanishing point, which they must do if the figure is correctly drawn. 

In any parallel square construct an oblique square from a givenpoint--given the parallel square at Fig. 150 B, and given point _n·_ onbase. Make _A·f·_ equal to _n·m·_, draw _f·S_ and _n·S_ to point ofsight. Where these lines cut the diagonal _AC_ draw horizontals to _P·_and _G·_, and so find the four points _G·E·P·n·_ through which to drawthe square. 

LXXXI

ON A GIVEN LINE PLACED AT AN ANGLE TO THE BASE DRAW A SQUARE IN ANGULARPERSPECTIVE, THE POINT OF SIGHT, AND DISTANCE, BEING GIVEN. 

 [Illustration: Fig. 151. ]

Let _AB_ be the given line, _S_ the point of sight, and _D_ the distance(Fig. 151, 1). Through _A_ draw _SC_ from point of sight to base (Fig. 151, 2 and 3). From _C_ draw _CD_ to point of distance. Draw _Ao_parallel to base till it cuts _CD_ at _o_, through _O_ draw _SP_, from_B_ mark off _BE_ equal to _CP_. From _E_ draw _ES_ intersecting _CD_ at_K_, from _K_ draw _KM_, thus completing the outer parallel square. Through _F_, where _PS_ intersects _MK_, draw _AV_ till it cuts thehorizon in _V_, its vanishing point. From _V_ draw _VB_ cutting side_KE_ of outer square in _G_, and we have the four points _AFGB_, whichare the four angles of the square required. Join _FG_, and the figure iscomplete. 

Any other side of the square might be given, such as _AF_. First through_A_ and _F_ draw _SC_, _SP_, then draw _Ao_, then through _o_ draw _CD_. From _C_ draw base of parallel square _CE_, and at _M_ through _F_ draw_MK_ cutting diagonal at _K_, which gives top of square. Now through _K_draw _SE_, giving _KE_ the remaining side thereof, produce _AF_ to _V_, from _V_ draw _VB_. Join _FG_, _GB_, and _BA_, and the square requiredis complete. 

The student can try the remaining two sides, and he will find they workout in a similar way. 

LXXXII

HOW TO DRAW SOLID FIGURES AT ANY ANGLE BY THE NEW METHOD

As we can draw planes by this method so can we draw solids, as shown inthese figures. The heights of the corners of the triangles are obtainedby means of the vanishing scales _AS_, _OS_, which have already beenexplained. 

 [Illustration: Fig. 152. ]

 [Illustration: Fig. 153. ]

In the same manner we can draw a cubic figure (Fig. 154)--a box, forinstance--at any required angle. In this case, besides the scale _AS_, _OS_, we have made use of the vanishing lines _DV_, _BV_, to corroboratethe scale, but they can be dispensed with in these simple objects, or wecan use a scale on each side of the figure as _a·o·S_, should bothvanishing points be inaccessible. Let it be noted that in the scale_AOS_, _AO_ is made equal to _BC_, the height of the box. 

 [Illustration: Fig. 154. ]

By a similar process we draw these two figures, one on the square, theother on the circle. 

 [Illustration: Fig. 155. ]

 [Illustration: Fig. 156. ]

LXXXIII

POINTS IN SPACE

The chief use of these figures is to show how by means of diagonals, horizontals, and perpendiculars almost any figure in space can be setdown. Lines at any slope and at any angle can be drawn by thisdescriptive geometry. 

The student can examine these figures for himself, and will understandtheir working from what has gone before. Here (Fig. 157) in thegeometrical square we have a vertical plane _AabB_ standing on its base_AB_. We wish to place a projection of this figure at a certain distanceand at a given angle in space. First of all we transfer it to the sideof the cube, where it is seen in perspective, whilst at its side isanother perspective square lying flat, on which we have to stand ourfigure. By means of the diagonal of this flat square, horizontals fromfigure on side of cube, and lines drawn from point of sight (as alreadyexplained), we obtain the direction of base line _AB_, and also by meansof lines _aa·_ and _bb·_ we obtain the two points in space _a·b·_. Join_Aa·_, _a·b·_ and _Bb·_, and we have the projection required, and whichmay be said to possess the third dimension. 

 [Illustration: Fig. 157. ]

In this other case (Fig. 158) we have a wedge-shaped figure standing ona triangle placed on the ground, as in the previous figure, its threecorners being the same height. In the vertical geometrical square wehave a ground-plan of the figure, from which we draw lines to diagonaland to base, and notify by numerals 1, 3, 2, 1, 3; these we transfer tobase of the horizontal perspective square, and then construct shadedtriangle 1, 2, 3, and raise to the height required as shown at1·, 2·, 3·. Although we may not want to make use of these specialfigures, they show us how we could work out almost any form or objectsuspended in space. 

 [Illustration: Fig. 158. ]

LXXXIV

THE SQUARE AND DIAGONAL APPLIED TO CUBES AND SOLIDS DRAWN THEREIN

 [Illustration: Fig. 159. ]

As we have made use of the square and diagonal to draw figures atvarious angles so can we make use of cubes either in parallel or angularperspective to draw other solid figures within them, as shown in thesedrawings, for this is simply an amplification of that method. Indeed wemight invent many more such things. But subjects for perspectivetreatment will constantly present themselves to the artist ordraughtsman in the course of his experience, and while I endeavour toshow him how to grapple with any new difficulty or subject that mayarise, it is impossible to set down all of them in this book. 

 [Illustration: Fig. 160. ]

LXXXV

TO DRAW AN OBLIQUE SQUARE IN ANOTHER OBLIQUE SQUAREWITHOUT USING VANISHING POINTS

It is not often that both vanishing points are inaccessible, still it iswell to know how to proceed when this is the case. We first draw thesquare _ABCD_ inside the parallel square, as in previous figures. Todraw the smaller square _K_ we simply draw a smaller parallel square _hh h h_, and within that, guided by the intersections of the diagonalstherewith, we obtain the four points through which to draw square _K_. To raise a solid figure on these squares we can make use of thevanishing scales as shown on each side of the figure, thus obtaining theupper square 1 2 3 4, then by means of the diagonal 1 3 and 2 4 andverticals raised from each corner of square _K_ to meet them we obtainthe smaller upper square corresponding to _K_. 

It might be said that all this can be done by using the two vanishingpoints in the usual way. In the first place, if they were as far off asrequired for this figure we could not get them into a page unless itwere three or four times the width of this one, and to use shorterdistances results in distortion, so that the real use of this system isthat we can make our figures look quite natural and with much lesstrouble than by the other method. 

 [Illustration: Fig. 161. ]

LXXXVI

SHOWING HOW A PEDESTAL CAN BE DRAWN BY THE NEW METHOD

This is a repetition of the previous problem, or rather the applicationof it to architecture, although when there are many details it may bemore convenient to use vanishing points or the centrolinead. 

 [Illustration: Fig. 162. ]

 [Illustration: Fig. 163. Honfleur. ]

LXXXVII

SCALE ON EACH SIDE OF THE PICTURE

As one of my objects in writing this book is to facilitate the workingof our perspective, partly for the comfort of the artist, and partlythat he may have no excuse for neglecting it, I will here show you howyou may, by a very simple means, secure the general correctness of yourperspective when sketching or painting out of doors. 

Let us take this example from a sketch made at Honfleur (Fig. 163), andin which my eye was my only guide, but it stands the test of the rule. First of all note that line _HH_, drawn from one side of the picture tothe other, is the horizontal line; below that is a wall and a pavementmarked _aV_, also going from one side of the picture to the other, andbeing lower down at _a_ than at _V_ it runs up as it were to meet thehorizon at some distant point. In order to form our scale I take firstthe length of _Ha_, and measure it above and below the horizon, alongthe side to our left as many times as required, in this case four orfive. I now take the length _HV_ on the right side of the picture andmeasure it above and below the horizon, as in the other case; and thenfrom these divisions obtain dotted lines crossing the picture from oneside to the other which must all meet at some distant point on thehorizon. These act as guiding lines, and are sufficient to give us thedirection of any vanishing lines going to the same point. For those thatgo in the opposite direction we proceed in the same way, as from _b_ onthe right to _V·_ on the left. They are here put in faintly, so as notto interfere with the drawing. In the sketch of Toledo (Fig. 164) thesame thing is shown by double lines on each side to separate the twosets of lines, and to make the principle more evident. 

 [Illustration: Fig. 164. Toledo. ]

LXXXVIII

THE CIRCLE

If we inscribe a circle in a square we find that it touches that squareat four points which are in the middle of each side, as at _a b c d_. Itwill also intersect the two diagonals at the four points _o_ (Fig. 165). If, then, we put this square and its diagonals, &c. , into perspective weshall have eight guiding points through which to trace the requiredcircle, as shown in Fig. 166, which has the same base as Fig. 165. 

 [Illustration: Fig. 165. ]

 [Illustration: Fig. 166. ]

LXXXIX

THE CIRCLE IN PERSPECTIVE A TRUE ELLIPSE

Although the circle drawn through certain points must be a freehanddrawing, which requires a little practice to make it true, it issufficient for ordinary purposes and on a small scale, but to bemathematically true it must be an ellipse. We will first draw an ellipse(Fig. 167). Let _ee_ be its long, or transverse, diameter, and _db_ itsshort or conjugate diameter. Now take half of the long diameter _eE_, and from point _d_ with _cE_ for radius mark on _ee_ the two points_ff_, which are the foci of the ellipse. At each focus fix a pin, thenmake a loop of fine string that does not stretch and of such a lengththat when drawn out the double thread will reach from _f_ to _e_. Nowplace this double thread round the two pins at the foci _ff·_ anddistend it with the pencil point until it forms triangle _fdf·_, thenpush the pencil along and right round the two foci, which being guidedby the thread will draw the curve, which is a true ellipse, and willpass through the eight points indicated in our first figure. This willbe a sufficient proof that the circle in perspective and the ellipse areidentical curves. We must also remember that the ellipse is an obliqueprojection of a circle, or an oblique section of a cone. The differencebetween the two figures consists in their centres not being in the sameplace, that of the perspective circle being at _c_, higher up than _e_the centre of the ellipse. The latter being a geometrical figure, itslong diameter is exactly in the centre of the figure, whereas the centre_c_ and the diameter of the perspective are at the intersection of thediagonals of the perspective square in which it is inscribed. 

 [Illustration: Fig. 167. ]

XC

FURTHER ILLUSTRATION OF THE ELLIPSE

In order to show that the ellipse drawn by a loop as in the previousfigure is also a circle in perspective we must reconstruct around it thesquare and its eight points by means of which it was drawn in the firstinstance. We start with nothing but the ellipse itself. We have to findthe points of sight and distance, the base, &c. Let us start with base_AB_, a horizontal tangent to the curve extending beyond it on eitherside. From _A_ and _B_ draw two other tangents so that they shall touchthe curve at points such as _TT·_ a little above the transverse diameterand on a level with each other. Produce these tangents till they meet atpoint _S_, which will be the point of sight. Through this point drawhorizontal line _H_. Now draw tangent _CD_ parallel to _AB_. Drawdiagonal _AD_ till it cuts the horizon at the point of distance, thiswill cut through diameter of circle at its centre, and so proceed tofind the eight points through which the perspective circle passes, whenit will be found that they all lie on the ellipse we have drawn with theloop, showing that the two curves are identical although their centresare distinct. 

 [Illustration: Fig. 168. ]

XCI

HOW TO DRAW A CIRCLE IN PERSPECTIVE WITHOUT A GEOMETRICAL PLAN

Divide base _AB_ into four equal parts. At _B_ drop perpendicular _Bn_, making _Bn_ equal to _Bm_, or one-fourth of base. Join _mn_ and transferthis measurement to each side of _d_ on base line; that is, make _df_and _df·_ equal to _mn_. Draw _fS_ and _f·S_, and the intersections ofthese lines with the diagonals of square will give us the four points _oo o o_. 

 [Illustration: Fig. 169. ]

The reason of this is that _ff·_ is the measurement on the base _AB_ ofanother square _o o o o_ which is exactly half of the outer square. Forif we inscribe a circle in a square and then inscribe a second square inthat circle, this second square will be exactly half the area of thelarger one; for its side will be equal to half the diagonal of thelarger square, as can be seen by studying the following figures. In Fig. 170, for instance, the side of small square _K_ is half the diagonal oflarge square _o_. 

 [Illustration: Fig. 170. ]

 [Illustration: Fig. 171. ]

In Fig. 171, _CB_ represents half of diagonal _EB_ of the outer squarein which the circle is inscribed. By taking a fourth of the base _mB_and drawing perpendicular _mh_ we cut _CB_ at _h_ in two equal parts, _Ch_, _hB_. It will be seen that _hB_ is equal to _mn_, one-quarter ofthe diagonal, so if we measure _mn_ on each side of _D_ we get _ff·_equal to _CB_, or half the diagonal. By drawing _ff_, _f·f_ passingthrough the diagonals we get the four points _o o o o_ through which todraw the smaller square. Without referring to geometry we can see at aglance by Fig. 172, where we have simply turned the square _o o o o_ onits centre so that its angles touch the sides of the outer square, thatit is exactly half of square _ABEF_, since each quarter of it, such asEoCo, is bisected by its diagonal _oo_. 

 [Illustration: Fig. 172. ]

 [Illustration: Fig. 173. ]

XCII

HOW TO DRAW A CIRCLE IN ANGULAR PERSPECTIVE

Let _ABCD_ be the oblique square. Produce _VA_ till it cuts the baseline at _G_. 

 [Illustration: Fig. 174. ]

Take _mD_, the fourth of the base. Find _mn_ as in Fig. 171, measure iton each side of _E_, and so obtain _Ef_ and _Ef·_, and proceed to draw_fV_, _EV_, _f·V_ and the diagonals, whose intersections with theselines will give us the eight points through which to draw the circle. Infact the process is the same as in parallel perspective, only instead ofmaking our divisions on the actual base _AD_ of the square, we make themon _GD_, the base line. 

To obtain the central line _hh_ passing through _O_, we can make use ofdiagonals of the half squares; that is, if the other vanishing point isinaccessible, as in this case. 

XCIII

HOW TO DRAW A CIRCLE IN PERSPECTIVE MORE CORRECTLY, BY USING SIXTEEN GUIDING POINTS

First draw square _ABCD_. From _O_, the middle of the base, drawsemicircle _AKB_, and divide it into eight equal parts. From eachdivision raise perpendiculars to the base, such as _2 O_, _3 O_, _5 O_, &c. , and from divisions _O_, _O_, _O_ draw lines to point of sight, and where these lines cut the diagonals _AC_, _DB_, draw horizontalsparallel to base _AB_. Then through the points thus obtained draw thecircle as shown in this figure, which also shows us how thecircumference of a circle in perspective may be divided into anynumber of equal parts. 

 [Illustration: Fig. 175. ]

XCIV

HOW TO DIVIDE A PERSPECTIVE CIRCLE INTO ANY NUMBER OF EQUAL PARTS

This is simply a repetition of the previous figure as far as itsconstruction is concerned, only in this case we have divided thesemicircle into twelve parts and the perspective into twenty-four. 

 [Illustration: Fig. 176. ]

 [Illustration: Fig. 177. ] We have raised perpendiculars from thedivisions on the semicircle, and proceeded as before to draw lines tothe point of sight, and have thus by their intersections with thecircumference already drawn in perspective divided it into the requirednumber of equal parts, to which from the centre we have drawn the radii. This will show us how to draw traceries in Gothic windows, columns in acircle, cart-wheels, &c. 

The geometrical figure (177) will explain the construction of theperspective one by showing how the divisions are obtained on the line_AB_, which represents base of square, from the divisions on thesemicircle _AKB_. 

XCV

HOW TO DRAW CONCENTRIC CIRCLES

 [Illustration: Fig. 178. ]

First draw a square with its diagonals (Fig. 178), and from its centre_O_ inscribe a circle; in this circle inscribe a square, and in thisagain inscribe a second circle, and so on. Through their intersectionswith the diagonals draw lines to base, and number them 1, 2, 3, 4, &c. ;transfer these measurements to the base of the perspective square (Fig. 179), and proceed to construct the circles as before, drawing lines fromeach point on the base to the point of sight, and drawing the curvesthrough the inter-sections of these lines with the diagonals. 

 [Illustration: Fig. 179. ]

Should it be required to make the circles at equal distances, as forsteps for instance, then the geometrical plan should be madeaccordingly. 

Or we may adopt the method shown at Fig. 180, by taking quarter base ofboth outer and inner square, and finding the measurement _mn_ on eachside of _C_, &c. 

 [Illustration: Fig. 180. ]

XCVI

THE ANGLE OF THE DIAMETER OF THE CIRCLE IN ANGULARAND PARALLEL PERSPECTIVE

The circle, whether in angular or parallel perspective, is always anellipse. In angular perspective the angle of the circle's diametervaries in accordance with the angle of the square in which it is placed, as in Fig. 181, _cc_ is the diameter of the circle and _ee_ the diameterof the ellipse. In parallel perspective the diameter of the circlealways remains horizontal, although the long diameter of the ellipsevaries in inclination according to the distance it is from the point ofsight, as shown in Fig. 182, in which the third circle is much elongatedand distorted, owing to its being outside the angle of vision. 

 [Illustration: Fig. 181. ]

 [Illustration: Fig. 182. ]

XCVII

HOW TO CORRECT DISPROPORTION IN THE WIDTH OF COLUMNS

 [Transcriber's Note: The column referred to as "1" in the text is marked "S" in both Figures. ]

The disproportion in the width of columns in Fig. 183 arises from thepoint of distance being too near the point of sight, or, in other words, taking too wide an angle of vision. It will be seen that column 3 ismuch wider than column 1. 

 [Illustration: Fig. 183. ]

 [Illustration: Fig. 184. ]

In our second figure (184) is shown how this defect is remedied, bydoubling the distance, or by counting the same distance as half, whichis easily effected by drawing the diagonal from _O_ to ½D, instead offrom _A_, as in the other figure, _O_ being at half base. Here thesquares lie much more level, and the columns are nearly the same width, showing the advantage of a long distance. 

XCVIII

HOW TO DRAW A CIRCLE OVER A CIRCLE OR A CYLINDER

First construct square and circle _ABE_, then draw square _CDF_ with itsdiagonals. Then find the various points _O_, and from these raiseperpendiculars to meet the diagonals of the upper square at points _P_, which, with the other points will be sufficient guides to draw thecircle required. This can be applied to towers, columns, &c. The size ofthe circles can be varied so that the upper portion of a cylinder orcolumn shall be smaller than the lower. 

 [Illustration: Fig. 185. ]

XCIX

TO DRAW A CIRCLE BELOW A GIVEN CIRCLE

Construct the upper square and circle as before, then by means of thevanishing scale _POV_, which should be made the depth required, dropperpendiculars from the various points marked _O_, obtained by thediagonals, making them the right depth by referring them to thevanishing scale, as shown in this figure. This can be used for drawinggarden fountains, basins, and various architectural objects. 

 [Illustration: Fig. 186. ]

C

APPLICATION OF PREVIOUS PROBLEM

That is, to draw a circle above a circle. In Fig. 187 can be seen how bymeans of the vanishing scale at the side we obtain the height of theverticals 1, 2, 3, 4, &c. , which determine the direction of the uppercircle; and in this second figure, how we resort to the same means todraw circular steps. 

 [Illustration: Fig. 187. ]

 [Illustration: Fig. 188. ]

CI

DORIC COLUMNS

It is as well for the art student to study the different orders ofarchitecture, whether architect or not, as he frequently has tointroduce them into his pictures, and at least must know theirproportions, and how columns diminish from base to capital, as shown inthis illustration. 

 [Illustration: Fig. 189. ]

CII

TO DRAW SEMICIRCLES STANDING UPON A CIRCLE AT ANY ANGLE

 [Illustration: Fig. 190. ]

Given the circle _ACBH_, on diagonal _AB_ draw semicircle _AKB_, and onthe same line _AB_ draw rectangle _AEFB_, its height being determined byradius _OK_ of semicircle. From centre _O_ draw _OF_ to corner ofrectangle. Through _f·_, where that line intersects the semicircle, draw_mn_ parallel to _AB_. This will give intersection _O·_ on the vertical_OK_, through which all such horizontals as _m·n·_, level with _mn_, must pass. Now take any other diameter, such as _GH_, and thereon raiserectangle _GghH_, the same height as the other. The manner of doing thisis to produce diameter _GH_ to the horizon till it finds its vanishingpoint at _V_. From _V_ through _K_ draw _hg_, and through _O·_ draw_n·m·_. From _O_ draw the two diagonals _og_ and _oh_, intersecting_m·n·_ at _O_, _O_, and thus we have the five points _GOKOH_ throughwhich to draw the required semicircle. 

CIII

A DOME STANDING ON A CYLINDER

 [Illustration: Fig. 191. ]

This figure is a combination of the two preceding it. A cylinder isfirst raised on the circle, and on the top of that we draw semicirclesfrom the different divisions on the circumference of the upper circle. This, however, only represents a small half-globular object. To draw thedome of a cathedral, or other building high above us, is another matter. From outside, where we can get to a distance, it is not difficult, butfrom within it will tax all our knowledge of perspective to give iteffect. 

We shall go more into this subject when we come to archways and vaultedroofs, &c. 

CIV

SECTION OF A DOME OR NICHE

 [Illustration: Fig. 192. ]

First draw outline of the niche _GFDBA_ (Fig. 193), then at its basedraw square and circle _GOA_, _S_ being the point of sight, and dividethe circumference of the circle into the required number of parts. Thendraw semicircle _FOB_, and over that another semicircle _EOC_. Themanner of drawing them is shown in Fig. 192. From the divisions on thecircle _GOA_ raise verticals to semicircle _FOB_, which will divide itin the same way. Divide the smaller semicircle _EOC_ into the samenumber of parts as the others, which divisions will serve as guidingpoints in drawing the curves of the dome that are drawn towards _D_, butthe shading must assist greatly in giving the effect of the recess. 

 [Illustration: Fig. 193. ]

In Fig. 192 will be seen how to draw semicircles in perspective. We first draw the half squares by drawing from centres _O_ of theirdiameters diagonals to distance-point, as _OD_, which cuts the vanishingline BS at _m_, and gives us the depth of the square, and in this wedraw the semicircle in the usual way. 

 [Illustration: Fig. 194. A Dome. ]

CV

A DOME

First draw a section of the dome ACEDB (Fig. 194) the shape required. Draw _AB_ at its base and _CD_ at some distance above it. Keeping theseas central lines, form squares thereon by drawing _SA_, _SB_, _SC_, _SD_, &c. , from point of sight, and determining their lengths bydiagonals _fh_, _f·h·_ from point of distance, passing through _O_. Having formed the two squares, draw perspective circles in each, anddivide their circumferences into twelve or whatever number of parts areneeded. To complete the figure draw from each division in the lowercircle curves passing through the corresponding divisions in the upperone, to the apex. But as these are freehand lines, it requires sometaste and knowledge to draw them properly, and of course in a largedrawing several more squares and circles might be added to aid thedraughtsman. The interior of the dome can be drawn in the same way. 

 [Illustration]

 [Illustration: Fig. 195. ]

CVI

HOW TO DRAW COLUMNS STANDING IN A CIRCLE

In Fig. 195 are sixteen cylinders or columns standing in a circle. Firstdraw the circle on the ground, then divide it into sixteen equal parts, and let each division be the centre of the circle on which to raise thecolumn. The question is how to make each one the right width inaccordance with its position, for it is evident that a near column mustappear wider than the opposite one. On the right of the figure is thevertical scale _A_, which gives the heights of the columns, and at itsfoot is a horizontal scale, or a scale of widths _B_. Now, according tothe line on which the column stands, we find its apparent width markedon the scale. Thus take the small square and circle at 15, without itscolumn, or the broken column at 16; and note that on each side of itscentre _O_ I have measured _oa_, _ob_, equal to spaces marked 3 on thesame horizontal in the scale _B_. Through these points _a_ and _b_ Ihave drawn lines towards point of sight _S_. Through their intersectionswith diagonal _e_, which is directed to point of distance, draw thefarther and nearer sides of the square in which to describe the circleand the cylinder or column thereon. I have made all the squares thusobtained in parallel perspective, but they do not represent the bases ofcolumns arranged in circles, which should converge towards the centre, and I believe in some cases are modified in form to suit that design. 

CVII

COLUMNS AND CAPITALS

This figure shows the application of the square and diagonal in drawingand placing columns in angular perspective. 

 [Illustration: Fig. 196. ]

CVIII

METHOD OF PERSPECTIVE EMPLOYED BY ARCHITECTS

The architects first draw a plan and elevation of the building to be putinto perspective. Having placed the plan at the required angle to thepicture plane, they fix upon the point of sight, and the distance fromwhich the drawing is to be viewed. They then draw a line _SP_ at rightangles to the picture plane _VV·_, which represents that distance sothat _P_ is the station-point. The eye is generally considered to bethe station-point, but when lines are drawn to that point from theground-plan, the station-point is placed on the ground, and is in factthe trace or projection exactly under the point at which the eye isplaced. From this station-point _P_, draw lines _PV_ and _PV·_ parallelto the two sides of the plan _ba_ and _ad_ (which will be at rightangles to each other), and produce them to the horizon, which they willtouch at points _V_ and _V·_. These points thus obtained will be thetwo vanishing points. 

 [Illustration: Fig. 197. A method of angular Perspective employed by architects. [_To face p. 171_] ]

The next operation is to draw lines from the principal points of theplan to the station-point _P_, such as _bP_, _cP_, _dP_, &c. , and wherethese lines intersect the picture plane (_VV·_ here represents it aswell as the horizon), drop perpendiculars _b·B_, _aA_, _d·D_, &c. , tomeet the vanishing lines _AV_, _AV·_, which will determine the points_A_, _B_, _C_, _D_, 1, 2, 3, &c. , and also the perspective lengths ofthe sides of the figure _AB_, _AD_, and the divisions _B_, 1, 2, &c. Taking the height of the figure _AE_ from the elevation, we measure iton _Aa_; as in this instance _A_ touches the ground line, it may be usedas a line of heights. 

I have here placed the perspective drawing under the ground plan to showthe relation between the two, and how the perspective is worked out, butthe general practice is to find the required measurements as here shown, to mark them on a straight edge of card or paper, and transfer them tothe paper on which the drawing is to be made. 

This of course is the simplest form of a plan and elevation. It is easyto see, however, that we could set out an elaborate building in the sameway as this figure, but in that case we should not place the drawingunderneath the ground-plan, but transfer the measurements to anothersheet of paper as mentioned above. 

CIX

THE OCTAGON

To draw the geometrical figure of an octagon contained in a square, takehalf of the diagonal of that square as radius, and from each cornerdescribe a quarter circle. At the eight points where they touch thesides of the square, draw the eight sides of the octagon. 

 [Illustration: Fig. 198. ]

 [Illustration: Fig. 199. ]

To put this into perspective take the base of the square _AB_ andthereon form the perspective square _ABCD_. From either extremity ofthat base (say _B_) drop perpendicular _BF_, draw diagonal _AF_, andthen from _B_ with radius _BO_, half that diagonal, describe arc _EOE_. This will give us the measurement _AE_. Make _GB_ equal to _AE_. Thendraw lines from _G_ and _E_ towards _S_, and by means of the diagonalsfind the transverse lines _KK_, _hh_, which will give us the eightpoints through which to draw the octagon. 

CX

HOW TO DRAW THE OCTAGON IN ANGULAR PERSPECTIVE

Form square _ABCD_ (new method), produce sides _BC_ and _AD_ to thehorizon at _V_, and produce _VA_ to _a·_ on base. Drop perpendicularfrom _B_ to _F_ the same length as _a·B_, and proceed as in the previousfigure to find the eight points on the oblique square through which todraw the octagon. 

 [Illustration: Fig. 200. ]

It will be seen that this operation is very much the same as in parallelperspective, only we make our measurements on the base line _a·B_ as wecannot measure the vanishing line _BA_ otherwise. 

CXI

HOW TO DRAW AN OCTAGONAL FIGURE IN ANGULAR PERSPECTIVE

In this figure in angular perspective we do precisely the same thing asin the previous problem, taking our measurements on the base line _EB_instead of on the vanishing line _BA_. If we wish to raise a figure onthis octagon the height of _EG_ we form the vanishing scale _EGO_, andfrom the eight points on the ground draw horizontals to _EO_ and thusfind all the points that give us the perspective height of each angle ofthe octagonal figure. 

 [Illustration: Fig. 201. ]

CXII

HOW TO DRAW CONCENTRIC OCTAGONS, WITH ILLUSTRATION OF A WELL

The geometrical figure 202 A shows how by means of diagonals _AC_ and_BD_ and the radii 1 2 3, &c. , we can obtain smaller octagons inside thelarger ones. Note how these are carried out in the second figure(202 B), and their application to this drawing of an octagonal well onan octagonal base. 

 [Illustration: Fig. 202 A. ]

 [Illustration: Fig. 202 B. ]

 [Illustration: Fig. 203. ]

CXIII

A PAVEMENT COMPOSED OF OCTAGONS AND SMALL SQUARES

To draw a pavement with octagonal tiles we will begin with an octagoncontained in a square _abcd_. Produce diagonal _ac_ to _V_. This will bethe vanishing point for the sides of the small squares directed towardsit. The other sides are directed to an inaccessible point out of thepicture, but their directions are determined by the lines drawn fromdivisions on base to V2 (see back, Fig. 133). 

 [Illustration: Fig. 204. ]

 [Illustration: Fig. 205. ]

I have drawn the lower figure to show how the squares which contain theoctagons are obtained by means of the diagonals, _BD_, _AC_, and thecentral line OV2. Given the square _ABCD_. From _D_ draw diagonal to_G_, then from _C_ through centre _o_ draw _CE_, and so on all the wayup the floor until sufficient are obtained. It is easy to see how othersquares on each side of these can be produced. 

CXIV

THE HEXAGON

The hexagon is a six-sided figure which, if inscribed in a circle, willhave each of its sides equal to the radius of that circle (Fig. 206). Ifinscribed in a rectangle _ABCD_, that rectangle will be equal in lengthto two sides of the hexagon or two radii of the circle, as _EF_, and itswidth will be twice the height of an equilateral triangle _mon_. 

 [Illustration: Fig. 206. ]

To put the hexagon into perspective, draw base of quadrilateral _AD_, divide it into four equal parts, and from each division draw lines topoint of sight. From _h_ drop perpendicular _ho_, and form equilateraltriangle _mno_. Take the height _ho_ and measure it twice along the basefrom _A_ to 2. From 2 draw line to point of distance, or from 1 to ½distance, and so find length of side _AB_ equal to A2. Draw _BC_, and _EF_ through centre _o·_, and thus we have the six points throughwhich to draw the hexagon. 

 [Illustration: Fig. 207. ]

CXV

A PAVEMENT COMPOSED OF HEXAGONAL TILES

In drawing pavements, except in the cases of square tiles, it isnecessary to make a plan of the required design, as in this figurecomposed of hexagons. First set out the hexagon as at _A_, then drawparallels 1 1, 2 2, &c. , to mark the horizontal ends of the tilesand the intermediate lines _oo_. Divide the base into the requirednumber of parts, each equal to one side of the hexagon, as 1, 2, 3, 4, &c. ; from these draw perpendiculars as shown in the figure, and also thediagonals passing through their intersections. Then mark with a strongline the outlines of the hexagonals, shading some of them; but thefigure explains itself. 

It is easy to put all these parallels, perpendiculars, and diagonalsinto perspective, and then to draw the hexagons. 

First draw the hexagon on _AD_ as in the previous figure, dividing _AD_into four, &c. , set off right and left spaces equal to these fourths, and from each division draw lines to point of sight. Produce sides _me_, _nf_ till they touch the horizon in points _V_, _V·_; these will be thetwo vanishing points for all the sides of the tiles that are recedingfrom us. From each division on base draw lines to each of thesevanishing points, then draw parallels through their intersections asshown on the figure. Having all these guiding lines it will not bedifficult to draw as many hexagons as you please. 

 [Illustration: Fig. 208. ]

Note that the vanishing points should be at equal distances from _S_, also that the parallelogram in which each tile is contained is oblong, and not square, as already pointed out. 

We have also made use of the triangle _omn_ to ascertain the length andwidth of that oblong. Another thing to note is that we have made use ofthe half distance, which enables us to make our pavement look flatwithout spreading our lines outside the picture. 

 [Illustration: Fig. 209. ]

CXVI

A PAVEMENT OF HEXAGONAL TILES IN ANGULAR PERSPECTIVE

This is more difficult than the previous figure, as we only make use ofone vanishing point; but it shows how much can be done by diagonals, asnearly all this pavement is drawn by their aid. First make a geometricalplan _A_ at the angle required. Then draw its perspective _K_. Divideline 4b into four equal parts, and continue these measurements allalong the base: from each division draw lines to _V_, and draw thehexagon _K_. Having this one to start with we produce its sides rightand left, but first to the left to find point _G_, the vanishing pointof the diagonals. Those to the right, if produced far enough, would meetat a distant vanishing point not in the picture. But the student shouldstudy this figure for himself, and refer back to Figs. 204 and 205. 

 [Illustration: Fig. 210. ]

CXVII

FURTHER ILLUSTRATION OF THE HEXAGON

 [Illustration: Fig. 211 A. ]

 [Illustration: Fig. 211 B. ]

To draw the hexagon in perspective we must first find the rectangle inwhich it is inscribed, according to the view we take of it. That at _A_we have already drawn. We will now work out that at _B_. Divide the base_AD_ into four equal parts and transfer those measurements to theperspective figure _C_, as at _AD_, measuring other equal spaces alongthe base. To find the depth _An_ of the rectangle, make _DK_ equal tobase of square. Draw _KO_ to distance-point, cutting _DO_ at _O_, andthus find line _LO_. Draw diagonal _Dn_, and through its intersectionswith the lines 1, 2, 3, 4 draw lines parallel to the base, and we shallthus have the framework, as it were, by which to draw the pavement. 

 [Illustration: Fig. 212. ]

CXVIII

ANOTHER VIEW OF THE HEXAGON IN ANGULAR PERSPECTIVE

 [Illustration: Fig. 213. ]

Given the rectangle _ABCD_ in angular perspective, produce side _DA_ to_E_ on base line. Divide _EB_ into four equal parts, and from eachdivision draw lines to vanishing point, then by means of diagonals, &c. , draw the hexagon. 

In Fig. 214 we have first drawn a geometrical plan, _G_, for the sake ofclearness, but the one above shows that this is not necessary. 

 [Illustration: Fig. 214. ]

To raise the hexagonal figure _K_ we have made use of the vanishingscale _O_ and the vanishing point _V_. Another method could be used bydrawing two hexagons one over the other at the required height. 

CXIX

APPLICATION OF THE HEXAGON TO DRAWING A KIOSK

 [Illustration: Fig. 215. ]

This figure is built up from the hexagon standing on a rectangular base, from which we have raised verticals, &c. Note how the jutting portionsof the roof are drawn from _o·_. But the figure explains itself, sothere is no necessity to repeat descriptions already given in theforegoing problems. 

CXX

THE PENTAGON

 [Illustration: Fig. 216. ]

The pentagon is a figure with five equal sides, and if inscribed in acircle will touch its circumference at five equidistant points. With anyconvenient radius describe circle. From half this radius, marked 1, drawa line to apex, marked 2. Again, with 1 as centre and 1 2 as radius, describe arc 2 3. Now with 2 as centre and 2 3 as radius describe arc3 4, which will cut the circumference at point 4. Then line 2 4 will beone of the sides of the pentagon, which we can measure round the circleand so produce the required figure. 

To put this pentagon into parallel perspective inscribe the circle inwhich it is drawn in a square, and from its five angles 4, 2, 4, &c. , drop perpendiculars to base and number them as in the figure. Then drawthe perspective square (Fig. 217) and transfer these measurements to itsbase. From these draw lines to point of sight, then by their aid and thetwo diagonals proceed to construct the pentagon in the same way that wedid the triangles and other figures. Should it be required to place thispentagon in the opposite position, then we can transfer our measurementsto the far side of the square, as in Fig. 218. 

 [Illustration: Fig. 217. ]

 [Illustration: Fig. 218. ]

Or if we wish to put it into angular perspective we adopt the samemethod as with the hexagon, as shown at Fig. 219. 

 [Illustration: Fig. 219. ]

Another way of drawing a pentagon (Fig. 220) is to draw an isoscelestriangle with an angle of 36° at its apex, and from centre of each sideof the triangle draw perpendiculars to meet at _o_, which will be thecentre of the circle in which it is inscribed. From this centre andwith radius _OA_ describe circle A 3 2, &c. Take base of triangle 1 2, measure it round the circle, and so find the five points through whichto draw the pentagon. The angles at 1 2 will each be 72°, double that at_A_, which is 36°. 

 [Illustration: Fig. 220. ]

CXXI

THE PYRAMID

Nothing can be more simple than to put a pyramid into perspective. Giventhe base (_abc_), raise from its centre a perpendicular (_OP_) of therequired height, then draw lines from the corners of that base to apoint _P_ on the vertical line, and the thing is done. These pyramidscan be used in drawing roofs, steeples, &c. The cone is drawn in thesame way, so also is any other figure, whether octagonal, hexangular, triangular, &c. 

 [Illustration: Fig. 221. ]

 [Illustration: Fig. 222. ]

 [Illustration: Fig. 223. ]

 [Illustration: Fig. 224. ]

CXXII

THE GREAT PYRAMID

This enormous structure stands on a square base of over thirteen acres, each side of which measures, or did measure, 764 feet. Its originalheight was 480 feet, each side being an equilateral triangle. Let us seehow we can draw this gigantic mass on our little sheet of paper. 

In the first place, to take it all in at one view we must put it veryfar back, and in the second the horizon must be so low down that wecannot draw the square base of thirteen acres on the perspective plane, that is on the ground, so we must draw it in the air, and also to a verysmall scale. 

Divide the base _AB_ into ten equal parts, and suppose each of theseparts to measure 10 feet, _S_, the point of sight, is placed on the leftof the picture near the side, in order that we may get a long line ofdistance, _S ½ D_; but even this line is only half the distance werequire. Let us therefore take the 16th distance, as shown in ourprevious illustration of the lighthouse (Fig. 92), which enables us tomeasure sixteen times the length of base _AB_, or 1, 600 feet. The base_ef_ of the pyramid is 1, 600 feet from the base line of the picture, andis, according to our 10-foot scale, 764 feet long. 

The next thing to consider is the height of the pyramid. We make a scaleto the right of the picture measuring 50 feet from _B_ to 50 at pointwhere _BP_ intersects base of pyramid, raise perpendicular _CG_ andthereon measure 480 feet. As we cannot obtain a palpable square on theground, let us draw one 480 feet above the ground. From _e_ and _f_raise verticals _eM_ and _fN_, making them equal to perpendicular _G_, and draw line _MN_, which will be the same length as base, or 764 feet. On this line form square _MNK_ parallel to the perspective plane, findits centre _O·_ by means of diagonals, and _O·_ will be the centralheight of the pyramid and exactly over the centre of the base. From thispoint _O·_ draw sloping lines _O·f_, _O·e_, _O·Y_, &c. , and the figureis complete. 

Note the way in which we find the measurements on base of pyramid and online _MN_. By drawing _AS_ and _BS_ to point of sight we find _Te_, which measures 100 feet at a distance of 1, 600 feet. We mark off sevenof these lengths, and an additional 64 feet by the scale, and so obtainthe required length. The position of the third corner of the base isfound by dropping a perpendicular from _K_, till it meets the line _eS_. 

Another thing to note is that the side of the pyramid that faces us, although an equilateral triangle, does not appear so, as its top angleis 382 feet farther off than its base owing to its leaning position. 

CXXIII

THE PYRAMID IN ANGULAR PERSPECTIVE

In order to show the working of this proposition I have taken a muchhigher horizon, which immediately detracts from the impression of thebigness of the pyramid. 

 [Illustration: Fig. 225. ]

We proceed to make our ground-plan _abcd_ high above the horizon insteadof below it, drawing first the parallel square and then the oblique one. From all the principal points drop perpendiculars to the ground and thusfind the points through which to draw the base of the pyramid. Findcentres _OO·_ and decide upon the height _OP_. Draw the sloping linesfrom _P_ to the corners of the base, and the figure is complete. 

CXXIV

TO DIVIDE THE SIDES OF THE PYRAMID HORIZONTALLY

Having raised the pyramid on a given oblique square, divide the verticalline OP into the required number of parts. From _A_ through _C_ draw_AG_ to horizon, which gives us _G_, the vanishing point of all thediagonals of squares parallel to and at the same angle as _ABCD_. From_G_ draw lines through the divisions 2, 3, &c. , on _OP_ cutting thelines _PA_ and _PC_, thus dividing them into the required parts. Throughthe points thus found draw from _V_ all those sides of the squares thathave _V_ for their vanishing point, as _ab_, _cd_, &c. Then join _bd_, _ac_, and the rest, and thus make the horizontal divisions required. 

 [Illustration: Fig. 226. ]

 [Illustration: Fig. 227. ]

The same method will apply to drawing steps, square blocks, &c. , asshown in Fig. 227, which is at the same angle as the above. 

CXXV

OF ROOFS

The pyramidal roof (Fig. 228) is so simple that it explains itself. Thechief thing to be noted is the way in which the diagonals are producedbeyond the square of the walls, to give the width of the eaves, according to their position. 

 [Illustration: Fig. 228. ]

Another form of the pyramidal roof is here given (Fig. 229). First drawthe cube _edcba_ at the required height, and on the side facing us, _adcb_, draw triangle _K_, which represents the end of a gable roof. Then draw similar triangles on the other sides of the cube (see Fig. 159, LXXXIV). Join the opposite triangles at the apex, and thus form twogable roofs crossing each other at right angles. From _o_, centre ofbase of cube, raise vertical _OP_, and then from _P_ draw sloping linesto each corner of base _a_, _b_, &c. , and by means of central linesdrawn from _P_ to half base, find the points where the gable roofsintersect the central spire or pyramid. Any other proportions can beobtained by adding to or altering the cube. 

 [Illustration: Fig. 229. ]

To draw a sloping or hip-roof which falls back at each end we must firstdraw its base, _CBDA_ (Fig. 230). Having found the centre _O_ andcentral line _SP_, and how far the roof is to fall back at each end, namely the distance _Pm_, draw horizontal line _RB_ through _m_. Thenfrom _B_ through _O_ draw diagonal _BA_, and from _A_ draw horizontal_AD_, which gives us point _n_. From these two points _m_ and _n_ raiseperpendiculars the height required for the roof, and from these drawsloping lines to the corners of the base. Join _ef_, that is, draw thetop line of the roof, which completes it. Fig. 231 shows a plan orbird's-eye view of the roof and the diagonal _AB_ passing through centre_O_. But there are so many varieties of roofs they would take almost abook to themselves to illustrate them, especially the cottages andfarm-buildings, barns, &c. , besides churches, old mansions, and others. There is also such irregularity about some of them that perspectiverules, beyond those few here given, are of very little use. So that thebest thing for an artist to do is to sketch them from the real wheneverhe has an opportunity. 

 [Illustration: Fig. 230. ]

 [Illustration: Fig. 231. ]

CXXVI

OF ARCHES, ARCADES, BRIDGES, &C. 

 [Illustration: Fig. 232. ]

For an arcade or cloister (Fig. 232) first set up the outer frame _ABCD_according to the proportions required. For round arches the height maybe twice that of the base, varying to one and a half. In Gothic archesthe height may be about three times the width, all of which proportionsare chosen to suit the different purposes and effects required. Dividethe base _AB_ into the desired number of parts, 8, 10, 12, &c. , eachpart representing 1 foot. (In this case the base is 10 feet and thehorizon 5 feet. ) Set out floor by means of ¼ distance. Divide it intosquares of 1 foot, so that there will be 8 feet between each column orpilaster, supposing we make them to stand on a square foot. Draw thefirst archway _EKF_ facing us, and its inner semicircle _gh_, with alsoits thickness or depth of 1 foot. Draw the span of the archway _EF_, then central line _PO_ to point of sight. Proceed to raise as many otherarches as required at the given distances. The intersections of thecentral line with the chords _mn_, &c. , will give the centres from whichto describe the semicircles. 

CXXVII

OUTLINE OF AN ARCADE WITH SEMICIRCULAR ARCHES

This is to show the method of drawing a long passage, corridor, orcloister with arches and columns at equal distances, and is worked inthe same way as the previous figure, using ¼ distance and ¼ base. The floor consists of five squares; the semicircles of the arches aredescribed from the numbered points on the central line _OS_, where itintersects the chords of the arches. 

 [Illustration: Fig. 233. ]

CXXVIII

SEMICIRCULAR ARCHES ON A RETREATING PLANE

First draw perspective square _abcd_. Let _ae·_ be the height of thefigure. Draw _ae·f·b_ and proceed with the rest of the outline. To drawthe arches begin with the one facing us, _Eo·F_ enclosed in thequadrangle _Ee·f·F_. With centre _O_ describe the semicircle and acrossit draw the diagonals _e·F_, _Ef·_, and through _nn_, where these linesintersect the semicircle, draw horizontal _KK_ and also _KS_ to point ofsight. It will be seen that the half-squares at the side are the samesize in perspective as the one facing us, and we carry out in them muchthe same operation; that is, we draw the diagonals, find the point _O_, and the points _nn_, &c. , through which to draw our arches. Seeperspective of the circle (Fig. 165). 

 [Illustration: Fig. 234. ]

If more points are required an additional diagonal from _O_ to _K_ maybe used, as shown in the figure, which perhaps explains itself. Themethod is very old and very simple, and of course can be applied to anykind of arch, pointed or stunted, as in this drawing of a pointed arch(Fig. 235). 

 [Illustration: Fig. 235. ]

CXXIX

AN ARCADE IN ANGULAR PERSPECTIVE

First draw the perspective square _ABCD_ at the angle required, by newmethod. Produce sides _AD_ and _BC_ to _V_. Draw diagonal _BD_ andproduce to point _G_, from whence we draw the other diagonals to _cfh_. Make spaces 1, 2, 3, &c. , on base line equal to _B 1_ to obtain sides ofsquares. Raise vertical _BM_ the height required. Produce _DA_ to _O_ onbase line, and from _O_ raise vertical _OP_ equal to _BM_. This lineenables us to dispense with the long vanishing point to the left; itsworking has been explained at Fig. 131. From _P_ draw _PRV_ to vanishingpoint _V_, which will intersect vertical _AR_ at _R_. Join _MR_, andthis line, if produced, would meet the horizon at the other vanishingpoint. In like manner make O2 equal to B2·. From 2 draw line to _V_, andat 2, its intersection with _AR_, draw line 2 2, which will also meetthe horizon at the other vanishing point. By means of the quarter-circle_A_ we can obtain the points through which to draw the semicirculararches in the same way as in the previous figure. 

 [Illustration: Fig. 236. ]

CXXX

A VAULTED CEILING

From the square ceiling _ABCD_ we have, as it were, suspended two archesfrom the two diagonals _DB_, _AC_, which spring from the four corners ofthe square _EFGH_, just underneath it. The curves of these arches, whichare not semicircular but elongated, are obtained by means of thevanishing scales _mS_, _nS_. Take any two convenient points _P_, _R_, oneach side of the semicircle, and raise verticals _Pm_, _Rn_ to _AB_, andon these verticals form the scales. Where _mS_ and _nS_ cut the diagonal_AC_ drop perpendiculars to meet the lower line of the scale at points1, 2. On the other side, using the other scales, we have droppedperpendiculars in the same way from the diagonal to 3, 4. These points, together with _EOG_, enable us to trace the curve _E 1 2 O 3 4 G_. Wedraw the arch under the other diagonal in precisely the same way. 

 [Illustration: Fig. 237. ]

The reason for thus proceeding is that the cross arches, althoughelongated, hang from their diagonals just as the semicircular arch _EKF_hangs from _AB_, and the lines _mn_, touching the circle at _PR_, arerepresented by 1, 2, hanging from the diagonal _AC_. 

 [Illustration: Fig. 238. ]

Figure 238, which is practically the same as the preceding onlydifferently shaded, is drawn in the following manner. Draw arch _EGF_facing us, and proceed with the rest of the corridor, but first findingthe flat ceiling above the square on the ground _ABcd_. Draw diagonals_ac_, _bd_, and the curves pending from them. But we no longer see theclear arch as in the other drawing, for the spaces between the curvesare filled in and arched across. 

CXXXI

A CLOISTER, FROM A PHOTOGRAPH

This drawing of a cloister from a photograph shows the correctness ofour perspective, and the manner of applying it to practical work. 

 [Illustration: Fig. 239. ]

CXXXII

THE LOW OR ELLIPTICAL ARCH

Let _AB_ be the span of the arch and _Oh_ its height. From centre _O_, with _OA_, or half the span, for radius, describe outer semicircle. Fromsame centre and _oh_ for radius describe the inner semicircle. Divideouter circle into a convenient number of parts, 1, 2, 3, &c. , to whichdraw radii from centre _O_. From each division drop perpendiculars. Where the radii intersect the inner circle, as at _gkmo_, drawhorizontals _op_, _mn_, _kj_, &c. , and through their intersections withthe perpendiculars _f_, _j_, _n_, _p_, draw the curve of the flattenedarch. Transfer this to the lower figure, and proceed to draw the tunnel. Note how the vanishing scale is formed on either side by horizontals_ba_, _fe_, &c. , which enable us to make the distant arches similar tothe near ones. 

 [Illustration: Fig. 240. ]

 [Illustration: Fig. 241. ]

CXXXIII

OPENING OR ARCHED WINDOW IN A VAULT

First draw the vault _AEB_. To introduce the window _K_, the upper partof which follows the form of the vault, we first decide on its width, which is _mn_, and its height from floor _Ba_. On line _Ba_ at the sideof the arch form scales _aa·S_, _bb·S_, &c. Raise the semicircular arch_K_, shown by a dotted line. The scale at the side will give the lengths_aa·_, _bb·_, &c. , from different parts of this dotted arch tocorresponding points in the curved archway or window required. 

 [Illustration: Fig. 242. ]

Note that to obtain the width of the window _K_ we have used thediagonals on the floor and width _m n_ on base. This method ofmeasurement is explained at Fig. 144, and is of ready application in acase of this kind. 

CXXXIV

STAIRS, STEPS, &C. 

Having decided upon the incline or angle, such as _CBA_, at which thesteps are to be placed, and the height _Bm_ of each step, draw _mn_ to_CB_, which will give the width. Then measure along base _AB_ this widthequal to _DB_, which will give that for all the other steps. Obtainlength _BF_ of steps, and draw _EF_ parallel to _CB_. These lines willaid in securing the exactness of the figure. 

 [Illustration: Fig. 243. ]

 [Illustration: Fig. 244. ]

CXXXV

STEPS, FRONT VIEW

In this figure the height of each step is measured on the vertical line_AB_ (this line is sometimes called the line of heights), and theirdepth is found by diagonals drawn to the point of distance _D_. The restof the figure explains itself. 

 [Illustration: Fig. 245. ]

CXXXVI

SQUARE STEPS

Draw first step _ABEF_ and its two diagonals. Raise vertical _AH_, andmeasure thereon the required height of each step, and thus form scale. Let the second step _CD_ be less all round than the first by _Ao_ or_Bo_. Draw _oC_ till it cuts the diagonal, and proceed to draw thesecond step, guided by the diagonals and taking its height from thescale as shown. Draw the third step in the same way. 

 [Illustration: Fig. 246. ]

CXXXVII

TO DIVIDE AN INCLINED PLANE INTO EQUAL PARTS--SUCH AS A LADDER PLACEDAGAINST A WALL

 [Illustration: Fig. 247. ]

Divide the vertical _EC_ into the required number of parts, and drawlines from point of sight _S_ through these divisions 1, 2, 3, &c. , cutting the line _AC_ at 1, 2, 3, &c. Draw parallels to _AB_, such as_mn_, from _AC_ to _BD_, which will represent the steps of the ladder. 

CXXXVIII

STEPS AND THE INCLINED PLANE

 [Illustration: Fig. 248. ]

In Fig. 248 we treat a flight of steps as if it were an inclined plane. Draw the first and second steps as in Fig. 245. Then through 1, 2, draw1V, _AV_ to _V_, the vanishing point on the vertical line _SV_. Thesetwo lines and the corresponding ones at _BV_ will form a kind ofvanishing scale, giving the height of each step as we ascend. It isespecially useful when we pass the horizontal line and we no longer seethe upper surface of the step, the scale on the right showing us how toproceed in that case. 

In Fig. 249 we have an example of steps ascending and descending. Firstset out the ground-plan, and find its vanishing point _S_ (point ofsight). Through _S_ draw vertical _BA_, and make _SA_ equal to _SB_. Setout the first step _CD_. Draw _EA_, _CA_, _DA_, and _GA_, for theascending guiding lines. Complete the steps facing us, at central line_OO_. Then draw guiding line _FB_ for the descending steps (see Rule 8). 

 [Illustration: Fig. 249. ]

CXXXIX

STEPS IN ANGULAR PERSPECTIVE

First draw the base _ABCD_ (Fig. 251) at the required angle by the newmethod (Fig. 250). Produce _BC_ to the horizon, and thus find vanishingpoint _V_. At this point raise vertical _VV·_. Construct first step_AB_, refer its height at _B_ to line of heights hI on left, and thusobtain height of step at _A_. Draw lines from _A_ and _F_ to _V·_. From_n_ draw diagonal through _O_ to _G_. Raise vertical at _O_ to representthe height of the next step, its height being determined by the scale ofheights at the side. From _A_ and _F_ draw lines to _V·_, and alsosimilar lines from _B_, which will serve as guiding lines to determinethe height of the steps at either end as we raise them to the requirednumber. 

 [Illustration: Fig. 250. ]

 [Illustration: Fig. 251. ]

CXL

A STEP LADDER AT AN ANGLE

 [Illustration: Fig. 252. ]

First draw the ground-plan _G_ at the required angle, using vanishingand measuring points. Find the height _hH_, and width at top _HH·_, anddraw the sides _HA_ and _H·E_. Note that _AE_ is wider than _HH·_, andalso that the back legs are not at the same angle as the front ones, andthat they overlap them. From _E_ raise vertical _EF_, and divide into asmany parts as you require rounds to the ladder. From these divisionsdraw lines 1 1, 2 2, &c. , towards the other vanishing point (not in thepicture), but having obtained their direction from the ground-plan inperspective at line _Ee_, you may set up a second vertical _ef_ at anypoint on _Ee_ and divide it into the same number of parts, which will bein proportion to those on _EF_, and you will obtain the same result bydrawing lines from the divisions on _EF_ to those on _ef_ as in drawingthem to the vanishing point. 

CXLI

SQUARE STEPS PLACED OVER EACH OTHER

 [Illustration: Fig. 253. ]

This figure shows the other method of drawing steps, which is simpleenough if we have sufficient room for our vanishing points. 

The manner of working it is shown at Fig. 124. 

CXLII

STEPS AND A DOUBLE CROSS DRAWN BY MEANS OF DIAGONALSAND ONE VANISHING POINT

Although in this figure we have taken a longer distance-point than inthe previous one, we are able to draw it all within the page. 

 [Illustration: Fig. 254. ]

Begin by setting out the square base at the angle required. Find point_G_ by means of diagonals, and produce _AB_ to _V_, &c. Mark height ofstep _Ao_, and proceed to draw the steps as already shown. Then by thediagonals and measurements on base draw the second step and the squareinside it on which to stand the foot of the cross. To draw the cross, raise verticals from the four corners of its base, and a line _K_ fromits centre. Through any point on this central line, if we draw adiagonal from point _G_ we cut the two opposite verticals of the shaftat _mn_ (see Fig. 255), and by means of the vanishing point _V_ we cutthe other two verticals at the opposite corners and thus obtain the fourpoints through which to draw the other sides of the square, which go tothe distant or inaccessible vanishing point. It will be seen bycarefully examining the figure that by this means we are enabled to drawthe double cross standing on its steps. 

 [Illustration: Fig. 255. ]

 [Illustration: Fig. 256. ]

CXLIII

A STAIRCASE LEADING TO A GALLERY

In this figure we have made use of the devices already set forth in theforegoing figures of steps, &c. , such as the side scale on the left ofthe figure to ascertain the height of the steps, the double lines drawnto the high vanishing point of the inclined plane, and so on; but theprincipal use of this diagram is to show on the perspective plane, whichas it were runs under the stairs, the trace or projection of the flightsof steps, the landings and positions of other objects, which will befound very useful in placing figures in a composition of this kind. It will be seen that these underneath measurements, so to speak, areobtained by the half-distance. 

CXLIV

WINDING STAIRS IN A SQUARE SHAFT

Draw square _ABCD_ in parallel perspective. Divide each side into four, and raise verticals from each division. These verticals will mark thepositions of the steps on each wall, four in number. From centre _O_raise vertical _OP_, around which the steps are to wind. Let _AF_ be theheight of each step. Form scale _AB_, which will give the height of eachstep according to its position. Thus at _mn_ we find the height at thecentre of the square, so if we transfer this measurement to the centralline _OP_ and repeat it upwards, say to fourteen, then we have theheight of each step on the line where they all meet. Starting then withthe first on the right, draw the rectangle _gD1f_, the height of _AF_, then draw to the central line _go_, f1, and 1 1, and thus complete thefirst step. On _DE_, measure heights equal to _D 1_. Draw 2 2 towardscentral line, and 2n towards point of sight till it meets the secondvertical _nK_. Then draw n2 to centre, and so complete the secondstep. From 3 draw 3a to third vertical, from 4 to fourth, and so on, thus obtaining the height of each ascending step on the wall to theright, completing them in the same way as numbers 1 and 2, when we cometo the sixth step, the other end of which is against the wall oppositeto us. Steps 6, 7, 8, 9 are all on this wall, and are therefore equal inheight all along, as they are equally distant. Step 10 is turned towardsus, and abuts on the wall to our left; its measurement is taken on thescale _AB_ just underneath it, and on the same line to which it isdrawn. Step 11 is just over the centre of base _mo_, and is thereforeparallel to it, and its height is _mn_. The widths of steps 12 and 13seem gradually to increase as they come towards us, and as they riseabove the horizon we begin to see underneath them. Steps 13, 14, 15, 16are against the wall on this side of the picture, which we may supposehas been removed to show the working of the drawing, or they might be anopen flight as we sometimes see in shops and galleries, although in thatcase they are generally enclosed in a cylindrical shaft. 

 [Illustration: Fig. 257. ]

 [Illustration: Fig. 258. ]

CXLV

WINDING STAIRS IN A CYLINDRICAL SHAFT

First draw the circular base _CD_. Divide the circumference into equalparts, according to the number of steps in a complete round, say twelve. Form scale _ASF_ and the larger scale _ASB_, on which is shown theperspective measurements of the steps according to their positions;raise verticals such as _ef_, _Gh_, &c. From divisions on circumferencemeasure out the central line _OP_, as in the other figure, and find theheights of the steps 1, 2, 3, 4, &c. , by the corresponding numbers inthe large scale to the left; then proceed in much the same way as in theprevious figure. Note the central column _OP_ cuts off a small portionof the steps at that end. 

In ordinary cases only a small portion of a winding staircase isactually seen, as in this sketch. 

 [Illustration: Fig. 259. Sketch of Courtyard in Toledo. ]

CXLVI

OF THE CYLINDRICAL PICTURE OR DIORAMA

 [Illustration: Fig. 260. ]

Although illusion is by no means the highest form of art, there is nopicture painted on a flat surface that gives such a wonderful appearanceof truth as that painted on a cylindrical canvas, such as thosepanoramas of 'Paris during the Siege', exhibited some years ago; 'TheBattle of Trafalgar', only lately shown at Earl's Court; and manyothers. In these pictures the spectator is in the centre of a cylinder, and although he turns round to look at the scene the point of sight isalways in front of him, or nearly so. I believe on the canvas thesepoints are from 12 to 16 feet apart. 

The reason of this look of truth may be explained thus. If we placethree globes of equal size in a straight line, and trace their apparentwidths on to a straight transparent plane, those at the sides, as _a_and _b_, will appear much wider than the centre one at _c_. Whereas, ifwe trace them on a semicircular glass they will appear very nearly equaland, of the three, the central one _c_ will be rather the largest, asmay be seen by this figure. 

We must remember that, in the first case, when we are looking at a globeor a circle, the visual rays form a cone, with a globe at its base. Ifthese three cones are intersected by a straight glass _GG_, and lookedat from point _S_, the intersection of _C_ will be a circle, as the coneis cut straight across. The other two being intersected at an angle, will each be an ellipse. At the same time, if we look at them from thestation point, with one eye only, then the three globes (or tracings ofthem) will appear equal and perfectly round. 

Of course the cylindrical canvas is necessary for panoramas; but wehave, as a rule, to paint our pictures and wall-decorations on flatsurfaces, and therefore must adapt our work to these conditions. 

In all cases the artist must exercise his own judgement both in thearrangement of his design and the execution of the work, for there isperspective even in the touch--a painting to be looked at from adistance requires a bold and broad handling; in small cabinet picturesthat we live with in our own rooms we look for the exquisite workmanshipof the best masters. 

BOOK FOURTH

CXLVII

THE PERSPECTIVE OF CAST SHADOWS

There is a pretty story of two lovers which is sometimes told as theorigin of art; at all events, I may tell it here as the origin ofsciagraphy. A young shepherd was in love with the daughter of a potter, but it so happened that they had to part, and were passing their lastevening together, when the girl, seeing the shadow of her lover'sprofile cast from a lamp on to some wet plaster or on the wall, took ametal point, perhaps some sort of iron needle, and traced the outline ofthe face she loved on to the plaster, following carefully the outline ofthe features, being naturally anxious to make it as like as possible. The old potter, the father of the girl, was so struck with it that hebegan to ornament his wares by similar devices, which gave themincreased value by the novelty and beauty thus imparted to them. 

Here then we have a very good illustration of our present subject andits three elements. First, the light shining on the wall; second, thewall or the plane of projection, or plane of shade; and third, theintervening object, which receives as much light on itself as itdeprives the wall of. So that the dark portion thus caused on the planeof shade is the cast shadow of the intervening object. 

We have to consider two sorts of shadows: those cast by a luminary along way off, such as the sun; and those cast by artificial light, suchas a lamp or candle, which is more or less close to the object. In thefirst case there is no perceptible divergence of rays, and the outlinesof the sides of the shadows of regular objects, as cubes, posts, &c. , will be parallel. In the second case, the rays diverge according to thenearness of the light, and consequently the lines of the shadows, instead of being parallel, are spread out. 

CXLVIII

THE TWO KINDS OF SHADOWS

In Figs. 261 and 262 is seen the shadow cast by the sun by parallelrays. 

Fig. 263 shows the shadows cast by a candle or lamp, where the raysdiverge from the point of light to meet corresponding diverging lineswhich start from the foot of the luminary on the ground. 

 [Illustration: Fig. 261. ]

 [Illustration: Fig. 262. ]

The simple principle of cast shadows is that the rays coming from thepoint of light or luminary pass over the top of the intervening objectwhich casts the shadow on to the plane of shade to meet the horizontaltrace of those rays on that plane, or the lines of light proceed fromthe point of light, and the lines of the shadow are drawn from the footor trace of the point of light. 

 [Illustration: Fig. 263. ]

 [Illustration: Fig. 264. ]

Fig. 264 shows this in profile. Here the sun is on the same plane as thepicture, and the shadow is cast sideways. 

Fig. 265 shows the same thing, but the sun being behind the object, casts its shadow forwards. Although the lines of light are parallel, they are subject to the laws of perspective, and are therefore drawnfrom their respective vanishing points. 

 [Illustration: Fig. 265. ]

CXLIX

SHADOWS CAST BY THE SUN

Owing to the great distance of the sun, we have to consider the rays oflight proceeding from it as parallel, and therefore subject to the samelaws as other parallel lines in perspective, as already noted. And forthe same reason we have to place the foot of the luminary on thehorizon. It is important to remember this, as these two things make thedifference between shadows cast by the sun and those cast by artificiallight. 

The sun has three principal positions in relation to the picture. In thefirst case it is supposed to be in the same plane either to the right orto the left, and in that case the shadows will be parallel with the baseof the picture. In the second position it is on the other side of it, or facing the spectator, when the shadows of objects will be thrownforwards or towards him. In the third, the sun is in front of thepicture, and behind the spectator, so that the shadows are thrown in theopposite direction, or towards the horizon, the objects themselves beingin full light. 

CL

THE SUN IN THE SAME PLANE AS THE PICTURE

Besides being in the same plane, the sun in this figure is at an angleof 45° to the horizon, consequently the shadows will be the same lengthas the figures that cast them are high. Note that the shadow of stepNo. 1 is cast upon step No. 2, and that of No. 2 on No. 3, the top ofeach of these becoming a plane of shade. 

 [Illustration: Fig. 266. ]

 [Illustration: Fig. 267. ]

 [Illustration: Fig. 268. ]

When the shadow of an object such as _A_, Fig. 268, which would fallupon the plane, is interrupted by another object _B_, then the outlineof the shadow is still drawn on the plane, but being interrupted by thesurface _B_ at _C_, the shadow runs up that plane till it meets the rays1, 2, which define the shadow on plane _B_. This is an important point, but is quite explained by the figure. 

Although we have said that the rays pass over the top of the objectcasting the shadow, in the case of an archway or similar figure theypass underneath it; but the same principle holds good, that is, we drawlines from the guiding points in the arch, 1, 2, 3, &c. , at the sameangle of 45° to meet the traces of those rays on the plane of shade, andso get the shadow of the archway, as here shown. 

 [Illustration: Fig. 269. ]

CLI

THE SUN BEHIND THE PICTURE

We have seen that when the sun's altitude is at an angle of 45° theshadows on the horizontal plane are the same length as the height of theobjects that cast them. Here (Fig. 270), the sun still being at 45°altitude, although behind the picture, and consequently throwing theshadow of _B_ forwards, that shadow must be the same length as theheight of cube _B_, which will be seen is the case, for the shadow _C_is a square in perspective. 

 [Illustration: Fig. 270. ]

To find the angle of altitude and the angle of the sun to the picture, we must first find the distance of the spectator from the foot of theluminary. 

 [Illustration: Fig. 271. ]

From point of sight _S_ (Fig. 270) drop perpendicular to _T_, thestation-point. From _T_ draw _TF_ at 45° to meet horizon at _F_. Withradius _FT_ make _FO_ equal to it. Then _O_ is the position of thespectator. From _F_ raise vertical _FL_, and from _O_ draw a line at 45°to meet _FL_ at _L_, which is the luminary at an altitude of 45°, and atan angle of 45° to the picture. 

Fig. 272 is similar to the foregoing, only the angles of altitude and ofthe sun to the picture are altered. 

_Note. _--The sun being at 50° to the picture instead of 45°, is nearerthe point of sight; at 90° it would be exactly opposite the spectator, and so on. Again, the elevation being less (40° instead of 45°) theshadow is longer. Owing to the changed position of the sun two sides ofthe cube throw a shadow. Note also that the outlines of the shadow, 1 2, 2 3, are drawn to the same vanishing points as the cube itself. 

It will not be necessary to mark the angles each time we make a drawing, as it must be seen we can place the luminary in any position that suitsour convenience. 

 [Illustration: Fig. 272. ]

CLII

SUN BEHIND THE PICTURE, SHADOWS THROWN ON A WALL

As here we change the conditions we must also change our procedure. Anupright wall now becomes the plane of shade, therefore as the principleof shadows must always remain the same we have to change the relativepositions of the luminary and the foot thereof. 

At _S_ (point of sight) raise vertical _SF·_, making it equal to _fL_. _F·_ becomes the foot of the luminary, whilst the luminary itself stillremains at _L_. 

 [Illustration: Fig. 273. ]

We have but to turn this page half round and look at it from the right, and we shall see that _SF·_ becomes as it were the horizontal line. Theluminary _L_ is at the right side of point _S_ instead of the left, andthe foot thereof is, as before, the trace of the luminary, as it is justunderneath it. We shall also see that by proceeding as in previousfigures we obtain the same results on the wall as we did on thehorizontal plane. Fig. B being on the horizontal plane is treated asalready shown. The steps have their shadows partly on the wall andpartly on the horizontal plane, so that the shadows on the wall areoutlined from _F·_ and those on the ground from _f_. Note shadow of roof_A_, and how the line drawn from _F·_ through _A_ is met by the linedrawn from the luminary _L_, at the point _P_, and how the lower line ofthe shadow is directed to point of sight _S_. 

 [Illustration: Fig. 274. ]

Fig. 274 is a larger drawing of the steps, &c. , in further illustrationof the above. 

CLIII

SUN BEHIND THE PICTURE THROWING SHADOW ON AN INCLINED PLANE

 [Illustration: Fig. 275. ]

The vanishing point of the shadows on an inclined plane is on a verticaldropped from the luminary to a point (_F_) on a level with the vanishingpoint (_P_) of that inclined plane. Thus _P_ is the vanishing point ofthe inclined plane _K_. Draw horizontal _PF_ to meet _fL_ (the linedrawn from the luminary to the horizon). Then _F_ will be the vanishingpoint of the shadows on the inclined plane. To find the shadow of _M_draw lines from _F_ through the base _eg_ to _cd_. From luminary _L_draw lines through _ab_, also to _cd_, where they will meet those drawnfrom _F_. Draw _CD_, which determines the length of the shadow _egcd_. 

CLIV

THE SUN IN FRONT OF THE PICTURE

 [Illustration: Fig. 276. ]

When the sun is in front of the picture we have exactly the oppositeeffect to that we have just been studying. The shadows, instead ofcoming towards us, are retreating from us, and the objects throwing themare in full light, consequently we have to reverse our treatment. Let ussuppose the sun to be placed above the horizon at _L·_, on the right ofthe picture and behind the spectator (Fig. 276). If we transport thelength _L·f·_ to the opposite side and draw the vertical downwards fromthe horizon, as at _FL_, we can then suppose point _L_ to be exactlyopposite the sun, and if we make that the vanishing point for the sun'srays we shall find that we obtain precisely the same result. As in Fig. 277, if we wish to find the length of _C_, which we may suppose to bethe shadow of _P_, we can either draw a line from _A_ through _O_ to_B_, or from _B_ through _O_ to _A_, for the result is the same. And aswe cannot make use of a point that is behind us and out of the picture, we have to resort to this very ingenious device. 

 [Illustration: Fig. 277. ]

In Fig. 276 we draw lines L1, L2, L3 from the luminary to the top of theobject to meet those drawn from the foot _F_, namely F1, F2, F3, in thesame way as in the figures we have already drawn. 

 [Illustration: Fig. 278. ]

Fig. 278 gives further illustration of this problem. 

CLV

THE SHADOW OF AN INCLINED PLANE

The two portions of this inclined plane which cast the shadow are firstthe side _fbd_, and second the farther end _abcd_. The points we have tofind are the shadows of _a_ and _b_. From luminary _L_ draw _La_, _Lb_, and from _F_, the foot, draw _Fc_, _Fd_. The intersection of these lineswill be at _a·b·_. If we join _fb·_ and _db·_ we have the shadow of theside _fbd_, and if we join _ca·_ and _a·b·_ we have the shadow of_abcd_, which together form that of the figure. 

 [Illustration: Fig. 279. ]

CLVI

SHADOW ON A ROOF OR INCLINED PLANE

To draw the shadow of the figure _M_ on the inclined plane _K_ (or achimney on a roof). First find the vanishing point _P_ of the inclinedplane and draw horizontal _PF_ to meet vertical raised from _L_, theluminary. Then _F_ will be the vanishing point of the shadow. From _L_draw L1, L2, L3 to top of figure _M_, and from the base of _M_ draw1F, 2F, 3F to _F_, the vanishing point of the shadow. Theintersections of these lines at 1, 2, 3 on _K_ will determine thelength and form of the shadow. 

 [Illustration: Fig. 280. ]

CLVII

TO FIND THE SHADOW OF A PROJECTION OR BALCONY ON A WALL

To find the shadow of the object _K_ on the wall _W_, drop verticals_OO_ till they meet the base line _B·B·_ of the wall. Then from thepoint of sight _S_ draw lines through _OO_, also drop verticals _Dd·_, _Cc·_, to meet these lines in _d·c·_; draw _c·F_ and _d·F_ to foot ofluminary. From the points _xx_ where these lines cut the base _B_ raiseperpendiculars _xa·_, _xb·_. From _D_, _A_, and _B_ draw lines to theluminary _L_. These lines or rays intersecting the verticals raised from_xx_ at _a·b·_ will give the respective points of the shadow. 

 [Illustration: Fig. 281. ]

The shadow of the eave of a roof can be obtained in the same way. Takeany point thereon, mark its trace on the ground, and then proceed asabove. 

CLVIII

SHADOW ON A RETREATING WALL, SUN IN FRONT

Let _L_ be the luminary. Raise vertical _LF_. _F_ will be the vanishingpoint of the shadows on the ground. Draw _Lf·_ parallel to _FS_. Drop_Sf·_ from point of sight; _f·_ (so found) is the vanishing point of theshadows on the wall. For shadow of roof draw _LE_ and _f·B_, giving us_e_, the shadow of _E_. Join _Be_, &c. , and so draw shadow of eave ofroof. 

 [Illustration: Fig. 282. ]

For shadow of _K_ draw lines from luminary _L_ to meet those from _f·_the foot, &c. 

The shadow of _D_ over the door is found in a similar way to that of theroof. 

 [Illustration: Fig. 283. ]

Figure 283 shows how the shadow of the old man in the preceding drawingis found. 

CLIX

SHADOW OF AN ARCH, SUN IN FRONT

Having drawn the arch, divide it into a certain number of parts, sayfive. From these divisions drop perpendiculars to base line. Fromdivisions on _AB_ draw lines to _F_ the foot, and from those on thesemicircle draw lines to _L_ the luminary. Their intersections will givethe points through which to draw the shadow of the arch. 

 [Illustration: Fig. 284. ]

CLX

SHADOW IN A NICHE OR RECESS

In this figure a similar method to that just explained is adopted. Dropperpendiculars from the divisions of the arch 1 2 3 to the base. Fromthe foot of each draw 1S, 2S, 3S to foot of luminary _S_, andfrom the top of each, A 1 2 3 B, draw lines to _L_ as before. Where theformer intersect the curve on the floor of the niche raise verticalsto meet the latter at P 1 2 B, &c. These points will indicate about theposition of the shadow; but the niche being semicircular and domed atthe top the shadow gradually loses itself in a gradated and somewhatserpentine half-tone. 

 [Illustration: Fig. 285. ]

CLXI

SHADOW IN AN ARCHED DOORWAY

 [Illustration: Fig. 286. ]

This is so similar to the last figure in many respects that I need notrepeat a description of the manner in which it is done. And surely anartist after making a few sketches from the actual thing will hardlyrequire all this machinery to draw a simple shadow. 

CLXII

SHADOWS PRODUCED BY ARTIFICIAL LIGHT

 [Illustration: Fig. 287. ]

Shadows thrown by artificial light, such as a candle or lamp, are foundby drawing lines from the seat of the luminary through the feet of theobjects to meet lines representing rays of light drawn from the luminaryitself over the tops or the corners of the objects; very much as in thecases of sun-shadows, but with this difference, that whereas the foot ofthe luminary in this latter case is supposed to be on the horizon aninfinite distance away, the foot in the case of a lamp or candle may beon the floor or on a table close to us. First draw the table and chair, &c. (Fig. 287), and let _L_ be the luminary. For objects on the tablesuch as _K_ the foot will be at _f_ on the table. For the shadows on thefloor, of the chair and table itself, we must find the foot of theluminary on the floor. Draw _So_, find trace of the edge of the table, drop vertical _oP_, draw _PS_ to point of sight, drop vertical from footof candlestick to meet _PS_ in _F_. Then _F_ is the foot of the luminaryon the floor. From this point draw lines through the feet or traces ofobjects such as the corners of the table, &c. , to meet other lines drawnfrom the point of light, and so obtain the shadow. 

CLXIII

SOME OBSERVATIONS ON REAL LIGHT AND SHADE

Although the figures we have been drawing show the principles on whichsun-shadows are shaped, still there are so many more laws to beconsidered in the great art of light and shade that it is better toobserve them in Nature herself or under the teaching of the real sun. Inthe study of a kitchen and scullery in an old house in Toledo (Fig. 288)we have an example of the many things to be considered besides the mereshapes of shadows of regular forms. It will be seen that the light isdispersed in all directions, and although there is a good deal ofhalf-shade there are scarcely any cast shadows except on the floor; butthe light on the white walls in the outside gallery is so reflected intothe cast shadows that they are extremely faint. The luminosity of thispart of the sketch is greatly enhanced by the contrast of the dark legsof the bench and the shadows in the roof. The warm glow of all thisportion is contrasted by the grey door and its frame. 

 [Illustration: Fig. 288. ]

Note that the door itself is quite luminous, and lighted up by thereflection of the sun from the tiled floor, so that the bars in theupper part throw distinct shadows, besides the mystery of colour thusintroduced. The little window to the left, though not admitting muchdirect sunlight, is evidence of the brilliant glare outside; for thereflected light is very conspicuous on the top and on the shutters oneach side; indeed they cast distinct shadows up and down, while someclear daylight from the blue sky is reflected on the window-sill. As tothe sink, the table, the wash-tubs, &c. , although they seem in stronglight and shade they really receive little or no direct light from asingle point; but from the strong reflected light re-reflected into themfrom the wall of the doorway. There are many other things in sucheffects as this which the artist will observe, and which can only bestudied from real light and shade. Such is the character of reflectedlight, varying according to the angle and intensity of the luminary anda hundred other things. When we come to study light in the open air weget into another region, and have to deal with it accordingly, and yetwe shall find that our sciagraphy will be a help to us even in thisbewilderment; for it will explain in a manner the innumerable shapes ofsun-shadows that we observe out of doors among hills and dales, showingup their forms and structure; its play in the woods and gardens, and itsvalue among buildings, showing all their juttings and abuttings, recesses, doorways, and all the other architectural details. Nor must weforget light's most glorious display of all on the sea and in the cloudsand in the sunrises and the sunsets down to the still and lovelymoonlight. 

These sun-shadows are useful in showing us the principle of light andshade, and so also are the shadows cast by artificial light; but theyare only the beginning of that beautiful study, that exquisite art oftone or _chiaro-oscuro_, which is infinite in its variety, is full ofthe deepest mystery, and is the true poetry of art. For this the studentmust go to Nature herself, must study her in all her moods from earlydawn to sunset, in the twilight and when night sets in. No mathematicalrules can help him, but only the thoughtful contemplation, the silentwatching, and the mental notes that he can make and commit to memory, combining them with the sentiments to which they in turn give rise. The_plein air_, or broad daylight effects, are but one item of the greatrange of this ever-changing and deepening mystery--from the hard realityto the soft blending of evening when form almost disappears, even to themerging of the whole landscape, nay, the whole world, into adream--which is felt rather than seen, but possesses a charm that almostdefies the pencil of the painter, and can only be expressed by the deepand sweet notes of the poet and the musician. For love and reverence arenecessary to appreciate and to present it. 

There is also much to learn about artificial light. For here, again, thestudy is endless: from the glare of a hundred lights--electric andotherwise--to the single lamp or candle. Indeed a whole volume could befilled with illustrations of its effects. To those who aim at producingintense brilliancy, refusing to acknowledge any limitations to theircapacity, a hundred or a thousand lights commend themselves; and eventhough wild splashes of paint may sometimes be the result, still theeffort is praiseworthy. But those who prefer the mysterious lighting ofa Rembrandt will find, if they sit contemplating in a room lit with onelamp only, that an endless depth of mystery surrounds them, full of darkrecesses peopled by fancy and sweet thought, whilst the most beautifulgradations soften the forms without distorting them; and at the sametime he can detect the laws of this science of light and shade athousand times repeated and endless in its variety. 

_Note. _--Fig. 288 must be looked upon as a rough sketch which only givesthe general effect of the original drawing; to render all the delicatetints, tones and reflections described in the text would require ahighly-finished reproduction in half-tone or in colour. 

As many of the figures in this book had to be re-drawn, not a lighttask, I must here thank Miss Margaret L. Williams, one of our Academystudents, for kindly coming to my assistance and volunteering hercareful co-operation. 

CLXIV

REFLECTION

 [Transcriber's Note: In this chapter, [R] represents "R" printed upside-down. ]

Reflections in still water can best be illustrated by placing somesimple object, such as a cube, on a looking-glass laid horizontally on atable, or by studying plants, stones, banks, trees, &c. , reflected insome quiet pond. It will then be seen that the reflection is thecounterpart of the object reversed, and having the same vanishing pointsas the object itself. 

 [Illustration: Fig. 289. ]

Let us suppose _R_ (Fig. 289) to be standing on the water or reflectingplane. To find its reflection make square [R] equal to the originalsquare _R_. Complete the reversed cube by drawing its other sides, &c. It is evident that this lower cube is the reflection of the one aboveit, although it differs in one respect, for whereas in figure _R_ thetop of the cube is seen, in its reflection [R] it is hidden, &c. Infigure A of a semicircular arch we see the underneath portion of thearch reflected in the water, but we do not see it in the actual object. However, these things are obvious. Note that the reflected line must beequal in length to the actual one, or the reflection of a square wouldnot be a square, nor that of a semicircle a semicircle. The apparentlengthening of reflections in water is owing to the surface being brokenby wavelets, which, leaping up near to us, catch some of the image ofthe tree, or whatever it is, that it is reflected. 

 [Illustration: Fig. 290. ]

In this view of an arch (Fig. 290) note that the reflection is obtainedby dropping perpendiculars from certain points on the arch, 1, 0, 2, &c. , to the surface of the reflecting plane, and then measuring the samelengths downwards to corresponding points, 1, 0, 2, &c. , in thereflection. 

CLXV

ANGLES OF REFLECTION

In Fig. 291 we take a side view of the reflected object in order to showthat at whatever angle the visual ray strikes the reflecting surface itis reflected from it at the same angle. 

 [Illustration: Fig. 291. ]

We have seen that the reflected line must be equal to the original line, therefore _mB_ must equal _Ma_. They are also at right angles to _MN_, the plane of reflection. We will now draw the visual ray passing from_E_, the eye, to _B_, which is the reflection of _A_; and justunderneath it passes through _MN_ at _O_, which is the point where thevisual ray strikes the reflecting surface. Draw _OA_. This linerepresents the ray reflected from it. We have now two triangles, _OAm_and _OmB_, which are right-angled triangles and equal, therefore angle_a_ equals angle _b_. But angle _b_ equals angle _c_. Therefore angle_EcM_ equals angle _Aam_, and the angle at which the ray strikes thereflecting plane is equal to the angle at which it is reflected from it. 

CLXVI

REFLECTIONS OF OBJECTS AT DIFFERENT DISTANCES

In this sketch the four posts and other objects are represented standingon a plane level or almost level with the water, in order to show theworking of our problem more clearly. It will be seen that the post _A_is on the brink of the reflecting plane, and therefore is entirelyreflected; _B_ and _C_ being farther back are only partially seen, whereas the reflection of _D_ is not seen at all. I have made all theposts the same height, but with regard to the houses, where the lengthof the vertical lines varies, we obtain their reflections by measuringfrom the points _oo_ upwards and downwards as in the previous figure. 

 [Illustration: Fig. 292. ]

Of course these reflections vary according to the position they areviewed from; the lower we are down, the more do we see of thereflections of distant objects, and vice versa. When the figures are ona higher plane than the water, that is, above the plane of reflection, we have to find their perspective position, and drop a perpendicular_AO_ (Fig. 293) till it comes in contact with the plane of reflection, which we suppose to run under the ground, then measure the same lengthdownwards, as in this figure of a girl on the top of the steps. Point_o_ marks the point of contact with the plane, and by measuringdownwards to _a·_ we get the length of her reflection, or as much as isseen of it. Note the reflection of the steps and the sloping bank, andthe application of the inclined plane ascending and descending. 

 [Illustration: Fig. 293. ]

CLXVII

REFLECTION IN A LOOKING-GLASS

I had noticed that some of the figures in Titian's pictures were onlyhalf life-size, and yet they looked natural; and one day, thinking Iwould trace myself in an upright mirror, I stood at arm's length from itand with a brush and Chinese white, I made a rough outline of my faceand figure, and when I measured it I found that my drawing was exactlyhalf as long and half as wide as nature. I went closer to the glass, butthe same outline fitted me. Then I retreated several paces, and stillthe same outline surrounded me. Although a little surprising at first, the reason is obvious. The image in the glass retreats or advancesexactly in the same measure as the spectator. 

 [Illustration: Fig. 294. ]

Suppose him to represent one end of a parallelogram _e·s·_, and hisimage _a·b·_ to represent the other. The mirror _AB_ is a perpendicularhalf-way between them, the diagonal _e·b·_ is the visual ray passingfrom the eye of the spectator to the foot of his image, and is thediagonal of a rectangle, therefore it cuts _AB_ in the centre _o_, and_AO_ represents _a·b·_ to the spectator. This is an experiment that anyone may try for himself. Perhaps the above fact may have something to dowith the remarks I made about Titian at the beginning of this chapter. 

 [Illustration: Fig. 295. ]

 [Illustration: Fig. 296. ]

CLXVIII

THE MIRROR AT AN ANGLE

If an object or line _AB_ is inclined at an angle of 45° to the mirror_RR_, then the angle _BAC_ will be a right angle, and this angle isexactly divided in two by the reflecting plane _RR_. And whatever theangle of the object or line makes with its reflection that angle willalso be exactly divided. 

 [Illustration: Fig. 297. ]

 [Illustration: Fig. 298. ]

Now suppose our mirror to be standing on a horizontal plane and on apivot, so that it can be inclined either way. Whatever angle the mirroris to the plane the reflection of that plane in the mirror will be atthe same angle on the other side of it, so that if the mirror _OA_ (Fig. 298) is at 45° to the plane _RR_ then the reflection of that plane inthe mirror will be 45° on the other side of it, or at right angles, andthe reflected plane will appear perpendicular, as shown in Fig. 299, where we have a front view of a mirror leaning forward at an angle of45° and reflecting the square _aob_ with a cube standing upon it, onlyin the reflection the cube appears to be projecting from an uprightplane or wall. 

 [Illustration: Fig. 299. ]

If we increase the angle from 45° to 60°, then the reflection of theplane and cube will lean backwards as shown in Fig. 300. If we place iton a level with the original plane, the cube will be standing uprighttwice the distance away. If the mirror is still farther tilted till itmakes an angle of 135° as at _E_ (Fig. 298), or 45° on the other side ofthe vertical _Oc_, then the plane and cube would disappear, and objectsexactly over that plane, such as the ceiling, would come into view. 

In Fig. 300 the mirror is at 60° to the plane _mn_, and the plane itselfat about 15° to the plane _an_ (so that here we are using angularperspective, _V_ being the accessible vanishing point). The reflectionof the plane and cube is seen leaning back at an angle of 60°. Note theway the reflection of this cube is found by the dotted lines on theplane, on the surface of the mirror, and also on the reflection. 

 [Illustration: Fig. 300. ]

CLXIX

THE UPRIGHT MIRROR AT AN ANGLE OF 45° TO THE WALL

In Fig. 301 the mirror is vertical and at an angle of 45° to the wallopposite the spectator, so that it reflects a portion of that wall asthough it were receding from us at right angles; and the wall with thepictures upon it, which appears to be facing us, in reality is on ourleft. 

 [Illustration: Fig. 301. ]

An endless number of complicated problems could be invented of theinclined mirror, but they would be mere puzzles calculated rather todeter the student than to instruct him. What we chiefly have to bear inmind is the simple principle of reflections. When a mirror is verticaland placed at the end or side of a room it reflects that room and givesthe impression that we are in one double the size. If two mirrors areplaced opposite to each other at each end of a room they reflect andreflect, so that we see an endless number of rooms. 

Again, if we are sitting in a gallery of pictures with a hand mirror, we can so turn and twist that mirror about that we can bring any picturein front of us, whether it is behind us, at the side, or even on theceiling. Indeed, when one goes to those old palaces and churches wherepictures are painted on the ceiling, as in the Sistine Chapel or theLouvre, or the palaces at Venice, it is not a bad plan to take a handmirror with us, so that we can see those elevated works of art incomfort. 

There are also many uses for the mirror in the studio, well known to theartist. One is to look at one's own picture reversed, when faults becomemore evident; and another, when the model is required to be at a longerdistance than the dimensions of the studio will admit, by drawing hisreflection in the glass we double the distance he is from us. 

The reason the mirror shows the fault of a work to which the eye hasbecome accustomed is that it doubles it. Thus if a line that should bevertical is leaning to one side, in the mirror it will lean to theother; so that if it is out of the perpendicular to the left, itsreflection will be out of the perpendicular to the right, making adouble divergence from one to the other. 

CLXX

MENTAL PERSPECTIVE

Before we part, I should like to say a word about mental perspective, for we must remember that some see farther than others, and some willendeavour to see even into the infinite. To see Nature in all hervastness and magnificence, the thought must supplement and must surpassthe eye. It is this far-seeing that makes the great poet, the greatphilosopher, and the great artist. Let the student bear this in mind, for if he possesses this quality or even a share of it, it will giveimmortality to his work. 

To explain in detail the full meaning of this suggestion is beyond theprovince of this book, but it may lead the student to think thisquestion out for himself in his solitary and imaginative moments, andshould, I think, give a charm and virtue to his work which he shouldendeavour to make of value, not only to his own time but to thegenerations that are to follow. Cultivate, therefore, this mentalperspective, without forgetting the solid foundation of the science Ihave endeavoured to impart to you. 

INDEX

 [Transcriber's Note: Index citations in the original book referred to page numbers. References to chapters (Roman numerals) or figures (Arabic numerals) have been added in brackets where possible. Note that the last two entries for "Toledo" are figure numbers rather than pages; these have not been corrected. ]

AAlbert Dürer, 2, 9. Angles of Reflection, 259 [CLXV]. Angular Perspective, 98 [XLIX] - 123 [LXXII], 133 [LXXX], 170. " " New Method, 133 [LXXX], 134 [LXXXI], 135 [LXXXII], 136 [LXXXIII]. Arches, Arcades, &c. , 198 [CXXVI], 200 [CXXVII] - 208 [CXXIII]. Architect's Perspective, 170 [CVIII], 171 [197]. Art Schools Perspective, 112 [LXII] - 118 [LXVI], 217 [CXLI]. Atmosphere, 1, 74 [XXX]. 

BBalcony, Shadow of, 246 [CLVII]. Base or groundline, 89 [XLI]. 

CCampanile Florence, 5, 59. Cast Shadows, 229 [CXLVII] - 253 [CLXII]. Centre of Vision, 15 [II]. Chessboard, 74 [XXXI]. Chinese Art, 11. Circle, 145 [LXXXVIII], 151 [XCII] - 156 [XCVI], 159 [XCIX]. Columns, 157 [XCVII], 159 [XCIX], 161 [CI], 169 [CVI], 170 [CVII]. Conditions of Perspective, 24 [VII], 25. Cottage in Angular Perspective, 116 [LXV]. Cube, 53 [XVII], 65 [XXIII], 115 [LXIV], 119 [LXVIII]. Cylinder, 158 [XCVIII], 159 [CXIX]. Cylindrical picture, 227 [CXLVI]. 

DDe Hoogh, 2, 62 [68], 73 [82]. Depths, How to measure by diagonals, 127 [LXXVI], 128 [LXXVII]. Descending plane, 92 [XLIV] - 95 [XLV]. Diagonals, 45, 124 [LXXIII], 125 [LXXIV], 126 [LXXV]. Disproportion, How to correct, 35, 118 [LXVII], 157 [XCVII]. Distance, 16 [III], 77 [XXXIII], 78 [XXXIV], 85 [XXXVII], 87 [XXXIX], 103 [LIV], 128 [LXXVII]. Distorted perspective, How to correct, 118 [LXVII]. Dome, 163 [CIII] - 167 [CV]. Double Cross, 218 [CXLII]. 

EEllipse, 145 [LXXXIX], 146 [XC], 147 [168]. Elliptical Arch, 207 [CXXXII]. 

FFarningham, 95 [103]. Figures on descending plane, 92 [XLIV], 93 [100], 94 [102], 95 [XLV]. " " an inclined plane, 88 [XL]. " " a level plane, 70 [79], 71 [XXVIII], 72 [81], 73 [82], 74 [XXX], 75 [XXXI]. " " uneven ground, 90 [XLII], 91 [XLIII]. 

GGeometrical and Perspective figures contrasted, 46 [XII] - 48. " plane, 99 [L]. Giovanni da Pistoya, Sonnet to, by Michelangelo, 60. Great Pyramid, 190 [CXXII]. 

HHexagon, 177 [CXIV], 183 [CXVII], 185 [CXIX]. Hogarth, 9. Honfleur, 83 [92], 142 [163]. Horizon, 3, 4, 15 [II], 20, 59 [XX], 60 [66]. Horizontal line, 13 [I], 15 [II]. Horizontals, 30, 31, 36. 

IInaccessible vanishing points, 77 [XXXII], 78 [XXXIII], 136, 140 - 144. Inclined plane, 33, 118, 213 [CXXXVIII], 244 [XLV], 245 [XLVI]. Interiors, 62 [XXI], 117 [LXVI], 118 [LXVII], 128. 

JJapanese Art, 11. Jesuit of Paris, Practice of Perspective by, 9. 

KKiosk, Application of Hexagon, 185 [XCIX]. Kirby, Joshua, Perspective made Easy (?), 9. 

LLadder, Step, 212 [CXXXVII], 216 [CXL]. Landscape Perspective, 74 [XXX]. Landseer, Sir Edwin, 1. Leonardo da Vinci, 1, 61. Light, Observations on, 253 [CLXIII]. Light-house, 84 [XXXVII]. Long distances, 85 [XXXVIII], 87 [XXXIX]. 

MMeasure distances by square and diagonal, 89 [XLI], 128 [LXXVII], 129. " vanishing lines, How to, 49 [XIV], 50 [XV]. Measuring points, 106 [LVII], 113. " point O, 108, 109, 110 [LX]. Mental Perspective, 269 [CLXX]. Michelangelo, 5, 57, 58, 60. 

NNatural Perspective, 12, 82 [91], 95 [103], 142 [163], 144 [164]. New Method of Angular Perspective, 133 [LXXX], 134 [LXXXI], 135 [LXXXII], 141 [LXXXVI], 215 [CXXXIX], 219. Niche, 164 [CIV], 165 [193], 250 [CLX]. 

OOblique Square, 139 [LXXXV]. Octagon, 172 [CIX] - 175 [202]. O, measuring point, 110 [LX]. Optic Cone, 20 [IV]. 

PParallels and Diagonals, 124 [LXXIII] - 128 [LXXVI]. Paul Potter, cattle, 19 [16]. Paul Veronese, 4. Pavements, 64 [XXII], 66 [XXIV], 176 [CXIII], 178 [CXV], 180 [209], 181 [CXVI], 183 [CXVII]. Pedestal, 141 [LXXXVI], 161 [CI]. Pentagon, 186 [CXX], 187 [217], 188 [219]. Perspective, Angular, 98 [XLIX] - 123 [LXXII]. " Definitions, 13 [I] - 23 [VI]. " Necessity of, 1. " Parallel, 42 - 97 [XLVII]. " Rules and Conditions of, 24 [VII] - 41. " Scientific definition of, 22 [VI]. " Theory of, 13 - 24 [VI]. " What is it? 6 - 12. Pictures painted according to positions they are to occupy, 59 [XX]. Point of Distance, 16 [III] - 21 [IV]. " " Sight, 12, 15 [II]. Points in Space, 129 [LXXVIII], 137 [LXXXIII]. Portico, 111 [122]. Projection, 21 [V], 137. Pyramid, 189 [CXXI], 190 [224], 191 [CXXII], 193 [CXXIII] - 196 [CXXV]. 

RRaphael, 3. Reduced distance, 77 [XXXIII], 78 [XXXIV], 79 [XXXV], 84 [90]. Reflection, 257 [CLXIV] - 268 [CLXIX]. Rembrandt, 59 [XX], 256. Reynolds, Sir Joshua, 9, 60. Rubens, 4. Rules of Perspective, 24 - 41. 

SScale on each side of Picture, 141 [LXXXVII], 142 [163] - 144 [164]. " Vanishing, 69 [XXVI], 71 [XXVII], 81 [XXXVI], 84 [90]. Serlio, 5, 126 [LXXV]. Shadows cast by sun, 229 [CXLVII] - 252 [CLXI]. " " " artificial light, 252 [CLXII]. Sight, Point of, 12, 15 [II]. Sistine Chapel, 60. Solid figures, 135 [LXXXII] - 140 [LXXXV]. Square in Angular Perspective, 105 [LVI], 106 [LVII], 109 [120], 112 [LXII], 114 [LXIII], 121 [LXX], 122 [LXXI], 123 [LXXII], 133 [LXXX], 134 [LXXXI], 139 [LXXXV]. " and diagonals, 125 [LXXIV], 138 [LXXXIV], 139 [LXXXV], 141 [LXXXVI]. " of the hypotenuse (fig. 170), 149 [170]. " in Parallel Perspective, 42 [IX], 43 [X], 50 [XV], 53 [XVII], 54 [XIX]. " at 45°, 64 [XXII] - 66 [XXIV]. Staircase leading to a Gallery, 221 [CXLIII]. Stairs, Winding, 222 [CXLIV], 225 [CXLV]. Station Point, 13 [I]. Steps, 209 [CXXXIV] - 218 [CXLII]. 

TTaddeo Gaddi, 5. Terms made use of, 48 [XIII]. Tiles, 176 [CXIII], 178 [CXV], 181 [CXVI]. Tintoretto, 4. Titian, 59 [XX], 262 [CLXVII]. Toledo, 96 [104], 144 [164], 259 [259], 288 [288]. Trace and projection, 21 [V]. Transposed distance, 53 [XVIII]. Triangles, 104 [LV], 106 [LVII], 132 [148], 135 [151], 138 [158]. Turner, 2, 87 [95]. 

UUbaldus, Guidus, 9. 

VVanishing lines, 49 [XIV]. " point, 119 [LXVIII]. " scale, 68 [XXV] - 72 [XXVIII], 74 [XXX], 77 [XXXII], 79 [XXXV], 84 [90]. Vaulted Ceiling, 203 [CXXX]. Velasquez, 59 [XX]. Vertical plane, 13 [I]. Visual rays, 20 [IV]. 

WWinding Stairs, 222 [CXLIV] - 225 [CXLV]. Water, Reflections in, 257 [CLXIV], 258 [CLXV], 260 [CLXVI], 261 [293]. 

 * * * * *

Errors and Anomalies:

Missing punctuation in the Index has been silently supplied. 

The name form "Albert Dürer" (for Albrecht) is used throughout. In all references to Kirby, _Perspective made Easy_ (?), the question mark is in the original text. 

Figure 66: _Caption missing, but number is given in text_ground plan of the required design, as at Figs. 73 and 74 _text reads "Figs. 74 and 75"_CV [Chapter head] _"C" invisible_

_Index_Dürer, Albert _umlaut missing_Taddeo Gaddi _text reads "Tadeo"_Titian _text reads Titien_