diff options
| author | Roger Frank <rfrank@pglaf.org> | 2025-10-15 01:19:41 -0700 |
|---|---|---|
| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-15 01:19:41 -0700 |
| commit | ad22660321889062bdd0c15887fe07bc8bd4af96 (patch) | |
| tree | d147d49e8340483b76fedb596398a5939f94199c /20165.txt | |
Diffstat (limited to '20165.txt')
| -rw-r--r-- | 20165.txt | 6770 |
1 files changed, 6770 insertions, 0 deletions
diff --git a/20165.txt b/20165.txt new file mode 100644 index 0000000..eddf209 --- /dev/null +++ b/20165.txt @@ -0,0 +1,6770 @@ +The Project Gutenberg eBook, The Theory and Practice of Perspective, by +George Adolphus Storey + + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + + + + +Title: The Theory and Practice of Perspective + + +Author: George Adolphus Storey + + + +Release Date: December 22, 2006 [eBook #20165] + +Language: English + +Character set encoding: ISO-646-US (US-ASCII) + + +***START OF THE PROJECT GUTENBERG EBOOK THE THEORY AND PRACTICE OF +PERSPECTIVE*** + + +E-text prepared by Louise Hope, Suzanne Lybarger, Jonathan Ingram, and the +Project Gutenberg Online Distributed Proofreading Team +(https://www.pgdp.net/c/) + + + +Note: Project Gutenberg also has an HTML version of this file which + includes the original 328 explanatory illustrations. + See 20165-h.htm or 20165-h.zip: + (https://www.gutenberg.org/dirs/2/0/1/6/20165/20165-h/20165-h.htm) + or + (https://www.gutenberg.org/dirs/2/0/1/6/20165/20165-h.zip) + + +Transcriber's Note: + + This 7-bit ASCII file is for readers who cannot use the "real" + (Latin-1) version of the text file or the html version (see above), + which is strongly recommended to the reader because of its + explanatory illustrations. Some substitutions have been made in + this ascii version: + raised dot (in diagram descriptions) is shown as ' + prime symbol (in diagram descriptions) is shown as " + degree sign is expanded to "deg" + + 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 Oxford +London, Edinburgh, New York +Toronto and Melbourne + +THE THEORY AND PRACTICE OF PERSPECTIVE + +by + +G. A. STOREY, A.R.A. + +Teacher of Perspective at the Royal Academy + + + + + + + +[Illustration: 'QUI FIT?'] + + +Oxford +At the Clarendon Press +1910 + +Oxford +Printed at the Clarendon Press +by Horace Hart, M.A. +Printer to the University + + + + + + DEDICATED + to + + SIR EDWARD J. POYNTER + Baronet + + President of the Royal Academy + + in Token of Friendship + and Regard + + + + +PREFACE + + +It is much easier to understand and remember a thing when a reason is +given for it, than when we are merely shown how to do it without being +told why it is so done; for in the latter case, instead of being +assisted by reason, our real help in all study, we have to rely upon +memory or our power of imitation, and to do simply as we are told +without thinking about it. The consequence is that at the very first +difficulty we are left to flounder about in the dark, or to remain +inactive till the master comes to our assistance. + +Now in this book it is proposed to enlist the reasoning faculty from the +very first: to let one problem grow out of another and to be dependent +on the foregoing, as in geometry, and so to explain each thing we do +that there shall be no doubt in the mind as to the correctness of the +proceeding. The student will thus gain the power of finding out any new +problem for himself, and will therefore acquire a true knowledge of +perspective. + + + + +CONTENTS + + +BOOK I + Page +THE NECESSITY OF THE STUDY OF PERSPECTIVE TO PAINTERS, + SCULPTORS, AND ARCHITECTS 1 +WHAT IS PERSPECTIVE? 6 +THE THEORY OF PERSPECTIVE: + I. Definitions 13 + II. The Point of Sight, the Horizon, and the Point + of Distance. 15 + III. Point of Distance 16 + IV. Perspective of a Point, Visual Rays, &c. 20 + V. Trace and Projection 21 + VI. Scientific Definition of Perspective 22 +RULES: + VII. The Rules and Conditions of Perspective 24 + VIII. A Table or Index of the Rules of Perspective 40 + +BOOK II + +THE PRACTICE OF PERSPECTIVE: + IX. The Square in Parallel Perspective 42 + X. The Diagonal 43 + XI. The Square 43 + XII. Geometrical and Perspective Figures Contrasted 46 + XIII. Of Certain Terms made use of in Perspective 48 + XIV. How to Measure Vanishing or Receding Lines 49 + XV. How to Place Squares in Given Positions 50 + XVI. How to Draw Pavements, &c. 51 + XVII. Of Squares placed Vertically and at Different + Heights, or the Cube in Parallel Perspective 53 + XVIII. The Transposed Distance 53 + XIX. The Front View of the Square and of the + Proportions of Figures at Different Heights 54 + XX. Of Pictures that are Painted according to the + Position they are to Occupy 59 + XXI. Interiors 62 + XXII. The Square at an Angle of 45 deg 64 + XXIII. The Cube at an Angle of 45 deg 65 + XXIV. Pavements Drawn by Means of Squares at 45 deg 66 + XXV. The Perspective Vanishing Scale 68 + XXVI. The Vanishing Scale can be Drawn to any Point + on the Horizon 69 + XXVII. Application of Vanishing Scales to Drawing Figures 71 + XXVIII. How to Determine the Heights of Figures + on a Level Plane 71 + XXIX. The Horizon above the Figures 72 + XXX. Landscape Perspective 74 + XXXI. Figures of Different Heights. The Chessboard 74 + XXXII. Application of the Vanishing Scale to Drawing + Figures at an Angle when their Vanishing + Points are Inaccessible or Outside the Picture 77 + XXXIII. The Reduced Distance. How to Proceed when the + Point of Distance is Inaccessible 77 + XXXIV. How to Draw a Long Passage or Cloister by Means + of the Reduced Distance 78 + XXXV. How to Form a Vanishing Scale that shall give + the Height, Depth, and Distance of any Object + in the Picture 79 + XXXVI. Measuring Scale on Ground 81 + XXXVII. Application of the Reduced Distance and the + Vanishing Scale to Drawing a Lighthouse, &c. 84 + XXXVIII. How to Measure Long Distances such as a Mile + or Upwards 85 + XXXIX. Further Illustration of Long Distances and + Extended Views. 87 + XL. How to Ascertain the Relative Heights of Figures + on an Inclined Plane 88 + XLI. How to Find the Distance of a Given Figure + or Point from the Base Line 89 + XLII. How to Measure the Height of Figures + on Uneven Ground 90 + XLIII. Further Illustration of the Size of Figures + at Different Distances and on Uneven Ground 91 + XLIV. Figures on a Descending Plane 92 + XLV. Further Illustration of the Descending Plane 95 + XLVI. Further Illustration of Uneven Ground 95 + XLVII. The Picture Standing on the Ground 96 + XLVIII. The Picture on a Height 97 + +BOOK III + + XLIX. Angular Perspective 98 + L. How to put a Given Point into Perspective 99 + LI. A Perspective Point being given, Find its + Position on the Geometrical Plane 100 + LII. How to put a Given Line into Perspective 101 + LIII. To Find the Length of a Given Perspective Line 102 + LIV. To Find these Points when the Distance-Point + is Inaccessible 103 + LV. How to put a Given Triangle or other + Rectilineal Figure into Perspective 104 + LVI. How to put a Given Square into Angular + Perspective 105 + LVII. Of Measuring Points 106 + LVIII. How to Divide any Given Straight Line into Equal + or Proportionate Parts 107 + LIX. How to Divide a Diagonal Vanishing Line into any + Number of Equal or Proportional Parts 107 + LX. Further Use of the Measuring Point O 110 + LXI. Further Use of the Measuring Point O 110 + LXII. Another Method of Angular Perspective, being that + Adopted in our Art Schools 112 + LXIII. Two Methods of Angular Perspective in one Figure 115 + LXIV. To Draw a Cube, the Points being Given 115 + LXV. Amplification of the Cube Applied to Drawing + a Cottage 116 + LXVI. How to Draw an Interior at an Angle 117 + LXVII. How to Correct Distorted Perspective by Doubling + the Line of Distance 118 + LXVIII. How to Draw a Cube on a Given Square, using only + One Vanishing Point 119 + LXIX. A Courtyard or Cloister Drawn with One Vanishing + Point 120 + LXX. How to Draw Lines which shall Meet at a Distant + Point, by Means of Diagonals 121 + LXXI. How to Divide a Square Placed at an Angle into + a Given Number of Small Squares 122 + LXXII. Further Example of how to Divide a Given Oblique + Square into a Given Number of Equal Squares, + say Twenty-five 122 + LXXIII. Of Parallels and Diagonals 124 + LXXIV. The Square, the Oblong, and their Diagonals 125 + LXXV. Showing the Use of the Square and Diagonals + in Drawing Doorways, Windows, and other + Architectural Features 126 + LXXVI. How to Measure Depths by Diagonals 127 + LXXVII. How to Measure Distances by the Square + and Diagonal 128 + LXXVIII. How by Means of the Square and Diagonal we can + Determine the Position of Points in Space 129 + LXXIX. Perspective of a Point Placed in any Position + within the Square 131 + LXXX. Perspective of a Square Placed at an Angle. + New Method 133 + LXXXI. On a Given Line Placed at an Angle to the Base + Draw a Square in Angular Perspective, the + Point of Sight, and Distance, being given 134 + LXXXII. How to Draw Solid Figures at any Angle + by the New Method 135 + LXXXIII. Points in Space 137 + LXXXIV. The Square and Diagonal Applied to Cubes + and Solids Drawn Therein 138 + LXXXV. To Draw an Oblique Square in Another Oblique + Square without Using Vanishing-points 139 + LXXXVI. Showing how a Pedestal can be Drawn + by the New Method 141 + LXXXVII. Scale on Each Side of the Picture 143 +LXXXVIII. The Circle 145 + LXXXIX. The Circle in Perspective a True Ellipse 145 + XC. Further Illustration of the Ellipse 146 + XCI. How to Draw a Circle in Perspective + Without a Geometrical Plan 148 + XCII. How to Draw a Circle in Angular Perspective 151 + XCIII. How to Draw a Circle in Perspective more + Correctly, by Using Sixteen Guiding Points 152 + XCIV. How to Divide a Perspective Circle + into any Number of Equal Parts 153 + XCV. How to Draw Concentric Circles 154 + XCVI. The Angle of the Diameter of the Circle + in Angular and Parallel Perspective 156 + XCVII. How to Correct Disproportion in the Width + of Columns 157 + XCVIII. How to Draw a Circle over a Circle or a Cylinder 158 + XCIX. To Draw a Circle Below a Given Circle 159 + C. Application of Previous Problem 160 + CI. Doric Columns 161 + CII. To Draw Semicircles Standing upon a Circle + at any Angle 162 + CIII. A Dome Standing on a Cylinder 163 + CIV. Section of a Dome or Niche 164 + CV. A Dome 167 + CVI. How to Draw Columns Standing in a Circle 169 + CVII. Columns and Capitals 170 + CVIII. Method of Perspective Employed by Architects 170 + CIX. The Octagon 172 + CX. How to Draw the Octagon in Angular Perspective 173 + CXI. How to Draw an Octagonal Figure in Angular + Perspective 174 + CXII. How to Draw Concentric Octagons, with + Illustration of a Well 174 + CXIII. A Pavement Composed of Octagons and Small Squares 176 + CXIV. The Hexagon 177 + CXV. A Pavement Composed of Hexagonal Tiles 178 + CXVI. A Pavement of Hexagonal Tiles in Angular + Perspective 181 + CXVII. Further Illustration of the Hexagon 182 + CXVIII. Another View of the Hexagon in Angular + Perspective 183 + CXIX. Application of the Hexagon to Drawing + a Kiosk 185 + CXX. The Pentagon 186 + CXXI. The Pyramid 189 + CXXII. The Great Pyramid 191 + CXXIII. The Pyramid in Angular Perspective 193 + CXXIV. To Divide the Sides of the Pyramid Horizontally 193 + CXXV. Of Roofs 195 + CXXVI. Of Arches, Arcades, Bridges, &c. 198 + CXXVII. Outline of an Arcade with Semicircular Arches 200 + CXXVIII. Semicircular Arches on a Retreating Plane 201 + CXXIX. An Arcade in Angular Perspective 202 + CXXX. A Vaulted Ceiling 203 + CXXXI. A Cloister, from a Photograph 206 + CXXXII. The Low or Elliptical Arch 207 + CXXXIII. Opening or Arched Window in a Vault 208 + CXXXIV. Stairs, Steps, &c. 209 + CXXXV. Steps, Front View 210 + CXXXVI. Square Steps 211 + CXXXVII. To Divide an Inclined Plane into Equal + Parts--such as a Ladder Placed against a Wall 212 +CXXXVIII. Steps and the Inclined Plane 213 + CXXXIX. Steps in Angular Perspective 214 + CXL. A Step Ladder at an Angle 216 + CXLI. Square Steps Placed over each other 217 + CXLII. Steps and a Double Cross Drawn by Means of + Diagonals and one Vanishing Point 218 + CXLIII. A Staircase Leading to a Gallery 221 + CXLIV. Winding Stairs in a Square Shaft 222 + CXLV. Winding Stairs in a Cylindrical Shaft 225 + CXLVI. Of the Cylindrical Picture or Diorama 227 + +BOOK IV + + CXLVII. The Perspective of Cast Shadows 229 + CXLVIII. The Two Kinds of Shadows 230 + CXLIX. Shadows Cast by the Sun 232 + CL. The Sun in the Same Plane as the Picture 233 + CLI. The Sun Behind the Picture 234 + CLII. Sun Behind the Picture, Shadows Thrown on a Wall 238 + CLIII. Sun Behind the Picture Throwing Shadow on + an Inclined Plane 240 + CLIV. The Sun in Front of the Picture 241 + CLV. The Shadow of an Inclined Plane 244 + CLVI. Shadow on a Roof or Inclined Plane 245 + CLVII. To Find the Shadow of a Projection or Balcony + on a Wall 246 + CLVIII. Shadow on a Retreating Wall, Sun in Front 247 + CLIX. Shadow of an Arch, Sun in Front 249 + CLX. Shadow in a Niche or Recess 250 + CLXI. Shadow in an Arched Doorway 251 + CLXII. Shadows Produced by Artificial Light 252 + CLXIII. Some Observations on Real Light and Shade 253 + CLXIV. Reflection 257 + CLXV. Angles of Reflection 259 + CLXVI. Reflections of Objects at Different Distances 260 + CLXVII. Reflection in a Looking-glass 262 + CLXVIII. The Mirror at an Angle 264 + CLXIX. The Upright Mirror at an Angle of 45 deg to + the Wall 266 + CLXX. Mental Perspective 269 + + + + +BOOK FIRST + +THE NECESSITY OF THE STUDY OF PERSPECTIVE +TO PAINTERS, SCULPTORS, AND ARCHITECTS + + +Leonardo da Vinci tells us in his celebrated _Treatise on Painting_ that +the young artist should first of all learn perspective, that is to say, +he should first of all learn that he has to depict on a flat surface +objects which are in relief or distant one from the other; for this is +the simple art of painting. Objects appear smaller at a distance than +near to us, so by drawing them thus we give depth to our canvas. The +outline of a ball is a mere flat circle, but with proper shading we make +it appear round, and this is the perspective of light and shade. + +'The next thing to be considered is the effect of the atmosphere and +light. If two figures are in the same coloured dress, and are standing +one behind the other, then they should be of slightly different tone, +so as to separate them. And in like manner, according to the distance of +the mountains in a landscape and the greater or less density of the air, +so do we depict space between them, not only making them smaller in +outline, but less distinct.'[1] + + [Footnote 1: Leonardo da Vinci's _Treatise on Painting_.] + +Sir Edwin Landseer used to say that in looking at a figure in a picture +he liked to feel that he could walk round it, and this exactly expresses +the impression that the true art of painting should make upon the +spectator. + +There is another observation of Leonardo's that it is well I should here +transcribe; he says: 'Many are desirous of learning to draw, and are +very fond of it, who are notwithstanding void of a proper disposition +for it. This may be known by their want of perseverance; like boys who +draw everything in a hurry, never finishing or shadowing.' This shows +they do not care for their work, and all instruction is thrown away upon +them. At the present time there is too much of this 'everything in a +hurry', and beginning in this way leads only to failure and +disappointment. These observations apply equally to perspective as to +drawing and painting. + +Unfortunately, this study is too often neglected by our painters, some +of them even complacently confessing their ignorance of it; while the +ordinary student either turns from it with distaste, or only endures +going through it with a view to passing an examination, little thinking +of what value it will be to him in working out his pictures. Whether the +manner of teaching perspective is the cause of this dislike for it, +I cannot say; but certainly most of our English books on the subject are +anything but attractive. + +All the great masters of painting have also been masters of perspective, +for they knew that without it, it would be impossible to carry out their +grand compositions. In many cases they were even inspired by it in +choosing their subjects. When one looks at those sunny interiors, those +corridors and courtyards by De Hooghe, with their figures far off and +near, one feels that their charm consists greatly in their perspective, +as well as in their light and tone and colour. Or if we study those +Venetian masterpieces by Paul Veronese, Titian, Tintoretto, and others, +we become convinced that it was through their knowledge of perspective +that they gave such space and grandeur to their canvases. + +I need not name all the great artists who have shown their interest and +delight in this study, both by writing about it and practising it, such +as Albert Duerer and others, but I cannot leave out our own Turner, who +was one of the greatest masters in this respect that ever lived; though +in his case we can only judge of the results of his knowledge as shown +in his pictures, for although he was Professor of Perspective at the +Royal Academy in 1807--over a hundred years ago--and took great pains +with the diagrams he prepared to illustrate his lectures, they seemed to +the students to be full of confusion and obscurity; nor am I aware that +any record of them remains, although they must have contained some +valuable teaching, had their author possessed the art of conveying it. + +However, we are here chiefly concerned with the necessity of this study, +and of the necessity of starting our work with it. + +Before undertaking a large composition of figures, such as the +'Wedding-feast at Cana', by Paul Veronese, or 'The School of Athens', +by Raphael, the artist should set out his floors, his walls, his +colonnades, his balconies, his steps, &c., so that he may know where to +place his personages, and to measure their different sizes according to +their distances; indeed, he must make his stage and his scenery before +he introduces his actors. He can then proceed with his composition, +arrange his groups and the accessories with ease, and above all with +correctness. But I have noticed that some of our cleverest painters will +arrange their figures to please the eye, and when fairly advanced with +their work will call in an expert, to (as they call it) put in their +perspective for them, but as it does not form part of their original +composition, it involves all sorts of difficulties and vexatious +alterings and rubbings out, and even then is not always satisfactory. +For the expert may not be an artist, nor in sympathy with the picture, +hence there will be a want of unity in it; whereas the whole thing, to +be in harmony, should be the conception of one mind, and the perspective +as much a part of the composition as the figures. + +If a ceiling has to be painted with figures floating or flying in the +air, or sitting high above us, then our perspective must take a +different form, and the point of sight will be above our heads instead +of on the horizon; nor can these difficulties be overcome without an +adequate knowledge of the science, which will enable us to work out for +ourselves any new problems of this kind that we may have to solve. + +Then again, with a view to giving different effects or impressions in +this decorative work, we must know where to place the horizon and the +points of sight, for several of the latter are sometimes required when +dealing with large surfaces such as the painting of walls, or stage +scenery, or panoramas depicted on a cylindrical canvas and viewed from +the centre thereof, where a fresh point of sight is required at every +twelve or sixteen feet. + +Without a true knowledge of perspective, none of these things can be +done. The artist should study them in the great compositions of the +masters, by analysing their pictures and seeing how and for what reasons +they applied their knowledge. Rubens put low horizons to most of his +large figure-subjects, as in 'The Descent from the Cross', which not +only gave grandeur to his designs, but, seeing they were to be placed +above the eye, gave a more natural appearance to his figures. The +Venetians often put the horizon almost on a level with the base of the +picture or edge of the frame, and sometimes even below it; as in 'The +Family of Darius at the Feet of Alexander', by Paul Veronese, and 'The +Origin of the "Via Lactea"', by Tintoretto, both in our National +Gallery. But in order to do all these things, the artist in designing +his work must have the knowledge of perspective at his fingers' ends, +and only the details, which are often tedious, should he leave to an +assistant to work out for him. + +We must remember that the line of the horizon should be as nearly as +possible on a level with the eye, as it is in nature; and yet one of the +commonest mistakes in our exhibitions is the bad placing of this line. +We see dozens of examples of it, where in full-length portraits and +other large pictures intended to be seen from below, the horizon is +placed high up in the canvas instead of low down; the consequence is +that compositions so treated not only lose in grandeur and truth, but +appear to be toppling over, or give the impression of smallness rather +than bigness. Indeed, they look like small pictures enlarged, which is a +very different thing from a large design. So that, in order to see them +properly, we should mount a ladder to get upon a level with their +horizon line (see Fig. 66, double-page illustration). + +We have here spoken in a general way of the importance of this study to +painters, but we shall see that it is of almost equal importance to the +sculptor and the architect. + +A sculptor student at the Academy, who was making his drawings rather +carelessly, asked me of what use perspective was to a sculptor. 'In the +first place,' I said, 'to reason out apparently difficult problems, and +to find how easy they become, will improve your mind; and in the second, +if you have to do monumental work, it will teach you the exact size to +make your figures according to the height they are to be placed, and +also the boldness with which they should be treated to give them their +full effect.' He at once acknowledged that I was right, proved himself +an efficient pupil, and took much interest in his work. + +I cannot help thinking that the reason our public monuments so often +fail to impress us with any sense of grandeur is in a great measure +owing to the neglect of the scientific study of perspective. As an +illustration of what I mean, let the student look at a good engraving or +photograph of the Arch of Constantine at Rome, or the Tombs of the +Medici, by Michelangelo, in the sacristy of San Lorenzo at Florence. And +then, for an example of a mistake in the placing of a colossal figure, +let him turn to the Tomb of Julius II in San Pietro in Vinculis, Rome, +and he will see that the figure of Moses, so grand in itself, not only +loses much of its dignity by being placed on the ground instead of in +the niche above it, but throws all the other figures out of proportion +or harmony, and was quite contrary to Michelangelo's intention. Indeed, +this tomb, which was to have been the finest thing of its kind ever +done, was really the tragedy of the great sculptor's life. + +The same remarks apply in a great measure to the architect as to the +sculptor. The old builders knew the value of a knowledge of perspective, +and, as in the case of Serlio, Vignola, and others, prefaced their +treatises on architecture with chapters on geometry and perspective. For +it showed them how to give proper proportions to their buildings and the +details thereof; how to give height and importance both to the interior +and exterior; also to give the right sizes of windows, doorways, +columns, vaults, and other parts, and the various heights they should +make their towers, walls, arches, roofs, and so forth. One of the most +beautiful examples of the application of this knowledge to architecture +is the Campanile of the Cathedral, at Florence, built by Giotto and +Taddeo Gaddi, who were painters as well as architects. Here it will be +seen that the height of the windows is increased as they are placed +higher up in the building, and the top windows or openings into the +belfry are about six times the size of those in the lower story. + + + + +WHAT IS PERSPECTIVE? + + + [Illustration: Fig. 1.] + +Perspective is a subtle form of geometry; it represents figures and +objects not as they are but as we see them in space, whereas geometry +represents figures not as we see them but as they are. When we have a +front view of a figure such as a square, its perspective and geometrical +appearance is the same, and we see it as it really is, that is, with all +its sides equal and all its angles right angles, the perspective only +varying in size according to the distance we are from it; but if we +place that square flat on the table and look at it sideways or at an +angle, then we become conscious of certain changes in its form--the side +farthest from us appears shorter than that near to us, and all the +angles are different. Thus A (Fig. 2) is a geometrical square and B is +the same square seen in perspective. + + [Illustration: Fig. 2.] + + [Illustration: Fig. 3.] + +The science of perspective gives the dimensions of objects seen in space +as they appear to the eye of the spectator, just as a perfect tracing of +those objects on a sheet of glass placed vertically between him and them +would do; indeed its very name is derived from _perspicere_, to see +through. But as no tracing done by hand could possibly be mathematically +correct, the mathematician teaches us how by certain points and +measurements we may yet give a perfect image of them. These images are +called projections, but the artist calls them pictures. In this sketch +_K_ is the vertical transparent plane or picture, _O_ is a cube placed +on one side of it. The young student is the spectator on the other side +of it, the dotted lines drawn from the corners of the cube to the eye of +the spectator are the visual rays, and the points on the transparent +picture plane where these visual rays pass through it indicate the +perspective position of those points on the picture. To find these +points is the main object or duty of linear perspective. + +Perspective up to a certain point is a pure science, not depending upon +the accidents of vision, but upon the exact laws of reasoning. Nor is it +to be considered as only pertaining to the craft of the painter and +draughtsman. It has an intimate connexion with our mental perceptions +and with the ideas that are impressed upon the brain by the appearance +of all that surrounds us. If we saw everything as depicted by plane +geometry, that is, as a map, we should have no difference of view, no +variety of ideas, and we should live in a world of unbearable monotony; +but as we see everything in perspective, which is infinite in its +variety of aspect, our minds are subjected to countless phases of +thought, making the world around us constantly interesting, so it is +devised that we shall see the infinite wherever we turn, and marvel at +it, and delight in it, although perhaps in many cases unconsciously. + + [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 takes +an endless variety of forms, whereas in geometry it has but one. Here +are three equal geometrical squares: they are all alike. Here are three +equal perspective squares, but all varied in form; and the same figure +changes in aspect as often as we view it from a different position. +A walk round the dining-room table will exemplify this. + +It is in proving that, notwithstanding this difference of appearance, +the figures do represent the same form, that much of our work consists; +and for those who care to exercise their reasoning powers it becomes not +only a sure means of knowledge, but a study of the greatest interest. + +Perspective is said to have been formed into a science about the +fifteenth century. Among the names mentioned by the unknown but pleasant +author of _The Practice of Perspective_, written by a Jesuit of Paris +in the eighteenth century, we find Albert Duerer, who has left us some +rules and principles in the fourth book of his _Geometry_; Jean Cousin, +who has an express treatise on the art wherein are many valuable things; +also Vignola, who altered the plans of St. Peter's left by Michelangelo; +Serlio, whose treatise is one of the best I have seen of these early +writers; Du Cerceau, Serigati, Solomon de Cause, Marolois, Vredemont; +Guidus Ubaldus, who first introduced foreshortening; the Sieur de +Vaulizard, the Sieur Dufarges, Joshua Kirby, for whose _Method of +Perspective made Easy_ (?) Hogarth drew the well-known frontispiece; and +lastly, the above-named _Practice of Perspective_ by a Jesuit of Paris, +which is very clear and excellent as far as it goes, and was the book +used by Sir Joshua Reynolds.[2] But nearly all these authors treat +chiefly of parallel perspective, which they do with clearness and +simplicity, and also mathematically, as shown in the short treatise +in Latin by Christian Wolff, but they scarcely touch upon the more +difficult problems of angular and oblique perspective. Of modern +books, those to which I am most indebted are the _Traite' Pratique +de Perspective_ of M. A. Cassagne (Paris, 1873), which is thoroughly +artistic, and full of pictorial examples admirably done; and to +M. Henriet's _Cours Rational de Dessin_. There are many other foreign +books of excellence, notably M. Thibault's _Perspective_, and some +German and Swiss books, and yet, notwithstanding this imposing array of +authors, I venture to say that many new features and original problems +are presented in this book, whilst the old ones are not neglected. As, +for instance, How to draw figures at an angle without vanishing points +(see p. 141, Fig. 162, &c.), a new method of angular perspective which +dispenses with the cumbersome setting out usually adopted, and enables +us to draw figures at any angle without vanishing lines, &c., and is +almost, if not quite, as simple as parallel perspective (see p. 133, +Fig. 150, &c.). How to measure distances by the square and diagonal, and +to draw interiors thereby (p. 128, Fig. 144). How to explain the theory +of perspective by ocular demonstration, using a vertical sheet of glass +with strings, placed on a drawing-board, which I have found of the +greatest use (see p. 29, Fig. 29). Then again, I show how all our +perspective can be done inside the picture; that we can measure any +distance into the picture from a foot to a mile or twenty miles (see p. +86, Fig. 94); how we can draw the Great Pyramid, which stands on +thirteen acres of ground, by putting it 1,600 feet off (Fig. 224), &c., +&c. And while preserving the mathematical science, so that all our +operations can be proved to be correct, my chief aim has been to make it +easy of application to our work and consequently useful to the artist. + + [Footnote 2: There is another book called _The Jesuit's Perspective_ + which I have not yet seen, but which I hear is a fine work.] + +The Egyptians do not appear to have made any use of linear perspective. +Perhaps it was considered out of character with their particular kind of +decoration, which is to be looked upon as picture writing rather than +pictorial art; a table, for instance, would be represented like a +ground-plan and the objects upon it in elevation or standing up. A row +of chariots with their horses and drivers side by side were placed one +over the other, and although the Egyptians had no doubt a reason for +this kind of representation, for they were grand artists, it seems to us +very primitive; and indeed quite young beginners who have never drawn +from real objects have a tendency to do very much the same thing as this +ancient people did, or even to emulate the mathematician and represent +things not as they appear but as they are, and will make the top of a +table an almost upright square and the objects upon it as if they would +fall off. + +No doubt the Greeks had correct notions of perspective, for the +paintings on vases, and at Pompeii and Herculaneum, which were either by +Greek artists or copied from Greek pictures, show some knowledge, though +not complete knowledge, of this science. Indeed, it is difficult to +conceive of any great artist making his perspective very wrong, for if +he can draw the human figure as the Greeks did, surely he can draw an +angle. + +The Japanese, who are great observers of nature, seem to have got at +their perspective by copying what they saw, and, although they are not +quite correct in a few things, they convey the idea of distance and make +their horizontal planes look level, which are two important things in +perspective. Some of their landscapes are beautiful; their trees, +flowers, and foliage exquisitely drawn and arranged with the greatest +taste; whilst there is a character and go about their figures and birds, +&c., that can hardly be surpassed. All their pictures are lively and +intelligent and appear to be executed with ease, which shows their +authors to be complete masters of their craft. + +The same may be said of the Chinese, although their perspective is more +decorative than true, and whilst their taste is exquisite their whole +art is much more conventional and traditional, and does not remind us of +nature like that of the Japanese. + +We may see defects in the perspective of the ancients, in the mediaeval +painters, in the Japanese and Chinese, but are we always right +ourselves? Even in celebrated pictures by old and modern masters there +are occasionally errors that might easily have been avoided, if a ready +means of settling the difficulty were at hand. We should endeavour then +to make this study as simple, as easy, and as complete as possible, to +show clear evidence of its correctness (according to its conditions), +and at the same time to serve as a guide on any and all occasions that +we may require it. + +To illustrate what is perspective, and as an experiment that any one can +make, whether artist or not, let us stand at a window that looks out on +to a courtyard or a street or a garden, &c., and trace with a +paint-brush charged with Indian ink or water-colour the outline of +whatever view there happens to be outside, being careful to keep the eye +always in the same place by means of a rest; when this is dry, place a +piece of drawing-paper over it and trace through with a pencil. Now we +will rub out the tracing on the glass, which is sure to be rather +clumsy, and, fixing our paper down on a board, proceed to draw the scene +before us, using the main lines of our tracing as our guiding lines. + +If we take pains over our work, we shall find that, without troubling +ourselves much about rules, we have produced a perfect perspective of +perhaps a very difficult subject. After practising for some little time +in this way we shall get accustomed to what are called perspective +deformations, and soon be able to dispense with the glass and the +tracing altogether and to sketch straight from nature, taking little +note of perspective beyond fixing the point of sight and the +horizontal-line; in fact, doing what every artist does when he goes out +sketching. + + [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 transparent +vertical plane through which the objects to be represented can be seen, +or on which they can be traced, such as the cube _C_. + + [Illustration: Fig. 7.] + +The line _HD_ is the +Horizontal-line+ or +Horizon+, the chief line in +perspective, as upon it are placed the principal points to which our +perspective lines are drawn. First, the +Point of Sight+ and next _D_, +the +Point of Distance+. The chief vanishing points and measuring points +are also placed on this line. + +Another important line is _AB_, the +Base+ or +Ground line+, as it is on +this that we measure the width of any object to be represented, such as +_ef_, the base of the square _efgh_, on which the cube _C_ is raised. +_E_ is the position of the eye of the spectator, being drawn in +perspective, and is called the +Station-point+. + +Note that the perspective of the board, and the line _SE_, is not the +same as that of the cube in the picture _AKB_, and also that so much of +the board which is behind the picture plane partially represents the ++Perspective-plane+, supposed to be perfectly level and to extend from +the base line to the horizon. Of this we shall speak further on. In +nature it is not really level, but partakes in extended views of the +rotundity of the earth, though in small areas such as ponds the +roundness is infinitesimal. + + [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_ are +represented as receding from us, and in that case are called vanishing +lines, a not quite satisfactory term. + +It is to be noted that the cube _C_ is placed close to the transparent +picture plane, indeed touches it, and that the square _fj_ faces the +spectator _E_, and although here drawn in perspective it appears to him +as in the other figure. Also, it is at the same time a perspective and a +geometrical figure, and can therefore be measured with the compasses. +Or in other words, we can touch the square _fj_, because it is on the +surface of the picture, but we cannot touch the square _ghmb_ at the +other end of the cube and can only measure it by the rules of +perspective. + + +II + +THE POINT OF SIGHT, THE HORIZON, AND THE POINT OF DISTANCE + + +There are three things to be considered and understood before we can +begin a perspective drawing. First, the position of the eye in front of +the picture, which is called the +Station-point+, and of course is not +in the picture itself, but its position is indicated by a point on the +picture which is exactly opposite the eye of the spectator, and is +called the +Point of Sight+, or +Principal Point+, or +Centre of +Vision+, but we will keep to the first of these. + + [Illustration: Fig. 9.] + + [Illustration: Fig. 10.] + +If our picture plane is a sheet of glass, and is so placed that we can +see the landscape behind it or a sea-view, we shall find that the +distant line of the horizon passes through that point of sight, and we +therefore draw a line on our picture which exactly corresponds with it, +and which we call the +Horizontal-line+ or +Horizon+.[3] The height of +the horizon then depends entirely upon the position of the eye of the +spectator: if he rises, so does the horizon; if he stoops or descends to +lower ground, so does the horizon follow his movements. You may sit in a +boat on a calm sea, and the horizon will be as low down as you are, or +you may go to the top of a high cliff, and still the horizon will be on +the same level as your eye. + + [Footnote 3: In a sea-view, owing to the rotundity of the earth, the + real horizontal line is slightly below the sea line, which is noted + in Chapter I.] + +This is an important line for the draughtsman to consider, for the +effect of his picture greatly depends upon the position of the horizon. +If you wish to give height and dignity to a mountain or a building, the +horizon should be low down, so that these things may appear to tower +above you. If you wish to show a wide expanse of landscape, then you +must survey it from a height. In a composition of figures, you select +your horizon according to the subject, and with a view to help the +grouping. Again, in portraits and decorative work to be placed high up, +a low horizon is desirable, but I have already spoken of this subject in +the chapter on the necessity of the study of perspective. + + +III + +POINT OF DISTANCE + +Fig. 11. The distance of the spectator from the picture is of great +importance; as the distortions and disproportions arising from too near +a view are to be avoided, the object of drawing being to make things +look natural; thus, the floor should look level, and not as if it were +running up hill--the top of a table flat, and not on a slant, as if cups +and what not, placed upon it, would fall off. + +In this figure we have a geometrical or ground plan of two squares at +different distances from the picture, which is represented by the line +_KK_. The spectator is first at _A_, the corner of the near square +_Acd_. If from _A_ we draw a diagonal of that square and produce it to +the line _KK_ (which may represent the horizontal-line in the picture), +where it intersects that line at _A'_ marks the distance that the +spectator is from the point of sight _S_. For it will be seen that line +_SA_ equals line _SA'_. In like manner, if the spectator is at _B_, his +distance from the point _S_ is also found on the horizon by means of the +diagonal _BB"_, so that all lines or diagonals at 45 deg are drawn to the +point of distance (see Rule 6). + +Figs. 12 and 13. In these two figures the difference is shown between +the effect of the short-distance point _A'_ and the long-distance point +_B'_; the first, _Acd_, does not appear to lie so flat on the ground as +the second square, _Bef_. + +From this it will be seen how important it is to choose the right point +of distance: if we take it too near the point of sight, as in Fig. 12, +the square looks unnatural and distorted. This, I may note, is a common +fault with photographs taken with a wide-angle lens, which throws +everything out of proportion, and will make the east end of a church or +a cathedral appear higher than the steeple or tower; but as soon as we +make our line of distance sufficiently long, as at Fig. 13, objects take +their right proportions and no distortion is noticeable. + + [Illustration: Fig. 11.] + + [Illustration: Fig. 12.] + + [Illustration: Fig. 13.] + +In some books on perspective we are told to make the angle of vision +60 deg, so that the distance _SD_ (Fig. 14) is to be rather less than the +length or height of the picture, as at _A_. The French recommend an +angle of 28 deg, and to make the distance about double the length of the +picture, as at _B_ (Fig. 15), which is far more agreeable. For we must +remember that the distance-point is not only the point from which we are +supposed to make our tracing on the vertical transparent plane, or a +point transferred to the horizon to make our measurements by, but it is +also the point in front of the canvas that we view the picture from, +called the station-point. It is ridiculous, then, to have it so close +that we must almost touch the canvas with our noses before we can see +its perspective properly. + + [Illustration: Fig. 14.] + +Now a picture should look right from whatever distance we view it, even +across the room or gallery, and of course in decorative work and in +scene-painting a long distance is necessary. + + [Illustration: Fig. 15.] + +We need not, however, tie ourselves down to any hard and fast rule, but +should choose our distance according to the impression of space we wish +to convey: if we have to represent a domestic scene in a small room, as +in many Dutch pictures, we must not make our distance-point too far off, +as it would exaggerate the size of the room. + + [Illustration: Fig. 16. Cattle. By Paul Potter.] + +The height of the horizon is also an important consideration in the +composition of a picture, and so also is the position of the point of +sight, as we shall see farther on. + +In landscape and cattle pictures a low horizon often gives space and +air, as in this sketch from a picture by Paul Potter--where the +horizontal-line is placed at one quarter the height of the canvas. +Indeed, a judicious use of the laws of perspective is a great aid to +composition, and no picture ever looks right unless these laws are +attended to. At the present time too little attention is paid to them; +the consequence is that much of the art of the day reflects in a great +measure the monotony of the snap-shot camera, with its everyday and +wearisome commonplace. + + + + +IV + +PERSPECTIVE OF A POINT, VISUAL RAYS, &C. + + +We perceive objects by means of the visual rays, which are imaginary +straight lines drawn from the eye to the various points of the thing we +are looking at. As those rays proceed from the pupil of the eye, which +is a circular opening, they form themselves into a cone called the ++Optic Cone+, the base of which increases in proportion to its distance +from the eye, so that the larger the view which we wish to take in, the +farther must we be removed from it. The diameter of the base of this +cone, with the visual rays drawn from each of its extremities to the +eye, form the angle of vision, which is wider or narrower according to +the distance of this diameter. + +Now let us suppose a visual ray _EA_ to be directed to some small object +on the floor, say the head of a nail, _A_ (Fig. 17). If we interpose +between this nail and our eye a sheet of glass, _K_, placed vertically +on the floor, we continue to see the nail through the glass, and it is +easily understood that its perspective appearance thereon is the point +_a_, where the visual ray passes through it. If now we trace on the +floor a line _AB_ from the nail to the spot _B_, just under the eye, and +from the point _o_, where this line passes through or under the glass, +we raise a perpendicular _oS_, that perpendicular passes through the +precise point that the visual ray passes through. The line _AB_ traced +on the floor is the horizontal trace of the visual ray, and it will be +seen that the point _a_ is situated on the vertical raised from this +horizontal trace. + + [Illustration: Fig. 17.] + + + + +V + +TRACE AND PROJECTION + + +If from any line _A_ or _B_ or _C_ (Fig. 18), &c., we drop +perpendiculars from different points of those lines on to a horizontal +plane, the intersections of those verticals with the plane will be on +a line called the horizontal trace or projection of the original line. +We may liken these projections to sun-shadows when the sun is in the +meridian, for it will be remarked that the trace does not represent the +length of the original line, but only so much of it as would be embraced +by the verticals dropped from each end of it, and although line _A_ is +the same length as line _B_ its horizontal trace is longer than that of +the other; that the projection of a curve (_C_) in this upright position +is a straight line, that of a horizontal line (_D_) is equal to it, and +the projection of a perpendicular or vertical (_E_) is a point only. +The projections of lines or points can likewise be shown on a vertical +plane, but in that case we draw lines parallel to the horizontal plane, +and by this means we can get the position of a point in space; and by +the assistance of perspective, as will be shown farther on, we can carry +out the most difficult propositions of descriptive geometry and of the +geometry of planes and solids. + + [Illustration: Fig. 18.] + +The position of a point in space is given by its projection on a +vertical 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 the +horizontal trace of the plane _fE_, and _e'f_ is the trace of the same +plane on the vertical plane _K_. + + + + +VI + +SCIENTIFIC DEFINITION OF PERSPECTIVE + + +The projections of the extremities of a right line which passes through +a vertical plane being given, one on either side of it, to find the +intersection of that line with the vertical plane. _AE_ (Fig. 20) is the +right line. The projection of its extremity _A_ on the vertical plane is +_a'_, the projection of _E_, the other extremity, is _e'_. _AS_ is the +horizontal trace of _AE_, and _a'e'_ is its trace on the vertical plane. +At point _f_, where the horizontal trace intersects the base _Bc_ of the +vertical plane, raise perpendicular _fP_ till it cuts _a'e'_ at point +_P_, which is the point required. For it is at the same time on the +given line _AE_ and the vertical plane _K_. + + [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 same +demonstration 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 the +object _A_ on the other side of the glass. Then if _E_ is the eye of the +spectator, its projection on the picture is _S_, the point of sight. + +If I draw a dotted line from _E_ to little _a_, this represents another +visual ray, and _o_, the point where it passes through the picture, is +the perspective of little _a_. I now draw another line from _g_ to _S_, +and thus form the shaded figure _ga'Po_, which is the perspective of +_aAa'g_. + +Let it be remarked that in the shaded perspective figure the lines _a'P_ +and _go_ are both drawn towards _S_, the point of sight, and that they +represent parallel lines _Aa'_ and _ag_, which are at right angles to +the picture plane. This is the most important fact in perspective, and +will be more fully explained farther on, when we speak of retreating or +so-called vanishing lines. + + + + +RULES + +VII + +THE RULES AND CONDITIONS OF PERSPECTIVE + + +The conditions of linear perspective are somewhat rigid. In the first +place, we are supposed to look at objects with one eye only; that is, +the visual rays are drawn from a single point, and not from two. Of this +we shall speak later on. Then again, the eye must be placed in a certain +position, as at _E_ (Fig. 22), at a given height from the ground, _S'E_, +and at a given distance from the picture, as _SE_. In the next place, +the picture or picture plane itself must be vertical and perpendicular +to the ground or horizontal plane, which plane is supposed to be as +level as a billiard-table, and to extend from the base line, _ef_, +of the picture to the horizon, that is, to infinity, for it does not +partake of the rotundity of the earth. + +We can only work out our propositions and figures in space with +mathematical precision by adopting such conditions as the above. But +afterwards the artist or draughtsman may modify and suit them to a more +elastic view of things; that is, he can make his figures separate from +one another, instead of their outlines coming close together as they do +when we look at them with only one eye. Also he will allow for the +unevenness of the ground and the roundness of our globe; he may even +move his head and his eyes, and use both of them, and in fact make +himself quite at his ease when he is out sketching, for Nature does all +his perspective for him. At the same time, a knowledge of this rigid +perspective is the sure and unerring basis of his freehand drawing. + + [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 the +same proportion as _AB_ (Fig. 24), the original line, and _a'b'_, the +perspective line, and if the one is divided at _O_ the other is divided +at _o'_ in the same way. + + [Illustration: Fig. 24.] + +It is not an uncommon error to suppose that the vertical lines of a high +building should converge towards the top; so they would if we stood at +the foot of that building and looked up, for then we should alter the +conditions of our perspective, and our point of sight, instead of being +on the horizon, would be up in the sky. But if we stood sufficiently far +away, so as to bring the whole of the building within our angle of +vision, and the point of sight down to the horizon, then these same +lines would appear perfectly parallel, and the different stories in +their true proportion. + + +RULE 3 + +Horizontals parallel to the base of the picture are also parallel to +that base in the picture. Thus _a'b'_ (Fig. 25) is parallel to _AB_, and +to _GL_, the base of the picture. Indeed, the same argument may be used +with regard to horizontal lines as with verticals. If we look at a +straight wall in front of us, its top and its rows of bricks, &c., are +parallel and horizontal; but if we look along it sideways, then we alter +the conditions, and the parallel lines converge to whichever point we +direct the eye. + + [Illustration: Fig. 25.] + + [Illustration: Fig. 26.] + +This rule is important, as we shall see when we come to the +consideration of the perspective vanishing scale. Its use may be +illustrated by this sketch, where the houses, walls, &c., are parallel +to the base of the picture. When that is the case, then objects exactly +facing us, such as windows, doors, rows of boards, or of bricks or +palings, &c., are drawn with their horizontal lines parallel to the +base; hence it is called parallel perspective. + + +RULE 4 + +All lines situated in a plane that is parallel to the picture plane +diminish in proportion as they become more distant, but do not undergo +any perspective deformation; and remain in the same relation and +proportion each to each as the original lines. This is called the front +view. + + [Illustration: Fig. 27.] + + +RULE 5 + +All horizontals which are at right angles to the picture plane are drawn +to the point of sight. + +Thus the lines _AB_ and _CD_ (Fig. 28) are horizontal or parallel to the +ground plane, and are also at right angles to the picture plane _K_. It +will be seen that the perspective lines _Ba'_, _Dc'_, must, according to +the laws of projection, be drawn to the point of sight. + +This is the most important rule in perspective (see Fig. 7 at beginning +of Definitions). + +An arrangement such as there indicated is the best means of illustrating +this rule. But instead of tracing the outline of the square or cube on +the glass, as there shown, I have a hole drilled through at the point +_S_ (Fig. 29), which I select for the point of sight, and through which +I pass two loose strings _A_ and _B_, fixing their ends at _S_. + + [Illustration: Fig. 28.] + + [Illustration: Fig. 29.] + +As _SD_ represents the distance the spectator is from the glass or +picture, I make string _SA_ equal in length to _SD_. Now if the pupil +takes this string in one hand and holds it at right angles to the glass, +that is, exactly in front of _S_, and then places one eye at the end _A_ +(of course with the string extended), he will be at the proper distance +from the picture. Let him then take the other string, _SB_, in the other +hand, and apply it to point _b"_ where the square touches the glass, and +he will find that it exactly tallies with the side _b"f_ of the square +_a'b"fe_. If he applies the same string to _a'_, the other corner of the +square, his string will exactly tally or cover the side _a'e_, and he +will thus have ocular demonstration of this important rule. + +In this little picture (Fig. 30) in parallel perspective it will be seen +that the lines which retreat from us at right angles to the picture +plane are directed to the point of sight _S_. + + [Illustration: Fig. 30.] + + +RULE 6 + +All horizontals which are at 45 deg, or half a right angle to the picture +plane, are drawn to the point of distance. + +We have already seen that the diagonal of the perspective square, if +produced to meet the horizon on the picture, will mark on that horizon +the distance that the spectator is from the point of sight (see +definition, p. 16). This point of distance becomes then the measuring +point for all horizontals at right angles to the picture plane. + +Thus in Fig. 31 lines _AS_ and _BS_ are drawn to the point of sight _S_, +and are therefore at right angles to the base _AB_. _AD_ being drawn to +_D_ (the distance-point), is at an angle of 45 deg to the base _AB_, and +_AC_ is therefore the diagonal of a square. The line 1C is made +parallel to _AB_, consequently A1CB is a square in perspective. The +line _BC_, therefore, being one side of that square, is equal to _AB_, +another side of it. So that to measure a length on a line drawn to the +point of sight, such as _BS_, we set out the length required, say _BA_, +on the base-line, then from _A_ draw a line to the point of distance, +and where it cuts _BS_ at _C_ is the length required. This can be +repeated any number of times, say five, so that in this figure _BE_ +is five times the length of _AB_. + + [Illustration: Fig. 31.] + + +RULE 7 + +All horizontals forming any other angles but the above are drawn to some +other points on the horizontal line. If the angle is greater than half a +right angle (Fig. 32), as _EBG_, the point is within the point of +distance, as at _V"_. If it is less, as _ABV""_, then it is beyond the +point of distance, and consequently farther from the point of sight. + + [Illustration: Fig. 32.] + +In Fig. 32, the dotted line _BD_, drawn to the point of distance _D_, +is at an angle of 45 deg to the base _AG_. It will be seen that the line +_BV"_ is at a greater angle to the base than _BD_; it is therefore drawn +to a point _V"_, within the point of distance and nearer to the point of +sight _S_. On the other hand, the line _BV""_ is at a more acute angle, +and is therefore drawn to a point some way beyond the other distance +point. + +_Note._--When this vanishing point is a long way outside the picture, +the architects make use of a centrolinead, and the painters fix a long +string at the required point, and get their perspective lines by that +means, which is very inconvenient. But I will show you later on how you +can dispense with this trouble by a very simple means, with equally +correct results. + + +RULE 8 + +Lines which incline upwards have their vanishing points above the +horizontal line, and those which incline downwards, below it. In both +cases they are on the vertical which passes through the vanishing point +(_S_) of their horizontal projections. + + [Illustration: Fig. 33.] + +This rule is useful in drawing steps, or roads going uphill and +downhill. + + [Illustration: Fig. 34.] + + +RULE 9 + +The farther a point is removed from the picture plane the nearer does +its perspective appearance approach the horizontal line so long as it is +viewed from the same position. On the contrary, if the spectator +retreats from the picture plane _K_ (which we suppose to be +transparent), the point remaining at the same place, the perspective +appearance of this point will approach the ground-line in proportion to +the distance of the spectator. + + [Illustrations: + Fig. 35. + Fig. 36. + The spectator at two different distances from the picture.] + +Therefore the position of a given point in perspective above the +ground-line or below the horizon is in proportion to the distance of the +spectator from the picture, or the picture from the point. + + [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 different +distances. In Fig. 38, where the distance is too short, there is a want +of proportion between the near and far objects, which is corrected in +Fig. 39 by taking a much longer distance. + + +RULE 10 + +Horizontals in the same plane which are drawn to the same point on the +horizon are parallel to each other. + + [Illustration: Fig. 40.] + +This is a very important rule, for all our perspective drawing depends +upon it. When we say that parallels are drawn to the same point on the +horizon it does not imply that they meet at that point, which would be a +contradiction; perspective parallels never reach that point, although +they appear to do so. Fig. 40 will explain this. + +Suppose _S_ to be the spectator, _AB_ a transparent vertical plane which +represents the picture seen edgeways, and _HS_ and _DC_ two parallel +lines, mark off spaces between these parallels equal to _SC_, the height +of the eye of the spectator, and raise verticals 2, 3, 4, 5, &c., +forming so many squares. Vertical line 2 viewed from _S_ will appear on +_AB_ but half its length, vertical 3 will be only a third, vertical 4 a +fourth, and so on, and if we multiplied these spaces _ad infinitum_ we +must keep on dividing the line _AB_ by the same number. So if we suppose +_AB_ to be a yard high and the distance from one vertical to another to +be also a yard, then if one of these were a thousand yards away its +representation at _AB_ would be the thousandth part of a yard, or ten +thousand yards away, its representation at _AB_ would be the +ten-thousandth part, and whatever the distance it must always be +something; and therefore _HS_ and _DC_, however far they may be produced +and however close they may appear to get, can never meet. + + [Illustration: Fig. 41.] + +Fig. 41 is a perspective view of the same figure--but more extended. It +will be seen that a line drawn from the tenth upright _K_ to _S_ cuts +off a tenth of _AB_. We look then upon these two lines _SP_, _OP_, as +the sides of a long parallelogram of which _SK_ is the diagonal, as +_cefd_, the figure on the ground, is also a parallelogram. + +The student can obtain for himself a further illustration of this rule +by placing a looking-glass on one of the walls of his studio and then +sketching himself and his surroundings as seen therein. He will find +that all the horizontals at right angles to the glass will converge to +his own eye. This rule applies equally to lines which are at an angle to +the picture plane as to those that are at right angles or perpendicular +to it, as in Rule 7. It also applies to those on an inclined plane, as +in Rule 8. + + [Illustration: Fig. 42. Sketch of artist in studio.] + +With the above rules and a clear notion of the definitions and +conditions of perspective, we should be able to work out any proposition +or any new figure that may present itself. At any rate, a thorough +understanding of these few pages will make the labour now before us +simple and easy. I hope, too, it may be found interesting. There is +always a certain pleasure in deceiving and being deceived by the senses, +and in optical and other illusions, such as making things appear far off +that are quite near, in making a picture of an object on a flat surface +to look as if it stood out and in relief by a kind of magic. But there +is, I think, a still greater pleasure than this, namely, in invention +and in overcoming difficulties--in finding out how to do things for +ourselves by our reasoning faculties, in originating or being original, +as it were. Let us now see how far we can go in this respect. + + +VIII + +A TABLE OR INDEX OF THE RULES OF PERSPECTIVE + +The rules here set down have been fully explained in the previous pages, +and this table is simply for the student's ready reference. + + +RULE 1 + +All straight lines remain straight in their perspective appearance. + + +RULE 2 + +Vertical lines remain vertical in perspective. + + +RULE 3 + +Horizontals parallel to the base of the picture are also parallel to +that base in the picture. + + +RULE 4 + +All lines situated in a plane that is parallel to the picture plane +diminish in proportion as they become more distant, but do not undergo +any perspective deformation. This is called the front view. + + +RULE 5 + +All horizontal lines which are at right angles to the picture plane are +drawn to the point of sight. + + +RULE 6 + +All horizontals which are at 45 deg to the picture plane are drawn to the +point of distance. + + +RULE 7 + +All horizontals forming any other angles but the above are drawn to some +other points on the horizontal line. + + +RULE 8 + +Lines which incline upwards have their vanishing points above the +horizon, and those which incline downwards, below it. In both cases they +are on the vertical which passes through the vanishing point of their +ground-plan or horizontal projections. + + +RULE 9 + +The farther a point is removed from the picture plane the nearer does it +appear to approach the horizon, so long as it is viewed from the same +position. + + +RULE 10 + +Horizontals in the same plane which are drawn to the same point on the +horizon are perspectively parallel to each other. + + + + +BOOK SECOND + +THE PRACTICE OF PERSPECTIVE + + +In the foregoing book we have explained the theory or science of +perspective; we now have to make use of our knowledge and to apply it to +the drawing of figures and the various objects that we wish to depict. + +The first of these will be a square with two of its sides parallel to +the picture plane and the other two at right angles to it, and which we +call + + +IX + +THE SQUARE IN PARALLEL PERSPECTIVE + +From a given point on the base line of the picture draw a line at right +angles to that base. Let _P_ be the given point on the base line _AB_, +and _S_ the point of sight. We simply draw a line along the ground to +the point of sight _S_, and this line will be at right angles to the +base, as explained in Rule 5, and consequently angle _APS_ will be equal +to angle _SPB_, although it does not look so here. This is our first +difficulty, but one that we shall soon get over. + + [Illustration: Fig. 43.] + +In like manner we can draw any number of lines at right angles to the +base, or we may suppose the point _P_ to be placed at so many different +positions, our only difficulty being to conceive these lines to be +parallel to each other. See Rule 10. + + [Illustration: Fig. 44.] + + +X + +THE DIAGONAL + +From a given point on the base line draw a line at 45 deg, or half a +right angle, to that base. Let _P_ be the given point. Draw a line from +_P_ to the point of distance _D_ and this line _PD_ will be at an angle +of 45 deg, or at the same angle as the diagonal of a square. See +definitions. + + [Illustration: Fig. 45.] + + +XI + +THE SQUARE + +Draw a square in parallel perspective on a given length on the base +line. Let _ab_ be the given length. From its two extremities _a_ and _b_ +draw _aS_ and _bS_ to the point of sight _S_. These two lines will be at +right angles to the base (see Fig. 43). From _a_ draw diagonal _aD_ to +point of distance _D_; this line will be 45 deg to base. At point _c_, +where it cuts _bS_, draw _dc_ parallel to _ab_ and _abcd_ is the square +required. + + [Illustration: Fig. 46.] + +We have here proceeded in much the same way as in drawing a geometrical +square (Fig. 47), by drawing two lines _AE_ and _BC_ at right angles to +a given line, _AB_, and from _A_, drawing the diagonal _AC_ at 45 deg +till it cuts _BC_ at _C_, and then through _C_ drawing _EC_ parallel +to _AB_. Let it be remarked that because the two perspective lines +(Fig. 48) _AS_ and _BS_ are at right angles to the base, they must +consequently be parallel to each other, and therefore are perspectively +equidistant, so that all lines parallel to _AB_ and lying between them, +such as _ad_, _cf_, &c., must be equal. + + [Illustration: Fig. 47.] + +So likewise all diagonals drawn to the point of distance, which are +contained between these parallels, such as _Ad_, _af_, &c., must be +equal. For all straight lines which meet at any point on the horizon are +perspectively parallel to each other, just as two geometrical parallels +crossing two others at any angle, as at Fig. 49. Note also (Fig. 48) +that all squares formed between the two vanishing lines _AS_, _BS_, and +by the aid of these diagonals, are also equal, and further, that any +number of squares such as are shown in this figure (Fig. 50), formed in +the same way and having equal bases, are also equal; and the nine +squares contained in the square _abcd_ being equal, they divide each +side of the larger square into three equal parts. + + [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 at +right angles to the base; and also how we can measure any width or +number of widths on a line such as _dc_, that is, parallel to the base +of the picture, however remote it may be from that base. + + [Illustration: Fig. 50.] + + + + +XII + +GEOMETRICAL AND PERSPECTIVE FIGURES CONTRASTED + + +As at first there may be a little difficulty in realizing the +resemblance between geometrical and perspective figures, and also about +certain expressions we make use of, such as horizontals, perpendiculars, +parallels, &c., which look quite different in perspective, I will here +make a note of them and also place side by side the two views of the +same figures. + + [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 verticals +only, but straight lines at right angles to other lines in any position. +Also in speaking of +lines+ a right or +straight line+ is to be +understood; or when we speak of +horizontals+ we mean all straight lines +that are parallel to the perspective plane, such as those on Fig. 52, no +matter what direction they take so long as they are level. They are not +to be confused with the horizon or horizontal-line. + + [Illustration: Fig. 52. Horizontals.] + +There are one or two other terms used in perspective which are not +satisfactory because they are confusing, such as vanishing lines and +vanishing points. The French term, _fuyante_ or _lignes fuyantes_, or +going-away lines, is more expressive; and _point de fuite_, instead of +vanishing point, is much better. I have occasionally called the former +retreating lines, but the simple meaning is, lines that are not parallel +to the picture plane; but a vanishing line implies a line that +disappears, and a vanishing point implies a point that gradually goes +out of sight. Still, it is difficult to alter terms that custom has +endorsed. All we can do is to use as few of them as possible. + + + + +XIV + +HOW TO MEASURE VANISHING OR RECEDING LINES + + +Divide a vanishing line which is at right angles to the picture plane +into any number of given measurements. Let _SA_ be the given line. From +_A_ measure off on the base line the divisions required, say five of +1 foot each; from each division draw diagonals to point of distance _D_, +and where these intersect the line _AC_ the corresponding divisions will +be found. Note that as lines _AB_ and _AC_ are two sides of the same +square they are necessarily equal, and so also are the divisions on _AC_ +equal to those on _AB_. + + [Illustration: Fig. 53.] + +The line _AB_ being the base of the picture, it is at the same time a +perspective line and a geometrical one, so that we can use it as a scale +for measuring given lengths thereon, but should there not be enough room +on it to measure the required number we draw a second line, _DC_, which +we divide in the same proportion and proceed to divide _cf_. This +geometrical figure gives, as it were, a bird's-eye view or ground-plan +of the above. + + [Illustration: Fig. 54.] + + + + +XV + +HOW TO PLACE SQUARES IN GIVEN POSITIONS + + +Draw squares of given dimensions at given distances from the base line +to the right or left of the vertical line, which passes through the +point of sight. + + [Illustration: Fig. 55.] + +Let _ab_ (Fig. 55) represent the base line of the picture divided into a +certain number of feet; _HD_ the horizon, _VO_ the vertical. It is +required to draw a square 3 feet wide, 2 feet to the right of the +vertical, and 1 foot from the base. + +First measure from _V_, 2 feet to _e_, which gives the distance from the +vertical. Second, from _e_ measure 3 feet to _b_, which gives the width +of the square; from _e_ and _b_ draw _eS_, _bS_, to point of sight. From +either _e_ or _b_ measure 1 foot to the left, to _f_ or _f'_. Draw _fD_ +to point of distance, which intersects _eS_ at _P_, and gives the +required distance from base. Draw _Pg_ and _B_ parallel to the base, and +we have the required square. + +Square _A_ to the left of the vertical is 2-1/2 feet wide, 1 foot from +the vertical and 2 feet from the base, and is worked out in the same +way. + +_Note._--It is necessary to know how to work to scale, especially in +architectural drawing, where it is indispensable, but in working out our +propositions and figures it is not always desirable. A given length +indicated by a line is generally sufficient for our requirements. To +work out every problem to scale is not only tedious and mechanical, but +wastes time, and also takes the mind of the student away from the +reasoning out of the subject. + + + + +XVI + +HOW TO DRAW PAVEMENTS, &C. + + +Divide a vanishing line into parts varying in length. Let _BS'_ be the +vanishing line: divide it into 4 long and 3 short spaces; then proceed +as in the previous figure. If we draw horizontals through the points +thus obtained and from these raise verticals, we form, as it were, the +interior of a building in which we can place pillars and other objects. + + [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'_, at +exactly the same distance from the point of sight as is the point of +distance on the horizontal line. + +It will be seen by examining this figure that the diagonals of the +squares in a vertical position are drawn to this vertical +distance-point, thus saving the necessity of taking the measurements +first on the base line, as at _CB_, which in the case of distant +objects, such as the farthest window, would be very inconvenient. Note +that the windows at _K_ are twice as high as they are wide. Of course +these or any other objects could be made of any proportion. + + [Illustration: Fig. 60.] + + + + +XIX + +THE FRONT VIEW OF THE SQUARE AND OF THE PROPORTIONS OF FIGURES +AT DIFFERENT HEIGHTS + + +According to Rule 4, all lines situated in a plane parallel to the +picture plane diminish in length as they become more distant, but remain +in the same proportions each to each as the original lines; as squares +or any other figures retain the same form. Take the two squares _ABCD_, +_abcd_ (Fig. 61), one inside the other; although moved back from square +_EFGH_ they retain the same form. So in dealing with figures of +different heights, such as statuary or ornament in a building, if +actually equal in size, so must we represent them. + + [Illustration: Fig. 61.] + + [Illustration: Fig. 62.] + +In this square _K_, with the checker pattern, we should not think of +making the top squares smaller than the bottom ones; so it is with +figures. + +This subject requires careful study, for, as pointed out in our opening +chapter, there are certain conditions under which we have to modify and +greatly alter this rule in large decorative work. + + [Illustration: Fig. 63.] + +In Fig. 63 the two statues _A_ and _B_ are the same size. So if traced +through a vertical sheet of glass, _K_, as at _c_ and _d_, they would +also be equal; but as the angle _b_ at which the upper one is seen is +smaller than angle _a_, at which the lower figure or statue is seen, it +will appear smaller to the spectator (_S_) both in reality and in the +picture. + + [Illustration: Fig. 64.] + +But if we wish them to appear the same size to the spectator who is +viewing them from below, we must make the angles _a_ and _b_ (Fig. 64), +at which they are viewed, both equal. Then draw lines through equal +arcs, as at _c_ and _d_, till they cut the vertical _NO_ (representing +the side of the building where the figures are to be placed). We shall +then obtain the exact size of the figure at that height, which will make +it look the same size as the lower one, _N_. The same rule applies to +the picture _K_, when it is of large proportions. As an example in +painting, take Michelangelo's large altar-piece in the Sistine Chapel, +'The Last Judgement'; here the figures forming the upper group, with our +Lord in judgement surrounded by saints, are about four times the size, +that is, about twice the height, of those at the lower part of the +fresco. The figures on the ceiling of the same chapel are studied not +only according to their height from the pavement, which is 60 ft., but +to suit the arched form of it. For instance, the head of the figure of +Jonah at the end over the altar is thrown back in the design, but owing +to the curvature in the architecture is actually more forward than the +feet. Then again, the prophets and sybils seated round the ceiling, +which are perhaps the grandest figures in the whole range of art, would +be 18 ft. high if they stood up; these, too, are not on a flat surface, +so that it required great knowledge to give them their right effect. + + [Illustration: Fig. 65.] + +Of course, much depends upon the distance we view these statues or +paintings from. In interiors, such as churches, halls, galleries, &c., +we can make a fair calculation, such as the length of the nave, if the +picture is an altar-piece--or say, half the length; so also with +statuary in niches, friezes, and other architectural ornaments. The +nearer we are to them, and the more we have to look up, the larger will +the upper figures have to be; but if these are on the outside of a +building that can be looked at from a long distance, then it is better +not to have too great a difference. + + + + + [Illustration: Fig. 66. 1909.] + + + +These remarks apply also to architecture in a great measure. Buildings +that can only be seen from the street below, as pictures in a narrow +gallery, require a different treatment from those out in the open, that +are to be looked at from a distance. In the former case the same +treatment as the Campanile at Florence is in some cases desirable, but +all must depend upon the taste and judgement of the architect in such +matters. All I venture to do here is to call attention to the subject, +which seems as a rule to be ignored, or not to be considered of +importance. Hence the many mistakes in our buildings, and the +unsatisfactory and mean look of some of our public monuments. + + + + +XX + +OF PICTURES THAT ARE PAINTED ACCORDING TO THE POSITION +THEY ARE TO OCCUPY + + +In this double-page illustration of the wall of a picture-gallery, +I have, as it were, hung the pictures in accordance with the style in +which they are painted and the perspective adopted by their painters. It +will be seen that those placed on the line level with the eye have their +horizon lines fairly high up, and are not suited to be placed any +higher. The Giorgione in the centre, the Monna Lisa to the right, and +the Velasquez and Watteau to the left, are all pictures that fit that +position; whereas the grander compositions above them are so designed, +and are so large in conception, that we gain in looking up to them. + +Note how grandly the young prince on his pony, by Velasquez, tells out +against the sky, with its low horizon and strong contrast of light and +dark; nor does it lose a bit by being placed where it is, over the +smaller pictures. + +The Rembrandt, on the opposite side, with its burgomasters in black hats +and coats and white collars, is evidently intended and painted for a +raised position, and to be looked up to, which is evident from the +perspective of the table. The grand Titian in the centre, an altar-piece +in one of the churches in Venice (here reversed), is also painted to +suit its elevated position, with low horizon and figures telling boldly +against the sky. Those placed low down are modern French pictures, with +the horizon high up and almost above their frames, but placed on the +ground they fit into the general harmony of the arrangement. + +It seems to me it is well, both for those who paint and for those who +hang pictures, that this subject should be taken into consideration. For +it must be seen by this illustration that a bigger style is adopted by +the artists who paint for high places in palaces or churches than by +those who produce smaller easel-pictures intended to be seen close. +Unfortunately, at our picture exhibitions, we see too often that nearly +all the works, whether on large or small canvases, are painted for the +line, and that those which happen to get high up look as if they were +toppling over, because they have such a high horizontal line; and +instead of the figures telling against the sky, as in this picture of +the 'Infant' by Velasquez, the Reynolds, and the fat man treading on a +flag, we have fields or sea or distant landscape almost to the top of +the frame, and all, so methinks, because the perspective is not +sufficiently considered. + + +_Note._--Whilst on this subject, I may note that the painter in his +large decorative work often had difficulties to contend with, which +arose from the form of the building or the shape of the wall on which he +had to place his frescoes. Painting on the ceiling was no easy task, and +Michelangelo, in a humorous sonnet addressed to Giovanni da Pistoya, +gives a burlesque portrait of himself while he was painting the Sistine +Chapel:-- + + _"I'ho gia' fatto un gozzo in questo stento."_ + + Now have I such a goitre 'neath my chin + That I am like to some Lombardic cat, + My beard is in the air, my head i' my back, + My chest like any harpy's, and my face + Patched like a carpet by my dripping brush. + Nor can I see, nor can I budge a step; + My skin though loose in front is tight behind, + And I am even as a Syrian bow. + Alas! methinks a bent tube shoots not well; + So give me now thine aid, my Giovanni. + +At present that difficulty is got over by using large strong canvas, on +which the picture can be painted in the studio and afterwards placed on +the wall. + +However, the other difficulty of form has to be got over also. A great +portion of the ceiling of the Sistine Chapel, and notably the prophets +and sibyls, are painted on a curved surface, in which case a similar +method to that explained by Leonardo da Vinci has to be adopted. + +In Chapter CCCI he shows us how to draw a figure twenty-four braccia +high upon a wall twelve braccia high. (The braccia is 1 ft. 10-7/8 in.). +He first draws the figure upright, then from the various points draws +lines to a point _F_ on the floor of the building, marking their +intersections on the profile of the wall somewhat in the manner we have +indicated, which serve as guides in making the outline to be traced. + + [Illustration: Fig. 67. + +'Draw upon part of wall _MN_ half the figure you mean to represent, and +the 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 be +removed or transparent. Indeed, in all our figures which represent +solids we suppose that we can see through them, and in most cases we +mark the hidden portions with dotted lines. So also with all those +imaginary lines which conduct the eye to the various vanishing points, +and which the old writers called 'occult'. + + [Illustration: Fig. 69.] + +When the cube is placed below the horizon (as in Fig. 59), we see the +top of it; when on the horizon, as in the above (Fig. 69), if the side +facing us is removed we see both top and bottom of it, or if a room, we +see floor and ceiling, but otherwise we should see but one side (that +facing us), or at most two sides. When the cube is above the horizon we +see underneath it. + +We shall find this simple cube of great use to us in architectural +subjects, such as towers, houses, roofs, interiors of rooms, &c. + +In this little picture by de Hoogh we have the application of the +perspective of the cube and other foregoing problems. + + + + +XXII + +THE SQUARE AT AN ANGLE OF 45 DEG. + + +When the square is at an angle of 45 deg to the base line, then its sides +are drawn respectively to the points of distance, _DD_, and one of its +diagonals which is at right angles to the base is drawn to the point of +sight _S_, and the other _ab_, is parallel to that base or ground line. + + [Illustration: Fig. 70.] + +To draw a pavement with its squares at this angle is but an +amplification of the above figure. Mark off on base equal distances, 1, +2, 3, &c., representing the diagonals of required squares, and from each +of these points draw lines to points of distance _DD"_. These lines will +intersect each other, and so form the squares of the pavement; to ensure +correctness, lines should also be drawn from these points 1, 2, 3, to +the point of sight _S_, and also horizontals parallel to the base, as +_ab_. + + [Illustration: Fig. 71.] + + + + +XXIII + +THE CUBE AT AN ANGLE OF 45 DEG. + + +Having drawn the square at an angle of 45 deg, as shown in the previous +figure, we find the length of one of its sides, _dh_, by drawing a line, +_SK_, through _h_, one of its extremities, till it cuts the base line at +_K_. Then, with the other extremity _d_ for centre and _dK_ for radius, +describe a quarter of a circle _Km_; the chord thereof _mK_ will be the +geometrical length of _dh_. At _d_ raise vertical _dC_ equal to _mK_, +which gives us the height of the cube, then raise verticals at _a_, _h_, +&c., their height being found by drawing _CD_ and _CD"_ to the two +points of distance, and so completing the figure. + + [Illustration: Fig. 72.] + + + + +XXIV + +PAVEMENTS DRAWN BY MEANS OF SQUARES AT 45 DEG. + + + [Illustration: Fig. 73.] + + [Illustration: Fig. 74.] + +The square at 45 deg will be found of great use in drawing pavements, +roofs, ceilings, &c. In Figs. 73, 74 it is shown how having set out one +square it can be divided into four or more equal squares, and any figure +or tile drawn therein. Begin by making a geometrical or ground plan of +the required design, as at Figs. 73 and 74, where we have bricks placed +at right angles to each other in rows, a common arrangement in brick +floors, or tiles of an octagonal form as at Fig. 75. + + [Illustration: Fig. 75.] + + + + +XXV + +THE PERSPECTIVE VANISHING SCALE + + +The vanishing scale, which we shall find of infinite use in our +perspective, is founded on the facts explained in Rule 10. We there find +that all horizontals in the same plane, which are drawn to the same +point on the horizon, are perspectively parallel to each other, so that +if we measure a certain height or width on the picture plane, and then +from each extremity draw lines to any convenient point on the horizon, +then all the perpendiculars drawn between these lines will be +perspectively equal, however much they may appear to vary in length. + + [Illustration: Fig. 76.] + +Let us suppose that in this figure (76) _AB_ and _A'B'_ each represent +5 feet. Then in the first case all the verticals, as _e_, _f_, _g_, _h_, +drawn between _AO_ and _BO_ represent 5 feet, and in the second case all +the horizontals _e_, _f_, _g_, _h_, drawn between _A'O_ and _B'O_ also +represent 5 feet each. So that by the aid of this scale we can give the +exact perspective height and width of any object in the picture, however +far it may be from the base line, for of course we can increase or +diminish our measurements at _AB_ and _A'B'_ to whatever length we +require. + +As it may not be quite evident at first that the points _O_ may be taken +at random, the following figure will prove it. + + + + +XXVI + +THE VANISHING SCALE CAN BE DRAWN TO ANY POINT ON THE HORIZON + + +From _AB_ (Fig. 77) draw _AO_, _BO_, thus forming the scale, raise +vertical _C_. Now form a second scale from _AB_ by drawing _AO' BO'_, +and therein raise vertical _D_ at an equal distance from the base. +First, then, vertical _C_ equals _AB_, and secondly vertical _D_ equals +_AB_, therefore _C_ equals _D_, so that either of these scales will +measure a given height at a given distance. + + [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 our +figures. _AB_ represents their average height, so we have made our scale +_AO_, _BO_, accordingly. From each point marked we draw a line parallel +to the base till it reaches the scale. From the point where it touches +the line _AO_, raise perpendicular as _a_, which gives the height +required at that distance, and must be referred back to the figure +itself. + + [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 have +substituted these schoolgirls for the vertical lines. If we wish to make +some taller than the others, and some shorter, we can easily do so, as +must be evident (see Fig. 79). + +Note that in this first case the scale is below the horizon, so that we +see over the heads of the figures, those nearest to us being the lowest +down. That is to say, we are looking on this scene from a slightly +raised platform. + + +_Second Case._ + +To draw figures at different distances when their heads are above the +horizon, or as they would appear to a person sitting on a low seat. The +height of the heads varies according to the distance of the figures +(Fig. 80). + + +_Third Case._ + +How to draw figures when their heads are about the height of the +horizon, or as they appear to a person standing on the same level or +walking among them. + +In this case the heads or the eyes are on a level with the horizon, and +we have little necessity for a scale at the side unless it is for the +purpose of ascertaining or marking their distances from the base line, +and their respective heights, which of course vary; so in all cases +allowance must be made for some being taller and some shorter than the +scale measurement. + + [Illustration: Fig. 81.] + + + + +XXIX + +THE HORIZON ABOVE THE FIGURES + + +In this example from De Hoogh the doorway to the left is higher up than +the figure of the lady, and the effect seems to me more pleasing and +natural for this kind of domestic subject. This delightful painter was +not only a master of colour, of sunlight effect, and perfect +composition, but also of perspective, and thoroughly understood the +charm it gives to a picture, when cunningly introduced, for he makes the +spectator feel that he can walk along his passages and courtyards. Note +that he frequently puts the point of sight quite at the side of his +canvas, as at _S_, which gives almost the effect of angular perspective +whilst it preserves the flatness and simplicity of parallel or +horizontal perspective. + + [Illustration: Fig. 82. Courtyard by De Hoogh.] + + + + +XXX + +LANDSCAPE PERSPECTIVE + + +In an extended view or landscape seen from a height, we have to consider +the perspective plane as in a great measure lying above it, reaching +from the base of the picture to the horizon; but of course pierced here +and there by trees, mountains, buildings, &c. As a rule in such cases, +we copy our perspective from nature, and do not trouble ourselves much +about mathematical rules. It is as well, however, to know them, so that +we may feel sure we are right, as this gives certainty to our touch and +enables us to work with freedom. Nor must we, when painting from nature, +forget to take into account the effects of atmosphere and the various +tones of the different planes of distance, for this makes much of the +difference between a good picture and a bad one; being a more subtle +quality, it requires a keener artistic sense to discover and depict it. +(See Figs. 95 and 103.) + +If the landscape painter wishes to test his knowledge of perspective, +let him dissect and work out one of Turner's pictures, or better still, +put his own sketch from nature to the same test. + + + + +XXXI + +FIGURES OF DIFFERENT HEIGHTS + +THE CHESSBOARD + + +In this figure the same principle is applied as in the previous one, but +the chessmen being of different heights we have to arrange the scale +accordingly. First ascertain the exact height of each piece, as _Q_, +_K_, _B_, which represent the queen, king, bishop, &c. Refer these +dimensions to the scale, as shown at _QKB_, which will give us the +perspective measurement of each piece according to the square on which +it is placed. + + [Illustration: Fig. 83. Chessboard and Men.] + +This is shown in the above drawing (Fig. 83) in the case of the white +queen and the black queen, &c. The castle, the knight, and the pawn +being about the same height are measured from the fourth line of the +scale marked _C_. + + [Illustration: Fig. 84.] + + + + +XXXII + +APPLICATION OF THE VANISHING SCALE TO DRAWING FIGURES AT AN ANGLE +WHEN THEIR VANISHING POINTS ARE INACCESSIBLE OR OUTSIDE THE PICTURE + + +This is exemplified in the drawing of a fence (Fig. 84). Form scale +_aS_, _bS_, in accordance with the height of the fence or wall to be +depicted. Let _ao_ represent the direction or angle at which it is +placed, draw _od_ to meet the scale at _d_, at _d_ raise vertical _dc_, +which gives the height of the fence at _oo'_. Draw lines _bo'_, _eo_, +_ao_, &c., and it will be found that all these lines if produced will +meet at the same point on the horizon. To divide the fence into spaces, +divide base line _af_ as required and proceed as already shown. + + + + +XXXIII + +THE REDUCED DISTANCE. HOW TO PROCEED WHEN THE POINT OF DISTANCE +IS INACCESSIBLE + + +It has already been shown that too near a point of distance is +objectionable on account of the distortion and disproportion resulting +from it. At the same time, the long distance-point must be some way out +of the picture and therefore inconvenient. The object of the reduced +distance 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 base +of 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_, and +determines the depth of the square _acob_. But we can find that same +point if we take half the base and draw a line from 1/2 base to 1/2 +distance. But even this 1/2 distance-point does not come inside the +picture, so we take a fourth of the base and a fourth of the distance +and draw a line from 1/4 base to 1/4 distance. We shall find that it +passes precisely through the same point _o_ as the other lines _aD_, &c. +We are thus able to find the required point _o_ without going outside +the picture. + +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 the +greatest ease. + +It will be seen in the next figure that without having to extend the +base, as is usually done, we can multiply that base to any amount by +making use of these reduced distances on the horizontal line. This is +quite a new method of proceeding, and it will be seen is mathematically +correct. + + + + +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 equal +parts, which may represent so many feet, so that A4 and _Bd_ being the +retreating sides of the square each represents 4 feet. But we found +point 1/4 D by drawing 3D from 1/4 base to 1/4 distance, and by +proceeding in 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 the whole base and the 1/4 distance we find point _O_, which is +distant four times the length of the base _AB_. We can multiply this +distance to any amount by drawing other diagonals to 8th distance, &c. +The same rule applies 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, we +can make _SB_, the lower line thereof, a measuring line for distances. +Let us first of all divide the base _AB_ into eight parts, each part +representing 5 feet. From each division draw lines to 8th distance; by +their intersections with _SB_ we obtain measurements of 40, 80, 120, +160, &c., feet. Now divide the side of the picture _BE_ in the same +manner as the base, which gives us the height of 40 feet. From the side +_BE_ draw lines 5S, 15S, &c., to point of sight, and from each +division on the base line also draw lines 5S, 10S, 15S, &c., to +point 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 the +height of the flagstaff, which is 140 feet high and 280 feet distant, is +obtained. So also any buildings or other objects can be measured, such +as 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 them +their 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 feet +each way, or one square yard. Taking this as our standard measure, we +find the door on the left is 10 feet high, that the archway at the end +is 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 similar +subject to Fig. 84, but the irregularity and freedom of the perspective +gives it a charm far beyond the rigid precision of the other, while it +conforms to its main laws. This sketch, however, is the real artist's +perspective, or what we might term natural perspective. + + + + +XXXVII + +APPLICATION OF THE REDUCED DISTANCE AND THE VANISHING SCALE TO 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 the +previous figure, but the spaces measure 5 feet instead of 3 feet: so +that 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, but +we require 800. So we again reduce the distance to a 16th. We thus +multiply the base by 16. Now let us take a base of 50 feet at _f_ and +draw line _fD_ to 16th distance; if we multiply 50 feet by 16 we obtain +the 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 from +the sea-level. At _L_ we raise a vertical _LM_, which shows the position +of the lighthouse. Then on that vertical measure the height required as +shown 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 serve +for 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 be +achieved by mere measurement, and indeed can only be properly done by +studying Nature and drawing her perspective as she presents it to us. At +the same time it is useful to be able to test and to set out distances +in arranging a composition. This latter, if neglected, often leads to +great difficulties and sometimes to repainting. + +To show the method of measuring very long distances we have to work with +a 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 1/4 distance, and we +obtain at 440 on line _BS_ four times the length of _AB_, or 110 feet +x 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 we +use 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 at +1,760, which is the same length in perspective as _AB_, we obtain 3,520 +feet, and then again using the base at 3,520 and proceeding in the same +way we obtain 5,280 feet, or one mile to the archway. The flags show +their heights at their respective distances from the base. By the scale +at the side of the picture, _BO_, we can measure any height above or any +depth below the perspective plane. + + [Illustration: Fig. 94.] + +_Note_.--This figure (here much reduced) should be drawn large by the +student, so that the numbering, &c., may be made more distinct. Indeed, +many of the other figures should be copied large, and worked out with +care, as lessons in perspective. + + + + +XXXIX + +FURTHER ILLUSTRATION OF LONG DISTANCES AND EXTENDED VIEWS + + +An extended view is generally taken from an elevated position, so that +the principal part of the landscape lies beneath the perspective plane, +as already noted, and we shall presently treat of objects and figures on +uneven ground. In the previous figure is shown how we can measure +heights and depths to any extent. But when we turn to a drawing by +Turner, such as the 'View from Richmond Hill', we feel that the only way +to accomplish such perspective as this, is to go and draw it from +nature, and even then to use our judgement, as he did, as to how much we +may 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 is +on a level with the perspective plane, while the river and surrounding +park and woods are hundreds of feet below us and stretch away for miles +into the distance. The contrasts obtained by this arrangement increase +the illusion of space, and the figures in the foreground give as it were +a standard of measurement, and by their contrast to the size of the +trees 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 on +level ground, and we measure them by the vanishing scale _aS_, _bS_. +Those to the left, which are repetitions of them, are on an inclined +plane, the vanishing point of which is _S'_; by the side of this plane +we have placed another vanishing scale _a'S'_, _b'S'_, by which we +measure the figures on that incline in the same way as on the level +plane. It will be seen that if a horizontal line is drawn from the foot +of one of these figures, say _G_, to point _O_ on the edge of the +incline, 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 also +that the other vanishing scale is the same width at that distance, so +that we can work from either one or the other. In the event of the +rising ground being uneven we can make use of the scale on the level +plane. + + + + +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 of +sight 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 perspective +plane; and as figure _P_ is standing on the far side of that square it +must be the distance _AB_, which is one side of it, from the base +line--or picture plane. For figures very far away it might be necessary +to 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 is +easy enough. We have now to represent some above and some below the +perspective plane. + + [Illustration: Fig. 98.] + +Form scale _bS_, _cS_; mark off distances 20 feet, 40 feet, &c. Suppose +figure _K_ to be 60 feet off. From point at his feet draw horizontal to +meet vertical _On_, which is 60 feet distant. At the point _m_ where +this line meets the vertical, measure height _mn_ equal to width of +scale at that distance, transfer this to _K_, and you have the required +height 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, namely +5 feet, and proceed in the usual way to find the height of the figures +on the sands, which are here supposed to be nearly on a level with the +sea, of course making allowance for different heights and various other +things. + + + + +XLIII + +FURTHER ILLUSTRATION OF THE SIZE OF FIGURES AT DIFFERENT DISTANCES +AND 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 on +the perspective plane. The figure marked _C_ is close to base, the group +of three is farther off (24 feet), and 6 feet higher up, so we measure +the height on the vanishing scale and also above it. The two girls +carrying fish are still farther off, and about 12 feet below. To tell +how 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 previous +problem, but as the road on the left is going down hill, the vanishing +point 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'_, that +on the incline. + +Fig. 101. This is an outline of above figure to show the working more +plainly. + +Note the wall to the left marked _W_ and the manner in which it appears +to drop at certain intervals, its base corresponding with the inclined +plane, but the upper lines of each division being made level are drawn +to the point of sight, or to their vanishing point on the horizon; it is +important to observe this, as it aids greatly in drawing a road going +down 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 be +descending till it reaches a tunnel which goes under a road or leads to +a river (like one leading out of the Strand near Somerset House). It is +drawn on the same principle as the foregoing figure. Of course to see +the road the spectator must get pretty near to it, otherwise it will be +out of sight. Also a level plane must be shown, as by its contrast to +the 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 that +descends about 45 feet. + + [Illustration: Fig. 103. Farningham.] + +In drawing a landscape such as Fig. 103 we have to bear in mind the +height of the horizon, which being exactly opposite the eye, shows us at +once which objects are below and which are above us, and to draw them +accordingly, especially roofs, buildings, walls, hedges, &c.; also it is +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 in +front of us. If stood on the ground our foreground will touch the base +line of the picture, and we can fix up a standard of measurement both on +the base and on the side as in this sketch, taking 6 feet as about the +height 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 a +window, 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 see +the near figures at _A_, and the nearest to come into view would be +those at _B_, so that a view from a window, &c., would be as it were +without a foreground. Note that the figures at _B_ would be (according +to this sketch) 30 feet from the picture plane and about 18 feet below +the base line. + + [Illustration: Fig. 105.] + + + + +BOOK THIRD + +XLIX + +ANGULAR PERSPECTIVE + + +Hitherto we have spoken only of parallel perspective, which is +comparatively easy, and in our first figure we placed the cube with one +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, nor +its diagonal to the point of distance, but to some other points on the +horizon, although the same rule holds good as regards their parallelism; +as for instance, in the case of _bc_ and _ad_, which, if produced, would +meet at _V_, a point on the horizon called a vanishing point. In this +figure only one vanishing point is seen, which is to the right of the +point of sight _S_, whilst the other is some distance to the left, and +outside the picture. If the cube is correctly drawn, it will be found +that the lines _ae_, _bg_, &c., if produced, will meet on the horizon at +this other vanishing point. This far-away vanishing point is one of the +inconveniences of oblique or angular perspective, and therefore it will +be 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 geometry +will 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 far +it is from the base line, and indeed in the exact position we wish it to +be in the picture. The geometrical plane is supposed to face us, to hang +down, as it were, from the base line _AB_, like the side of a table, the +top of which represents the perspective plane. It is to that perspective +plane 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 now +the same length as _mP_. As point _P_ is opposite point _m_, so must it +be in the perspective, therefore we draw a line at right angles to the +base, that is to the point of sight, and somewhere on this line will be +found the required point _P'_. We now have to find how far from _m_ must +that 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 at +an angle of 45 deg, or half a right angle, makes _mP_' the perspective +length of _mn_ by its intersection with _mS_, and thus gives us the +point _P'_, which is the perspective of the original point. + + + + +LI + +A PERSPECTIVE POINT BEING GIVEN, FIND ITS POSITION +ON THE GEOMETRICAL PLANE + + +To do this we simply reverse the foregoing problem. Thus let _P_ be the +given 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 then +with centre _m_ and radius _mn_ describe arc, and where it cuts that +perpendicular is the required point _P'_. We often have to make use of +this 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 the +two points represent the two extremities of the line. Thus we have to +find 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 to +find one point, namely _b_. We also find the perspective of the angle +_mAB_, namely the shaded triangle mAb. Note also that the perspective +triangle equals the geometrical triangle. + + [Illustration: Fig. 111.] + +When the line required is parallel to the base line of the picture, then +the 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 the +angle at which it is placed. + +This is simply the reverse of the previous problem. Let _AB_ be the +given line. From distance _D_ through _A_ draw _DC_, and from _S_, point +of 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 length +of _AB_. + +This problem is useful in finding the position of any given line or +point 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 same +result can be obtained by using the half distance and half base, as +already shown. + +From _a_, half of _mP_', draw quadrant _ab_, from _b_ (half base), draw +line from _b_ to half Dist., which intersects _Sm_ at _P_, precisely the +same 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 given +triangle _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 any +number can be drawn by means of the same point of sight and the same +point 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 to +the 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 angle +that we wish. We then proceed to find its perspective on the picture by +finding the perspective of the four points _ABCD_ as already shown. Note +that the two sides of the perspective square _dc_ and _ab_ being +produced, meet at point _V_ on the horizon, which is their vanishing +point, 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, which +by this method is not necessary. + + [Illustration: Fig. 115.] + + + + +LVII + +OF MEASURING POINTS + + +We now have to find certain points by which to measure those vanishing +or retreating lines which are no longer at right angles to the picture +plane, as in parallel perspective, and have to be measured in a +different 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 an +isosceles triangle. So likewise if we join _m'b_ in the perspective +figure will m'Ab be the same isosceles triangle in perspective. Continue +line _m'b_ till it cuts the horizon in _m_, which point will be the +measuring point for the vanishing line _AbV_. For if in an isosceles +triangle we draw lines across it, parallel to its base from one side to +the other, we divide both sides in exactly the same quantities and +proportions, so that if we measure on the base line of the picture the +spaces we require, such as 1, 2, 3, on the length _Am'_, and then +from these divisions draw lines to the measuring point, these lines +will intersect the vanishing line _AbV_ in the lengths and proportions +required. To find a measuring point for the lines that go to the other +vanishing point, we proceed in the same way. Of course great accuracy +is necessary. + +Note that the dotted lines 1,1, 2,2, &c., are parallel in the +perspective, as in the geometrical figure. In the former the lines are +drawn 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 divide +into five equal parts. Draw _AC_ at any convenient angle, and measure +off five equal parts with the compasses thereon, as 1, 2, 3, 4, 5. From +5C 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 number +of equal parts. + + + + +LIX + +HOW TO DIVIDE A DIAGONAL VANISHING LINE INTO ANY NUMBER +OF EQUAL OR PROPORTIONAL PARTS + + +In a previous figure (Fig. 116) we have shown how to find a measuring +point when the exact measure of a vanishing line is required, but if it +suffices 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 on +the ground line the five equal spaces required. From 5, the point to +which these measures extend (as they are taken at random), draw a line +through _b_ till it cuts the horizon at _O_. Then proceed to draw lines +from each division on the base to point _O_, and they will intersect and +divide _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 various +proportions, 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 any +number of other squares equal to it by means of this point _O_ and the +diagonals. + +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 of +equal spaces of any convenient length, as 1, 2, 3, &c.; from 1, through +corner 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 will +divide the vanishing line _DV_ into the required number of parts equal +to _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_ draw +diagonals to point _G_: their intersections with the other vanishing +line _AV_ will determine the direction of the cross-lines which form the +bases of other squares without the necessity of drawing them to the +other vanishing point, which in this case is some distance to the left +of the picture. If we produce these cross-lines to the horizon we shall +find that they all meet at the other vanishing point, to which of course +it 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 to +do without it. + +Figure 121 corroborates the above by showing the two vanishing points +and 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 to +draw a balcony, a series of windows, or columns, or what not, or, in +other 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 line +through _B_ till it cuts the horizon at _O_. Then proceed to draw the +other lines from each division to _O_, and thus divide the vanishing +line _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 between +them, making twenty-five divisions. The required number of parts to draw +the columns can be obtained in the same way. + + + + +LXII + +ANOTHER METHOD OF ANGULAR PERSPECTIVE, BEING THAT ADOPTED +IN OUR ART SCHOOLS + + +In the previous method we have drawn our squares by means of a +geometrical plan, putting each point into perspective as required, and +then by means of the perspective drawing thus obtained, finding our +vanishing and measuring points. In this method we proceed in exactly the +opposite way, setting out our points first, and drawing the square (or +other figure) afterwards. + + [Illustration: Fig. 124.] + +Having drawn the horizontal and base lines, and fixed upon the position +of the point of sight, we next mark the position of the spectator by +dropping a perpendicular, _S ST_, from that point of sight, making it +the same length as the distance we suppose the spectator to be from the +picture, and thus we make _ST_ the station-point. + +To understand this figure we must first look upon it as a ground-plan or +bird's-eye view, the line V2V1 or horizon line representing the picture +seen edgeways, because of course the station-point cannot be in the +picture itself, but a certain distance in front of it. The angle at +_ST_, that is the angle which decides the positions of the two vanishing +points V1, V2, is always a right angle, and the two remaining angles +on that side of the line, called the directing line, are together equal +to a right angle or 90 deg. So that in fixing upon the angle at which +the square or other figure is to be placed, we say 'let it be 60 deg and +30 deg, or 70 deg and 20 deg', &c. Having decided upon the station-point +and the angle at which the square is to be placed, draw TV1 and TV2, +till they cut the horizon at V1 and V2. These are the two vanishing +points to which the sides of the figure are respectively drawn. But +we still want the measuring points for these two vanishing lines. We +therefore take first, V1 as centre and V1T as radius, and describe arc +of circle till it cuts the horizon in M1, which is the measuring point +for all lines drawn to V1. Then with radius V2T describe arc from centre +V2 till it cuts the horizon in M2, which is the measuring point for all +vanishing lines drawn to V2. We have now set out our points. Let us +proceed to draw the square _Abcd_. From _A_, the nearest angle (in this +instance touching the base line), measure on each side of it the equal +lengths _AB_ and _AE_, which represent the width or side of the square. +Draw EM2 and BM1 from the two measuring points, which give us, by their +intersections with the vanishing lines AV1 and AV2, the perspective +lengths of the sides of the square _Abcd_. Join _b_ and V1 and dV2, +which intersect each other at _C_, then _Adcb_ is the square required. + +This method, which is easy when you know it, has certain drawbacks, the +chief one being that if we require a long-distance point, and a small +angle, such as 10 deg on one side, and 80 deg on the other, then the size +of the diagram becomes so large that it has to be carried out on the +floor of the studio with long strings, &c., which is a very clumsy and +unscientific way of setting to work. The architects in such cases make +use of the centrolinead, a clever mechanical contrivance for getting +over the difficulty of the far-off vanishing point, but by the method +I have shown you, and shall further illustrate, you will find that you +can dispense with all this trouble, and do all your perspective either +inside 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 student +is apt to take it on trust, and not to trouble about the reasons for its +construction: but to show that it is equally correct, I will draw the +two 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 the +horizon, as the result is the same. In Fig. 125 it is placed above, as +the lower part of the figure is occupied with the geometrical plan of +the 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 method +we work out this perspective, the result is the same, so that both are +correct: the great advantage of the first or geometrical system being, +that we can place the square at any angle, as it is drawn without +reference 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 of +angular perspective, we need now merely transfer them to the horizontal +line in this figure, as in this case they will answer our purpose +perfectly 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 meet +eV1, 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 have +taken 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 the +roof. Note also the use of the measuring-points in connexion with the +measurements 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 the +addition of the point _G_, which is the vanishing point of the diagonals +of 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 a +diagonal till it cuts the horizon at _G_. From _G_ draw diagonals +through _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 lines +drawn in the direction of _Ab_, _dc_ and _Ad_, _bc_, draw lines to the +other vanishing point V1, thus completing the squares, and so cover +the 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, and +those on wall 2 to V2. + +In order to see this drawing properly, the eye should be placed about +3 inches from it, and opposite the point of sight; it will then stand +out like a stereoscopic picture, and appear as actual space, but +otherwise the perspective seems deformed, and the angles exaggerated. +To make this drawing look right from a reasonable distance, the point of +distance should be at least twice as far off as it is here, and this +would mean altering all the other points and sending them a long way out +of the picture; this is why artists use those long strings referred to +above. I would however, advise them to make their perspective drawing on +a 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 double +the distance, so that the perspective is not so violent and the objects +are 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 book +without a folding plate, but it comes to the same thing if we double the +distances between the various points. Thus, if from _S_ to _G_ in the +small 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 the +drawing you wish to enlarge. Make _AB_ the length that you wish to +enlarge to, draw _CB_, and then from each division on _AC_ draw lines +parallel to _CB_, and _AB_ will be divided in the same proportions, as I +have already shown (Fig. 117). + +There is no doubt that it is easier to work direct from the vanishing +points themselves, especially in complicated architectural work, but at +the same time I will now show you how we can dispense with, at all +events, 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_ equal +to 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_ raise +vertical _EF_, making it equal to _Aa_. From _F_ draw _FV_. Raise _Dd_ +and _Cc_, their heights being determined by the line _FV_. Join _da_ and +the cube is complete. It will be seen that the verticals raised at each +corner of the square are equal perspectively, as they are drawn between +parallels which start from equal heights, namely, from _EF_ and _Aa_ to +the 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 this +book which have been called up by the wish to facilitate the work of the +artist, 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 cuts +the horizon at _G_. From this same point _G_ draw lines through the +other 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 give +the diagonals of all the squares on the pavement. Produce sides of +square _A_, and where these lines are intersected by the diagonals _Go_ +draw lines from the vanishing point _V_ to base. These will give us the +outlines of the squares lying between them and also guiding points that +will enable us to draw as many more as we please. These again will give +us our measurements for the widths of the arches, &c., or between the +columns. 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 proportionate +measurement 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, and +therefore needs no further explanation. + + [Illustration: Fig. 133.] + +I must, however, point out the importance of the point _G_. In angular +perspective it in a measure takes the place of the point of distance in +parallel perspective, since it is the vanishing point of diagonals at +45 deg drawn between parallels such as _AV_, _DV_, drawn to a vanishing +point _V_. The method of dividing line _AV_ into a number of parts equal +to _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 NUMBER +OF 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 base +at _E_. Divide _EA_ into four equal parts. From each division draw lines +to vanishing point _V_. Draw diagonals _BD_ and _AC_, and produce the +latter till it cuts the horizon in _G_. Draw the three cross-lines +through 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 also +shows how the long vanishing point can be dispensed with, and the +perspective drawing brought within the picture. + + + + +LXXII + +FURTHER EXAMPLE OF HOW TO DIVIDE A GIVEN OBLIQUE SQUARE +INTO A GIVEN NUMBER OF EQUAL SQUARES, SAY TWENTY-FIVE + + +Having drawn the square _ABCD_, which is enclosed, as will be seen, in a +dotted square in parallel perspective, I divide the line _EA_ into five +equal parts instead of four (Fig. 135), and have made use of the device +for that purpose by measuring off the required number on line _EF_, &c. +Fig. 136 is introduced here simply to show that the square can be +divided into any number of smaller squares. Nor need the figure be +necessarily 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 any +direction 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 but +to draw its two diagonals, and their intersection will give us the +centre of the figure (see 137 A). We do the same with perspective +figures, as at B. In Fig. C is shown how a diagonal, drawn from one +angle of a square _B_ through the centre _O_ of the opposite side of the +square, will enable us to find a second square lying between the same +parallels, then a third, a fourth, and so on. At figure _K_ lying on the +ground, I have divided the farther side of the square _mn_ into 1/4, +1/3, 1/2. If I draw a diagonal from _G_ (at the base) through the half +of 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 the +vanishing line _FS_, and so on. In Fig. 137 D is shown how easily any +number 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 and +139, if we draw a third line through that centre at a given angle and +then at each of its extremities draw perpendiculars _AB_, _DC_, we +divide that square or oblong into three parts, the two outer portions +being equal to each other, and the centre one either larger or smaller +as desired; as, for instance, in the triumphal arch we make the centre +portion larger than the two outer sides. When certain architectural +details and spaces are to be put into perspective, a scale such as that +in Fig. 123 will be found of great convenience; but if only a ready +division of the principal proportions is required, then these diagonals +will 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 what +excellent proportion can be obtained by the square and diagonals. The +width of the door is one-third of the base of square, the height +two-thirds. As a further illustration we have drawn the same figure in +perspective. + + [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_, and +from _g_ raise a perpendicular till it cuts the diagonal _AB_ in _O_, +then from _O_ draw horizontal _Og'_, we form a square AgOg', and thus +measure on one side of the square the distance or depth _Ag'_. So can we +measure 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 shown +in 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 the +picture. + + + + +LXXVII + +HOW TO MEASURE DISTANCES BY THE SQUARE AND DIAGONAL + + +It may sometimes be convenient to have a ready method by which to +measure the width and length of objects standing against the wall of a +gallery, without referring to distance-points, &c. + + [Illustration: Fig. 144.] + +In Fig. 144 the floor is divided into two large squares with their +diagonals. 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 to +be in the picture; draw _eo_ and _fo_ to point of sight, and proceed as +in the previous figure by drawing parallels from _Oo_, &c. + +Let it be observed that the great advantage of this method is, that we +can use it to measure such distant objects as _XY_ just as easily as +those near to us. + +There is, however, a still further advantage arising from it, and that +is that it introduces us to a new and simpler method of perspective, to +which I have already referred, and it will, I hope, be found of infinite +use to the artist. + +_Note._--As we have founded many of these figures on a given square in +angular perspective, it is as well to have a ready and certain means of +drawing that square without the elaborate setting out of a geometrical +plan, as in the first method, or the more cumbersome and extended system +of the second method. I shall therefore show you another method equally +correct, but much simpler than either, which I have invented for our +use, and which indeed forms one of the chief features of this book. + + + + +LXXVIII + +HOW BY MEANS OF THE SQUARE AND DIAGONAL WE CAN DETERMINE +THE POSITION OF POINTS IN SPACE + + +Apart from the aid that perspective affords the draughtsman, there is a +further value in it, in that it teaches us almost a new science, which +we might call the mystery of aspect, and how it is that the objects +around us take so many different forms, or rather appearances, although +they themselves remain the same. And also that it enables us, with, +I think, great pleasure to ourselves, to fathom space, to work out +difficult problems by simple reasoning, and to exercise those inventive +and critical faculties which give strength and enjoyment to mental life. + +And now, after this brief excursion into philosophy, let us come down to +the 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 geometrical +square _A_, which is facing us, and the perspective square _B_, which we +suppose to lie flat on the table, or rather on the perspective plane. +Line _A'C'_ is the perspective of line _AC_. On the geometrical square +we can make what measurements we please with the compasses, but on the +perspective square _B'_ the only line we can actually measure is the +base line. In both figures this base line is the same length. Suppose we +want to find the perspective of point _P_ (Fig. 146), we make use of the +diagonal _CA_. From _P_ in the geometrical square draw _PO_ to meet the +diagonal 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_, draw +horizontal _OP'_, which gives us the point required. In the same way we +can find the perspective of any number of points on any side of the +square. + + + + +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 have +but to repeat the process of the previous problem, making use of our +measurements on the base, the diagonals, &c. + + [Illustration: Fig. 147.] + +Indeed these figures are so plain and evident that further description +of them is hardly necessary, so I will here give two drawings of +triangles which explain themselves. To put a triangle into perspective +we have but to find three points, such as _fEP_, Fig. 148 A, and then +transfer 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 same +way, which is not only the simplest method, but it carries its +mathematical 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 square +in 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 necessary +to construct the geometrical figure, as I will show presently. It is +here 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 they +will meet at _V_, their vanishing point, and again produce the other +sides _n'G'_ and _P'E'_ till they meet on the horizon at the other +vanishing point, which they must do if the figure is correctly drawn. + +In any parallel square construct an oblique square from a given +point--given the parallel square at Fig. 150 B, and given point _n'_ on +base. Make _A'f'_ equal to _n'm'_, draw _f'S_ and _n'S_ to point of +sight. 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 draw +the square. + + + + +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. + + + [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 the +horizon in _V_, its vanishing point. From _V_ draw _VB_ cutting side +_KE_ of outer square in _G_, and we have the four points _AFGB_, which +are the four angles of the square required. Join _FG_, and the figure is +complete. + +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 required +is complete. + +The student can try the remaining two sides, and he will find they work +out 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 in +these figures. The heights of the corners of the triangles are obtained +by means of the vanishing scales _AS_, _OS_, which have already been +explained. + + [Illustration: Fig. 152.] + + [Illustration: Fig. 153.] + +In the same manner we can draw a cubic figure (Fig. 154)--a box, for +instance--at any required angle. In this case, besides the scale _AS_, +_OS_, we have made use of the vanishing lines _DV_, _BV_, to corroborate +the scale, but they can be dispensed with in these simple objects, or we +can use a scale on each side of the figure as _a'o'S_, should both +vanishing 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, the +other 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 set +down. Lines at any slope and at any angle can be drawn by this +descriptive geometry. + +The student can examine these figures for himself, and will understand +their working from what has gone before. Here (Fig. 157) in the +geometrical square we have a vertical plane _AabB_ standing on its base +_AB_. We wish to place a projection of this figure at a certain distance +and at a given angle in space. First of all we transfer it to the side +of the cube, where it is seen in perspective, whilst at its side is +another perspective square lying flat, on which we have to stand our +figure. By means of the diagonal of this flat square, horizontals from +figure on side of cube, and lines drawn from point of sight (as already +explained), we obtain the direction of base line _AB_, and also by means +of 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 which +may be said to possess the third dimension. + + [Illustration: Fig. 157.] + +In this other case (Fig. 158) we have a wedge-shaped figure standing on +a triangle placed on the ground, as in the previous figure, its three +corners being the same height. In the vertical geometrical square we +have a ground-plan of the figure, from which we draw lines to diagonal +and to base, and notify by numerals 1, 3, 2, 1, 3; these we transfer to +base of the horizontal perspective square, and then construct shaded +triangle 1, 2, 3, and raise to the height required as shown at +1', 2', 3'. Although we may not want to make use of these special +figures, they show us how we could work out almost any form or object +suspended 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 at +various angles so can we make use of cubes either in parallel or angular +perspective to draw other solid figures within them, as shown in these +drawings, for this is simply an amplification of that method. Indeed we +might invent many more such things. But subjects for perspective +treatment will constantly present themselves to the artist or +draughtsman in the course of his experience, and while I endeavour to +show him how to grapple with any new difficulty or subject that may +arise, it is impossible to set down all of them in this book. + + [Illustration: Fig. 160.] + + + + +LXXXV + +TO DRAW AN OBLIQUE SQUARE IN ANOTHER OBLIQUE SQUARE +WITHOUT USING VANISHING POINTS + + +It is not often that both vanishing points are inaccessible, still it is +well to know how to proceed when this is the case. We first draw the +square _ABCD_ inside the parallel square, as in previous figures. To +draw the smaller square _K_ we simply draw a smaller parallel square _h +h h h_, and within that, guided by the intersections of the diagonals +therewith, we obtain the four points through which to draw square _K_. +To raise a solid figure on these squares we can make use of the +vanishing scales as shown on each side of the figure, thus obtaining the +upper square 1 2 3 4, then by means of the diagonal 1 3 and 2 4 and +verticals raised from each corner of square _K_ to meet them we obtain +the smaller upper square corresponding to _K_. + +It might be said that all this can be done by using the two vanishing +points in the usual way. In the first place, if they were as far off as +required for this figure we could not get them into a page unless it +were three or four times the width of this one, and to use shorter +distances results in distortion, so that the real use of this system is +that we can make our figures look quite natural and with much less +trouble 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 application +of it to architecture, although when there are many details it may be +more 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 working +of our perspective, partly for the comfort of the artist, and partly +that he may have no excuse for neglecting it, I will here show you how +you may, by a very simple means, secure the general correctness of your +perspective when sketching or painting out of doors. + +Let us take this example from a sketch made at Honfleur (Fig. 163), and +in 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 to +the other, is the horizontal line; below that is a wall and a pavement +marked _aV_, also going from one side of the picture to the other, and +being lower down at _a_ than at _V_ it runs up as it were to meet the +horizon at some distant point. In order to form our scale I take first +the length of _Ha_, and measure it above and below the horizon, along +the side to our left as many times as required, in this case four or +five. I now take the length _HV_ on the right side of the picture and +measure it above and below the horizon, as in the other case; and then +from these divisions obtain dotted lines crossing the picture from one +side to the other which must all meet at some distant point on the +horizon. These act as guiding lines, and are sufficient to give us the +direction of any vanishing lines going to the same point. For those that +go in the opposite direction we proceed in the same way, as from _b_ on +the right to _V'_ on the left. They are here put in faintly, so as not +to interfere with the drawing. In the sketch of Toledo (Fig. 164) the +same thing is shown by double lines on each side to separate the two +sets 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 square +at four points which are in the middle of each side, as at _a b c d_. It +will also intersect the two diagonals at the four points _o_ (Fig. 165). +If, then, we put this square and its diagonals, &c., into perspective we +shall have eight guiding points through which to trace the required +circle, 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 freehand +drawing, which requires a little practice to make it true, it is +sufficient for ordinary purposes and on a small scale, but to be +mathematically true it must be an ellipse. We will first draw an ellipse +(Fig. 167). Let _ee_ be its long, or transverse, diameter, and _db_ its +short 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, then +make a loop of fine string that does not stretch and of such a length +that when drawn out the double thread will reach from _f_ to _e_. Now +place this double thread round the two pins at the foci _ff'_ and +distend it with the pencil point until it forms triangle _fdf'_, then +push the pencil along and right round the two foci, which being guided +by the thread will draw the curve, which is a true ellipse, and will +pass through the eight points indicated in our first figure. This will +be a sufficient proof that the circle in perspective and the ellipse are +identical curves. We must also remember that the ellipse is an oblique +projection of a circle, or an oblique section of a cone. The difference +between the two figures consists in their centres not being in the same +place, that of the perspective circle being at _c_, higher up than _e_ +the centre of the ellipse. The latter being a geometrical figure, its +long diameter is exactly in the centre of the figure, whereas the centre +_c_ and the diameter of the perspective are at the intersection of the +diagonals 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 previous +figure is also a circle in perspective we must reconstruct around it the +square and its eight points by means of which it was drawn in the first +instance. We start with nothing but the ellipse itself. We have to find +the points of sight and distance, the base, &c. Let us start with base +_AB_, a horizontal tangent to the curve extending beyond it on either +side. From _A_ and _B_ draw two other tangents so that they shall touch +the curve at points such as _TT'_ a little above the transverse diameter +and on a level with each other. Produce these tangents till they meet at +point _S_, which will be the point of sight. Through this point draw +horizontal line _H_. Now draw tangent _CD_ parallel to _AB_. Draw +diagonal _AD_ till it cuts the horizon at the point of distance, this +will cut through diameter of circle at its centre, and so proceed to +find the eight points through which the perspective circle passes, when +it will be found that they all lie on the ellipse we have drawn with the +loop, showing that the two curves are identical although their centres +are 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 transfer +this 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 of +these lines with the diagonals of square will give us the four points _o +o o o_. + + [Illustration: Fig. 169.] + +The reason of this is that _ff'_ is the measurement on the base _AB_ of +another square _o o o o_ which is exactly half of the outer square. For +if we inscribe a circle in a square and then inscribe a second square in +that circle, this second square will be exactly half the area of the +larger one; for its side will be equal to half the diagonal of the +larger 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 of +large square _o_. + + [Illustration: Fig. 170.] + + [Illustration: Fig. 171.] + +In Fig. 171, _CB_ represents half of diagonal _EB_ of the outer square +in 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 of +the 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_ passing +through the diagonals we get the four points _o o o o_ through which to +draw the smaller square. Without referring to geometry we can see at a +glance by Fig. 172, where we have simply turned the square _o o o o_ on +its centre so that its angles touch the sides of the outer square, that +it is exactly half of square _ABEF_, since each quarter of it, such as +EoCo, 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 base +line at _G_. + + [Illustration: Fig. 174.] + +Take _mD_, the fourth of the base. Find _mn_ as in Fig. 171, measure it +on each side of _E_, and so obtain _Ef_ and _Ef'_, and proceed to draw +_fV_, _EV_, _f'V_ and the diagonals, whose intersections with these +lines will give us the eight points through which to draw the circle. In +fact the process is the same as in parallel perspective, only instead of +making our divisions on the actual base _AD_ of the square, we make them +on _GD_, the base line. + +To obtain the central line _hh_ passing through _O_, we can make use of +diagonals of the half squares; that is, if the other vanishing point is +inaccessible, 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, draw +semicircle _AKB_, and divide it into eight equal parts. From each +division 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 horizontals +parallel to base _AB_. Then through the points thus obtained draw the +circle as shown in this figure, which also shows us how the +circumference of a circle in perspective may be divided into any +number 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 its +construction is concerned, only in this case we have divided the +semicircle into twelve parts and the perspective into twenty-four. + + [Illustration: Fig. 176.] + + [Illustration: Fig. 177.] We have raised perpendiculars from the +divisions on the semicircle, and proceeded as before to draw lines to +the point of sight, and have thus by their intersections with the +circumference already drawn in perspective divided it into the required +number 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 a +circle, cart-wheels, &c. + +The geometrical figure (177) will explain the construction of the +perspective one by showing how the divisions are obtained on the line +_AB_, which represents base of square, from the divisions on the +semicircle _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 this +again inscribe a second circle, and so on. Through their intersections +with 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 from +each point on the base to the point of sight, and drawing the curves +through the inter-sections of these lines with the diagonals. + + [Illustration: Fig. 179.] + +Should it be required to make the circles at equal distances, as for +steps for instance, then the geometrical plan should be made +accordingly. + +Or we may adopt the method shown at Fig. 180, by taking quarter base of +both outer and inner square, and finding the measurement _mn_ on each +side of _C_, &c. + + [Illustration: Fig. 180.] + + + + +XCVI + +THE ANGLE OF THE DIAMETER OF THE CIRCLE IN ANGULAR +AND PARALLEL PERSPECTIVE + + +The circle, whether in angular or parallel perspective, is always an +ellipse. In angular perspective the angle of the circle's diameter +varies 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 diameter +of the ellipse. In parallel perspective the diameter of the circle +always remains horizontal, although the long diameter of the ellipse +varies in inclination according to the distance it is from the point of +sight, as shown in Fig. 182, in which the third circle is much elongated +and 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 the +point 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 is +much wider than column 1. + + [Illustration: Fig. 183.] + + [Illustration: Fig. 184.] + +In our second figure (184) is shown how this defect is remedied, by +doubling the distance, or by counting the same distance as half, which +is easily effected by drawing the diagonal from _O_ to 1/2-D, instead +of from _A_, as in the other figure, _O_ being at half base. Here the +squares 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 its +diagonals. Then find the various points _O_, and from these raise +perpendiculars to meet the diagonals of the upper square at points _P_, +which, with the other points will be sufficient guides to draw the +circle required. This can be applied to towers, columns, &c. The size of +the circles can be varied so that the upper portion of a cylinder or +column 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 the +vanishing scale _POV_, which should be made the depth required, drop +perpendiculars from the various points marked _O_, obtained by the +diagonals, making them the right depth by referring them to the +vanishing scale, as shown in this figure. This can be used for drawing +garden 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 by +means of the vanishing scale at the side we obtain the height of the +verticals 1, 2, 3, 4, &c., which determine the direction of the upper +circle; and in this second figure, how we resort to the same means to +draw circular steps. + + [Illustration: Fig. 187.] + + [Illustration: Fig. 188.] + + + + +CI + +DORIC COLUMNS + + +It is as well for the art student to study the different orders of +architecture, whether architect or not, as he frequently has to +introduce them into his pictures, and at least must know their +proportions, and how columns diminish from base to capital, as shown in +this 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 on +the same line _AB_ draw rectangle _AEFB_, its height being determined by +radius _OK_ of semicircle. From centre _O_ draw _OF_ to corner of +rectangle. 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 raise +rectangle _GghH_, the same height as the other. The manner of doing this +is to produce diameter _GH_ to the horizon till it finds its vanishing +point 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_ through +which 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 is +first raised on the circle, and on the top of that we draw semicircles +from the different divisions on the circumference of the upper circle. +This, however, only represents a small half-globular object. To draw the +dome 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, but +from within it will tax all our knowledge of perspective to give it +effect. + +We shall go more into this subject when we come to archways and vaulted +roofs, &c. + + + + +CIV + +SECTION OF A DOME OR NICHE + + + [Illustration: Fig. 192.] + +First draw outline of the niche _GFDBA_ (Fig. 193), then at its base +draw square and circle _GOA_, _S_ being the point of sight, and divide +the circumference of the circle into the required number of parts. Then +draw semicircle _FOB_, and over that another semicircle _EOC_. The +manner of drawing them is shown in Fig. 192. From the divisions on the +circle _GOA_ raise verticals to semicircle _FOB_, which will divide it +in the same way. Divide the smaller semicircle _EOC_ into the same +number of parts as the others, which divisions will serve as guiding +points in drawing the curves of the dome that are drawn towards _D_, but +the 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 their +diameters diagonals to distance-point, as _OD_, which cuts the vanishing +line BS at _m_, and gives us the depth of the square, and in this we +draw 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 these +as central lines, form squares thereon by drawing _SA_, _SB_, _SC_, +_SD_, &c., from point of sight, and determining their lengths by +diagonals _fh_, _f'h'_ from point of distance, passing through _O_. +Having formed the two squares, draw perspective circles in each, and +divide their circumferences into twelve or whatever number of parts are +needed. To complete the figure draw from each division in the lower +circle curves passing through the corresponding divisions in the upper +one, to the apex. But as these are freehand lines, it requires some +taste and knowledge to draw them properly, and of course in a large +drawing several more squares and circles might be added to aid the +draughtsman. 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. First +draw 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 the +column. The question is how to make each one the right width in +accordance with its position, for it is evident that a near column must +appear wider than the opposite one. On the right of the figure is the +vertical scale _A_, which gives the heights of the columns, and at its +foot is a horizontal scale, or a scale of widths _B_. Now, according to +the line on which the column stands, we find its apparent width marked +on the scale. Thus take the small square and circle at 15, without its +column, or the broken column at 16; and note that on each side of its +centre _O_ I have measured _oa_, _ob_, equal to spaces marked 3 on the +same horizontal in the scale _B_. Through these points _a_ and _b_ I +have drawn lines towards point of sight _S_. Through their intersections +with diagonal _e_, which is directed to point of distance, draw the +farther and nearer sides of the square in which to describe the circle +and the cylinder or column thereon. I have made all the squares thus +obtained in parallel perspective, but they do not represent the bases of +columns 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 drawing +and 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 put +into perspective. Having placed the plan at the required angle to the +picture plane, they fix upon the point of sight, and the distance from +which the drawing is to be viewed. They then draw a line _SP_ at right +angles to the picture plane _VV'_, which represents that distance so +that _P_ is the station-point. The eye is generally considered to be +the station-point, but when lines are drawn to that point from the +ground-plan, the station-point is placed on the ground, and is in fact +the trace or projection exactly under the point at which the eye is +placed. From this station-point _P_, draw lines _PV_ and _PV'_ parallel +to the two sides of the plan _ba_ and _ad_ (which will be at right +angles to each other), and produce them to the horizon, which they will +touch at points _V_ and _V'_. These points thus obtained will be the +two 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 the +plan to the station-point _P_, such as _bP_, _cP_, _dP_, &c., and where +these lines intersect the picture plane (_VV'_ here represents it as +well as the horizon), drop perpendiculars _b'B_, _aA_, _d'D_, &c., to +meet the vanishing lines _AV_, _AV'_, which will determine the points +_A_, _B_, _C_, _D_, 1, 2, 3, &c., and also the perspective lengths of +the 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 it +on _Aa_; as in this instance _A_ touches the ground line, it may be used +as a line of heights. + +I have here placed the perspective drawing under the ground plan to show +the relation between the two, and how the perspective is worked out, but +the 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 to +the paper on which the drawing is to be made. + +This of course is the simplest form of a plan and elevation. It is easy +to see, however, that we could set out an elaborate building in the same +way as this figure, but in that case we should not place the drawing +underneath the ground-plan, but transfer the measurements to another +sheet of paper as mentioned above. + + + + +CIX + +THE OCTAGON + + +To draw the geometrical figure of an octagon contained in a square, take +half of the diagonal of that square as radius, and from each corner +describe a quarter circle. At the eight points where they touch the +sides 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_ and +thereon form the perspective square _ABCD_. From either extremity of +that base (say _B_) drop perpendicular _BF_, draw diagonal _AF_, and +then from _B_ with radius _BO_, half that diagonal, describe arc _EOE_. +This will give us the measurement _AE_. Make _GB_ equal to _AE_. Then +draw lines from _G_ and _E_ towards _S_, and by means of the diagonals +find the transverse lines _KK_, _hh_, which will give us the eight +points 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 the +horizon at _V_, and produce _VA_ to _a'_ on base. Drop perpendicular +from _B_ to _F_ the same length as _a'B_, and proceed as in the previous +figure to find the eight points on the oblique square through which to +draw the octagon. + + [Illustration: Fig. 200.] + +It will be seen that this operation is very much the same as in parallel +perspective, only we make our measurements on the base line _a'B_ as we +cannot 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 as +in 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 on +this octagon the height of _EG_ we form the vanishing scale _EGO_, and +from the eight points on the ground draw horizontals to _EO_ and thus +find all the points that give us the perspective height of each angle of +the 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 the +larger ones. Note how these are carried out in the second figure +(202 B), and their application to this drawing of an octagonal well on +an 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 octagon +contained in a square _abcd_. Produce diagonal _ac_ to _V_. This will be +the vanishing point for the sides of the small squares directed towards +it. The other sides are directed to an inaccessible point out of the +picture, but their directions are determined by the lines drawn from +divisions 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 the +octagons are obtained by means of the diagonals, _BD_, _AC_, and the +central 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 way +up the floor until sufficient are obtained. It is easy to see how other +squares on each side of these can be produced. + + + + +CXIV + +THE HEXAGON + + +The hexagon is a six-sided figure which, if inscribed in a circle, will +have each of its sides equal to the radius of that circle (Fig. 206). If +inscribed in a rectangle _ABCD_, that rectangle will be equal in length +to two sides of the hexagon or two radii of the circle, as _EF_, and its +width 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 to +point of sight. From _h_ drop perpendicular _ho_, and form equilateral +triangle _mno_. Take the height _ho_ and measure it twice along the base +from _A_ to 2. From 2 draw line to point of distance, or from 1 to +1/2 distance, and so find length of side _AB_ equal to A2. Draw _BC_, +and _EF_ through centre _o'_, and thus we have the six points through +which 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 is +necessary to make a plan of the required design, as in this figure +composed of hexagons. First set out the hexagon as at _A_, then draw +parallels 1 1, 2 2, &c., to mark the horizontal ends of the tiles +and the intermediate lines _oo_. Divide the base into the required +number 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 the +diagonals passing through their intersections. Then mark with a strong +line the outlines of the hexagonals, shading some of them; but the +figure explains itself. + +It is easy to put all these parallels, perpendiculars, and diagonals +into 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 the +two vanishing points for all the sides of the tiles that are receding +from us. From each division on base draw lines to each of these +vanishing points, then draw parallels through their intersections as +shown on the figure. Having all these guiding lines it will not be +difficult 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 and +width of that oblong. Another thing to note is that we have made use of +the half distance, which enables us to make our pavement look flat +without 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 of +one vanishing point; but it shows how much can be done by diagonals, as +nearly all this pavement is drawn by their aid. First make a geometrical +plan _A_ at the angle required. Then draw its perspective _K_. Divide +line 4b into four equal parts, and continue these measurements all +along the base: from each division draw lines to _V_, and draw the +hexagon _K_. Having this one to start with we produce its sides right +and left, but first to the left to find point _G_, the vanishing point +of the diagonals. Those to the right, if produced far enough, would meet +at a distant vanishing point not in the picture. But the student should +study 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 in +which 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 the +perspective figure _C_, as at _AD_, measuring other equal spaces along +the base. To find the depth _An_ of the rectangle, make _DK_ equal to +base of square. Draw _KO_ to distance-point, cutting _DO_ at _O_, and +thus find line _LO_. Draw diagonal _Dn_, and through its intersections +with the lines 1, 2, 3, 4 draw lines parallel to the base, and we shall +thus 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 each +division 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 of +clearness, 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 vanishing +scale _O_ and the vanishing point _V_. Another method could be used by +drawing 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 portions +of the roof are drawn from _o'_. But the figure explains itself, so +there is no necessity to repeat descriptions already given in the +foregoing problems. + + + + +CXX + +THE PENTAGON + + + [Illustration: Fig. 216.] + +The pentagon is a figure with five equal sides, and if inscribed in a +circle will touch its circumference at five equidistant points. With any +convenient radius describe circle. From half this radius, marked 1, draw +a 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 arc +3 4, which will cut the circumference at point 4. Then line 2 4 will be +one of the sides of the pentagon, which we can measure round the circle +and so produce the required figure. + +To put this pentagon into parallel perspective inscribe the circle in +which 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 draw +the perspective square (Fig. 217) and transfer these measurements to its +base. From these draw lines to point of sight, then by their aid and the +two diagonals proceed to construct the pentagon in the same way that we +did the triangles and other figures. Should it be required to place this +pentagon in the opposite position, then we can transfer our measurements +to 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 same +method as with the hexagon, as shown at Fig. 219. + + [Illustration: Fig. 219.] + +Another way of drawing a pentagon (Fig. 220) is to draw an isosceles +triangle with an angle of 36 deg at its apex, and from centre of each +side of the triangle draw perpendiculars to meet at _o_, which will be +the centre of the circle in which it is inscribed. From this centre and +with 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 which +to draw the pentagon. The angles at 1 2 will each be 72 deg, double that +at _A_, which is 36 deg. + + [Illustration: Fig. 220.] + + + + +CXXI + +THE PYRAMID + + +Nothing can be more simple than to put a pyramid into perspective. Given +the base (_abc_), raise from its centre a perpendicular (_OP_) of the +required height, then draw lines from the corners of that base to a +point _P_ on the vertical line, and the thing is done. These pyramids +can be used in drawing roofs, steeples, &c. The cone is drawn in the +same 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 original +height was 480 feet, each side being an equilateral triangle. Let us see +how 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 very +far back, and in the second the horizon must be so low down that we +cannot 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 very +small scale. + +Divide the base _AB_ into ten equal parts, and suppose each of these +parts to measure 10 feet, _S_, the point of sight, is placed on the left +of the picture near the side, in order that we may get a long line of +distance, _S 1/2 D_; but even this line is only half the distance we +require. Let us therefore take the 16th distance, as shown in our +previous illustration of the lighthouse (Fig. 92), which enables us to +measure 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, and +is, according to our 10-foot scale, 764 feet long. + +The next thing to consider is the height of the pyramid. We make a scale +to the right of the picture measuring 50 feet from _B_ to 50 at point +where _BP_ intersects base of pyramid, raise perpendicular _CG_ and +thereon measure 480 feet. As we cannot obtain a palpable square on the +ground, 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, find +its centre _O'_ by means of diagonals, and _O'_ will be the central +height of the pyramid and exactly over the centre of the base. From this +point _O'_ draw sloping lines _O'f_, _O'e_, _O'Y_, &c., and the figure +is complete. + +Note the way in which we find the measurements on base of pyramid and on +line _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 seven +of these lengths, and an additional 64 feet by the scale, and so obtain +the required length. The position of the third corner of the base is +found 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 angle +is 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 much +higher horizon, which immediately detracts from the impression of the +bigness of the pyramid. + + [Illustration: Fig. 225.] + +We proceed to make our ground-plan _abcd_ high above the horizon instead +of below it, drawing first the parallel square and then the oblique one. +From all the principal points drop perpendiculars to the ground and thus +find the points through which to draw the base of the pyramid. Find +centres _OO'_ and decide upon the height _OP_. Draw the sloping lines +from _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 vertical +line OP into the required number of parts. From _A_ through _C_ draw +_AG_ to horizon, which gives us _G_, the vanishing point of all the +diagonals of squares parallel to and at the same angle as _ABCD_. From +_G_ draw lines through the divisions 2, 3, &c., on _OP_ cutting the +lines _PA_ and _PC_, thus dividing them into the required parts. Through +the points thus found draw from _V_ all those sides of the squares that +have _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., as +shown 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. The +chief thing to be noted is the way in which the diagonals are produced +beyond 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 draw +the 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 two +gable roofs crossing each other at right angles. From _o_, centre of +base of cube, raise vertical _OP_, and then from _P_ draw sloping lines +to each corner of base _a_, _b_, &c., and by means of central lines +drawn from _P_ to half base, find the points where the gable roofs +intersect the central spire or pyramid. Any other proportions can be +obtained 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 first +draw its base, _CBDA_ (Fig. 230). Having found the centre _O_ and +central line _SP_, and how far the roof is to fall back at each end, +namely the distance _Pm_, draw horizontal line _RB_ through _m_. Then +from _B_ through _O_ draw diagonal _BA_, and from _A_ draw horizontal +_AD_, which gives us point _n_. From these two points _m_ and _n_ raise +perpendiculars the height required for the roof, and from these draw +sloping lines to the corners of the base. Join _ef_, that is, draw the +top line of the roof, which completes it. Fig. 231 shows a plan or +bird'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 a +book to themselves to illustrate them, especially the cottages and +farm-buildings, barns, &c., besides churches, old mansions, and others. +There is also such irregularity about some of them that perspective +rules, beyond those few here given, are of very little use. So that the +best thing for an artist to do is to sketch them from the real whenever +he 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 may +be twice that of the base, varying to one and a half. In Gothic arches +the height may be about three times the width, all of which proportions +are chosen to suit the different purposes and effects required. Divide +the base _AB_ into the desired number of parts, 8, 10, 12, &c., each +part representing 1 foot. (In this case the base is 10 feet and the +horizon 5 feet.) Set out floor by means of 1/4 distance. Divide it into +squares of 1 foot, so that there will be 8 feet between each column or +pilaster, supposing we make them to stand on a square foot. Draw the +first archway _EKF_ facing us, and its inner semicircle _gh_, with also +its 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 other +arches as required at the given distances. The intersections of the +central line with the chords _mn_, &c., will give the centres from which +to describe the semicircles. + + + + +CXXVII + +OUTLINE OF AN ARCADE WITH SEMICIRCULAR ARCHES + + +This is to show the method of drawing a long passage, corridor, or +cloister with arches and columns at equal distances, and is worked in +the same way as the previous figure, using 1/4 distance and 1/4 base. +The floor consists of five squares; the semicircles of the arches are +described from the numbered points on the central line _OS_, where it +intersects 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 the +figure. Draw _ae'f'b_ and proceed with the rest of the outline. To draw +the arches begin with the one facing us, _Eo'F_ enclosed in the +quadrangle _Ee'f'F_. With centre _O_ describe the semicircle and across +it draw the diagonals _e'F_, _Ef'_, and through _nn_, where these lines +intersect the semicircle, draw horizontal _KK_ and also _KS_ to point of +sight. It will be seen that the half-squares at the side are the same +size in perspective as the one facing us, and we carry out in them much +the same operation; that is, we draw the diagonals, find the point _O_, +and the points _nn_, &c., through which to draw our arches. See +perspective of the circle (Fig. 165). + + [Illustration: Fig. 234.] + +If more points are required an additional diagonal from _O_ to _K_ may +be used, as shown in the figure, which perhaps explains itself. The +method is very old and very simple, and of course can be applied to any +kind 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 new +method. Produce sides _AD_ and _BC_ to _V_. Draw diagonal _BD_ and +produce 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 of +squares. Raise vertical _BM_ the height required. Produce _DA_ to _O_ on +base line, and from _O_ raise vertical _OP_ equal to _BM_. This line +enables us to dispense with the long vanishing point to the left; its +working has been explained at Fig. 131. From _P_ draw _PRV_ to vanishing +point _V_, which will intersect vertical _AR_ at _R_. Join _MR_, and +this line, if produced, would meet the horizon at the other vanishing +point. In like manner make O2 equal to B2'. From 2 draw line to _V_, and +at 2, its intersection with _AR_, draw line 2 2, which will also meet +the horizon at the other vanishing point. By means of the quarter-circle +_A_ we can obtain the points through which to draw the semicircular +arches 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 arches +from the two diagonals _DB_, _AC_, which spring from the four corners of +the square _EFGH_, just underneath it. The curves of these arches, which +are not semicircular but elongated, are obtained by means of the +vanishing scales _mS_, _nS_. Take any two convenient points _P_, _R_, on +each side of the semicircle, and raise verticals _Pm_, _Rn_ to _AB_, and +on these verticals form the scales. Where _mS_ and _nS_ cut the diagonal +_AC_ drop perpendiculars to meet the lower line of the scale at points +1, 2. On the other side, using the other scales, we have dropped +perpendiculars 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_. We +draw the arch under the other diagonal in precisely the same way. + + [Illustration: Fig. 237.] + +The reason for thus proceeding is that the cross arches, although +elongated, hang from their diagonals just as the semicircular arch _EKF_ +hangs from _AB_, and the lines _mn_, touching the circle at _PR_, are +represented by 1, 2, hanging from the diagonal _AC_. + + [Illustration: Fig. 238.] + +Figure 238, which is practically the same as the preceding only +differently shaded, is drawn in the following manner. Draw arch _EGF_ +facing us, and proceed with the rest of the corridor, but first finding +the flat ceiling above the square on the ground _ABcd_. Draw diagonals +_ac_, _bd_, and the curves pending from them. But we no longer see the +clear arch as in the other drawing, for the spaces between the curves +are filled in and arched across. + + + + +CXXXI + +A CLOISTER, FROM A PHOTOGRAPH + + +This drawing of a cloister from a photograph shows the correctness of +our 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. From +same centre and _oh_ for radius describe the inner semicircle. Divide +outer circle into a convenient number of parts, 1, 2, 3, &c., to which +draw radii from centre _O_. From each division drop perpendiculars. +Where the radii intersect the inner circle, as at _gkmo_, draw +horizontals _op_, _mn_, _kj_, &c., and through their intersections with +the perpendiculars _f_, _j_, _n_, _p_, draw the curve of the flattened +arch. 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 to +the 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 part +of 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 side +of 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 to +corresponding points in the curved archway or window required. + + [Illustration: Fig. 242.] + +Note that to obtain the width of the window _K_ we have used the +diagonals on the floor and width _m n_ on base. This method of +measurement is explained at Fig. 144, and is of ready application in a +case of this kind. + + + + +CXXXIV + +STAIRS, STEPS, &C. + + +Having decided upon the incline or angle, such as _CBA_, at which the +steps 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 width +equal to _DB_, which will give that for all the other steps. Obtain +length _BF_ of steps, and draw _EF_ parallel to _CB_. These lines will +aid 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 their +depth is found by diagonals drawn to the point of distance _D_. The rest +of the figure explains itself. + + [Illustration: Fig. 245.] + + + + +CXXXVI + +SQUARE STEPS + + +Draw first step _ABEF_ and its two diagonals. Raise vertical _AH_, and +measure 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 the +second step, guided by the diagonals and taking its height from the +scale 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 PLACED +AGAINST A WALL + + + [Illustration: Fig. 247.] + +Divide the vertical _EC_ into the required number of parts, and draw +lines 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, draw +1V, _AV_ to _V_, the vanishing point on the vertical line _SV_. These +two lines and the corresponding ones at _BV_ will form a kind of +vanishing scale, giving the height of each step as we ascend. It is +especially useful when we pass the horizontal line and we no longer see +the upper surface of the step, the scale on the right showing us how to +proceed in that case. + +In Fig. 249 we have an example of steps ascending and descending. First +set out the ground-plan, and find its vanishing point _S_ (point of +sight). Through _S_ draw vertical _BA_, and make _SA_ equal to _SB_. Set +out the first step _CD_. Draw _EA_, _CA_, _DA_, and _GA_, for the +ascending 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 new +method (Fig. 250). Produce _BC_ to the horizon, and thus find vanishing +point _V_. At this point raise vertical _VV'_. Construct first step +_AB_, refer its height at _B_ to line of heights hI on left, and thus +obtain 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 represent +the height of the next step, its height being determined by the scale of +heights at the side. From _A_ and _F_ draw lines to _V'_, and also +similar lines from _B_, which will serve as guiding lines to determine +the height of the steps at either end as we raise them to the required +number. + + [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 vanishing +and measuring points. Find the height _hH_, and width at top _HH'_, and +draw the sides _HA_ and _H'E_. Note that _AE_ is wider than _HH'_, and +also that the back legs are not at the same angle as the front ones, and +that they overlap them. From _E_ raise vertical _EF_, and divide into as +many parts as you require rounds to the ladder. From these divisions +draw lines 1 1, 2 2, &c., towards the other vanishing point (not in the +picture), but having obtained their direction from the ground-plan in +perspective at line _Ee_, you may set up a second vertical _ef_ at any +point on _Ee_ and divide it into the same number of parts, which will be +in proportion to those on _EF_, and you will obtain the same result by +drawing lines from the divisions on _EF_ to those on _ef_ as in drawing +them 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 simple +enough 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 DIAGONALS +AND ONE VANISHING POINT + + +Although in this figure we have taken a longer distance-point than in +the 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 of +step _Ao_, and proceed to draw the steps as already shown. Then by the +diagonals and measurements on base draw the second step and the square +inside 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_ from +its centre. Through any point on this central line, if we draw a +diagonal from point _G_ we cut the two opposite verticals of the shaft +at _mn_ (see Fig. 255), and by means of the vanishing point _V_ we cut +the other two verticals at the opposite corners and thus obtain the four +points through which to draw the other sides of the square, which go to +the distant or inaccessible vanishing point. It will be seen by +carefully examining the figure that by this means we are enabled to draw +the 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 the +foregoing figures of steps, &c., such as the side scale on the left of +the figure to ascertain the height of the steps, the double lines drawn +to the high vanishing point of the inclined plane, and so on; but the +principal use of this diagram is to show on the perspective plane, which +as it were runs under the stairs, the trace or projection of the flights +of steps, the landings and positions of other objects, which will be +found very useful in placing figures in a composition of this kind. +It will be seen that these underneath measurements, so to speak, are +obtained 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 the +positions 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 the +height of each step. Form scale _AB_, which will give the height of each +step according to its position. Thus at _mn_ we find the height at the +centre of the square, so if we transfer this measurement to the central +line _OP_ and repeat it upwards, say to fourteen, then we have the +height of each step on the line where they all meet. Starting then with +the 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 the +first step. On _DE_, measure heights equal to _D 1_. Draw 2 2 towards +central line, and 2n towards point of sight till it meets the second +vertical _nK_. Then draw n2 to centre, and so complete the second +step. 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 the +right, completing them in the same way as numbers 1 and 2, when we come +to the sixth step, the other end of which is against the wall opposite +to us. Steps 6, 7, 8, 9 are all on this wall, and are therefore equal in +height all along, as they are equally distant. Step 10 is turned towards +us, and abuts on the wall to our left; its measurement is taken on the +scale _AB_ just underneath it, and on the same line to which it is +drawn. Step 11 is just over the centre of base _mo_, and is therefore +parallel to it, and its height is _mn_. The widths of steps 12 and 13 +seem gradually to increase as they come towards us, and as they rise +above the horizon we begin to see underneath them. Steps 13, 14, 15, 16 +are against the wall on this side of the picture, which we may suppose +has been removed to show the working of the drawing, or they might be an +open flight as we sometimes see in shops and galleries, although in that +case 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 equal +parts, according to the number of steps in a complete round, say twelve. +Form scale _ASF_ and the larger scale _ASB_, on which is shown the +perspective measurements of the steps according to their positions; +raise verticals such as _ef_, _Gh_, &c. From divisions on circumference +measure out the central line _OP_, as in the other figure, and find the +heights of the steps 1, 2, 3, 4, &c., by the corresponding numbers in +the large scale to the left; then proceed in much the same way as in the +previous figure. Note the central column _OP_ cuts off a small portion +of the steps at that end. + +In ordinary cases only a small portion of a winding staircase is +actually 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 no +picture painted on a flat surface that gives such a wonderful appearance +of truth as that painted on a cylindrical canvas, such as those +panoramas of 'Paris during the Siege', exhibited some years ago; 'The +Battle of Trafalgar', only lately shown at Earl's Court; and many +others. 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 is +always in front of him, or nearly so. I believe on the canvas these +points are from 12 to 16 feet apart. + +The reason of this look of truth may be explained thus. If we place +three globes of equal size in a straight line, and trace their apparent +widths 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, if +we trace them on a semicircular glass they will appear very nearly equal +and, of the three, the central one _c_ will be rather the largest, as +may be seen by this figure. + +We must remember that, in the first case, when we are looking at a globe +or a circle, the visual rays form a cone, with a globe at its base. If +these three cones are intersected by a straight glass _GG_, and looked +at from point _S_, the intersection of _C_ will be a circle, as the cone +is 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 the +station point, with one eye only, then the three globes (or tracings of +them) will appear equal and perfectly round. + +Of course the cylindrical canvas is necessary for panoramas; but we +have, as a rule, to paint our pictures and wall-decorations on flat +surfaces, and therefore must adapt our work to these conditions. + +In all cases the artist must exercise his own judgement both in the +arrangement of his design and the execution of the work, for there is +perspective even in the touch--a painting to be looked at from a +distance requires a bold and broad handling; in small cabinet pictures +that we live with in our own rooms we look for the exquisite workmanship +of the best masters. + + + + +BOOK FOURTH + +CXLVII + +THE PERSPECTIVE OF CAST SHADOWS + + +There is a pretty story of two lovers which is sometimes told as the +origin of art; at all events, I may tell it here as the origin of +sciagraphy. 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 last +evening together, when the girl, seeing the shadow of her lover's +profile cast from a lamp on to some wet plaster or on the wall, took a +metal point, perhaps some sort of iron needle, and traced the outline of +the face she loved on to the plaster, following carefully the outline of +the 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 he +began to ornament his wares by similar devices, which gave them +increased value by the novelty and beauty thus imparted to them. + +Here then we have a very good illustration of our present subject and +its three elements. First, the light shining on the wall; second, the +wall or the plane of projection, or plane of shade; and third, the +intervening object, which receives as much light on itself as it +deprives the wall of. So that the dark portion thus caused on the plane +of shade is the cast shadow of the intervening object. + +We have to consider two sorts of shadows: those cast by a luminary a +long way off, such as the sun; and those cast by artificial light, such +as a lamp or candle, which is more or less close to the object. In the +first case there is no perceptible divergence of rays, and the outlines +of the sides of the shadows of regular objects, as cubes, posts, &c., +will be parallel. In the second case, the rays diverge according to the +nearness 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 parallel +rays. + +Fig. 263 shows the shadows cast by a candle or lamp, where the rays +diverge from the point of light to meet corresponding diverging lines +which 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 the +point of light or luminary pass over the top of the intervening object +which casts the shadow on to the plane of shade to meet the horizontal +trace of those rays on that plane, or the lines of light proceed from +the point of light, and the lines of the shadow are drawn from the foot +or 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 the +picture, 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 drawn +from 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 of +light proceeding from it as parallel, and therefore subject to the same +laws as other parallel lines in perspective, as already noted. And for +the same reason we have to place the foot of the luminary on the +horizon. It is important to remember this, as these two things make the +difference between shadows cast by the sun and those cast by artificial +light. + +The sun has three principal positions in relation to the picture. In the +first case it is supposed to be in the same plane either to the right or +to the left, and in that case the shadows will be parallel with the base +of 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 thrown +forwards or towards him. In the third, the sun is in front of the +picture, and behind the spectator, so that the shadows are thrown in the +opposite direction, or towards the horizon, the objects themselves being +in 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 angle +of 45 deg to the horizon, consequently the shadows will be the same +length as the figures that cast them are high. Note that the shadow of +step No. 1 is cast upon step No. 2, and that of No. 2 on No. 3, the top +of each 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 fall +upon the plane, is interrupted by another object _B_, then the outline +of the shadow is still drawn on the plane, but being interrupted by the +surface _B_ at _C_, the shadow runs up that plane till it meets the rays +1, 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 object +casting the shadow, in the case of an archway or similar figure they +pass underneath it; but the same principle holds good, that is, we draw +lines from the guiding points in the arch, 1, 2, 3, &c., at the same +angle of 45 deg to meet the traces of those rays on the plane of shade, +and so 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 deg the +shadows on the horizontal plane are the same length as the height of the +objects that cast them. Here (Fig. 270), the sun still being at 45 deg +altitude, although behind the picture, and consequently throwing the +shadow of _B_ forwards, that shadow must be the same length as the +height 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 the +luminary. + + [Illustration: Fig. 271.] + +From point of sight _S_ (Fig. 270) drop perpendicular to _T_, the +station-point. From _T_ draw _TF_ at 45 deg to meet horizon at _F_. With +radius _FT_ make _FO_ equal to it. Then _O_ is the position of the +spectator. From _F_ raise vertical _FL_, and from _O_ draw a line at +45 deg to meet _FL_ at _L_, which is the luminary at an altitude of +45 deg, and at an angle of 45 deg to the picture. + +Fig. 272 is similar to the foregoing, only the angles of altitude and of +the sun to the picture are altered. + +_Note._--The sun being at 50 deg to the picture instead of 45 deg, is +nearer the point of sight; at 90 deg it would be exactly opposite the +spectator, and so on. Again, the elevation being less (40 deg instead of +45 deg) the shadow is longer. Owing to the changed position of the sun +two sides of the 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 suits +our 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. An +upright wall now becomes the plane of shade, therefore as the principle +of shadows must always remain the same we have to change the relative +positions 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 still +remains 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. The +luminary _L_ is at the right side of point _S_ instead of the left, and +the foot thereof is, as before, the trace of the luminary, as it is just +underneath it. We shall also see that by proceeding as in previous +figures we obtain the same results on the wall as we did on the +horizontal plane. Fig. B being on the horizontal plane is treated as +already shown. The steps have their shadows partly on the wall and +partly on the horizontal plane, so that the shadows on the wall are +outlined 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 line +drawn from the luminary _L_, at the point _P_, and how the lower line of +the shadow is directed to point of sight _S_. + + [Illustration: Fig. 274.] + +Fig. 274 is a larger drawing of the steps, &c., in further illustration +of 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 vertical +dropped from the luminary to a point (_F_) on a level with the vanishing +point (_P_) of that inclined plane. Thus _P_ is the vanishing point of +the inclined plane _K_. Draw horizontal _PF_ to meet _fL_ (the line +drawn from the luminary to the horizon). Then _F_ will be the vanishing +point 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 drawn +from _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 opposite +effect to that we have just been studying. The shadows, instead of +coming towards us, are retreating from us, and the objects throwing them +are in full light, consequently we have to reverse our treatment. Let us +suppose the sun to be placed above the horizon at _L'_, on the right of +the picture and behind the spectator (Fig. 276). If we transport the +length _L'f'_ to the opposite side and draw the vertical downwards from +the horizon, as at _FL_, we can then suppose point _L_ to be exactly +opposite the sun, and if we make that the vanishing point for the sun's +rays 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 be +the 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 as +we 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 the +object to meet those drawn from the foot _F_, namely F1, F2, F3, in the +same 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 first +the side _fbd_, and second the farther end _abcd_. The points we have to +find 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 lines +will be at _a'b'_. If we join _fb'_ and _db'_ we have the shadow of the +side _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 a +chimney on a roof). First find the vanishing point _P_ of the inclined +plane and draw horizontal _PF_ to meet vertical raised from _L_, the +luminary. 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_ draw +1F, 2F, 3F to _F_, the vanishing point of the shadow. The +intersections of these lines at 1, 2, 3 on _K_ will determine the +length 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 the +point 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 of +luminary. From the points _xx_ where these lines cut the base _B_ raise +perpendiculars _xa'_, _xb'_. From _D_, _A_, and _B_ draw lines to the +luminary _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. Take +any point thereon, mark its trace on the ground, and then proceed as +above. + + + + +CLVIII + +SHADOW ON A RETREATING WALL, SUN IN FRONT + + +Let _L_ be the luminary. Raise vertical _LF_. _F_ will be the vanishing +point 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 the +shadows 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 of +roof. + + [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 the +roof. + + [Illustration: Fig. 283.] + +Figure 283 shows how the shadow of the old man in the preceding drawing +is found. + + + + +CLIX + +SHADOW OF AN ARCH, SUN IN FRONT + + +Having drawn the arch, divide it into a certain number of parts, say +five. From these divisions drop perpendiculars to base line. From +divisions on _AB_ draw lines to _F_ the foot, and from those on the +semicircle draw lines to _L_ the luminary. Their intersections will give +the 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. Drop +perpendiculars from the divisions of the arch 1 2 3 to the base. From +the foot of each draw 1S, 2S, 3S to foot of luminary _S_, and +from the top of each, A 1 2 3 B, draw lines to _L_ as before. Where the +former intersect the curve on the floor of the niche raise verticals +to meet the latter at P 1 2 B, &c. These points will indicate about the +position of the shadow; but the niche being semicircular and domed at +the top the shadow gradually loses itself in a gradated and somewhat +serpentine 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 not +repeat a description of the manner in which it is done. And surely an +artist after making a few sketches from the actual thing will hardly +require 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 found +by drawing lines from the seat of the luminary through the feet of the +objects to meet lines representing rays of light drawn from the luminary +itself over the tops or the corners of the objects; very much as in the +cases of sun-shadows, but with this difference, that whereas the foot of +the luminary in this latter case is supposed to be on the horizon an +infinite distance away, the foot in the case of a lamp or candle may be +on 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 table +such as _K_ the foot will be at _f_ on the table. For the shadows on the +floor, of the chair and table itself, we must find the foot of the +luminary 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 foot +of candlestick to meet _PS_ in _F_. Then _F_ is the foot of the luminary +on the floor. From this point draw lines through the feet or traces of +objects such as the corners of the table, &c., to meet other lines drawn +from 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 which +sun-shadows are shaped, still there are so many more laws to be +considered in the great art of light and shade that it is better to +observe them in Nature herself or under the teaching of the real sun. In +the 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 mere +shapes of shadows of regular forms. It will be seen that the light is +dispersed in all directions, and although there is a good deal of +half-shade there are scarcely any cast shadows except on the floor; but +the light on the white walls in the outside gallery is so reflected into +the cast shadows that they are extremely faint. The luminosity of this +part of the sketch is greatly enhanced by the contrast of the dark legs +of the bench and the shadows in the roof. The warm glow of all this +portion is contrasted by the grey door and its frame. + + [Illustration: Fig. 288.] + +Note that the door itself is quite luminous, and lighted up by the +reflection of the sun from the tiled floor, so that the bars in the +upper part throw distinct shadows, besides the mystery of colour thus +introduced. The little window to the left, though not admitting much +direct sunlight, is evidence of the brilliant glare outside; for the +reflected light is very conspicuous on the top and on the shutters on +each side; indeed they cast distinct shadows up and down, while some +clear daylight from the blue sky is reflected on the window-sill. As to +the sink, the table, the wash-tubs, &c., although they seem in strong +light and shade they really receive little or no direct light from a +single point; but from the strong reflected light re-reflected into them +from the wall of the doorway. There are many other things in such +effects as this which the artist will observe, and which can only be +studied from real light and shade. Such is the character of reflected +light, varying according to the angle and intensity of the luminary and +a hundred other things. When we come to study light in the open air we +get into another region, and have to deal with it accordingly, and yet +we shall find that our sciagraphy will be a help to us even in this +bewilderment; for it will explain in a manner the innumerable shapes of +sun-shadows that we observe out of doors among hills and dales, showing +up their forms and structure; its play in the woods and gardens, and its +value among buildings, showing all their juttings and abuttings, +recesses, doorways, and all the other architectural details. Nor must we +forget light's most glorious display of all on the sea and in the clouds +and in the sunrises and the sunsets down to the still and lovely +moonlight. + +These sun-shadows are useful in showing us the principle of light and +shade, and so also are the shadows cast by artificial light; but they +are only the beginning of that beautiful study, that exquisite art of +tone or _chiaro-oscuro_, which is infinite in its variety, is full of +the deepest mystery, and is the true poetry of art. For this the student +must go to Nature herself, must study her in all her moods from early +dawn to sunset, in the twilight and when night sets in. No mathematical +rules can help him, but only the thoughtful contemplation, the silent +watching, 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 great +range of this ever-changing and deepening mystery--from the hard reality +to the soft blending of evening when form almost disappears, even to the +merging of the whole landscape, nay, the whole world, into a +dream--which is felt rather than seen, but possesses a charm that almost +defies the pencil of the painter, and can only be expressed by the deep +and sweet notes of the poet and the musician. For love and reverence are +necessary to appreciate and to present it. + +There is also much to learn about artificial light. For here, again, the +study is endless: from the glare of a hundred lights--electric and +otherwise--to the single lamp or candle. Indeed a whole volume could be +filled with illustrations of its effects. To those who aim at producing +intense brilliancy, refusing to acknowledge any limitations to their +capacity, a hundred or a thousand lights commend themselves; and even +though wild splashes of paint may sometimes be the result, still the +effort is praiseworthy. But those who prefer the mysterious lighting of +a Rembrandt will find, if they sit contemplating in a room lit with one +lamp only, that an endless depth of mystery surrounds them, full of dark +recesses peopled by fancy and sweet thought, whilst the most beautiful +gradations soften the forms without distorting them; and at the same +time he can detect the laws of this science of light and shade a +thousand times repeated and endless in its variety. + +_Note._--Fig. 288 must be looked upon as a rough sketch which only gives +the general effect of the original drawing; to render all the delicate +tints, tones and reflections described in the text would require a +highly-finished reproduction in half-tone or in colour. + +As many of the figures in this book had to be re-drawn, not a light +task, I must here thank Miss Margaret L. Williams, one of our Academy +students, for kindly coming to my assistance and volunteering her +careful 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 some +simple object, such as a cube, on a looking-glass laid horizontally on a +table, or by studying plants, stones, banks, trees, &c., reflected in +some quiet pond. It will then be seen that the reflection is the +counterpart of the object reversed, and having the same vanishing points +as the object itself. + + [Illustration: Fig. 289.] + +Let us suppose _R_ (Fig. 289) to be standing on the water or reflecting +plane. To find its reflection make square [R] equal to the original +square _R_. Complete the reversed cube by drawing its other sides, &c. +It is evident that this lower cube is the reflection of the one above +it, although it differs in one respect, for whereas in figure _R_ the +top of the cube is seen, in its reflection [R] it is hidden, &c. In +figure A of a semicircular arch we see the underneath portion of the +arch 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 be +equal in length to the actual one, or the reflection of a square would +not be a square, nor that of a semicircle a semicircle. The apparent +lengthening of reflections in water is owing to the surface being broken +by wavelets, which, leaping up near to us, catch some of the image of +the 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 obtained +by dropping perpendiculars from certain points on the arch, 1, 0, 2, +&c., to the surface of the reflecting plane, and then measuring the same +lengths downwards to corresponding points, 1, 0, 2, &c., in the +reflection. + + + + +CLXV + +ANGLES OF REFLECTION + + +In Fig. 291 we take a side view of the reflected object in order to show +that at whatever angle the visual ray strikes the reflecting surface it +is 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 just +underneath it passes through _MN_ at _O_, which is the point where the +visual ray strikes the reflecting surface. Draw _OA_. This line +represents 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 the +reflecting 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 standing +on a plane level or almost level with the water, in order to show the +working of our problem more clearly. It will be seen that the post _A_ +is on the brink of the reflecting plane, and therefore is entirely +reflected; _B_ and _C_ being farther back are only partially seen, +whereas the reflection of _D_ is not seen at all. I have made all the +posts the same height, but with regard to the houses, where the length +of the vertical lines varies, we obtain their reflections by measuring +from the points _oo_ upwards and downwards as in the previous figure. + + [Illustration: Fig. 292.] + +Of course these reflections vary according to the position they are +viewed from; the lower we are down, the more do we see of the +reflections of distant objects, and vice versa. When the figures are on +a 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 length +downwards, 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 measuring +downwards to _a'_ we get the length of her reflection, or as much as is +seen of it. Note the reflection of the steps and the sloping bank, and +the 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 only +half life-size, and yet they looked natural; and one day, thinking I +would trace myself in an upright mirror, I stood at arm's length from it +and with a brush and Chinese white, I made a rough outline of my face +and figure, and when I measured it I found that my drawing was exactly +half as long and half as wide as nature. I went closer to the glass, but +the same outline fitted me. Then I retreated several paces, and still +the same outline surrounded me. Although a little surprising at first, +the reason is obvious. The image in the glass retreats or advances +exactly in the same measure as the spectator. + + [Illustration: Fig. 294.] + +Suppose him to represent one end of a parallelogram _e's'_, and his +image _a'b'_ to represent the other. The mirror _AB_ is a perpendicular +half-way between them, the diagonal _e'b'_ is the visual ray passing +from the eye of the spectator to the foot of his image, and is the +diagonal of a rectangle, therefore it cuts _AB_ in the centre _o_, and +_AO_ represents _a'b'_ to the spectator. This is an experiment that any +one may try for himself. Perhaps the above fact may have something to do +with 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 deg to the mirror +_RR_, then the angle _BAC_ will be a right angle, and this angle is +exactly divided in two by the reflecting plane _RR_. And whatever the +angle of the object or line makes with its reflection that angle will +also be exactly divided. + + [Illustration: Fig. 297.] + + [Illustration: Fig. 298.] + +Now suppose our mirror to be standing on a horizontal plane and on a +pivot, so that it can be inclined either way. Whatever angle the mirror +is to the plane the reflection of that plane in the mirror will be at +the same angle on the other side of it, so that if the mirror _OA_ (Fig. +298) is at 45 deg to the plane _RR_ then the reflection of that plane in +the mirror will be 45 deg on the other side of it, or at right angles, +and the reflected plane will appear perpendicular, as shown in Fig. 299, +where we have a front view of a mirror leaning forward at an angle of +45 deg and reflecting the square _aob_ with a cube standing upon it, only +in the reflection the cube appears to be projecting from an upright +plane or wall. + + [Illustration: Fig. 299.] + +If we increase the angle from 45 deg to 60 deg, then the reflection of the +plane and cube will lean backwards as shown in Fig. 300. If we place it +on a level with the original plane, the cube will be standing upright +twice the distance away. If the mirror is still farther tilted till it +makes an angle of 135 deg as at _E_ (Fig. 298), or 45 deg on the other +side of the vertical _Oc_, then the plane and cube would disappear, and +objects exactly over that plane, such as the ceiling, would come into +view. + +In Fig. 300 the mirror is at 60 deg to the plane _mn_, and the plane +itself at about 15 deg to the plane _an_ (so that here we are using +angular perspective, _V_ being the accessible vanishing point). The +reflection of the plane and cube is seen leaning back at an angle of +60 deg. Note the way the reflection of this cube is found by the dotted +lines on the plane, on the surface of the mirror, and also on the +reflection. + + [Illustration: Fig. 300.] + + + + +CLXIX + +THE UPRIGHT MIRROR AT AN ANGLE OF 45 DEG. TO THE WALL + + +In Fig. 301 the mirror is vertical and at an angle of 45 deg to the wall +opposite the spectator, so that it reflects a portion of that wall as +though it were receding from us at right angles; and the wall with the +pictures upon it, which appears to be facing us, in reality is on our +left. + + [Illustration: Fig. 301.] + +An endless number of complicated problems could be invented of the +inclined mirror, but they would be mere puzzles calculated rather to +deter the student than to instruct him. What we chiefly have to bear in +mind is the simple principle of reflections. When a mirror is vertical +and placed at the end or side of a room it reflects that room and gives +the impression that we are in one double the size. If two mirrors are +placed opposite to each other at each end of a room they reflect and +reflect, 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 picture +in front of us, whether it is behind us, at the side, or even on the +ceiling. Indeed, when one goes to those old palaces and churches where +pictures are painted on the ceiling, as in the Sistine Chapel or the +Louvre, or the palaces at Venice, it is not a bad plan to take a hand +mirror with us, so that we can see those elevated works of art in +comfort. + +There are also many uses for the mirror in the studio, well known to the +artist. One is to look at one's own picture reversed, when faults become +more evident; and another, when the model is required to be at a longer +distance than the dimensions of the studio will admit, by drawing his +reflection 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 has +become accustomed is that it doubles it. Thus if a line that should be +vertical is leaning to one side, in the mirror it will lean to the +other; so that if it is out of the perpendicular to the left, its +reflection will be out of the perpendicular to the right, making a +double 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 will +endeavour to see even into the infinite. To see Nature in all her +vastness and magnificence, the thought must supplement and must surpass +the eye. It is this far-seeing that makes the great poet, the great +philosopher, 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 give +immortality to his work. + +To explain in detail the full meaning of this suggestion is beyond the +province of this book, but it may lead the student to think this +question out for himself in his solitary and imaginative moments, and +should, I think, give a charm and virtue to his work which he should +endeavour to make of value, not only to his own time but to the +generations that are to follow. Cultivate, therefore, this mental +perspective, without forgetting the solid foundation of the science I +have 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.] + + +A +Albert Durer, 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]. + +B +Balcony, Shadow of, 246 [CLVII]. +Base or groundline, 89 [XLI]. + +C +Campanile 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]. + +D +De 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]. + +E +Ellipse, 145 [LXXXIX], 146 [XC], 147 [168]. +Elliptical Arch, 207 [CXXXII]. + +F +Farningham, 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]. + +G +Geometrical and Perspective figures contrasted, 46 [XII] - 48. + " plane, 99 [L]. +Giovanni da Pistoya, Sonnet to, by Michelangelo, 60. +Great Pyramid, 190 [CXXII]. + +H +Hexagon, 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. + +I +Inaccessible 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. + +J +Japanese Art, 11. +Jesuit of Paris, Practice of Perspective by, 9. + +K +Kiosk, Application of Hexagon, 185 [XCIX]. +Kirby, Joshua, Perspective made Easy (?), 9. + +L +Ladder, 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]. + +M +Measure 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. + +N +Natural 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]. + +O +Oblique Square, 139 [LXXXV]. +Octagon, 172 [CIX] - 175 [202]. +O, measuring point, 110 [LX]. +Optic Cone, 20 [IV]. + +P +Parallels 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]. + +R +Raphael, 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. + +S +Scale 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 deg, 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]. + +T +Taddeo 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]. + +U +Ubaldus, Guidus, 9. + +V +Vanishing 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]. + +W +Winding 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 Duerer" (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_ +Durer, Albert + _umlaut missing_ +Taddeo Gaddi + _text reads "Tadeo"_ +Titian + _text reads Titien_ + + + +***END OF THE PROJECT GUTENBERG EBOOK THE THEORY AND PRACTICE OF +PERSPECTIVE*** + + +******* This file should be named 20165.txt or 20165.zip ******* + + +This and all associated files of various formats will be found in: +https://www.gutenberg.org/dirs/2/0/1/6/20165 + + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. Special rules, +set forth in the General Terms of Use part of this license, apply to +copying and distributing Project Gutenberg-tm electronic works to +protect the PROJECT GUTENBERG-tm concept and trademark. Project +Gutenberg is a registered trademark, and may not be used if you +charge for the eBooks, unless you receive specific permission. If you +do not charge anything for copies of this eBook, complying with the +rules is very easy. You may use this eBook for nearly any purpose +such as creation of derivative works, reports, performances and +research. They may be modified and printed and given away--you may do +practically ANYTHING with public domain eBooks. Redistribution is +subject to the trademark license, especially commercial +redistribution. + + + +*** START: FULL LICENSE *** + +THE FULL PROJECT GUTENBERG LICENSE +PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK + +To protect the Project Gutenberg-tm mission of promoting the free +distribution of electronic works, by using or distributing this work +(or any other work associated in any way with the phrase "Project +Gutenberg"), you agree to comply with all the terms of the Full Project +Gutenberg-tm License (available with this file or online at +https://www.gutenberg.org/license). + + +Section 1. General Terms of Use and Redistributing Project Gutenberg-tm +electronic works + +1.A. By reading or using any part of this Project Gutenberg-tm +electronic work, you indicate that you have read, understand, agree to +and accept all the terms of this license and intellectual property +(trademark/copyright) agreement. If you do not agree to abide by all +the terms of this agreement, you must cease using and return or destroy +all copies of Project Gutenberg-tm electronic works in your possession. +If you paid a fee for obtaining a copy of or access to a Project +Gutenberg-tm electronic work and you do not agree to be bound by the +terms of this agreement, you may obtain a refund from the person or +entity to whom you paid the fee as set forth in paragraph 1.E.8. + +1.B. "Project Gutenberg" is a registered trademark. It may only be +used on or associated in any way with an electronic work by people who +agree to be bound by the terms of this agreement. There are a few +things that you can do with most Project Gutenberg-tm electronic works +even without complying with the full terms of this agreement. See +paragraph 1.C below. There are a lot of things you can do with Project +Gutenberg-tm electronic works if you follow the terms of this agreement +and help preserve free future access to Project Gutenberg-tm electronic +works. See paragraph 1.E below. + +1.C. The Project Gutenberg Literary Archive Foundation ("the Foundation" +or PGLAF), owns a compilation copyright in the collection of Project +Gutenberg-tm electronic works. Nearly all the individual works in the +collection are in the public domain in the United States. If an +individual work is in the public domain in the United States and you are +located in the United States, we do not claim a right to prevent you from +copying, distributing, performing, displaying or creating derivative +works based on the work as long as all references to Project Gutenberg +are removed. Of course, we hope that you will support the Project +Gutenberg-tm mission of promoting free access to electronic works by +freely sharing Project Gutenberg-tm works in compliance with the terms of +this agreement for keeping the Project Gutenberg-tm name associated with +the work. You can easily comply with the terms of this agreement by +keeping this work in the same format with its attached full Project +Gutenberg-tm License when you share it without charge with others. + +1.D. The copyright laws of the place where you are located also govern +what you can do with this work. Copyright laws in most countries are in +a constant state of change. If you are outside the United States, check +the laws of your country in addition to the terms of this agreement +before downloading, copying, displaying, performing, distributing or +creating derivative works based on this work or any other Project +Gutenberg-tm work. The Foundation makes no representations concerning +the copyright status of any work in any country outside the United +States. + +1.E. Unless you have removed all references to Project Gutenberg: + +1.E.1. The following sentence, with active links to, or other immediate +access to, the full Project Gutenberg-tm License must appear prominently +whenever any copy of a Project Gutenberg-tm work (any work on which the +phrase "Project Gutenberg" appears, or with which the phrase "Project +Gutenberg" is associated) is accessed, displayed, performed, viewed, +copied or distributed: + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + +1.E.2. If an individual Project Gutenberg-tm electronic work is derived +from the public domain (does not contain a notice indicating that it is +posted with permission of the copyright holder), the work can be copied +and distributed to anyone in the United States without paying any fees +or charges. If you are redistributing or providing access to a work +with the phrase "Project Gutenberg" associated with or appearing on the +work, you must comply either with the requirements of paragraphs 1.E.1 +through 1.E.7 or obtain permission for the use of the work and the +Project Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or +1.E.9. + +1.E.3. If an individual Project Gutenberg-tm electronic work is posted +with the permission of the copyright holder, your use and distribution +must comply with both paragraphs 1.E.1 through 1.E.7 and any additional +terms imposed by the copyright holder. Additional terms will be linked +to the Project Gutenberg-tm License for all works posted with the +permission of the copyright holder found at the beginning of this work. + +1.E.4. Do not unlink or detach or remove the full Project Gutenberg-tm +License terms from this work, or any files containing a part of this +work or any other work associated with Project Gutenberg-tm. + +1.E.5. Do not copy, display, perform, distribute or redistribute this +electronic work, or any part of this electronic work, without +prominently displaying the sentence set forth in paragraph 1.E.1 with +active links or immediate access to the full terms of the Project +Gutenberg-tm License. + +1.E.6. You may convert to and distribute this work in any binary, +compressed, marked up, nonproprietary or proprietary form, including any +word processing or hypertext form. However, if you provide access to or +distribute copies of a Project Gutenberg-tm work in a format other than +"Plain Vanilla ASCII" or other format used in the official version +posted on the official Project Gutenberg-tm web site (www.gutenberg.org), +you must, at no additional cost, fee or expense to the user, provide a +copy, a means of exporting a copy, or a means of obtaining a copy upon +request, of the work in its original "Plain Vanilla ASCII" or other +form. Any alternate format must include the full Project Gutenberg-tm +License as specified in paragraph 1.E.1. + +1.E.7. Do not charge a fee for access to, viewing, displaying, +performing, copying or distributing any Project Gutenberg-tm works +unless you comply with paragraph 1.E.8 or 1.E.9. + +1.E.8. You may charge a reasonable fee for copies of or providing +access to or distributing Project Gutenberg-tm electronic works provided +that + +- You pay a royalty fee of 20% of the gross profits you derive from + the use of Project Gutenberg-tm works calculated using the method + you already use to calculate your applicable taxes. The fee is + owed to the owner of the Project Gutenberg-tm trademark, but he + has agreed to donate royalties under this paragraph to the + Project Gutenberg Literary Archive Foundation. Royalty payments + must be paid within 60 days following each date on which you + prepare (or are legally required to prepare) your periodic tax + returns. Royalty payments should be clearly marked as such and + sent to the Project Gutenberg Literary Archive Foundation at the + address specified in Section 4, "Information about donations to + the Project Gutenberg Literary Archive Foundation." + +- You provide a full refund of any money paid by a user who notifies + you in writing (or by e-mail) within 30 days of receipt that s/he + does not agree to the terms of the full Project Gutenberg-tm + License. You must require such a user to return or + destroy all copies of the works possessed in a physical medium + and discontinue all use of and all access to other copies of + Project Gutenberg-tm works. + +- You provide, in accordance with paragraph 1.F.3, a full refund of any + money paid for a work or a replacement copy, if a defect in the + electronic work is discovered and reported to you within 90 days + of receipt of the work. + +- You comply with all other terms of this agreement for free + distribution of Project Gutenberg-tm works. + +1.E.9. If you wish to charge a fee or distribute a Project Gutenberg-tm +electronic work or group of works on different terms than are set +forth in this agreement, you must obtain permission in writing from +both the Project Gutenberg Literary Archive Foundation and Michael +Hart, the owner of the Project Gutenberg-tm trademark. Contact the +Foundation as set forth in Section 3 below. + +1.F. + +1.F.1. Project Gutenberg volunteers and employees expend considerable +effort to identify, do copyright research on, transcribe and proofread +public domain works in creating the Project Gutenberg-tm +collection. Despite these efforts, Project Gutenberg-tm electronic +works, and the medium on which they may be stored, may contain +"Defects," such as, but not limited to, incomplete, inaccurate or +corrupt data, transcription errors, a copyright or other intellectual +property infringement, a defective or damaged disk or other medium, a +computer virus, or computer codes that damage or cannot be read by +your equipment. + +1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right +of Replacement or Refund" described in paragraph 1.F.3, the Project +Gutenberg Literary Archive Foundation, the owner of the Project +Gutenberg-tm trademark, and any other party distributing a Project +Gutenberg-tm electronic work under this agreement, disclaim all +liability to you for damages, costs and expenses, including legal +fees. YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT +LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE +PROVIDED IN PARAGRAPH F3. YOU AGREE THAT THE FOUNDATION, THE +TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE +LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR +INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH +DAMAGE. + +1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a +defect in this electronic work within 90 days of receiving it, you can +receive a refund of the money (if any) you paid for it by sending a +written explanation to the person you received the work from. If you +received the work on a physical medium, you must return the medium with +your written explanation. The person or entity that provided you with +the defective work may elect to provide a replacement copy in lieu of a +refund. If you received the work electronically, the person or entity +providing it to you may choose to give you a second opportunity to +receive the work electronically in lieu of a refund. If the second copy +is also defective, you may demand a refund in writing without further +opportunities to fix the problem. + +1.F.4. Except for the limited right of replacement or refund set forth +in paragraph 1.F.3, this work is provided to you 'AS-IS', WITH NO OTHER +WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO +WARRANTIES OF MERCHANTIBILITY OR FITNESS FOR ANY PURPOSE. + +1.F.5. Some states do not allow disclaimers of certain implied +warranties or the exclusion or limitation of certain types of damages. +If any disclaimer or limitation set forth in this agreement violates the +law of the state applicable to this agreement, the agreement shall be +interpreted to make the maximum disclaimer or limitation permitted by +the applicable state law. The invalidity or unenforceability of any +provision of this agreement shall not void the remaining provisions. + +1.F.6. INDEMNITY - You agree to indemnify and hold the Foundation, the +trademark owner, any agent or employee of the Foundation, anyone +providing copies of Project Gutenberg-tm electronic works in accordance +with this agreement, and any volunteers associated with the production, +promotion and distribution of Project Gutenberg-tm electronic works, +harmless from all liability, costs and expenses, including legal fees, +that arise directly or indirectly from any of the following which you do +or cause to occur: (a) distribution of this or any Project Gutenberg-tm +work, (b) alteration, modification, or additions or deletions to any +Project Gutenberg-tm work, and (c) any Defect you cause. + + +Section 2. Information about the Mission of Project Gutenberg-tm + +Project Gutenberg-tm is synonymous with the free distribution of +electronic works in formats readable by the widest variety of computers +including obsolete, old, middle-aged and new computers. It exists +because of the efforts of hundreds of volunteers and donations from +people in all walks of life. + +Volunteers and financial support to provide volunteers with the +assistance they need, is critical to reaching Project Gutenberg-tm's +goals and ensuring that the Project Gutenberg-tm collection will +remain freely available for generations to come. In 2001, the Project +Gutenberg Literary Archive Foundation was created to provide a secure +and permanent future for Project Gutenberg-tm and future generations. +To learn more about the Project Gutenberg Literary Archive Foundation +and how your efforts and donations can help, see Sections 3 and 4 +and the Foundation web page at https://www.gutenberg.org/fundraising/pglaf. + + +Section 3. Information about the Project Gutenberg Literary Archive +Foundation + +The Project Gutenberg Literary Archive Foundation is a non profit +501(c)(3) educational corporation organized under the laws of the +state of Mississippi and granted tax exempt status by the Internal +Revenue Service. The Foundation's EIN or federal tax identification +number is 64-6221541. Contributions to the Project Gutenberg +Literary Archive Foundation are tax deductible to the full extent +permitted by U.S. federal laws and your state's laws. + +The Foundation's principal office is located at 4557 Melan Dr. S. +Fairbanks, AK, 99712., but its volunteers and employees are scattered +throughout numerous locations. Its business office is located at +809 North 1500 West, Salt Lake City, UT 84116, (801) 596-1887, email +business@pglaf.org. Email contact links and up to date contact +information can be found at the Foundation's web site and official +page at https://www.gutenberg.org/about/contact + +For additional contact information: + Dr. Gregory B. Newby + Chief Executive and Director + gbnewby@pglaf.org + +Section 4. Information about Donations to the Project Gutenberg +Literary Archive Foundation + +Project Gutenberg-tm depends upon and cannot survive without wide +spread public support and donations to carry out its mission of +increasing the number of public domain and licensed works that can be +freely distributed in machine readable form accessible by the widest +array of equipment including outdated equipment. Many small donations +($1 to $5,000) are particularly important to maintaining tax exempt +status with the IRS. + +The Foundation is committed to complying with the laws regulating +charities and charitable donations in all 50 states of the United +States. Compliance requirements are not uniform and it takes a +considerable effort, much paperwork and many fees to meet and keep up +with these requirements. We do not solicit donations in locations +where we have not received written confirmation of compliance. To +SEND DONATIONS or determine the status of compliance for any +particular state visit https://www.gutenberg.org/fundraising/donate + +While we cannot and do not solicit contributions from states where we +have not met the solicitation requirements, we know of no prohibition +against accepting unsolicited donations from donors in such states who +approach us with offers to donate. + +International donations are gratefully accepted, but we cannot make +any statements concerning tax treatment of donations received from +outside the United States. U.S. laws alone swamp our small staff. + +Please check the Project Gutenberg Web pages for current donation +methods and addresses. Donations are accepted in a number of other +ways including checks, online payments and credit card donations. +To donate, please visit: +https://www.gutenberg.org/fundraising/donate + + +Section 5. General Information About Project Gutenberg-tm electronic +works. + +Professor Michael S. Hart was the originator of the Project Gutenberg-tm +concept of a library of electronic works that could be freely shared +with anyone. For thirty years, he produced and distributed Project +Gutenberg-tm eBooks with only a loose network of volunteer support. + +Project Gutenberg-tm eBooks are often created from several printed +editions, all of which are confirmed as Public Domain in the U.S. +unless a copyright notice is included. Thus, we do not necessarily +keep eBooks in compliance with any particular paper edition. + +Most people start at our Web site which has the main PG search facility: + + https://www.gutenberg.org + +This Web site includes information about Project Gutenberg-tm, +including how to make donations to the Project Gutenberg Literary +Archive Foundation, how to help produce our new eBooks, and how to +subscribe to our email newsletter to hear about new eBooks. + |