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

THE THEORY AND PRACTICE OF PERSPECTIVE

by

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

Teacher of Perspective at the Royal Academy







[Illustration: 'QUÎ 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°                       64
   XXIII. The Cube at an Angle of 45°                         65
    XXIV. Pavements Drawn by Means of Squares at 45°          66
     XXV. The Perspective Vanishing Scale                     68
    XXVI. The Vanishing Scale can be Drawn to any Point
              on the Horizon                                  69
   XXVII. Application of Vanishing Scales to Drawing Figures  71
  XXVIII. How to Determine the Heights of Figures
              on a Level Plane                                71
    XXIX. The Horizon above the Figures                       72
     XXX. Landscape Perspective                               74
    XXXI. Figures of Different Heights. The Chessboard        74
   XXXII. Application of the Vanishing Scale to Drawing
              Figures at an Angle when their Vanishing
              Points are Inaccessible or Outside the Picture  77
  XXXIII. The Reduced Distance. How to Proceed when the
              Point of Distance is Inaccessible               77
   XXXIV. How to Draw a Long Passage or Cloister by Means
              of the Reduced Distance                         78
    XXXV. How to Form a Vanishing Scale that shall give
              the Height, Depth, and Distance of any Object
              in the Picture                                  79
   XXXVI. Measuring Scale on Ground                           81
  XXXVII. Application of the Reduced Distance and the
              Vanishing Scale to Drawing a Lighthouse, &c.    84
 XXXVIII. How to Measure Long Distances such as a Mile
              or Upwards                                      85
   XXXIX. Further Illustration of Long Distances and
              Extended Views.                                 87
      XL. How to Ascertain the Relative Heights of Figures
              on an Inclined Plane                            88
     XLI. How to Find the Distance of a Given Figure
              or Point from the Base Line                     89
    XLII. How to Measure the Height of Figures
              on Uneven Ground                                90
   XLIII. Further Illustration of the Size of Figures
              at Different Distances and on Uneven Ground     91
    XLIV. Figures on a Descending Plane                       92
     XLV. Further Illustration of the Descending Plane        95
    XLVI. Further Illustration of Uneven Ground               95
   XLVII. The Picture Standing on the Ground                  96
  XLVIII. The Picture on a Height                             97

BOOK III

    XLIX. Angular Perspective                                 98
       L. How to put a Given Point into Perspective           99
      LI. A Perspective Point being given, Find its
              Position on the Geometrical Plane              100
     LII. How to put a Given Line into Perspective           101
    LIII. To Find the Length of a Given Perspective Line     102
     LIV. To Find these Points when the Distance-Point
              is Inaccessible                                103
      LV. How to put a Given Triangle or other
              Rectilineal Figure into Perspective            104
     LVI. How to put a Given Square into Angular
              Perspective                                    105
    LVII. Of Measuring Points                                106
   LVIII. How to Divide any Given Straight Line into Equal
              or Proportionate Parts                         107
     LIX. How to Divide a Diagonal Vanishing Line into any
              Number of Equal or Proportional Parts          107
      LX. Further Use of the Measuring Point O               110
     LXI. Further Use of the Measuring Point O               110
    LXII. Another Method of Angular Perspective, being that
              Adopted in our Art Schools                     112
   LXIII. Two Methods of Angular Perspective in one Figure   115
    LXIV. To Draw a Cube, the Points being Given             115
     LXV. Amplification of the Cube Applied to Drawing
              a Cottage                                      116
    LXVI. How to Draw an Interior at an Angle                117
   LXVII. How to Correct Distorted Perspective by Doubling
              the Line of Distance                           118
  LXVIII. How to Draw a Cube on a Given Square, using only
              One Vanishing Point                            119
    LXIX. A Courtyard or Cloister Drawn with One Vanishing
              Point                                          120
     LXX. How to Draw Lines which shall Meet at a Distant
              Point, by Means of Diagonals                   121
    LXXI. How to Divide a Square Placed at an Angle into
              a Given Number of Small Squares                122
   LXXII. Further Example of how to Divide a Given Oblique
              Square into a Given Number of Equal Squares,
              say Twenty-five                                122
  LXXIII. Of Parallels and Diagonals                         124
   LXXIV. The Square, the Oblong, and their Diagonals        125
    LXXV. Showing the Use of the Square and Diagonals
              in Drawing Doorways, Windows, and other
              Architectural Features                         126
   LXXVI. How to Measure Depths by Diagonals                 127
  LXXVII. How to Measure Distances by the Square
              and Diagonal                                   128
 LXXVIII. How by Means of the Square and Diagonal we can
              Determine the Position of Points in Space      129
   LXXIX. Perspective of a Point Placed in any Position
              within the Square                              131
    LXXX. Perspective of a Square Placed at an Angle.
              New Method                                     133
   LXXXI. On a Given Line Placed at an Angle to the Base
              Draw a Square in Angular Perspective, the
              Point of Sight, and Distance, being given      134
  LXXXII. How to Draw Solid Figures at any Angle
              by the New Method                              135
 LXXXIII. Points in Space                                    137
  LXXXIV. The Square and Diagonal Applied to Cubes
              and Solids Drawn Therein                       138
   LXXXV. To Draw an Oblique Square in Another Oblique
              Square without Using Vanishing-points          139
  LXXXVI. Showing how a Pedestal can be Drawn
              by the New Method                              141
 LXXXVII. Scale on Each Side of the Picture                  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° 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 Dürer 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 Dürer, 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 _Traité 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° 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°, 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°, 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°, 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° 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° 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° 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°, 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°,
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° 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° 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 2½ 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 già fatto un gozzo in questo stento."_

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

At present that difficulty is got over by using large strong canvas, 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°


When the square is at an angle of 45° 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°


Having drawn the square at an angle of 45°, 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°


  [Illustration: Fig. 73.]

  [Illustration: Fig. 74.]

The square at 45° will be found of great use in drawing pavements,
roofs, ceilings, &c. In Figs. 73, 74 it is shown how having set out 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 ½ base to ½
distance. But even this ½ 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 ¼ base to ¼ 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 ¼D by drawing 3D from ¼ base to ¼ 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 ¼ 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 ¼ distance, and we
obtain at 440 on line _BS_ four times the length of _AB_, or 110 feet
× 4 = 440 feet. Again, taking the whole base and drawing a line from _S_
to 8th distance we obtain eight times 110 feet or 880 feet. If now 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°, 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°. So that in fixing upon the angle at which the
square or other figure is to be placed, we say 'let it be 60° and 30°,
or 70° and 20°', &c. Having decided upon the station-point and the 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° on one side, and 80° 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° 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 ¼,
1/3, ½. 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 ½D, 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 ½
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° 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°, double that at
_A_, which is 36°.

  [Illustration: Fig. 220.]




CXXI

THE PYRAMID


Nothing can be more simple than to put a pyramid into perspective. 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 ½ 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 ¼ 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 ¼ distance and ¼ 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° 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° 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° 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°
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° 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°
to meet _FL_ at _L_, which is the luminary at an altitude of 45°, and at
an angle of 45° 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° to the picture instead of 45°, is nearer
the point of sight; at 90° it would be exactly opposite the spectator,
and so on. Again, the elevation being less (40° instead of 45°) 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° 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° to the plane _RR_ then the reflection of that plane in
the mirror will be 45° 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° 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° to 60°, 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° as at _E_ (Fig. 298), or 45° 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° to the plane _mn_, and the plane itself
at about 15° 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°. 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° TO THE WALL


In Fig. 301 the mirror is vertical and at an angle of 45° 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 Dürer, 2, 9.
Angles of Reflection, 259 [CLXV].
Angular Perspective, 98 [XLIX] - 123 [LXXII], 133 [LXXX], 170.
   "         "       New Method, 133 [LXXX],
                       134 [LXXXI], 135 [LXXXII], 136 [LXXXIII].
Arches, Arcades, &c., 198 [CXXVI], 200 [CXXVII] - 208 [CXXIII].
Architect's Perspective, 170 [CVIII], 171 [197].
Art Schools Perspective, 112 [LXII] - 118 [LXVI], 217 [CXLI].
Atmosphere, 1, 74 [XXX].

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

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

_Index_
Dürer, Albert
  _umlaut missing_
Taddeo Gaddi
  _text reads "Tadeo"_
Titian
  _text reads Titien_