The Theory and Practice of Perspective

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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: 'QUI FIT?']

Oxford
At the Clarendon Press
1910

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

DEDICATED
to

SIR EDWARD J. POYNTER
Baronet

President of the Royal Academy

in Token of Friendship
and Regard

PREFACE

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

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

CONTENTS

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

BOOK II

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

BOOK III

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

BOOK IV

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

BOOK FIRST

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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