CXLVII. The Perspective of Cast Shadows 229
CXLVIII. The Two Kinds of Shadows 230
CXLIX. Shadows Cast by the Sun 232
CL. The Sun in the Same Plane as the Picture 233
CLI. The Sun Behind the Picture 234
CLII. Sun Behind the Picture, Shadows Thrown on a Wall 238
CLIII. Sun Behind the Picture Throwing Shadow on
an Inclined Plane 240
CLIV. The Sun in Front of the Picture 241
CLV. The Shadow of an Inclined Plane 244
CLVI. Shadow on a Roof or Inclined Plane 245
CLVII. To Find the Shadow of a Projection or Balcony
on a Wall 246
CLVIII. Shadow on a Retreating Wall, Sun in Front 247
CLIX. Shadow of an Arch, Sun in Front 249
CLX. Shadow in a Niche or Recess 250
CLXI. Shadow in an Arched Doorway 251
CLXII. Shadows Produced by Artificial Light 252
CLXIII. Some Observations on Real Light and Shade 253
CLXIV. Reflection 257
CLXV. Angles of Reflection 259
CLXVI. Reflections of Objects at Different Distances 260
CLXVII. Reflection in a Looking-glass 262
CLXVIII. The Mirror at an Angle 264
CLXIX. The Upright Mirror at an Angle of 45 deg to
the Wall 266
CLXX. Mental Perspective 269
BOOK FIRST
THE NECESSITY OF THE STUDY OF PERSPECTIVE
TO PAINTERS, SCULPTORS, AND ARCHITECTS
Leonardo da Vinci tells us in his celebrated _Treatise on Painting_ that
the young artist should first of all learn perspective, that is to say,
he should first of all learn that he has to depict on a flat surface
objects which are in relief or distant one from the other; for this is
the simple art of painting. Objects appear smaller at a distance than
near to us, so by drawing them thus we give depth to our canvas. The
outline of a ball is a mere flat circle, but with proper shading we make
it appear round, and this is the perspective of light and shade.
'The next thing to be considered is the effect of the atmosphere and
light. If two figures are in the same coloured dress, and are standing
one behind the other, then they should be of slightly different tone,
so as to separate them. And in like manner, according to the distance of
the mountains in a landscape and the greater or less density of the air,
so do we depict space between them, not only making them smaller in
outline, but less distinct.'[1]
[Footnote 1: Leonardo da Vinci's _Treatise on Painting_.]
Sir Edwin Landseer used to say that in looking at a figure in a picture
he liked to feel that he could walk round it, and this exactly expresses
the impression that the true art of painting should make upon the
spectator.
There is another observation of Leonardo's that it is well I should here
transcribe; he says: 'Many are desirous of learning to draw, and are
very fond of it, who are notwithstanding void of a proper disposition
for it. This may be known by their want of perseverance; like boys who
draw everything in a hurry, never finishing or shadowing.' This shows
they do not care for their work, and all instruction is thrown away upon
them. At the present time there is too much of this 'everything in a
hurry', and beginning in this way leads only to failure and
disappointment. These observations apply equally to perspective as to
drawing and painting.
Unfortunately, this study is too often neglected by our painters, some
of them even complacently confessing their ignorance of it; while the
ordinary student either turns from it with distaste, or only endures
going through it with a view to passing an examination, little thinking
of what value it will be to him in working out his pictures. Whether the
manner of teaching perspective is the cause of this dislike for it,
I cannot say; but certainly most of our English books on the subject are
anything but attractive.
All the great masters of painting have also been masters of perspective,
for they knew that without it, it would be impossible to carry out their
grand compositions. In many cases they were even inspired by it in
choosing their subjects. When one looks at those sunny interiors, those
corridors and courtyards by De Hooghe, with their figures far off and
near, one feels that their charm consists greatly in their perspective,
as well as in their light and tone and colour. Or if we study those
Venetian masterpieces by Paul Veronese, Titian, Tintoretto, and others,
we become convinced that it was through their knowledge of perspective
that they gave such space and grandeur to their canvases.
I need not name all the great artists who have shown their interest and
delight in this study, both by writing about it and practising it, such
as Albert Duerer and others, but I cannot leave out our own Turner, who
was one of the greatest masters in this respect that ever lived; though
in his case we can only judge of the results of his knowledge as shown
in his pictures, for although he was Professor of Perspective at the
Royal Academy in 1807--over a hundred years ago--and took great pains
with the diagrams he prepared to illustrate his lectures, they seemed to
the students to be full of confusion and obscurity; nor am I aware that
any record of them remains, although they must have contained some
valuable teaching, had their author possessed the art of conveying it.
However, we are here chiefly concerned with the necessity of this study,
and of the necessity of starting our work with it.
Before undertaking a large composition of figures, such as the
'Wedding-feast at Cana', by Paul Veronese, or 'The School of Athens',
by Raphael, the artist should set out his floors, his walls, his
colonnades, his balconies, his steps, &c., so that he may know where to
place his personages, and to measure their different sizes according to
their distances; indeed, he must make his stage and his scenery before
he introduces his actors. He can then proceed with his composition,
arrange his groups and the accessories with ease, and above all with
correctness. But I have noticed that some of our cleverest painters will
arrange their figures to please the eye, and when fairly advanced with
their work will call in an expert, to (as they call it) put in their
perspective for them, but as it does not form part of their original
composition, it involves all sorts of difficulties and vexatious
alterings and rubbings out, and even then is not always satisfactory.
For the expert may not be an artist, nor in sympathy with the picture,
hence there will be a want of unity in it; whereas the whole thing, to
be in harmony, should be the conception of one mind, and the perspective
as much a part of the composition as the figures.
If a ceiling has to be painted with figures floating or flying in the
air, or sitting high above us, then our perspective must take a
different form, and the point of sight will be above our heads instead
of on the horizon; nor can these difficulties be overcome without an
adequate knowledge of the science, which will enable us to work out for
ourselves any new problems of this kind that we may have to solve.
Then again, with a view to giving different effects or impressions in
this decorative work, we must know where to place the horizon and the
points of sight, for several of the latter are sometimes required when
dealing with large surfaces such as the painting of walls, or stage
scenery, or panoramas depicted on a cylindrical canvas and viewed from
the centre thereof, where a fresh point of sight is required at every
twelve or sixteen feet.
Without a true knowledge of perspective, none of these things can be
done. The artist should study them in the great compositions of the
masters, by analysing their pictures and seeing how and for what reasons
they applied their knowledge. Rubens put low horizons to most of his
large figure-subjects, as in 'The Descent from the Cross', which not
only gave grandeur to his designs, but, seeing they were to be placed
above the eye, gave a more natural appearance to his figures. The
Venetians often put the horizon almost on a level with the base of the
picture or edge of the frame, and sometimes even below it; as in 'The
Family of Darius at the Feet of Alexander', by Paul Veronese, and 'The
Origin of the "Via Lactea"', by Tintoretto, both in our National
Gallery. But in order to do all these things, the artist in designing
his work must have the knowledge of perspective at his fingers' ends,
and only the details, which are often tedious, should he leave to an
assistant to work out for him.
We must remember that the line of the horizon should be as nearly as
possible on a level with the eye, as it is in nature; and yet one of the
commonest mistakes in our exhibitions is the bad placing of this line.
We see dozens of examples of it, where in full-length portraits and
other large pictures intended to be seen from below, the horizon is
placed high up in the canvas instead of low down; the consequence is
that compositions so treated not only lose in grandeur and truth, but
appear to be toppling over, or give the impression of smallness rather
than bigness. Indeed, they look like small pictures enlarged, which is a
very different thing from a large design. So that, in order to see them
properly, we should mount a ladder to get upon a level with their
horizon line (see Fig. 66, double-page illustration).
We have here spoken in a general way of the importance of this study to
painters, but we shall see that it is of almost equal importance to the
sculptor and the architect.
A sculptor student at the Academy, who was making his drawings rather
carelessly, asked me of what use perspective was to a sculptor. 'In the
first place,' I said, 'to reason out apparently difficult problems, and
to find how easy they become, will improve your mind; and in the second,
if you have to do monumental work, it will teach you the exact size to
make your figures according to the height they are to be placed, and
also the boldness with which they should be treated to give them their
full effect.' He at once acknowledged that I was right, proved himself
an efficient pupil, and took much interest in his work.
I cannot help thinking that the reason our public monuments so often
fail to impress us with any sense of grandeur is in a great measure
owing to the neglect of the scientific study of perspective. As an
illustration of what I mean, let the student look at a good engraving or
photograph of the Arch of Constantine at Rome, or the Tombs of the
Medici, by Michelangelo, in the sacristy of San Lorenzo at Florence. And
then, for an example of a mistake in the placing of a colossal figure,
let him turn to the Tomb of Julius II in San Pietro in Vinculis, Rome,
and he will see that the figure of Moses, so grand in itself, not only
loses much of its dignity by being placed on the ground instead of in
the niche above it, but throws all the other figures out of proportion
or harmony, and was quite contrary to Michelangelo's intention. Indeed,
this tomb, which was to have been the finest thing of its kind ever
done, was really the tragedy of the great sculptor's life.
The same remarks apply in a great measure to the architect as to the
sculptor. The old builders knew the value of a knowledge of perspective,
and, as in the case of Serlio, Vignola, and others, prefaced their
treatises on architecture with chapters on geometry and perspective. For
it showed them how to give proper proportions to their buildings and the
details thereof; how to give height and importance both to the interior
and exterior; also to give the right sizes of windows, doorways,
columns, vaults, and other parts, and the various heights they should
make their towers, walls, arches, roofs, and so forth. One of the most
beautiful examples of the application of this knowledge to architecture
is the Campanile of the Cathedral, at Florence, built by Giotto and
Taddeo Gaddi, who were painters as well as architects. Here it will be
seen that the height of the windows is increased as they are placed
higher up in the building, and the top windows or openings into the
belfry are about six times the size of those in the lower story.
WHAT IS PERSPECTIVE?
[Illustration: Fig. 1.]
Perspective is a subtle form of geometry; it represents figures and
objects not as they are but as we see them in space, whereas geometry
represents figures not as we see them but as they are. When we have a
front view of a figure such as a square, its perspective and geometrical
appearance is the same, and we see it as it really is, that is, with all
its sides equal and all its angles right angles, the perspective only
varying in size according to the distance we are from it; but if we
place that square flat on the table and look at it sideways or at an
angle, then we become conscious of certain changes in its form--the side
farthest from us appears shorter than that near to us, and all the
angles are different. Thus A (Fig. 2) is a geometrical square and B is
the same square seen in perspective.
[Illustration: Fig. 2.]
[Illustration: Fig. 3.]
The science of perspective gives the dimensions of objects seen in space
as they appear to the eye of the spectator, just as a perfect tracing of
those objects on a sheet of glass placed vertically between him and them
would do; indeed its very name is derived from _perspicere_, to see
through. But as no tracing done by hand could possibly be mathematically
correct, the mathematician teaches us how by certain points and
measurements we may yet give a perfect image of them. These images are
called projections, but the artist calls them pictures. In this sketch
_K_ is the vertical transparent plane or picture, _O_ is a cube placed
on one side of it. The young student is the spectator on the other side
of it, the dotted lines drawn from the corners of the cube to the eye of
the spectator are the visual rays, and the points on the transparent
picture plane where these visual rays pass through it indicate the
perspective position of those points on the picture. To find these
points is the main object or duty of linear perspective.
Perspective up to a certain point is a pure science, not depending upon
the accidents of vision, but upon the exact laws of reasoning. Nor is it
to be considered as only pertaining to the craft of the painter and
draughtsman. It has an intimate connexion with our mental perceptions
and with the ideas that are impressed upon the brain by the appearance
of all that surrounds us. If we saw everything as depicted by plane
geometry, that is, as a map, we should have no difference of view, no
variety of ideas, and we should live in a world of unbearable monotony;
but as we see everything in perspective, which is infinite in its
variety of aspect, our minds are subjected to countless phases of
thought, making the world around us constantly interesting, so it is
devised that we shall see the infinite wherever we turn, and marvel at
it, and delight in it, although perhaps in many cases unconsciously.
[Illustration: Fig. 4.]
[Illustration: Fig. 5.]
In perspective, as in geometry, we deal with parallels, squares,
triangles, cubes, circles, &c.; but in perspective the same figure takes
an endless variety of forms, whereas in geometry it has but one. Here
are three equal geometrical squares: they are all alike. Here are three
equal perspective squares, but all varied in form; and the same figure
changes in aspect as often as we view it from a different position.
A walk round the dining-room table will exemplify this.
It is in proving that, notwithstanding this difference of appearance,
the figures do represent the same form, that much of our work consists;
and for those who care to exercise their reasoning powers it becomes not
only a sure means of knowledge, but a study of the greatest interest.
Perspective is said to have been formed into a science about the
fifteenth century. Among the names mentioned by the unknown but pleasant
author of _The Practice of Perspective_, written by a Jesuit of Paris
in the eighteenth century, we find Albert Duerer, who has left us some
rules and principles in the fourth book of his _Geometry_; Jean Cousin,
who has an express treatise on the art wherein are many valuable things;
also Vignola, who altered the plans of St. Peter's left by Michelangelo;
Serlio, whose treatise is one of the best I have seen of these early
writers; Du Cerceau, Serigati, Solomon de Cause, Marolois, Vredemont;
Guidus Ubaldus, who first introduced foreshortening; the Sieur de
Vaulizard, the Sieur Dufarges, Joshua Kirby, for whose _Method of
Perspective made Easy_ (?) Hogarth drew the well-known frontispiece; and
lastly, the above-named _Practice of Perspective_ by a Jesuit of Paris,
which is very clear and excellent as far as it goes, and was the book
used by Sir Joshua Reynolds.[2] But nearly all these authors treat
chiefly of parallel perspective, which they do with clearness and
simplicity, and also mathematically, as shown in the short treatise
in Latin by Christian Wolff, but they scarcely touch upon the more
difficult problems of angular and oblique perspective. Of modern
books, those to which I am most indebted are the _Traite' Pratique
de Perspective_ of M. A. Cassagne (Paris, 1873), which is thoroughly
artistic, and full of pictorial examples admirably done; and to
M. Henriet's _Cours Rational de Dessin_. There are many other foreign
books of excellence, notably M. Thibault's _Perspective_, and some
German and Swiss books, and yet, notwithstanding this imposing array of
authors, I venture to say that many new features and original problems
are presented in this book, whilst the old ones are not neglected. As,
for instance, How to draw figures at an angle without vanishing points
(see p. 141, Fig. 162, &c.), a new method of angular perspective which
dispenses with the cumbersome setting out usually adopted, and enables
us to draw figures at any angle without vanishing lines, &c., and is
almost, if not quite, as simple as parallel perspective (see p. 133,
Fig. 150, &c.). How to measure distances by the square and diagonal, and
to draw interiors thereby (p. 128, Fig. 144). How to explain the theory
of perspective by ocular demonstration, using a vertical sheet of glass
with strings, placed on a drawing-board, which I have found of the
greatest use (see p. 29, Fig. 29). Then again, I show how all our
perspective can be done inside the picture; that we can measure any
distance into the picture from a foot to a mile or twenty miles (see p.
86, Fig. 94); how we can draw the Great Pyramid, which stands on
thirteen acres of ground, by putting it 1,600 feet off (Fig. 224), &c.,
&c. And while preserving the mathematical science, so that all our
operations can be proved to be correct, my chief aim has been to make it
easy of application to our work and consequently useful to the artist.
[Footnote 2: There is another book called _The Jesuit's Perspective_
which I have not yet seen, but which I hear is a fine work.]
The Egyptians do not appear to have made any use of linear perspective.
Perhaps it was considered out of character with their particular kind of
decoration, which is to be looked upon as picture writing rather than
pictorial art; a table, for instance, would be represented like a
ground-plan and the objects upon it in elevation or standing up. A row
of chariots with their horses and drivers side by side were placed one
over the other, and although the Egyptians had no doubt a reason for
this kind of representation, for they were grand artists, it seems to us
very primitive; and indeed quite young beginners who have never drawn
from real objects have a tendency to do very much the same thing as this
ancient people did, or even to emulate the mathematician and represent
things not as they appear but as they are, and will make the top of a
table an almost upright square and the objects upon it as if they would
fall off.
No doubt the Greeks had correct notions of perspective, for the
paintings on vases, and at Pompeii and Herculaneum, which were either by
Greek artists or copied from Greek pictures, show some knowledge, though
not complete knowledge, of this science. Indeed, it is difficult to
conceive of any great artist making his perspective very wrong, for if
he can draw the human figure as the Greeks did, surely he can draw an
angle.
The Japanese, who are great observers of nature, seem to have got at
their perspective by copying what they saw, and, although they are not
quite correct in a few things, they convey the idea of distance and make
their horizontal planes look level, which are two important things in
perspective. Some of their landscapes are beautiful; their trees,
flowers, and foliage exquisitely drawn and arranged with the greatest
taste; whilst there is a character and go about their figures and birds,
&c., that can hardly be surpassed. All their pictures are lively and
intelligent and appear to be executed with ease, which shows their
authors to be complete masters of their craft.
The same may be said of the Chinese, although their perspective is more
decorative than true, and whilst their taste is exquisite their whole
art is much more conventional and traditional, and does not remind us of
nature like that of the Japanese.
We may see defects in the perspective of the ancients, in the mediaeval
painters, in the Japanese and Chinese, but are we always right
ourselves? Even in celebrated pictures by old and modern masters there
are occasionally errors that might easily have been avoided, if a ready
means of settling the difficulty were at hand. We should endeavour then
to make this study as simple, as easy, and as complete as possible, to
show clear evidence of its correctness (according to its conditions),
and at the same time to serve as a guide on any and all occasions that
we may require it.
To illustrate what is perspective, and as an experiment that any one can
make, whether artist or not, let us stand at a window that looks out on
to a courtyard or a street or a garden, &c., and trace with a
paint-brush charged with Indian ink or water-colour the outline of
whatever view there happens to be outside, being careful to keep the eye
always in the same place by means of a rest; when this is dry, place a
piece of drawing-paper over it and trace through with a pencil. Now we
will rub out the tracing on the glass, which is sure to be rather
clumsy, and, fixing our paper down on a board, proceed to draw the scene
before us, using the main lines of our tracing as our guiding lines.
If we take pains over our work, we shall find that, without troubling
ourselves much about rules, we have produced a perfect perspective of
perhaps a very difficult subject. After practising for some little time
in this way we shall get accustomed to what are called perspective
deformations, and soon be able to dispense with the glass and the
tracing altogether and to sketch straight from nature, taking little
note of perspective beyond fixing the point of sight and the
horizontal-line; in fact, doing what every artist does when he goes out
sketching.
[Illustration: Fig. 6.
This is a much reduced reproduction of a drawing made on my studio
window in this way some twenty years ago, when the builder started
covering the fields at the back with rows and rows of houses.]
THE THEORY OF PERSPECTIVE
DEFINITIONS
I
Fig. 7. In this figure, _AKB_ represents the picture or transparent
vertical plane through which the objects to be represented can be seen,
or on which they can be traced, such as the cube _C_.
[Illustration: Fig. 7.]
The line _HD_ is the +Horizontal-line+ or +Horizon+, the chief line in
perspective, as upon it are placed the principal points to which our
perspective lines are drawn. First, the +Point of Sight+ and next _D_,
the +Point of Distance+. The chief vanishing points and measuring points
are also placed on this line.
Another important line is _AB_, the +Base+ or +Ground line+, as it is on
this that we measure the width of any object to be represented, such as
_ef_, the base of the square _efgh_, on which the cube _C_ is raised.
_E_ is the position of the eye of the spectator, being drawn in
perspective, and is called the +Station-point+.
Note that the perspective of the board, and the line _SE_, is not the
same as that of the cube in the picture _AKB_, and also that so much of
the board which is behind the picture plane partially represents the
+Perspective-plane+, supposed to be perfectly level and to extend from
the base line to the horizon. Of this we shall speak further on. In
nature it is not really level, but partakes in extended views of the
rotundity of the earth, though in small areas such as ponds the
roundness is infinitesimal.
[Illustration: Fig. 8.]
Fig. 8. This is a side view of the previous figure, the picture plane
_K_ being represented edgeways, and the line _SE_ its full length.
It also shows the position of the eye in front of the point of sight
_S_. The horizontal-line _HD_ and the base or ground-line _AB_ are
represented as receding from us, and in that case are called vanishing
lines, a not quite satisfactory term.
It is to be noted that the cube _C_ is placed close to the transparent
picture plane, indeed touches it, and that the square _fj_ faces the
spectator _E_, and although here drawn in perspective it appears to him
as in the other figure. Also, it is at the same time a perspective and a
geometrical figure, and can therefore be measured with the compasses.
Or in other words, we can touch the square _fj_, because it is on the
surface of the picture, but we cannot touch the square _ghmb_ at the
other end of the cube and can only measure it by the rules of
perspective.
II
THE POINT OF SIGHT, THE HORIZON, AND THE POINT OF DISTANCE
There are three things to be considered and understood before we can
begin a perspective drawing. First, the position of the eye in front of
the picture, which is called the +Station-point+, and of course is not
in the picture itself, but its position is indicated by a point on the
picture which is exactly opposite the eye of the spectator, and is
called the +Point of Sight+, or +Principal Point+, or +Centre of
Vision+, but we will keep to the first of these.
[Illustration: Fig. 9.]
[Illustration: Fig. 10.]
If our picture plane is a sheet of glass, and is so placed that we can
see the landscape behind it or a sea-view, we shall find that the
distant line of the horizon passes through that point of sight, and we
therefore draw a line on our picture which exactly corresponds with it,
and which we call the +Horizontal-line+ or +Horizon+.[3] The height of
the horizon then depends entirely upon the position of the eye of the
spectator: if he rises, so does the horizon; if he stoops or descends to
lower ground, so does the horizon follow his movements. You may sit in a
boat on a calm sea, and the horizon will be as low down as you are, or
you may go to the top of a high cliff, and still the horizon will be on
the same level as your eye.
[Footnote 3: In a sea-view, owing to the rotundity of the earth, the
real horizontal line is slightly below the sea line, which is noted
in Chapter I.]
This is an important line for the draughtsman to consider, for the
effect of his picture greatly depends upon the position of the horizon.
If you wish to give height and dignity to a mountain or a building, the
horizon should be low down, so that these things may appear to tower
above you. If you wish to show a wide expanse of landscape, then you
must survey it from a height. In a composition of figures, you select
your horizon according to the subject, and with a view to help the
grouping. Again, in portraits and decorative work to be placed high up,
a low horizon is desirable, but I have already spoken of this subject in
the chapter on the necessity of the study of perspective.
III
POINT OF DISTANCE
Fig. 11. The distance of the spectator from the picture is of great
importance; as the distortions and disproportions arising from too near
a view are to be avoided, the object of drawing being to make things
look natural; thus, the floor should look level, and not as if it were
running up hill--the top of a table flat, and not on a slant, as if cups
and what not, placed upon it, would fall off.
In this figure we have a geometrical or ground plan of two squares at
different distances from the picture, which is represented by the line
_KK_. The spectator is first at _A_, the corner of the near square
_Acd_. If from _A_ we draw a diagonal of that square and produce it to
the line _KK_ (which may represent the horizontal-line in the picture),
where it intersects that line at _A'_ marks the distance that the
spectator is from the point of sight _S_. For it will be seen that line
_SA_ equals line _SA'_. In like manner, if the spectator is at _B_, his
distance from the point _S_ is also found on the horizon by means of the
diagonal _BB"_, so that all lines or diagonals at 45 deg are drawn to the
point of distance (see Rule 6).
Figs. 12 and 13. In these two figures the difference is shown between
the effect of the short-distance point _A'_ and the long-distance point
_B'_; the first, _Acd_, does not appear to lie so flat on the ground as
the second square, _Bef_.
From this it will be seen how important it is to choose the right point
of distance: if we take it too near the point of sight, as in Fig. 12,
the square looks unnatural and distorted. This, I may note, is a common
fault with photographs taken with a wide-angle lens, which throws
everything out of proportion, and will make the east end of a church or
a cathedral appear higher than the steeple or tower; but as soon as we
make our line of distance sufficiently long, as at Fig. 13, objects take
their right proportions and no distortion is noticeable.
[Illustration: Fig. 11.]
[Illustration: Fig. 12.]
[Illustration: Fig. 13.]
In some books on perspective we are told to make the angle of vision
60 deg, so that the distance _SD_ (Fig. 14) is to be rather less than the
length or height of the picture, as at _A_. The French recommend an
angle of 28 deg, and to make the distance about double the length of the
picture, as at _B_ (Fig. 15), which is far more agreeable. For we must
remember that the distance-point is not only the point from which we are
supposed to make our tracing on the vertical transparent plane, or a
point transferred to the horizon to make our measurements by, but it is
also the point in front of the canvas that we view the picture from,
called the station-point. It is ridiculous, then, to have it so close
that we must almost touch the canvas with our noses before we can see
its perspective properly.
[Illustration: Fig. 14.]
Now a picture should look right from whatever distance we view it, even
across the room or gallery, and of course in decorative work and in
scene-painting a long distance is necessary.
[Illustration: Fig. 15.]
We need not, however, tie ourselves down to any hard and fast rule, but
should choose our distance according to the impression of space we wish
to convey: if we have to represent a domestic scene in a small room, as
in many Dutch pictures, we must not make our distance-point too far off,
as it would exaggerate the size of the room.
[Illustration: Fig. 16. Cattle. By Paul Potter.]
The height of the horizon is also an important consideration in the
composition of a picture, and so also is the position of the point of
sight, as we shall see farther on.
In landscape and cattle pictures a low horizon often gives space and
air, as in this sketch from a picture by Paul Potter--where the
horizontal-line is placed at one quarter the height of the canvas.
Indeed, a judicious use of the laws of perspective is a great aid to
composition, and no picture ever looks right unless these laws are
attended to. At the present time too little attention is paid to them;
the consequence is that much of the art of the day reflects in a great
measure the monotony of the snap-shot camera, with its everyday and
wearisome commonplace.
IV
PERSPECTIVE OF A POINT, VISUAL RAYS, &C.
We perceive objects by means of the visual rays, which are imaginary
straight lines drawn from the eye to the various points of the thing we
are looking at. As those rays proceed from the pupil of the eye, which
is a circular opening, they form themselves into a cone called the
+Optic Cone+, the base of which increases in proportion to its distance
from the eye, so that the larger the view which we wish to take in, the
farther must we be removed from it. The diameter of the base of this
cone, with the visual rays drawn from each of its extremities to the
eye, form the angle of vision, which is wider or narrower according to
the distance of this diameter.
Now let us suppose a visual ray _EA_ to be directed to some small object
on the floor, say the head of a nail, _A_ (Fig. 17). If we interpose
between this nail and our eye a sheet of glass, _K_, placed vertically
on the floor, we continue to see the nail through the glass, and it is
easily understood that its perspective appearance thereon is the point
_a_, where the visual ray passes through it. If now we trace on the
floor a line _AB_ from the nail to the spot _B_, just under the eye, and
from the point _o_, where this line passes through or under the glass,
we raise a perpendicular _oS_, that perpendicular passes through the
precise point that the visual ray passes through. The line _AB_ traced
on the floor is the horizontal trace of the visual ray, and it will be
seen that the point _a_ is situated on the vertical raised from this
horizontal trace.
[Illustration: Fig. 17.]
V
TRACE AND PROJECTION
If from any line _A_ or _B_ or _C_ (Fig. 18), &c., we drop
perpendiculars from different points of those lines on to a horizontal
plane, the intersections of those verticals with the plane will be on
a line called the horizontal trace or projection of the original line.
We may liken these projections to sun-shadows when the sun is in the
meridian, for it will be remarked that the trace does not represent the
length of the original line, but only so much of it as would be embraced
by the verticals dropped from each end of it, and although line _A_ is
the same length as line _B_ its horizontal trace is longer than that of
the other; that the projection of a curve (_C_) in this upright position
is a straight line, that of a horizontal line (_D_) is equal to it, and
the projection of a perpendicular or vertical (_E_) is a point only.
The projections of lines or points can likewise be shown on a vertical
plane, but in that case we draw lines parallel to the horizontal plane,
and by this means we can get the position of a point in space; and by
the assistance of perspective, as will be shown farther on, we can carry
out the most difficult propositions of descriptive geometry and of the
geometry of planes and solids.
[Illustration: Fig. 18.]
The position of a point in space is given by its projection on a
vertical and a horizontal plane--
[Illustration: Fig. 19.]
Thus _e'_ is the projection of _E_ on the vertical plane _K_, and
_e''_ is the projection of _E_ on the horizontal plane; _fe''_ is the
horizontal trace of the plane _fE_, and _e'f_ is the trace of the same
plane on the vertical plane _K_.
VI
SCIENTIFIC DEFINITION OF PERSPECTIVE
The projections of the extremities of a right line which passes through
a vertical plane being given, one on either side of it, to find the
intersection of that line with the vertical plane. _AE_ (Fig. 20) is the
right line. The projection of its extremity _A_ on the vertical plane is
_a'_, the projection of _E_, the other extremity, is _e'_. _AS_ is the
horizontal trace of _AE_, and _a'e'_ is its trace on the vertical plane.
At point _f_, where the horizontal trace intersects the base _Bc_ of the
vertical plane, raise perpendicular _fP_ till it cuts _a'e'_ at point
_P_, which is the point required. For it is at the same time on the
given line _AE_ and the vertical plane _K_.
[Illustration: Fig. 20.]
This figure is similar to the previous one, except that the extremity
_A_ of the given line is raised from the ground, but the same
demonstration applies to it.
[Illustration: Fig. 21.]
And now let us suppose the vertical plane _K_ to be a sheet of glass,
and the given line _AE_ to be the visual ray passing from the eye to the
object _A_ on the other side of the glass. Then if _E_ is the eye of the
spectator, its projection on the picture is _S_, the point of sight.
If I draw a dotted line from _E_ to little _a_, this represents another
visual ray, and _o_, the point where it passes through the picture, is
the perspective of little _a_. I now draw another line from _g_ to _S_,
and thus form the shaded figure _ga'Po_, which is the perspective of
_aAa'g_.
Let it be remarked that in the shaded perspective figure the lines _a'P_
and _go_ are both drawn towards _S_, the point of sight, and that they
represent parallel lines _Aa'_ and _ag_, which are at right angles to
the picture plane. This is the most important fact in perspective, and
will be more fully explained farther on, when we speak of retreating or
so-called vanishing lines.
RULES
VII
THE RULES AND CONDITIONS OF PERSPECTIVE
The conditions of linear perspective are somewhat rigid. In the first
place, we are supposed to look at objects with one eye only; that is,
the visual rays are drawn from a single point, and not from two. Of this
we shall speak later on. Then again, the eye must be placed in a certain
position, as at _E_ (Fig. 22), at a given height from the ground, _S'E_,
and at a given distance from the picture, as _SE_. In the next place,
the picture or picture plane itself must be vertical and perpendicular
to the ground or horizontal plane, which plane is supposed to be as
level as a billiard-table, and to extend from the base line, _ef_,
of the picture to the horizon, that is, to infinity, for it does not
partake of the rotundity of the earth.
We can only work out our propositions and figures in space with
mathematical precision by adopting such conditions as the above. But
afterwards the artist or draughtsman may modify and suit them to a more
elastic view of things; that is, he can make his figures separate from
one another, instead of their outlines coming close together as they do
when we look at them with only one eye. Also he will allow for the
unevenness of the ground and the roundness of our globe; he may even
move his head and his eyes, and use both of them, and in fact make
himself quite at his ease when he is out sketching, for Nature does all
his perspective for him. At the same time, a knowledge of this rigid
perspective is the sure and unerring basis of his freehand drawing.
[Illustration: Fig. 22.]
[Illustration: Fig. 23. Front view of above figure.]
RULE 1
All straight lines remain straight in their perspective appearance.[4]
[Footnote 4: Some will tell us that Nature abhors a straight line,
that all long straight lines in space appear curved, &c., owing to
certain optical conditions; but this is not apparent in short straight
lines, so if our drawing is small it would be wrong to curve them; if
it is large, like a scene or diorama, the same optical condition which
applies to the line in space would also apply to the line in the
picture.]
RULE 2
Vertical lines remain vertical in perspective, and are divided in the
same proportion as _AB_ (Fig. 24), the original line, and _a'b'_, the
perspective line, and if the one is divided at _O_ the other is divided
at _o'_ in the same way.
[Illustration: Fig. 24.]
It is not an uncommon error to suppose that the vertical lines of a high
building should converge towards the top; so they would if we stood at
the foot of that building and looked up, for then we should alter the
conditions of our perspective, and our point of sight, instead of being
on the horizon, would be up in the sky. But if we stood sufficiently far
away, so as to bring the whole of the building within our angle of
vision, and the point of sight down to the horizon, then these same
lines would appear perfectly parallel, and the different stories in
their true proportion.
RULE 3
Horizontals parallel to the base of the picture are also parallel to
that base in the picture. Thus _a'b'_ (Fig. 25) is parallel to _AB_, and
to _GL_, the base of the picture. Indeed, the same argument may be used
with regard to horizontal lines as with verticals. If we look at a
straight wall in front of us, its top and its rows of bricks, &c., are
parallel and horizontal; but if we look along it sideways, then we alter
the conditions, and the parallel lines converge to whichever point we
direct the eye.
[Illustration: Fig. 25.]
[Illustration: Fig. 26.]
This rule is important, as we shall see when we come to the
consideration of the perspective vanishing scale. Its use may be
illustrated by this sketch, where the houses, walls, &c., are parallel
to the base of the picture. When that is the case, then objects exactly
facing us, such as windows, doors, rows of boards, or of bricks or
palings, &c., are drawn with their horizontal lines parallel to the
base; hence it is called parallel perspective.
RULE 4
All lines situated in a plane that is parallel to the picture plane
diminish in proportion as they become more distant, but do not undergo
any perspective deformation; and remain in the same relation and
proportion each to each as the original lines. This is called the front
view.
[Illustration: Fig. 27.]
RULE 5
All horizontals which are at right angles to the picture plane are drawn
to the point of sight.
Thus the lines _AB_ and _CD_ (Fig. 28) are horizontal or parallel to the
ground plane, and are also at right angles to the picture plane _K_. It
will be seen that the perspective lines _Ba'_, _Dc'_, must, according to
the laws of projection, be drawn to the point of sight.
This is the most important rule in perspective (see Fig. 7 at beginning
of Definitions).
An arrangement such as there indicated is the best means of illustrating
this rule. But instead of tracing the outline of the square or cube on
the glass, as there shown, I have a hole drilled through at the point
_S_ (Fig. 29), which I select for the point of sight, and through which
I pass two loose strings _A_ and _B_, fixing their ends at _S_.
[Illustration: Fig. 28.]
[Illustration: Fig. 29.]
As _SD_ represents the distance the spectator is from the glass or
picture, I make string _SA_ equal in length to _SD_. Now if the pupil
takes this string in one hand and holds it at right angles to the glass,
that is, exactly in front of _S_, and then places one eye at the end _A_
(of course with the string extended), he will be at the proper distance
from the picture. Let him then take the other string, _SB_, in the other
hand, and apply it to point _b"_ where the square touches the glass, and
he will find that it exactly tallies with the side _b"f_ of the square
_a'b"fe_. If he applies the same string to _a'_, the other corner of the
square, his string will exactly tally or cover the side _a'e_, and he
will thus have ocular demonstration of this important rule.
In this little picture (Fig. 30) in parallel perspective it will be seen
that the lines which retreat from us at right angles to the picture
plane are directed to the point of sight _S_.
[Illustration: Fig. 30.]
RULE 6
All horizontals which are at 45 deg, or half a right angle to the picture
plane, are drawn to the point of distance.
We have already seen that the diagonal of the perspective square, if
produced to meet the horizon on the picture, will mark on that horizon
the distance that the spectator is from the point of sight (see
definition, p. 16). This point of distance becomes then the measuring
point for all horizontals at right angles to the picture plane.
Thus in Fig. 31 lines _AS_ and _BS_ are drawn to the point of sight _S_,
and are therefore at right angles to the base _AB_. _AD_ being drawn to
_D_ (the distance-point), is at an angle of 45 deg to the base _AB_, and
_AC_ is therefore the diagonal of a square. The line 1C is made
parallel to _AB_, consequently A1CB is a square in perspective. The
line _BC_, therefore, being one side of that square, is equal to _AB_,
another side of it. So that to measure a length on a line drawn to the
point of sight, such as _BS_, we set out the length required, say _BA_,
on the base-line, then from _A_ draw a line to the point of distance,
and where it cuts _BS_ at _C_ is the length required. This can be
repeated any number of times, say five, so that in this figure _BE_
is five times the length of _AB_.
[Illustration: Fig. 31.]
RULE 7
All horizontals forming any other angles but the above are drawn to some
other points on the horizontal line. If the angle is greater than half a
right angle (Fig. 32), as _EBG_, the point is within the point of
distance, as at _V"_. If it is less, as _ABV""_, then it is beyond the
point of distance, and consequently farther from the point of sight.
[Illustration: Fig. 32.]
In Fig. 32, the dotted line _BD_, drawn to the point of distance _D_,
is at an angle of 45 deg to the base _AG_. It will be seen that the line
_BV"_ is at a greater angle to the base than _BD_; it is therefore drawn
to a point _V"_, within the point of distance and nearer to the point of
sight _S_. On the other hand, the line _BV""_ is at a more acute angle,
and is therefore drawn to a point some way beyond the other distance
point.
_Note._--When this vanishing point is a long way outside the picture,
the architects make use of a centrolinead, and the painters fix a long
string at the required point, and get their perspective lines by that
means, which is very inconvenient. But I will show you later on how you
can dispense with this trouble by a very simple means, with equally
correct results.
RULE 8
Lines which incline upwards have their vanishing points above the
horizontal line, and those which incline downwards, below it. In both
cases they are on the vertical which passes through the vanishing point
(_S_) of their horizontal projections.
[Illustration: Fig. 33.]
This rule is useful in drawing steps, or roads going uphill and
downhill.
[Illustration: Fig. 34.]
RULE 9
The farther a point is removed from the picture plane the nearer does
its perspective appearance approach the horizontal line so long as it is
viewed from the same position. On the contrary, if the spectator
retreats from the picture plane _K_ (which we suppose to be
transparent), the point remaining at the same place, the perspective
appearance of this point will approach the ground-line in proportion to
the distance of the spectator.
[Illustrations:
Fig. 35.
Fig. 36.
The spectator at two different distances from the picture.]
Therefore the position of a given point in perspective above the
ground-line or below the horizon is in proportion to the distance of the
spectator from the picture, or the picture from the point.
[Illustration: Fig. 37.]
[Illustrations:
The picture at two different distances from the point.
Fig. 38.
Fig. 39.]
Figures 38 and 39 are two views of the same gallery from different
distances. In Fig. 38, where the distance is too short, there is a want
of proportion between the near and far objects, which is corrected in
Fig. 39 by taking a much longer distance.
RULE 10
Horizontals in the same plane which are drawn to the same point on the
horizon are parallel to each other.
[Illustration: Fig. 40.]
This is a very important rule, for all our perspective drawing depends
upon it. When we say that parallels are drawn to the same point on the
horizon it does not imply that they meet at that point, which would be a
contradiction; perspective parallels never reach that point, although
they appear to do so. Fig. 40 will explain this.
Suppose _S_ to be the spectator, _AB_ a transparent vertical plane which
represents the picture seen edgeways, and _HS_ and _DC_ two parallel
lines, mark off spaces between these parallels equal to _SC_, the height
of the eye of the spectator, and raise verticals 2, 3, 4, 5, &c.,
forming so many squares. Vertical line 2 viewed from _S_ will appear on
_AB_ but half its length, vertical 3 will be only a third, vertical 4 a
fourth, and so on, and if we multiplied these spaces _ad infinitum_ we
must keep on dividing the line _AB_ by the same number. So if we suppose
_AB_ to be a yard high and the distance from one vertical to another to
be also a yard, then if one of these were a thousand yards away its
representation at _AB_ would be the thousandth part of a yard, or ten
thousand yards away, its representation at _AB_ would be the
ten-thousandth part, and whatever the distance it must always be
something; and therefore _HS_ and _DC_, however far they may be produced
and however close they may appear to get, can never meet.
[Illustration: Fig. 41.]
Fig. 41 is a perspective view of the same figure--but more extended. It
will be seen that a line drawn from the tenth upright _K_ to _S_ cuts
off a tenth of _AB_. We look then upon these two lines _SP_, _OP_, as
the sides of a long parallelogram of which _SK_ is the diagonal, as
_cefd_, the figure on the ground, is also a parallelogram.
The student can obtain for himself a further illustration of this rule
by placing a looking-glass on one of the walls of his studio and then
sketching himself and his surroundings as seen therein. He will find
that all the horizontals at right angles to the glass will converge to
his own eye. This rule applies equally to lines which are at an angle to
the picture plane as to those that are at right angles or perpendicular
to it, as in Rule 7. It also applies to those on an inclined plane, as
in Rule 8.
[Illustration: Fig. 42. Sketch of artist in studio.]
With the above rules and a clear notion of the definitions and
conditions of perspective, we should be able to work out any proposition
or any new figure that may present itself. At any rate, a thorough
understanding of these few pages will make the labour now before us
simple and easy. I hope, too, it may be found interesting. There is
always a certain pleasure in deceiving and being deceived by the senses,
and in optical and other illusions, such as making things appear far off
that are quite near, in making a picture of an object on a flat surface
to look as if it stood out and in relief by a kind of magic. But there
is, I think, a still greater pleasure than this, namely, in invention
and in overcoming difficulties--in finding out how to do things for
ourselves by our reasoning faculties, in originating or being original,
as it were. Let us now see how far we can go in this respect.
VIII
A TABLE OR INDEX OF THE RULES OF PERSPECTIVE
The rules here set down have been fully explained in the previous pages,
and this table is simply for the student's ready reference.
RULE 1
All straight lines remain straight in their perspective appearance.
RULE 2
Vertical lines remain vertical in perspective.
RULE 3
Horizontals parallel to the base of the picture are also parallel to
that base in the picture.
RULE 4
All lines situated in a plane that is parallel to the picture plane
diminish in proportion as they become more distant, but do not undergo
any perspective deformation. This is called the front view.
RULE 5
All horizontal lines which are at right angles to the picture plane are
drawn to the point of sight.
RULE 6
All horizontals which are at 45 deg to the picture plane are drawn to the
point of distance.
RULE 7
All horizontals forming any other angles but the above are drawn to some
other points on the horizontal line.
RULE 8
Lines which incline upwards have their vanishing points above the
horizon, and those which incline downwards, below it. In both cases they
are on the vertical which passes through the vanishing point of their
ground-plan or horizontal projections.
RULE 9
The farther a point is removed from the picture plane the nearer does it
appear to approach the horizon, so long as it is viewed from the same
position.
RULE 10
Horizontals in the same plane which are drawn to the same point on the
horizon are perspectively parallel to each other.
BOOK SECOND
THE PRACTICE OF PERSPECTIVE
In the foregoing book we have explained the theory or science of
perspective; we now have to make use of our knowledge and to apply it to
the drawing of figures and the various objects that we wish to depict.
The first of these will be a square with two of its sides parallel to
the picture plane and the other two at right angles to it, and which we
call
IX
THE SQUARE IN PARALLEL PERSPECTIVE
From a given point on the base line of the picture draw a line at right
angles to that base. Let _P_ be the given point on the base line _AB_,
and _S_ the point of sight. We simply draw a line along the ground to
the point of sight _S_, and this line will be at right angles to the
base, as explained in Rule 5, and consequently angle _APS_ will be equal
to angle _SPB_, although it does not look so here. This is our first
difficulty, but one that we shall soon get over.
[Illustration: Fig. 43.]
In like manner we can draw any number of lines at right angles to the
base, or we may suppose the point _P_ to be placed at so many different
positions, our only difficulty being to conceive these lines to be
parallel to each other. See Rule 10.
[Illustration: Fig. 44.]
X
THE DIAGONAL
From a given point on the base line draw a line at 45 deg, or half a
right angle, to that base. Let _P_ be the given point. Draw a line from
_P_ to the point of distance _D_ and this line _PD_ will be at an angle
of 45 deg, or at the same angle as the diagonal of a square. See
definitions.
[Illustration: Fig. 45.]
XI
THE SQUARE
Draw a square in parallel perspective on a given length on the base
line. Let _ab_ be the given length. From its two extremities _a_ and _b_
draw _aS_ and _bS_ to the point of sight _S_. These two lines will be at
right angles to the base (see Fig. 43). From _a_ draw diagonal _aD_ to
point of distance _D_; this line will be 45 deg to base. At point _c_,
where it cuts _bS_, draw _dc_ parallel to _ab_ and _abcd_ is the square
required.
[Illustration: Fig. 46.]
We have here proceeded in much the same way as in drawing a geometrical
square (Fig. 47), by drawing two lines _AE_ and _BC_ at right angles to
a given line, _AB_, and from _A_, drawing the diagonal _AC_ at 45 deg
till it cuts _BC_ at _C_, and then through _C_ drawing _EC_ parallel
to _AB_. Let it be remarked that because the two perspective lines
(Fig. 48) _AS_ and _BS_ are at right angles to the base, they must
consequently be parallel to each other, and therefore are perspectively
equidistant, so that all lines parallel to _AB_ and lying between them,
such as _ad_, _cf_, &c., must be equal.
[Illustration: Fig. 47.]
So likewise all diagonals drawn to the point of distance, which are
contained between these parallels, such as _Ad_, _af_, &c., must be
equal. For all straight lines which meet at any point on the horizon are
perspectively parallel to each other, just as two geometrical parallels
crossing two others at any angle, as at Fig. 49. Note also (Fig. 48)
that all squares formed between the two vanishing lines _AS_, _BS_, and
by the aid of these diagonals, are also equal, and further, that any
number of squares such as are shown in this figure (Fig. 50), formed in
the same way and having equal bases, are also equal; and the nine
squares contained in the square _abcd_ being equal, they divide each
side of the larger square into three equal parts.
[Illustration: Fig. 48.]
[Illustration: Fig. 49.]
From this we learn how we can measure any number of given lengths,
either equal or unequal, on a vanishing or retreating line which is at
right angles to the base; and also how we can measure any width or
number of widths on a line such as _dc_, that is, parallel to the base
of the picture, however remote it may be from that base.
[Illustration: Fig. 50.]
XII
GEOMETRICAL AND PERSPECTIVE FIGURES CONTRASTED
As at first there may be a little difficulty in realizing the
resemblance between geometrical and perspective figures, and also about
certain expressions we make use of, such as horizontals, perpendiculars,
parallels, &c., which look quite different in perspective, I will here
make a note of them and also place side by side the two views of the
same figures.
[Illustration: Fig. 51 A. The geometrical view.]
[Illustration: Fig. 51 B. The perspective view.]
[Illustration: Fig. 51 C. A geometrical square.]
[Illustration: Fig. 51 D. A perspective square.]
[Illustration: Fig. 51 E. Geometrical parallels.]
[Illustration: Fig. 51 F. Perspective parallels.]
[Illustration: Fig. 51 G. Geometrical perpendicular.]
[Illustration: Fig. 51 H. Perspective perpendicular.]
[Illustration: Fig. 51 I. Geometrical equal lines.]
[Illustration: Fig. 51 J. Perspective equal lines.]
[Illustration: Fig. 51 K. A geometrical circle.]
[Illustration: Fig. 51 L. A perspective circle.]
XIII
OF CERTAIN TERMS MADE USE OF IN PERSPECTIVE
Of course when we speak of +Perpendiculars+ we do not mean verticals
only, but straight lines at right angles to other lines in any position.
Also in speaking of +lines+ a right or +straight line+ is to be
understood; or when we speak of +horizontals+ we mean all straight lines
that are parallel to the perspective plane, such as those on Fig. 52, no
matter what direction they take so long as they are level. They are not
to be confused with the horizon or horizontal-line.
[Illustration: Fig. 52. Horizontals.]
There are one or two other terms used in perspective which are not
satisfactory because they are confusing, such as vanishing lines and
vanishing points. The French term, _fuyante_ or _lignes fuyantes_, or
going-away lines, is more expressive; and _point de fuite_, instead of
vanishing point, is much better. I have occasionally called the former
retreating lines, but the simple meaning is, lines that are not parallel
to the picture plane; but a vanishing line implies a line that
disappears, and a vanishing point implies a point that gradually goes
out of sight. Still, it is difficult to alter terms that custom has
endorsed. All we can do is to use as few of them as possible.
XIV
HOW TO MEASURE VANISHING OR RECEDING LINES
Divide a vanishing line which is at right angles to the picture plane
into any number of given measurements. Let _SA_ be the given line. From
_A_ measure off on the base line the divisions required, say five of
1 foot each; from each division draw diagonals to point of distance _D_,
and where these intersect the line _AC_ the corresponding divisions will
be found. Note that as lines _AB_ and _AC_ are two sides of the same
square they are necessarily equal, and so also are the divisions on _AC_
equal to those on _AB_.
[Illustration: Fig. 53.]
The line _AB_ being the base of the picture, it is at the same time a
perspective line and a geometrical one, so that we can use it as a scale
for measuring given lengths thereon, but should there not be enough room
on it to measure the required number we draw a second line, _DC_, which
we divide in the same proportion and proceed to divide _cf_. This
geometrical figure gives, as it were, a bird's-eye view or ground-plan
of the above.
[Illustration: Fig. 54.]
XV
HOW TO PLACE SQUARES IN GIVEN POSITIONS
Draw squares of given dimensions at given distances from the base line
to the right or left of the vertical line, which passes through the
point of sight.
[Illustration: Fig. 55.]
Let _ab_ (Fig. 55) represent the base line of the picture divided into a
certain number of feet; _HD_ the horizon, _VO_ the vertical. It is
required to draw a square 3 feet wide, 2 feet to the right of the
vertical, and 1 foot from the base.
First measure from _V_, 2 feet to _e_, which gives the distance from the
vertical. Second, from _e_ measure 3 feet to _b_, which gives the width
of the square; from _e_ and _b_ draw _eS_, _bS_, to point of sight. From
either _e_ or _b_ measure 1 foot to the left, to _f_ or _f'_. Draw _fD_
to point of distance, which intersects _eS_ at _P_, and gives the
required distance from base. Draw _Pg_ and _B_ parallel to the base, and
we have the required square.
Square _A_ to the left of the vertical is 2-1/2 feet wide, 1 foot from
the vertical and 2 feet from the base, and is worked out in the same
way.
_Note._--It is necessary to know how to work to scale, especially in
architectural drawing, where it is indispensable, but in working out our
propositions and figures it is not always desirable. A given length
indicated by a line is generally sufficient for our requirements. To
work out every problem to scale is not only tedious and mechanical, but
wastes time, and also takes the mind of the student away from the
reasoning out of the subject.
XVI
HOW TO DRAW PAVEMENTS, &C.
Divide a vanishing line into parts varying in length. Let _BS'_ be the
vanishing line: divide it into 4 long and 3 short spaces; then proceed
as in the previous figure. If we draw horizontals through the points
thus obtained and from these raise verticals, we form, as it were, the
interior of a building in which we can place pillars and other objects.
[Illustration: Fig. 56.]
Or we can simply draw the plan of the pavement as in this figure.
[Illustration: Fig. 57.]
[Illustration: Fig. 58.]
And then put it into perspective.
XVII
OF SQUARES PLACED VERTICALLY AND AT DIFFERENT HEIGHTS,
OR THE CUBE IN PARALLEL PERSPECTIVE
On a given square raise a cube.
[Illustration: Fig. 59.]
_ABCD_ is the given square; from _A_ and _B_ raise verticals _AE_, _BF_,
equal to _AB_; join _EF_. Draw _ES_, _FS_, to point of sight; from _C_
and _D_ raise verticals _CG_, _DH_, till they meet vanishing lines _ES_,
_FS_, in _G_ and _H_, and the cube is complete.
XVIII
THE TRANSPOSED DISTANCE
The transposed distance is a point _D'_ on the vertical _VD'_, at
exactly the same distance from the point of sight as is the point of
distance on the horizontal line.
It will be seen by examining this figure that the diagonals of the
squares in a vertical position are drawn to this vertical
distance-point, thus saving the necessity of taking the measurements
first on the base line, as at _CB_, which in the case of distant
objects, such as the farthest window, would be very inconvenient. Note
that the windows at _K_ are twice as high as they are wide. Of course
these or any other objects could be made of any proportion.
[Illustration: Fig. 60.]
XIX
THE FRONT VIEW OF THE SQUARE AND OF THE PROPORTIONS OF FIGURES
AT DIFFERENT HEIGHTS
According to Rule 4, all lines situated in a plane parallel to the
picture plane diminish in length as they become more distant, but remain
in the same proportions each to each as the original lines; as squares
or any other figures retain the same form. Take the two squares _ABCD_,
_abcd_ (Fig. 61), one inside the other; although moved back from square
_EFGH_ they retain the same form. So in dealing with figures of
different heights, such as statuary or ornament in a building, if
actually equal in size, so must we represent them.
[Illustration: Fig. 61.]
[Illustration: Fig. 62.]
In this square _K_, with the checker pattern, we should not think of
making the top squares smaller than the bottom ones; so it is with
figures.
This subject requires careful study, for, as pointed out in our opening
chapter, there are certain conditions under which we have to modify and
greatly alter this rule in large decorative work.
[Illustration: Fig. 63.]
In Fig. 63 the two statues _A_ and _B_ are the same size. So if traced
through a vertical sheet of glass, _K_, as at _c_ and _d_, they would
also be equal; but as the angle _b_ at which the upper one is seen is
smaller than angle _a_, at which the lower figure or statue is seen, it
will appear smaller to the spectator (_S_) both in reality and in the
picture.
[Illustration: Fig. 64.]
But if we wish them to appear the same size to the spectator who is
viewing them from below, we must make the angles _a_ and _b_ (Fig. 64),
at which they are viewed, both equal. Then draw lines through equal
arcs, as at _c_ and _d_, till they cut the vertical _NO_ (representing
the side of the building where the figures are to be placed). We shall
then obtain the exact size of the figure at that height, which will make
it look the same size as the lower one, _N_. The same rule applies to
the picture _K_, when it is of large proportions. As an example in
painting, take Michelangelo's large altar-piece in the Sistine Chapel,
'The Last Judgement'; here the figures forming the upper group, with our
Lord in judgement surrounded by saints, are about four times the size,
that is, about twice the height, of those at the lower part of the
fresco. The figures on the ceiling of the same chapel are studied not
only according to their height from the pavement, which is 60 ft., but
to suit the arched form of it. For instance, the head of the figure of
Jonah at the end over the altar is thrown back in the design, but owing
to the curvature in the architecture is actually more forward than the
feet. Then again, the prophets and sybils seated round the ceiling,
which are perhaps the grandest figures in the whole range of art, would
be 18 ft. high if they stood up; these, too, are not on a flat surface,
so that it required great knowledge to give them their right effect.
[Illustration: Fig. 65.]
Of course, much depends upon the distance we view these statues or
paintings from. In interiors, such as churches, halls, galleries, &c.,
we can make a fair calculation, such as the length of the nave, if the
picture is an altar-piece--or say, half the length; so also with
statuary in niches, friezes, and other architectural ornaments. The
nearer we are to them, and the more we have to look up, the larger will
the upper figures have to be; but if these are on the outside of a
building that can be looked at from a long distance, then it is better
not to have too great a difference.
[Illustration: Fig. 66. 1909.]
These remarks apply also to architecture in a great measure. Buildings
that can only be seen from the street below, as pictures in a narrow
gallery, require a different treatment from those out in the open, that
are to be looked at from a distance. In the former case the same
treatment as the Campanile at Florence is in some cases desirable, but
all must depend upon the taste and judgement of the architect in such
matters. All I venture to do here is to call attention to the subject,
which seems as a rule to be ignored, or not to be considered of
importance. Hence the many mistakes in our buildings, and the
unsatisfactory and mean look of some of our public monuments.
XX
OF PICTURES THAT ARE PAINTED ACCORDING TO THE POSITION
THEY ARE TO OCCUPY
In this double-page illustration of the wall of a picture-gallery,
I have, as it were, hung the pictures in accordance with the style in
which they are painted and the perspective adopted by their painters. It
will be seen that those placed on the line level with the eye have their
horizon lines fairly high up, and are not suited to be placed any
higher. The Giorgione in the centre, the Monna Lisa to the right, and
the Velasquez and Watteau to the left, are all pictures that fit that
position; whereas the grander compositions above them are so designed,
and are so large in conception, that we gain in looking up to them.
Note how grandly the young prince on his pony, by Velasquez, tells out
against the sky, with its low horizon and strong contrast of light and
dark; nor does it lose a bit by being placed where it is, over the
smaller pictures.
The Rembrandt, on the opposite side, with its burgomasters in black hats
and coats and white collars, is evidently intended and painted for a
raised position, and to be looked up to, which is evident from the
perspective of the table. The grand Titian in the centre, an altar-piece
in one of the churches in Venice (here reversed), is also painted to
suit its elevated position, with low horizon and figures telling boldly
against the sky. Those placed low down are modern French pictures, with
the horizon high up and almost above their frames, but placed on the
ground they fit into the general harmony of the arrangement.
It seems to me it is well, both for those who paint and for those who
hang pictures, that this subject should be taken into consideration. For
it must be seen by this illustration that a bigger style is adopted by
the artists who paint for high places in palaces or churches than by
those who produce smaller easel-pictures intended to be seen close.
Unfortunately, at our picture exhibitions, we see too often that nearly
all the works, whether on large or small canvases, are painted for the
line, and that those which happen to get high up look as if they were
toppling over, because they have such a high horizontal line; and
instead of the figures telling against the sky, as in this picture of
the 'Infant' by Velasquez, the Reynolds, and the fat man treading on a
flag, we have fields or sea or distant landscape almost to the top of
the frame, and all, so methinks, because the perspective is not
sufficiently considered.
_Note._--Whilst on this subject, I may note that the painter in his
large decorative work often had difficulties to contend with, which
arose from the form of the building or the shape of the wall on which he
had to place his frescoes. Painting on the ceiling was no easy task, and
Michelangelo, in a humorous sonnet addressed to Giovanni da Pistoya,
gives a burlesque portrait of himself while he was painting the Sistine
Chapel:--
_"I'ho gia' fatto un gozzo in questo stento."_
Now have I such a goitre 'neath my chin
That I am like to some Lombardic cat,
My beard is in the air, my head i' my back,
My chest like any harpy's, and my face
Patched like a carpet by my dripping brush.
Nor can I see, nor can I budge a step;
My skin though loose in front is tight behind,
And I am even as a Syrian bow.
Alas! methinks a bent tube shoots not well;
So give me now thine aid, my Giovanni.
At present that difficulty is got over by using large strong canvas, on
which the picture can be painted in the studio and afterwards placed on
the wall.
However, the other difficulty of form has to be got over also. A great
portion of the ceiling of the Sistine Chapel, and notably the prophets
and sibyls, are painted on a curved surface, in which case a similar
method to that explained by Leonardo da Vinci has to be adopted.
In Chapter CCCI he shows us how to draw a figure twenty-four braccia
high upon a wall twelve braccia high. (The braccia is 1 ft. 10-7/8 in.).
He first draws the figure upright, then from the various points draws
lines to a point _F_ on the floor of the building, marking their
intersections on the profile of the wall somewhat in the manner we have
indicated, which serve as guides in making the outline to be traced.
[Illustration: Fig. 67.
'Draw upon part of wall _MN_ half the figure you mean to represent, and
the other half upon the cove above (_MR_).' Leonardo da Vinci's
_Treatise on Painting_.]
XXI
INTERIORS
[Illustration: Fig. 68. Interior by de Hoogh.]
To draw the interior of a cube we must suppose the side facing us to be
removed or transparent. Indeed, in all our figures which represent
solids we suppose that we can see through them, and in most cases we
mark the hidden portions with dotted lines. So also with all those
imaginary lines which conduct the eye to the various vanishing points,
and which the old writers called 'occult'.
[Illustration: Fig. 69.]
When the cube is placed below the horizon (as in Fig. 59), we see the
top of it; when on the horizon, as in the above (Fig. 69), if the side
facing us is removed we see both top and bottom of it, or if a room, we
see floor and ceiling, but otherwise we should see but one side (that
facing us), or at most two sides. When the cube is above the horizon we
see underneath it.
We shall find this simple cube of great use to us in architectural
subjects, such as towers, houses, roofs, interiors of rooms, &c.
In this little picture by de Hoogh we have the application of the
perspective of the cube and other foregoing problems.
XXII
THE SQUARE AT AN ANGLE OF 45 DEG.
When the square is at an angle of 45 deg to the base line, then its sides
are drawn respectively to the points of distance, _DD_, and one of its
diagonals which is at right angles to the base is drawn to the point of
sight _S_, and the other _ab_, is parallel to that base or ground line.
[Illustration: Fig. 70.]
To draw a pavement with its squares at this angle is but an
amplification of the above figure. Mark off on base equal distances, 1,
2, 3, &c., representing the diagonals of required squares, and from each
of these points draw lines to points of distance _DD"_. These lines will
intersect each other, and so form the squares of the pavement; to ensure
correctness, lines should also be drawn from these points 1, 2, 3, to
the point of sight _S_, and also horizontals parallel to the base, as
_ab_.
[Illustration: Fig. 71.]
XXIII
THE CUBE AT AN ANGLE OF 45 DEG.
Having drawn the square at an angle of 45 deg, as shown in the previous
figure, we find the length of one of its sides, _dh_, by drawing a line,
_SK_, through _h_, one of its extremities, till it cuts the base line at
_K_. Then, with the other extremity _d_ for centre and _dK_ for radius,
describe a quarter of a circle _Km_; the chord thereof _mK_ will be the
geometrical length of _dh_. At _d_ raise vertical _dC_ equal to _mK_,
which gives us the height of the cube, then raise verticals at _a_, _h_,
&c., their height being found by drawing _CD_ and _CD"_ to the two
points of distance, and so completing the figure.
[Illustration: Fig. 72.]
XXIV
PAVEMENTS DRAWN BY MEANS OF SQUARES AT 45 DEG.
[Illustration: Fig. 73.]
[Illustration: Fig. 74.]
The square at 45 deg will be found of great use in drawing pavements,
roofs, ceilings, &c. In Figs. 73, 74 it is shown how having set out one
square it can be divided into four or more equal squares, and any figure
or tile drawn therein. Begin by making a geometrical or ground plan of
the required design, as at Figs. 73 and 74, where we have bricks placed
at right angles to each other in rows, a common arrangement in brick
floors, or tiles of an octagonal form as at Fig. 75.
[Illustration: Fig. 75.]
XXV
THE PERSPECTIVE VANISHING SCALE
The vanishing scale, which we shall find of infinite use in our
perspective, is founded on the facts explained in Rule 10. We there find
that all horizontals in the same plane, which are drawn to the same
point on the horizon, are perspectively parallel to each other, so that
if we measure a certain height or width on the picture plane, and then
from each extremity draw lines to any convenient point on the horizon,
then all the perpendiculars drawn between these lines will be
perspectively equal, however much they may appear to vary in length.
[Illustration: Fig. 76.]
Let us suppose that in this figure (76) _AB_ and _A'B'_ each represent
5 feet. Then in the first case all the verticals, as _e_, _f_, _g_, _h_,
drawn between _AO_ and _BO_ represent 5 feet, and in the second case all
the horizontals _e_, _f_, _g_, _h_, drawn between _A'O_ and _B'O_ also
represent 5 feet each. So that by the aid of this scale we can give the
exact perspective height and width of any object in the picture, however
far it may be from the base line, for of course we can increase or
diminish our measurements at _AB_ and _A'B'_ to whatever length we
require.
As it may not be quite evident at first that the points _O_ may be taken
at random, the following figure will prove it.
XXVI
THE VANISHING SCALE CAN BE DRAWN TO ANY POINT ON THE HORIZON
From _AB_ (Fig. 77) draw _AO_, _BO_, thus forming the scale, raise
vertical _C_. Now form a second scale from _AB_ by drawing _AO' BO'_,
and therein raise vertical _D_ at an equal distance from the base.
First, then, vertical _C_ equals _AB_, and secondly vertical _D_ equals
_AB_, therefore _C_ equals _D_, so that either of these scales will
measure a given height at a given distance.
[Illustration: Fig. 77.]
(See axioms of geometry.)
[Illustration: Fig. 79. Schoolgirls.]
[Illustration: Fig. 80. Cavaliers.]
XXVII
APPLICATION OF VANISHING SCALES TO DRAWING FIGURES
In this figure we have marked off on a level plain three or four points
_a_, _b_, _c_, _d_, to indicate the places where we wish to stand our
figures. _AB_ represents their average height, so we have made our scale
_AO_, _BO_, accordingly. From each point marked we draw a line parallel
to the base till it reaches the scale. From the point where it touches
the line _AO_, raise perpendicular as _a_, which gives the height
required at that distance, and must be referred back to the figure
itself.
[Illustration: Fig. 78.]
XXVIII
HOW TO DETERMINE THE HEIGHTS OF FIGURES ON A LEVEL PLANE
_First Case._
This is but a repetition of the previous figure, excepting that we have
substituted these schoolgirls for the vertical lines. If we wish to make
some taller than the others, and some shorter, we can easily do so, as
must be evident (see Fig. 79).
Note that in this first case the scale is below the horizon, so that we
see over the heads of the figures, those nearest to us being the lowest
down. That is to say, we are looking on this scene from a slightly
raised platform.
_Second Case._
To draw figures at different distances when their heads are above the
horizon, or as they would appear to a person sitting on a low seat. The
height of the heads varies according to the distance of the figures
(Fig. 80).
_Third Case._
How to draw figures when their heads are about the height of the
horizon, or as they appear to a person standing on the same level or
walking among them.
In this case the heads or the eyes are on a level with the horizon, and
we have little necessity for a scale at the side unless it is for the
purpose of ascertaining or marking their distances from the base line,
and their respective heights, which of course vary; so in all cases
allowance must be made for some being taller and some shorter than the
scale measurement.
[Illustration: Fig. 81.]
XXIX
THE HORIZON ABOVE THE FIGURES
In this example from De Hoogh the doorway to the left is higher up than
the figure of the lady, and the effect seems to me more pleasing and
natural for this kind of domestic subject. This delightful painter was
not only a master of colour, of sunlight effect, and perfect
composition, but also of perspective, and thoroughly understood the
charm it gives to a picture, when cunningly introduced, for he makes the
spectator feel that he can walk along his passages and courtyards. Note
that he frequently puts the point of sight quite at the side of his
canvas, as at _S_, which gives almost the effect of angular perspective
whilst it preserves the flatness and simplicity of parallel or
horizontal perspective.
[Illustration: Fig. 82. Courtyard by De Hoogh.]
XXX
LANDSCAPE PERSPECTIVE
In an extended view or landscape seen from a height, we have to consider
the perspective plane as in a great measure lying above it, reaching
from the base of the picture to the horizon; but of course pierced here
and there by trees, mountains, buildings, &c. As a rule in such cases,
we copy our perspective from nature, and do not trouble ourselves much
about mathematical rules. It is as well, however, to know them, so that
we may feel sure we are right, as this gives certainty to our touch and
enables us to work with freedom. Nor must we, when painting from nature,
forget to take into account the effects of atmosphere and the various
tones of the different planes of distance, for this makes much of the
difference between a good picture and a bad one; being a more subtle
quality, it requires a keener artistic sense to discover and depict it.
(See Figs. 95 and 103.)
If the landscape painter wishes to test his knowledge of perspective,
let him dissect and work out one of Turner's pictures, or better still,
put his own sketch from nature to the same test.
XXXI
FIGURES OF DIFFERENT HEIGHTS
THE CHESSBOARD
In this figure the same principle is applied as in the previous one, but
the chessmen being of different heights we have to arrange the scale
accordingly. First ascertain the exact height of each piece, as _Q_,
_K_, _B_, which represent the queen, king, bishop, &c. Refer these
dimensions to the scale, as shown at _QKB_, which will give us the
perspective measurement of each piece according to the square on which
it is placed.
[Illustration: Fig. 83. Chessboard and Men.]
This is shown in the above drawing (Fig. 83) in the case of the white
queen and the black queen, &c. The castle, the knight, and the pawn
being about the same height are measured from the fourth line of the
scale marked _C_.
[Illustration: Fig. 84.]
XXXII
APPLICATION OF THE VANISHING SCALE TO DRAWING FIGURES AT AN ANGLE
WHEN THEIR VANISHING POINTS ARE INACCESSIBLE OR OUTSIDE THE PICTURE
This is exemplified in the drawing of a fence (Fig. 84). Form scale
_aS_, _bS_, in accordance with the height of the fence or wall to be
depicted. Let _ao_ represent the direction or angle at which it is
placed, draw _od_ to meet the scale at _d_, at _d_ raise vertical _dc_,
which gives the height of the fence at _oo'_. Draw lines _bo'_, _eo_,
_ao_, &c., and it will be found that all these lines if produced will
meet at the same point on the horizon. To divide the fence into spaces,
divide base line _af_ as required and proceed as already shown.
XXXIII
THE REDUCED DISTANCE. HOW TO PROCEED WHEN THE POINT OF DISTANCE
IS INACCESSIBLE
It has already been shown that too near a point of distance is
objectionable on account of the distortion and disproportion resulting
from it. At the same time, the long distance-point must be some way out
of the picture and therefore inconvenient. The object of the reduced
distance is to bring that point within the picture.
[Illustration: Fig. 85.]
In Fig. 85 we have made the distance nearly twice the length of the base
of the picture, and consequently a long way out of it. Draw _Sa_, _Sb_,
and from _a_ draw _aD_ to point of distance, which cuts _Sb_ at _o_, and
determines the depth of the square _acob_. But we can find that same
point if we take half the base and draw a line from 1/2 base to 1/2
distance. But even this 1/2 distance-point does not come inside the
picture, so we take a fourth of the base and a fourth of the distance
and draw a line from 1/4 base to 1/4 distance. We shall find that it
passes precisely through the same point _o_ as the other lines _aD_, &c.
We are thus able to find the required point _o_ without going outside
the picture.
Of course we could in the same way take an 8th or even a 16th distance,
but the great use of this reduced distance, in addition to the above,
is that it enables us to measure any depth into the picture with the
greatest ease.
It will be seen in the next figure that without having to extend the
base, as is usually done, we can multiply that base to any amount by
making use of these reduced distances on the horizontal line. This is
quite a new method of proceeding, and it will be seen is mathematically
correct.
XXXIV
HOW TO DRAW A LONG PASSAGE OR CLOISTER BY MEANS OF THE REDUCED DISTANCE
[Illustration: Fig. 86.]
In Fig. 86 we have divided the base of the first square into four equal
parts, which may represent so many feet, so that A4 and _Bd_ being the
retreating sides of the square each represents 4 feet. But we found
point 1/4 D by drawing 3D from 1/4 base to 1/4 distance, and by
proceeding in the same way from each division, _A_, 1, 2, 3, we mark off
on _SB_ four spaces each equal to 4 feet, in all 16 feet, so that by
taking the whole base and the 1/4 distance we find point _O_, which is
distant four times the length of the base _AB_. We can multiply this
distance to any amount by drawing other diagonals to 8th distance, &c.
The same rule applies to this corridor (Fig. 87 and Fig. 88).
[Illustration: Fig. 87.]
[Illustration: Fig. 88.]
XXXV
HOW TO FORM A VANISHING SCALE THAT SHALL GIVE THE HEIGHT, DEPTH,
AND DISTANCE OF ANY OBJECT IN THE PICTURE
If we make our scale to vanish to the point of sight, as in Fig. 89, we
can make _SB_, the lower line thereof, a measuring line for distances.
Let us first of all divide the base _AB_ into eight parts, each part
representing 5 feet. From each division draw lines to 8th distance; by
their intersections with _SB_ we obtain measurements of 40, 80, 120,
160, &c., feet. Now divide the side of the picture _BE_ in the same
manner as the base, which gives us the height of 40 feet. From the side
_BE_ draw lines 5S, 15S, &c., to point of sight, and from each
division on the base line also draw lines 5S, 10S, 15S, &c., to
point of sight, and from each division on _SB_, such as 40, 80, &c.,
draw horizontals parallel to base. We thus obtain squares 40 feet wide,
beginning at base _AB_ and reaching as far as required. Note how the
height of the flagstaff, which is 140 feet high and 280 feet distant, is
obtained. So also any buildings or other objects can be measured, such
as those shown on the left of the picture.
[Illustration: Fig. 89.]
XXXVI
MEASURING SCALE ON GROUND
A simple and very old method of drawing buildings, &c., and giving them
their right width and height is by means of squares of a given size,
drawn on the ground.
[Illustration: Fig. 90.]
In the above sketch (Fig. 90) the squares on the ground represent 3 feet
each way, or one square yard. Taking this as our standard measure, we
find the door on the left is 10 feet high, that the archway at the end
is 21 feet high and 12 feet wide, and so on.
[Illustration: Fig. 91. Natural Perspective.]
[Illustration: Fig. 92. Honfleur.]
Fig. 91 is a sketch made at Sandwich, Kent, and shows a somewhat similar
subject to Fig. 84, but the irregularity and freedom of the perspective
gives it a charm far beyond the rigid precision of the other, while it
conforms to its main laws. This sketch, however, is the real artist's
perspective, or what we might term natural perspective.
XXXVII
APPLICATION OF THE REDUCED DISTANCE AND THE VANISHING SCALE TO DRAWING A
LIGHTHOUSE, &C.
[Above illustration:
Perspective of a lighthouse 135 feet high at 800 feet distance.]
[Illustration: Fig. 93. Key to Fig. 92, Honfleur.]
In the drawing of Honfleur (Fig. 92) we divide the base _AB_ as in the
previous figure, but the spaces measure 5 feet instead of 3 feet: so
that taking the 8th distance, the divisions on the vanishing line _BS_
measure 40 feet each, and at point _O_ we have 400 feet of distance, but
we require 800. So we again reduce the distance to a 16th. We thus
multiply the base by 16. Now let us take a base of 50 feet at _f_ and
draw line _fD_ to 16th distance; if we multiply 50 feet by 16 we obtain
the 800 feet required.
The height of the lighthouse is found by means of the vanishing scale,
which is 15 feet below and 15 feet above the horizon, or 30 feet from
the sea-level. At _L_ we raise a vertical _LM_, which shows the position
of the lighthouse. Then on that vertical measure the height required as
shown in the figure.
The 800 feet could be obtained at once by drawing line _fD_, or 50 feet,
to 16th distance. The other measurements obtained by 8th distance serve
for nearer buildings.
XXXVIII
HOW TO MEASURE LONG DISTANCES SUCH AS A MILE OR UPWARDS
The wonderful effect of distance in Turner's pictures is not to be
achieved by mere measurement, and indeed can only be properly done by
studying Nature and drawing her perspective as she presents it to us. At
the same time it is useful to be able to test and to set out distances
in arranging a composition. This latter, if neglected, often leads to
great difficulties and sometimes to repainting.
To show the method of measuring very long distances we have to work with
a very small scale to the foot, and in Fig. 94 I have divided the base
_AB_ into eleven parts, each part representing 10 feet. First draw _AS_
and _BS_ to point of sight. From _A_ draw _AD_ to 1/4 distance, and we
obtain at 440 on line _BS_ four times the length of _AB_, or 110 feet
x 4 = 440 feet. Again, taking the whole base and drawing a line from _S_
to 8th distance we obtain eight times 110 feet or 880 feet. If now we
use the 16th distance we get sixteen times 110 feet, or 1,760 feet,
one-third of a mile; by repeating this process, but by using the base at
1,760, which is the same length in perspective as _AB_, we obtain 3,520
feet, and then again using the base at 3,520 and proceeding in the same
way we obtain 5,280 feet, or one mile to the archway. The flags show
their heights at their respective distances from the base. By the scale
at the side of the picture, _BO_, we can measure any height above or any
depth below the perspective plane.
[Illustration: Fig. 94.]
_Note_.--This figure (here much reduced) should be drawn large by the
student, so that the numbering, &c., may be made more distinct. Indeed,
many of the other figures should be copied large, and worked out with
care, as lessons in perspective.
XXXIX
FURTHER ILLUSTRATION OF LONG DISTANCES AND EXTENDED VIEWS
An extended view is generally taken from an elevated position, so that
the principal part of the landscape lies beneath the perspective plane,
as already noted, and we shall presently treat of objects and figures on
uneven ground. In the previous figure is shown how we can measure
heights and depths to any extent. But when we turn to a drawing by
Turner, such as the 'View from Richmond Hill', we feel that the only way
to accomplish such perspective as this, is to go and draw it from
nature, and even then to use our judgement, as he did, as to how much we
may emphasize or even exaggerate certain features.
[Illustration: Fig. 95. Turner's View from Richmond Hill.]
Note in this view the foreground on which the principal figures stand is
on a level with the perspective plane, while the river and surrounding
park and woods are hundreds of feet below us and stretch away for miles
into the distance. The contrasts obtained by this arrangement increase
the illusion of space, and the figures in the foreground give as it were
a standard of measurement, and by their contrast to the size of the
trees show us how far away those trees are.
XL
HOW TO ASCERTAIN THE RELATIVE HEIGHTS OF FIGURES ON AN INCLINED PLANE
[Illustration: Fig. 96.]
The three figures to the right marked _f_, _g_, _b_ (Fig. 96) are on
level ground, and we measure them by the vanishing scale _aS_, _bS_.
Those to the left, which are repetitions of them, are on an inclined
plane, the vanishing point of which is _S'_; by the side of this plane
we have placed another vanishing scale _a'S'_, _b'S'_, by which we
measure the figures on that incline in the same way as on the level
plane. It will be seen that if a horizontal line is drawn from the foot
of one of these figures, say _G_, to point _O_ on the edge of the
incline, then dropped vertically to _o'_, then again carried on to _o''_
where the other figure _g_ is, we find it is the same height and also
that the other vanishing scale is the same width at that distance, so
that we can work from either one or the other. In the event of the
rising ground being uneven we can make use of the scale on the level
plane.
XLI
HOW TO FIND THE DISTANCE OF A GIVEN FIGURE OR POINT FROM THE BASE LINE
[Illustration: Fig. 97.]
Let _P_ be the given figure. Form scale _ACS_, _S_ being the point of
sight and _D_ the distance. Draw horizontal _do_ through _P_. From _A_
draw diagonal _AD_ to distance point, cutting _do_ in _o_, through _o_
draw _SB_ to base, and we now have a square _AdoB_ on the perspective
plane; and as figure _P_ is standing on the far side of that square it
must be the distance _AB_, which is one side of it, from the base
line--or picture plane. For figures very far away it might be necessary
to make use of half-distance.
XLII
HOW TO MEASURE THE HEIGHT OF FIGURES ON UNEVEN GROUND
In previous problems we have drawn figures on level planes, which is
easy enough. We have now to represent some above and some below the
perspective plane.
[Illustration: Fig. 98.]
Form scale _bS_, _cS_; mark off distances 20 feet, 40 feet, &c. Suppose
figure _K_ to be 60 feet off. From point at his feet draw horizontal to
meet vertical _On_, which is 60 feet distant. At the point _m_ where
this line meets the vertical, measure height _mn_ equal to width of
scale at that distance, transfer this to _K_, and you have the required
height of the figure in black.
For the figures under the cliff 20 feet below the perspective plane,
form scale _FS_, _GS_, making it the same width as the other, namely
5 feet, and proceed in the usual way to find the height of the figures
on the sands, which are here supposed to be nearly on a level with the
sea, of course making allowance for different heights and various other
things.
XLIII
FURTHER ILLUSTRATION OF THE SIZE OF FIGURES AT DIFFERENT DISTANCES
AND ON UNEVEN GROUND
[Illustration: Fig. 99.]
Let _ab_ be the height of a figure, say 6 feet. First form scale _aS_,
_bS_, the lower line of which, _aS_, is on a level with the base or on
the perspective plane. The figure marked _C_ is close to base, the group
of three is farther off (24 feet), and 6 feet higher up, so we measure
the height on the vanishing scale and also above it. The two girls
carrying fish are still farther off, and about 12 feet below. To tell
how far a figure is away, refer its measurements to the vanishing scale
(see Fig. 96).
XLIV
FIGURES ON A DESCENDING PLANE
In this case (Fig. 100) the same rule applies as in the previous
problem, but as the road on the left is going down hill, the vanishing
point of the inclined plane is below the horizon at point _S'_; _AS_,
_BS_ is the vanishing scale on the level plane; and _A'S'_, _B'S'_, that
on the incline.
Fig. 101. This is an outline of above figure to show the working more
plainly.
Note the wall to the left marked _W_ and the manner in which it appears
to drop at certain intervals, its base corresponding with the inclined
plane, but the upper lines of each division being made level are drawn
to the point of sight, or to their vanishing point on the horizon; it is
important to observe this, as it aids greatly in drawing a road going
down hill.
[Illustration: Fig. 100.]
[Illustration: Fig. 101.]
[Illustration: Fig. 102.]
XLV
FURTHER ILLUSTRATION OF THE DESCENDING PLANE
In the centre of this picture (Fig. 102) we suppose the road to be
descending till it reaches a tunnel which goes under a road or leads to
a river (like one leading out of the Strand near Somerset House). It is
drawn on the same principle as the foregoing figure. Of course to see
the road the spectator must get pretty near to it, otherwise it will be
out of sight. Also a level plane must be shown, as by its contrast to
the other we perceive that the latter is going down hill.
XLVI
FURTHER ILLUSTRATION OF UNEVEN GROUND
An extended view drawn from a height of about 30 feet from a road that
descends about 45 feet.
[Illustration: Fig. 103. Farningham.]
In drawing a landscape such as Fig. 103 we have to bear in mind the
height of the horizon, which being exactly opposite the eye, shows us at
once which objects are below and which are above us, and to draw them
accordingly, especially roofs, buildings, walls, hedges, &c.; also it is
well to sketch in the different fields figures of men and cattle, as
from the size of these we can judge of the rest.
XLVII
THE PICTURE STANDING ON THE GROUND
Let _K_ represent a frame placed vertically and at a given distance in
front of us. If stood on the ground our foreground will touch the base
line of the picture, and we can fix up a standard of measurement both on
the base and on the side as in this sketch, taking 6 feet as about the
height of the figures.
[Illustration: Fig. 104. Toledo.]
XLVIII
THE PICTURE ON A HEIGHT
If we are looking at a scene from a height, that is from a terrace, or a
window, or a cliff, then the near foreground, unless it be the terrace,
window-sill, &c., would not come into the picture, and we could not see
the near figures at _A_, and the nearest to come into view would be
those at _B_, so that a view from a window, &c., would be as it were
without a foreground. Note that the figures at _B_ would be (according
to this sketch) 30 feet from the picture plane and about 18 feet below
the base line.
[Illustration: Fig. 105.]
BOOK THIRD
XLIX
ANGULAR PERSPECTIVE
Hitherto we have spoken only of parallel perspective, which is
comparatively easy, and in our first figure we placed the cube with one
of its sides either touching or parallel to the transparent plane. We
now place it so that one angle only (_ab_), touches the picture.
[Illustration: Fig. 106.]
Its sides are no longer drawn to the point of sight as in Fig. 7, nor
its diagonal to the point of distance, but to some other points on the
horizon, although the same rule holds good as regards their parallelism;
as for instance, in the case of _bc_ and _ad_, which, if produced, would
meet at _V_, a point on the horizon called a vanishing point. In this
figure only one vanishing point is seen, which is to the right of the
point of sight _S_, whilst the other is some distance to the left, and
outside the picture. If the cube is correctly drawn, it will be found
that the lines _ae_, _bg_, &c., if produced, will meet on the horizon at
this other vanishing point. This far-away vanishing point is one of the
inconveniences of oblique or angular perspective, and therefore it will
be a considerable gain to the draughtsman if we can dispense with it.
This can be easily done, as in the above figure, and here our geometry
will come to our assistance, as I shall show presently.
L
HOW TO PUT A GIVEN POINT INTO PERSPECTIVE
Let us place the given point _P_ on a geometrical plane, to show how far
it is from the base line, and indeed in the exact position we wish it to
be in the picture. The geometrical plane is supposed to face us, to hang
down, as it were, from the base line _AB_, like the side of a table, the
top of which represents the perspective plane. It is to that perspective
plane that we now have to transfer the point _P_.
[Illustration: Fig. 107.]
From _P_ raise perpendicular _Pm_ till it touches the base line at _m_.
With centre _m_ and radius _mP_ describe arc _Pn_ so that _mn_ is now
the same length as _mP_. As point _P_ is opposite point _m_, so must it
be in the perspective, therefore we draw a line at right angles to the
base, that is to the point of sight, and somewhere on this line will be
found the required point _P'_. We now have to find how far from _m_ must
that point be. It must be the length of _mn_, which is the same as _mP_.
We therefore from _n_ draw _nD_ to the point of distance, which being at
an angle of 45 deg, or half a right angle, makes _mP_' the perspective
length of _mn_ by its intersection with _mS_, and thus gives us the
point _P'_, which is the perspective of the original point.
LI
A PERSPECTIVE POINT BEING GIVEN, FIND ITS POSITION
ON THE GEOMETRICAL PLANE
To do this we simply reverse the foregoing problem. Thus let _P_ be the
given perspective point. From point of sight _S_ draw a line through _P_
till it cuts _AB_ at _m_. From distance _D_ draw another line through
_P_ till it cuts the base at _n_. From _m_ drop perpendicular, and then
with centre _m_ and radius _mn_ describe arc, and where it cuts that
perpendicular is the required point _P'_. We often have to make use of
this problem.
[Illustration: Fig. 108.]
LII
HOW TO PUT A GIVEN LINE INTO PERSPECTIVE
This is simply a question of putting two points into perspective,
instead of one, or like doing the previous problem twice over, for the
two points represent the two extremities of the line. Thus we have to
find the perspective of _A_ and _B_, namely _a'b'_. Join those points,
and we have the line required.
[Illustration: Fig. 109.]
[Illustration: Fig. 110.]
If one end touches the base, as at _A_ (Fig. 110), then we have but to
find one point, namely _b_. We also find the perspective of the angle
_mAB_, namely the shaded triangle mAb. Note also that the perspective
triangle equals the geometrical triangle.
[Illustration: Fig. 111.]
When the line required is parallel to the base line of the picture, then
the perspective of it is also parallel to that base (see Rule 3).
LIII
TO FIND THE LENGTH OF A GIVEN PERSPECTIVE LINE
A perspective line _AB_ being given, find its actual length and the
angle at which it is placed.
This is simply the reverse of the previous problem. Let _AB_ be the
given line. From distance _D_ through _A_ draw _DC_, and from _S_, point
of sight, through _A_ draw _SO_. Drop _OP_ at right angles to base,
making it equal to _OC_. Join _PB_, and line _PB_ is the actual length
of _AB_.
This problem is useful in finding the position of any given line or
point on the perspective plane.
[Illustration: Fig. 112.]
LIV
TO FIND THESE POINTS WHEN THE DISTANCE-POINT IS INACCESSIBLE
[Illustration: Fig. 113.]
If the distance-point is a long way out of the picture, then the same
result can be obtained by using the half distance and half base, as
already shown.
From _a_, half of _mP_', draw quadrant _ab_, from _b_ (half base), draw
line from _b_ to half Dist., which intersects _Sm_ at _P_, precisely the
same point as would be obtained by using the whole distance.
LV
HOW TO PUT A GIVEN TRIANGLE OR OTHER RECTILINEAL FIGURE INTO PERSPECTIVE
Here we simply put three points into perspective to obtain the given
triangle _A_, or five points to obtain the five-sided figure at _B_.
So can we deal with any number of figures placed at any angle.
[Illustration: Fig. 114.]
Both the above figures are placed in the same diagram, showing how any
number can be drawn by means of the same point of sight and the same
point of distance, which makes them belong to the same picture.
It is to be noted that the figures appear reversed in the perspective.
That is, in the geometrical triangle the base at _ab_ is uppermost,
whereas in the perspective _ab_ is lowermost, yet both are nearest to
the ground line.
LVI
HOW TO PUT A GIVEN SQUARE INTO ANGULAR PERSPECTIVE
Let _ABCD_ (Fig. 115) be the given square on the geometrical plane,
where we can place it as near or as far from the base and at any angle
that we wish. We then proceed to find its perspective on the picture by
finding the perspective of the four points _ABCD_ as already shown. Note
that the two sides of the perspective square _dc_ and _ab_ being
produced, meet at point _V_ on the horizon, which is their vanishing
point, but to find the point on the horizon where sides _bc_ and _ad_
meet, we should have to go a long way to the left of the figure, which
by this method is not necessary.
[Illustration: Fig. 115.]
LVII
OF MEASURING POINTS
We now have to find certain points by which to measure those vanishing
or retreating lines which are no longer at right angles to the picture
plane, as in parallel perspective, and have to be measured in a
different way, and here geometry comes to our assistance.
[Illustration: Fig. 116.]
Note that the perspective square _P_ equals the geometrical square _K_,
so that side _AB_ of the one equals side _ab_ of the other. With centre
_A_ and radius _AB_ describe arc _Bm'_ till it cuts the base line at
_m'_. Now _AB_ = _Am'_, and if we join _bm'_ then triangle _BAm'_ is an
isosceles triangle. So likewise if we join _m'b_ in the perspective
figure will m'Ab be the same isosceles triangle in perspective. Continue
line _m'b_ till it cuts the horizon in _m_, which point will be the
measuring point for the vanishing line _AbV_. For if in an isosceles
triangle we draw lines across it, parallel to its base from one side to
the other, we divide both sides in exactly the same quantities and
proportions, so that if we measure on the base line of the picture the
spaces we require, such as 1, 2, 3, on the length _Am'_, and then
from these divisions draw lines to the measuring point, these lines
will intersect the vanishing line _AbV_ in the lengths and proportions
required. To find a measuring point for the lines that go to the other
vanishing point, we proceed in the same way. Of course great accuracy
is necessary.
Note that the dotted lines 1,1, 2,2, &c., are parallel in the
perspective, as in the geometrical figure. In the former the lines are
drawn to the same point _m_ on the horizon.
LVIII
HOW TO DIVIDE ANY GIVEN STRAIGHT LINE INTO EQUAL OR PROPORTIONATE PARTS
[Illustration: Fig. 117.]
Let _AB_ (Fig. 117) be the given straight line that we wish to divide
into five equal parts. Draw _AC_ at any convenient angle, and measure
off five equal parts with the compasses thereon, as 1, 2, 3, 4, 5. From
5C draw line to 5B. Now from each division on _AC_ draw lines 4,4, 3,3,
&c., parallel to 5,5. Then _AB_ will be divided into the required number
of equal parts.
LIX
HOW TO DIVIDE A DIAGONAL VANISHING LINE INTO ANY NUMBER
OF EQUAL OR PROPORTIONAL PARTS
In a previous figure (Fig. 116) we have shown how to find a measuring
point when the exact measure of a vanishing line is required, but if it
suffices merely to divide a line into a given number of equal parts,
then the following simple method can be adopted.
We wish to divide _ab_ into five equal parts. From _a_, measure off on
the ground line the five equal spaces required. From 5, the point to
which these measures extend (as they are taken at random), draw a line
through _b_ till it cuts the horizon at _O_. Then proceed to draw lines
from each division on the base to point _O_, and they will intersect and
divide _ab_ into the required number of equal parts.
[Illustration: Fig. 118.]
[Illustration: Fig. 119.]
The same method applies to a given line to be divided into various
proportions, as shown in this lower figure.
[Illustration: Fig. 120.]
[Illustration: Fig. 121.]
LX
FURTHER USE OF THE MEASURING POINT O
One square in oblique or angular perspective being given, draw any
number of other squares equal to it by means of this point _O_ and the
diagonals.
Let _ABCD_ (Fig. 120) be the given square; produce its sides _AB_, _DC_
till they meet at point _V_. From _D_ measure off on base any number of
equal spaces of any convenient length, as 1, 2, 3, &c.; from 1, through
corner of square _C_, draw a line to meet the horizon at _O_, and from
_O_ draw lines to the several divisions on base line. These lines will
divide the vanishing line _DV_ into the required number of parts equal
to _DC_, the side of the square. Produce the diagonal of the square _DB_
till it cuts the horizon at _G_. From the divisions on line _DV_ draw
diagonals to point _G_: their intersections with the other vanishing
line _AV_ will determine the direction of the cross-lines which form the
bases of other squares without the necessity of drawing them to the
other vanishing point, which in this case is some distance to the left
of the picture. If we produce these cross-lines to the horizon we shall
find that they all meet at the other vanishing point, to which of course
it is easy to draw them when that point is accessible, as in Fig. 121;
but if it is too far out of the picture, then this method enables us to
do without it.
Figure 121 corroborates the above by showing the two vanishing points
and additional squares. Note the working of the diagonals drawn to point
_G_, in both figures.
LXI
FURTHER USE OF THE MEASURING POINT O
Suppose we wish to divide the side of a building, as in Fig. 123, or to
draw a balcony, a series of windows, or columns, or what not, or, in
other words, any line above the horizon, as _AB_. Then from _A_ we draw
_AC_ parallel to the horizon, and mark thereon the required divisions 5,
10, 15, &c.: in this case twenty-five (Fig. 122). From _C_ draw a line
through _B_ till it cuts the horizon at _O_. Then proceed to draw the
other lines from each division to _O_, and thus divide the vanishing
line _AB_ as required.
[Illustration: Fig. 122 is a front view of the portico, Fig. 123.]
[Illustration: Fig. 123.]
In this portico there are thirteen triglyphs with twelve spaces between
them, making twenty-five divisions. The required number of parts to draw
the columns can be obtained in the same way.
LXII
ANOTHER METHOD OF ANGULAR PERSPECTIVE, BEING THAT ADOPTED
IN OUR ART SCHOOLS
In the previous method we have drawn our squares by means of a
geometrical plan, putting each point into perspective as required, and
then by means of the perspective drawing thus obtained, finding our
vanishing and measuring points. In this method we proceed in exactly the
opposite way, setting out our points first, and drawing the square (or
other figure) afterwards.
[Illustration: Fig. 124.]
Having drawn the horizontal and base lines, and fixed upon the position
of the point of sight, we next mark the position of the spectator by
dropping a perpendicular, _S ST_, from that point of sight, making it
the same length as the distance we suppose the spectator to be from the
picture, and thus we make _ST_ the station-point.
To understand this figure we must first look upon it as a ground-plan or
bird's-eye view, the line V2V1 or horizon line representing the picture
seen edgeways, because of course the station-point cannot be in the
picture itself, but a certain distance in front of it. The angle at
_ST_, that is the angle which decides the positions of the two vanishing
points V1, V2, is always a right angle, and the two remaining angles
on that side of the line, called the directing line, are together equal
to a right angle or 90 deg. So that in fixing upon the angle at which
the square or other figure is to be placed, we say 'let it be 60 deg and
30 deg, or 70 deg and 20 deg', &c. Having decided upon the station-point
and the angle at which the square is to be placed, draw TV1 and TV2,
till they cut the horizon at V1 and V2. These are the two vanishing
points to which the sides of the figure are respectively drawn. But
we still want the measuring points for these two vanishing lines. We
therefore take first, V1 as centre and V1T as radius, and describe arc
of circle till it cuts the horizon in M1, which is the measuring point
for all lines drawn to V1. Then with radius V2T describe arc from centre
V2 till it cuts the horizon in M2, which is the measuring point for all
vanishing lines drawn to V2. We have now set out our points. Let us
proceed to draw the square _Abcd_. From _A_, the nearest angle (in this
instance touching the base line), measure on each side of it the equal
lengths _AB_ and _AE_, which represent the width or side of the square.
Draw EM2 and BM1 from the two measuring points, which give us, by their
intersections with the vanishing lines AV1 and AV2, the perspective
lengths of the sides of the square _Abcd_. Join _b_ and V1 and dV2,
which intersect each other at _C_, then _Adcb_ is the square required.
This method, which is easy when you know it, has certain drawbacks, the
chief one being that if we require a long-distance point, and a small
angle, such as 10 deg on one side, and 80 deg on the other, then the size
of the diagram becomes so large that it has to be carried out on the
floor of the studio with long strings, &c., which is a very clumsy and
unscientific way of setting to work. The architects in such cases make
use of the centrolinead, a clever mechanical contrivance for getting
over the difficulty of the far-off vanishing point, but by the method
I have shown you, and shall further illustrate, you will find that you
can dispense with all this trouble, and do all your perspective either
inside the picture or on a very small margin outside it.
Perhaps another drawback to this method is that it is not self-evident,
as in the former one, and being rather difficult to explain, the student
is apt to take it on trust, and not to trouble about the reasons for its
construction: but to show that it is equally correct, I will draw the
two methods in one figure.
LXIII
TWO METHODS OF ANGULAR PERSPECTIVE IN ONE FIGURE
[Illustration: Fig. 125.]
It matters little whether the station-point is placed above or below the
horizon, as the result is the same. In Fig. 125 it is placed above, as
the lower part of the figure is occupied with the geometrical plan of
the other method.
In each case we make the square _K_ the same size and at the same angle,
its near corner being at _A_. It must be seen that by whichever method
we work out this perspective, the result is the same, so that both are
correct: the great advantage of the first or geometrical system being,
that we can place the square at any angle, as it is drawn without
reference to vanishing points.
We will, however, work out a few figures by the second method.
LXIV
TO DRAW A CUBE, THE POINTS BEING GIVEN
As in a previous figure (124) we found the various working points of
angular perspective, we need now merely transfer them to the horizontal
line in this figure, as in this case they will answer our purpose
perfectly well.
[Illustration: Fig. 126.]
Let _A_ be the nearest angle touching the base. Draw AV1, AV2. From
_A_, raise vertical _Ae_, the height of the cube. From _e_ draw eV1,
eV2, from the other angles raise verticals _bf_, _dh_, _cg_, to meet
eV1, eV2, fV2, &c., and the cube is complete.
LXV
AMPLIFICATION OF THE CUBE APPLIED TO DRAWING A COTTAGE
[Illustration: Fig. 127.]
Note that we have started this figure with the cube _Adhefb_. We have
taken three times _AB_, its width, for the front of our house, and twice
_AB_ for the side, and have made it two cubes high, not counting the
roof. Note also the use of the measuring-points in connexion with the
measurements on the base line, and the upper measuring line _TPK_.
LXVI
HOW TO DRAW AN INTERIOR AT AN ANGLE
Here we make use of the same points as in a previous figure, with the
addition of the point _G_, which is the vanishing point of the diagonals
of the squares on the floor.
[Illustration: Fig. 128.]
From _A_ draw square _Abcd_, and produce its sides in all directions;
again from _A_, through the opposite angle of the square _C_, draw a
diagonal till it cuts the horizon at _G_. From _G_ draw diagonals
through _b_ and _d_, cutting the base at _o_, _o_, make spaces _o_, _o_,
equal to _Ao_ all along the base, and from them draw diagonals to _G_;
through the points where these diagonals intersect the vanishing lines
drawn in the direction of _Ab_, _dc_ and _Ad_, _bc_, draw lines to the
other vanishing point V1, thus completing the squares, and so cover
the floor with them; they will then serve to measure width of door,
windows, &c. Of course horizontal lines on wall 1 are drawn to V1, and
those on wall 2 to V2.
In order to see this drawing properly, the eye should be placed about
3 inches from it, and opposite the point of sight; it will then stand
out like a stereoscopic picture, and appear as actual space, but
otherwise the perspective seems deformed, and the angles exaggerated.
To make this drawing look right from a reasonable distance, the point of
distance should be at least twice as far off as it is here, and this
would mean altering all the other points and sending them a long way out
of the picture; this is why artists use those long strings referred to
above. I would however, advise them to make their perspective drawing on
a small scale, and then square it up to the size of the canvas.
LXVII
HOW TO CORRECT DISTORTED PERSPECTIVE BY DOUBLING THE LINE OF DISTANCE
Here we have the same interior as the foregoing, but drawn with double
the distance, so that the perspective is not so violent and the objects
are truer in proportion to each other.
[Illustration: Fig. 129.]
To redraw the whole figure double the size, including the station-point,
would require a very large diagram, that we could not get into this book
without a folding plate, but it comes to the same thing if we double the
distances between the various points. Thus, if from _S_ to _G_ in the
small diagram is 1 inch, in the larger one make it 2 inches. If from _S_
to M2 is 2 inches, in the larger make it 4, and so on.
Or this form may be used: make _AB_ twice the length of _AC_ (Fig. 130),
or in any other proportion required. On _AC_ mark the points as in the
drawing you wish to enlarge. Make _AB_ the length that you wish to
enlarge to, draw _CB_, and then from each division on _AC_ draw lines
parallel to _CB_, and _AB_ will be divided in the same proportions, as I
have already shown (Fig. 117).
There is no doubt that it is easier to work direct from the vanishing
points themselves, especially in complicated architectural work, but at
the same time I will now show you how we can dispense with, at all
events, one of them, and that the farthest away.
[Illustration: Fig. 130.]
LXVIII
HOW TO DRAW A CUBE ON A GIVEN SQUARE, USING ONLY ONE VANISHING POINT
_ABCD_ is the given square (Fig. 131). At _A_ raise vertical _Aa_ equal
to side of square _AB'_, from _a_ draw _ab_ to the vanishing point.
Raise _Bb_. Produce _VD_ to _E_ to touch the base line. From _E_ raise
vertical _EF_, making it equal to _Aa_. From _F_ draw _FV_. Raise _Dd_
and _Cc_, their heights being determined by the line _FV_. Join _da_ and
the cube is complete. It will be seen that the verticals raised at each
corner of the square are equal perspectively, as they are drawn between
parallels which start from equal heights, namely, from _EF_ and _Aa_ to
the same point _V_, the vanishing point. Any other line, such as _OO'_,
can be directed to the inaccessible vanishing point in the same way as
_ad_, &c.
_Note._ This is only one of many original figures and problems in this
book which have been called up by the wish to facilitate the work of the
artist, and as it were by necessity.
[Illustration: Fig. 131.]
LXIX
A COURTYARD OR CLOISTER DRAWN WITH ONE VANISHING POINT
[Illustration: Fig. 132.]
In this figure I have first drawn the pavement by means of the diagonals
_GA_, _Go_, _Go_, &c., and the vanishing point _V_, the square at _A_
being given. From _A_ draw diagonal through opposite corner till it cuts
the horizon at _G_. From this same point _G_ draw lines through the
other corners of the square till they cut the ground line at _o_, _o_.
Take this measurement _Ao_ and mark it along the base right and left of
_A_, and the lines drawn from these points _o_ to point _G_ will give
the diagonals of all the squares on the pavement. Produce sides of
square _A_, and where these lines are intersected by the diagonals _Go_
draw lines from the vanishing point _V_ to base. These will give us the
outlines of the squares lying between them and also guiding points that
will enable us to draw as many more as we please. These again will give
us our measurements for the widths of the arches, &c., or between the
columns. Having fixed the height of wall or dado, we make use of _V_
point to draw the sides of the building, and by means of proportionate
measurement complete the rest, as in Fig. 128.
LXX
HOW TO DRAW LINES WHICH SHALL MEET AT A DISTANT POINT,
BY MEANS OF DIAGONALS
This is in a great measure a repetition of the foregoing figure, and
therefore needs no further explanation.
[Illustration: Fig. 133.]
I must, however, point out the importance of the point _G_. In angular
perspective it in a measure takes the place of the point of distance in
parallel perspective, since it is the vanishing point of diagonals at
45 deg drawn between parallels such as _AV_, _DV_, drawn to a vanishing
point _V_. The method of dividing line _AV_ into a number of parts equal
to _AB_, the side of the square, is also shown in a previous figure
(Fig. 120).
LXXI
HOW TO DIVIDE A SQUARE PLACED AT AN ANGLE INTO A GIVEN NUMBER
OF SMALL SQUARES
_ABCD_ is the given square, and only one vanishing point is accessible.
Let us divide it into sixteen small squares. Produce side _CD_ to base
at _E_. Divide _EA_ into four equal parts. From each division draw lines
to vanishing point _V_. Draw diagonals _BD_ and _AC_, and produce the
latter till it cuts the horizon in _G_. Draw the three cross-lines
through the intersections made by the diagonals and the lines drawn to
_V_, and thus divide the square into sixteen.
[Illustration: Fig. 134.]
This is to some extent the reverse of the previous problem. It also
shows how the long vanishing point can be dispensed with, and the
perspective drawing brought within the picture.
LXXII
FURTHER EXAMPLE OF HOW TO DIVIDE A GIVEN OBLIQUE SQUARE
INTO A GIVEN NUMBER OF EQUAL SQUARES, SAY TWENTY-FIVE
Having drawn the square _ABCD_, which is enclosed, as will be seen, in a
dotted square in parallel perspective, I divide the line _EA_ into five
equal parts instead of four (Fig. 135), and have made use of the device
for that purpose by measuring off the required number on line _EF_, &c.
Fig. 136 is introduced here simply to show that the square can be
divided into any number of smaller squares. Nor need the figure be
necessarily a square; it is just as easy to make it an oblong, as _ABEF_
(Fig. 136); for although we begin with a square we can extend it in any
direction we please, as here shown.
[Illustration: Fig. 135.]
[Illustration: Fig. 136.]
LXXIII
OF PARALLELS AND DIAGONALS
[Illustration: Fig. 137 A.]
[Illustration: Fig. 137 B.]
[Illustration: Fig. 137 C.]
To find the centre of a square or other rectangular figure we have but
to draw its two diagonals, and their intersection will give us the
centre of the figure (see 137 A). We do the same with perspective
figures, as at B. In Fig. C is shown how a diagonal, drawn from one
angle of a square _B_ through the centre _O_ of the opposite side of the
square, will enable us to find a second square lying between the same
parallels, then a third, a fourth, and so on. At figure _K_ lying on the
ground, I have divided the farther side of the square _mn_ into 1/4,
1/3, 1/2. If I draw a diagonal from _G_ (at the base) through the half
of this line I cut off on _FS_ the lengths or sides of two squares;
if through the quarter I cut off the length of four squares on the
vanishing line _FS_, and so on. In Fig. 137 D is shown how easily any
number of objects at any equal distances apart, such as posts, trees,
columns, &c., can be drawn by means of diagonals between parallels,
guided by a central line _GS_.
[Illustration: Fig. 137 D.]
LXXIV
THE SQUARE, THE OBLONG, AND THEIR DIAGONALS
[Illustration: Fig. 138.]
[Illustration: Fig. 139.]