When you begin to read this book, sit down very near the window, and
shut the window. I hope the view out of it is pretty; but, whatever the
view may be, we shall find enough in it for an illustration of the first
principles of perspective (or, literally, of “looking through”).
Every pane of your window may be considered, if you choose, as a glass
picture; and what you see through it, as painted on its surface.
And if, holding your head still, you extend your hand to the glass, you
may, with a brush full of any thick color, trace, roughly, the lines of
the landscape on the glass.
But, to do this, you must hold your head very still. Not only you must
not move it sideways, nor up and down, but it must not even move
backwards or forwards; for, if you move your head forwards, you will see
_more_ of the landscape through the pane; and, if you move it backwards,
you will see _less_: or considering the pane of glass as a picture, when
you hold your head near it, the objects are painted small, and a great
many of them go into a little space; but, when you hold your head some
distance back, the objects are painted larger upon the pane, and fewer
of them go into the field of it.
But, besides holding your head still, you must, when you try to trace
the picture on the glass, shut one of your eyes. If you do not, the
point of the brush appears double; and, on farther experiment, you will
observe that each of your eyes sees the object in a different place on
the glass, so that the tracing which is true to the sight of the right
eye is a couple of inches (or more, according to your distance from the
pane,) to the left of that which is true to the sight of the left.
Thus, it is only possible to draw what you see through the window
rightly on the surface of the glass, by fixing one eye at a given point,
and neither moving it to the right nor left, nor up nor down, nor
backwards nor forwards. Every picture drawn in true perspective may be
considered as an upright piece of glass,[3] on which the objects seen
through it have been thus drawn. Perspective can, therefore, only be
quite right, by being calculated for one fixed position of the eye of
the observer; nor will it ever appear _deceptively_ right unless seen
precisely from the point it is calculated for. Custom, however, enables
us to feel the rightness of the work on using both our eyes, and to be
satisfied with it, even when we stand at some distance from the point it
is designed for.
Supposing that, instead of a window, an unbroken plate of crystal
extended itself to the right and left of you, and high in front, and
that you had a brush as long as you wanted (a mile long, suppose), and
could paint with such a brush, then the clouds high up, nearly over your
head, and the landscape far away to the right and left, might be traced,
and painted, on this enormous crystal field.[4] But if the field were so
vast (suppose a mile high and a mile wide), certainly, after the picture
was done, you would not stand as near to it, to see it, as you are now
sitting near to your window. In order to trace the upper clouds through
your great glass, you would have had to stretch your neck quite back,
and nobody likes to bend their neck back to see the top of a picture. So
you would walk a long way back to see the great picture—a quarter of a
mile, perhaps,—and then all the perspective would be wrong, and would
look quite distorted, and you would discover that you ought to have
painted it from the greater distance, if you meant to look at it from
that distance. Thus, the distance at which you intend the observer to
stand from a picture, and for which you calculate the perspective,
ought to regulate to a certain degree the size of the picture. If you
place the point of observation near the canvas, you should not make the
picture very large: _vice versâ_, if you place the point of observation
far from the canvas, you should not make it very small; the fixing,
therefore, of this point of observation determines, as a matter of
convenience, within certain limits, the size of your picture. But it
does not determine this size by any perspective law; and it is a mistake
made by many writers on perspective, to connect some of their rules
definitely with the size of the picture. For, suppose that you had what
you now see through your window painted actually upon its surface, it
would be quite optional to cut out any piece you chose, with the piece
of the landscape that was painted on it. You might have only half a
pane, with a single tree; or a whole pane, with two trees and a cottage;
or two panes, with the whole farmyard and pond; or four panes, with
farmyard, pond, and foreground. And any of these pieces, if the
landscape upon them were, as a scene, pleasantly composed, would be
agreeable pictures, though of quite different sizes; and yet they would
be all calculated for the same distance of observation.
In the following treatise, therefore, I keep the size of the picture
entirely undetermined. I consider the field of canvas as wholly
unlimited, and on that condition determine the perspective laws. After
we know how to apply those laws without limitation, we shall see what
limitations of the size of the picture their results may render
advisable.
But although the size of the _picture_ is thus independent of the
observer’s distance, the size of the _object represented_ in the picture
is not. On the contrary, that size is fixed by absolute mathematical
law; that is to say, supposing you have to draw a tower a hundred feet
high, and a quarter of a mile distant from you, the height which you
ought to give that tower on your paper depends, with mathematical
precision, on the distance at which you intend your paper to be placed.
So, also, do all the rules for drawing the form of the tower, whatever
it may be.
Hence, the first thing to be done in beginning a drawing is to fix, at
your choice, this distance of observation, or the distance at which you
mean to stand from your paper. After that is determined, all is
determined, except only the ultimate size of your picture, which you may
make greater, or less, not by altering the size of the things
represented, but by _taking in more, or fewer_ of them. So, then, before
proceeding to apply any practical perspective rule, we must always have
our distance of observation marked, and the most convenient way of
marking it is the following:
[Illustration: Fig. 1. PLACING OF THE SIGHT-POINT, SIGHT-LINE,
STATION-POINT, AND STATION-LINE.]
I. THE SIGHT-POINT.—Let _ABCD_, Fig. 1., be your sheet of paper, the
larger the better, though perhaps we may cut out of it at last only a
small piece for our picture, such as the dotted circle _NOPQ_. This
circle is not intended to limit either the size or shape of our picture:
you may ultimately have it round or oval, horizontal or upright, small
or large, as you choose. I only dot the line to give you an idea of
whereabouts you will probably like to have it; and, as the operations of
perspective are more conveniently performed upon paper underneath the
picture than above it, I put this conjectural circle at the top of the
paper, about the middle of it, leaving plenty of paper on both sides and
at the bottom. Now, as an observer generally stands near the middle of a
picture to look at it, we had better at first, and for simplicity’s
sake, fix the point of observation opposite the middle of our
conjectural picture. So take the point _S_, the center of the circle
_NOPQ_;—or, which will be simpler for you in your own work, take the
point _S_ at random near the top of your paper, and strike the circle
_NOPQ_ round it, any size you like. Then the point _S_ is to represent
the point _opposite_ which you wish the observer of your picture to
place his eye, in looking at it. Call this point the “Sight-Point.”
II. THE SIGHT-LINE.—Through the Sight-point, _S_, draw a horizontal
line, _GH_, right across your paper from side to side, and call this
line the “Sight-Line.”
This line is of great practical use, representing the level of the eye
of the observer all through the picture. You will find hereafter that if
there is a horizon to be represented in your picture, as of distant sea
or plain, this line defines it.
III. THE STATION-LINE.—From _S_ let fall a perpendicular line, _SR_, to
the bottom of the paper, and call this line the “Station-Line.”
This represents the line on which the observer stands, at a greater or
less distance from the picture; and it ought to be _imagined_ as drawn
right out from the paper at the point s. Hold your paper upright in
front of you, and hold your pencil horizontally, with its point against
the point _S_, as if you wanted to run it through the paper there, and
the pencil will represent the direction in which the line _SR_ ought to
be drawn. But as all the measurements which we have to set upon this
line, and operations which we have to perform with it, are just the same
when it is drawn on the paper itself, below _S_, as they would be if it
were represented by a wire in the position of the leveled pencil, and as
they are much more easily performed when it is drawn on the paper, it is
always in practice, so drawn.
IV. THE STATION-POINT.—On this line, mark the distance _ST_ at your
pleasure, for the distance at which you wish your picture to be seen,
and call the point T the “Station-Point.”
[Illustration: Fig. 2.]
In practice, it is generally advisable to make the distance _ST_ about
as great as the diameter of your intended picture; and it should, for
the most part, be more rather than less; but, as I have just stated,
this is quite arbitrary. However, in this figure, as an approximation to
a generally advisable distance, I make the distance _ST_ equal to the
diameter of the circle _NOPQ_. Now, having fixed this distance, _ST_,
all the dimensions of the objects in our picture are fixed likewise, and
for this reason:—
Let the upright line _AB_, Fig. 2., represent a pane of glass placed
where our picture is to be placed; but seen at the side of it,
edgeways; let _S_ be the Sight-point; _ST_ the Station-line, which, in
this figure, observe, is in its true position, drawn out from the paper,
not down upon it; and _T_ the Station-point.
Suppose the Station-line _ST_ to be continued, or in mathematical
language “produced,” through _S_, far beyond the pane of glass, and let
_PQ_ be a tower or other upright object situated on or above this line.
Now the _apparent_ height of the tower _PQ_ is measured by the angle
_QTP_, between the rays of light which come from the top and bottom of
it to the eye of the observer. But the _actual_ height of the _image_ of
the tower on the pane of glass _AB_, between us and it, is the distance
_P′Q′_ between the points where the rays traverse the glass.
Evidently, the farther from the point _T_ we place the glass, making
_ST_ longer, the larger will be the image; and the nearer we place it to
_T_, the smaller the image, and that in a fixed ratio. Let the distance
_DT_ be the direct distance from the Station-point to the foot of the
object. Then, if we place the glass _AB_ at one-third of that whole
distance, _P′Q′_ will be one-third of the real height of the object; if
we place the glass at two-thirds of the distance, as at _EF_, _P″Q″_
(the height of the image at that point) will be two-thirds the height[5]
of the object, and so on. Therefore the mathematical law is that _P′Q′_
will be to _PQ_ as _ST_ to _DT_. I put this ratio clearly by itself that
you may remember it:
_P′Q′_ ∶ _PQ_ ∷ _ST_ ∶ _DT_
or in words:
_P_ dash _Q_ dash is to _PQ_ as _ST_ to _DT_
In which formula, recollect that _P′Q′_ is the height of the appearance
of the object on the picture; _PQ_ the height of the object itself; _S_
the Sight-point; _T_ the Station-point; _D_ a point at the direct
distance of the object; though the object is seldom placed actually on
the line _TS_ produced, and may be far to the right or left of it, the
formula is still the same.
For let _S_, Fig. 3., be the Sight-point, and _AB_ the glass—here seen
looking _down_ on its _upper edge_, not sideways;—then if the tower
(represented now, as on a map, by the dark square), instead of being at
_D_ on the line _ST_ produced, be at _E_, to the right (or left) of the
spectator, still the apparent height of the tower on _AB_ will be as
_S′T_ to _ET_, which is the same ratio as that of _ST_ to _DT_.
[Illustration: Fig. 3.]
Now in many perspective problems, the position of an object is more
conveniently expressed by the two measurements _DT_ and _DE_, than by
the single oblique measurement _ET_.
I shall call _DT_ the “direct distance” of the object at _E_, and _DE_
its “lateral distance.” It is rather a license to call _DT_ its “direct”
distance, for _ET_ is the more direct of the two; but there is no other
term which would not cause confusion.
Lastly, in order to complete our knowledge of the position of an object,
the vertical height of some point in it, above or below the eye, must be
given; that is to say, either _DP_ or _DQ_ in Fig. 2.[6]: this I shall
call the “vertical distance” of the point given. In all perspective
problems these three distances, and the dimensions of the object, must
be stated, otherwise the problem is imperfectly given. It ought not to
be required of us merely to draw _a_ room or _a_ church in perspective;
but to draw _this_ room from _this_ corner, and _that_ church on _that_
spot, in perspective. For want of knowing how to base their drawings on
the measurement and place of the object, I have known practiced students
represent a parish church, certainly in true perspective, but with a
nave about two miles and a half long.
It is true that in drawing landscapes from nature the sizes and
distances of the objects cannot be accurately known. When, however, we
know how to draw them rightly, if their size were given, we have only to
_assume a rational approximation_ to their size, and the resulting
drawing will be true enough for all intents and purposes. It does not in
the least matter that we represent a distant cottage as eighteen feet
long, when it is in reality only seventeen; but it matters much that we
do not represent it as eighty feet long, as we easily might if we had
not been accustomed to draw from measurement. Therefore, in all the
following problems the measurement of the object is given.
The student must observe, however, that in order to bring the diagrams
into convenient compass, the measurements assumed are generally very
different from any likely to occur in practice. Thus, in Fig. 3., the
distance _DS_ would be probably in practice half a mile or a mile, and
the distance _TS_, from the eye of the observer to the paper, only two
or three feet. The mathematical law is however precisely the same,
whatever the proportions; and I use such proportions as are best
calculated to make the diagram clear.
Now, therefore, the conditions of a perspective problem are the
following:
The Sight-line _GH_ given, Fig. 1.;
The Sight-point _S_ given;
The Station-point _T_ given; and
The three distances of the object,[7] direct, lateral, and vertical,
with its dimensions, given.
The size of the picture, conjecturally limited by the dotted circle, is
to be determined afterwards at our pleasure. On these conditions I
proceed at once to construction.
[3] If the glass were not upright, but sloping, the objects might
still be drawn through it, but their perspective would then be
different. Perspective, as commonly taught, is always calculated
for a vertical plane of picture.
[4] Supposing it to have no thickness; otherwise the images would be
distorted by refraction.
[5] I say “height” instead of “magnitude,” for a reason stated in
Appendix I., to which you will soon be referred. Read on here at
present.
[6] _P_ and _Q_ being points indicative of the place of the tower’s
base and top. In this figure both are above the sight-line; if the
tower were below the spectator both would be below it, and
therefore measured below _D_.
[7] More accurately, “the three distances of any point, either in the
object itself, or indicative of its distance.”