INTRODUCTION.

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, [p2] 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,[Footnote 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.[Footnote 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 [p3] 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 [p4] 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:

PLACING OF THE SIGHT-POINT, SIGHT-LINE, STATION-POINT, AND STATION-LINE.
[Geometric diagram]
Fig.1.

I. The Sight-Point.—

Let ABCD, Fig.1., be your sheet [p5] 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 [p6] 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.”

[Geometric diagram]
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 [p7] 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, PQ (the height of the image at that point) will be two-thirds the height[Footnote 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:

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 [p8] 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.

[Geometric diagram]
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.[Footnote 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 [p9] 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,[Footnote 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.