The Elements of Perspective arranged for the use of schools and intended to be read in connection with the first three books of Euclid

Part 1

Chapter 13,726 wordsPublic domain

+--------------------------------------------------------------------+ | | | Transcriber’s Note | | | | In the original book the author used lowercase italic letters and | | lowercase small capitals to label the geometric diagrams. These | | have been here transcribed as lowercase and uppercase italic | | letters respectively (_aA_). | | | | This file should be read using a font that supports the following | | Unicode characters: | | ∠ angle, ∶ ratio, ∷ proportion, ∴ therefore, ′ prime, and | | ″ double prime. | | | +--------------------------------------------------------------------+

THE ELEMENTS OF PERSPECTIVE

ARRANGED FOR THE USE OF SCHOOLS

AND INTENDED TO BE READ IN CONNECTION WITH THE FIRST THREE BOOKS OF EUCLID.

CONTENTS.

PAGE Preface ix

Introduction 1

PROBLEM I. To fix the Position of a given Point 10

PROBLEM II. To draw a Right Line between two given Points 13

PROBLEM III. To find the Vanishing-Point of a given Horizontal Line 17

PROBLEM IV. To find the Dividing-Points of a given Horizontal Line 23

PROBLEM V. To draw a Horizontal Line, given in Position and Magnitude, by means of its Sight-Magnitude and Dividing-Points 24

PROBLEM VI. To draw any Triangle, given in Position and Magnitude, in a Horizontal Plane 27

PROBLEM VII. To draw any Rectilinear Quadrilateral Figure, given in Position and Magnitude, in a Horizontal Plane 29

PROBLEM VIII. To draw a Square, given in Position and Magnitude, in a Horizontal Plane 31

PROBLEM IX. To draw a Square Pillar, given in Position and Magnitude, its Base and Top being in Horizontal Planes 34

PROBLEM X. To draw a Pyramid, given in Position and Magnitude, on a Square Base in a Horizontal Plane 36

PROBLEM XI. To draw any Curve in a Horizontal or Vertical Plane 38

PROBLEM XII. To divide a Circle drawn in Perspective into any given Number of Equal Parts 42

PROBLEM XIII. To draw a Square, given in Magnitude, within a larger Square given in Position and Magnitude; the Sides of the two Squares being Parallel 45

PROBLEM XIV. To draw a Truncated Circular Cone, given in Position and Magnitude, the Truncations being in Horizontal Planes, and the Axis of the Cone vertical 47

PROBLEM XV. To draw an Inclined Line, given in Position and Magnitude 50

PROBLEM XVI. To find the Vanishing-Point of a given Inclined Line 53

PROBLEM XVII. To find the Dividing-Points of a given Inclined Line 55

PROBLEM XVIII. To find the Sight-Line of an Inclined Plane in which Two Lines are given in Position 57

PROBLEM XIX. To find the Vanishing-Point of Steepest Lines in an Inclined Plane whose Sight-Line is given 59

PROBLEM XX. To find the Vanishing-Point of Lines perpendicular to the Surface of a given Inclined Plane 61

APPENDIX.

I. Practice and Observations on the preceding Problems 69

II. Demonstrations which could not conveniently be included in the Text 99

PREFACE.

For some time back I have felt the want, among Students of Drawing, of a written code of accurate Perspective Law; the modes of construction in common use being various, and, for some problems, insufficient. It would have been desirable to draw up such a code in popular language, so as to do away with the most repulsive difficulties of the subject; but finding this popularization would be impossible, without elaborate figures and long explanations, such as I had no leisure to prepare, I have arranged the necessary rules in a short mathematical form, which any schoolboy may read through in a few days, after he has mastered the first three and the sixth books of Euclid.

Some awkward compromises have been admitted between the first-attempted popular explanation, and the severer arrangement, involving irregular lettering and redundant phraseology; but I cannot for the present do more, and leave the book therefore to its trial, hoping that, if it be found by masters of schools to answer its purpose, I may hereafter bring it into better form.[1]

An account of practical methods, sufficient for general purposes of sketching, might indeed have been set down in much less space: but if the student reads the following pages carefully, he will not only find himself able, on occasion, to solve perspective problems of a complexity greater than the ordinary rules will reach, but obtain a clue to many important laws of pictorial effect, no less than of outline. The subject thus examined becomes, at least to my mind, very curious and interesting; but, for students who are unable or unwilling to take it up in this abstract form, I believe good help will be soon furnished, in a series of illustrations of practical perspective now in preparation by Mr. Le Vengeur. I have not seen this essay in an advanced state, but the illustrations shown to me were very clear and good; and, as the author has devoted much thought to their arrangement, I hope that his work will be precisely what is wanted by the general learner.

Students wishing to pursue the subject into its more extended branches will find, I believe, Cloquet’s treatise the best hitherto published.[2]

[1] Some irregularities of arrangement have been admitted merely for the sake of convenient reference; the eighth problem, for instance, ought to have been given as a case of the seventh, but is separately enunciated on account of its importance.

Several constructions, which ought to have been given as problems, are on the contrary given as corollaries, in order to keep the more directly connected problems in closer sequence; thus the construction of rectangles and polygons in vertical planes would appear by the Table of Contents to have been omitted, being given in the corollary to Problem IX.

[2] Nouveau Traité Élémentaire de Perspective. Bachelier, 1823.

THE ELEMENTS OF PERSPECTIVE.

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, 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:

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.”

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_.

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.