Produced by Juliet Sutherland, David Wilson and the Online Distributed Proofreading Team at http://www.pgdp.net +--------------------------------------------------------------------+ | | | Transcriberâs 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: | | â angle, â¶ ratio, â· proportion, â´ therefore, â² prime, and | | â³ double prime. | | | +--------------------------------------------------------------------+ +----------------------------------+ | | | Library Edition | | | +----------------------------------+ | | | | | THE COMPLETE WORKS | | OF | | JOHN RUSKIN | | | | | | ELEMENTS OF DRAWING AND | | PERSPECTIVE | | THE TWO PATHS | | UNTO THIS LAST | | MUNERA PULVERIS | | SESAME AND LILIES | | ETHICS OF THE DUST | | | | | +----------------------------------+ | NATIONAL LIBRARY ASSOCIATION | | NEW YORK CHICAGO | +----------------------------------+ 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.Â 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, Cloquetâs 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 ProblemÂ IX. [2] Nouveau TraitÃ© ÃlÃ©mentaire 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 âlooking throughâ). 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 pictureâa quarter of a mile, perhaps,âand 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 versÃ¢_, 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 observerâs 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.Â 1. PLACING OF THE SIGHT-POINT, SIGHT-LINE, STATION-POINT, AND STATION-LINE.] I. THE SIGHT-POINT.âLet _ABCD_, Fig.Â 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 simplicityâs sake, fix the point of observation opposite the middle of our conjectural picture. So take the point _S_, the center of the circle _NOPQ_;âor, 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 âSight-Point.â II. THE SIGHT-LINE.âThrough the Sight-point, _S_, draw a horizontal line, _GH_, right across your paper from side to side, and call this line the âSight-Line.â 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.âFrom _S_ let fall a perpendicular line, _SR_, to the bottom of the paper, and call this line the âStation-Line.â 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.âOn 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 âStation-Point.â [Illustration: Fig.Â 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:â Let the upright line _AB_, Fig.Â 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 âproduced,â 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 _Pâ²Qâ²_ 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, _Pâ²Qâ²_ 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_, _Pâ³Qâ³_ (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 _Pâ²Qâ²_ will be to _PQ_ as _ST_ to _DT_. I put this ratio clearly by itself that you may remember it: _Pâ²Qâ²_ â¶ _PQ_ â· _ST_ â¶ _DT_ or in words: _P_ dash _Q_ dash is to _PQ_ as _ST_ to _DT_ In which formula, recollect that _Pâ²Qâ²_ 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.Â 3., be the Sight-point, and _AB_ the glassâhere seen looking _down_ on its _upper edge_, not sideways;âthen 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 _Sâ²T_ to _ET_, which is the same ratio as that of _ST_ to _DT_. [Illustration: Fig.Â 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 âdirect distanceâ of the object at _E_, and _DE_ its âlateral distance.â It is rather a license to call _DT_ its âdirectâ 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.Â 2.[6]: this I shall call the âvertical distanceâ 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.Â 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.Â 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 âheightâ instead of âmagnitude,â for a reason stated in AppendixÂ 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 towerâs 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, âthe three distances of any point, either in the object itself, or indicative of its distance.â PROBLEM I. TO FIX THE POSITION OF A GIVEN POINT.[8] Let _P_, Fig.Â 4., be the given point. [Illustration: Fig.Â 4.] Let its direct distance be _DT_; its lateral distance to the left, _DC_; and vertical distance _beneath_ the eye of the observer, _CP_. [Let _GH_ be the Sight-line, _S_ the Sight-point, and _T_ the Station-point.][9] It is required to fix on the plane of the picture the position of the point P. Arrange the three distances of the object on your paper, as in Fig.Â 4.[10] Join _CT_, cutting _GH_ in _Q_. From _Q_ let fall the vertical line _QPâ²_. Join _PT_, cutting _QP_ in _Pâ²_. _Pâ²_ is the point required. If the point _P_ is _above_ the eye of the observer instead of below it, _CP_ is to be measured upwards from _C_, and _QPâ²_ drawn upwards from _Q_. The construction will be as in Fig.Â 5. [Illustration: Fig.Â 5.] And if the point _P_ is to the right instead of the left of the observer, _DC_ is to be measured to the right instead of the left. The figures 4. and 5., looked at in a mirror, will show the construction of each, on that supposition. Now read very carefully the examples and notes to this problem in AppendixÂ I. (page 69). I have put them in the Appendix in order to keep the sequence of following problems more clearly traceable here in the text; but you must read the first Appendix before going on. [8] More accurately, âTo fix on the plane of the picture the apparent position of a point given in actual position.â In the headings of all the following problems the words âon the plane of the pictureâ are to be understood after the words âto draw.â The plane of the picture means a surface extended indefinitely in the direction of the picture. [9] The sentence within brackets will not be repeated in succeeding statements of problems. It is always to be understood. [10] In order to be able to do this, you must assume the distances to be small; as in the case of some object on the table: how large distances are to be treated you will see presently; the mathematical principle, being the same for all, is best illustrated first on a small scale. Suppose, for instance, _P_ to be the corner of a book on the table, seven inches below the eye, five inches to the left of it, and a foot and a half in advance of it, and that you mean to hold your finished drawing at six inches from the eye; then _TS_ will be six inches, _TD_ a foot and a half, _DC_ five inches, and _CP_ seven. PROBLEM II. TO DRAW A RIGHT LINE BETWEEN TWO GIVEN POINTS. [Illustration: Fig.Â 6.] Let _AB_, Fig.Â 6., be the given right line, joining the given points _A_ and _B_. Let the direct, lateral, and vertical distances of the point _A_ be _TD_, _DC_, and _CA_. Let the direct, lateral, and vertical distances of the point _B_ be _TDâ²_, _DCâ²_, and _Câ²B_. Then, by ProblemÂ I., the position of the point _A_ on the plane of the picture is _a_. And similarly, the position of the point _B_ on the plane of the picture is _b_. Join _ab_. Then _ab_ is the line required. COROLLARY I. If the line _AB_ is in a plane parallel to that of the picture, one end of the line _AB_ must be at the same direct distance from the eye of the observer as the other. Therefore, in that case, _DT_ is equal to _Dâ²T_. Then the construction will be as in Fig.Â 7.; and the student will find experimentally that _ab_ is now parallel to _AB_.[11] [Illustration: Fig.Â 7.] And that _ab_ is to _AB_ as _TS_ is to _TD_. Therefore, to draw any line in a plane parallel to that of the picture, we have only to fix the position of one of its extremities, _a_ or _b_, and then to draw from _a_ or _b_ a line parallel to the given line, bearing the proportion to it that _TS_ bears to _TD_. COROLLARY II. If the line _AB_ is in a horizontal plane, the vertical distance of one of its extremities must be the same as that of the other. Therefore, in that case, _AC_ equals _BCâ²_ (Fig.Â 6.). And the construction is as in Fig.Â 8. [Illustration: Fig.Â 8.] In Fig.Â 8. produce _ab_ to the sight-line, cutting the sight-line in _V_; the point _V_, thus determined, is called the VANISHING-POINT of the line _AB_. Join _TV_. Then the student will find experimentally that _TV_ is parallel to _AB_.[12] COROLLARY III. If the line _AB_ produced would pass through some point beneath or above the station-point, _CD_ is to _DT_ as _Câ²Dâ²_ is to _Dâ²T_; in which case the point _c_ coincides with the point _câ²_, and the line _ab_ is vertical. Therefore every vertical line in a picture is, or may be, the perspective representation of a horizontal one which, produced, would pass beneath the feet or above the head of the spectator.[13] [11] For by the construction _AT_ â¶ _aT_ â· _BT_ â¶ _bT_; and therefore the two triangles _ABT_, _abT_, (having a common angle _ATB_,) are similar. [12] The demonstration is in AppendixÂ II. ArticleÂ I. [13] The reflection in water of any luminous point or isolated object (such as the sun or moon) is therefore, in perspective, a vertical line; since such reflection, if produced, would pass under the feet of the spectator. Many artists (Claude among the rest) knowing something of optics, but nothing of perspective, have been led occasionally to draw such reflections towards a point at the center of the base of the picture. PROBLEM III. TO FIND THE VANISHING-POINT OF A GIVEN HORIZONTAL LINE. [Illustration: Fig.Â 9.] Let _AB_, Fig.Â 9., be the given line. From _T_, the station-point, draw _TV_ parallel to _AB_, cutting the sight-line in _V_. _V_ is the Vanishing-point required.[14] COROLLARY I. As, if the point _b_ is first found, _V_ may be determined by it, so, if the point _V_ is first found, _b_ may be determined by it. For let _AB_, Fig.Â 10., be the given line, constructed upon the paper as in Fig.Â 8.; and let it be required to draw the line _ab_ without using the point _Câ²_. [Illustration: Fig.Â 10.] Find the position of the point _A_ in _a_. (ProblemÂ I.) Find the vanishing-point of _AB_ in _V_. (ProblemÂ III.) Join _aV_. Join _BT_, cutting _aV_ in _b_. Then _ab_ is the line required.[15] COROLLARY II. We have hitherto proceeded on the supposition that the given line was small enough, and near enough, to be actually drawn on our paper of its real size; as in the example given in AppendixÂ I. We may, however, now deduce a construction available under all circumstances, whatever may be the distance and length of the line given. [Illustration: Fig.Â 11.] From Fig.Â 8. remove, for the sake of clearness, the lines _Câ²Dâ²_, _bV_, and _TV_; and, taking the figure as here in Fig.Â 11., draw from _a_, the line _aR_ parallel to _AB_, cutting _BT_ in _R_. Then _aR_ is to _AB_ as _aT_ is to _AT_. ---- ---- as _cT_ is to _CT_. ---- ---- as _TS_ is to _TD_. That is to say, _aR_ is the sight-magnitude of _AB_.[16] [Illustration: Fig.Â 12.] Therefore, when the position of the point _A_ is fixed in _a_, as in Fig.Â 12., and _aV_ is drawn to the vanishing-point; if we draw a line _aR_ from _a_, parallel to _AB_, and make _aR_ equal to the sight-magnitude of _AB_, and then join _RT_, the line _RT_ will cut _aV_ in _b_. So that, in order to determine the length of _ab_, we need not draw the long and distant line _AB_, but only _aR_ parallel to it, and of its sight-magnitude; which is a great gain, for the line _AB_ may be two miles long, and the line _aR_ perhaps only two inches. COROLLARY III. In Fig.Â 12., altering its proportions a little for the sake of clearness, and putting it as here in Fig.Â 13., draw a horizontal line _aRâ²_ and make _aRâ²_ equal to _aR_. Through the points _R_ and _b_ draw _Râ²M_, cutting the sight-line in _M_. Join _TV_. Now the reader will find experimentally that _VM_ is equal to _VT_.[17] [Illustration: Fig.Â 13.] Hence it follows that, if from the vanishing-point _V_ we lay off on the sight-line a distance, _VM_, equal to _VT_; then draw through _a_ a horizontal line _aRâ²_, make _aRâ²_ equal to the sight-magnitude of _AB_, and join _Râ²M_; the line _Râ²M_ will cut _aV_ in _b_. And this is in practice generally the most convenient way of obtaining the length of _ab_. COROLLARY IV. Removing from the preceding figure the unnecessary lines, and retaining only _Râ²M_ and _aV_, as in Fig.Â 14., produce the line _aRâ²_ to the other side of _a_, and make _aX_ equal to _aRâ²_. Join _Xb_, and produce _Xb_ to cut the line of sight in _N_. [Illustration: Fig.Â 14.] Then as _XRâ²_ is parallel to _MN_, and _aRâ²_ is equal to _aX_, _VN_ must, by similar triangles, be equal to _VM_ (equal to _VT_ in Fig.Â 13.). Therefore, on whichever side of _V_ we measure the distance _VT_, so as to obtain either the point _M_, or the point _N_, if we measure the sight-magnitude _aRâ²_ or _aX_ on the opposite side of the line _aV_, the line joining _Râ²M_ or _XN_ will equally cut _aV_ in _b_. The points _M_ and _N_ are called the âDIVIDING-POINTSâ of the original line _AB_ (Fig.Â 12.), and we resume the results of these corollaries in the following three problems. [14] The student will observe, in practice, that, his paper lying flat on the table, he has only to draw the line _TV_ on its horizontal surface, parallel to the given horizontal line _AB_. In theory, the paper should be vertical, but the station-line _ST_ horizontal (see its definition above, page 5); in which case _TV_, being drawn parallel to _AB_, will be horizontal also, and still cut the sight-line in _V_. The construction will be seen to be founded on the second Corollary of the preceding problem. It is evident that if any other line, as _MN_ in Fig.Â 9., parallel to _AB_, occurs in the picture, the line _TV_, drawn from _T_, parallel to _MN_, to find the vanishing-point of _MN_, will coincide with the line drawn from _T_, parallel to _AB_, to find the vanishing-point of _AB_. Therefore _AB_ and _MN_ will have the same vanishing-point. Therefore all parallel horizontal lines have the same vanishing-point. It will be shown hereafter that all parallel _inclined_ lines also have the same vanishing-point; the student may here accept the general conclusionââ_All parallel lines have the same vanishing-point._â It is also evident that if _AB_ is parallel to the plane of the picture, _TV_ must be drawn parallel to _GH_, and will therefore never cut _GH_. The line _AB_ has in that case no vanishing-point: it is to be drawn by the construction given in Fig.Â 7. It is also evident that if _AB_ is at right angles with the plane of the picture, _TV_ will coincide with _TS_, and the vanishing-point of _AB_ will be the sight-point. [15] I spare the student the formality of the _reductio ad absurdum_, which would be necessary to prove this. [16] For definition of Sight-Magnitude, see AppendixÂ I. It ought to have been read before the student comes to this problem; but I refer to it in case it has not. [17] The demonstration is in AppendixÂ II. ArticleÂ II. p.Â 101. PROBLEM IV. TO FIND THE DIVIDING-POINTS OF A GIVEN HORIZONTAL LINE. [Illustration: Fig.Â 15.] Let the horizontal line _AB_ (Fig.Â 15.) be given in position and magnitude. It is required to find its dividing-points. Find the vanishing-point _V_ of the line _AB_. With center _V_ and distance _VT_, describe circle cutting the sight-line in _M_ and _N_. Then _M_ and _N_ are the dividing-points required. In general, only one dividing-point is needed for use with any vanishing-point, namely, the one nearest _S_ (in this case the point _M_). But its opposite _N_, or both, may be needed under certain circumstances. PROBLEM V. TO DRAW A HORIZONTAL LINE, GIVEN IN POSITION AND MAGNITUDE, BY MEANS OF ITS SIGHT-MAGNITUDE AND DIVIDING-POINTS. [Illustration: Fig.Â 16.] Let _AB_ (Fig.Â 16.) be the given line. Find the position of the point _A_ in _a_. Find the vanishing-point _V_, and most convenient dividing-point _M_, of the line _AB_. Join _aV_. Through _a_ draw a horizontal line _abâ²_ and make _abâ²_ equal to the sight-magnitude of _AB_. Join _bâ²M_, cutting _aV_ in _b_. Then _ab_ is the line required. COROLLARY I. [Illustration: Fig.Â 17.] Supposing it were now required to draw a line _AC_ (Fig.Â 17.) twice as long as _AB_, it is evident that the sight-magnitude _acâ²_ must be twice as long as the sight-magnitude _abâ²_; we have, therefore, merely to continue the horizontal line _abâ²_, make _bâ²câ²_ equal to _abâ²_, join _cMâ²_, cutting _aV_ in _c_, and _ac_ will be the line required. Similarly, if we have to draw a line _AD_, three times the length of _AB_, _adâ²_ must be three times the length of _abâ²_, and, joining _dâ²M_, _ad_ will be the line required. The student will observe that the nearer the portions cut off, _bc_, _cd_, etc., approach the point _V_, the smaller they become; and, whatever lengths may be added to the line _AD_, and successively cut off from _aV_, the line _aV_ will never be cut off entirely, but the portions cut off will become infinitely small, and apparently âvanishâ as they approach the point _V_; hence this point is called the âvanishingâ point. COROLLARY II. It is evident that if the line _AD_ had been given originally, and we had been required to draw it, and divide it into three equal parts, we should have had only to divide its sight-magnitude, _adâ²_, into the three equal parts, _abâ²_, _bâ²câ²_, and _câ²dâ²_, and then, drawing to _M_ from _bâ²_ and _câ²_, the line _ad_ would have been divided as required in _b_ and _c_. And supposing the original line _AD_ be divided _irregularly into any number_ of parts, if the line _adâ²_ be divided into a similar number in the same proportions (by the construction given in AppendixÂ I.), and, from these points of division, lines are drawn to _M_, they will divide the line _ad_ in true perspective into a similar number of proportionate parts. The horizontal line drawn through _a_, on which the sight-magnitudes are measured, is called the âMEASURING-LINE.â And the line _ad_, when properly divided in _b_ and _c_, or any other required points, is said to be divided âIN PERSPECTIVE RATIOâ to the divisions of the original line _AD_. If the line _aV_ is above the sight-line instead of beneath it, the measuring-line is to be drawn above also: and the lines _bâ²M_, _câ²M_, etc., drawn _down_ to the dividing-point. Turn Fig.Â 17. upside down, and it will show the construction. PROBLEM VI. TO DRAW ANY TRIANGLE, GIVEN IN POSITION AND MAGNITUDE, IN A HORIZONTAL PLANE. [Illustration: Fig.Â 18.] Let _ABC_ (Fig.Â 18.) be the triangle. As it is given in position and magnitude, one of its sides, at least, must be given in position and magnitude, and the directions of the two other sides. Let _AB_ be the side given in position and magnitude. Then _AB_ is a horizontal line, in a given position, and of a given length. Draw the line _AB_. (ProblemÂ V.) Let _ab_ be the line so drawn. Find _V_ and _Vâ²_, the vanishing-points respectively of the lines _AC_ and _BC_. (ProblemÂ III.) From _a_ draw _aV_, and from _b_, draw _bVâ²_, cutting each other in _c_. Then _abc_ is the triangle required. If _AC_ is the line originally given, _ac_ is the line which must be first drawn, and the line _Vâ²b_ must be drawn from _Vâ²_ to _c_ and produced to cut _ab_ in _b_. Similarly, if _BC_ is given, _Vc_ must be drawn to _c_ and produced, and _ab_ from its vanishing-point to _b_, and produced to cut _ac_ in _a_. PROBLEM VII. TO DRAW ANY RECTILINEAR QUADRILATERAL FIGURE, GIVEN IN POSITION AND MAGNITUDE, IN A HORIZONTAL PLANE. [Illustration: Fig.Â 19.] Let _ABCD_ (Fig.Â 19.) be the given figure. Join any two of its opposite angles by the line _BC_. Draw first the triangle _ABC_. (ProblemÂ VI.) And then, from the base _BC_, the two lines _BD_, _CD_, to their vanishing-points, which will complete the figure. It is unnecessary to give a diagram of the construction, which is merely that of Fig.Â 18. duplicated; another triangle being drawn on the line _AC_ or _BC_. COROLLARY. It is evident that by this application of ProblemÂ VI. any given rectilinear figure whatever in a horizontal plane may be drawn, since any such figure may be divided into a number of triangles, and the triangles then drawn in succession. More convenient methods may, however, be generally found, according to the form of the figure required, by the use of succeeding problems; and for the quadrilateral figure which occurs most frequently in practice, namely, the square, the following construction is more convenient than that used in the present problem. PROBLEM VIII. TO DRAW A SQUARE, GIVEN IN POSITION AND MAGNITUDE, IN A HORIZONTAL PLANE. [Illustration: Fig.Â 20.] Let _ABCD_, Fig.Â 20., be the square. As it is given in position and magnitude, the position and magnitude of all its sides are given. Fix the position of the point _A_ in _a_. Find _V_, the vanishing-point of _AB_; and _M_, the dividing-point of _AB_, nearest _S_. Find _Vâ²_, the vanishing-point of _AC_; and _N_, the dividing-point of _AC_, nearest _S_. Draw the measuring-line through _a_, and make _abâ²_, _acâ²_, each equal to the sight-magnitude of _AB_. (For since _ABCD_ is a square, _AC_ is equal to _AB_.) Draw _aVâ²_ and _câ²N_, cutting each other in _c_. Draw _aV_, and _bâ²M_, cutting each other in _b_. Then _ac_, _ab_, are the two nearest sides of the square. Now, clearing the figure of superfluous lines, we have _ab_, _ac_, drawn in position, as in Fig.Â 21. [Illustration: Fig.Â 21.] And because _ABCD_ is a square, _CD_ (Fig.Â 20.) is parallel to _AB_. And all parallel lines have the same vanishing-point. (Note to ProblemÂ III.) Therefore, _V_ is the vanishing-point of _CD_. Similarly, _Vâ²_ is the vanishing-point of _BD_. Therefore, from _b_ and _c_ (Fig.Â 22.) draw _bVâ²_, _cV_, cutting each other in _d_. Then _abcd_ is the square required. COROLLARY I. It is obvious that any rectangle in a horizontal plane may be drawn by this problem, merely making _abâ²_, on the measuring-line, Fig.Â 20., equal to the sight-magnitude of one of its sides, and _acâ²_ the sight-magnitude of the other. COROLLARY II. Let _abcd_, Fig.Â 22., be any square drawn in perspective. Draw the diagonals _ad_ and _bc_, cutting each other in _C_. Then _C_ is the center of the square. Through _C_, draw _ef_ to the vanishing-point of _ab_, and _gh_ to the vanishing-point of _ac_, and these lines will bisect the sides of the square, so that _ag_ is the perspective representation of half the side _ab_; _ae_ is half _ac_; _ch_ is half _cd_; and _bf_ is half _bd_. [Illustration: Fig.Â 22.] COROLLARY III. Since _ABCD_, Fig.Â 20., is a square, _BAC_ is a right angle; and as _TV_ is parallel to _AB_, and _TVâ²_ to _AC_, _Vâ²TV_ must be a right angle also. As the ground plan of most buildings is rectangular, it constantly happens in practice that their angles (as the corners of ordinary houses) throw the lines to the vanishing-points thus at right angles; and so that this law is observed, and _VTVâ²_ is kept a right angle, it does not matter in general practice whether the vanishing-points are thrown a little more or a little less to the right or left of _S_: but it matters much that the relation of the vanishing-points should be accurate. Their position with respect to _S_ merely causes the spectator to see a little more or less on one side or other of the house, which may be a matter of chance or choice; but their rectangular relation determines the rectangular shape of the building, which is an essential point. PROBLEM IX. TO DRAW A SQUARE PILLAR, GIVEN IN POSITION AND MAGNITUDE, ITS BASE AND TOP BEING IN HORIZONTAL PLANES. Let _AH_, Fig.Â 23., be the square pillar. Then, as it is given in position and magnitude, the position and magnitude of the square it stands upon must be given (that is, the line _AB_ or _AC_ in position), and the height of its side _AE_. [Illustration: Fig.Â 23.] [Illustration: Fig.Â 24.] Find the sight-magnitudes of _AB_ and _AE_. Draw the two sides _ab_, _ac_, of the square of the base, by ProblemÂ VIII., as in Fig.Â 24. From the points _a_, _b_, and _c_, raise vertical lines _ae_, _cf_, _bg_. Make _ae_ equal to the sight-magnitude of _AE_. Now because the top and base of the pillar are in horizontal planes, the square of its top, _FG_, is parallel to the square of its base, _BC_. Therefore the line _EF_ is parallel to _AC_, and _EG_ to _AB_. Therefore _EF_ has the same vanishing-point as _AC_, and _EG_ the same vanishing-point as _AB_. From _e_ draw _ef_ to the vanishing-point of _ac_, cutting _cf_ in _f_. Similarly draw _eg_ to the vanishing-point of _ab_, cutting _bg_ in _g_. Complete the square _gf_ in _h_, by drawing _gh_ to the vanishing-point of _ef_, and _fh_ to the vanishing-point of _eg_, cutting each other in _h_. Then _aghf_ is the square pillar required. COROLLARY. It is obvious that if _AE_ is equal to _AC_, the whole figure will be a cube, and each side, _aefc_ and _aegb_, will be a square in a given vertical plane. And by making _AB_ or _AC_ longer or shorter in any given proportion, any form of rectangle may be given to either of the sides of the pillar. No other rule is therefore needed for drawing squares or rectangles in vertical planes. Also any triangle may be thus drawn in a vertical plane, by inclosing it in a rectangle and determining, in perspective ratio, on the sides of the rectangle, the points of their contact with the angles of the triangle. And if any triangle, then any polygon. A less complicated construction will, however, be given hereafter.[18] [18] See page 96 (note), after you have read ProblemÂ XVI. PROBLEM X. TO DRAW A PYRAMID, GIVEN IN POSITION AND MAGNITUDE, ON A SQUARE BASE IN A HORIZONTAL PLANE. [Illustration: Fig.Â 25.] Let _AB_, Fig.Â 25., be the four-sided pyramid. As it is given in position and magnitude, the square base on which it stands must be given in position and magnitude, and its vertical height, _CD_.[19] [Illustration: Fig.Â 26.] Draw a square pillar, _ABGE_, Fig.Â 26., on the square base of the pyramid, and make the height of the pillar _AF_ equal to the vertical height of the pyramid _CD_ (ProblemÂ IX.). Draw the diagonals _GF_, _HI_, on the top of the square pillar, cutting each other in _C_. Therefore _C_ is the center of the square _FGHI_. (Prob.Â VIII. Cor.Â II.) [Illustration: Fig.Â 27.] Join _CE_, _CA_, _CB_. Then _ABCE_ is the pyramid required. If the base of the pyramid is above the eye, as when a square spire is seen on the top of a church-tower, the construction will be as in Fig.Â 27. [19] If, instead of the vertical height, the length of _AD_ is given, the vertical must be deduced from it. See the Exercises on this Problem in the Appendix, p.Â 79. PROBLEM XI. TO DRAW ANY CURVE IN A HORIZONTAL OR VERTICAL PLANE. [Illustration: Fig.Â 28.] Let _AB_, Fig.Â 28., be the curve. Inclose it in a rectangle, _CDEF_. Fix the position of the point _C_ or _D_, and draw the rectangle. (ProblemÂ VIII. Coroll.Â I.)[20] Let _CDEF_, Fig.Â 29., be the rectangle so drawn. [Illustration: Fig.Â 29.] If an extremity of the curve, as _A_, is in a side of the rectangle, divide the side _CE_, Fig.Â 29., so that _AC_ shall be (in perspective ratio) to _AE_ as _AC_ is to _AE_ in Fig.Â 28. (Prob.Â V. Cor.Â II.) Similarly determine the points of contact of the curve and rectangle _e_, _f_, _g_. If an extremity of the curve, as _B_, is not in a side of the rectangle, let fall the perpendiculars _Ba_, _Bb_ on the rectangle sides. Determine the correspondent points _a_ and _b_ in Fig.Â 29., as you have already determined _A_, _B_, _e_, and _f_. From _b_, Fig.Â 29., draw _bB_ parallel to _CD_,[21] and from _a_ draw _aB_ to the vanishing-point of _DF_, cutting each other in _B_. Then _B_ is the extremity of the curve. Determine any other important point in the curve, as _P_, in the same way, by letting fall _Pq_ and
{'title': 'The Elements of Perspective by John Ruskin', 'source': 'Project Gutenberg', 'lang': 'en'}