ir Digitized by the Internet Archive in 2015 https://archive.org/details/amateursperspectOOdave AMATEUR'S PERSPECTIVE; BEING AN ATTEMPT TO PRESENT THE THEORY IN THE SIMPLEST FORM ; AND SO TO METHODIZE AND ARRANGE THE SUBJECT, AS TO RENDER THE PRACTICE FAMILIARLY INTELLIGIBLE TO THE UNINITIATED IN A FEW HOURS OF STUDY. By RICHARD DAVENPORT, Esq. LON PRINTED FOR THE AUTHOR: AND SOLD EGERTON, WHITEHALL; AND DON: BY J. HATCHARD AND SON, PICCADILLY; COLNAGHI, COCKSPUR STREET. 1828. LONDON: PRINTED BY RICHARD TAYLOR, RED LION COURT, FLEET STREET. TO MISS C A My Dear Lady, THE following Treatise belongs to you. I have not forgotten {and you will probably have sometimes recollected) the promise made concerning it, during our tour in Switzerland and Italy, for so large a part of the pleasure of tvhich I and mine are indebted to you. I very well remember {perhaps you do not^ the good humour with ivhich you received criticisins of mine on draivings executed tvith a nicety very far beyond tvhat I could pretend to. It is more easy to point out errors iti others' poformances, than to avoid them oneself ; and the greater the value of the diamond, the more we 2vish to see it without a flaiv. My promise was, that if you would give me your attention for one week, at the rate of one hour a day, the Treatise I would write for you should put you in possession of all the rules of Perspective necessary for an Amateur. It was long delayed, but I trust is now a 2 iv DEDICATION. fulfilled ; and if you find your skv hours' attention usefully repaid, I shall think my sioc months' occupation requited. At the time I allude to, I had no thoughts of offering my lucu- brations to the public ; but during my labours, I began to think I could make them more generally useful : and that amongst the many who now travel for pleasure, and to whom a portion of that pleasure consists in the recollection of the scenes they ham em'oyed, not a fetv might be found whom it would assist in recording them; and I trust you will not think it unfair that I should so dispose of my work. If you find that I have dwelt tvith unnecessary and even tedious minuteness on some points, which you will think I might well imagine you to be perfectly acquainted with, you will recollect that it is to go before those tvho do not know half so much as you do ; and these i)tstructions and eaplanations ichich to you are not needful, it may be necessary to give to them. Your obliged and obedient Friend and Servant, July, 1828. THE AUTHOR. PREFACE It cannot but be a matter of surprise, that in a confined branch or department of geometry, there should be much variety of system ; that the teaching- of Perspective, whicli is purely geometrical, should not, long ago, have been reduced to a concise and generally adopted me- thod, and that systems and treatises should have succeeded each other for centuries, and books on the subject should still be multiplying. The Author of the present Treatise must tlierefore apologize for adding another to the heap ; and his apology is this : viz. that having looked into most that have been mentioned to him both of old and modern treatises, including three Encyclopaedias, he has not found one that combines all the requisites he deems essential. Some he has found (to himself at least) absolutely unintelligible. Some, involving most unnecessary intricacies. Some, omitting links in the chain of pro- gressive instruction, leaving the scholar in difficulties which can be remedied only by reference to other treatises, or by requiring the assistance of masters ; and in general, the rules and probleins given without the rationale of the system, or demonstration of the correctness of the solutions, for want of which they appear to the scholar as so many arbitrary rules, which frighten him at the outset by their appa- rent perplexity, and escape from his memory afterwards. Even the terms themselves are not sufficiently precise, being used in more senses than one. The " point of sight" is defined "the place of the eye of the spectator a point within the picture " represents it." The line of sight, or imaginary horizontal line terminating in the eye of the spectator, is a line drawn from top to bottom in the middle of the picture. The visual rays that are defined " lines proceeding from the objects to the eye," run to a point in the picture. The line of distance vi PREFACE. in front, is placed above or below, or at the side. No notice is taken how or why these lines " represent" the lines in front, and the whole remains a mystery. If the arbitrary rules are carefully followed, the scholar in despair finds his figure turned upside down, (supposing that to be the system he lias first fallen in with,) and he cannot conceive why. In the " Jesuit's Perspective," some of the diagrams invert the figure, some present it direct. One essential defect in many books of the kind, is the reference to letters arbitrarily disposed in the diagrams, of which the construction is not previously described. The reader's eye wanders about in search of them, and he spends more time in barely finding the letters of reference, than would be required to construct the diagram. If one is misplaced, or an erratum occurs in the letter-press, the study of the diagram is in vain. " Pozzo's Prospettiva" is scientific, probably correct, really splendid in its execution and appearance, and to an architect must be valuable ; and it is not difficult to follow : but its initiatory rules are very meagre, and a beginner will not find it easy to gain from it the first rules he will want, nor indeed much of general principle ; though if he pursues his study, he may learn to delineate frieze, cornice, and architrave, &c. or even a grand deceptive cathedral dome. The " Jesuit's Perspective" is far famed, and handed down as a clas- sical work ; but it does too much and too little. It gives a multitude of problems for putting in perspective a tedious number of figures, which probably the amateur will never want to practise, still without laying down principle applicable to general cases. Dr. Brook Taylor's work is well known, and somewhat dreaded : and Highmore's exposition of the Doctor's system is akin to it. Not that they are without value, far from it ; and in the education of a pro- fessional man, the study of them may be well bestowed : but the undertaking is formidable to one who takes up landscape-drawing for amusement. On the Encyclopaedias the Author scarcely dares to comment. They PREFACE. vii are in every body's hands ; and those who can understand them, and find the information they want, will of course be satisfied without looking- further. The Author of the present Treatise plainly confesses that he cannot. ' : ■ Of modern single books, he has no right to say that none are excellent or sufficient, for he does not pretend that he has seen all ; but he can say that he has searched with some diligence for the assist- ance he wanted himself, and has not found it. He cannot doubt that there are sterling works, and he knows that there are admirable practitioners of the art : but on the whole he does believe that the art is rather traditional than written, and that viva voce instruction, the study of mathematics generally, and professional practice, and tlie in- vention of rules by the individual for his own use, are the constituents of the skill exhibited in their performances. As to such books as are written to assist the teacher, and will not convey the required knowledge without one, they must be deemed nuisances, except such as in their title-page profess that purpose only. These let the teachers use, and all is fair. Whatever good modern books there may be, it cannot be denied that there is abundance of what must be called catch-penny. Children are deemed fair game ; and the trash that is written for them has no other view. Neither reader nor writer understands the page. In the following, it will be found that the Author has made an attempt to simplify the theory, and show how the visible lines directed to be drawn within the picture, do represent the imaginary lines defined in the system. This part may be studied by those who find it interesting. It will certainly facilitate their knowledge of the practical part, enabling them not only more readily to comprehend and re- member the rules, but to apply them universally. Those who would save themselves the trouble of studying the theory, and wish only for a present rule to perform a present operation in practice, may omit it ; and it is hoped that they will find what they want in its proper place in the practical part of the work. The Author has not been studious altogether to avoid repetition, not viii PREFACE. only as mindful of the adage " hrevis esse laboro, obsctirus Jio," but as believing really that time is saved by it ; that the same instruction repeated on a second occasion, and put in a somewhat different point of view, strikes more readily on the imagination, and one part explains another. Besides, very many searchers after practical instruction, like better to read more and think less. Still (lest he should carry this system too far,) he has spent a good deal of time in abridging what first fell from his pen. Some persons object to the study of Perspective, thinking that it confines genius, gives a stiffness of execution, and tends to produce such pictures only, as, if mathematically correct, are yet unpleasing and uninteresting. It is possible that persons, of whose education it has formed a part, may have been so long accustomed to straight lines, that their eye requires them, and their hands habitually form them ; but such effect is by no means the necessary consequence of the study of correct Per- spective. It is probable that the objectors are prejudiced by the unpleasing appearance of the illustrations in books of Perspective, but they should consider them as illustration only, not as examples. In these illustrations generally the horizon-line is made higher, and the line of distance shorter, than the designer of them recommends in his book ; but this is purposely done for the sake of making plain the thing taught, i. e. making it more visible ; for if diagrams were pictures, the parts described would be too minute and crowded, to be intelligible. . ; Again ; diagrams must necessarily be described by actual lines. There are no actual lines in nature, therefore the diagram is displeasing: besides, (and this is an important ingredient of the difference between a diagram and a picture,) visible lines have actual thickness and intensity ; and the same must necessarily be extended to the distance represented ; whereas that thickness and intensity ought to vanish in the true perspective distance. The disgust therefore arises from the diagram (the representative of the picture), not the picture drawn according to its rules. PIIEF ACE. ix But some objectors say, " If one draws what he sees, is not that enough ; why study rules ?" We answer, it is enough. It is all that is wanted, if one draws what he sees : but the rules are given to enable him to " draw what he sees." But he never saw a man slandinp; on the same level with himself, and a mountain or a cottage, or even another man brought by the distance below his elbow, or even shoulder. How often do we see it in pictures.^ He never saw the great window in front of a cathedral directly, or nearly directly, facing him, and the side of the structure at the same time : but many a drawing exhibits them so. He never saw the head or the stern of a boat and its side in profile at the same time. Half the number of boats on paper (\\ol to say canvass,) are so represented. There is a large and voluminous work containing fifty and more fine views of the antiquities and ruins of Rome and its environs. The spectator will there see ranges of columns receding from the eye, equally distant one from the other in the original, and equally distant one from the other in the picture. The like absurdities run through the work ; and whether the artist is so ignorant of Perspective as to think he has " drawn what he has seen," or has affected a slovenly contempt of rules, the worth of his work is annihilated. Even small objects put in in the foreground of a sketch, insignificant of themselves, if out of perspective, destroy the levels, and put the rest of the sketch, as it were, awry : and though some persons have a naturally correct and accurate eye, and will describe " what they see" rightly, without study or instruction, such instances are rare ; and it is worth while to those who draw for amusement, to gain the little know- ledge requisite for their purpose. P.S. When this little work was nearly finished, the drawings for the plates in hand, and the Author undecided whether to engrave tiiem and venture on the publication, — a work just come out was put into his hands, which at first sight checked him, and made him pause. Some paragraphs are so like his own, that if the book had come out after his, he must have suspected them to have been taken from it ; and on b X PREFACE. the other hand, he would find it difficult to vindicate himself against the charge of plagiarism, if friends had not seen his MS. in part two years ago. The work in question has the appearance of great simplicity and correctness ; but when the process detailed is followed through, it is not found so short as at first sight it appeared to be : for no time is really saved by the putting three processes into one rule ; and in fact it would require a great deal of study in a learner, to apply the rules to all the variety of cases in common drawings, without more diagrams and more instruction. Still the work seemed to pre-occupy a part, though only a part, of the road which the following sheets are travelling, and caused the Author's resolution to waver till he came to a note in express contradiction to his first proposition, — a proposition which he conceived to be so obvious, that, except for the rule which he has laid down, he should have left as a point that would be con- ceded of course ; and this proposition is in fact a part of the foundation on which all the rules of Perspective must rest. In the note alluded to, it is asserted, that " no horizontal lines can appear horizontal in the picture, except those which are on a level with the observer's eye and therefore, that " all the horizontal lines in the face of a building, except those which are on a level with the eye, must appear to be curved instead of straight /" In the same note it is asserted, that " it is absurd in anv case to draw the top and bottom of one side of a building parallel to the horizon, and make the other side vanish towards a point in tlie liorizon opposite to the observer's eye," which contradicts another proposition laid down by the author as a rule without any exception ; viz. that " all right lines in the original parallel to each other and parallel to the plane of the picture, will also be parallel to each other in the picture." This note, subversive of the fundamental rules of all Perspective, fortified the Author of the present Treatise, and tended to cure his irresolution as to its publication ; for seeing that the writer of the book and notes in question, (who is a mathematician, as may be inferred from his lan- guage,} if he had attempted to demonstrate these assertions, would have PREFACE. XI demonstrated their fallacy, he more strongly saw the necessity of de- monstrating, instead of authoritatively laying down even that which to his own imagination appears the most evident. It is easy to see how the writer in question has fallen into the error; for he has confused " the distance of objects from the eye of the spec- tato7\'" with their " distance from the "plane of the picture,'''' and has forgotten that the poiiits he alludes to in the perspective plane are also further from the eye, in the same proportion as the corresponding points in the original. It is no disadvantage to the present Treatise, that the note in question has prompted the Author to add a few propositions and diagrams il- lustrative of that part of the subject. He has however, in a slight degree, varied the form of his first theoretical part, having for the sake of reference divided the text into nvmibered sections, and put most of his propositions into that form ; considering explanation by super- position and measurement more popular than abstract demonstration, and leaving only two dry propositions, which the reader may follow through for liis own satisfaction, or may pass over without breaking the chain of the theory. CONTENTS. PART I. THE THEORY. GENERAL PRINCIPLES AND DEFINITIONS, page 1 to 27. PART II. PRACTICAL RULES, AND APPLICATION page 29. Chapter I. Recapitulation of Definitions page 29 to 33. Chapter II. Problems. — To find the Perspective of Points on the Ground Plane, p, 30 ; — Perspective of a Line oblique to Plane of Picture, 37 ; — Of a Line perpendicular to same, 37 ; — parallel, 39 ; — Perspective of Triangles, 39 ; — Of Parallelograms, 40 to 42 ; — Perspective of Divisions of Lines, equal or proportional, 48; — Perspective of Curves, 44 ; — Of Circles, 45 and 46; — ft'i/t^'i^.-i Curvilinear Lincy generally, 49 ; — Volutes and Spirals, 51 ; — Inscriptions, ibid. — For finding Perspectives without inversion of original, 50 and 51 ; — Of the general bearings of Squares and Parallelograms, with their Dia- gonals, 52 and 53. Chapter III. Of Solids^ p. 54 ; — Right-Angled Quadrangular Structures, 55 to 59 ; — Their Faces and Sides with symmetrical Breaks and Divisions, 59 to 61 ; — Of Planes oblique to Horizon-Plane, 61. 63; — Elevations on Circular Planes, (as Columns, &c.) 63 ; — Paradoxical appearance of equal Circles and Cylinders ; those at the sides larger than the central, 63 to 65 ; — In per- spective, diameter not the longest line, 64. — Sections of Solids, horizontal, C6, 67 ; — Vertical, 67, 68, 69 ; — Relative Magnitudes and Heights of Ob- jects, 71 to 74. Chapter IV. Summary, and additional Remarks, p. 74 ;— Various methods of Di- viding Surfaces ; by Diagonals, 77 ; — by proportional Triangle, 79; — Scale for Division extended beyond the Picture, 79 ; — Contrivance for Ruling or Proving a multitude of Lines vanishing together, 80 ; — Shorter Method of putting Circles in Perspective, 81 ; — Geometrical Plan for Circles of all sizes, ibid. — Of Mechanical Instruments ; — Camera Lucida, 82 ; — Divided Frame for choosing a view, 83 ; — Further consideration of Horizon-line, — Point of Distance, — Field of Vision, — Necessity of placing the Zero Point in centre of Picture, — Perspective of very high Vertical Lines and Spaces, —Difficulty of Down-Hill Views, and of Levels much below the Eye, — postponed. Conclusion, 84. THE AMATEUR S PERSPECTIVE. PART I. THE THEORY, Th E science of Perspective, is, the collection and arrangement of rules, for describing on one or more surfaces visible to the eye placed at a given point, such lines or figures as would be described on surfaces so visible, by the intersection of right lines (as rays) proceeding from the objects to be repre sented, and passing through such surfaces towards the eye so placed. The surfaces may be curved, angular, or plane. An instance of the representation on Different surfaces, (interposed one before another), is exhibited at theatres, on the side and back scenes. On Curved surfaces. Spherical, as in a dome. Cylindrical, as in a panorama. In various forms, as in vaulted roofs, niches, &c. On Plane surfaces. As in a picture : and this last may be called Plane Perspective, which is the subject now to be treated of. Section I. A picture may be supposed to represent a pane of glass througli which the objects to be drawn are viewed. B 2 General Principles and Definitions, [Part 1- Points and lines drawn upon that pane of glass, so as to intercept all those objects, would form a picture in perfect Perspective. It is pre-supposed that the eye must be viewing- the objects directly, not ob- liquely, from a fixed point, at a certain distance. The art of Plane Perspective, then, is, to find on the board, paper, or other surface on which we would draw, the positions of points and lines correspond- ing to such supposed picture on the pane of glass. The experiment is easily made ; and it may be worth the while of a beginner to try it, as it will tend to impress first principles on his mind. Take some upright that will stand of itself; such as a light screen-pole, or a point on the back of a chair. Set it before a window, through which build- ings or other objects may be seen, and opposite to the central vertical line of some pane, at a distance somewhat greater than the width of the pane, and at not more than half that height above the bottom of the pane. Shut one eye, and bring the other close to the point of the screen-pole, and thus observe the landscape and objects through the glass. Observe the parts of the glass that severally intercept the lines and objects on the landscape ; and having prepared some opake mixture, (such as whiting mixed up with water and gum,) with a small camel's hair brush dipped into the mixture, trace the objects on the glass, as they appear to the eye viewing them from the top of the pole- This tracing will be the perspective representation of the scene viewed through the window. As the screen-pole or upright used, may not chance to be of a convenient height, take a common card and cut two slits in it, as at 1 — 1, and 2 — 2; and slipping it over the top of the pole, bring it down to a convenient height. First bore a clean hole through it as at a, for the eye. Having found the proper height for sitting or standing, as most convenient, tie a piece of packthread underneath, to prevent its slipping down ; and grasping the pole with the left hand, reach the right hand forward to draw on the glass. If this experiment be carefull}' tried by one not used to drawing, he will be astonished to find how good a representation he has made. He will have a perfect idea of what Perspective is. He wiil see that its rules are not arbi- trary, nor those of an invented art, but that it is a true representation of the objects and view desired : and seeing how good a representation is elFected by a few clumsy lines, by their being drawn in their true places, he will judge how wanting- in value must be the most nicely executed drawing without that truth of design which tells so much in so humble an etfort. ; /— r-i — / a O 2 1— J 2 Part I.] and Detnonstrations of the Truth of the Theory. 3 The author is cautious as to recommending a frequent exercise of this experi- ment, as it tries the sight too much. The eye adapts itself by a different effort to the inspection of different distances; and since in this experiment it is directed to distant objects and to the near glass at the same time, or alternately in quick succession, the rapid change of adaptation fatigues, or the failure of the adaptation distresses it. While it is practised, however, the eye may be relieved at intervals by being covered, or turned to other objects and brought back again to the spot, the screen-pole remaining, to insure a recurrence to the exact point where the eye must be when viewing the sketch*. After one or two trials of this kind, a less fatiguing and more convenient me- thod may be tried. Mark a line only across the pane at the height of the eye, and another verti- cally through the centre ; and draw two similar lines on a sheet of paper. Bring the eye to the screen-pole : and having a pair of compasses, place one foot on some object, and measure the distance from the horizontal line, and from the vertical line ; and then make a corresponding mark on the paper. Proceed with other objects in like manner, and draw lines on the paper con- necting them in the forms you will observe in nature. Section II. In the following pages, the natural landscape or objects so copied, will be called indiscriminately, "the lines in nature" — "the landscape" — "the sub- ject" — or "the original": and the lines drawn, will be called "the picture" — "the representation" — or " the perspective". The points on this glass, where lines drawn from points in the landscape intersect the plane of the glass, are the perspectives of those points : and it will be shown that the lines formed by the junction of those points^ and tlie figures comprised within those lines, are the perspectives of the corresponding- lines and figures in the original. Section III. Having thus settled the definition, and obtained a knowledge of the prin- ciple of perspective, we are to proceed to consider in what way we shall be able to find and lay down on a paper, board, canvass, or other plane surface, such lines as would be in this manner drawn on the supposed pane of glass. *The late learned and ingenious Professor Playfair used, in his Lectures on Optics, to recommend to all persons who frequently used telescopes, to acquire a habit of looking through them without shutting either eye. The practice is a little difficult at first, but soon becomes easy and habitual ; and the sight is much less liable to injury. B 2 4 General Principles and Definitions, [Part I. It will be necessary to begin with supposing objects to be lying on a per- fectly flat extended plane : and when their perspective is understood^ proceed to objects raised on the plane^ and to different levels. Let A A A A represent a level plain. Let F F, B B represent the exterior or frame of the glass or picture, supposed transparent, and standing upright on the plane. (B B being the hottom or base line). Let st be the station, or foot of a pole placed upright at a proper distance from the glass. And e, the eye of the spectator. No. 1. o', and o', are objects lying ou the ground plane ; and the dotted lines are rays proceeding from those objects passing through the glass, and meeting at the eye of the spectator. We shall now call such lines, the Visual Rays of those objects. A A A A the Level Plane, or Ground Plane, though confined in extent in practice, must in theory be supposed to extend indefinitely every way. In like manner F F, B B (the glass or picture) must be supposed in a plane indefinitely extended every way ; and this supposed plane is called the Plane OF THE Picture. And the visible bottom line of this glass or picture, is called the Base (or Fundamental) Line. e, the situation of the eye, is vertical to the point of station, and of course at the same distance from the plane of the picture : and it must be at a due height above the ground plane. The comparative advantages of different heights to be chosen, will be discussed as we proceed. For the present, sup- pose it to be about equal to ^rd of the base line. st, the foot of the screen-pole^ must be placed at a due distance from the plane of the picture_, on the ground plane : and the spot where it stands is called the Point of Station. The line from that point to the base (or the line from the picture to the eye, they being necessarily equal) is called the Line of Distance. Part l.J and Demonstrations of the Truth of the Theory. 5 If the distance be too small, the objects in the picture will be crowded, and the outer ones will appear distorted. If too great^ the picture will com- prise too little. A convenient distance, is that in which the base line, and the two right lines drawn from the point to the extremes of the base, form an equilateral triangle : making the angle 60° of a circle. But the effect is more pleasing when a greater distance is taken : perhaps the most generally adapted to landscapes, is, when the line of distance is about equal to the length of the base line. For easy illustration of this part of the subject, let the student recur to the window and screen-pole. If he places this very near the window, and ob- serves the objects it takes in, and then removes it further, and compares the difference, he will see the comparative advantages and defects. If he sets it too near, the eye cannot take in all that the pane would comprehend : — if too far, the pane will not compreliend so much as is desirable. (These points will be discussed more at length in a subsequent Part.) The quantity,, or sweep of landscape comprehended in the pane of glass under the circumstances thus described, is called the Field of Vision. Section IV. Now, to ascertain the points on the glass or picture through which the visual rays pass, (the points which would intercept the objects from the eye, — where their impressions would be left, if by magic we could fix them), is the whole of Perspective. But many intermediate steps must be taken before we can arrive at the solution of this main and ultimate problem. First, then, let us suppose objects on the ground plane, and lines drawn through them perpendicularly to the base line of the picture; As the dotted lines m m drawn through the objects o' and o% in the following diagram. v Suppose other lines drawn through the same objects parallel to the base; As the dotted lines p p. ' -. No. 2. i i i U p i — 6^- p i i in m The dotted lines of this diagram may be compared to the meridians and 6 General Principles and Definitions, [Part I. parallels on a globe or map ; and may therefore be not inaptly called^ — the former the Longitude^ the second the Latitude^ (of course giving the true place) of the object on the ground plane : and if we find in the picture the perspectives of these meridians and parallels^ their intersections will give the Place of the object in the picture^ or the Perspective of that object. Section V. In viewing through the glass, distant objects lying on the ground plane, it will be seen that the visual rays from the most distant, will intersect the glass (or plane of the picture) in higher points than the visual rays of nearer objects ; i. e. higher above the base line, or (to continue the simile,) in higher parallels of latitude. No. 3. F F B is a side section, or profile of the glass or picture. A A, a section of the ground plane, o', o^, o^, &c. objects lying on it; and, The dotted lines terminating in e, (the eye of the spectator) are visual rays proceeding from the same objects, &c. : and also from others supposed be- yond the diagram at indefinite distances, hut in the same plane. (They are in part discontinued in the diagram, to avoid confusion.) Now, a line or visual ray drawn from 1 to the eye, will cut the plane of the picture at a certain spot. A line from object 2, will cut that plane at a point somewhat higher ; and one from object 3, higher still : and lines from inde- finite distances will cut it at the height of the eye, and will be parallel to the ground plane. It will also be seen, that, supposing these objects to be at equal distances, - one beyond the other, their perspective points on the plane of the picture will not be at equal heights one above the other, but each one will become suc- cessively nearer to the preceding one. Thus, the visual ray of 3, is nearer to that of 2, than 2 is to that of 1. The successive intervals between the perspectives of the objects become less and less, till at length the interval vanishes as it were, and becomes wholly im- perceptible. Part I.] and Demonstrations of the Truth of the Theory. 7 In strict mathematical language, The highest visual ray will not be absolutely parallel to the ground plane : but, practically, it may be called so, as the dif- ference is wholly imperceptible in a picture. (This also will be further discussed in another part.) Section VI. Proposition I. — The perspective of a right line, will always be a right line. And if a right line be drawn connecting any two or more points in the ori- ginal, and the perspectives of these points be found in the picture, a right line connecting such points in the picture, will be the perspective of the line connecting such points in the original. Demonstration. — For, the visual rays from any two points in the original, together with the line connecting such two points, form a triangle whose plane intersects the plane of the picture : and all intersections of planes are right lines. The line connecting the two objects, is the base of the triangle, and the apex is at the eye. The visual rays, the two sides. No. 4. or. Now, a line is composed of an infinite number of points ; and visual rays from any of these points, (being part of the same triangular, plane,) must be in the same intersection ; and, therefore, the infinite number of points of which the line joining the two perspective points is composed, are the perspectives of the infinite number of points, of which the original line is composed ; or in other words, the perspective of that line. Section VII. In the diagram No. 3, the objects are supposed lying one beyond the other in a right line, and directly opposite to the central vertical line of the picture ; and the perspectives of these objects w iH appear on the plane of the picture, at their proper relative heights above the base ; or in what has here been called, their proper latitudes : but the visual rays from objects on the ground plane at equal distances with those from the base of the picture, but wide of that central line, will cut the plane of the picture at equal heights with these. Their perspectives, then, will be in the same parallel. Draw that parallel 8 General Principles and Definitions, [Part I. then ; and such parallel is the perspective of a line passing through points on the ground plane, equally distant one with the other, from the plane of the j)icture ; or in other words, it is the perspective of a line on the ground plane, ])arallel to the base of the picture. No. 3. Section VIII. The highest of the visual rays in the diagram No. 3, is, as has been said, practically parallel to the ground plane ; and as it terminates in the eye of the spectator, the point where it cuts the plane of the picture, must be on a level with the eye. It also proceeds from the extreme distance, or horizon. There- fore, a line in the picture drawn through the perspective point of that visual ray, parallel to the base of the picture, is the perspective of the line of the horizon in nature, and is therefore called the Horizon-line*. In every species of design, the truth of delineation depends on the conformity of all parts to the supposed place of this line in the picture. Section IX. A line drawn from the eye, perpendicular to the plane of the picture, im- pinges on the horizon-line in the centre ; and the point where it meets, is, in treatises on Perspective, generally called the Point of Sight. This leads to confusion ; for, properly speaking, the situation of the eye of the spectator (the point of the screen-pole in our first experiment. Section I), is the Point of Sight. The central point of the horizon-line, or the point where that line and the central vertical line cut each other, is truly the perspective of the central point of the real horizon : but as it must be very frequently mentioned, and this cir- * CaUed in the books frequently the horizontal line : but every line parallel to the ground plane is horizontal. ■ Part I.] and Demonstrations of the Truth of the Theory. 9 cumlocution would be inconvenient, the reader must permit the introduction of a new term, which it is hoped, when so defined, will be intelligible : and it will be called the Zero Point. Note on the word " Perpendiculaii." The word ^^perpendicular'''' is not always understood; and, to confess the truth, it is not used by mathematicians exactly according to its etymology. If a weight be suspended by a string, the line on which it hangs, (^perpends,) is an upright. If a line proceeds from the earth in the direction of a radius from the centre of the earth, it is also " uprighV or " vertical'''' : and thus the words '■^upright,'''' '■^vertical,'''' and perpendicular,'''' should seem to have exactly the same meaning: but the word ^^perpendicular'''' is always con- sidered as relative ; i. e. perpendicular to some other line or surface. "Per- pendicular" to the horizon, is the same as "upright" or "vertical:" but "perpendicular" to a line or surface, denotes its meeting that line or surface in such a direction, as that the angles on each side are equal to one another : i. e. each of them a right angle. d b In each of these figures, c c? is perpendicular to a b. Section X. The height, (or, as it has been called, the latitude) of the perspectives of objects on the ground plane at all distances between the base of the picture and the horizon, having been theoretically proved, it remains to find. The perspective distance from the central vertical line of the picture, (or the longitude) of objects on the ground plane, lying wide of that vertical line. (This line is called in the treatises, the "Prime vertical," and the term will be adopted here.) Let B B represent the base of the transparent picture. Let st mark the distance of the station of the spectator. Let s« be the central line of the ground plane, the perspective of which is, the central vertical line in the picture. c 10 General Principles and Definitions, [Part I. Let 1 2 3 4 5 be objects on the plane, all in the same parallel, (?. e. equally distant from the base,) but not in the direction of the central line. 12 3 4 5 No. 6. - r St Now, as the station is at the same distance from the base line B that the eye of the spectator will be from the zero point in the picture, the dotted lines running from the objects to the station, will intersect the base line at distances from its centre, {i. e. their longitudes) exactly equal to the lateral distances of the intersections of the visual rays of the same objects : and as it has been before shown (Sect. VII. diag. 5), that the visual rays of ob- jects on the ground plane, at equal distances from the base, intersect the picture at equal heights, it follows that these intersections must be in the same perspective parallel, or in other words, at equal heights above the base of the picture. Then, all the intersections (or perspective points) of these visual rays, must be in points vertical to the intersections on the base : and if vertical lines be drawn to connect them, such must be of equal lengths, thus : No. 7. A And the parallel line that connects them, is the perspective of the line in which they he (as before in the diagram No. 5). It follows then, that lines drawn from the objects to the station, or to any Part I.] and Demonstrations of the Truth of the Theory. 11 point vertical to it^ give at their intersections, the distances of their perspectives from the prime vertical : and their distances from the base, determine the height in the picture at which the visual rays intersect it. Thus we have in theory the measure both of the height and the central distance, or as it has been called, the latitude and longitude, or true place. This diagram, and the diagrams No. 1 and No. 5, cannot be made otherwise than in oblique view : so the lines and angles must not be measured mathe- matically, but considered as a perspective representation, and merely to assist in elucidating, not as demonstrations of, the propositions. The upper set of dotted lines is discontinued beyond the transparent picture, for the sake of avoiding confusion of lines : but it will be seen that they are in the direction of visual rays from the objects. The lines on the ground plane are continued at length, and upright lines drawn from the intersections at the base to those in the perspective parallel. These uprights cannot be of equal heights in the diagram, for the reasons given above : but they are per- spective representations of uprights, that would be equal in a direct view. Section XI. If the objects No. 1 2 3 4 5, in diagram No. 6, be placed at equal distances from each other, and at corresponding distances from the prime vertical, but nearer to the base of the picture, (as below in diagram No. 8,) the intersec- tions of their visual rays with the plane of the picture, will be further from the central vertical, and further from each other : and on the other hand, if placed further from the base, (as in No. 9,) the intersections will be nearer to the centre, and nearer to each other. J 2 ^ 4 S Will . No.8. \ I / / \ \ I / / Wl// W f St St And, as in the former case, (Sect. V. diag. 4,) the intersections of the visual rays rose in the plane of the picture, until the extreme became on a level with the eye — so here, the intersections of the rays from more distant c2 \2 General Principles and Definitions, [Part I. objects will continue to approach; and such as proceed from extreme distances, will apparently intersect the zero, or central point, in the picture. J • Section XII. In the last Section, the objects in each several diagram are placed on the ground plane, equally distant from each other, and at corresponding distances from the central vertical line, though further from the base of the picture. Therefore a right line connecting them, will be a right line parallel to that base. Its perspective also, will be a right line parallel to the same. But the objects in the 2nd diagram (No. 9), are further distant from the base than those of No. 8. The perspectives of those lines then, will be shorter than that of the former, as proved by those diagrams, and higher, as proved in Sect. V. diag. 3. The ground plan of similar lines passing through objects at successive distances from the picture, would be as those described in the following dia- gram No. 10, and their perspectives, as in diagram No. 11. . . ■ ■ - F . ■ ' f/ No. 10. No. 11. And the lines in No. 1 1 will be the perspectives of lines on the ground plane, parallel to the plane and base of the picture. Section XIII. It remains to find the perspective of lines 07i the ground plane, perpen- dicular to the plane of the picture. (See note to Sect. IX.) To recur to our window or pane of glass, (Sect. I.) Imagine certain right lines in the landscape to run directly from the window. Such would be the top and the base of a wall ; the sides of an avenue of trees ; the lines of a gravel walk, or the furrows of a plough running in that direction. Supposing Part I.] and Demonstrations of the Truth of the Theory. 13 the screen-pole placed as in Sect. I. and the tracing's of these lines marked as therein directed ; it will be founds that although these lines lie parallel to each other in the original, their perspective lines all converge towards the central point or zero : and we know that such lines in nature (the lines of an avenue notoriously) do appear to converge, and, at an extreme distance, to meet in a point. Imagine then a right line on the ground plane, running perpendicularly from the base of the picture. Assume any two points on that line. Imagine a visual ray proceeding from each of these points ; and where these visual rays intersect the plane of the picture, (^. e. at tiie perspectives of these two points, see Sect. X. diag. 7,) draw a right line to connect them. That line will be the perspective of so much of the line in the original as lies between the points. Now it has been shown that the intersection of the distant point will be nearer to the central vertical line of the picture, and also higher in the plane of the picture, than that of the nearer one : and if the extreme point be taken in the horizon, the visual ray from that point will intersect at the height of the eye and horizon, as shown Sect. V. diag. 3 ; and in the central line, as shown Sect. XI. ; in other words, at the zero : for it is there the central vertical line and the horizon-line meet ; and the visual ray from the nearer point will intersect the picture at a point nearer the base, and wide of the central line, by same theorems. Join these points by a right line, and such line is the perspective of the per- pendicular line ; and if the nearest of the objects is at the base of the picture, the perspective line reaches from the base to the zero. F F V R Section XIV. If a line be drawn any where on the ground plane perpendicular to the base line of the picture, and a line (a visual ray) be drawn from the point where such line meets the base ; and a visual ray from any other point on the same line be also drawn, both meeting in the eye of the spectator, — then these three lines form an obtuse-angled triangle, one side of which is on the ground plane. 14 General Principles and Definitions, [Part I. The plane of the triangle intersects the plane of the picture^ and the line of intersection of the two planes^ runs from the base towards the zero. In order to form a clear idea of the intersection of planes, take two common cards. Mark a line across one, from side to side, representing the horizon- line. Mark the centre of that line. Make a slit in the card from that point, nearly down to one part of the base. As at fig. 1 in following diagram. ' Cut the other card down to an obtuse-angled triangle, as at e 6 c?, fig 2. Set the first card on its base, and put the triangle through the slit, so that the angle e be brought up to the supposed eye of the spectator, and the side b d be on the ground plane ; the angle b at the base of the picture, and the angle d at the distant point. An oblique view of the cards will be as at fig 3. Fig. 2. It follows that the perspective of the line b d running perpendicularly to the base, is in the oblique line of fig. 1, (the slit in the card,) and if the ground line b dhe extended to the horizon (or in other words, if a visual ray in that perpendicular proceeds from a point in the horizon), the ray from such di- stant point will cut the picture at the zero point, (as shown Sect. XIII.) and the whole of that oblique line (or slit) will be the perspective of that whole indefinite perpendicular. Of oblique lines on the ground plane. It has been shown that the representation of lines on the ground plane, parallel to the base, are parallel on the picture. It is evident that the perspective of a line not parallel to the base, must tend towards the distance, and meet the horizon- line : and since nothing can be visible beyond the horizon, it must terminate there. The perpendiculars terminate (as has been shown) in the centre of that line, (the %ero) ; and oblique lines must terminate wide of that point : and as the horizon- Part I] and Demonstrations of the Truth of the Theory. 15 line is to be supposed of indefinite extent, the point to which they tend may be within or without the picture : and such of them as are parallel to each other vanish in the same point. Section XV. Having- arrived at the theoretical conception of the perspective representa- tion of points and right lines, (which, as all figures are bounded by lines will extend generally to rectilinear figures,) the next step will be, to describe the means of application of the theory, and to demonstrate the truth of the appli- cation. This step will be prepared by two more definitions, ^ If, for the purpose of putting in perspective a certain object on the ground plane, at any distance from the base, lines be drawn from its several points perpendicular to the base of the picture, such lines are called the Lines of Incidence of these points, and the point on the base on which any such line impinges, is called its Point of Incidence. . i . . : ; ^ A line drawn from the point of incidence to or towards the zero point, is the perspective of such line of incidence, and in the following pages will be called the Radius of that point. It is the perspective of the Meridian of the point from which it is drawn, ac- cording to the nomenclature adopted, Sect. IV. diag. No. 2. Section XVI. The plan on the ground plane must of course represent the figures in their true geometrical forms and proportions. For this reason, the ground plane having such figures described upon it, is called the Geometric Plane, in con- tradistinction to the perspective plane, where the forms of tlie same figures are quite different. (See Sect. XII. diag. 10 and 11.) Section XVII. From certain points in this geometrical representation, (as the angles of a rectilinear figure,) lines of incidence are to be drawn to the base ; and from the points of incidence, radii are to be drawn to the zero. (See Sect. XV.) The perspectives of these points must lie in those radii (or meridians) : and we have now to construct another set of lines for finding the latitudes (or parallels), i. e. their heights on these radii. 16 General Principles and Definitions, [Part I. Section XVIII. But the student objects_, that his picture is not transparent; and since the objects are (of course) beyond the picture, he is stopped at the outset of his work. Let him then, instead of the transparent pane in his first experiment, imagine a plain mirror set opposite to his landscape : and he now turns his back on the objects, and imagines his tracings on the looking-glass. Now it is well known that on this glass the same view may be had that would be seen through the transparent pane : the self-same view, seen by reflection of the visual rays, instead of their transmission ; and it will appear altogether to the spectator, as if the scene and objects were placed on the other side of the glass : and there will be an image as of another spectator, the place of whose eye will appear to be exactly at the same distance as his own, and in the same horizontal and vertical lines. This point, then, may now be supposed the eye of the spectator, (or "^"^ place of the eye,") and the visual rays proceed towards that, exactly as before represented towards the real " place of the eye," in Sect. I. diag. 1 : and they will be reflected towards the eye of the real spectator, from that point in the mirror which they would have intersected on the transparent pane ; and in exactly the same direction that would be, if they proceeded from objects behind the glass. No. 13. Fig. 1. Fig. 2. F A A A A the ground plane : B B the base of the picture : a} 6' objects on the ground plane : and the same reflected : e the eye : e a and e b visual rays ; which by reflection appear to proceed from and b^. The dotted lines on the ground plane from a and b to the picture, are lines of incidence ; which also are reflected, and appear to proceed from and 6^ The dotted lines in the picture are the meridian lines, or radii from the points of incidence towards the zero, and are i\ie perspectives of the lines of incidence. Part L] a7id Demonstrations of the Truth of the Theory. 17 A line joining them in the picture where the visual rays touch the picture and are reflected, would be the perspective of a line joining ah on the ground plane. For the angle of reflection is equal to the angle of incidence. Rays striking perpendicularly on a reflecting surface, are reflected in the same perpendicular ; and of course no angle is formed with that surface, either by the incident or the reflected part of the rays : but when a ray impinges on a reflecting surface with more or less obliquity, it is reflected at its point of contact : and the angle between the reflected ray and the surface, is equal to the angle between the incident ray and the surface. Thus, let F B (fig. 2, diag. 13) represent the side section of a plane mirror. Let a be a point from which a ray proceeds towards it. Let p be the point where the ray impinges, and from which it is reflected towards e. Then the angle between the lines e p and p is equal to the angle between the lines a p and p B ; and if a p and e p were both prolonged, the prolonged lines would be exact representations of the original ones. So are the lines in fig. 1 prolonged to and 6^ Section XIX. But the student will still object, that his picture is supposed to be vertical to the ground plane. How then can he construct his geometrical plan on the same paper ? This difficulty is easily removed : for the geometric plane and plan are wanted for the purpose of measurement only : he may therefore lay them down separately, and transfer the measures from the geometrical plane to his picture : or, the geometric plane may be laid down underneath the base line of the picture, wiiile the principles of perspective are the object of study, though it would not be convenient in general drawing. In fact, the lines of incidence run along the ground plane, and mark on the base the poiiits of incidence, whether the picture stand vertically on the ground plane, or be itself in the same plane with it. Let him then measure from the centre of the base, a distance equal to the intended distance of the eye from the perspective, (as in his first experiment on the pane of glass) ; then draw a parallel line across his picture at the height intended for the horizon, thus : F F, B B, (diag. 14.) the picture. a and b, objects on the ground plane, now laid down geometrically under- neath the picture. 18 General Principles and Definitions, [Part I. The dotted lines of incidence run to the base, and dotted radii from the points of incidence to the zero. st the station at the proposed distance. It has been sufficiently shown in the previous propositions, that the height of the objects in the picture depends on their distance on the ground plane ; and also on the supposed distance of the eye from the picture. But now a new difficulty occurs. Although the distance of station is equal to that of the eye, still, as the place of the eye is the point where the visual rays meet, and this is the level in which the horizon-line is, the point of station will not answer the purpose ; because its line will cut the base instead of the horizon-line : and to draw the line of sight in front of the picture is impossible. The zero point is the section of that line. We must therefore, for the purpose of mea- surement, lay down elsewhere a line equal to it, measured from the zero. Suppose the vertical line v c (diag. 15), to represent a side section of the prime vertical line of the picture. o the zero, and c the eye : the horizontal line e o being equal to the distance of the point of sight. a, a distance from c on the base line, equal to the supposed distance of an ob- ject on the geometric plane in front of the picture. The line a e will intersect the vertical, and that intersection will give the height in perspective of the supposed object. No. 14. No. 15. Then, if the height of this intersection be transferred to the central vertical line of the picture, a parallel drawn through it will intersect the radii, and give the height (or latitude) of all objects equally distant from the base of the picture ; and thus a practical method is given for ascertaining the perspective of any point on the original ground plane. Part!,] and Demonstrations of the Truth of the Theory. 19 Section XX. Instead of drawing- lines for the above purpose on a separate paper, it may save trouble to use the lines already drawn in the picture, and to prolong the horizon-line on each side beyond the frame line of the picture, to a length (measured from the zero) equal to the line of distance, and mark the point of distance there : and a length equal to that of the line of incidence may be measured with compasses on the base line, commencing from the bottom of the vertical, and marked as at a in the last diagram : then the edge of a ruler (or a thread) laid from that mark to the side point of distance in the pro- longed horizon-line, will cut the vertical at the height of the parallel sought. This is in fact merely laying down the last diagram in the picture, instead of a separate paper : but a method may be practised requiring Jewcr lines ; and the above is given principally for the purpose of leading to the demonstra- tion of it. Section XXI. Instead of finding the parallel by the foregoing method, viz. by intersection of the vertical, it may be found on the perspective of the line of incidence, (i. e. on any radius, or meridian). Having drawn a line of incidence and its radius, mark off on the base line a distance from the point of incidence of the object, equal to the length of the line of incidence. From that mark draw a line to the side point of distance, and it will intersect the radius in the proper parallel. V c, the central (or prime) vertical line. 0, the zero. h d, the horizontal distance from the zero, viz. the point on the prolonged horizon-line, at an equal distance with that of the eye in front. a, an object on the geometric plane ; the dotted perpendicular is its line of incidence. And its radius is also dotted. 1, a point marking a distance on the base from the point of incidence, equal to the length of the line of incidence. (The dotted curve belongs to the dia- gram, no otherwise than as indicating the equal measure, as if drawn by compasses.) This point will now be called the Quadrant Point. 2, is an equal measure from the bottom of the vertical c ; (it is the same as that of a in the former diagram 15). A line is drawn from I to h d, cutting the radius. D 2 20 General Principles and Definitions, [Part T. And another from 2 to the same point, cutting the vertical. p p, a line drawn througli these intersections. And it will be found that such line will be parallel to the base line of the picture : so that a line drawn through the radius from the quadrant point of any line of incidence to the side point of distance, is equivalent to a similar line drawn through the vertical, as in the last diagram ; and will give, at its intersection with the radius, the parallel sought, and therefore the place of the object in perspective; and is a readier method of finding it. No. 16." a ■ [ Note] The truth of this may be so far apparent, as to satisfy the reader that it will be sufficient in practice ; but according to the plan professed by the author in his preface, its sufficiency ought to be demonstrated : and therefore he gives the following proposition and demonstration, which the reader may study or omit at his option, without interrupting the chain of the general theory. Proposition. — If lines be drawn diverging from a given point in a right line, to any num- ber of given points in a second line parallel to the first, and from other points on the second line, severally at equal distances from those points, other lines be drawn crossing the said diverging lines, and meeting in any other given point on the first line, the several intersections of each of tliese last described (or converging) lines, with its corresponding one of the diverging lines, will be in the same parallel. o c and a are two points on the lower or second line, to which lines are drawn diver- ging from the point o in the upper or first line. 2 and 1 are other points in the same lower line ; 2 being at the same distance from c, that 1 is at from a. From 2 and 1 draw lines severally crossing o c and o «, and converging in d, (another point in the upper Une). Part I.] and Demonstrations of the Truth of the Theory. 21 I say, the intersection of 1 d with a o, is in the same parallel with the intersection of 2 d, with c o. (q is the first intersection, and p the second.) Dem. For as the bases of the two proportional triangles I q a, and o q taken together, are to the base of the former, 1 a, so are the altitudes of the same triangles, taken together, to the altitude of the same. Also, as the bases of the triangles 2 p c, and o p d taken together, are to the base of the former, so are their altitudes, taken together, to the altitude of the same : for these triangles also are opposite, equi-angular, and proportional. Now the altitudes of one pair of opposite triangles taken together, are equal to the altitudes of the other pair ; for by the terms of the proposition they are between parallel lines. Also the line o is a base common to each pair of opposite triangles, and the other two bases are by the terms of the proposition equal ; so that a line equal to u d and 1 «, taken together, is equal also to o d and 2 c taken together : and Such line I call H. Also a line equal to either of the 2 equal bases 1 a and 2 c, I call I. And a line equal to the tvro altitudes of either pair, I call K. And the altitude of 1 g a, I call alt. 1, or the first altitude sought. And that of 2 p c, I call alt. 2, or the second altitude sought. Then it follows that as H is to I, so is K to alt. 1. Also as II is to I, so is K to alt. 2 ; and therefore the altitude of 1 g a is equal to the altitude of 2 p c ; and since their bases are on the same right line, a line drawn from the apex of one to the apex of the other, must be parallel to that line. Section XXII. Ag-ain the student objects, that his picture^ in point of fact, does not stand on the ground plane; so that neither reflected nor transmitted hues of inci- dence can be drawn on the ground plane so as to touch the base. This is true ; but the difficulty is easily removed. When a painter would represent on a wall, or very large vertical plane, a scene on the scale of nature, or the original scene, exterior or interior, such as may be supposed to exist there, the base of his picture is on the plane of station of the spectator; and such situation of the picture is usually given in books by way of example and illustration of the rules of perspec- tive ; but as easel pictures and drawings are not so placed (nor can produce the effect, unless they be of a size to comprehend figures as large as life, and space above them), the illustration is imperfect. It serves well enough, however, as an introduction ; and after the student has entered into the principle of the theory by the foregoing- detail, he has only to change the scale in his imagination, — to suppose the picture, proportionably diminished in height and width, and brought nearer to the eye, — and the ground plane, with the objects and geometric plane and lines of incidence, raised in the same proportion. 22 General Principles and Definitions, {Part I. Let F F, B B, represent the frame of the picture, as in foregoing diagrams, e, the place of the eye. F The four dotted lines, visual rays from the four angles of the picture, meeting at the eye. ff,bb, the smaller picture, raised and bi'ought nearer to the eye. It must be diminished in size, so that the dotted visual rays shall cut its four angles at ff^bb; and then at whatever intermediate distance it shall be placed, these visual rays will determine its true proportion and height. A, St, B, is a section of the ground plane on the same scale as the larger picture. fl, st, b, is a section of the supposed ground plane on the diminished scale. The two frames, the two sets of visual rays, the height of the eye, the lines of distance, the ground planes, and all figures geometrically drawn on the ground plane, and their perspectives in the smaller picture, will be pro- portional ; and the nearer ones (supposing them drawn on transparent sur- faces) will exactly intercept the larger. Section XXIII. . The theory of the perspective of lines on other planes besides the ground plane, is perfectly analogous, and will therefore require but little explanation. Of Horizontal Planes, or Planes parallel to the Ground Plane. Take first such as are above the eye ; as the ceiling- of an apartment. The geometric plane might be laid down above the picture. The points, lines, and figures, drawn on it; the lines of incidence in like manner, and their perspectives, would meet in or tend towards the same zero point. The parallels of the distances would be found by taking on the upper line of the picture, the quadrant distances and finders, which still meet in the same side- distance-point on the prolonged horizon-line. This process will seldom be necessary or expedient, as there may be found readier modes of measurement ; and principally by supposing vertical lines from the perspectives on theground plane, which will touch the superior plane. Part I.] and Demonstrations of the Truth of the Theory. 23 Section XXIV. Three horizontal planes have been mentioned and defined. The ground plane, which is the lowest and most extensive visible plane in the picture, and occupies the greatest part, if not the whole, of the base line, and vanishes, in the distance at the horizon. The plane of the eye, into which the ground plane vanishes at the horizon, and which is therefore called the horizon- plane. The highest plane (or the plane, whether visible or imaginary, in which is the highest visible point on the picture), as the sky, or a ceiling, &c. which also vanishes in the horizon-plane. But the number of horizontal planes that may require partial representation is indefinite : for instance, one part of the picture may consist of flat land, on whicli the spectator stands; another part, of water, which of course is lower; and many diiferent levels occur both below and above the eye. The reader will be no longer detained by the consideration of them here, as the determi- nation of the height of the horizon line (to which subject this is incident), is reserved for a subsequent part of this treatise : but we are now led to the definition of another term, which, to avoid circumlocution, will be used in the practical part ; and we now call horizontal planes above the eye, Superior Horizontal Planes. And such as are below the eye. Inferior Horizontal Planes. They all vanish into the horizon plane, and constitute there the horizon- line. (See Sect. V. and VIII.) Section XXV. Of the Perspective of Lines on the several Horizontal Planes. On superior planes, lines parallel to the upper line of the picture, remain so in the perspective, in like manner as in the ground plane, and other inferior planes ; for lines parallel to the upper fine, must also be parallel to the base line ; and such have been proved before (Sect. VII.) ; and of lines perpendi- cular to the upper line, the perspectives are radii meeting in the zero on the horizon-line : and oblique lines follow the same law as those on the ground plane (Sect. XIV.) ; and such of them as are parallel to each other, meet in, or tend towards the same point in the horizon-line in or out of the picture; which, as before said, is therefore called their vanishing point. 24 General Principles and Definitions, [Part I. Section XXVI. Of Vertical Planes, or Planes perpendicular to the Ground Plane. These may be either perpendicular or oblique to the plane of the picture (Sect. Ill and IX. and infinite in number. Those which are perpendicular, all meet at (or vanish in) the prime vertical line (Sect. X.), in the extreme distance. They are, in fact, the planes of the radii ; and vertical lines standing on the infinite number of points, of which a radius on the ground plane is composed, would constitute the plane of that radius. The planes themselves intersect the horizontal planes at right angles ; but the perspectives of such vertical lines will be limited in height by the per- spectives of the highest horizontal lines represented : those in the front of the picture will reach from the base line to the upper line, and the succeeding ones will diminish in height till they vanish to a point at the zero. No. 19. No. 20. After the descriptions of former diagrams, this cannot need any explanation, further than to say that the horizontal lines represent the parallels of equi- distant lines on the superior and inferior horizontal planes; and the vertical lines stand on the extreme points of the lower parallels, and are limited in height by touching the superior parallels. Such planes might be represented standing on any radius on the ground plane, and have place frequently in pictures ; as in the sides of buildings, &c. The two principal ones are the two side planes, which the above diagrams represent, and which are the subject of the following Section. Section XXVIt. Of the Side Planes. The vertical planes (in fact all the planes) of a picture are imaginary only, except when some solid flat surface is to be represented ; and when these are Part I.] and Demonstrations of the Truth of the Theory. 25 partial^ the plane is truncated. Diag-. 19. represents the two vertical side planes, and the highest and lowest horizontal planes carried to the utmost visible distance^ and vanishing in the zero. Diag-. 20, two side walls, a ceiling and a floor, terminated by a fiat wall ; so that both the vertical and hori- zontal planes are truncated (cut olF) by a distant plane parallel to the plane of the picture. Still the intersections of the two pairs of planes are radii con- verging- to the zero ; and so would be lines on any of them perpendicular to that plane. Section XXVIII. Objects on the Side Planes (such as the sides of a rectangular apartment,) might also be put in perspective, by the construction of geometric planes at the side of the picture, in the same manner as that underneath the ground plane : but the process is entirely unnecessary ; and to practise it, would be in fact very nearly doing the business twice over : for as a side plane is perpen- dicular to the ground plane, it is obvious that lines parallel to one, are per- pendicular to the other. Now, of lines parallel to the ground plane, the per- spectives have been already proved (Sect. VII, X. and XII.) ; and lines per- pendicular to the ground plane (vertical lines), must be vertical to some point on that plane : so that vertical lines parallel to the sides of the picture, are vertical to the points whose perspective has been already demonstrated ; and such of them as are on the side planes, must be vertical to points in the intersections of the side and ground planes; i. e. in the two outer radii which extend from the extremes of the base line to the zero point, as in diagrams 19. and 20 : and such as are perpendicular to the side lines of the picture, are themselves radii converging to the zero. No. 2J. No. 22. h- h- -h In both these diagrams, the converging lines are the perspectives of lines on the side planes perpendicular to the side lines. E 36 General Principles and Definitions, [Part I. Section XXIX. fAe Field OF Vision. It has been shown in Sect. III. that the extent of the field of vision is de- termined by the distance of the eye from the plane of the picture ; (in front of the pane of glass. Sect. I.) and that tlie shorter this line of distance is, the more sweep of landscape will be opened to the eye. Lines from the eye, extending through the plane between the sides of the picture, form a certain angle at the eye ; and the horizon-line constitutes the base, and completes the triangle. The geometrical form of this great indefinitely extended triangle Avill be, of course, on the ground plane beyond the picture ; and according to the sy- stem here adopted, must be represented by reflection in front of the picture, (in like manner as the ground plane in Sect. XVIII. diag. 13.) F F No. 23. — 1 — B B, the io^e of ^/jp^jj/c^^re prolonged indefinitely- st the station, or point of distance. The lines st B are reflected towards the " required, at distances from one another equal to the divisions on the original line. These will be the quadrant points of the divisions, and the finders will intersect the perspective line at the points sought (as in fig. 1. of this problem). The points will be nearer and nearer in the perspective, and the remote in- tervals will vanish. Parts of buildings equally distant from each other, on a face or plane per- pendicular to the plane of the picture, equidistant columns running in that di- rection, equal parts of an avenue through which the spectator looks directly, will be thus measured in perspective. [Prob. IX. fig. 2.] If the line be parallel to the plane of the picture, there will be as many lines of incidence as there are divisions. G 2 44 Practical Rules, and [Part II. These also will find their points of incidence on the base, at intervals equal to those on the original line ; and therefore may be at once marked by mea- surement, as ], ^, 3, &c. on the base in the diagram. The radii then will cut the perspective line in divisions smaller than those of the original line, but proportional; and of course equal to each other, as 1, 2, 3, &c. in the picture. Such are equidistant windows^ columns, trees, &c. in lines parallel to the plane of the picture. [Prob. IX. fig-. 3.] If the line is oblique, it will have (like the latter) as many lines of incidence as there are divisions ; and they will fall on the base at equal distances one from the other, but less than those in the original. The line on the ground plane being divided, as at 1, 2, 3, 4, &c. fig. 3. Draw its perspective (by Prob. II.). In the figure it is made to cut the side line of the picture, which will take in so much of it as lies within the field of vision. Were it nearer to a perpendicular, it would cut the horizon-line. Draw the fnders, which will cut the perspective line itself at the points sought ; which will not be equal to one another like those of fig. 2. but at successively diminishing intervals ; and the ratio of diminution will depend on the degree of obliquity. If the perspective cuts the horizon-line, the intervals vanish in the distance. This problem is of almost universal practical utility ; and may be extended not only to equal, but to any known proportional divisions or intervals of the buildings, trees, columns, &c. to be represented. If, for instance, the fa9ade of a palace has windows equally distant, then a space, niche, portico, or co- lumns, &c. and between these, other windows, and the proportions of the spaces are apparent to the eye, mark the similar proportionate measurements on the base line, and their radii (or finders, as the case may be) will give the perspectives of the proportionate, as well as those of equal measurements. Of the Perspective of CvRYEs. It is impossible to put curves in strict perspective by commensuration with right lines. Lines consist of an infinite number of points. It has been proved, (Sect. I. p. 7. of the Theory) that the perspectives of right lines are right lines ; and the perspectives of any two points being found, there is no diffi- culty in joining them by a right line; and that line will be the perspective sought. Not so in curves ; for if the perspective of two points in the curve be found. Part II.] Applicatio7i of the foregoing Theory. 45 there is no rule by which we can describe the curve that joins them. All that can be accomplished,, is, an approximation. Take at pleasure any number of points in the curve on the geometric plane; and having" put those points in perspective^ join them by curves falling easily into one another. The greater the number of points taken, the nearer will the curve be to the truth ; but the difference between them can never be an- nihilated. Problem X. To put a Circle in Perspective. Describe the circle on the geometric plane. Through its centre draw the first diameter a b, in a direction perpendicular to the plane of the picture. Also the second diameter c 6/,prtrrt//eHo the same plane. These will divide the circle into four equal parts. Describe a square that shall inclose the circle, touching it at the points abed. Put the square in perspective, with its two diameters (by Prob. III. IV. and VI.), and mark the points on the four sides of the square, where they touch the diameters a b c d (a being the nearest to the base, according to the rule under Prob. III.). NoW;, since the original circle touches the original square at those four pointS;, so will the intended perspective circle touch the perspective square in the corresponding points. - In the original circle, draw further the two diagonals cutting the circumference at ef, and g h ; (e and g marking the two intersections nearest to the picture.) Do the same in the picture. The diagonals in the picture will be the perspectives of the original diagonals. Again in the original, from // through e, and from / through g, draw lines per- pendicular to the picture (which will be lines of incidence of those four points). In the picture draw their meridians, which will be the perspectives of those perpendiculars. NoWj the diagonals of the original square divide the quarter circles equally ; and of course *the whole circle is divided into eight equal parts : and these perpendiculars cut the diagonals at corresponding points. T\\e perspectives, then, of these perpendiculars (viz. the meridians) cut the perspective diagonals at corresponding points. 46 Practical Rules, and [Part II. We have, therefore, the perspectives of the eight points in the circumference of the intended circle ; which being joined by curves falHng easily into each other, as nearly as the eye can decide, will be a tolerable approximation to the perspective of the circle. The circle is a form continually wanted ; as in the representation of a well in a landscape ; the base of a round tower, or its top, or intermediate lines around it ; or the base of a column, or its summit under the capital ; or even a common hoop or tub. For these purposes, the method above described is sufficient ; but if the circle is large, and to be put in on a large scale, it will be difficult, even with an experienced eye, to proportion the swell of the in- termediate curves between the eight points laid down ; every one of which curves differs in form as well as magnitude from the others. The interior of a regular amphitheatre may be wanted, or a circle on the pavement of a church may constitute a large and important proportion of the representa- tion ; and a want of truth in such parts of the picture may destroy much of the general effect. In order, then, to obtain a nearer approximation (for, as has been said, that is all that can be attained), a greater number of points must be determined in the circumference of the original. The equality of the divisions is not ne- cessary ; and certain divisions may be obtained very easily, by dividing either of the diameters into equal parts, and drawing parallel lines through those di- visions, which will cut the circumference in certain^ though ?/wejwa/ portions. But, for reasons which it seems unnecessary to detain the reader upon, it is thought best to make the divisions on the circumference equal : and, as the most ready way of dividing is to bisect, let the eight arcs (or parts of the circle) described in the above problem be divided equally, and we have sixteen divisions. Of these the next problem treats. Problem XI. To put a Circle in Perspective with more precision, hy means of a greater number of divisions of its circumference. Describe the original circle with its four diameters and circumscribing squares, and divide it in eight points, as in the last problem, marked in the same order, a, g, d,f, b, h, c, e : mark these points successively, 2, 4, 6, 8, 10, 12, 14, 16, and put the whole in perspective as in the last diagram. We are now to find eight intermediate divisions. Draw a line (a chord) from a to g, and bisect* it; and through the point of * Though by etymology this word signifies merely "cut in two," it is always understood to express "in equal parts." Part II. J Application of the foregoing Theory. 47 bisection and the centre of the circle, draw a line, (a diameter,) reaching the two opposite points in the circumference. Mark the point near the new chord 1, and its opposite point 9. Draw the chord from g to d (2 to 4), and bisect it; and through the bisection draw the diameter 3, 11. Then on the other side of the upper part of the circumfei'ence, draw the chords 16 to 14, and 14 to 12; and bisect them, and draw their diameters in like manner, viz. 15 to 7, and 13 to 5. The circle is now divided into sixteen equal parts. Now draw parallel lines joining 1 and 15, 2 and 14, S and 13, (4 and 12 is done before,) 5 and 11, 6 and 10 : and where these parallels cut the perpendicular diameter, mark on it the divisions 1, 2, 3, 4, 5, 6, 7. The perpendicular diameter, and its two extreme points a and h, (now called 16 and 8,) are already in their places in the picture ; therefore by quadrants from !, 2, and 3, on that line in the original, and their fnders, find the corre- sponding divisions on the perspective diameter, and draw parallels through them to the sides of the square in the picture. As these are the parallels (latitudes) of 1, 15 — 2, 14 — and 3, 13 in the original circumference, so will they be in the picture. The perspectives of 2, 14, were known before. The longitudes of the other two pairs are wanted. Their four lines of incidence, viz. 1, 3, 15 and 13, and their perspective tneridians, will intersect the parallels at their per- spective points. There are now four points ascertained, in addition to the eight of the former problem. Mark them accordingly, and number the former eight in the per- spective in correspondence with the numbers in the original : and thus twelve points of the circumference are put in perspective, viz. 1, 2, 3, 4 — 6—8 — 10 — 12, 13, 14, 15, 16. For the remaining four, (viz. points 5, 7, 9, 11,) their longitudes are already ascertained : 5, being on the merrdian of 3 ; 7, in that of 1 ; 9, in that of 15 ; and 11, in that of 13. Then, since the quadrant lines of the distant points would be long, and the finders would run up in the contracted space of the distant perspective, where the lines of latitude would crowd the figure, it seems better to find the remaining four points by the following method. Observe, that a diameter drawn from 1 through the centre, will cut the meri- dian of 45 in the parallel of 9 in the original. So must it in i\ie jjerspective. The diameter of 15, cuts the meridian of 1 at the point of 7. That of 13, cuts the meridian of 3 at the point of 5. And that of 3, cuts the meridian of 5 at the point of 11. Draw these diameters accordingly in the perspective, and put the marks where they cut the perspective meridians {radii of the zero). 48 Practical Rules, and [Part II. Draw the sixteen arcs to join them^ and the circle is complete. Now it is evident^ that if the circle is to be very large in the picture^ and the painter wishes to be still more exacts he may bisect these sixteen angles in the original, and find thirty-two equidistant points in the circumference. Or if, perceiving that the curves of the several arcs in the perspective cannot be uniform, (every arc having a swell in one part peculiar to itself,) he finds his eye dissatisfied with it, he has always this resource, viz. to mark another point in the original, in that part of the circumference of which he finds the per- spective most difficult, and putting that point in perspective, curve his line through it. In the Jesuit's perspective is given a rule (called Serlio's) for putting in perspective a circle divided into sixteen equal parts ; proposing a semicircle (of which the diameter is the base of the picture), and that semicircle divided into eight equal parts. The author of ihe present treatise would vindicate himself against the charge of affected singularity in rejecting an accepted and sufficient rule, and substituting another problem for the same purpose. He has more reasons than one for preferring the one he has proposed above. 1st, That although Serlio's problem is both simple and sufficient, it is not de- monstrated, nor reduced in tertns to previously demonstrated principles : whereas the problem above given, carries on the same chain of principle that was begun with. Serlio also (or his quoter) contents himself with referring to the figure, and directing the reader to "make points from angle to angle, and connect them by crooked or circular lines now for this, he must rely on his diagram to explain the explanation ; for there are sixty-four angles, of which he wants only sixteen : but it has been the system in the present treatise to make the diagramyro/w the instruction. Again, in Serlio's, no instructions are given to the student, enabling him to '^'^ divide his circle into any number of equal parts." He wrote for mathematicians, who could not need such in- structions ; but an amateur who may not have studied geometry at all, or a mechanic taking up an elementary book, has a right to expect full instructions to enable him to perform the operations as therein explained. Again, the rule and diagram in question enable the reader only to put the perspective circle in one place in the picture ; and so far it is insufficient. On the whole, if all necessary instructions were included in the rule as quoted from Serlio, his diagram would include a greater number of lines, and his rule require more words than the problem and explanation given above. Part II.] Application of the foregoing Theory. 49 Problem XII. To put in Perspective other Curvilinear Figures and Curved Lines GENERA LLYj ichcthcr regular or not. In the last problems the advantage was shown of dividing- the curve into corresponding parts ; so that after the perspective of any one part was de- termined^ that of other parts might be ascertained by their relative positions : and the same principle is easily extended by analogy to other regular curvi- linear figures. An oval, or ellipse, for example, may be inclosed in a parallelogram, of which two diameters and two diagonals may be drawn, which shall divide it into eight (not equal, but) correspondent parts; and the whole will be easily put in perspective in perfect analogy to the practice on the circle : but with irregularly curved lines, spirals, volutes, scrolls, &c. &c. the points must be put each severally in perspective. A considerable assistance however may be obtained by inclosing the whole in a rectangular figure, touching as many parts of the curve as it can ; and dividing that again into smaller rectangles that inclose and touch smaller parts of the curve ; and afterwards, if necessary, drawing parallel and perpendicular lines cutting the curves on intermediate parts. Take the spiral of fig. 1. on the diagram of this problem. Plate VI. Draw the rectangle A B C D inclosing the whole figure, and touching the highest and lowest parts, and touching also the two opposite sides, and put this rectangle in perspective. Then draw the two parallel horizontal lines e e and f f, touching the highest and lowest swell of the inner part of the spiral ; and also the two perpen- dicular lines g g and h //, touching the two outer points of the same. The figure now consists of a rectangle, and two perpendicular lines or me- ridians within it ; and also of two horizontal lines or lines of latitude, all which are easily placed in the picture, and may be marked with corresponding letters. Then at the beginning of the spiral line where it is cut by the perpendicular B D, m'ark I . Follow the spiral, and at every point of contact put a point; and proceeding throughout, mark the points in succession, 2, 3, 4, &c. the last in the dia- gram being 9. Make corresponding marks at the points of contact in the picture, and carry the perspective curve through all the points as numbered in succession. 11 50 Practical Rules, and [Part II. If some of the spaces appear too long, so that the eye cannot easily esti- mate the swells of the intermediate portions of the spiral,, draw more lines on the original. Such as i i and k k, each of which cuts the spiral in two or more points. The points where they cut the spiral being put in perspective and marked in the picture, will assist in describing- the curve. The Perspective o/" Rectilinear Polygons on the Ground Plane is easily made out by the application of the general principles detailed above. They must consist of lines parallel, perpendicular, or oblique to the plane of the picture, and to each other, and are therefore all included in the above rules ; a repetition of which, and a multiplication of analogous problems, is unnecessary. It may now be time to relieve the student from the apprehension that he will be required whenever he is disposed to make a drawing, to delineate pre- viously a reversed plan on the geometrical plane. It has been more than once intimated, that the method propounded is intended to give him a clear idea of the principle, by comparing the operation, first, with the tracing on a trans- parent plane, which his picture is to represent ; and secondly, by the like tracing on a rejlecting plane ; a contrivance devised as a substitute for the former, because the subject cannot be placed beyond the intended picture. Understanding now the effect produced by that contrivance, he may make use of a less complex practice, which the diagrams in Plate VI. are intended to illustrate. It has been shown that lines in the original, parallel to the plane of the picture, remain parallel in the perspective. That such of them as are parallel to each other in the original, retain that parallelism ; and such as are oblique to each other, retain the same degree of obliquity. In a word, superficial figures bounded by those lines, retain in the picture the same forms and pro- portions ; and their perspective differs from the original in magnitude only ; diminishing proportiorialli^ with their distance from the eye. (Let it be ob- served, by the way, that this law extends to superficies only ; as solids present different yaces according to their position in the picture: and their 02i^/me therefore varies with that position.) When the solid, as a building for in- stance, presents one plane face parallel to the plane of the picture, such face comes within the rule. Part II.] Application of the foregoing Theory. 51 It follows now, that the picture of any such plane (i. e. of a vertical plane parallel to the plane of the picture, or in other words, of a plane directly before the eye) will be exactly the same thing as the geometrical drawing which has hitherto been supposed on the ground plane, except that in the system here adopted it has been reversed ; and as such reversion is (in some cases, but not in all) difficult and troublesome, it is expedient to find a method whereby it may be avoided. In circles or in squares, or other rectangles, having two sides parallel to the base, there is no difference between the reverted and direct drawing ; in ob- lique right lines, there is no difficulty or trouble in inverting the obliquity ; but in varying forms, the reversion is troublesome. The subject taken for illustration in Plate VI. fig. 2. is a literal inscrip- tion, which is chosen as a conspicuous and familiar instance ; and it is sup- posed Avritten on a tomb-stone flat on the ground plane. The geometric drawing of it is, what a direct copy would be, ot an inscription on a wall before the eye. In a former diagram (Plate II.) it is inverlecl, and placed under the supposed picture : in the present diagram it is written direclli/^ and placed at the side of the intended picture. i - We suppose the stone itself first put in perspective by the rules given above. Determine the part whereon the inscription shall be in the perspective, and draw a line accordingly, on which the letters shall stand. The radius in the diagram is to be drawn now from the zero through the be- ginning of that line, till it reaches the base line of the picture. And the fnder is to be drawn from the side distance point, cutting that radius at the same point, and reaching the base. Thus finding the point of incidence and quadrant point from the zero, and side distance instead of beginning with them ; because we have now deter- mined the place in the picture first, and from that are to find a corresponding distance or latitude of the original. Measure then the distance from the quadrant point to the point of inci- dence, (the quadrant line, Def. 29.) and place the subject at the side of the picture at a height equal to the length of that line. Draw 2i parallel there, and on that place the subject geometrically drawn, as the words in the plate ; which are in a latitude corresponding with that in the picture. Draw another /j«rrt//e/ line at the top of the subject, and the writing will be inclosed between the two parallel lines. H 2 - 52 . Practical Rules, and [Part II. The latitude of the lowest of these in the perspective^ is, where the radius and finders cut each other^ and is drawn already : and that of the highest will be determined by another finder drawn from a point on the basC;, at a distance from the first quadrant point equal to the space between the parallel lines in the original. This gives the space in which the perspective inscription is to lie ; and per- haps at the first trial the student will be surprised to find how small it is. He must proportion his original according to what he wants in the picture. Now draw perpendicular lines from the foot of each several letter ; (or rather from little parallelograms inclosing the letters in the original,,) (As 1, 2, 3, &c. in the plate ;) and marking on the base line^ points at equal distances with (hem^ draw radii through the parallel lines, and they will give the spaces in the picture in which corresponding letters must be inscribed ; and also the sloping of the uprights of the letters themselves^ which will have various degrees of obliquity and dif- ferent directions. Further, the reader will see that he might draw his original on a separate slip of paper, and laying it on the picture as if written there, so that the foot of the letters shall rest on the base line, mark oflf the points there, instead of measuring and transferring lines of incidence. Another figure is also given in the plate, representing the spiral of fig. 1. placed at the side without inversion, and put in perspective in the same manner as the inscription. Before quitting the ground plane, some further remarks are offered of the same kind, which may afford facilities to the student in practice. For, by observing the relation which certain descriptions of lines bear towards each other, as to the place in the picture, he may avoid the necessity of going out of the picture for measurement or guide. If a square having one side parallel to the plane and base of the picture be put in perspective in any part of the ground plane, a diagonal extended through it and prolonged, will reach the side point of distance ; (as in fig. 1. Plate Vll.) and if there are other squares in any part of the ground plane still parallel, their diagonals, if prolonged, will meet in the same point : for if the sides of any square be parallel to those of another, their diagonals are also parallel, and vanish together. If the base be divided by points at equai distances apart, and radii be drawn from them to the zero, one line drawn diagonally to the side point of di- Part II.] Application of the foregoing Theory. 63 stance, («. e. a finder to the measuring point, Def. 26. and 31.) will give, at its intersections with the radii, the latitudes of all equal contiguous squares. If, then, parallel lines be drawn through these intersections, the whole ground plane may be filled with contiguous and equal squares ; of which the parallels will be the bases, and the radii the sides. (Fig. 2.) .The base so divided may be of indefinite extent. A contrivance for finding within the picture such radii as come from distant points on the base, will be given in a future part. The diagram shows the effect produced. If the diagonals vanish in any other point of the horizon-line, not being the side distance point, the figure will be the perspective, not of a square, but a rectangle ; and if the vanishing point be bej/ond the side distance point, the figures will be perspectives of rectangles having their parallel bases longer than their perpendicular sides. If short of the side distance, the contrary. (Fig. 3.) which describes the two cases on the opposite sides. If the square be set obliquely, so that one of its diagonals be a meridian, all the corresponding diagonals will be radii to the zero, and the sides will vanish at the side distance (or measuring) point ; (fig. 4.) and in this case the other dia- gonals would be parallels. In the diagrams the alternate squares, &c. are shaded to give a more distinct view -of the places and proportions. It is easy to see that diirerent degrees of obliquity may be given, by the different obliquities of the two first lines that are made to cross each other ; and the rules and diagrams need not to be multiplied for that purpose. The reader will be no longer detained on the ground plane. If he begins to be weary of it, let him reflect that the ground plane (without any play upon the word) is the foundation of perspective. Every solid, every structure, covers a ground plan. Every vertical plane, face of a building, wall, row of trees, «&c. &c. stands on a line on the ground plane : even an object suspended, is vertical to a point on the ground plane ; to determine the latitude and longitude of which, has been the object of the foregoing pages : so that all locality in picture is to be referred to the Per^ spective of the Ground Pl.ine. 54 Practical Rules, and [Part II. CHAPTER III. Of the Perspectives of Solids, and of Objects in general : including Figures on Vertical Planes, — on the Superior and Inferior Horizontal Planes, — and on Planes Oblique to the Horizon. All solids are comprehended in, and bounded by, plane or curved surfaces. If the bounding surfaces are angular or irregular, this is only a multiplication of them. The parts are distinct surfaces, still plane or curved. Since all objects, and all parts of objects at rest, rest either on, or over the ground plane, so the painter in representing them must have in his mind a ground plan, on or over which he places his superstructure. At the same time he will remember^ that in very few cases, more than half, in not many quite half, of that plan can be occupied by the visible part of the superstructure. Where the ground plan has less than four sides, there may be either one or two sides visible. Thus the base of a three-sided prism or pyramidj is a triangle. Two of the angles must be acute ; all the three mat/ be. In either case, two sides only can be seen from any point in front of the picture. If the side of the triangle nearest the base of the picture is parallel to it, one side only of the structure is visible. If oblique, (i. e. if sufficiently so) two are seen, and both oblique. When the ground plan of the structure has four sides, not more than two of its sides can be presented to the eye ; (except, however, where the base con- sists of one very long, and three short sides, making two contiguous obtuse angles, as at fig. I. Plate VIII. in which case it may be considered rather as a part of a polygon ; and is to be treated of hereafter.) Of this quadrangular figure, the opposite sides will be parallel to each other : (or nearly so, for we are not limited to perfectbj regular figures.) Such is perhaps the most common of all the forms of buildings ; and with this the illustration of the practice shall begin. On reference to any of the diagrams of quadrangles on the ground plane, it will be seen that the perspectives of the right angles nearest to the eye, are always obtuse : therefore the only visible angle in the perspective of such a quadrangular superstructure as is now in question, will be obtuse. Part II.] Application of the foregoing Theorij. 55 Problem XIII. To put in Perspective a Right Angled Quadrangular Structure^, its four Sides being of equal Heights, and one Side parallel to the Plane of the Picture. . .' The picture being prepared, as in all the foregoing problems, i. e. with the horizon-line, zero, and points of distance determined. Let the student place a point in whatever part of the picture he pleases ; and on that point, erect a vertical line of such height as he intends for the near part of the structure. Supposing it to be for a building higher than a man, and not viewed from an eminence, it must reach so much above the horizon-line. Intending that it shall face the front, (according to the terms of the problem proposed,) let him, from the foot of his vertical line, draw the base line of the building, parallel to the base of the picture ; and at the end of that, erect another vertical at the same height as the first, and draw another parallel from top to top, as at fig. 2. Now if these parallel lines cross ihe prime vertical line of the picture, (see Def. 11.) or, in other words, if any part of the front to be represented crosses the middle of the picture, no other side of the building can be seen. If it does not cross, a second side will be seen ; but in either case the perspective of this face will be an upright geometrical copy of the original. The per- spectives of the parallel lines will remain parallel ; and all the proportions of the parts will be as in the original. \_Note.'\ It must not be considered as an exception to this rule, that there is in drawings of elevations generally a difference between the geometric and the perspective representation of the front of a piece of architecture. The reason of this, is, that in all architectural faces, there are recesses, or projecting parts, greater or smaller : as the architraves and sills of windows and doors, &c. but then they are parts of other planes. The outlines only are in the principal plane here spoken of ; and ^/ie_i/ are geometrical. Supposing now that this face do not reach the middle of the picture, then that side which is nearest to the prime vertical must also be presented to the eye ; more or less obliquely, according as it approaches to it more or less. Proceed as before with two vertical lines and base, and parallel top-line. (Fig. 3.) Then from the foot of the first vertical line, let a line be drawn on the ground plane towards the zero, and its length determined in the same manner as the length of a side of a parallelogram, in Prob. VI. 66 Practical Rules, and [Part II. On the end, erect a third vertical line, and determine its height by a line from the top of the first vertical line, as in the last case. Now all the visible sides are complete. The diagram exhibits figures in different parts of the picture ; and if the reader refers to the former problems and diagrams where the ground plane has many parallelograms, &c. he will see that the angles of these figures are similar to those that are in corresponding parts of the ground plane in the present picture. Problem XIV. To put in Perspective a Quadrangular Structure similar to that in the last pro- blem, but of ivhich the Sides are all oblique to the Plane of the Picture. Determine at will^ the first point of the first vertical line^ as in the former problem. Make a temporary ground plan, (in front, or at the side of the picture, or on a separate paper,) in order to determine the degree of obliquity intended, and put the first oblique line in perspective by Prob. II. (Or such line may be placed at once in the picture, arbitrarily as the eye directs, and the geometric form be taken from it, as in fig. 2. and 3. Plate VI.) The second line must be put in perspective by rule, or the angle will not be correct, and the figure will be distorted. The rule will be. Place on the geometric plane the first ground line, and from the end nearest the base of the picture, draw another at right angles with it, tending from the base of the picture, (see Prob. VII.) and let this line bear, in length, the same proportion to the first, as you intend that side of the building should bear to its other side, on its true (not perspective) ground plan. Put this second line in perspective by the former rules. At the extremity of the two perspective lines, erect the second and third ver- tical lines ; and their height will be determined by the vanishing lines of the (op and base, as in the last problem. i. e. prolong each of these base lines towards the point in the horizon-line, which it would meet in or out of the picture ; and draw lines from the top of the first vertical line towards the same two points. These are the vanishing points of the top lines and base lines. (See the Rule in Sect. XXV. and Def. S3.) Erect the second and third vertical lines to touch these vanishing lines, and the two visible sides of the structure are complete. Part 11] Application of the foregoing Theory. 57 So that in order to put a four-sided structure in perspective^ (provided it is liigh enough to appear above the horizon-line,) nothing more is neces- sary than to erect three vertical lines ; one at the visible angle, and two at tlie ends of the nearest sides of a parallelogram on the ground plane ; and deter- mine the heights of the two further ones by lines from the top of the first, va- nishing in the same points with those of the ground lines, as in this problem, or remaining parallel, if these are parallel, as in the preceding problem. The structures described in the two last problems are for the present sup- posed to be rectangular buildings. If an additional height is to be given to them, (as another story,) raise the nearest vertical line, and from it draw vanishing lines towards the same points as the former, for tliey are parallel to them in the original subject; and there is no exception to the rule, " That all lines parallel to each other, and not to the plane of the picture, vanish in perspective towards the same point;" and if they are in planes parallel to the horizon, (which these are,) that vanishing point is in the horizon-line. Or if a sloping roof is to be represented, we must, for the theory of it, recur again to the ground plane. To illustrate this, the simplest form will be chosen : — that of a sloping roof with two upright gable ends. Of course the face of only one of these gable ends can be seen ; but the peak of the other is the termination of the ridge of the roof, and must be duly placed, or the whole will appear crooked. Now it is plain that if such roof is meant to be regular, and of equal sides, the ridge will run along in a plane vertical to a line drawn through the ground plan, parallel to, and equidistant from, both the sides; i. e. a central line from end to end of the ground plan. Two ground plans are drawn of buildings similar to those of the two last pro- blems. The one with one of tlie visible faces parallel to the plane of the picture ; the other with both oblique. Put these in perspective by Prob. VI. and VII. as in the figure. The lines which will be visible when the solid is erected, are drawn plainly ; the others dotted only ; as is the central line also. Erect the structures as in the two last problems. On the nearest end of the dotted central ground line, set up a vertical line to the height intended for the near peak of the roof This also is dotted in the diagrams. From this peak carry another line towards the vanishing point of the corre- sponding base line. This will be the line of the ridge of the roof 58 Practical Rules, and [Part II. ■ ' ' On the further end of the same central ground line, set up another vertical to reach this ridge line, where it will determine the further peak. : (It is scarcely necessary to observe, that i\\e perspective of the central line will not be equidistant from the side ground-lines.) Now draw two sloping lines from the near peak to the shoulders of the building, and one from the further peak to the one visible distant shoulder, and thus the roof is completed. If one of the sides of the building, not under the gable end_, be in the front, parallel to the plane of the picture, and crossing the prime vertical line so that no other side is seen, the gable end of the roof will not be seen ; (see page 55.) and a sloping side only is to be put in perspective. Now this sloping side of the roof is part of a plane oblique to the horizon plane, rising as it recedes from the plane of the picture. It is bounded above and below, by lines parallel to each other, and to the plane of the picture ; which lines are therefore parallel in perspective. It is bounded at each end by lines also parallel to each other, but not to the plane of the picture ; which therefore converge in perspective. These lines are not parallel to the horizontal planes, therefore they will not vanish in the horizon-line. The plane they lie in, rises towards the distance, therefore its vanishing line will be above the horizon-line. (So far the description of planes not parallel nor perpendicular to the horizon- plane, is anticipated ; but the parts of the subject are so much connected and involved together, that it must not be deemed quite out of its place.) It remains to find the degree of convergence in the perspective of the pa- rallel rising ends of this sloping roof, and ascertaining their vanishing point above the horizon-line ; which may be done by imagining a vertical plane in which the highest ends shall terminate, and finding the ground-line on which that plane stands. Put the ground plan in perspective with a central line, as at fig. 3. The central line is of course shorter in the perspective than the near line. Erect the side lines of the visible face of the building on the two nearest angles, and draw the upper line to connect them, and finish the face. Erect also vertical lines on the ends of the central line, (dotted in the diagram,) to the height required. The ridge of the roof will be a line from point to point parallel to the base, and will be shorter than the lower edge of the roof, in the same degree as the central line in the ground plane in the perspective is shorter than the near line of the base. The representations following the diagrams above described, are merely fillings up of their outlines. They may be supposed to be barns on ground plans like those Part II.] Application of the foregoing T/ieoiy. 59 of the same problems, and similar and equal to each other in construction and size, equally near to the plane of the picture, but differing only in their position with re- spect to the prime vertical. Fig. a crosses the prime vertical or centre of the picture, showing one face only. Fig. b is on the same parallel, but wide of the centre, showing two faces, of which one has the gable end. Fig. c — still of same size and proportion, showing the gable end in front, pa- rallel to the plane of the picture, and the real front perpendicular to such plane, and therefore seen. All its longest sides are in the direction of a radius to the zero. Fig. d is still on an equal and similar ground plan, but oblique to the plane of the picture, and as it were the filling up of the outlines of fig. 2. in the same problem. The reader will now be led a step further in construction ; and buildings will be represented consisting- of a greater number of parts ; such as doors^ windows^ chimneys, columns, &c. And first we must again recur to the g-round plane, and the mode of "dividing- lines equally or proportionally. Problem XV. To describe in Perspective the Faces and Sides of Buildings, with any Sym- metrical Breaks or Divisions on the Surfaces represented. Let the representation be that of an oblong house, showing- a long- fa9ade obliquely ; with windows at equal distances from each other, and one central door. Let a front sketch be made of it, as in fig. 1 . Plate IX. Mark the divisions at its base, corresponding with the spaces between the windows, &c. as in same fig. Determine the form and proportion of the building, and its size according to its intended place in the picture. Put in perspective (by Prob. VII. fig. 2.) the parallelogram in which it is to stand, or rather the two nearer sides of it, for the other two will not be seen ; as a 6, a c, in fig. 2. * The base of its front obliques off in the figure, vanishing towards the mea- suring point on the left. This will be the perspective of an obliquity of forty-five degrees, or half way * Another way of putting in perspective any determined angle, will be proposed in a subse. quent part : but as the business now is to describe the boundaries of solids, the chain of propo- sitions is not here interrupted for that purpose. I 2 60 Practical Rules, and [Part II. between a meridian and a parallel ; and of course the geometric plan of it will form that angle at the base of the picture. On the geometric line, mark the same divisions that appear on the base of the sketch. Then by lines of incidence and Jinders, (as in Prob. IX. fig. 3.) mark the cor- responding divisions on the perspective ground line. Raise the vertical line on the nearest point, to such height as seems to belong to it in the picture, and from the highest point, draw a line vanishing towards the same point where the ground line vanishes ; viz. the left measuring j^oint. Do the like on the other side; the lines of which vanish towards the meo^wrmo- point there. Raise verticals on the two distant points of the ground line, as in Prob. XIV. and the outlines of front and side are complete. Determine the height of the lower and upper horizontal lines of both rows of win- dows, and draw lines vanishing towards the left distance point, (for such lines being parallel to the top and bottom lines of that facade, vanish in the same point,) and then upright lines springing from the divisions on the ground line will form the sides of the windows, which are thus complete. The upper and lower line of the door-way vanish in the same point ; and the sides are vertical to the divisions on the base. With respect to the roof and chimneys, little needs to be said. The vanishing points of their lines may be found by easy analogy to those of the former lines. It will be seen that the perspectives of the windows are not equally wide, nor equally high, nor the spaces between them ; but that they diminish as they re- cede, in like manner as the divisions on the line in Prob. IX. fig*. 3. The author takes the liberty to remark, that the eflfect of many very nice sketches by amateurs, is destroyed by want of attention to this part of perspective. If the building- is in the near part of the picture, and of magnitude sufficient to show the panes or other divisions, they also must partake of the same dimi- nution, as they recede from the eye; and it will be in general better to mark the perspective centres of them on the ground line. This centre can never be equidistant from the sides in perspective. Observe, these windows, &c. are figures on vertical planes, which the title of the present chapter comprehends. The next figure is of the same building on the same scale, but with the end parallel, and the facade perpendicular to the plane of the picture. The method of putting it in perspective is exactly the same ; only as the fa- 9ade is a radius (or meridian), it must be divided as the line in fig. 1 . Prob. IX. The end being parallel to the plane of the picture, is drawn in its direct posi- tion, and divided equally. When the solid to be drawn does not rise so high as the horizon-line, the Part II.] Application of the foregoing Theory. 61 upper part of it becomes visible. This will take place either when the spec- tator is raised above the structure or other large object, or when the object itself is small. Fig. 4, 5, and 6, in Plate IX. represent rectangular solids, concerning Avliich little needs to be said in the way of explanation or direction, as the principles above detailed are to be followed in strict and easy analogy. The lines parallel to each other vanish together in the zero., or in the side measuring poinl, or nearer, or more distant accidental vanishing point on the horizon-line, according to the degree of their obliquity to the plane of tlie picture; or, being parallel to that plane, do not vanish at all. Of Planes oblique to the Horizon. The mode of describing a rectangular figure upon a plane rising from the eye, and directly before it, is particularized nearly enough in the example of the sloping roof of the house in Prob. XIV. pages 57 and 58. The rising plane itself (such as the side of a hill,) vanishes above the hori- zon-line. The radii (if they may now be so called, for it is a slight variation from the definition,) terminate in a point in the prime vertical, but above the zero. Rules for determining that height cannot be given ; or rather it may be said, that that heiglit gives the rule. However, all lines on the plane parallel to each other, vanish in some point in a line parallel to the horizon-line, but higher in the picture. The lines describing the parallels of latitude, will be further apart than those on the horizontal ground plane. Problem XVI. To put Rising Planes in Perspective. Fig. 1. Plate X. represents the perspective of a rising plane, (such as might be the plane of the edges of the steps of a stair-case, or a boarded platform laid on them ;) and the reader is referred to the diagrams 19. and 20. (in Sect. XXVI. of the Theorj/,) which represent superior and inferior horizontal planes between two side planes, that he may observe the .difference in the perspectives of horizontal planes, and planes oblique to them. The letters h //, at the sides, mark the horizon-line. The dot in the centre is the zero. The dot above, is the vanishing point of lines on the rising plane parallel to each other, but running direcilj/ from the plane of the picture. 62 Practical Rules, and [Part II. (These lines are in fact formed on the rising plane, by the intersections of vertical planes standing on meridians. If the reader has not quite made him- self acquainted with the theory of the intersection of planes, he is referred again to the experiment of the slits in the cards, Sect. XIV. of the Theory ; and he may draw meridians on one card, slits in the one representing the oblique plane, and place (or imagine) upright cards passing through them, which will be a pi'actical illustration of the present position.) Fig. 2. is intended to make this more evident, by the perspective delineation of equal squares on the plane ; their sides vanishing in the point above. And fig. 3. shows the difference between the perspectives of parallels of lati- tude on a horizontal plane, and the like on the rising plane on the same scale ; the one set being dotted, the other marked as lines. Fig. 4. represents a rectangular parallelogram (such as a flat slab of marble) set up on its end on the ground plane, (or it might be a book on a table) supported at the other end in a position oblique with respect to the plane of the picture, as well as to the plane of the horizon. To put this last object in perspective^ determine at will^ the length and the degree of obliquity of the first line ; i. e. the line on which the end on the ground shall stands as 1, 2 ; 2 being the nearer to the base of the picture. Then determine the length and slope of the nearest rising side of the object; as from the foot 2, to its high end ; which mark, 6. From the foot 2 draw a Jinder to the measuring point d. From the end of the rising line 6, let fall a vertical line to touch the finder^ and call the point of contact 4. Now the two lines 1 2, and 2 4, are the two visible sides of the parallelogram, on and over which the object is supposed to be placed. Complete this parallelogram by prolonging the line 2 1 till it reaches the horizon-line ; and mark that point I ; and by drawing another line from 4 to the same point /. Then draw a line from 1 towards the measuring point . ^ M^- 'Jt/.U^ i7^;%t. iK-K^JL. e/ ' ' -/fuuv Ma^t. fc/qiA/ c ^vi ncMiiA- ^ '^o ,711x0'^ 'fAt 'loi/i'i^/' -tS I ^ ^ •' ' — / 3 ^ At * - * - - 3 1^' yly>-&yA^ <: ^A^l^ ( lJ/z<^^ y-i-^i^Aw: /An.. /lyfiy^yAU^iti, , / • OAj- /l^t^H^ fc. in, -AAt- i>vAi; U,<:i-A, 'j^c'uj eeiJ^--( 'i. . if. -f- ^./leioitii '^'> i^/i^ ■ ^ Ay?t^,J^c^'!t ^ /tni/cfyy^'^ ^'-^ c:Aj./'i^ia^ 'f/lX. ^I'^AA'^ ^T^-^- ^i^'Tt^ArayH^^ c^-u iyL ^t-t. ^!---f-syv^^A y^Ci>n-£^. O^t*. i^ie- 2, /. ^ A/iinA Ar!^ A rL- -Ax-i^ m/- ctyy^A^^T-nyi^i Ai/ fy.- '^'icyi^ /^i>c- yyu/~eyyucCe '■'^ ov&clAA>.^i ■'/Attn AAivt.- '}^y/it Ci i^y^^ •<^A*iAtc-7'^£i^'^ f^'- A^''-^- ^'/i^vii^'L, -^ftTfmj oAcrA- u /lAti-c^eC, i:t'l^^A^^A77c£(^Wtl^,^/<*/.^^ii ac^-i^/ti^T^ -^^'i^i'^ cA^ OL- ^Jft^f^'^^iA^, 2, Al' A /ASl yt^'^-^/ittitAv-c i^ACi- '^rtr«//eZ to each other. 2ndly. They may be parallel to each other, and not parallel to the plane of the picture; and in this case they may he perpendicular to the plane of the picture, or oblique to that plane. The lines formed by the meeting of the side walls and the ceiling, or the side walls and the floor, lines drawn on the ceiling or floor directly towards the end wall, the side edges of a table standing lengthwise, the upper and lower 76 Practical Rules, and [Part II. lines of pictures against a side wall, — all are lines parallel to each other, parallel to the plane of the horizon, and perpendicular to the plane of the picture. Parallel lines drawn on the floor or ceiling- in a sloping direction towards the end wall, or the side edges of a table placed aslant in the room, the opposite sides of diamond-shaped objects laid on the table, or of diamond-shaped pavements, — are parallel to each other, parallel to the plane of the horizon, and OBLIQUE to the plane of the picture. None of this second class of lines will be represented by lines parallel to each other in the picture, but thei/ will all converge. Those that are perpendicular to the plane of the picture will tend towards the centre of the horizon-line (here called the zero point). Those which are oblique to that plane, but parallel to the natural horizon, (such as those described on the ceiling, floor, or flat table,) will tend towards some point in the horizon-line within or without the picture, more or less distant from the zero. 3rdly. Lines may he parallel to each other, and oblique to the plane of the picture, and neither vertical nor parallel to the natural horizon. Such would be the parallel sides of a ladder set up against the end wall, or a parallel-sided oblique plane rising or descending before the eye. This class of parallel lines also will converge in the perspective^ and meet not in, but above or below the horizon-line. Thus our two universal rules are estabhshed and explained; viz. I. That lines parallel to each other, and to the plane of the picture, will be described by lines parallel to each other in the picture. II. That lines parallel to each other, but not to the plane of the picture, will never be parallel to each other in the perspective, but always convergent*. In a note to Prob. XV. a readier way of putting in perspective any determined angle is promised. If a right line be drawn, cutting through a circle in any part, the portion of the line between the intersections is a chord, and the portion of the * If this repetition appears prolix, the apology offered for it is, that the crying sin of amateur drawings is, putting lines parallel to each other in the picture, (the upper and lower lines of straight buildings, rows of windows, ranges of equal columns, the parallel strata of rocks. Sec.) because they are seen to be parallel in nature, or in the subject, when they are intended to be placed obliquely in the picture. It sets the whole drawing askew. Amateur drawings have been spoken of, but professors and even teachers may take the hint ; for the Author has seen a drawing-book, published not very long ago, consisting principally of buildings, all put in false directions as grossly as if they were intended to instance the fault complained of. Part II.] Application of the foregoing Theory. 77 circumference so cut is an arc of that circle; and if any point, or number of points, be chosen on the arc, and two right lines be drawn from the extremities of the chord, meeting- at any of those points, the angles so formed at those points will be equal to each other, and proportioned to the arc of the circle, as at fig. 1. and 2. Plate XIV. These arcs contain triangles, of each of which the angle at the apex is equal to that of the others in the same arc ; /. c a is equal to ft, and also to c in the same figure. If each portion be a semicircle, the angle will be a right angle. If less than a semicircle, it will be obtuse ; if greater, acute. If then such portion of a circle be put in perspective in any part of the picture, an angle so formed within that perspective arc will be the perspective of an equal angle in the original. If the original be a semicircle and right angle, and the perspective of it be intended for the angle at the base of any solid structure, sucli will be the perspective of the base of a rectangular structure. Or if the ground-line of the picture be divided in equal parts, and tfie squares put in according to the easy and short process represented on Plate VII. either parallel or oblique (the line of distance and height of horizon- line being made equal to those of the intended picture), the student will find the perspective of a right angle in whatever part of the ground plane he wishes, and may transfer it by measure to the corresponding part of his picture. He will also see the relative niagtiilude of equal squares or paralle- lograms, serving as foundation-lines for his structure in any pait of the picture; and may also, by counting the squares (if they are subdivided), determine the relative proportions as well as oblicpiitj/ of the two visible sides of his quadrangular structure. Square numbers, as 4, 9, 16, 25, 36, 49, 64, 81, 100, of contiguous squares (the two sides being equal in number) on t!ie ground plane, form the perspective of one square. Four for one side, and two for the other; or six for one, and three for the other, make the double square. 3 -f- 2, or 4 + 3, &c. make square and half square, and so on ; and thus by simply counting perspective squares in the divided ground plan, and raising in their appropriate places the three vertical lines that include the two visible faces of the intended structure, it may be put in in its due proportioti in any part of the picture without the process of fjuadratit lines and fuiders . For equal perspective divisions of quadrangular surfaces, diagonals furnisli the readiest way. Take a geometric square, and draw the diagonals. 78 Practical Rules, and [Prtit II. The intersection is the centre. A perpendicular through it divides it into two equal parallelograms. Diagonals of the two parallelograms give the centres of them, and their perpendiculars divide the whole scjuare into four ecjual parallelograms; which again may be divided by diagonals and perpen- diculars in like manner as often as desired. See fig-. 3. and 4. Plate XIV. And the perspective, whether parallel or oblique, may be divided in the same way within the picture without the aid of lines of incidence, C|uadrants, &c. This furnishes an easy and certain division of the perspective, provided the number of parts required are as the powers of 2 ; viz. 2, 4, 8, 16, 32, &c, but not otherwise. If, besides the perpendicular line, a horizontal line be also drawn through the centre of the square, it is divided into four squares. Each of these may be divided in like manner by four, and thus sixteen squares are given. These again divided give sixty-four, or the chess-board : and if the whole square be j)ut in perspective either directly or obliquely, it may in like manner be divided within the picture. So may the whole ground plane, or such part of it as may be convenient for any particular purpose of measurement. For dividing in any other jiroportions, the sector given in all cases of instruments, and also the instrument called proportional compasses, are expressly adapted ; but a very convenient substitute for them may be made in a minute when wanted for use. Mark an upright line of any accidental length on a sheet of paper : at right angles with that line, draw another of any greater or lesser lengtli. In the present instance let it be supposed greater. a b, fig. 5, the shorter upright line. h f, the longer horizontal line. Join a and c by a diagonal, forming with the others a right-angled triangle. Now if a line be placed anywhere (as at e, f,) perpendicularly on the horizontal line b c, cutting off a part of it, and reaching to the line a c, such vertical line will bear the same proportion to the part so cut off, as the vertical ine a h bears to the whole of the line be. Suppose then the horizontal line b c, to be three times the length of the vertical line a b, then the new vertical line e /is equal to ^rd of the remaining part, or f c. Take then the measure of any line that is required to be divided into 3rds. Mark on the horizontal line an equal measure from the point c; erect a Part II.] Application of the foregoing T^heory. 79 vertical line to touch the oblique line above^ and that vertical line is the measure required ; viz. ^rd of the line that was to be divided. Thus the division is gained into Srds, of any line or space in the original (as the surface of a building, &c.) ; and if such divisions are marked on the base line of the picture, the fnders (or radii, as the case may be) will divide the perspective accordingly : and these again may be divided into halves, 4ths, &c. by diagonals, which will give the 6ths, 12ths, &c. intermediate between the powers of two, as in the last paragraph. This scale furnishes a most convenient method for enlarging or reducing a picture or object in copying. Make the upright line equal to the base of the smaller picture, and the horizontal equal to that of the larger. Measure any object of the smaller picture, and set it upright on the horizontal line wherever it will reach the oblique line, and the part of the horizontal cut oif will give the magnitude of the object on the larger picture. Or measure the object in the larger picture, and lay it down on the horizontal line beginning from the point c ; and where it ends, erect a vertical to touch the oblique line, and that will give the magnitude of the object in the smaller picture. If the size of the picture to be copied is too great for a convenient scale, take ^, jth, or ^th, or other fraction of its base for the horizontal line, and the same fraction of the base of the smaller for the upright line. (This is on the principle of the sector, but it is more convenient to measure from. Proportional compasses are the most convenient for forming such divisions, but the substitute here proposed is always at hand.) . When some line in the picture is to be divided into parts representip.g- certain divisions of the original line, as for a range of columns, trees, or equal or proportional divisions of the face of a building, it will frequently happen that the divisions on the base line of the picture from which the finders run, would extend beyond the drawing itself, or board on which it is laid. In that case, any higher parallel may be prolonged through the side-line of the picture, and divisions may be marked on it ec[ual to those which the finders mark on that parallel in tlie picture. As at fig. 6. Plate XIV. B B, the base. a b, the perspective line to be divided. d, the side distance or measuring point. 1, 2, 3, 4, &c. equal divisions on the base line. 6, 7, 8, 9, &c. equal distances on the prolongation of the parallel. These are smaller than those on the base line, but equal to those on the parallel within the picture. 80 Practical Rules, and [Part II. Or if the drawing- be large enough, the scale may be put at first on a high parallel, near and below the measuring point, and continued to the distance wanted. The first division may be determined by finders from the base, and the rest marked equal to it on the parallel. As at t', w, X, &c. on the high parallel in the same figure. Method of proving the direction of Lines that should vanish together. It has been said above, that a strained thread is better adapted than the edge of a ruler for ascertaining the direction of a right line ; and it will be found so, because the space is visible on both sides at the same time. And the employment of it will be found extremely convenient in drawing; for if one end be held firmly on the ccro or measuring point, or other vanishing point of the representatives of parallel lines, and the strained thread be swept over the picture, as a carrying radius, it will coincide with all the lines in the picture that vanish in that point, and thereby prove whether their direction is right in the perspective. Or it will give the painter the direction of any short line he wants to put in as the representative of a line parallel to one already in the picture. A mark should always be made on the side-lines of the picture, denoting the height of the horizon-line (as at h in many of the preceding diagrams), and a line may be drawn on the board, against which this mark on the draw- ing should be laid. The tight thread may be laid on any oblicjue line in the picture, (as the top-line of a house,) and where the thread reaches the hori- zon-line, be held or pinned down ; and sweeping the line across the picture, I he artist may mark as coinciding w ith it, the upper and lower lines of the windows, the foundation line of the house, the chimneys, and also the lines of any other buildings, &c. in other parts of the picture representing buildings parallel to the first. To lessen the tediousness of putting in the many short vanishing lines of recesses in windows, &c. this contrivance is particularly useful. If the fixed point is at the zero, it will prove, not only all lines perpendicular to the plane of the picture, but the relative heights of all objects, men, &c. by coinciding with the standards (see Def. in Prob. XXII.) on the side lines. See fig. 7. Plate XIV. (The thread may be pinned down on the board, and the pin may be moved and refixed as wanted, or a small weight may be sewed in a cloth to which the thread is attached ; and in the latter case it can be laid in the picture). Part II.] Application of the foregoing Theory. 81 Shorter Method of putting Circles in Perspective. The reader will perhaps have observed, that in the diagram of Prob. XIX. a circle is put in perspective by a shorter process than either of the two de- scribed in Prob. X. and XI. and this shall now be shown him : — but to have given these before a systematic method was established, would have been teaching abbreviations before writing at length ; whereas one who has a system, easily gets abbreviations, and finds the use of them. Describe a semicircle on the geometric plane, the diameter being a part of the base line of the picture. Fig. 8. Plate XIV. Inclose it in half a square, and draw radii from the extreme points to the zero ; and put the square in perspective in such parallel of the picture as may be desired between these radii. Draw the diagonals in the perspective square, and a meridian through its centre ; and then a parallel diameter through the same centre. This meridian and parallel give in the square four points for the circle to touch. In the geometric half square draw lines from the two lower angles towards the centre of the semicircle, just cutting the circumference; and from the points so cut draw two lines of incidence, (Def. 27.) and their radii will cut the diagonals at four other points : thus eight points are given for the circle, similar to those of Prob. X. For sixteen points, similar to those of Prob. XI. draw the four chords of the semicircle, and bisect them, as in that problem, and follow up the process in the manner there described. For a Substitute for a Geometric Plan, adapted to Circles of any Size required. Take a line divided by the points of incidence of a circle divided into eight or sixteen equal parts (or, which is the same thing, those of a semicircle divided into four or eight equal parts). a 0, and b c, fig. 9, are two such lines ; the first being the fac-simile of the upper side of the square of Prob. X. the other that of Prob. XI. Those points marked on the base line of the picture, will of course furnish rflfZw sufficient for the perspective ; and two parallels will finish the circum- scribing square ; and the diagonals and diameters will give all the eight or sixteen points. But it is required to magnify or diminish the circle for pictures on different scales. Draw a line perpendicularly from the centre of the divided line, and radii from all the points of divisions meeting in one point on that perpendicular, as at X in same fig. M 82 Practical Rules, and [Part II. Then draw, anywhere, another line parallel to the first, and cutting those radii. Such line will be divided proportionally to the first, and diminished in any degree required according to its distance from it. If it be required to magnify the circle, draw the parallel line on the opposite side of the first line, and prolong the radii through it. Then it will be cut proportionally, and in larger divisions. The diagram cannot need further explanation, and the scale is applicable to the base line of the picture in any part. Of Mechanical Instruments for tracing Perspective Lines. Many have from time to time been proposed, but none probably have ever come much into use. It is easy to vary them on the principle of the pantagraph, or of the pro- portional rod and scioptic ball; but at best they must be cumbrous, and perliaps rather adapted to lectures than to actual practice. The camera lucida, however, is a beautiful invention; it is easily portable, and may be made of much use, not only for the purpose of drawing the individual scene desired, but in improving the eye, and introducing a facility of taking correct views of natural scenes or artificial objects when the eye is unassisted. There are difficulties in the use of it; so there are in the use of a telescope, in which a person entirely unaccustomed to it has a difficulty in finding his object, a difficulty in keeping it in the field of vision, and a diffi- culty in adjusting the focus to his own eye : but a few trials remove them, and he is repaid by the pleasure he finds. An attention in placing the circular hole of the brass slider of the camera, partialb/ over the edge of the prisn), and moving the eye a little this way and that way, a little further or nearer, will in a few trials remove the first difficulty, provided the spectator is not too near-sighted, nor too far-sighted to see his pencilled lines at the distance required. In these cases he must use a concave or convex glass, such as his eye requires on other occasions. Sometimes a bright sun on the polished brass dazzles, — it may be remedied by covering it with dark silk or cloth : or the sun is too much on the paper, — it may be screened : or there is too little light on the landscape ; in this case, for the time, it fails. There are other difficulties not so easily overcome. That of finding a level and firm station for the instrument and for the drawing ; and in a boat, any tracing with it is impossible. Then the philosopher that invented it has not discovered a method of enabling it to penerate a gentleman's hat, or a lady's bonnet; and these prove sometimes decisive impediments. Still it may always Part 1 1.] Application of the foregoing Theori/. 83 be of some use. Let the amateur merely hold the prism in his fingers when he cannot procure a convenient station, and looking through it down on his paper, he will there see a beautiful and perfect picture, with the relative pro- portions of the parts and the harmony of the whole. He may then draw a little from his eye alone, and compare his drawing with the phantom, even though unable to throw the image on the same spot ; and he will discover his errors at the time, and regulate the remainder of his sketch. Artists affect to despise and discourage the use of this and all artificial helps for learners. Some of them, because it exposes their faults. Others, forgetting the long practised steps and process by which they have acquired those habits that now appear natural to them, the progressive means by which they have attained the skill they possess and enjoy, imagine that the road they have travelled must be the best for all, and refuse to encourage the helps themselves have not needed. Some think that relying on helps in progress, renders them necessary ever after. But if a person would learn to shoot accurately with a rifle, — would he set up his mark where the ball would leave no trace to prove whether it passed above or below ? or would he endeavour to improve his aim by placing it where he can inspect the vestige of every shot, and thus attain the desired accuracy ? And do we not find amateurs relying on their eye in copying nature, carrying, through their lives, the same faults, the same disproportions ?— spreading out the field of vision as if for a panorama rather than a single view, and carrying up their mountains as high as the moon, without any chance of discovering that this is not a repre- sentation of a scene that ever existed in nature ? Others again object, not only to helps, but to rules. They will improve on nature; and taking a hint or two from her, will not confine their genius by rules, but make their picture in great measure from their imagination. As if one person could represent to the eye of another what he imagines, before he has learnt to represent what he sees! Accuracy must precede freedom. Ars est celare artem. Ease and freedom are, habitual accuracy. As a help that may be easily procured, and may be used in all situations, the Author proposes the following. Take a piece of card-paper or thin stiff pasteboard, of a size that will lie within the compass of a small octavo book. Cut out all the inner part so as to leave only a frame. Punch holes in that frame, and with moderately thin string (the material called bobbin the best) interlace it from hole to hole, so as to form six squares in the length, and three in the height ; as in fig. 10. Plate XIV. Cut another slip of the same card, of a length equal to the width of the 84 Practical Rules, and Application of the foregoing Theory. [Part II. frame, with a small notch at the end into which the central thread may hitch, and put its other end to one eye, the other eye being shut, and observe the landscape through the lattice-work. It will divide it into exact proportions, and give a true perspective to copy. : , . > < Observe first the central object, and two side ones ; and when the frame is removed and replaced, bring them to the same points again. The most experienced artist needs not to despise this help ; for it enables the spectator to choose readily his aspect, field of vision, and height of hori- zon-line ; presenting to him his picture, as it were, ready framed, and enables him to compare the effect of different positions instantaneously. It remains now to speak further of the horizon-line, the point of distance, and the field of vision ; and the Author is bold enough to maintain that neither " Pozzo's Prospettiva de' Pittori ed Architetti," nor " the Jesuit's Practice of Perspective," nor " Highmore on the Principle of Brook Taylor," nor "Chambers," nor the " Encyclopasdia Britannica," nor any of the modern treatises he has met with (and he has searched not a few) have given anything like a just Theory. The question whether the zero point must be in the centre, he ventures to decide in the affirmative, aware that some high authorities are against him. The question whether very high parallel vertical lines whose intervals are diminished in apparent magnitude must be represented as parallel, strange to say, is sometimes disputed. It is self-evident, or else some space must be annihilated. The question why we cannot easily paint down-hill views, or levels much below that of the plane of station, together with many considerations of this kind, have afforded the Author a good deal of interest in the investigation, and he was about to add the result here ; but he has already filled many more pages than he intended ; and as no sight is so unpicturesque in the eye of an author as a pile of his own sheets lumbering in the warehouse of the printer, he hesitates to encounter the danger of adding to its bulk. Whether he shall venture on a second volume, or only on an appendix, he must leave doubtful till the reception these sheets meet with encourages or forbids it; and he now resolutely writes the word FINIS. Printed by Richard Taylor, Red Lion Court, Fleet Street, )LDOUT BLANK IT. ^ * Iv % ^ PERSPECTIVE J? F h 0 F t Prol 7U run n,.ti TV. 7 Prcl Ilfig 3 riato mi M xrm. '^'^9^ V z f/aU H/u GETTY RESEARCH INSTITUTE