NRLF B 3 1ME REESE LIBRARY UNIVERSITY OF CALIFORNIA. Received.. U/M> Accessions No.<&&/^ Shelf Nu. . WORKS ON DESCRIPTIVE GEOMETRY, AND ITS APPLICATIONS TO ENGINEERING, MECHANICAL AND OTHER INDUSTRIAL DRAWING. By S. EDWARD WARREN, C.E. I. ELEMENTARY WORKS. 1. PRIMARY GEOMETRY. An introduction to geometry as usually presented ; and designed, first, to facilitate an earlier beginning of the subject, and, second, to lead to its graphical applications in manual and other elementary schools. With numerous practical examples and cuts. Large 12mo, cloth, 80c. 2. FREE-HAND GEOMETRICAL DRAWING, widely and variously useful in training the eye and hand in accurate sketching of plane and solid figures, lettering, etc. 12 folding plates, many cuts. Large 12mo, cloth, $1.00. 3. DRAFTING INSTRUMENTS AND OPERATIONS. A full descrip- tion of drawing instruments and materials, with applications to useful examples ; tile work, wall and arch faces, ovals, etc. 7 folding plates, many cuts. Large 12mo, cloth, $1.25. 4. ELEMENTARY PROJECTION DRAWING. Fully explaining, in six divisions, the principles and practice of elementary plan and elevation drawing of simple solids ; constructive details ;~ shadows ; isometrical drawing ; elements of machines ; simple structures. 24 folding plates, numerous cuts. Large 12mo, cloth, $1.50. This and No. 3 are especially adapted to scientific, preparatory, and manual-training industrial schools and classes, and to all mechanics for self-instruction. 5. ELEMENTARY PERSPECTIVE. With numerous practical examples, and every step fully explained. Numerous cuts. Large 12mo, cloth, $1.00. 6. PLANE PROBLEMS on the Point, Straight Line, and Circle. 225 problems. Many on Tangencies, and other useful or curious ones. 150 woodcuts, and plates. Large 12mo, cloth, $1.25. II. HIGHER WORKS. 1. THE ELEMENTS OF DESCRIPTIVE GEOMETRY, SHADOWS AND PERSPECTIVE, with brief treatment of Trehedrals ; Trans- versals; and Spherical, Axonometric, and Oblique Projections; and many examples for practice. 24 folding plates. 8vo, cloth, $3.50. 2. PROBLEMS, THEOREMS, AND EXAMPLES IN DESCRIPTIVE GEOMETRY. Entirely distinct from the last, with 115 problems, embracing many useful constructions ; 52 theorems, including examples of the demonstration of geometrical properties by the method of projections ; and many examples for practice. 24 fold- ing plates. 8vo, cloth, $2.50. 3. GENERAL PROBLEMS IN SHADES AND SHADOWS, with practical examples, and including every variety of surface. 15 folding plates. 8vo, cloth, $3.00. 4. GENERAL PROBLEMS IN THE LINEAR PERSPECTIVE OF FORM, SHADOW, AND EEFLECTION. A complete treatise on the principles and practice of perspective by various older and recent methods ; in 98 problems, 24 theorems, and with 17 large plates. Detailed contents, and numbered and titled topics in the larger problems, facilitate study and class use. Revised edition, correc- tions, changes and additions. 17 folding plates. 8vo, cloth, $3.50. 5. ELEMENTS OF MACHINE CONSTRUCTION AND DRAWING. 73 practical examples drawn to scale and of great variety ; besides 30 problems and 31 theorems relating to gearing, belting, valve- motions, screw-propellers, etc. 2 vols., 8vo, cloth, one of text, one of 34 folding plates. $7.50. 6. PROBLEMS IN STONE CUTTING. 20 problems, with exam- ples for practice under them, arranged according to dominant surface (plane, developable, warped or double-curved) in each, and embracing every variety of structure ; gateways, stairs, arches, domes, winding passages, etc. Elegantly printed at the Riverside Press. 10 folding plates. 8vo, cloth, $2.50. SCIENCE X> I? ' A. W I INT O- A MANUAu OF ELEMENTAKY PROBLEMS IN THE LINEAR PERSPECTIVE OP m atifc jlbatooto: C ' OR THE REPRESENTATION OF OBJECTS AS THEY APPEAR, MADE FROM THE REPRESENTATION OF OBJECTS AS THEY ARE In PART I. PRIMITIVE METHODS) WITH AN INTRODUCTION. PART II. DERIVATIVE METHODS) WITH SOME NOTES ON AERIAL PERSPECTIVE. BY S. EDWARD WARREN, C. K, PBOFES8OR OF DESCRIPTIVE OEOMKTRY, ETC., IN THE REN8SELAEE POLYTECHNIC INSTITUTE AND AUTUOE OF TUB " DRAFTSMAN'S MANUAL ;" AND " GENERAL PROBLEMS O DESCRIPTIVE GEOMETRY." NEW YORK : JOHN WILEY & SONS, 15 ASTOR PLACE. 1888. Entered according to Act of Congress, fc; the year eighteen hundred and sixty-three, by 8. EDWAED WAKBEN, In the CWK s Office of the District Court c f the United States for the No-thern District -Jl New York. CONTENTS. PAG I PREFACE . . vi INTRODUCTION. CHAPTER I. Instruments and Materials 9 Paper 9 Support of Paper 9 Pencils ? 9 Rulers '. 9 Compasses % 10 Use of Compasses 10 Irregular Curves 10 Indian Ink 10 CHAPTER II. Preliminary Principles and Explanations 12 PART I. PRIMITIVE METHODS. CHAPTER I. Definitions and General Principles I " II. The Elements of Projections 1 " III. The Construction of the Perspectives of Objects from their Pro- jections 24 " IV. Real Projections, and Perspectives made from them 28 Perspectives of Geometrical Solids, Art. (68.) 32 Example 1. To Find the Perspective of a Vertical Square Prism 2 " 2. To Find the Perspective of a Triangular Pyramid 85 IV CONTENTS. PA en CHAPTER Y Removal of Practical Difficulties, arising from the confusion of Projections and Perspectives 36 I. First Method. Translation forward of the Perspective Plane.. 36 Example 3. To Find the Perspective of a Cube, etc 38 II. Second Method. Use of Three Planes 38 Example 4. To Find the Perspective of an Obelisk, etc. . . 41 VI. Projections and Perspectives of Circles, and of Bodies having partly or wholly curved boundaries 4 Example 5. To Find the Perspective of a Circle, lying in the horizontal plane 43 Of Planes, Arts. (77-84.) 45-46 " 6. To Find the Perspective of a Cylinder, stand- ing on the horizontal plane 47 44 7. To Find the Perspective of a Cone, standing on the horizontal plane 50 u 8. Do. of a Cone whose axis is parallel to the ground line 52 9. Do. of a Cone whose axis is parallel to the vertical plane only 64 ;< 10. Do. of a Cone whose axis is oblique to both planes of projection 55 " 11. To Find the Perspective of a Sphere 58 First Method of finding the apparent contour. 58 Second Method " " " 61 " 12. To Find the Perspective of a Concave Cupola Roof. 63 CHAPTER VII. Perspectives of Shadows 65 General Principles and Illustrations, Arts. (89-99.). 65 Example 13. To Find the Perspective of the Shadow of a Square Abacus on a Square Pillar. 67 14. Do. of a Triangular Pyramid upon the Hori- zontal Plane 69 tt 15. Do. of a Dormer Window upon a Roof. 71 PART II. DERIVATIVE METHODS. Oe AFTER I. General Principles and Illustrations 75 Example 1. To Find the Vanishing Point of Telegraph Wires, etc 78 CONTENTS. V PAG1 Example 2. To Find the Vanishing Point of a Perpendi- cular and of a Diagonal 79 Particular Derivative Methods,Arts. (1 18-121.) 81 3. To Find the Perspective of a Straight Line, in any position oblique to both planes of pro- jection, etc 82 14 4-. Do. of a Tower and Spire 84 Practical Remarks, Art. (122.) 86 5. To Find the Perspective of a Cross and Pedestal 89 CHAPTER II. Perspectives of Shadows 92 Example 6. To Find the Vanishing Poiat of Rays, and of their Horizontal Projections 92 7. TO Find the Perspective of the Shadow of any Vertical Line upon the Horizontal Plane. 93 CHAPTER III. Miscellaneous Problems 96 Example 8. To Find the Perspective of a Pavement of Squares, whose sides are parallel to the ground line 96 * 9. Do. of a Pavement of Hexagons, whose sides make angles of 30 and 90 with the ground line 98 10. Do. of an Interior 98 11. Do. of the Shadows in an Interior 102 12. Do. of a Cabin 104 " 13. Do. of the Shadow of a Chimney on a Roof. . 108 CHAPTER IV. Pictures, and Aerial Perspective 110 Landscape Outlines 110 Landscape Details. Trees Ill Hills Ill Valleys Ill Ascent and Descent 112 Level of the Eye 112 Reflections in "Water. 113 Location of the Centre of the Picture. 113 Do. of the Perspective Plane 114 Shadows of Trees, and other Vertical Objects 114 Time of a Given Aspect 114 Light and Shade 115 Edges 116 Color.. . 11.6 PREFACE. FOR several years past, while teaching a comparatively advanced course on Perspective, embracing some of its higher problems, I have cherished a purpose to compose an elementary perspec- tive, for genera] use ; which should be clearly demonstrative at every step, and also, if possible, interesting to its readers ; which should, in fact, be truly popular, without being empirical ; and, on the other hand, perfectly demonstrative, without being too elevated. In other words, I have sought to make my work elementary, not in the sense of merely stating perspective facts without adequate explanation, but in the sense of selecting sim- ple, yet widely and always useful examples, and then fully explaining, in easy order, the few plain principles necessary to the solution of such examples. An exact knowledge of perspective is indispensable to those who would make exact representations, for industrial purposes, of architectural or mechanical structures as they appear. It is highly useful to those, even, who practise perspective as an ornamental art, in the making of pictures; inasmuch as it nables them to know scientifically, as well as feel sensibly, whether their drawings are correct or incorrect. It is also interesting to the amateur judges and admirers of pictures, as well as to their makers ; and, finally, it is useless to none who are in any manner engaged with the arts of graphical representation or design. It is a part of the price of truth, whereby we discover its worth, that we must discover the truth concerning propriety oj Vlll PREFACE. arrangement in any subject, as an indispensable condition for its successful treatment. A course of research, which in any degree ignores certain elements on which it is based, cannot but become proportionately involved in bewildering, and, both to the student and critic, comfortless confusion and intricacy. It is equally true, that natural progress from a proper starting point cannot fail to be effectual and agreeable. As the truth about primary industrial utilities for daily work- ing life, is more elementary and fundamental than truth about beauty and other higher utilities for the adornment of higher life, it results, in the present instance, that " Projections," which, usually for industrial purposes, represent objects as they are, in form and size, naturally precede, in a course of exact study, "Perspective," which, usually for pictorial effect, represents objects as they appear. Perspectives, or drawings of objects as they appear, are made, then, from Projections, or drawings of objects as they are ; and which, therefore, are competent representatives of those objects. The study of projections thus properly preceding that of perspective, as its natural foundation, disadvantages will unavoidably arise from attempts to treat of exact perspective, without a formal preliminary treatment of ppojections. Hence, this work, while complete in itself, is the natural successor, for those who use both, of my " Manual of Elementary Geometrical Drawing of Three Dimensions," in which objects are shown in projection only. It is unfortunate for learners, that a subject so simple, useful, and attractive, as Perspective is, when properly treated, should come to be regarded with aversion, merely owing to defects in its treatment ; the chief of which defects is, perhaps, the failure fully to exhibit its foundation in "projections." The present volume is an attempt to expressly present Per- spective, as founded on "Projections," and in this respect it differs, more or less noticeably, from numerous elementary works on the subject. I accordingly hope for such results, in PREFACE. respect to ready and interested understanding of the subject, as the improved treatment of it, which I have endeavored to give, leads me to anticipate. The construction of the perspective of a shadow is, from first to last, a problem of more tediousness and complexity, espe cially as applied to complex objects and positions, than falls within the scope of an elementary work like the present. Hence, only a few, and quite simple, problems in perspectives of shadows have been inserted. The simplest conception, and resulting definition, of the per- spective of a point is, that it is where the "visual ray" through the point pierces the plane of the picture, i.e. the " perspective plane." The method of construction just indicated, and here adopted for PART I., does away with the whole machinery of " vanish- ing points," "perpendiculars," "diagonals," etc.; and, accord- ingly, these, with their advantages, are briefly explained and illustrated, in PART II., as incidental matters, giving rise to derivative methods of construction, and tending to aid the reader in understanding the methods usually employed by writers on perspective. In a proposed future general work on perspective, I hope to exhibit more fully a systematic arrangement of all its methods, and interesting peculiarities and details. LINEAR PERSPECTIVE. INTRODUCTION. CHAPTER I. INSTRUMENTS AND MATERIALS. 1. PAPER. For elementary practice, thick unruled writing, or tough printing paper, will answer. For nicer work, German car- toon, or English smooth drawing paper, will be convenient ; and for exact constructions, in lines or tints of Indian Ink, Whatman's drawing paper will be best. 2. Support of the Paper. For slight pencil or ink sketches, the paper may lie flat on an atlas, or a few thicknesses of smooth paper, or any firm, but not rigid surface. For larger and exact ink drawings, the paper must be well wet, and then fastened round the edges with gum-arabic, to a smooth board. Then, when dry, it will be found to be tightly stretched. Great care should be taken to keep paper flat and smooth, when not stretched as just described. 3. PENCILS. For sketching, use a hard pencil, as No. 4, or 5, of Faber's, and make only the faintest lines. For finishing up pencil drawings, use softer and blacker pencils, as No. 3 for well defined objects, and Nos. 1 and 2 for shadows, foliage, &c. In pencilling a drawing which is to be inked, use a pencil sharpened on a fine file, to a thin edge, rather than a round point, since it will thus keep sharp much longer. 4. RULERS. For drawing on stretched paper, use a T rule, foi drawing all lines from right to left ; and a right angled triangle, for drawing lines perpendicular to these. With loose paper, use a common ruler and right angled triangle. 10 LINEAR PERSPECTIVE. 5. To draw parallels in any oblique position, by a ruler and tri angle. To draw through p, for example, a parallel to ab. Place FIG. 1. a side of the triangle on ab, and bring up the ruler ac, as shown in Fig. 1. Hold the ruler fast, and slide down the triangle to the position (2) when pd will be parallel to ab. 6. To draw perpendiculars in oblique positions. Slide the tri- angle, as before, till de passes through p, then d being a right angle, a line can be drawn through p, and perpendicular to ab. 7. COMPASSES. For drawing ink or pencil circles, the com- passes should have movable legs, which may be replaced by a drawing pen, or pencil-holder. 8. In using the compasses, hold them by the joint, with the thumb and forefinger. Then, in setting off distances on a line, turn them, alternately, on one side and the other of the line, never taking both points at once from the paper, till the operation is finished. This method is most expeditious and accurate. Like wise, in describing a circle, the whole can be accomplished with quick and uninterrupted motion. 9. Irregular Curves. For drawing other curves than circles, points of which have been previously constructed, use the thin plate of wood with variously curved edges and openings, and called an irregular curve. 10. INDIAN INK. This color, when of good quality, is of a brownish black, and is prepared in polished or gilded cakes, fine- grained, and usually scented with musk or camphor. It is prepared for use, like other water colors, by touching the end to water, an< q INSTRUMENTS AND MATERIALS. 1] rubbing on an earthen plate or tile. When enough has been ground off, wipe the cake dry to prevent its crumbling. 11. This ink, when thick, may be applied in a drawing-pen, or brush, so as to make black lines, or surfaces. When diluted with a quantity of water, tints of any degree of lightness may be quickly laid on the paper (stretched) by a rapid use of a goose-quill sized, or larger camel's hair brush. 12 LINEAR PERSPECTIVE. CHAPTER II. PTHT.TJMINARY PRINCIPLES AND EXPLANATIONS. 12. Sitting by a window, you may fix your attention on all that you see through one of its panes buildings and parts thereof, trees, roads, fields, woods, streams and clouds. As soon, however, as you give attention, both to the pane and to what you see through it, you will find that, by looking with each eye separately, you will see partly different sights through the same pane. Hence, to see definitely both 'the pane and what you see through it, you must close one eye. 13. This being done, you might paint upon the glass everything that you see through it, just where you see it, and of the same shade and color. A perfect picture, in every respect, of all seen through the glass, from one point of sight, might thus be made on the pane. Such a picture would be called the perspective of the view seen through the pane. 14. I. Hence & perspective is a picture which shows one or more objects just as they appear, in respect both to form and color, and as seen from one fixed point of sight. 15. If seated quite near the window, you will observe that you cannot see all that is 'to be seen through it, without turning the head ; while each new direction of sight gives you at least a partly new view. Also each new position of the eye gives, evidently, a different view through the same pane, 16. II. Hence any single perspective drawing should embrace no more than one view, that is, no more than can really be seen vhen looking in one direction from one fixed point of sight. 17. The chief exceptions to this rule are in panoramic and architectural interior scene painting, which, being intended to please large assemblies, are painted from several points of sight, or from one quite remote one. 18. All this being understood, suppose you are in a field, and dewing a distant tree through a framed pane of glass, held at a .axed distance from the eye. As yon approach the tree it appears to occupy a larger and larger portion of the glass ; while, as you recede from it, a contrary effect is produced. PRELIMINARY PRINCIPLES AND EXPLANATIONS. 13 III. Hence the size of the object, in a picture, depends on its distance from the eye. 19. Again; if your distance from the tree is fixed, the nearer the pane is carried to the tree the more completely will the view of the tree fill it. That is IY. The comparative size of an object in the picture, and th whole picture, depends also on the distance of the picture plan from the object. 20. Further, if two trees, at equal distances, and of different sizes, be viewed at once through the same pane, and from the same fixed position, the larger one will cover a larger space on the pane, as seen through it. Y. Hence, other things being the same, the size of an object, in the picture, depends on its actual size. 21. Once more, by moving, together with the pane, from side to side, or up and down, the tree will be seen through different portions of the pane, when seen from the different positions so taken. YI. That is, the place of an object in a given picture, its size and distance being also given, depends on its direction from the observer. 22. VII. From the last four particulars, we now conclude that, in order to represent a given object truly, its dimensions, distance from the picture, distance from the eye, and direction must all be known. In other words, the relative position of the eye, the picture, and the object, and the size of the latter, must be known. 23. Returning now to the picture painted on the window pane, each point of that picture is in a straight line, from the point repre- sented, to the eye. Such a line is called a visual ray. VIII. Hence the perspective of any point is where the visual ray from that point meets the surface of the picture. Finally, the following general principles may serve to connect; his introductory sketch, which embraces the primary facts of per spective, given by the testimony of the senses, with the more exact treatment of the subject, which succeeds, and in which the prin- ciples of perspective, based upon these facts, are demonstrated. 24. Science is a complete body of truth, whose parts are naturally related to each other ; and hence may be expressed by a systematic and connected statement of successive particulars, proceeding in natural order from primary elements to complete results. Perspective science is such a body of truth, relating to the 14 LINEAR PERSPECTIVE. manner of representing objects as they appear. This science IP founded on the simple facts of vision already described, and which are learned by observing what and how we see. As in making a picture itself, its outlines and most conspicuous objects, alone, may be represented, or all its peculiarities and details may also be included; so a science may be presented in its outlines only, or in entire completeness. This work aims to exhibit little more than the outlines of the subject of perspective, but yet fully enough to assist any one, who desires to draw ordinary objects as a business or pleasure, to do so intelligently and accurately. We now proceed to unfold the elements of Perspective from the preceding simple facts of vision, and to apply those elements to practical exercises in perspective drawing. DEFINITIONS AND GENERAL PRINCIPLES. Iff PART I PRIMITIVE METHODS. CHAPTER I. DEFINITIONS AND GENERAL PRINCIPLES. 25. The complete perspective of an object, is a picture of it, which, when viewed from a certain point, produces the same image upon the eye that the object itself does, when viewed from the game point. Each point and line of such a picture, must, when suitably placed between the eye and the object, exactly cover and conceal from view the corresponding points and lines of that object. It must also, as truly as art will allow, present, at each point, the same shade and color that is exhibited by the same object. 26. Hence perspective embraces two branches: the perspective of form, called linear perspective / and the perspective of color and gradations of shade, called aerial perspective. 27. Aerial perspective is an imitative art, founded on extensive observation of nature, and on the science of optics. Linear perspective is either an imitative art, or an art of exact geometrical construction, according as the outlines of pictures of given objects are traced by the eye, or constructed with instru- ments, according to geometrical principles. 28. In point of fact, linear perspective is practised as a construc- tive art, chiefly in its application to regular objects. It is practised as an imitative art, mainly in the drawing of irregular, or pic turesque objects, such as trees, animals, hills, streams, and old buildings. We will next inquire into the natural principles, which lead to the exact construction of the perspectives of objects. 29. The eyes are so related, that in attempting to see, with both of them together, objects at different instances, distinctly and at once, we see these objects partly double (12). Hence in making an exact picture of any object, we suppose it 1C LINEAR PERSPECTIVE. to be viewed with one eye, or that the two eyes are reduced to a single seeing point, called the point of sight (14). 30. Objects become visible by means of rays of light, reflected from them to the eye, and called visual rays (23). 31. Kays of light proceed in straight lines; as is proved by the fact that we can see nothing through an opaque bent tube. 32. The visible boundary of an object is called its apparent con- tour. The perspective of this contour, is the linear perspective of the object. 33. Any body, having a vertex, and plane sides, is, in a genera 1 sense, called a Pyramid. Any curved surface, having a vertex, and therefore containing straight lines drawn through that vertex, is a Cone, in the general sense of the term. Hence visual rays, from all points of the apparent contour of an object to the eye, form a pyramid, or a cone whose vertex is the point of sight according as the object is bounded by straight or by curved lines. This being understood, this pyramid and cone are, for the sake of brevity, called indifferently the visual cone. 34. Next, conceive a plane to intersect the visual cone, anywhere between its vertex, that is the eye, and its base, that is the object. This plane will cut from the cone a figure which will exactly conceal from the eye at its vertex, the apparent contour of the original object. That is (25) this figure will be the linear perspective (27) of that contour, and hence of the object (32). 35. In like manner, the intersection of the visual ray (23) from any one point of the given object, with the given plane, is the perspective of the point from which that ray proceeded. 36. The given plane is therefore called the perspective plane / and is understood to be vertical, unless the contrary is mentioned. Illustration. In Fig. 2, let E be the position of the eye, ABC the object, as a wood or paper triangle, to be represented ; and PQ, the perspective plane. Then AE, BE, and CE, represent visual rays from the corners or vertices of the given triangle. Now let , b, and c represent the points in which these visual rays pierce the perspective plane PQ, then dbc will be the perspective of ABC. 37. It now clearly appears, that, in order to find the perspective of an object, three things must be given ; the object itself, the posi- tion of the eye, and the perspective plane (22). Observe here, also, that, as lines from the visible points of the object to the eye are visual rays, these rays become known as soon as the positions of the eye and of the object are given. DEFINITIONS AND GENERAL PlilNdPLES. If either of the lines, as a#, of the perspective, were prolonged either way, or both ways, it would be called the indefinite perspec- tive of the original line as AB. 38. Any angle, as AEC, formed at the eye by two visual rays FIG. 2. is called a visual angle. It now appears that any line, as AC, and its perspective, ac, subtend the same visual angle. The reason, therefore, ;why an object and its perspective present the same appearance (25) to the eye, is, that they subtend the same visual angle; for the apparent size of any object depends on the size of the visual angle which includes it. Hence if two equal lines be in parallel positions, but at unequal distances from the eye, the further one will subtend the smaller visual angle and will therefore appear the shorter (18). Also, if a line or a surface be viewed obliquely, instead of directly, it will appear of diminished size, and is said to be foreshortened. 39. The position, E, of the eye, and the form and position of the bject ABC remaining fixed, there will be as many different sizes and forms of the perspective, 5c, as there may be different dis- tances and positions of the perspective plane, between E and ABC And these various forms and sizes of abc will all be true perspec- tives of ABC. To understand this completely, it is only necessary to remember, 1: That all these forms of abc are sections of the same visual pyramid ABC E (34), and 2 : That the definition of a perspective is not, a figure that is as the. original object appears; but, only, one that appears as that object does, when viewed from the sane point (25). 2 LINEAR PERSPECTIVE. CHAPTER H. THE ELEMENTS OF PROJECTIONS. 40. It is evident from a consideration of Fig. 2, that we cannot, practically, find a perspective picture directly according to (35) i. e. directly from objects themselves. Visual rays are invisible and intangible, and we cannot conveniently substitute for them, threads from every point of an object, as a house, to a fixed point, as the top of a stake, taken to represent the place of the eye, and then find where all the threads pierce a paper plane, set- up between the object and the place of the eye. 41. What then can be used in place of the actual object, from which to make its perspective, as truly as if found mechanically, as above described ? We employ auxiliary drawings, which show the positions, forms, and dimensions of the original objects, just as they really are, and from such drawings, with similar representa- tions of the visual rays, we construct the perspectives, which show those objects as they appear. 42. These auxiliary drawings, which show the given object, and its visual rays, as they really are, in respect to form and relative position, are called projections. To the expla- nation and construction of projections, we therefore turn, as the next thing in r order. 43. Illustration. Let HH', Fig. 3, represent a level plane, called the hori- zontal plane, and W an upright plane, at right an- gles to HH', and hence called the vertical plane. FIG. 3. The floor and any wall of a room, would be such a horizontal and THE ELEMENTS OF PROJECTIONS. 19 vertical plane. HV, the intersection of these planes, is called the ground line. Next, let P be any point in the open angular space between these two planes. Then let Pp be a straight line from P, perpendicular to the horizontal plane, HH', and meeting it at some point repre- sented by JP. Likewise, let Pp r be a line from P, perpendicular to the vertical plane, VV, and meeting it at p '. Then P is called the horizontal projection of P, and p' is the vertical projection of P. 44. Observe now, according to (41) that the two projections of a point are an adequate representative of the real position of the point. For p'q, the vertical height of the vertical projection,^', above the ground line HV, is equal to the real height, Pp, of the point P above the horizontal plane. Likewise pq, the perpendicu- lar distance of the horizontal projection,^, from the ground line, is equal to the real distance, Pp', of the point in space, P, from the vertical plane. Hence a point is named by naming its projections / thus, we describe P as the point J0p'. 45. If a point, as S, is in the horizontal plane, it coincides with its horizontal projection, s, and its vertical projection, s', must be in the ground line HV. Likewise, if a point, as R, lies in the vertical plane, it is its own vertical projection, r ', and its horizontal projec- tion, r, must be in the ground line. 46. By considering the explanations just given, we are led to the following additional practical particulars. First: The forms of bodies are indicated by the positions of the points which compose or limit their bounding lines. Hence, if the distinguishing points of the boundaries of an object be projected in the simple manner just explained, and if the projections of these points be connected, in each plane, the projections of the object will be formed. Second: p represents the real point, P, as it would appear if seen from above, in the vertical direction Pp. Likewise p' represents the same point as seen in looking in the direction Pp'. Third: In order to view all the points of an object simultaneously in the sam direction, the eye must be at an indefinitely great distance from it Hence projections represent objects as they would appear, if visible from an indefinitely great distance, and viewed in a direction per- pendicular to each plane of projection in succession. 47. Illustration. Fig. 4. GL is the ground line ; GH, the hori- zontal plane ; and GV, the vertical plane. ABC-D is a triangular prism, placed with its edges parallel to the ground line. In view- ing this prism from a great distance above it, so as to look at all parts at once, in the parallel directions Aa, F/*, ' is the visual ray from the vertex vv', and it pierces the perspective plane at V, the intersection of AY and t/E', giving V as the perspective of vv f . Joining the points now found, ABC V is the perspective of the given pyramid abc-v a'b'c' -v', as seen from EE'. Remark. The student should construct other figures by the above method, till quite familiar with it. LINEAR PERSPECTIVE. CHAPTER V. BtiMOVAL OF PEACTICAL DIFFICULTIES ARISING FROM THE CONFCJ SIGN OF PROJECTIONS AND PERSPECTIVES. I. First Method. Translation, forward, of the Perspective, Plane. 70. The perspective plane being between the eye and the given object, the plan of that object must lie behind the ground line. Also, as the perspective plane contains both the vertical projection and the perspective of the object, these two must both fall upon the plan, when the perspective plane is revolved back into the horizon- tal plane ; as seen in the last two examples. The confusion of lines arising from this source is sufficiently ap- parent from Figs. 16 and 17, though they embrace very simple objects, and remove the perspectives as far as possible from the projections, by placing the eye considerably to one side of the projections. 71. Hence, before proceeding further with practical constructions, we shall present a simple method of obviating the difficulty just mentioned. This method consists in transferring the perspective plane, with all the points in it, directly forward, far enough to allow it to be revolved back so as to lodge the figures in it entirely below, or in front of, the plan. A r P' a :' -*r FIG. 18. -E This method is illustrated in Fig. 18. A is a point whose projec- tions are a and a', on planes seen edgewise and in their real posi- tions at right angles to each other, at aGG' and GP. E is the place of the eye. Then X represents die trsi>e<,tive of aa'. REMOVAL OF PRACTICAL DIFFICULTIES. 37 When, now, the perspective plane GP is revolved back as shown by the arrows, carrying a' and X to a," and X', a,X' and a" will be crowded together. But suppose the perspective plane to be first moved forward carrying along the points a' and X to a new position G'P', and then to be revolved. The perspective, X", will then appear at X'", free from the plan ; and it may also be freed from the elevation, in practice, by erasing portions of the latter from time to time, as the construction of the perspective progresses, or by transferring only the perspective points. The elementary examples of the last chap or are here continued, according to the method just explained. EXAMPLE 3. To find the Perspective of a Cube, which stands obliquely with respect to the perspective plane. See Fig. 19. aceg is the plan of a cube thus situated, and a'Vc'f is its elevation. The ground line GL indicates the first position of the perspective plane, and G'L/ shows its position after translation forward. E is the horizontal projection of the point of sight. Being in the hori- zontal plane, its position is not affected by the translation of the perspective plane. E' is the vertical projection of the point of sight, shown only on the second position of the perspective plane, since it is used only there. For a similar reason, the vertical pro- jections of the visual rays are shown only on the second position of the perspective plane. #E is the horizontal projection of the visual rays from the two points oaf and afi (Ex. 1.). By making b"a"=b'a', and in a!l>' produced, we find the projections of of and V upon the second position of the perspective plane. Likewise we find/", e", c", etc. Then, for example, E and a"E' are the projections, employed, of the visual ray from a, a" ; or, more briefly (58) aE a"E' is the visual ray from a,a". This ray pierces the perspective plane at A, the intersection of a"E' with the perpendicular to GL, at A, where the horizontal projection, aE, of the ray meets the real, that is the original position of the ground line (57-8). Then A is the required perspective of a,af f . Other points as B,F, etc., of the perspective of the cube may be found in a precisely similar manner. The construction of some of the points is therefore omitted, to avoid unnecessary confusion of the figure. Thus, the perspective of the point, c,c?" will be at the intersection of a line c?"E' with tLe perpendicular to GL at n. The perspective of the back upper corner g,g" is likewise at the inter- 38 LINEAR PERSPECTIVE. section of a ray from g" to 2?', with the perpendicular to GL at o. To avoid the acute intersections, as at B, by the method of two planes, without setting E,E' far to one side, as in Fig. 1 7, trans- late as the points, aj)", of the given object, only, to one side in a direction parallel to the ground line, and then find their per- spectives, as B' (not shown) which will be well defined. Then a parallel to the ground line, through B', will in- tersect either "E', or AB, giving B by a well defined in- tersection. Observe that E,E' is not moved. Remark. The perspectives of other plane-sided objects, in various positions, should be construct- ed by the learner, by the method just explained. For ex- ample, let Fig. 17 be re-constructed according to the method of Fig. 19. II. Second Method. Use of three Planes. 72. The confusion of the diagrams, arising from the confounding together of the perspective with either or both of the projections REMOVAL OF PRACTICAL DIFFICULTIES. 39 of the given object, may be still further avoided by making the perspective plane a third plane, separate from both of .the planes of projection, and at right angles to both of them. This is accomplished in the manner illustrated in Fig. 20. FIG. 20. OHH is the horizontal plane of projection ; VV, the vertical plane of projection, and OLQ the perspective plane. P is a point in space, whose perspective is to be found, p represents its hori- zontal, and p' its vertical projection. E is the position of the eye, e its horizontal, and e' its vertical projection. Then PE represents the visual ray, whose intersection with OLQ will be the perspec- tive of P. pe is the horizontal, and p' e' the vertical projection of this ray. The perspective plane OLQ is perpendicular to both of the other given planes, and LQ is its intersection with the ver- tical plane of projection. LQ is called the trace of OLQ upon the vertical plane of projection. Then, as in previous cases, Pi, the perspective of P, is in the line n P l5 perpendicular to the ground line OL at n. Likewise it is obviously in the line r Pj, perpendicu- lar to the trace LQ at r. Hence P is at the intersection of n Pi and r P,. 73. Now in order to bring all three of these planes into a single surface, as is done in practical drawing, the perspective plane may be revolved about its trace LQ till it coincides with the vertical plane W, which may then be revolved back as usual around t-Le principal ground line, HLj. But by such a proceeding, the per- spective of an object would by revolution fall upon the vertical 40 LINEAR PERSPECTIVE. projection of that object. Hence the perspective plane is moved towards the eye, and parallel to its first position to some con- venient new position as n^r^ before being revolved. Then, as every point of the perspective plane moves parallel to the ground line, n will appear at n^ and r at / and after revolution in the di- rection n\n^ the vertical line ?iP } will appear at n^P^ and th horizontal linerP,, at rfz. Hence P will be the perspective of P, after the translation and first revolution of the perspective plane. 74. The perspective of a point by the method of three planes, \ .--* u. FIG. 21. shown pictorially in Fig. 20, is shown as an actual construction in Fig. 21. The former figure is exactly transformed into the latter by making the corresponding distances equal in both, and by letter- ing the same points with the same letters, so far as shown at all. pp is the given point, given by its projections, ee is likewise the point of sight, 9iLr the first, and n^L^ the second position of the perspective plane, thus indicated as at right angles to both planes of projection, pep'e' is the visual ray from pp', which pierces the perspective plane nLr at a point whose projections are n and r. After translating this plane, parallel to the ground line, to the position n^L[r^ these points appear at n 1 and r t . Then, by revolving the perspective plane from n{L\r\ into the vertical plane of projection, the point n\r\ describes a horizontal arc about the point L,, TI as a centre. The projections of this arc are n^ and *,P , and P 2 thus appears as the perspective of pp'. REMOVAL OF PllACTICAL DIFFICULTIES. 41 Remarks. a. The perspective plane must, in Fig. 21, be trans- lated to the right so as to revolve to the left, in order that the right hand of the perspective may continue to correspond with the right hand of the object drawn. This will be obvious on inspection in the succeeding examples, wherever three planes shall be used. b. Either of the methods of disposing of the perspective plane, explained in this chapter, will be used at pleasure in the solutions which follow. The student is advised to solve the subsequent problems, on three planes, when two are used by the author, and vice versa. To assist therefore in becoming more familiar with the use of three planes, the following practical problem is giveo. Fia. 22. EXAMPLE 4. To find the Perspective of an Obelisk, com- posed of a frustum of a long pyramid, capped oy a snort pyramid. 42 LINEAR PERSPECTIVE. Let the square acbdc'd'. Fig. 22, be the horizontal and verti- cal projections of the base of the obelisk ; and vfnot v'-n't' the projections of the cap pyramid. Let PQP' be the first and real position of the perspective plane, at right angles to botli planes of projection. Let PjQiP'j, be its second position, parallel to the first, from which it is revolved round P'iQ h its intersection with the vertical plane, until it coin cides with that plane. EE' is the point of sight. To find the perspective of any point, as aa', of the base. E and a'E' are the projections of the visual ray from this point. This ray pierces the perspective plane at gg'. This point, after translation, appears at g\.g^ found by drawing gg^ and g'gl, parallel to the ground line. After its revolution through the horizontal quarter circle whose projections are g^ and g^ A, it appears at A, the inter- section of ffi'A with <7 2 A, perpendicular to the ground line QQi. In like manner C and B, the perspectives of cc' and lib' are found. Note that W, the invisible corner of the base as seen in ver- tical projection, is the right hand corner, to the eye at EE' looking in the direction Ew. To find the perspective of any point of the cap pyramid, we also proceed just as before. Thus, oE o'E' is the visual ray from the corner oo'. This ray pierces the perspective plane PQP' at pp \ which is translated to >$>/, and from that position revolved in a horizontal arc, as before, to O, the perspective of oo'. Remarks. a. Every point of the perspective being thus found in precisely the same manner, the construction of several of them is left to be made by the student. b. Observe also, that as the operations in Figs. 21 and 22 are precisely similar, the perspective of any object, by the method of three planes, is simply, and only, a continued repetition of the con. Btruction of the perspective of a single point, as in Fig. 21. c. Practice is required, however, to enable the learner to under stand readily tine form and position of any given object from it* projections^ and to determine easily, by mere inspection, the pro- jections of those points which are seen from the given point of sight. Hence, again, the student is advised to construct the per- spectives of various other simple objects, from their projections, as in this example. l J iiOJECTIONS AND PERSPECTIVES QF CIRCULAR BODIES. 43 CHAPTER VI. PROJECTIONS AND PERSPECTIVES OF CIRCLES, AND OF BODIES HAYING PARTLY OR WHOLLY CIRCULAR BOUNDARIES. 75. The outlines of almost all artificial objects will be found, by analyzing them, to consist of straight lines and circular lines. Having now shown how to find the perspectives of points, straight lines, and plane sided figures, both pictorially and by actual con- struction, we next proceed to explain the construction of the per- spectives of circles, and of various bodies bounded in part, at least, by circles. EXAMPLE 5. To find the Perspective of a Circle lying in the horizontal plane. The method by two planes, with the vertical or perspective plane translated forward before being revolved back into the horizontal plane (71) is here employed. See Fig. 23. Let acde be the horizontal projection of the given circle. As this circle lies in the horizontal plane, its vertical projection, a'd', must lie in the ground line LL (45). Now let the perspective plane, which is perpendicular to the paper at LL, be translated forward to the parallel position L'L', and then, as usual, revolved backwards into the horizontal plane, or plane of the paper. Then take EE' as the point of sight, and let all the vertical projections be shown on the translated position of the perspective plane. Accordingly, a"d" will be the new vertical projection of the given circle. 5E is the horizontal, and/^E' the vertical projection of the visual ray from the point ,/" in the circle. The point n, where the horizontal projection &E meets the ground line, LL, is the horizontal projection of that point of the ray itself in which it pierces the perspective plane (58). The latter point is at once in the perpendicular, riB, to the ground line, and in the vertical pro- jection /"'E' of the same ray. Hence the desired point is B, which is the perspective of 5, f. [This being a new form of example, the construction of the per- spective of one point is explained as minutely as if it had not been 44 LINEAR PERSPECTIVE, fully explained already. The details of the explanation will there fore be omitted in future similar constructions.] FIG. 23. The ray cE-c ff E' pierces the perspective plane at C, which is therefore the perspective of c,c". In like manner the perspectives of any other points can be found. By inspection of the vertical projection, a"d r> ', it appears that the extreme visual rays, as a"E', as seen in vertical projection, are those which proceed from the opposite ends of the diameter whose horizontal projection is at. Hence rays from points, as g, before that diameter, or #, behind it, find their vertical projections, g"E' and/"'E' within a"E'. Hence no point of the perspective of the PROJECTIONS AND PERSPECTIVES OF CIRCULAR BODIES. 45 circle can appear outside of the ray a"E', and therefore the per- spective must be tangent to e&"E', at A, the perspective of aa". The similar result, at the perspective of tf, is not shown, as it could not appear distinctly on account of the position of EE'. The rays whose horizontal projections are tangent to the plan at g and d, include the other rays between them. Hence all points of the perspective are between the perpendiculars mD and AC, and the perspective is tangent to these perpendiculars at D and G. The perspectives of tangents, parallel to the ground line, will be tangents to the perspective and parallel to I/I/. Having now six tangents with their points of contact, besides other points, the perspective curve can be very accurately sketched. 76. In the previous perspectives ol plane-sided figures, which are distinguished by well denned edges and corners, however viewed, it will be observed that it can be determined, by simple inspection, which edges will be visible from the point of sight. But, in the case of objects bounded partly or wholly by continuous curved surfaces, the consequent partial or total absence of limiting edges makes it necessary to discover the visible boundaries by more or less of preliminary construction. Hence, a few additional defini- tions and principles are introduced here for use in the following problems : 77. Other planes than the planes of projection, go by the general name of auxiliary planes. Their positions are indicated by their intersections with the planes of projections, called their traces. Each of these traces takes its name from the plane of projection in which it is found. 78. The point where either trace meets the ground line is where the plane cuts the ground line ; hence both traces of a plane must meet the ground line at the same point, if they meet it at all. The traces of a plane will meet the ground line unless the plane is parallel to that line. 79. If, as in Fig. 24, a plane is vertical, but oblique to the ver- tical plane of projection, its verti- cal trace, VT, will be perpendi- Fig. 24. cular to the ground line, G L. 80. If, as in Fig. 25, a plane is perpendicular to the vertical 46 LINEAR PERSPECTIVE. plane of projection, its horizontal trace, H T, will be perpendicular to the ground line, G L. If a plane is perpendicular to both of the planes of projection, both of its traces will be perpendicular to the ground line, as we have seen in (72-74). FIG. 25. 81. Again; when a plane is vertical, that is, perpendicular to the horizontal plane, all points and lines in it ave horizontally pro-, jected in its horizontal trace ; since the horizontal projections of points and lines are vertically under the points and lines themselves. Likewise, when a plane is perpendicular to the vertical plane of projection, all points and lines in it find their vertical projections in its vertical trace. 82. Any plane containing the point of sight, contains an indefi- nite number of visual rays, whose directions radiate in all direc- tions, in that plane, and from the eye. Hence such a plane is called a visual plane. 83. A visual plane being thus composed of visual rays, if such a plane be passed through a line whose perspective is to be found, the trace of that visual plane on the perspective plane will be the perspective of the given line. See Ex. 1, Rem. , also Fig. 10, where the plane triangle EAB serves to mark the visual plane of indefinite extent, and containing the line AB. A BI is a portion of the trace of this plane on the perspective plane, and is, therefore, the perspective of AB. 84. For the reason just given (82), the point or line at which a visual plane is tangent to a curved surface, is a point or line of the PROJECTIONS AND PERSPECTIVES OF CIRCULAR BODIES. 47 visible contour of that surface. The perspective of this visible contour or boundary, is the boundary of the required perspec- tive. (32.) Ex. 6. To find the Perspective of a Cylinder, standing on the horizontal plane. A cylinder, seen from above, as it stands on the horizontal plane, appears only as a circle. As seen looking forward at it, perpendicularly to the vertical plane, its diameter and height are visible. Hence the circle afm, Fig. 26, and the rectangle n'o'p'r' are the projections of a cylinder in the given position. JTL FIG. 26. This being established, EA, tangent to the horizontal projection of the cylinder, is the horizontal trace of a vertical visual plune, tangent to the cylinder along a vertical line of its convex surface at 48 LINEAB PERSPECTIVE. h. Likewise, Ea is the horizontal trace of a similar plane, tangent to the cylinder along a vertical line at a. The vertical projections of these lines are h'k' and a'b', and they are the projections of the visible boundaries of the convex surface, as seen from EE'. The tangent planes being vertical, their horizontal traces, as EA, are the horizontal projections of both of the visual rays, asEA-E'A' and EA-E'&', from the lower and upper extremities of the lines of con tact, as h-h'k'. (81.) This being understood, nothing peculiar remains in the con struction of the perspective, ABFK, of the cylinder. Thus, the perspective of the point aa' of the lower base, is found by drawing the visual ray, aE-a'E', which pierces the perspective plane at A, the intersection of a'E' and #A, perpendicular to GL at q. Like wise, K, the perspective of the point A, k r of the upper base, is at the intersection of &'E' and sK, &'E' being the vertical projection of the visual ray from A, &', and sK the perpendicular to GL from the intersection of GL with AE, the horizontal projection of the same ray. The perspective bases are tangent, as at A and B, to the extreme visible elements, as AB ; for the visual plane containing such ex- treme element, as a a'V is tangent to the visual cone from either base. Therefore, the intersections, as AB, of the visual plane, and AtfF, or BK, of the visual cone, with the perspective plane are tangent to each other, as at A. (See Art. 85.) Remarks. a. Since there will, even when great care is taken, often be slight instrumental errors in the construction of points, the curves in the perspective can be more advantageously drawn by carefully connecting a few carefully constructed points by easy curves, than by finding many points in those curves. b. The figures in this book being designed for purposes of instruc- tion, necessarily show the lines of construction much more fully than is necessary in practice. For example, in finding the point B, Fig. 26, it is not necessary actually to draw either E, #'E', or ^B, but only to mark the point q in the line aE, then to draw little frag- ments of #'E' and $J3, just at their intersection B. Likewise in Fig. 22, all that is essential in finding O, for example, after drawing oE o'E', is to make p^O equal to Q^>, and in the horizontal liuep'pi'. c. Another matter of still greater practical importance, is the order in -which the lines of construction should be drawn. All the lines necessary for finding the perspective of one point, should be drawn, before proceeding to draw those by which a new PROJECTIONS AND PERSPECTIVES OF CIRCULAR BODIES. 49 point is found. Thus in Fig. 26, draw E, a'E', and om the ground line LL. The equal circle whose centre is O', i he vertical projection of the same sphere, and the height of O above LL is the height of the centre of the sphere above the hori- zontal plane. E and E' are the projections of the point of sight, and the vertical plane is taken also as the perspective plane. ing objects shall have been found, the two following problems may be omitted at the discretion of the teacher. * This method being chiefly valuable as an intellectual exercise in the conception of positions and motions in space, it may be omitted at the discretion of the teacher PROJECTIONS AND PERSPECTIVES OF CIRCULAR BODIES. 59 It is now evident that, if a vertical visual plane EO# be drawn through the centre of the sphere, it will cut a vertical great circle I/ $" 7" FIG. 32. from the sphere, to which two tangent visual rays may be drawr. The points of tangency of these visual rays will, as such, be points of the apparent contour of the sphere. But to show these rays, the plane EO must be revolved to a position parallel to one of the planes of projection. Let it be revolved about the horizontal diameter, ab, of the sphere, till it becomes parallel to the horizontal plane of projection. The vertical great circle will evidently then appear in the great circle hc"b, on ab as a diameter. The eye is at the vertical distance E'n below the level, O'w, of the centre of the sphere ; hence, if the highest point of the vertical circle revolve to the right, as shown by the arrow, EE' will be found after revolu 60 LINEAR PERSPECTIVE. tion at e*, at the left of the axis of revolution aE, and at a pei pendicular distance from it equal to E'^. This done, e"c" and e"d" are the revolved positions of the desired tangent visual rays, and their points of tangency, c" and d" , with the circle d"c"b^ the revolved positions of two points of the appa- rent contour. By revolving the plane containing c" and d" back to ts vertical position Ea5, c" and d" will revolve back about db as an xis in arcs whose horizontal projections are c"c and d"d, perpendi cular to ah, and will give c and d as the horizontal projections of these two points of the apparent contour. To find the vertical projection of c, for example. Consider, first, that it must be in a perpendicular to LL, through c, and that it is at a height equal to cc" above the level O'n of the centre of the sphere. Hence on c-c' make c'k = c"c, and c' will be the vertical projection of the point of apparent contour whose horizontal pro- jection is c. The vertical projection of d, not shown, in order to simplify the diagram, will be below O'n at a distance equal to d"d. To find any other points of apparent contour. Intersect the sphere by any other vertical visual plane as E^ 1 , which cuts from it the small circle whose revolved position, about pq as an axis, is f'g"q. In this revolution, EE' will appear at e'" ; Eg'" being equal to ~Ei'n and perpendicular to Ether points of the apparent contour may be similarly found. As the eye is placed, in this problem, in the horizontal plane 'E', through the centre of the sphere, the apparent contour of tli sphere is evidently a vertical circle, hence nm, a straight lino through t and u, and which will be perpendicular to YE, is its hori- zontal projection, n and w, being on the horizontal great circle ot the sphere, are vertically projected at ri and m'. Two points are horizontally projected at c, at t and at u. Those at c are on the vertical great circle A'B'<#', and are vertically projected at c r and d' . r' and s' are in the line's t-t' and u-u' and as far below the line mW, as t' and u' are above them. Remarks. a. This second method is, from its nature, called the method of tangent visual planes. b. In order to familiarize the learner more effectually with this beautiful method of tangent visual planes, located by the use of auxiliary tangent cones, we will now apply it to an object of another and very different form, and with the use of three planes of projection. Among the comparatively few large double curved surfaces occur- ring in the mechanic arts, whose perspectives need to be accurately constructed, are Domes, and concave Spires, &c., whose perspectives may be found as follows. EXAMPLE 12. To find the Perspective of a concave Cupola-Roof. Let the figures with centre A and vertex A', Fig. 34, be the projec- tions of the cupola roof; PQP' the original, or real position of the perspective plane, and EE' the point of sight. The construction of A' is, after previous similar constructions, sufficiently indicated in the figure. Then assume BtfT-B'C' as the circle of contact of an auxiliary cone, tangent to the inside of the cupola. Drawing CV tangent to the cupola at C', we find v r the vertex of this cone. Then AE- v'E', the visual ray from this vertex, pierces the plane of the cone's base at R'R (Ex. 11), hence Rtf and RT are the traces, on this plane, of two planes which are tangent to this cone on elements At and AT (not drawn) and hence to the cupola at the points tt' and TT' (Ex.11 ; 6). The perspectives of these points are t" and T", found by making c*T* =Qc, &c. (Ex. 6, Rem. t>.) 64 LINEAR PERSPECTIVE. At the base of the cupola is a round edged band, three points of \vhich, mm' / nn' and r' it is sufficient to find in perspective, as at p' A ,/' m ^kJ--*/_-J CLE> FIG. 34. m*, n ff y and r". Through these points the perspective of the cupola can be sketched. PERSPECTIVES OF SHADOWS. 65 CHAPTER VII. PERSPECTIVES OF SHADOWS. General Principles and Illustrations. 89. Perspectives of shadows, like those of objects, are readily found from their projections, by the method of visual rays already explained. But shadows, being obviously not independent of the bodies cast- ing them, require a little separate preliminary study, to show how they are found when those bodies are given. 90. The shadow of a body on any surface, is that portion of that surface from which light is excluded by the body. A shadow is known when its bounding edge, called the line of shadow, is known. 91. Rays of light from a very distant source, as the sun, fall upon any terrestrial object in parallel straight lines. 92. Any ray which is intercepted by the given body will, evi- dently, if produced through the body, pierce the shadow within its boundary. Any ray, not intercepted by the body, will evidently pierce the supposed surface containing the shadow, beyond the edge of the shadow. Hence the line of shadow is the shadow of that line on the given body, at all points of which the rays are tangent to the oody. This line of contact of rays, separates the illuminated from the miilluminated portion of the given body, and is called the line of shade. 93. Since the line of shadow is thus the shadow of the line of shade, the latter must always be found first. The line of shade, from which shadows are determined, is found in the same general manner as the line of apparent contour, from which perspectives are determined ; viz. by inspection on most plane sided bodies, and by the aid of tangent rays of light, or tan- gent planes of rays of light, on curved surfaces. 5 LINEAR PERSPECTIVE. 94. Practically, shadows are found, a point at a time, and any one point in a line of shadow is where a ray of light, from some point in the line of shade of a given body, pierces the surface receiving the shadow. Hence it is obvious that the form of a shadow will depend both on the form of the body casting it and that of the surface receiving it, and also on the direction of the light ; while the method of find ing it will depend only on the form of the surface receiving it. 95. Finally: To find a shadow, we must have given, by their projections, 1st. The body casting it ; 2nd. The surface receiving it ; 3rd. The direction of the light. These given, we may then con- struct, 1st. The line of shade on the given body ; 2nd. The shadow determined by that line of shade. This done, we can at last con- struct the perspective of the shadow. Problems of perspectives of shadows being thus obviously some- what tedious and complex, only a few simple and generally useful ones are here inserted, as an introduction to the subject. The shadows which most frequently occur in perspective drawings such as are made largely for pictorial effect, are the shadows cast by lines in various positions, on the ground, and on the walls and roofs of buildings. The following principles and examples, therefore, give elementary illustrations of the operations necessary in finding such shadows. 96. Let AB, Fig. 35, be a slender vertical rod or wire, and let LR represent a ray of light drawn through its upper extremity, B, and piercing the horizontal plane GH at R. Then R will be the shadow of the point B. But the point, as A, in which a line meets a surface, is a point of the sha- dow of that line on that sur- face. Hence AR is the sha- dow of AB on the horizontal plane GH. But BA being vertical, AR is also the hori- zontal projection of the ray BR. Hence, the shadow of a vertical line on the horizontal plane is in the direction of the horizontal projection of the light. 97. By operating in a precisely similar manner upon a line per- pendicular to the vertical plane GV, it will be found that the sha- FIG. 35. PERSPECTIVES OF SHADOWS. 67 dow of such a line upon the vertical plane, will be in the direction of the vertical projection of the rays of light. 98. Since the rays of light are parallel, it is clear that the shadow of a vertical line on the vertical plane, will, itself, be a vertical line ; likewise, the shadow of a horizontal line on a horizontal plane, will be parallel to the line, and, generally, for the same reason, if a line be parallel to any plane, its shadow on that plane will be parallel to the line itself. Hence, also, the shadows of parallel lines on the same plane, will be parallel to each other. 99. If the same line casts a shadow on both planes of projection, the shadows on the two planes must meet the ground line at the same point. Thus in Fig. 36, let AC be a vertical line, long enough to cast its shadow partly on each plane of projection. Then R, where the shadow AR leaves the horizontal plane GH, must be the beginning of the shadow RT on the vertical plane. For all the rays CT, BR, &c., from points on AC, being parallel, form a plane called a plane of rays, of which AR and RT are the traces, and it has already been shown (78), that the two traces of the same plane must meet the ground line at the same point. FIG. 36. Finally, in general, if the sha- dow of any line, as the top edge of a roof, falls on both of any two intersecting surfaces, as the front and side of another building, these two shadows will meet at a common point on the edge dividing those surfaces. Hence, when we have the complete shadow on one such surface, this common point gives us one point of the shadow on the other surface. EXAMPLE 13. To find the Perspective of the Shadow of a Square Abacus upon a Square Pillar. The method of two planes is here employed, GL, Fig. 37, being the first, and G'L' the second position of the ground line. The construction of the perspective of the pillar and its cap (abacus) is not shown, it being exactly like many previous constructions. Also, no more of the vertical projection of the object is made than is necessary in finding its perspective, and shadows. Rays of light, like other lines, being indicated, in position, by their projections, let Al and A'L' be the projections of the ray through the front right hand upper corner, AA', of the abacus 68 LINEAR PERSPECTIVE. This ray pierces the horizontal plane at /, whoso vertical projection which must be in the ground line (45), is L'. Therefore the point FIG. 37. tself is at , which is therefore the shadow of AA' on the horizontal lane. The shadow of the lower front edge, aA-a'b', of the abacus, upon the front of the pillar, will be parallel to itself (98). When the direction of a line is known, one point in it is sufficient to determine it ; hence, to find this shadow, it is only necessary to pass a ray, as cb-c'd', through any point, as cc', of the lower front edge of the abacus, and to find where this ray pierces the front face of the pillar, as at b,d'. In this case, by drawing the ray through , in plan, the point of shadow, b,d, is made to fall on the right hand vertical edge, b-b'f, of the pillar. A line through d', and PERSPECTIVES OF SHADOWS. 69 parallel to a'b', will be the vertical projection of the shadow of #A- a'b'. Drawing the visual ray J'E', its intersection, D. with DF, the perspective of #'/"', will be the perspective of d! ; and as the shadow is parallel to the perspective plane, its perspective will be parallel to itself (69, a), that is a horizontal line through D. Next, drawing the ray ah-a'h\ we find M', the shadow of ad the lower, front, left hand corner of the abacus, on the side surface. gn, of the pillar, whose vertical projection is a line through h r equal and parallel to b'f. The visual ray, AE-A'E', from hh' gives its perspective, e. Then eo is the shadow of a small portion of aA- a'b' upon the left side of the pillar. Now for the shadow on the horizontal plane. The shadow of the point AA' is , where the ray A-A'L' pierces that plane. The ver- tical projection of I is I/ (45), and the visual ray, E-L'E', therefore gives M as the perspective of the point IL'. In the same way, find the shadows of the points A,#' and bb', and join the latter shadow with F. The shadow of A' AK will be parallel to that line, and will begin at I. Hence, as will fully appear on making the construc- tion, L'E' will also be the vertical projection of the visual ray from the shadow of K. Hence ME', up to DF, is the visible portion of the perspective of the shadow of A'-AK. EXAMPLE 14. To find the Perspective of the Shadow of any triangular Pyramid upon the Horizontal Plane. In this problem we shall employ the simple principles, that the shadow of the point where any number of lines meet is the point where the shadows of those lines meet ; and that the point in which a line pierces the horizontal plane is a point of its shadow on that plane. The method by two planes is employed, and the construction of the perspective of the pyramid, being the same as in many previous problems, is briefly indicated in the diagram, only, Fig. 38. Let ABC be the plan of the base of the pyramid, and V, that of its vertex. V-A'B'C' is the vertical projection of the pyramid This vertical projection, being shown in full on the original positioi of the vertical, or perspective plane, only its points, A'B'C* and V ", are shown, in the same relative position, on the translated position of the same plane, whose ground line is G'L'. In fact, after becom- ing quite at home in the subject of perspective, the student will see that A'B'C'-V might have been omitted altogether ; and, in gene- ral, that often only points, and not lines, of the projections of objects need be shown, in order to find their perspectives. Having, as in previous problems, found v-abc, the perspective of 70 LINEAR PERSPECTIVE. the pyramid, draw the ray of light VR-VR' which pierces the horizontal plane at R, projected back from R' in the ground line. C" B" FIG. 38. Then R is the shadow of W on the horizontal plane, and A and PERSPECTIVES OF SHADOWS. 71 C being their own shadows (96 and the second principle above stated )RA and RC are the shadows of VA and VC. Then, drawing the visual ray RE-R"E', we find r for the perspective of RR" ; R" being the vertical projection, R', in its second position. Hence ra and re are the perspectives of the shadows RA and RC, which limit the shadow whose perspective was required. . EXAMPLE 15. To find the Perspective of the Shadow of Dormer Window upon a Roof. In this concluding example of shadows, found by primitive methods, we will, for further variety, employ the method by three planes. Again, this example involves the shadows of lines in three different positions, upon a slanting surface, and affords the most instructive variety with the fewest lines. Moreover, as the sha- dows of lines are determined by the shadows of points in them, and as the shadow of a point is the same and similarly found whether the point be on a straight line or curve, a careful study of this and the two preceding examples should enable the student to find the projections and perspective of any ordinary shadow. First, now, in Fig. 39, to find the projections of the windows and roof. To avoid unnecessary lines, only a small portion of one slope of a roof is shown, of which ABCD is the plan, CDC'"D'" is the auxiliary elevation, showing the true size of the front of the dor- mer, and A'B'C'D'-C"D", found as in Ex. 10, is the principal elevation. G and F are the plans of the vertical edges of the dormer, whose true heights I"G" and J"F" appear in vertical pro- jection at I'G' and J'F'. The vertical projection, E', of the peak of the gable, FEG, is found on the projecting line ENE', by laying off NE' equal to its height, N"E", above the horizontal plane. Then draw E'G' and E'F'. NK' parallel to B'C", is evidently the trace on the roof, of a vertical plane through the ridge EK-E'K', which therefore meets the roof in this trace at K', whose horizontal projection is K. The points HH' and LI/ are imilarly found, and then joined with KK'. Next, to find the projections of the shadows on the roof. Le FP and F'P' be the projections of a ray of light to which all tho other rays are parallel. The shadow of the vertical edge, F-F'J', will fall in FS-J'S', the trace on the roof of a vertical plane of rays (99) through that edge. The ray FP-F'P' meets this trace at P', which is then horizon- tally projected at P, and FP-J' P' is the shadow of F-J'F'. Then PP' being the shadow of FF' (94), and LI/ being its own shadow (96), LP-LT' is (geometrically, for this shadow is unreal), the sha- 72 LINEAR PERSPECTIVE. dow of FL-F'L'. The shadows of parallel lines, on the same plane, FIG. 39. PERSPECTIVES OF SHADOWS. 7S being parallel (98), KR-K'R' the shadow of EK-E'K', is parallel to LP-L'P' and is limited at R by the ray ER. R' is then projected from R. Finally, by drawing RP-R'P', we have the shadow of EF-E'F'. Lastly, to find the perspective of the roof and shadows, whose projections have just been completed. Let b"A's be the original, and a'd' the translated position of the perspective plane ; and let | OO' be the point of sight. This construction scarcely needs any explanation, exactly similar ones having been often fully explained already. One or two points only are mentioned, to acquaint the learner with the abbreviations which are made in the construction of the figure. To find e, for example. Draw the ray EO-E'O' and from its intersection, n, with the perspective plane, draw nn\ parallel to the ground line ; then make rie=n"A.', and e will be the perspective of EE', since this is obviously equivalent to translating n" to ri", and revolving it, as in the previous unabridged constructions. Having found d, the perspective of DD", in the same way, also^, and <7, the lines dd",fj and gi can immediately be drawn perpendi- cular to the ground line, since they are the perspectives of vertical lines, dd" is limited at d", simply by drawing D'O' to u, and ud\ parallel to the ground line, fj and gi are limited by their intersec- tion with ab. a is the perspective of the point AA', which is its own perspective, it being in the perspective plane, and a'a=A'A. The perspectives of the points of shadow are found in the same manner. Thus, to find >, draw the ray PO-P'O' to qq f , and the line of translation q 1 p f , and make p'p A'q, which will give jt>, the perspective of PP'. Drawing jp, finding r, the perspective of RR', as p was just found, then drawing JOT, and rJc, we sh?ll have the complete perspective of the shadow on the roo GENERAL PRINCIPLES AND ILLUSTRATIONS. 76 PART II. DERIVATIVE METHODS CHAPTER I. GENERAL PRINCIPLES AND ILLUSTRATIONS. 100. In all the problems of PART I., we have found the perspec- tive of every point by one and the same primitive and natural method, which consists in finding where a visual ray (actually represented) through any given point, pierces the perspective plane. This method is primitive, and peculiarly the natural one, because it manifestly embodies the simplest geometrical definition of the perspective of a given point, viz. that it is where a visual ray from that point pierces the perspective plane (35). It is true, that in the practical application of this method, having revolved the perspective plane directly back into the horizontal plane, a difficulty arose, as in Figs. 16 and IV, from the confound- ing together of the projections and the perspective in one place on the paper. This difficulty led,^^: to the translation forward of the perspective plane, till it could be revolved back into the hori- zontal plane so as to bring the perspective below the projections, as in Fig. 19; second: to the use of three distinct planes, as in Figs. 22, etc., where the difficulty of confusion of figures was obviated nost completely. But these merely particular graphical methods f applying the method of visual rays, evidently do "not alter the method itself, and we repeat, that all the problems of PART I. were solved by the primitive method of finding where visual rays, actu- ally represented, through given points, pierced the perspective plane. 101. All problems whatever, in perspective, might be readily solved in this simple and beautiful manner ; but by inspection of the perspectives thus found, certain peculiarities may be discovered, which, on examination, lead to other methods, hence called deriva- 76 LINEAR PERSPECTIVE. tive ; or, because the visual rays are no longer represented, com paratively, artificial methods. The advantages of knowing several methods, which will soon appear, are chiefly two : 1 Abbreviation of the operations of con- struction. 2* Provision of checks upon inaccuracy. It has already been shown by experimental proof, in previous constructions, that any lines, whether vertical or horizontal, which are parallel to the perspective plane, have their perspectives parallel to each other, and to the lines themselves. Also that the perspec- tives of all lines which are perpendicular to the perspective plane, meet at the vertical projection of the eye. (Fig. 16.) These two results have also already been separately proved to be true, (Ex. 1. Hems. a,b.) but by now considering them in connexion with a few others, we shall arrive at a body of principles by which perspec- tives of objects can be found by the derivative methods, which it is the object of this second " PART " to explain. 102. In standing on a vast plane, such as a natural plain, its remotest visible limit appears as a horizontal line on a level with the eye. The reason of this is evident from Fig. 40. Let E be the jp~~ b <*- H FIG. 40. place of the observer's eye, looking forward in the direction ER, parallel to the ground HP. In taking successive points on the ground, as , #, and P, at greater and greater distances from the ob- server, standing at H, the visual rays aE .... PE, &c., become more and more nearly horizontal, and finally, when a ray comes from an indefinitely remote point on the ground, its direction cannot be dis- tinguished from that of the horizontal visual ray ER. Hence, as the apparent position of objects depends on the direction of the visual rays entering the eye from them, the very remote limit of any level plane appears as a horizontal line, on a level with the eye. 103. The indefinitely remote limit of a natural plain, or horizon- tal geometrical plane, is called the horizon / hence a line parallel to the ground line, and through the vertical projection of the point of sight, is the perspective of the horizon. GENERAL PRINCIPLES AND ILLUSTRATIONS. 77 Such a line is called the horizontal line, or the horizon of the pic- ture. 104. It follows from this, that the remotest visible limit of all lines in such a plane, that is of all horizontal lines, will appear to be in the " horizontal " line, which represents the remotest limit of this plane. 105. Now the remotest visible limit of the plane supposed, is literally its vanishing line, and, likewise, the remotest visible point of any line in that plane, is its vanishing point. The representation of this line, or point, on the perspective plane, is the perspective of such line or point, and is, according to the last two articles, a line or point on the perspective plane, and at the height of the eye. 106. The perspective of a vanishing line or point being of con- stant use in the construction of perspectives, while the original indefinitely distant real vanishing line, or point, is not, the former is, for brevity, itself termed the vanishing line, or point. Hence we have these principles : 1. The vanishing line of any horizontal plane, is a horizontal line, drawn on the perspective plane and at the height of the eye. 2. Any horizontal line has its vanishing point in the "horizontal" line. 107. Similar reasoning might be applied, and with corresponding conclusions, to vertical, or oblique planes, but as we do not find in Nature real planes of indefinite extent, everyway, in these positions, it will be sufficient to consider the vanishing points of lines, only, in any direction. 108. The visual ray from the indefinitely distant, or remotest visible limit of an unlimited line, will evidently appear to be parallel to that line, and the intersection of this ray with the perspective plane, is the perspective of that remote or real vanishing point. This intersection itself (106) is practically called the vanishing point, in making perspective drawings ; and will be so called in the following pages. Also, a visual ray which is parallel to one line, is arallel to all others, which are parallel to that one. Hence to find the vanishing point of any line or group of parallel lines, we have the following rule. Find where a visual ray, parallel to the given lines, pierces the perspective plane ; the point thus found will be the required vanishing point. 109. Illustration. Let PP, Fig. 41, represent the perspective plane, and L,L,L, three parallel lines in any direction. These lines will appa rently meet, and so maybe considered as meeting, at an indefinitely great distance, and the visual ray YE from their distant apparent intersection, will, for any short distance, as EV, be sensibly parallel LINEAR PERSPECTIVE. to them. But V, the intersection of this ray with the perspective plane, is the perspective of that intersection. That is, V is the vanishing point of L,L,L, FIG. 41. Observe finally, that as parallel lines themselves appear to meet at their indefinitely remote point, so their perspectives will meet at their vanishing point on the perspective plane, which is the perspec- tive of their real vanishing point in space. Thus, if aV, 5V, and cV are the perspectives of L,L,L, they will meet at V. 110. In general, if any number of lines meet at any point, their perspectives will evidently meet at the perspective of that point. EXAMPLE 1. Let it be required to find the vanishing point of several Telegraph Wires which go over a hill. In Fig. 42 let AA' and BB' be two successive poles, carrying two wires. AB is the plan of both of these wires. Let CC' and DD' be another pair of poles, of a line of single wire, and let EE' be the position of the eye. Then EY, parallel to AB or CD, and E'V, parallel to A'B' or C'D', are the projections of the visual ray, parallel to these wires, and therefore giving the perspective of an indefinitely remote point upon them. This ray meets the perspec- tive plane at V (58), which is therefore the vanishing point at which the perspectives of the wires will meet. 111. From the general case just considered, in illustration of the general principle of (108) let us proceed to find the location of the vanishing points of groups of parallels, having particular positions with respect to the perspective plane. First. It follows directly from the rule (108), that all lines which are parallel to the perspective plane have no vanishing point. Hence their perspectives will be parallel to themselves. That is, the perspectives of vertical lines, for example, will be vertical, as GENERAL PRINCIPLES AND ILLUSTRATIONS. 79 seen in (Fig. 16, etc., PART I.) Also, if lines are parallel to the ground line, their perspectives will be parallel to the ground line, as also seen in Fig. 16. FIG. 42. Second. It also follows from (108), that all horizontal lines nave their vanishing points in the horizontal line, or horizon (103). 112. In particular, among horizontal lines, we notice those which are also perpendicular to the perspective plane ; and those which make an angle of 45 with the perspective plane. The former are called perpendiculars, and the latter, diagonals. EXAMPLE 2. To find the vanishing point of a Perpendicu- lar, and of a Diagonal. See Fig. 43, where DC is the ground line, EE' the point of sight, and D'A the horizontal line. By (48), when a line is perpendicular to the vertical plane, its vertical projection is a point, and its horizontal projection, a line, perpendicular to the ground line. Therefore db is the horizontal, and a' the vertical projection of a perpendicular, at the height aa' above the horizontal plane. Likewise Ee is the horizontal, and E' the vertical projection of a visual ray, parallel to db-a' . This visual ray pierces the perspective, or vertical plane at E', which is therefore the vanishing point of db-a' and of all perpendiculars (108) while EE' remains as the place of the eye. 80 LINEAR PERSPECTIVE. 113. The point E', the vertical projection of the point of sight, is usually known among artists as the centre of the picture / since in a picture of equal interest throughout, it should be in the centre, of the horizontal width, at least, of the canvas. Therefore we say that the vanishing point of perpendiculars is at the centre of the picture. E is often called the station point. 114. To return now to the diagonal. By (51), when a line is parallel to the horizontal plane only, its vertical projection is paral- lel to the ground line, hence (112), making ac = ab, abc acb = 45, and be will be the horizontal projection of a diagonal through the point 5, a' and a'c' will be its vertical projection. Then ED- ET)' is the parallel visual ray which pierces the perspective plane (58), at D'. Hence D' is the vanishing point of bc-a'c f and of all other diagonals (108). D' I E' V A 1 { e a \ D \ \ \ \. FIG. 43. 115. Observe in Fig. 43, that Ee = eD = E'D', that is, the dis- tance from the centre of the picture to the vanishing point of dia- gonals is equal to the distance of the eye from the perspective plane. Hence, having either E or D' given, with E', we can find the other of these points. Thus, having E' and D,' make eE = E'D' which gives E ; and having E and E' given, make E'D' = Ee, which gives D'. 116. The point in which a line pierces the perspective plane, is a point of its perspective ; for the visual ray from that point pierces the perspective plane at its outset. Also, as follows from (109), the vanishing point of a line is a point of its perspective. Moreover, two points determine a straight line, hence the perspective of a straight line is a line joining its van ishing point with the point where it pierces the perspective plane. GENERAL PRINCIPLE AND 81 Thus, in Fig. 44, the perpendicular a5-a'"pierces the perspective plane at a'/ and the diagonal, at c' / hence, if we draw a'E', it will be the perspective of this perpendicular, and if we draw c'D', it will be the perspective of the diagonal, bc-a'c'. \ :m FIG. 44. See also a pictorial illustration in Fig. 41, for lines in any direc- tion. There the three parallels meet the perspective plane at a, b, and c, and Y being their vanishing point (109) aV, 5Y, and cV are their perspectives. 117. It follows from (110) that if two lines intersect at a point, their perspectives will intersect at the perspective of that point, that is, the intersection of the perspectives of two lines, is the per- spective of the intersection of the lines themselves. Hence in Fig. 44, B, the intersection of the perspectives of the perpendicular ab- a' and diagonal bc-a'c', is the perspective of the point b, a' from which both of these lines originated. Particular Derivative Methods. 118. It is now apparent that, by the principles of (116) and (117) the perspective of any point, and hence of any object, can be found without the use of any visual rays. Derivative methods, then, consist in substituting for the visual ray from any given point, any two lines containing that point ; and in finding their perspectives, by joining their intersections with the perspective plane, with their vanishing points (116). The intersec- tion of the perspectives of these lines will then be the perspective of the given point (HV). 119. Since all parallel lines have the same vanishing point (10S) 82 LINEAR PERSPECTIVE. it will obviously abridge the constructions to use auxiliary lines in parallel sets. This being clear, it further appears, that no auxiliary lines are so universally simple and convenient as diagonals and per- pendiculars ; first: because the centre of the picture, which is always given, being the vanishing point of perpendiculars, no vanishing point need be constructed for them ; second, because the distance D'E' from the centre of the picture to the vanishing point of diagonals is equal to the distance, eE, of the eye from the per- spective plane, Fig. 43 ; so that if the latter is given, the former is immediately known, and if it is not given, E'D' can be assumed at pleasure. Foremost therefore among derivative methods, is the method of diagonals and perpendiculars, as explained and illustrated in (112 to 117) all of which is based on (110). 120. The only other derivative method, which need be mentioned, is one which is applicable to bodies bounded by straight lines, which are arranged in parallel groups, as in a square prism. In this case, the lines of the object itself may be put in perspective by (116). The intersections of their perspectives will then be the perspectives of the corners of the object. Derivative methods, exclusively, are generally used in connection with two planes, only, of projection. 121. We will now close this chapter with three fundamental illus- trative examples, showing,^?-^, how to find the perspective of any line whatever by its vanishing point and point of intersection with the perspective plane ; second, how to find the perspective of any object by the method of diagonals and perpendiculars ; and, third, how to find the perspective of a plane sided object by finding the vanishing and intersection points of its own edges. EXAMPLE 3. To find the Perspective of a Straight Line in any position, oblique to both planes of projection, by ts vanishing point and intersection with the perspective lane. Figs. 45 and 46. To familiarize the student more fully with this problem, and so to render the conception of positions in space, cor- responding to given projections, more easy, two different lines have been taken in the above figures, while for more ready comparison, like points are lettered with the same letters in both figures. Accordingly, ab-a'b', in both cases, is a line behind the perspective plane, as usual. Its extremity aa' is at the distance ac behind the perspective plane, and height, a'c above the horizontal plane, b, being in the ground line, is the horizontal projection of that point GENERAL PRINCIPLES AND ILLUSTRATIONS. 83 of the given line, which is in the vertical plane, that is, the point b'. That is ab-a'b' pierces the vertical, or perspective plane at V which is therefore one point of the perspective of this line (116). a FIG. 45. Again: Ew-EV, parallel to ab-a'b' , is the parallel visual ray from the infinitely distant point of ab-a'b' (108). Hence v f , where this ray pierces the perspective plane, is the perspective of that infi- nitely remote point. That is, v' is the vanishing point (106) of ab~ a'b'. Hence by (116) v'b' is the perspective of ab-a'b'. Remarks, a. As v' is the perspective of an infinitely distant 84 LINEAR PERSPECTIVE. point on aba'b', v'b' is the perspective ofab-a'b f produced to a& infinite length, from W, back from the perspective plane. b. As a further exercise, let the student take lines in othei positions. Thus in Fig. 45 let the given line have such a position that a'b' shall be its horizontal, and ab its vertical projection, and then find its perspective, as before. EXAMPLE 4. To construct the Perspective of a Tower and. Spire, by diagonals and perpendiculars. FIG. 47. Fig. 47, let PBE be the plan of the tower, and FGN of the spire whose vertex is A. Let LL be the ground line, taken through the GENERAL PRINCIPLES AND ILLUSTRATIONS. 8 corner B, which indicates the real position of the perspective plane, L'L' indicates the position of the perspective plane after translation forward. CD is the horizon, and C the centre of the picture. The vanishing point of diagonals is assumed on CD, at the left of C, and, in this case, beyond the limits of the picture. [This preliminary explanation is substantially common to most of the following problems, and is therefore to be understood though not repeated.] Since the vertical edge at B is in the perspective plane, it is its own perspective, hence its vertical projection, B'C', which shows the true height of the tower, is also its perspective. To find the perspective of either of the other visible vertical edges, as the one at E, draw the diagonal E5 from the corner, E, of the base of the tower. E'&' is the vertical projection of this diagonal, since it is in the horizontal plane ; it pierces the perspective plane at 5', and b'D (D meaning the vanishing point of diagonals, not shown) is its perspective. Em, the perpendicular from the same point E, pierces the perspective plane at E', and E'C is its perspective. Hence 6, the intersection of #'D and E'C, is the perspective of E, considered as in the lower base of the tower, p is found from P in a precisely similar manner. Then draw jB' and We. To find any point, as A, of the top of the tower. E&, considered as the diagonal from the point E in the top of the tower, pierces the perspective plane at the true height of the tower, that is at c', in the horizontal line through C', since diagonals are always horizontal lines (112). Then c'D is the perspective of this diagonal. The perspective of the perpendicular from the upper point, E, is not needed, since the perspective of the vertical edge at E is known to be a vertical line through e. Hence A, the intersection of eh, drawn perpendicular to the ground line, with c'D, is the perspective of the top corner of the tower at E. q is found-in the same manner Then draw qC r and C'A. To find any point, as/, in the perspective of the base of the spire f is the perspective of F, and the plane of the base of the spire is the same as that of the top of the tower ; hence the diagonal, Frf, and the perpendicular, F^, pierce the perspective plane in the hori- zontal line through C', at d' and F', respectively. Then d'D is the perspective of Fc?, and F'C, that of Yg ; and/, the intersection of these perspectives, is the perspective of F. The top of the towei being above CD, the level of the eye, the base of the spire is invi- sible. The perspectives of G and N are found in the manner just described. 80 LINEAR PERSPECTIVE. Finally, the height of K'm' above the ground line, represents the height of the top of the spire. Then the diagonal, Am, and per- pendicular, An, pierce the perspective plane at m' and A'. m'D is the perspective of Am, and A'C, that of An. Hence a is the per spective of A. Now join a with f, and the other points of the base of the spire, limiting the lines thus drawn by qC' and C'A, and the required perspective will be complete. 122. Various miscellaneous points, which naturally arise in the mind of a beginner, are most conveniently disposed of hqre, after the progress thus far made with primitive and derivative methods. They are therefore discussed in the following Jtemarks. a. The statements in (101) can now be made more intelligible. First. Derivative methods abridge the labor of con- struction: first : through the partial omission of the projections^ as seen in the above example, where the vertical projection was not required, because the auxiliary lines, being horizontal, will always pierce the perspective plane at the height of the points from w r hich they are drawn, and these heights can always be indicated by set- ting them off, as b'c' was, equal to B'C', the real known height of the top of, the tower ; second : by the provision of common van- ishing points for all parallel lines, so that, 1, only one other point in each indefinite line, need be found ; and 2, so that any particular point, as h, on such a line, can be found by a single auxi- liary, as c'D. Second. Derivative methods also conduce to accuracy / first : by providing against errors arising from very acute intersections in the lines of construction. See PART I., Fig. 19, where, though the intersection at A is well defined, that at F, and especially at the perspectives of c" and d" (not shown) are not. Whenever, there- fore, another method fails to give well-defined intersections, that of diagonals and perpendiculars will be generally found available. Second : by the provision against distortion of apparent propor- tions, which is afforded by vanishing points. It is a matter of familiar experience, that all receding parallels in the same group appear to vanish at the same point, and in a drawing, where vanishing points are employed, their perspectives will likewise vanish. But if no vanishing points are used, so that the perspec- tive of each line of a parallel set is found hidependently of the others, by finding two points in it, it may happen that these per- spectives, if produced, will not meet at one point. Errors in the true relative direction of perspectives are far more offensive to the eye than the less obvious errors in the absolute place of single points. GENERAL PRINCIPLES AND ILLUSTRATIONS. 87 b. The disuse of vertical projections which the method of diago- nals and perpendiculars allows, is another advantage of that method over those in which the auxiliary lines might not be horizontal. c. The question naturally occurs to a practical inquirer, " how shall I represent an object of given dimensions, viewed from a given distance, and in a given direction" See Fig. 47. If in practice the distances from P and E to the observer be measured, the exact relative position of the tower and the observer will be known, and so can be laid down on paper. This done, we can, from a given position, look straight forward towards the centre 01 an object, as shown by the line CA in the figure, or we can turn and look towards the right or left of the centre so as to see the object partially by a sidewise glance of the eye. The clearest view is obtained in the former case, but in any case the perspective plane is supposed to be perpendicular to the direc- tion of vision. Thus, if the spectator at E, Fig. 48, observe O, among other things, while looking in the direction Ee, the perspective plane PQ should be perpendicular to Eey simply because this is its simplest and most natu- ral position. This being understood, make eD = eE, to find the vanishing point of diagonals FIG. 48. (115) or, in Fig. 47, lay off from n, on AnC produced, a distance equal to CD, to find the position of the observer, or horizontal projection of the point of sight, often called the station point(ll 3). d. But, further, in representing large objects truly, all these dimensions and distances, just spoken of, must be reduced uniformly, so as to be shown at all, and in true proportion, on paper. In other words they must be drawn to a scale. For example, let it be supposed that in Fig. 47, all given parts are to be shown on a uniform scale of five feet to an inch, i.e. Jive feet on the real object, to one inch on the drawing. On any straight line as XY, lay off two or more inches, and divide each inch, as shown, into five equal parts. Each of these parts will therefore represent one foot, and hence, in connexion with the drawing, may be called one foot. Let the left hand one of these feet be subdivided into twelfths (fourths only are shown) which will be inches. Any other scale is made in a similar manner. Having such a scale, its zero point is at the right hand end of the divided foot. If then, the tower is 5 ft. 9 ins. square, as at BP and 88 LINEAR PERSPECTIVE. BE, extend the dividers from the point marked five to the 3 inch point between 6 and 12, which will be five feet nine inches, on a scale of five feet to one inch. So, if the object be 11 ft. 6 ins. high, make the line A.'m' at this perpendicular distance from the ground line L'L'. It thus appears that in using any scale, thus constructed and numbered, no calculations need be made, since we take up in th dividers the same number of scale feet and inches, that there art of real feet and inches in any given line to be represented. The question of Hem. c is thus fully answered. e. It is a familiar fact that the apparent size of an object decreases with its increased distance from the eye, but the term apparent size is really a little ambiguous, on account of the interfer- ence of knowledge with sense impressions. Thus, when I see a whole house through one window pane, I perceive that the appa- rent size of the house is less than that of the pane, and it is so because the image of the house on the retina of the eye, which is what determines its real apparent size to simple sense, is less than that of the pane. But I know that the house is much larger than many panes, and this knowledge is so far controlling, that the sight of the house affords a mental impression of an object much larger than the pane, though the merely sense impression is, that it is smaller. In relation to the distinction here explained, a completely natural artist is one, who sees things, only, and just as his sense of sight sees, without any interference from thought or knowledge of real relative sizes ; and who draws objects just as his eye sees them. Such a one will spontaneously conform to the principles of per spective, which, in relation to him, will only be the natural history of his natural performances. In proportion, however, as knowledge of the real sizes of objects warps the judgment, as to their real apparent size to the eye alone, does a scientific knowledge and practice of perspective become necessary as a guard against errors in drawing. f. According to (14-16) and all the preceding constructions, a perspective drawing should be viewed from the precise point from which the object represented is supposed to be viewed. Thus, Fig. 47 should be viewed by the eye placed in a perpendicular to the paper at E, and five inches ( = CD) from that point. The per spective will then make identically the same image on the retina that would be made by the original object in its full size, and 25 ft, (the distance by scale) from n, on the perpendicular AnC. GENERAL PRINCIPLES AND ILLUSTRATIONS. 89 In a picture, properly so called, where the sensible effect ia greatly assisted by shade and color, if it be viewed through a tube, so as to exclude the surrounding objects which warp the judgment when compared with the small size of the picture, the illusion may be made complete, by abandoning the mind to the picture exclu sively, and we really seem to look up through extended valleys, winding among great hills, and overhung by a real far distant sky. g. The principal exception to this rule for the position of the eye, is in viewing decorative wall paintings of interiors, which may be painted as if seen from a great distance, or otherwise modified so as not to be offensively distorted to beholders in any ordinary position within the building. h. In connexion with oblique vision of an object, as mentioned in (c), the question occurs, " to what extent is such vision admissi- ble.'' In other words, what is the practical limit of the visual angle. We can examine objects with the greatest minuteness only a point at a time or in the line of but a single visual ray at a time. On the other hand, we can be conscious of the existence of objects within a range of 180, either vertically or horizontally. Where, now, between these limits, is the greatest visual angle which will allow of a clear and pleasing general effect ? It is usually supposed to vary from 45 to 60. Accordingly, in Fig. 47, by laying off five inches, = CD, in front of w, to obtain the station point (113) (115), and from this point drawing lines to P and E, it will appear that a small visual angle is formed. Hence when Fig. 47 is viewed, as directed in (/), it will be very clearly seen. i. This clear view is also due to looking directly, in the line CA, at the centre of the object. Thus Fig. 47 is much more satisfactory than Figs. 16 and 17, PART I., where the eye is placed considerably to one side of the given object, partly to avoid the confounding of plans and perspectives, and partly to avoid the very acute intersec- tions of lines of construction that would have occurred had the point EE' been placed directly in front of the objects. The last consideration points to another disadvantage of the method of visual rays^ especially as employed in connection with two planes only. EXAMPLE 5. To find the Perspective of a Cross and Pedestal. This problem is chosen as one embracing numerous lines arranged in parallel sets. In Fig. 49, let ABD be the plan of the pedestal, EFG, of 90 LINEAR PERSPECTIVE. the horizontal arm of the cross, and HIK, that of its vertical arm. LL is the ground line which indicates the first, and L'L' the one which indicates the second position of the perspective plane. VV is the level of the eye, and therefore by (105) contains the vanishing points of all the horizontal lines of the object. S is the station point (113) taken in a perpendicular to the perspective plane through the centre of the object (I22c). - m,' FIG. 49. Then, drawing SL, parallel to AB, and LV perpendicular to LI^ we find V, the vanishing point of all lines in the direction of AB. In a similar manner V is found. Other points in the indefinite perspectives of the horizontal lines, are where those lines pierce the perspective plane. Accordingly, as 'shown by the figure, and (I22a) and assuming ca as the height of the pedestal, MB, produced, meets the perspec- tive plane at m', where m'm" = ac / and BA meets the perspective plane at a. Then ra'V is the indefinite perspective of BM, and aV, that of AB. Hence #, their intersection, is the perspective of B. From #, draw be, perpendicular to the ground line and limited GENERAL PRINCIPLES AND ILLUSTRATIONS. 91 by cV, and one face of the pedestal will be represented. The con- struction of its other visible surfaces is similar to the foregoing, as is seen in the figure. To find the foot of the vertical arm. Kp and K^ pierce the per- spective plane at p' and Rays from such a luminous point diverge in every direction, hence if in Fig. 51, we suppose AA' to be the luminous point, a is its perspective ; and, not aV only, but any line from a will be the perspective of some ray. The completion of the construction thus far explained, is left as an exercise lor the student LTNKAK PERSPECTIVE. CHAPTER MISCELLANEOUS PROBLEMS. 128. The following problems are added, not to illustrate any ne~W principles, but to familiarize the student more fully with the appli- cation of those already explained, to practical problems. Premising that the drawing of exterior and interior views of buildings, with their accompaniments ; arcades, pavements, and furniture, is perhaps the chief exact application of perspective, this chapter is occupied with examples of this character, the execution of which will enable the draftsman to proceed with the perspective drawing of Avenues, Bridges, Machines, etc., and with the correct additions of features of natural scenery to the geometrical portions of his drawings. EXAMPLE 8. To find the Perspective of a Pavement of Squares, whose sides are parallel to the ground line. Let GB, Fig. 52, be the ground line, DC the horizontal line, or horizon, and AGK a group of twelve squares, lying in the horizon- tal plane, and with one side, GK, taken as the ground line. Operating by the method of diagonals and perpendiculars, let C be the centre of the picture (113) and D the vanishing point of diago- nals. C is the vanishing point of perpendiculars (112) and these perpendiculars, as LK, AH, etc., pierce the perspective plane at K, H, etc., hence (116) KC, HC, etc., are their perspectives. AB is MISCELLANEOUS PROBLEMS. 97 the diagonal from A and BD is its perspective. Therefore a is the perspective of A. Likewise e is the perspective of E, and /, of F. Drawing parallels to GK, through a, e, and/, (111) they will inter FIG. 53. feect the perspectives of the perpendiculars, so as to complete the required perspective GmlK. 7 9S LINEAR PERSPECTIVE EXAMPLE 9. To find the Perspective of a Pavement of Hexagons, -whose sides make angles of 30 and 90 with the ground line. Let HE be the ground line, Fig. 53. Construct an equilateral triangle, as ABD, with one of its sides perpendicular to the ground line. Divide either of its sides, as AB, into any convenient num ber of equal parts. Through each of the points of division, as Q, draw indefinite lines, as QN and QP, parallel to the remaining sides, DB arid DA, of the triangle. Portions of these lines, together with perpendiculars, as OP, joining the proper intersections, which will be obvious on inspection, will form a group of regular hexagons. These may be limited at pleasure, as by the rectangle HEFG. Now let LL be the ground line, after translation, CV the hori- zontal line, C the centre of the picture, and S the station point, taken in this case, for variety, at one side of the middle of the figure. The sides of the hexagons, forming parallel groups, are taken as lines of construction. Their vanishing points beyond the limits of the figure are found by drawing lines at S (partly shown) parallel to HK and IG, till they meet HE. From the latter points^ perpendiculars to CV produced, will meet CV in the vanishing points of HK and IG and of all lines parallel to them. The re- maining lines of the hexagons are perpendiculars, and C is their vanishing point. Observing that the plans and perspectives of the same points have the same letters, the remainder of the construction needs no further explanation. HemarJc. If, in Ex. 8, the squares had been placed with their sides making angles of 45 with the ground line, those sides would all have been diagonals, instead of parallels and perpendiculars. In the last example, if AB had been taken in the ground line, c he sides of the hexagons would have made angles of 60 with '' he ground line, except those which would have been parallel to it. Hence, SR remaining the same, the vanishing points of the inclin- ed sides would have been nearer to C. The student should re- construct these examples under these new conditions. EXAMPLE 10. To find the Perspective of an Interior. Preliminary explanation. According to (12 2 A) a person stand- ing against one wall of a room, can be conscious of the entire in- terior, though the whole cannot be distinctly recognized. If, then, Fig. 54, a person stand at E, seeing clearly everything within the MISCELLANEOUS PROBLEMS. 99 visual angle AEB, only the portion of the room be yond AB can be represented in a picture. Hence, A if a larger portion, or the whole of the interior is , to be represented, the near wall A^B" must be sup posed to be removed, so that E', or E", may be the position of the observer, from which all beyond A'B', or A"B", will be visible. Construction of Fig. 55. In this example, let the near wall be removed, and let the whole interior be seen under a visual angle of 45. ABGL is the plan of the room, with an elliptically arched passage, of the width EF, opening out of it on the right, and with a door, HK, in the left wall. Let the observer stand opposite the point X, at one third the width of the room from G. We have then to construct S, the ver- tex of an angle of 45, whose base is GL, and placed opposite to X. Draw GT and LT, each making an angle of 45 with GL. Draw, at X, a perpendicular to GL, and with T as a centre and TG as a radius, describe a small arc, intersecting this perpendicular at S, which will be the station point as required. Now let G'L' be the ground line, indicating the second position of the perspective plane, and let CD be the horizontal line. This line must, if the observer is supposed to stand on the floor, be about five feet above G'L', on the same scale on which the plan, AG, is drawn. Note that C is in the perpendicular XS, produced to meet CD. Observing, now, that a diagonal from A will meet LG at a dis- tance to the left of L, equal to LA, and so for other points, the diagonals themselves need not be drawn. Thus, make CD = SX (115) and D will be a vanishing point of diagonals. Then make L'A ==LA and A'D will be the perspective of the diagonal from 4. The perspective of the perpendicular LA is L'C, hence a is the perspective of the right hand back corner, A, of the floor. Draw 6, parallel to L'G', till it meets G'C, the perspective of GB, and L'G' db will be the perspective of the floor. The front wall of the room, GL, being taken as the perspective plane, the intersection of the room with that plane will be its own perspective, in full size and real form. Hence make L'L" and G'G" equal to the height of the walls \ and, supposing the coiling to be semicircular, describe a semicircle on L"G" as a diameter. As an example of a simple cornice, in perspective, make the small rectangles at L" and G", as sections of it in the perspective 100 LINEAR PERSPECTIVE. MISCELLANEOUS PKOBLEMb. 101 plane. Then draw its edges towards C, limiting it by a Horizontal and vertical line where its lower back edge, on each wall, meets the vertical lines from a and b. QC is the perspective of a perpendicular through the centre of the floor. Hence qs is the perspective of the centre line of the fur- ther wall. Where qs meets a's, a' being the intersection of aa r nd L"C, is the centre of the semicircular boundary of the further end of the ceiling ; which is a semicircle in perspective, because it is parallel to the perspective plane. If there be a round topped window in the centre of the further wall, lay off its half width, QR=QI, each side of the middle point Q, draw RC and 1C, and per- pendiculars to ab as at r. Then make L'v equal to the height of the base above the floor, draw vC, and v'r' paralled to ab, and the semicircular top, with s as a centre, to have it concentric with the ceiling. This will complete the outline of the window. To draw the opening HK (which is very wide in order to show the construction more plainly). Draw HS and KS, horizontal pro- jections of visual rays, or horizontal traces of vertical visual planes, through the sides of the opening. Then the intersections of these planes with the perspective plane, at hw and k, drawn from H' and K', will be the perspectives of the vertical doorway lines at II and K. Make G'W equal to the height of the door, and WC will limit the inside of the top of the door. Next draw the edges in the thickness of the doorway as H, parallel to G'L', the vertical line It, and from t the line towards C, which completes the doorway. To draw the archway EF. Make L'E'=LE, and L'F'=LF. Draw E'D and F'D, which will give e, and/*, the perspectives of E and F. Make I/O" equal to the height of the vertical portion of the archway, and limit the vertical lines at e and f, by O"C. To find the perspective of the highest point, draw the semi-ellipse, FPE, representing the elliptical top of the arch as revolved round EF, till parallel to the horizontal plane. Lines, as OP, in this semi-ellipse, are called ordinates. Take the longest ordinate, OP, set it off at O"P" and draw P"C. Make L'O'=LO, draw O'D, and op, then p will be the perspective of O, that is of P. Any point in the ellipse may be similarly found. Thus, take MN any where, and parallel to OP. Make O"N'=MN, and draw N'C. Also make L'M'=LM, draw M'D and mn, then n, the intersection of mn with N'C, will be the perspective of N. After finding one or two more points in like manner, the perspective ellipse, fpne' can be sketched. The horizontal line at /will then complete the archway, and the whole figure. 102 LINEAR PERSPECTIVE. EXAMPLE 11. To find the Perspectives of the Shadows in an Interior. In order not to confuse figure 55, the constructions of the re- quired shadows are made on the following enlarged copies oi detached portions. First y to find the shadow of the edges of the doorway, Fig. 56. Supposing no particular direction of the light to be given, assume H as the vanishing point of horizontal projections of rays, and R, FIG. 56. as the vanishing point of rays. It is readily apparent on considera tion, that ka, tc and cd are those edges of the doorway, parts of which, at least, will cast shadows. MI is the perspective of the horizontal projection of all rays through the vertical edge Tea. That is, it is the horizontal trace of a vertical plane of rays (99) through Jca. It is therefore the shadow of lea on the floor, as far as m, where it meets the further wall ABD. Thence, this plane being vertical, its trace, and shadow of ka on ABD, is vertical, as seen at mE. To find E, consider that ct will partly cast a shadow on the surface atk. This surface being parallel to the perspective plane, and ct perpendicular to it, the shadow of ct on atk will be parallel MISCELLANEOUS PROBLEMS. 10b to the vertical projection, CR, of a ray of light (97), and will begin at , where ct meets atk. Hence fe, parallel to CR (HI) is the shadow of ct on atk. Therefore e is the highest point of ha that can cast a shadow. Hence draw the ray eR, and E, its intersection with mE, will be the limit of the shadow of ka. The remainder of the construction is now evident from the figure. Since light streams through the doorway, the area within th shadow of its edges is light, as indicated by the partial shade line of the figure. Second. To find the shadow in the archway, Fig. 57. This shadow is in four parts ; first, the shadow of the edge ee' upon the floor ; second, that of the same edge on the wall Qff ; third, that of the curve e'pf on the same wall ; and fourth, that of the same curve upon the cylindrical surface of the archway, above the hori- zontal plane through e' and/'. Let H be the vanishing point of horizontal projections of rays, and R, the vanishing point of rays. Then eR is the perspective of Cu~,,, FIG. 57. the horizontal trace of a vertical plane of rays through ee\ and eG is the shadow of this edge on the floor. By drawing the ray RG, and producing it to g, we learn that eg is the precise portion of ee' which casts a shadow on the floor. From G, the shadow of ge 1 is 104 LINEAR PERSPECTIVE. the vertical line GE, limited at E by the. ray e'R. Above E, the shadow is cast by the arch curve, and is found as follows. Assume any point, q\ and draw the vertical line q'q, which is the trace of a vertical plane of rays through q\ upon the side of the room. Then qH is the perspective of the trace of this plane upon the floor, and the vertical line from the intersection of qH. with /"G, is its trace on the wall of the arch. This plane contains the ray #'R, which meets the vertical line, just named, at Q, which is therefore the shadow 01 q'. D, the shadow of d' is found in like manner. T, the point of con- tact of the ray TR, with the arch curve/^?e', is the upper end of the shadow, which may be sketched by joining the points already found. In this figure, the shadow on the cylindrical surface of the arch is so small, that no points in it have been found, except T. The most elementary method of finding points of this shadow, is the following indirect one, which is so fully indicated that the student will probably find no difficulty in applying it for himself. Assume any point as h quite near to /", on I/a, and draw through it lines parallel to /G arid ff, which will represent a plane parallel to the wall Off. This plane will cut a line from the arch, parallel to /G, and beginning where the vertical line from h meets the arch curve. Next find a few points of shadow on this plane, just as DQE was found. Then the intersection of this auxiliary shadow with the horizontal line cut from the arch, will be a point of shadow on the arch (99) and by drawing a ray from R through this point, we can find the precise point onfpe' which casts this point of shadow. EXAMPLE 12. To find the Perspective of a Cabin. In this example, a variety of methods will be employed, by way of review ; also some special operations, suited to the construction of particular points. Let ABD, Fig. 58, be the plan of the cabin walls, EF of its roof ridge, and H"HI of Its chimney. Let the perspective plane be taken at GK, through the corner A, and let G'K' be its ground line after translation and revolution into the plane of the paper. Let W be the horizon, C the centre of the picture, and S the station point. The perpendicular to the ground line, and contain- ing S and C passes through *, the centre of the plan (122 i). The edge at A, being in the perspective plane, is its own perspec- tive, and appears in its real height at aa'. The visual ray BS L'C pierces the perspective plane at #, the perspective of the lower corner at B. Make L"B"=aa', then BS B"C is the visual ray from the upper corner at B, and b' is the perspective of that corner. Draw db and a'b'. MISCELLANEOUS PROBLEMS. 10ft FIG. 88. 106 LINEAR PERSPECTIVE. The vanishing point of all lines parallel to AB, can now be found in either of two ways. In the usual way, it would be found by drawing through S a line parallel to AB, till it meets GK, whence drop a perpendicular to VV (108). Or, produce db and a'b' till they meet VV, in the same vanishing point ; which, being out of the paper, is indicated by V". Likewise find V, the vanishing point of all lines parallel to AD, in the usual way, if before finding dd, or as just explained, if after finding dd', as shown in the figure. Having found the end, add' , of the cabin, the intersection, e, of its diagonals ad and da ', is the perspective centre of that end, over which, in the vertical line ee', the peak of the roof is found as follows. Lay off the real height, projected from E, at E" ; then E"C, the perspective of a perpen- dicular from E, will intersect ee' at e', the perspective of EE". Now draw a'e' and d'e', the perspectives of the left end lines of the roof. These lines are in the same vertical plane with ad, hence their vanishing points are in the perpendicular, GG', to G'K' and through V (125-6). Hence produce a'e' to meet GG' in R, which will j|be the vanishing point of all lines parallel, in space, to a'e. Also e'd, produced to T, makes T the vanishing point of all lines parallel to e'd. To find R and T by the usual process, consider that a'E" and D"E" are the vertical projections of AE and DE, and then find where lines through the point of sight C, S, and parallel to a'E" AE and D"E" DE, pierce the perspective plane, which will be, as before, at R and T. Next draw e'f to V", and b'fto R, which will complete the perspective of the roof. To find the perspective of the chimney, and first of its base. Produce IH to J and, drawing the visual ray JS, project J' into the edge of the roof at j. Then draw jh through V"; or, by ele- mentary geometry, draw b'f" parallel to a'e', and limit it by e'f produced, then divide a'e' and b'f" proportionally at j and f and draw jf (For example, if e'j is J of e'a', then make/' /7 t /"==^ of / "'#'). Find h by the visual ray HS, whose intersection with the perspective plane at H' is projected into jf at h. Find u in the same way from I. To find i, set up the full height of the chimney top from the ground at i', projected from I, and draw the perspec- tive perpendicular i'C to limit the vertical edge ui at i. Other wise: (Ex. 5.) produce the right hand side of the chimney to Y , and set up its height, projected from I", at i", and limit ui by t'V. Then limit hh' by ih' drawn through V", and draw A'V. Draw AR until the ridge is met, thence a line towards T, limited as follows. Draw/V and note/, its intersection with e'd', whence MISCELLANEOUS PROBLEMS. 107 draw a line, j'h", to V", limiting an edge of the chimney at A", whence draw this edge, which is limited by h'V. In finding the door and window, further special constructions will be used, as proposed. If lines be drawn, parallel to any line as BK, AB and AK will be similarly, that is, proportionally divided, and if AK=AB, these similar parts will be equal, and in the same order. Hence make K'=AB, and K'# will be the perspective of KB, and V", its in- tersection with W, will be the vanishing point of all parallels to KB, (106). Then make aP and aO equal to the distances of the two sides of the door from a, that is from A, and draw PV" and OV" which will meet db at p and o, the perspectives of the sides of the threshold. Set off the true height of the door from a on aa' and draw a line to V", which will complete the door by limit- ing the verticals at p and o. In like manner, a window in the front of the cabin could be put in perspective. To find the perspective of the end window. According to the method just explained, make G' AD (G f accidentally falls on the perpendicular RT (Ex. 5. Rem. c.) and let NM be the true relative width and place of the window. Draw GWV, analogous to K'#, also NV and MV. At n and m draw vertical lines, and having made aa" equal to the height of the window seat, limit them by a" V. Make aQ equal to the thickness of the cabin wall, and draw QV", noting q, its intersection with db. Then draw qq\ limited by a"V", and from q' draw