2715 UJl CORNELL UNIVERSITY LIBRARY FINE ARTS LIBRARY The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924015429362 Shades and Shadows WITH APPLICATIONS TO ARCHITECTURAL DETAILS AND EXERCISES IN DRAWING THEM WITH THE BRUSH AND PEN Cornell University Library NA2715.W26 ^Shades and shadows, with aPPJicaJJS '" 3 1924 015 429 362 PART I RECTILINEAR FIGU WITH AN APPENDIX UPON PROJECTIONS A AND A NOTE UPON TRIGONOMETRIC By WILLIAM R. WARE FORMERLY PROFESSOR OF ARCHITECTURE IN THE MASSACHUSETTS EMERITUS PROFESSOR OF ARCHITECTURE IN COLUMB DATE DUE PRINTED IN U.SA SCRANTON INTERNATIONAL TEXTBOOK COMPANY 1912 '^ 1 11 fL-^ Gelt: I Copyright, 1912, by International Textbook Company All rights reserved Entered at Stationers' Hall, London PREFACE THE student is supposed to be familiar with the elements of Geometry and to be able to follow common Algebraical processes. Some acquaintance with Projection and Perspective is also assumed, and many of the subjects are illustrated both by plans and elevations, drawn in orthographical projection, and by perspectives and perspective plans, and in the explanations the language of Trigonometry is freely employed. In one or two cases the methods of the Differential Calculus are cited. For the convenience of students is added, at the end of Part I, an Appendix in which the principles of Projections and of Perspective are set forth, so far at least as they relate to this work ; to this is added a note explaining some common Trigonometrical terms. Part I is preceded by an introductory chapter in which the phenomena of Ught and shade, including those of diffused light and reflected light, are set forth in general terms, and the methods to be adopted in the solution of the geometrical problems they present are explained and illustrated. Where there are several ways of solving a problem, it has seemed best to give them all, not only for the sake of making the discussion more complete, but so that the stiident may be able to vary his procedure according to circumstances, and in so doing may clearly understand just which principle he is applying. It is the object of this Introduction to make the student familiar with these methods and with these phenomena, and by accustoming him to exercise his imagination in terms of three dimensions, to make it easier for him, in the chapters that follow, to see the objects under discussion, the rays of light that fall upon them, and the shades and shadows that ensue, in their real relations. Of the twelve chapters which constitute the substance of this book, the four which are comprised in the first volume relate to the shades upon rectilinear solids and the shadows cast by them upon other rectilinear solids. In the eight chapters which are comprised in the second volume are taken up the shades and shadows of curved hnes and of solids of revolution, including cylinders, cones, and spheres, solid and hollow, the torus, scotia, and spindle, and the combinations of these figures in some less simple shapes, all of which, however, are of frequent occurrence in the composition of architectural details. Examples of such details are given in illustration of each of the chapters, so that by the time the student has completed the study of his Shades and Shadows he will have incidentally become acquainted with most of the conventional details commonly employed in classical architecture, and not only with their forms and propor- tions, but with the shades and shadows which habitually accompany them in architectural drawings. Most of these examples are shown, with their shadows, both in their orthographical projections and in perspective. But in some of them, the perspectives and even the shadows are left for the student to supply. Several branches of elementary study are thus combined in one. Indeed, it is practicable, and it may sometimes be convenient, also to use the architectural applications here given as exercises in draftsmanship, with the pen and brush, either copying the figures just as they stand, or, preferably, varying the propor- tions from those here given. iii IV PREFACE It is in that case a useful exercise, as experience has frequently shown, to repeat a drawing several times at a continually smaller scale. Repetition is, of course, the best way to get famiHar with processes and with their results, while a change of scale not only makes each task easier than the last, thus avoiding the tedium of doing exactly the same thing twice, but introduces variations in the handling and technical manipulations which are of themselves both interesting and instructive. These discussions are calculated to secure for the study of Shades and Shadows, as thus presented, the educational and disciplinary advantages which are often sought to be obtained by the study of the more indirect and artificial procedures of Descriptive Geometry. The basis of this work is some notes upon Shades and Shadows which I printed about the year 1879 for the use of my classes at the Institute of Technology, in which it was pointed out that the problems presented in architectural drawings are few and comparatively simple, and are of such frequent occurrence, under almost identical conditions, that, they can advantageously be solved once for all, and the results formulated for further use. They need not be solved anew every time they come up. But in the further development of this idea I have been indebted, as is ' every one, to Mr. Fillet's admirable publications and especially, and chiefly, for the use of the auxiliary 45-degree plane of projection, which I conceive to be his own invention. But his exhaustive discussion of the intensity of illumination upon curved surfaces has not seemed to me of enough interest or practical value to make it worth while to cite his conclusions in these pages. The figures and architectural examples presented in the Plates or embodied in the text have been drawn out from sketches, partly by the draftsmen in the service of the publishers, partly by others. Of these, I am especially indebted to Mr. Henry W. Frohne, and Mr. Philip R. Whitney, and especially to Mr. George Lawrence Smith, not only for an intelUgent and sympathetic interpretation of the sketches, but for helpful suggestions in regard to both the form and the substance of the text itself. WILLIAM R. WARE Milton, Massachusetts July, 1912 CONTENTS (PART 1) PAGE Preface iii Introduction — Figures A-Z Light, Shade, and Shadow; Architectural Shades and Shadows 1 The Shades AND Shadows OF Points AND Lines; Plane AND Solid Figures .... 6 The Shades and Shadows of Polyhedra . . 7 The Shades and Shadows of Surfaces of Revolution 8 1. The Method of Projected Tangent Rays 9- 2. The Method of Revolved Tangent Rays . .... . . 10 3. The Method of Slicing 12 4. The Method of Tangent Cylinders 13 5. The Method of Tangent Cones 13 6. The Method of Tangent Spheres 14 7. The Method of Envelopes 14 8. The Method of Points . . 15 Shadows on Irregular Surfaces 16 9. The Method of Parallel Planes ' . . 10 Shadows on Oblique Planes 19 Special Methods . 19 Shading. Diffused and Reflected Light; Reflected Shadows; Double Shadows . . 23 Chapter I — Figures 1-5 The Shadow of a Point 24 Chapter II — Figures 6-1-4 The Shadows of Right Lines 26 Principal Lines 28 Inclined Lines 29 The Diagonals of a Cube 29 The Diagonals of the Faces of a Cube '. 30 Other Inclined Lines 30 Chapter III — Figures 15-25 The Shadows of Plane Rectilinear Figures . 31 The Square 31 Parallelograms ..... 32 Crosses 32 Octagons 33 The Auxiliary Plane at 45° 34 V VI CONTENTS Chapter IV— Figures 26-38 The Shades and Shadows of Solid Rectilinear Figures Rectangular Solids The Cube The Vertical Cube The Parallelopiped The Octagonal Prism Polyhedra Pyramids . Shadows Upon Rectangular Solids 35 35 35 36 37 37 38 38 39 APPLICATIONS TO ARCHITECTURAL DETAILS Chapter I. Points No Architectural Details Consist Merely of Points ... Chapter II. Right Lines — Example I Example I. A Post and Wall 40 Chapter III. Plane Rectilinear Figures — Examples II-IV Example II. A Square Tower or Beacon 43 III. Five Square Lamp Posts 42 Chapter IV. Solid Rectilinear Figures — Examples IV-XXXV Example IV. A Dentil 43 V. A Block Cornice and Frieze 44 VI. A Fillet and Three Dentils 45 VII. A Fillet and Dentils From Pompeii 47 ■ VIII. A Square Niche 47 IX. An Entablature, With Piers and Pedestals 47 X. An Entablature With 45° Moldings ... 47 XI. An Entablature 4g XII. A Doric Gutta, or Drop 49 XIII. Two GuTTiE and Fillet 5q XIV. A Roman Doric Architrave _ _ 5q XV. A Roman Doric Frieze g^ XVI. Steps, With One Parapet g9 XVII. Steps, With Two Parapets go XVIII. Steps, With Ramps \ go . XIX. A Square Abacus on an Angular Post g4 XX. A Square Abacus on an Octagonal Post ka XXI. An Octagonal Abacus on a Square Post ch XXII. An Octagonal Abacus on an Octagonal Post g^ XXIII. A Stone Bench -c 56 XXIV. A Bracket, With Inclined Face c^. XXV. The Same, Side Elevation .... c-r. XXVI. A Roof and Chimney _„ 56 CONTENTS vu Chapter lY^-Continued Example XXVII. XXVIII. XXIX. XXX. XXXI. XXXII. XXXIII. XXXV. An Octagonal Bracket and Canopy . ... A Portal A SauARE Tower and Octagonal Steeple .... A Triangular Lamp Post With Hexagonal Cubical Lantern A Square Lamp Post With Octagonal Lantern . Pyramidal Voussoirs and XXXIV. Gables . . Pediments 58 58 58 58 59, 59 61 62 APPENDIX Projections — Figures AA-QQ Problems in Solid Geometry Perspective — Figures RR-YY 45° Perspective . . . 90° Perspective Trigonometrical Terms — Figure ZZ 65 66 73 73 74 77 SHADES AND SHADOWS PART 1 INTRODUCTION Lijiht aiiii Slta,ii\—\( n solid object is exposed to sinili,nlit, one Kido is illuminated axid is said to he in JJ^^Iit. The sun's rays do not, fall upon the other side, which is said to be in Slhhit\ and it would bo invisible, like the dark side of I lie moon, if other light wore not rolleeted upon it from other objects, or from the earth or the sky. Thus, .at the time of the new moon, the dark side of the moon is often visible by light reflected from tho earth. Line of l.ii^hl i.\7 Slhuio;<ht ami partly in shadow, like the screen in Fix- H, so that only part of its .surf.'ice is darkened, this portion is called the Ciisl Sliiuio;(\ or ]'isible Slhhlo;<\ of tho object that casts it, or simply its Sliihiou\ Line of Lii^lit mul Shtuiow, or Line of Sh is the shadow of Ute Line oj Slhuie. ELEVATION Fill, .\ Pre,B 1 SHADES AND SHADOWS Fig. C 1 Pig. C2 Fig. D Surfaces of Light and Shade. — If a plane surface stands edgewise to the light, one element of the surface being parallel to the direction of the rays of light, these rays do not strike it but glance along it, and the surface is neither in light nor in shade (Fig. CI). It is nevertheless called a Plane of Light and Shade, since, like a Line of Light and Shade, it lies, if it occurs upon a solid object, between the portions of the sur- face that are in light and those that are in shade. It is as if a Line of Light and Shade were expanded into a plane. The outline or perimeter of such a plane is a Line of Light and Shade, the half of which that is nearest the sun throwing its shadow along the plane exactly upon the other half. The two halves of this line and the surface that they enclose all have the same Invisible Shadow, which is a plane, and is an extension of the Plane of Light and Shade, and that plane and the line cast the same Visible Shadow upon any surface upon which it falls. This Shadow, when it falls upon a plane, is a Straight Line, as in the figure. The same is true of a cylindrical surface the elements of which are parallel to the rays of light (Fig. C 2). It is a Surface of Light and Shade. Its invisible shadow is the exten- sion of the surface itself, and it is also the invisible shadow of the lines that terminate the cylinder at either end, one of which exactly throws its shadow upon the other. These lines and the cylindrical surface between them cast the same Visible Shadow. This Shadow, when it falls upon a plane, is a Curved Line, as in the figure. It is obvious that surfaces turned toward the light are either in Light or in Shadow, and that neither light nor shadow can occur upon the side of an object that is turned away from the light; also, that surfaces turned away from the light are always in Shade, and that shade cannot occur on the side of an object that is turned toward the light. When a rectangular object, such as a building, is seen in elevation, the front being parallel to the plane of projection, all the surfaces that are visible are turned toward the sun. Such elevations may accordingly exhibit lights and shadows, but no shades. Re-entrant Angles: Line of Shade and Shadow. — Fig. D exhibits these same phenomena of Light, Shade, and Shadow, in the case of an angular object bounded by plane surfaces. But besides the Lines of Shade that separate the surfaces in Ught from those in shade, and that now occur on the external angles, and the Lines of Shadow, which lie between the Shadows and the illuminated portions of the surfaces on which they fall, this figure shows, in the internal, or re-entrant angles, lines of a third kind, lying between the Shades and the Shadows. These, which have no recognized name, may be called Lines of Shade and Shadow, SHADES AND SHADOWS IN HOLLOW BODIES Fig. E 1 Fig. E2 Fig. £ 3 SHADES AND SHADOWS IN HOLLOW BODIES If a body is hollow, like a pail, round or square (Fig. E 1, 2, and 3), the interior surface, as well as the exterior, is divided into two parts, one of which is turned toward the Light, and the other turned away from . it. But while on the outside the half nearest the siin is in light, and the half farthest from it is turned away and is in shade, on the inside, the half nearest the sun is turned away from it, and is in shade, and the half that is farthest from the sun faces the sun and is in light, except where the shadow cast by the first half falls upon it. The interior surface, therefore, exhibits in itself, without the intervention of any other body, all the phenomena of Light, Shade, and Cast Shadow. The half next to the light is in shade and counts as the body that casts the shadow, the other half is illuminated and counts as the body that receives the shadow. Half of the exterior surface and half of the interior surface are, accordingly, in Shade, being turned away from the light; and the other half of each, being turned toward the sun, is either in Light or in Shadow. On the outside, the Line of _Light and Shade, or Line of Shade, divides the light from the shade; on the inside, the Line of Shade and Shadow divides the shade from the shadow. But both lines lie between the surface that is turned toward the light and that which is turned away from it. On the interior surface, the Line of Shade that casts the shadow, the Line of Shade and Shadow, between the shade and the shadow, and the Line of Shadow, all three, meet at the same point a, and at a similar point b on the opposite side of the rim. These relations are exhibited in Fig. E, and also in the shades and shadows of the Wall, Gable, and Chimney in Fig. D. It follows that if two of these lines are known, the points where they meet will give points of the third line. This principle is made use of in Fig. 105, in Chapter X, to find the Line of Shadow in an Oblique Hemisphere, and in Fig. 142, in Chapter XII, to find the Line of Shadow in an Oblique Niche. The Line of Shade and the Line of Shade and Shadow being known, the points where they intersect are points also of the Line of Shadow, which can then, under the given circumstances, be easily drawn through them. If the two surfaces meet at an angle, as in Figs. E 3 and D, the Line of Shade and Shadow is plainly marked. But if the Concave Surface is rounded, as in Fig. E 1, the shadow grows lighter and the shade grows darker, until, at the point where the surfaces coalesce, the two tints are the same. The two surfaces there have the same illumination, and the Line of Shade and Shadow disappears, as in the figure. But the shade is, as a whole, lighter than the shadow, being exposed to reflected light, and if both are put in in flat tints, as in Fig. E 2, the Line of Shade and Shadow is visible between them, separating the Shade from the Shadow. A similar effect is produced upon a Convex Surface when, as in Fig. E 1, a shadow is cast upon the portion of the exterior surface that is in light. This shadow grows lighter in tone, gradually receiving more and more reflected light, as the surface turns away, until, at the Line of Shade, it is of the same tint as the Shade, and the Line of Shade disappears. But here again, as appears in Fig. E 2, the Shade is, as a whole, lighter than the Shadow, and if both are put in in flat tints the Line of Shade is seen, separating the Cast Shadow from the Shade, as in this figure. SHADES AND. SHADOWS Fig. F Points of Flight. — It sometimes happens that the second object intercepts only a part of the invisible shadow cast by the first object, the rest of it passing on iintil it strikes a third object, or surfacg. The second object will, of course, also cast its shadow upon this surface (Fig. F). In Fig. F the Line of Shade on the farther side of the first sphere, between the points /i and /i, casts its shadow upon the second sphere, between the points f^ and f^. The rest of the Shadow falls upon the plane surface beyond it, where it inter- sects the shadow cast by the Line of Shade of the second sphere. The Line of Shade of the larger sphere between the points /j and /j lies between Shadow and Shade, like the Line of Shade on the convex surface in Fig. E 1 and 2„not between Light and Shade. But, in these cases, the Line of Shade still divides the portion of the surface that is turned toward the sun from the portion that is turned away. from it. , So much of the shadow of the first sphere as falls upon the second is bounded partly by the shadow of its own Line of Shade and partly by the Line of Shade of the second sphere. The two points /j and /j upon the surface of the second sphere, where the Line of Shadow cuts this Line of Shade, are called Points of Flight. They are, as it were, "jumping-of£ places." From between these points, the shadow of the first sphere jumps, alighting upon the surface of the third object, between /g and f^. The shadow cast upon this surface is bounded partly by the shadow of the first Line of Shade and partly by the shadow of the second Line of Shade. The points ^ and /g, where these two shadows intersect, are the shadows of the two Points of Flight. The rays that pass through them are tangent to the second sphere at the Points of Flight f^ and f.^ and to the first sphere at /i and f^. If the two shadows upon the third surface are constructed independently, the Points of Flight fa and f^ and the points /i and f^ can easily be found by tracing the rays of shadow back from the points of intersection /g and /g upon the third object, to the Lines of Shade upon the other two objects. Notation. — The French call the Shade, on the side of an object away from the light, the Ombre Propre; the Cast Shadow the Ombre Portie; the Invisible Shadow, the Prism (or Cylindre) D'Omhre; the Line of Shade, the Ligne D'Ombre Propre; the outline of the Cast Shadow, the Ligne D'Omhre PortSe; and the Points of Flight, the Points de Perte. They apparently have no distinctive term for the line that, in a re-entering angle or on a concave surface, lies between the portion of the surface that is turned away from the sun and that which faces it; that is to say, between the Shade and the Shadow. But,. although this line is geometrically the same as the Line of Shade that, upon a similar convex surface, lies between the Light and the Shade, the character and relations of the two. lines are so unlike that it is convenient to have different names for them- calling the line upon the Convex Surface, a Line of Light and Shade, or simply a Line of Shade, and the line upon the Concave Surface, the Line of Shade and Shadow, as in Fig. D. When, as in Fig. E 1 and 2 and in Fig. F, a Shadow is cast upon a convex surface in such a way that the Shadow and Shade meet upon the Line of Shade, there is no inconvenience in still using this term for the line of meeting. Architectural Shades and Shadows. — Drawings are often made without any indication of light and shade. In Oriental Art, and in the kind of work often seen in posters, and to some extent in pictures executed in stained glass, and in some kinds of Mosaic,. both Shades and Shadows are omitted, only outline and color being employed, and the surfaces shown look fiat. European Art, on the contrary, often omits color, while using Shadows, and, more generally, Shading, as a means of giving relief, or modeling, thus showing the'third dimen- sion of the objects depicted. In architectural drawings, especially, since orthographic elevations show only two dimensions of an object. Shades and carefully constructed Shadows are employed to indicate the third dimension or depth. They make a plan, or a second elevation, or an isometric or perspective drawing, the less necessary! Shades and Shadows, as a branch of Graphics, is an application of the methods of Projections and Per- spective to determine, first, the Shade upon any object, that is to say, the shape of the Line of Shade that divides its illuminated side from its dark side; secondly, the Shadow cast by this object upon the surface of ARCHITECTURAL SHADES AND SHADOWS Fig. G1 any other object, that is to say, the shape of the Line of Shadow, which is the Shadow of the Line of Shade; and thirdly, the Projections, or Perspectives, of these Lines of Shade and of Shadow. The results obviously depend, first, on the shapes of the objects in question and their attitude, or the way they face ; secondly, on their position, relatively to one another; and thirdly, on the direction of the light. As all these may vary to any extent, the shapes of Shades and Shadows, in general, are difficult to determine, and it is still more difficult to know how to draw them, since their appearance varies with the varying position of the spectator. But in Architectural Drawings, the task is comparatively easy; since the conditions are greatly simplified. In the first place, architectural details are, for the most part, composed either of regular and irregular polyhedrons, guch as Cubes, Parallelopipeds, Prisms, and Pyramids, or of Cylinders, Spheres, Cones, and other solids of revolution, all so placed that their edges, or axes, are either parallel or perpendicular to one of the principal planes of projection. In the second place, the surfaces upon which shadows are cast are also, for the most part, either plane surfaces, which are generally parallel to the planes of projection, or are the surfaces of other geometrical solids, the shapes and positions of which are also easily determined. In the third place, the direction of the Ught is always supposed either to be parallel to the ver,tical plane of projection or, more generally, to have the direction of the diagonal . of a cube whose faces are parallel and perpendicular to the planes of projection. The angle made by this Diagonal with the planes of projection is known as the angle 0. It measures 35° 15' 52". Fig. G shows such a cube, both in perspective at 1 and in plan and elevation at 2. Unless something is said to the contrary, the light is supposed to come from the left, and to be parallel to the diagonal drawn from the upper left-hand front corner of the cube to the lower right-hand back comer, as in this figure. The projection of the diagonal of such a cube upon either plane of projection, and of the rays of light or rays of shadow, makes accordingly an angle of 45 degrees with the ground line, as in Fig. G 2. Finally, when plans, elevations, and sections are employed, the position of the spectator is supposed to be always the same, and the same as in orthographic projections; namely, opposite the plane of projection and at an infinite distance from it, the scale of the drawing being increased accordingly. But in a perspective drawing at 45 degrees, the Station Point S, where the spectator stands, is supposed to be directly opposite to the Center of the picture, or Point of Sight, V^, which is half way between the right- and the left-hand Vanishing Points, and to be as far from the Center as the Center itself is from those points. All the Perspective Draw- ings in these Volumes are drawn under these conditions. (See Example IV B, page 44, and Fig. RR, page 71.) The problems of Architectural Shades and Shadows are thus-limited to determining the projection, on either plane of projection, of the Lines of Shade of certain Geometrical Solids and objects composed of those solids, and of the outlines of the shadows that these bodies cast upon other similar solids and upon horizontal and vertical planes. Since architectural details are, for the most part, composed of these few elements, and the same problems are constantly recurring, and in the same combinations, it is the object of the following chapters merely to explain the principles and methods involved in the solution of these few problems, and to make the results so intelligible and so familiar that they may be freely used, as occasion requires, without the necessity of constantly repeating the geometrical processes by which they are obtained. SHADES AND SHADOWS THE SHADES AND SHADOWS OF POINTS, LINES, AND SURFACES Points. — ^The invisible shadow of a point is a Right Line. The visible, or cast, shadow of a point is a Point. (Fig. H, 1.) Lines. — The invisible shadow of a line, straight or curved, is a Surface, plane or cylindrical. The visible, or cast, shadow of a line is a Line. (Fig. H, 2-6.) The invisible shadow of a right line is a Plane. The visible shadow cast by a right line upon a plane is a Right Line lying between the shadows of the points at its extremities. (Fig. H, 2). The invisible shadow cast by a curved line is, in general, a portion of a Cylinder and its visible shadow is a Curve (Fig. H, 4 and 6) ; but if the curve lies in a plane that stands edgewise to the light, its invisible shadow is a Plane, and its visible shadow a Right Line, as in Fig. H, 6. If a Hne is parallel to the plane on which the shadow falls, the shadow is equal and similar to the line; and if this plane is parallel to the plane of projection, the projection of the line and the projection of the shadow are equal and similar, both to one another and to the line itself (Fig. H, 3 and 6). Surfaces. — The invisible shadow of a surface is a Solid Cylinder, or Prism, and its visible shadow is a Surface, being the shadow of the portion of the surface that is in light. Plane Figures. — If one side of a Plane Figure is illuminated, the other side is in Shade. Its perimeter, or edge, is a Line of Light and Shade (Fig.' I). If, as here, the shadow falls upon a plane parallel to the figure, the shadow is equal and similar to the figure; and if this plane is parallel to the plane of projection, both the projection of the figure and its shadow are like the figure. Surfaces Parallel to the Light. — If any element of a plane figure is parallel to the rays of light, the plane is parallel to the light, and it is a Plane of Light and Shade, as has already been said (Fig. C 1). Its invisible shadow is a plane that is the extension of the plane of the figure; and its visible shadow, or cast shadow when it falls upon a plane, is a right line that coincides with the shadow of the perimeter of the figure. The half of this perimeter that is nearest the sun throws its shadow exactly upon the other half and both halves have the same visible and invisible shadows, which are those of the surface they enclose. If therefore a polygon stands edgewise to the light, so that it is a Surface of Light and Shade, the edges upon which the rays of light fall are in Light and the others are in Shade (Fig. C 1). So, also, as has already been seen in Fig. C 2, if a hollow cylinder or prism stands endwise to the light, its elements being parallel to the rays of light so that it is a Surface of Light and Shade, its invisible shadow is not a solid but a Surface, and it may be regarded as the invisible shadow of the closed line that bounds the given surface at either end, for the edge that is toward the sun and is illuminated casts its shadow exactly upon the other edge, and the shadow cast by one end coincides with that cast by the other. Both ends accordingly cast the same shadow, which is also the shadow of the surface that lies between them. The invisible shadow is an extension of that surface and lies in the extension of the cylinder. Dimensions of Shadows. — While the invisible shadow is thus always of a higher dimension than the thing that casts it, the visible shadow is of the same dimension ; that is to say, the invisible shadow of a point is a Line the visible shadow is a Point; the invisible shadow of a line is a Surface, its visible shadow, a Line; and the invis- ible shadow of a surface is a SoUd, the visible shadow, a Surface ; the cast shadows all being intersections of the invisible shadows by the surfaces on which they are cast. But the cast shadow of a solid object is also a Surface and its shadow in space a Solid, since they are the shadows of the illuminated surface of the object. Fig. I SHADES AND SHADOWS OF POLYHEDRA THE SHADES AND SHADOWS OF SOLID OBJECTS Besides Planes and Plane Figures, the only subjects treated of in Shades and Shadows are Polyhedra, regular and irregular, and Surfaces of Revolution. I. POLYHEDRA Shades and Lines of Shade. — The Line of Shade upon a Polyhedron is a broken line composed of the edges, or solid angles, that divide its illuminated faces from those in shade. To find which faces are in. light and which are in shade, and thus discover the Line of Shade, it suffices to pass planes through the Poly- hedron parallel to the rays of light, to find the resulting polygon of intersection, and then to see which of its edges are in light ; for the faces of the Polyhedron in which these edges lie are also in light, since if any line of a plane is in light the whole plane is illuminated. In Fig. J 1, the light falls upon a regular Dodecagon, in a direction parallel to the vertical plane of projection, making the angle (P with the horizontal plane. Of the four faces shown in the elevation, it appears that A and C are in light and B and D are in shade, and the illustration is shaded accordingly. In Fig. J 2, the light falls upon the same object in the direction of the diagonal of a cube, making the angle with both planes of projec- tion, and its projections upon those planes lie at 45 degrees. It appears that all four of the faces visible in the elevation are now in light, so that the vertical projection of the Line of Shade now coincides with the outline. Shadows and Lines of Shadow. — ^When the Shadow of a Polyhedron is cast upon a plane surface, the Line of Shadow is a Polygon, which is the shadow of those edges of the polyhedron which are Lines of Light and Shade, as in these fig-ures. The Angles of the Polygon are the shadows of the solid angles in which the edges meet. In Fig. J 1, the shadow is cast upon the horizontal plane; in Fig. J 2, upon the vertical plane. The dodecagon being a regular symmet- rical figure, six of its twelve faces are in light and the six that are in shade are exactly opposite to them. The polygons of shadow have ten sides. Fig. Jl Fig. J 2 SHADES AND SHADOWS II. SURFACES OF REVOLUTION If a line, straight or curved, is revolved about a straight line as an axis, the resulting surface is called a Surface of Revolution. The given line may not lie in the same plane as the Axis, and indeed may not be a plane figure at all ; but the resulting surface will be the same as if generated by the plane curve that is obtained when the Surface of Revolution is cut by a plane drawn through the axis. This curve may be regarded as the generatrix of the Surface. The directrix, then, is a circle whose center is on the axis, and which lies in a plane perpendicular to the axis, and the generatrix, in moving along this circle, keeps always in a plane passing through the axis. Shades and Lines of Shade. — If a Surface of Revolution is exposed to the light of the sun, the half that is in light is separated from the half in shade by a Line of Light and Shade. The rays of light are tangent to the surface along this line.. Conversely, any points on the surface at which the rays of light are tangent to it are Points of the Line of Shade, which can be determined by finding a series of such points. Shadows and Lines of Shadow. — ^The Line of Shadow that bounds the Shadow of a Surface of Revolution is the shadow of its Line of Shade. It passes through the shadows of any points given or taken in the Line of Shade. Tangent Planes. — If through any point of the Line of Shade of a Surface of Revolution, a plane is passed tangent to the surface, it will stand edgewise to the light. The shadow in space, or Invisible Shadow, of this tangent plane, will be an extension of the plane itself, and the shadow cast by it upon any plane surface will be a right line, tangent to the shadow of the surface of revolution. Right Lines. — ^The shadows of all straight lines that lie in such a tangent plane coincide with the shadow of the plane. If these lines pass through the point of tangency, they are tangent to the surface of revolution, and, conversely, all right lines that pass through such a point of tangency, that are tangent to the surface, lie in the tangent plane and have their shadows in its shadow. The ray of light that is tangent to the given surface at the given point is such a line. It pierces the surface upon which the shadow falls at a point in this Line of Shadow that is the shadow of the point of tangency. A right line that does not lie in the tangent plane, though it passes through the point of tangency, pierces both the tangent plane and the Surface of Revolution, and its shadow cuts the shadow of the tangent plane and the shadows of all lines lying in it. Curved Lines. — ^The shadows, also, of Curved Lines that lie in the Tangent Plane coincide with its shadow, and if cast upon a plane surface are Right. Lines. If they pass through the point of tangency, they are tangent to the surface of revolution at that point. But curved lines lying in a plane that is not tangent to the given surface, but intersects it, may still be tangent to the surface at this point. In this case, their shadows are tangent to the shadow of the tangent plane. Such curved lines pass through the point of tangency but instead of being tangent to the surface, pierce it. Their shadows intersect the shadow of the tangent plane and of all the lines that are tangent to it. Tangent Surfaces. — If the Surface of Revolution is embraced, or enclosed, by a Cylinder, Cone, Sphere, or other surface, as is shown in Fig. K 1, 2, and 3, so that the two surfaces are in contact along a continuous curve, or oval, as a line of contact common to both, then this hne of contact will meet both Lines of Shade that of the surface in question and that of the enclosing surface, at the same point, as at a in these figures. A tangent ray of light will also pass through this point. If then, the Line of Contact is known and also the Line of Shade upon one of the tangent surfaces, a point in the Line of Shade upon the other surface is determined. Just as the Line of Shade of the given surface, the Line of Shade of the enclosing surface, the Line of Con- tact of the two surfaces, and the Tangent Plane, are all tangent to one another and to the tangent ray at the point of tangency, a, so their shadows are all tangent to one another at the shadow of that point, a^ which is the point where the tangent ray pierces the surface on which the shadows fall. SURFACES OF REVOLUTION II 'i'li'iii' Lines of Shade and Lines of Shadow. — There are six ways of finding the Line of Shade upon a Surface of Revolution: (1) the Method of Projected Tangent Rays; (2) the Method of Revolved Tangent Rays; (3) the Method of Slicing; (4) the Method of Tangent Cylinders; (5) the Method of Tangent Cones; and (6) the Method of Tangent Spheres. The last four methods give also the Line of Shadow cast upon a plane. This can also be obtained by (7) the Metliod of Envelopes and by (8) the Method of Points. Shadows cast upon Polyhedra or upon Surfaces of Revo- lution can be obtained either by Slicing or by (9) the Method of Parallel. Planes. I. The Method of Projected Tangent Rays. — Since the Line of Shade is the line in which rays of light are tangent to the Surface of Revolution, any points upon the surface at which these rays are thus tangent are points of the Line of Shade. Two such points occur upon the outline of every projection of the surface, being the points where the projections of the rays of light are tangent to the projec- tion of the outline. In Fig. L, the four projections 1,2,3, and .4 thus give eight points of the Line of Shade, at a, a' ; h, V; c,c! ; and d,d' , respectively. These eight points can all be found in Fig. L, 1, b and b' being on the " corners " of the " equator," or largest horizontal circle; c and c' being on the axis, at the same height as a and a' ; and d and d' being on the comers of the circles to which the rays are tangent. In Fig. L 4, the light falls parallel to the vertical plane, making the angle (?with the horizontal plane. 1—2 10 SHADES AND SHADOWS Fig. M 3 Fig. M 1 Fig. M 2 II. The Method of Revolved Tangent Rays. — If, when the light is taken as parallel to the Vertical Plane of Projection, a tangent ray is drawn at any point of the Line of Shade, as at I', Fig. M 1, and the surface is revolved about its axis until the point of tangency comes upon the outline of the projection, the projection of the ray will then be tangent to the outline, as at I in the figure. Conversely, if a point I is taken upon the outline, and a tangent ray is drawn at that point, then, if the object is revolved back until the projection of the ray is parallel to the direction of the light, the point I' thus found will be a point in the Line of Shade. The Light Parallel to the Vertical Plane of Projection. — Let /, Fig. M 2, be such a point, the light being parallel to the vertical plane and making the angle with the horizontal plane. Let ^ be a point on the axis of revo- lution at the same level as I. Draw pi; draw pn making with p I the angle 0; and draw the vertical line / n, meeting ^w at the point n. If now a horizontal plane is passed through the point n, the segment of the tangent ray cut off. by it will be projected in the line Im; and if this ray is revolved about the vertical line in until it is parallel with the vertical plane, it will then be projected in plan and elevation in the line 1 1. The horizontal angle through which it is thus revolved will be the angle 6, as shown in the plan. If now, in the plan, the figure plmt is revolved about the axis of revolution at p by the angle 0,the ray Im will become parallel to the vertical plane. It will be projected at I' m' in plan and elevation, and V will be the projection of a point in the Line of Shade of the Surface of Revolution, being a point at which a tangent ray SURFACES OF REVOLUTION 11 /f\ 1 \ \ m'" "" '"if llili^^ Fig. M 4 Fig. M 5 Fig. M Fig. M 7 of light falling at the angle is parallel to the vertical plane. It will also be the projection of a similar point, I' on the further side of the surface, as appeajs from the plan. Since, by construction, the triangles Itnt, I'm't', and pi I', in the plan, are equal and similar, since in the elevation pi equals nt, it follows that in the plan p q equals n s, and in the elevation p I' equals n m. (The Unes p I' and n m are, indeed, two sides of a parallelogram, of which pn and I' m are the other two sides.) The point V, on the vertical projection of the Line of Shade, can accordingly be found by laying off the distance nm upon the line pi. Since also, in the plan, the line pn equals I'm', it follows that pn + ns equals ^g + /'w', and that the vertical projection of the point m' coincides with that of m, as in the figure. Light in Plane at 46 Degrees With the Vertical Plane of Projection. — If the figure is now revolved about its vertical axis until the light, still making the angle with the horizontal plane, makes an angle of 45 degrees with the vertical plane, as in Fig. M 3, the figure pql' I' will appear in the plan as shown at pql" I", and the points I" and I", in the vertical projection of the Line of Shade, now also revolved, are easily determined from them. These procedures, which are entirely independent of the value of the angle <^, are summarized in Fig. M 4, which also illustrates the application of the Method of Revolved Tangent Rays to points in the lower part of the Surface of Revolution. Light in a Plane at 90 Degrees With the Vertical Plane of Projection. — Fig. M 5 shows that the same process of revolving the lines of the plan about the axis — ^by which, in Fig. M 4, the Line of Shade is obtained when the light is at 45 degrees with the plane of projection — is applicable to obtain the Line of Shade when the light is parallel to the 90-degree plane. Either of these three projections can be obtained from either one of the others by revolving the plan one way or the other. It makes no difference, of course, by what method the first projection is determined. The Direction of Tangent Lines. — The accuracy of these results depends, of course, upon the accuracy with which lines can be drawn tangent to a given curve at given points. This is an easy matter in the case of Arcs of Circles, and not difficult with Arcs of Ellipses. With other curves, the best way is to take two points upon the curve, near the given point and equidistant from it on either side, and then to draw the tangent line parallel to the line connecting them, as. in Fig. M 6. The nearer the points, the more accurate the result. A series of such tangents can be drawn with considerable accuracy by means of the device illustrated in Fig. M 7. This figure shows, above the given curve, a horizontal line., on the upper side of which are the ends of lines drawn, as just described, through a series of points taken at equal distances apart on the given curve. 12 SHADES AND SHADOWS The irregularity in the spacing of these lines shows that their positions are only approximately correct, and the new positions given them, below the horizontal line, are obviously more accurate. The spaces are made gradually wider, toward the left, as they plainly should be. Lines drawn through the assumed points to the points thus determined are almost exactly tangent to the curve. III. The Method of Slicing ; The Line of Shade. — ^When a plane figure stands edgewise to the light, as has been illustrated in Fig. C 1, half of its perimeter is in light and half in shade, and the two halves are separated by points of Light and Shade, at which the rays of light are tangent to the perimeter. If now a solid object is cut by a plane parallel to the light, the surface of intersection may be regarded as an infinitely thin slice, constituting a plane figure standing edgewise to the light. The line of intersection will be the Perimeter of this figure, and it will have two points of Light and Shade, which will be points of the Line of Shade upon the given object. Fig. N2 Fig. N 1 shows a Surface of Revolution cut by three planes parallel to the light, the tangent rays making the angle (P with the honzontal plane, and their projections making 45 degrees with both planes of projection This gives SIX pomts of the Line of Shade, 1 , 1. ; 2, 2^ ; and 3, 3^ ; two for each slice. Other slices would give other points In order to find the outline of each slice, which is the line in which the secant plane cuts the Surface of Revolution, there must be drawn upon the surface a series of horizontal circles, projected on the plan as Circles, and m the elevation as Horizontal Lines, as is done in Fig. LL, in the Appendix page 67 The points at which the vertical secant planes cut the circles in the plan, give, when projected into the elevation points on the outlines of the slices, as in that Figure. ' The Points of Light and Shade, which are points in the Line of Light and Shade, are found by drawine 45-degree rays tangent to the projections of the curves of intersection, as in Figure N. SURFACES OF REVOLUTION 13 The Line of Shadow. — The Shadows of the points thus ascertained give corresponding points in the Line of Shadow. The sHces being parallel to the light, their shadows are Right Lines, upon which the shadows of the points of tangency are easily deter- mined upon the vertical plane of projection, as in Fig. N 1 at 1., 2„ 3,. Fig. N 2 shows the shadows of these points, and of the given surface, cast upon the horizontal plane of projection. The shadows of the secant planes are, again. Right Lines. It is generally most convenient, as is illustrated in the figure, to take these secant planes (parallel to the Hght) perpendicular to the Horizontal Plane, thus making angles of 45° both with the Vertical Plane of Projection and with the Vertical Normal plane. But it is sometimes more serviceable to take them per- pendicular to the Vertical Plane of Projection, and at 45° with both the Horizontal Plane and the Noirmal Plane ; or sometimes even to take them perpendicular to the Normal Plane, and at 45° with the Planes of Projection, as is done in Example XCV, in Part II. IV. The Method of Tangent Cylinders. — Fig. K 1, 2, and 3 has already exhibited a Cylinder, a Cone, and a Sphere, tangent to a Surface of Revo- lution, and shown that the points at which the Line of Contact between the two surfaces is met by the Line of Shade of the Cylinder, Cone, or Sphere, are points also on the Line of Shade of the given surface. This affords three addi- tional ways of finding the Line of Shade on a surface of revolution, all of which give also the Line of Shadow, as follows: Fig. O shows three cylinders perpendicular to the three planes of projection, all tangent to the Surface of Revolu- tion, -one of which, B, is Vertical, and two, A and C, are Horizontal. The circular vertical cylinder B is tangent to the given surface along its Equator, which is cut by the Lines of Shade of the cylinder at the points b and b^. The oval horizontal cylinders A and C give, in like manner, points at a and a^, and at c and q. The Line of Shade being tangent to the Lines of Shade of the three cylinders at these six points, its Line of Shadow is tangent to the Lines of Shadow of the three cylinders at the Shadows of the six points, as in the figure. This gives six points of the required Line of Shadow, and also the direction of the curve at these points. V. The Method of Tangent Cones: The Line of Shade. If, as in Fig. P 1, a cone is drawn tangent to a Surface of Revolution, having the same axis as this surface, the line of contact is a circle, and the two points at which this circle is intersected by the Line of Shade upon the cone give two points of the required Line of Shade upon the given surface, as at e and e^ in the figure. Lines of Shadow. — Since the vertical and horizontal projections both of the vertex of the cone E, and of the two points e and e^ are known, their shadows upon the vertical plane at E', ^, and e^, are easily determined. The Line of Shadow of the Cone is the shadow of its Line of Shade, and may be drawn, as in the figure, through these three points of Shadow. The Line of Shadow of the Surface of Revolution is tangent to the Line of Shadow of the Cone at the points e and e'. 14 SHADES AND SHADOWS Fig. P 1 Fio. P 2 In Fig. P 2, the same process is applied to finding the shadow cast by a Surface of Revolutioii upon a hori- zontal plane. The result is the same as that shown in Fig. N 2. Any required number of points in the Line of Shade and the Line of Shadow of the Surface of Revolution can "be obtained by means of additional cones. Each cone will give two points of the Line of Shade and two points of the Line of Shadow, with the tangents at those points which show the direction of the curve. But the three cones whose elements make with their bases the angles 0, 45° and 90°, the first of which gives the highest and lowest points of the Line of Shade, and the last of which is a cylinder, and gives two points upon the Equator of the given surface, generally suffice for practical purposes. If steeper cones are employed, their vertexes are likely to come inconveniently far away, and the Method of Revolved Tangent Rays, which is more compact, is to be preferred. The different ways of determining the Line of Shade upon a Cone are described in Chapter VIII. VI. The Method of Tangent Spheres. — ^The different ways of determining the Line of Shade upon a Sphere and its Line of Shadow are described in Chapter IX. As they are comparatively laborious, and as the results reached are only approximate, the Method of Tangent Spheres is of no practical importance. VII. The Method of Envelopes. — ^Since a plane figure, parallel to a Plane of Projection, casts upon that plane a shadow similar to itself, as was illustrated in Fig. I , it follow^ that if a Surface of Revolution is cut by planes parallel to a plane of projection and the shadows of the successive sections are cast upon that plane, the Lines of Shadow will be similar to the lines of intersection. A line circumscribing these shadows will then be the Line of Shadow of the given surface. This way of finding the Shadow is called the Method of Envelopes. Fig. Q 1 shows its application to finding the shadow upon a plane parallel to the axis of revolution, and Fig. Q 2, upon a plane perpendicular to the axis. This result is the same as that obtained by Tangent Cones shown in Fig. P 2; but in that figure the position of the successive points in the Line of Shadow is more precisely given, and the direction of the curve at each point is also determined. SURFACES OF REVOLUTION 15 Fig. Q 1 Fig. 2 Although the Method of Envelopes is thus useful for obtaining the Line of Shadow, it is not available for getting the Line of Shade, for a line drawn through the points where 45-degree lines are tangent to the con- tours of these sections would not coincide with the projection of the required Line of Shade, either in plan or in elevation. In the elevation it would be the projection of the Line of Shade that would result from light parallel to the vertical plane and at 45 degrees with the horizontal plane, as shown by the dotted line in Fig. Q 1. Rays which, as usual, made the angle with the horizontal plane, their projections making angles of 45 degrees with both planes, would not be tangent to the Surface of Revolution at these points, but would pierce it. VIII. The Method of Points. — Since the shadow of any surface is the shadow of its Line of Shade, it must pass through the shadows of all the points in the Line of Shade that have in any way been determined, whether by means of Tangent Rays, Tangent Cylinders, Tangent Cones, by Slicing, or, as in Fig. A, by simple inspec- tion. When, accordingly, the Line of Shade is, by virtue of the geometrical properties of the surface, given as a continuous line, straight or curved, or, as upon a polyhedron, is a broken line, the Line of Shadow can be obtained by taking points in it at pleasure, finding their shadows, and drawing the Line of Shadow through them. The Shadows in Fig. J 1 and J 2 were obtained in this way. 16 SHADES AND SHADOWS SHADOWS ON IRREGULAR SURFACES The Shadow cast by one solid figure upon another must, in general, be obtained by Slicing. But in some cases the Method of Parallel Planes can be employed. Slicing. — If a secant plane, parallel to the light, is passed through both objects, the plane figure that is bounded by the line of intersection upon the object nearest the light, casts its shadow exactly upon the edge of the corresponding figure in the .other object, and the two points of shade upon the first give two points of shadow upon the second. As many such secant planes can be used as may be desired. Lines drawn through the points of shade and points of shadow thus obtained, give the required Line of Shade and Line of Shadow, as already shown in Figs. N 1 and N 2. Figs. R 1 and R 2 illustrate the ^application of the Method of Slicing to finding the shadow of a Cube and ©f a Circle upon a Pyramid. In both cases, 45-degree vertical secant planes, and their intersections with both the given surfaces, are first drawn upon the plan. These lines of intersection are then transferred to the elevations. Points upon the object that casts the shadow have their shadows in the corresponding lines of intersection upon the pyramid. Lines drawn in the elevation connecting the points of shadow thus obtained are the Lines of Shadow required. These are then transferred to the plan. J 3 g Fig. R 1 Fig. R2 SHADOWS ON IRREGULAR SURFACES 17 Fig. R 3 shows the shadow cast by one sphere upon another, the light still being projected at 45 degrees upon both planes of projection. Since, in this example, the line joining the centers of the two spheres is parallel to the rays of light, the Line of Shadow is a Small Circle of the largest sphere, and is, in both plan and elevation, projected as an Ellipse. The lines of intersection in both spheres are also Circles, and their vertical projections are Ellipses. The points where the 45-degree rays are tangent to the three slices upon the small sphere, are points in its Line and Shade, and the points on the corresponding slices of the large sphere are points of the required Line of Shadow. These lines are then transferred from the elevation to the plan. Fig. R 4 shows the same two spheres, with the light falling parallel to the vertical plane of projection. The Line of Shade in the elevation, is seen edgewise, and is proj^ted accordingly in a Right Line, as in Fig. A. So is also its Line of Shadow. In the plan both lines are projected as ellipses. FiQ. R3 Fig. R4 18 SHADES AND SHADOWS IX. The Method op Parallel Planes. — It sometimes happens that the surface upon which a shadow is to fall is easily cut by planes parallel to the Plane of Projection. This is the case with polyhedra, in which the lines of intersection are right lines, and with Surfaces of Revolution whose axes are perpendicular to the plane of projection, in which case the lines of intersection are circles. The shadow cast upon such a surface by a plane or solid figure is easily found, for it will throw upon each of the successive secant planes a shadow similar to that which it casts upon the plane of projection to which they are parallel. But, just as the line of intersection in each such secant plane is the only portion of that plane which has any real existence, so the points in that line at which it is crossed by the Line of Shadow of the given figure are the only points of the required Line of Shadow that have any real existence. The required Line of Shadow can then be drawn through these points. Fig. S 1 shows the shadow of a circle cast upon two faces of a square vertical pyramid by the Method of Parallel Planes. Seven planes, numbered from 1 to 7, cut the two faces of the pyramid in lines parallel to the vertical plane of projection. The center of the given circle casts its shadow successively upon these seven planes at the points c^, c^, c^, c^, c^, c„ and Cj. These points are first obtained in the Plan and then transferred to the Elevation. Then, in the Elevation, circles described about these centers are the shadows of the given circle cast upon these planes, and the points in which they cut the secant lines numbered from 1 to 7, are points in the required ellipses of shadow. The points thus attained in the Elevation can then be transferred to the Plan, Fig. S 1 Fig. S2 SHADOWS ON IRREGULAR SURFACES 19 If the two faces of the pyramid are supposed to be extended across the edge in which they meet, and the lines of intersection extended accordingly, the whole of the ellipses can be determined, as "in the upper figure. The Line of Shadow in Fig. S 1 is seen to be the same as that obtained in Fig. R 2 by the Method of Slicing. Fig. S 2 shows how the shadow cast by one sphere upon another is obtained by the Method of Parallel Planes. The conditions are the same as in Fig. R 3, and the result is the same. As is explained in Chapter IX, the shadow of the small sphere, when cast upon the vertical plane of projection, is an Ellipse, like that shown in Fig. S 3. It is circumscribed about two equilateral triangles whose sides are equal to the diameter of the circle. Two of these ellipses, lying in the planes numbered 1 and 2, have their centers at the points c^ and Cj in those planes. The lines in which these planes intersect the surface of the large sphere are Circles. The two points at which each of these ellipses of shadow cut these circles, are points in the required Line of Shadow. Any number of additional points can be obtained in the same way, by drawing additional ellipses and circles. This, again, is seen to be the same shadow as that obtained in Fig. R 3 by the- Method of Slicing. SHADOWS ON OBLIQUE PLANES When shadows are cast, not upon a principal plane, but upon a plane oblique to the planes of projection, two cases of interest are presented. 1. If a plane slopes back toward the vertical plane of projection, its horizontal element being parallel to the Ground Line, the shadow cast upon it by a vertical line has, in projection, the same slope as the plane itself. See Fig. 12, Chapter II, and Example XXVI, among the Architectural Applications. 2. If a vertical plane makes an angle of 45 degrees with the vertical plane of projection the projection of the Shadow cast upon it by a horizontal square, two of whose sides are parallel to the vertical planes of pro- jection, is a square "diamond" half the size of the given square, as is shown in Fig. 24, Chapter III. In this figure the real size of the Shadow is given in broken lines. The shadow cast upon it by a horizontal circle is projected upon that plane as a circle, half as large as the given circle, its diameter being half the diagonal of the given diameter. See Fig. 47, Chapter V, in Part II. 3. But, in general, Shadows cast upon oblique planes are determined by Slicing, as in Figs. R 1 and 2, or by means of Parallel Planes, as in S 1 and 2. SPECIAL METHODS The ways of getting Lines of Shade and Lines of Shadow here described give only series of points through which the desired lines can be drawn. But in the case of Circles and Ellipses, and of Cylinders and Cones with circular or elliptical bases, and of Spheres and Prolate and Oblate ElUpsoids, the Lines of Shade and of Shadow are in general either Right Lines that can easily be determined, or Ellipses, the elements of which can be ascertained by special processes based upon the special geometrical properties of these figures. These methods are explained, as occasion requires, in the chapters in Part II, which treat of Circles, Cylinders, Cones and Spheres. As constantly happens in Mathematics, these special methods, though of limited applicatien, are simpler than those based upon more general principles. 20 SHADES AND SHADOWS (1) C4) Fig. T .Fig. U 1 Fig. U 2 SHADING. DIFFUSED AND REFLECTED LIGHT Rough and Smooth Surfaces. — A convex surface grows less luminous as it turns away from the sun. But the eye does not accurately estimate these grada- tions, and on a smooth cylinder, such as a marble or limestone column, the surface in light appears to be of nearly uniform brightness quite up to the Line of Shade, and should be so represented (Figs. T 1 and W 2). If, on the other hand, the surface is rough, as is often the case with sand- stones and granites, the diminution of light is more marked. But this comes not so much from the decrease in the illumination as from the increase in the length of the little shadows that are cast by the irregularities of the surface (Fig. T 2). If the surface is so smooth as to be polished, a bright line of reflected light is seen (Fig. T 3). This occurs not on the "comer," that is, at the line of 45 degrees, but at the point half way between the front of the shaft and the comer, at the line of 22 degrees and 30 minutes. This point is, in the eleva- tion, about two-fifths of the way from the axis of the shaft to the edge. This bright line is very conspicuous if the polished shaft is of a dark color, and it is either white, or of the color of the light that falls upon it, not showing the cotor of the material unless the column is of metal. Diffused Light. — Objects are often illuminated not by the direct rays of the sun, but by a Diffused Light, as if the sun were behind a cloud. In this case, there are no definite Lines of Light and Shade, and no well-defined Shadows. (Fig. T 4). In almost all drawings and paintings, objects are modeled as if thus illuminated by diffused light, as is to be seen in portraits and in most drawings and photographs of the htunan figure, from the cast or from life. The surfaces of the objects are, in this case, more or less illuminated accord- ing as they are tumed toward the light or away from it, gradations of tint that hardly exist in broad sunlight, becoming conspicuous (Fig. T 4). Diffused light is accordingly well adapted to bring out the forms of architectural mold- ings and other details with simplicity and clearness (Fig. U 1 and 2). It is sometimes strong enough to produce not only Shade, but Shadow, though ill defined. But in drawing details effects of shadow are generally' neglected, and surfaces are made lighter or darker according as they are turned towards the Hght or away from it, as in these figures. Indoors, things are generally seen by diffused light. But buildings being out of doors, are generally drawn with Cast Shadows, as if in the sunlight Reflected Light.— Surfaces in shade or in shadow would, as has been said be mvisible; that is to say, they would look black, if light were not reflected upon them from the earth and sky, or from other objects. This light is in general, of the nature of a diffused light, coming in a direction opposite to that of the Sim's rays. The sky, if without clouds, is brightest about the sun, so that the light reflected upon any object from a clear sky comes, virtually, in the same direction as the direct sunhght. But the light reflected from other objects, such as build- ings comes mainly from such as are farther from the sun than the object in question, and which accordingly present to it th^ir illuminated faces Objects SHADING. DIFFUSED AND REFLECTED LIGHT 21 Fig. VI Fio. V2 nearer the sun present their shaded faces, and thus furnish only light that is doubly reflected, and is too faint to be noticeable. The same is true of clouds and of the ground, especially if it is rough, only the stones on the side away from the sun presenting their illuminated faces. The general direction of the light reflected from the ground is upwards, and of that reflected from surrounding objects forwards. The total result is accordingly directed forwards and upwards in a direction just opposite to the sun's rays, which go downwards and backwards. The projections of the rays of reflected light, both in plan and in elevation, are thus considered to lie at 45 degrees, in directions exactly opposite to the projections of the sun's rays. This reflected light is naturally strongest near the ground, and it often sensibly diminishes the depth of. the shades and shadows that are exposed to it. Even in the case of Diffused Light, the Shades are somewhat affected by it. Reflected Light Upon Surfaces in Shadow. — Shadows are, accordingly, in architectural drawings, generally graded down from the top, that is, they are made darker at the top and lighter at the ground (Fig. V 1). But walls in shadow behind colonnades and arcades are sometimes graded the other way, from the ground up, that is, darkest below, as if modified by light doubly reflected from the ceiling (Fig. V 2). Reflected Light Upon Surfaces in Shade. — If, as in Fig. W 1, the object exposed to the light is a vertical prism, with a polygonal base, one side of which directly faces the sun, making an' angle, of 45 degrees with the plane of projection, that side is the best illuminated and looks the brightest. The other planes are more or less illuminated according to the angle at which they stand, the difference being more marked the rougher the surface. If the surface is smooth the difference is not noticeable (Fig. W 1). The corresponding face on the other side, also at 45 degrees with the picture, receives neither direct nor reflected light, being parallel to the sun's rays. But the surfaces on the side away from the sun that are inclined to the picture more than 45 degrees are in shade, and are more or less exposed to reflected light. Thus, the surface at 45 degrees is the darkest, as in the figure. The same is true of an isolated vertical cylinder, as in Fig. W 2, which may be regarded as a prism of an infinite nutnber of sides. The element best lighted by reflected light is at the farther "comer" of the cylinder, where the radius makes an angle of 45 degrees with the vertical plane of projection, a point turned directly away from the suUj and not visible in the figure. The curved surface of shade, as has already been illustrated in Fig. E and Fig. T, receives more or less reflected light according as it is more or less directly turned away from the sun. Hence, although the tone along the Line of Shade of a cylinder is of only moderate depth, the shade is darker there than in any other part; for the rays of direct light are tangent along this line, and go by without touching, and so does the beam of reflected light, moving in the contrary direction. It thus receives less light of either sort than any part of the portion in shade, though more than any part of the shadow, the Shadow, as a whole, being always darker than the Shade. Moreover, it looks a little darker than it really is, lying as it does in sharp contrast with the lighted surface. But this contrast, though conspicuous on a smooth and light-colored surface, is less noticeable on a rough and dark one, as has already been pointed out. Fig. Wl Fig. W 2 22 SHADES AND SHADOWS Reflected Light Upon Cylindrical Surfaces. — ^Figs. X 1 and 2 again show the effect of diffused reflected light in modifying shades and shadows upon cylindrical surfaces, both solid and hollow, that is to say, whether convex or concave. The portions of the surface, whether convex or concave, that are turned away from the sun are in Shade, those turned toward it are either in Light or in Shadow. Fig. X 2 shows moldings partly in sunlight and partly in shadow. The shadow cast across the moldings from above affects of course, only their illuminated portions, and their own shades and shadows pass through it unchanged. When the shades and shadows are continuous, as in Fig. X 2 , where the cast shadow crosses the moldings, the only light that falls upon the surfaces is a diffused reflected light coming in a direction just opposite to that of the sun's rays. It follows that while in direct diffused light, so to speak, surfaces are lighter in proportion as they are turned toward the source of light, as in Fig. U 1 and 2, surfaces that are in shade and shadow, and are illuminated only by reflected diffused light, as in Fig. X 1 and 2, are darkest where they face the sun, and brightest where they are turned away. . On the concave surfaces, the shadow is lightest along the Line of Shade and Shadow, being there most exposed to reflected light, and the shade is darkest, being there most turned away from it, so that although the shadow is, as a whole, darker than the shade, one melts into the other, as at a and a. On the convex surfaces, the shadow is light- est where it meets the Line of Light and Shade, and the shade is darkest, so that here, again, they run together and the Line of Light and Shade disappears, as at b and b. This has already been illustrated in Fig. E 1 and 2. Fig. Y 1 and 2 illustrate the same points by the example of a square Abacus throwing its shadow upon the capital of a column. * In Fig. Y 1, both the shade and the shadow are graded, as they would really be, and the Line of Light and Shade that separates them disappears, as in nature. The portion of the shaft that is in Shadow is darkest at the point nearest the sun, just below the abacus, where it receives the least reflected light. It is lighter on the left-hand margin and on the axis, at which places it is equally illuminated, and is lightest on the right-hand margin. The point farthest away from the sun, where it is most exposed to the reflected light, and is consequently the lightest, is of course out of sight. It is customary, in diagrams and in drawings made to a small scale, to put in both the shade and the shadow in flat tints. This is exemplified in Figs. Y 2 and Y 3, in which the Line of Shade is clearly apparent, separa- ting the Shadow from the Shade, which is luminous with reflected light. Since surfaces in shade are all ttimed more or less toward the reflected light and those in shadow are all turned more or less away from it, and as no other light falls upon them, it follows that the shades are lighter, as a whole, than the shadows, as has been already said. This is illustrated in Figs. E 2, W 2, and Y 2, where the Line of Light and Shade is seen to divide them. Fig. Y1 Fig. Y3 Fig. Y2 Fig. X 2 SHADING. DIFFUSED AND REFLECTED LIGHT 23 Fig. Z 1 Reflected Shadows. — The reflected light is sometimes sufficiently strong and definite in direction to cast a well-defined shadow, and in architectural drawings, by a pardonable exaggeration, it is frequently represented as doing so (Fig. Z 1). Shadows cast by reflected light are usually called, for brevity, Reflected Shadows. These secondary shadows serve to define and give relief to objects and details within the main shadow, just as the main shadows themselves do for objects in the sunlight. Moreover, they give luminousness and transparency to the primary shadows, and these effects are all enhanced if, as has been suggested, the primary shadows are graded up instead of being graded down, as in Fig. Z 2. Although, in the case of an isolated building, the main direction of the reflected light is from the back forwards, just opposite to the direction of the sun's rays, the reflected light that falls upon the shadows which, as in Fig. Z 1 and 2, occur in planes parallel to the plane of projection, such as ordinarily present themselves in the elevations of buildings, comes only from the portion of the ground, and from the adjacent objects, which are in front of the fajade. The general direction of the reflected light is still upwards, but it is now backwards instead of forwards. In elevation, its projection is still opposite to that of the sun's rays; but in the plan, it is at right angles to that of the sun's rays. It would seem to follow that the shaded portion of a column, or of a half or three-quarter column attached to such a wall, should also be mod'eled in reflected light coming from the front, and not, as in the case of a full column, standing alone, from behind. But, in fact, the adjacent portion of the wall from which such shafts project throws upon them a reflected light which comes in just the same direction as the light cast by a more distant object behind it, so that here, too, the darkest part of the shade is found at the Line of Shade. Fig. Z2 Fig. Z 3 Double Shadows. — If an object is exposed to two shadows, the one cast by the brightest Hght being because the strong light is cut off, and the shadow not cut off. Where the two shadows cross, the shadow is not and if there is no third source of light it will be presence of a gas burner and a small lantern. This is illustrated in the figure, in which it is the shadows do not cross, rather than the amounts sources of light at once, as in Fig. Z 3, there will be two the darkest. The space occupied by this shadow is dark, cast by the faint light is weak, because the strong light is the sum of the other two, but is darker than both together, perfectly black, as may be seen out of doors at night in plain that the amounts of light are added together where of darkness where they do. 24 SHADES AND SHADOWS Chapter I THE SHADOW OF A POINT 1. The Invisible Shadow of a point, that is to say, its shadow in space, is a line, namely, the ray of shadow cast by the point. The Visible Shadow, or cast shadow, is a point, namely, the point in which the ray of invisible shadow strikes the surface upon which it falls. 2. Since the light is, in orthographic projections, taken as falling in the direction of the diagonal of a cube, generally from left to right, a point and its visible shadow may be held to occupy the opposite comers of an imagi- nary cube, as in Fig. 1, which is the same as Fig. G 1, in the Introduction, the ray of invisible shadow passing from the given point, at the nearer upper left-hand comer of the cube, to its visible shadow, at the farther lower right-hand comer. 3. If the edge of the cube is taken as equal to 1, the diagonal of the cube, which is the length of the ray of shadow, is equal to the square root of 3, and the projection of this ray on each face of the cube, or on either plane of projection, is equal to the diagonal of the square face of the cube, that is to say to the "diagonal" of the edge, and is equal to the square root of 2. By the "Diagonal" of a line is meant the diagonal of a square of which the given line is one side. Its length is equal to that of the given line multiplied by the square root of 2. A diagonal section through the cube, normal to either plane of projection (Fig. 2 A) is a rectangular parallelogram, the width of which is 1, its length is the square root of 2, and the diagonal is the square root of 3. This diagonal makes an angle of 35 degrees, 15 minutes, and 52 seconds with the long side, and an angle of 54 degrees, 44 minutes, and 8 seconds with the short side, each angle being the complement of the other. 4. This angle of 35 degrees, 15 minutes, and 52 seconds, which the diagonal of a cube makes with the faces of the cube, and which is the angle made by a ray of Ught with either plane of projection, is known, in Shades and Shadows, as the ^ angle (p (Fig. 2 B). The \^ ?*• • ■( angles and 90° — are complementary, and their natural tangents are recip- rocals of one another, being one-half the square root of 2 and the square root of 2, respectively. At each of its • extremities, the ray of shadow thus makes an angle of about 54 degrees and Fig. 1 \ /' \^'. •Os^' *p>v / '/V / ! i '*XV''""'' I 1 -^ "^Nk .» VsS;;. S "O^. t •■>. f - -^ - -H \ "v y-;;/^'^^ ,.x;ir^o-z^s*' x^ ■— -- \. y k '^^.35- is'sa" ^/\^ j-t'-^+'a" -■•V2: Fig. 2 A Tan. ^ a-yS-yj" Tan.90-$-V«. Fig. 2 B 44 minutes with the three edges which it meets, and the complementary angle of about 35 degrees and 16 minutes with its three faces (that is to say, with its projections upon the faces, which are the diagonals of the faces). 5. The projections of the four diagonals of the cube, upon either of the six squares which form its faces make right angles with each other, since they fall upon the diagonals of the squares. But the four diagonals of the cube themselves make with one another angles not of 90 degrees, as one might carelessly fancy but THE SHADOW OF A POINT 25 of 70 degrees, 31 minutes, and 44 seconds (or about 70 degrees and 32 minutes), being twice the angle (Fig. 2B). Only three lines, of course, can meet at angles of 90 degrees. 6. If the shadow of a point falls upon a principal plane parallel to a plane of projection, as in Fig. 3, the distance from the point to the plane gives the length of one edge of this imaginary cube. The projection of the ray of invisible shadow upon such a plane is a line lying at 45 degrees with the Ground Line and equal in length to the diagonal of the distance from the point to the plane. This line connects the projection of the given point on the plane with its shadow on the plane, as does the line a" a" in the figure. 7. When a point is given by its projection upon the plane that receives its shadow, its distance from that plane is generally shown by its projection upon another plane at right angles to the first, as in Fig. 4. A line joining the two projections of a point is always, of course, at right angles with the ground line. If the point is nearer to the horizontal plane than to the vertical one, as is point 1, the shadow falls on the horizontal plane; and vice versa, as is the case with point 3. If the point is equally distant from both planes, as is point 2, its shadow falls just upon the ground line. In both planes, the projections of the rays of shadow are drawn at 45 degrees to the ground line. 8. But in all cases, the line joining the projection of a point upon a plane with the shadow of the point on the same plane is a line, lying at 45 degrees, which is equal in length to the diagonal of the distance of the point from the plane. Therefore, if the projection of a point upon a plane is given and the distance of the point from the plane is known, the shadow of the point upon that plane can be found at once, without the necessity of showing the projection of the point on another plane of projection, by drawing a 45-degree \ tk.. M Fio. 3 ■L" A?i- Fig. 4 +. y* Fig. 5 line whose length is equal to the diagonal of this distance (Fig. 3) ; or, what comes to the same thing, the distance itself may be laid off twice, first in a direction parallel to the ground line toward the right, and then in a direction at right angles to it, as is done in Figs. 3 and 4. When both projections are given, the easiest way to find the shadow of a point is to draw a 45-degree line from the projection nearest the ground line, toward the right, until it intersects the ground line; and from this point of intersection to draw a line at right angles to the ground line, and from the other projection of the point to draw a 45-degree line intersecting this last line. The point of intersection of these lines is the shadow of the point, as at V and S', Fig. 4. 9. Auxiliary Plane of 45 Degrees. — If the shadow of a point falls not upon a principal plane, parallel to the plane of projection, but, as in Fig. 5, upon a diagonal plane at 45 degrees with the given plane of projection and perpendicular to the other, the projection of the line of invisible shadow upon the given plane will be a 45-degree line equal in length to the distance of this point from the diagonal plane, as shown. The figure shows the traces in which the diagonal plane cuts the planes of projection (or lines of intersection). The point of visible shadow may be readily found in a manner similar to that shown in Fig. 4, as may be seen in Fig. 5. The only difference is that the first 45-degree line is drawn to the trace of the auxiliary 45-degree plane instead of to the ground line. 10. But, if the shadow of a point falls, not upon a plane parallel to a principal plane of projection, but upon an inclined plane or upon an irregular surface, it is necessary first to pass an auxiliary Hne through the point, find the shadow of this line upon the given surface, and then find the shadow of the point upon the line of shadow, as is shown in Fig. 14, Chapter II. For, if a point lies in a line, its shadow must lie in the shadow of the line. 1—3 26 SHADES AND SHADOWS Chapter II Fig. 6 A THE SHADOWS OF RIGHT LINES 11. Right Lines.— The Invisible Shadow of a right line is an invisible plane of shadow, one element of which is the line itself, which serves as a directrix, and the other elements are the invisible shadows of the successive points in the line, any one of which elements may be regarded as a generatrix (Fig. 6). The surface of the shadow is composed of the successive shadows of the successive points of the line. Its Visible Shadow is the line in which this plane of invisible shadow intersects the surface upon which it falls. If this surface is a plane, the visible shadow is a right line lying between the shadows of the points which are at the extremi- ties of the given line, and the shadow is determined when the shadows of these terminal points are found. Fig. 7 A illustrates the case in which this shadow falls entirely on the vertical plane of projection; in Fig. 7 B, it falls entirely on the horizontal plane. But it is often more convenient to regard the given line as a segment if a line of indefinite length, as is done in Fig. 7 C; and to find the point, 3, where this line pierces the plane on which the shadow falls. Its shadow will pass from this point through the shadow of any other point in the line, and the shadow of the segment can be cut from it by 45-degree lines drawn through the projections of its extremities, as in the figure. In Fig. 7 D it falls partly on one plane and partly on the other. The segment 1-3 throws its shadow on the hori- zontal plane from Ish to 3s; the segment 3-^ upon the vertical plane, from 3 s to 2 s v. If the vertical plane were transparent and the horizontal plane were extended behind it, the shadow of the line would fall wholly on the horizontal plane, forming the right line 1 sh-3s-2sh. If the reverse were true, the horizontal plane being transparent, the shadow would fall wholly on the vertical plane in the right line 2 s v-3 s-1 s v. These two shadows intersect the ground line at the shadow of the point 3. 12. If the given line is parallel to the plane on which the shadow falls, the shadow, whether the line is horizontal, vertical, or inclined, is a line equal and parallel to the given line (Fig. 8, A, B, and C). Since a curved line may be regarded as composed of an infinite number of right lines, it follows that if a plane curve or any plane figure is parallel to a plane its shadow upon that plane is equal and similar to itself (Fig. 8 D). In this case, and in all cases of lines or figures parallel to the plane of projection, it suffices to obtain the shadow of a single point. THE SHADOWS OF RIGHT LINES 21 13. Figs. 9 A and 9B illustrate the application of Perspective and of Orthographic Projection to the representation of the shadow of a straight line, cast upon horizontal and vertical surfaces. In Fig. 9 A, the horizontal and vertical planes of projection are shown in Perspective, with an inclined line A B lying between them. Its projection on the horizontal plane is shown at A b and on the vertical plane at a B, the points a and b being the projections of the points A and B. The direction of the light, which is supposed to be sunlight, is shown by the arrows, and is, as generally '. L b* X ahb" D ■'f\u Fig. 9 A Fig, 8 in these perspectives, taken parallel to the plane of the picture and making the angle Q with the planes of projection, its projection on each plane making 45 degrees with the other, and with the ground Une. 14. The invisible shadow, or shadow in space, of the line A B, is, of course, a plane passing down through the air. The heavy line A ^ B, in which this plane of invisible shadow cuts the planes of projection, is the shadow cast by the given line upon those planes. Tliis, as has been said, is called the Line of Cast Shadow, or Visible Shadow, or simply the Line of Shadow. 15. This shadow falls partly on the horizontal plane at A-^, and partly in the vertical plane at S-r. Of the five points in the line which are marked A 1, 2,3, and B, A and B are their own shadows, and the points 1, 2, and 3, have their shadows cast at 1', ^, and 3". The point 1, being near the horizontal plane, casts its shadow upon that plane as 3 does upon the vertical plane. The point 2, which is equally distant from both planes, casts its shadow upon the grotuid line. The line A B and its projections are drawn double; its rays of invisible shadow 1-1', 2-2; 3-3% parallel to the Sun's rays, are drawn as broken lines; and their projections upon the horizontal and vertical planes of projection are shown by dotted lines. The construction lines are also dotted. 16 Fig 9 B shows all these things by the methods of Orthographic Projection, giving the projections, on both' planes of the given line A B, and of the invisible shadows of the points 1, 2, and 3. It does not show any of these lines themselves. Their position has to be inferred, as always in Projections and in Descriptive Fig. 9 B ' S ■• .V 28 SHADES AND SHADOWS Geometry, from their projections. If these Hnes terminate in the horizontal plane, their projections upon the vertical plane terminate in the Ground Line; so also, if they terminate in. the vertical plane, their pro- jections upon the horizontal plane terminate in the Ground Line. The figure shows the line of visible shadow. A-S" B, which is the line in which, the plane of invisible shadow intersects the planes of projection. It is, in the language of Descriptive Geometry, the trace of this plane in the two planes of projection. 17. The projections of these lines and- points upon the vertical plane come, of course, in the drawing, directly above their projections upon- the horizontal plane. B bV o.^'V L G N L ^^ to" I -*b° b"* Fig. 10 Fig. 11 a Fig. 11 B 18. Normal Lines. — If a line is perp'endicular to either plane of projection, its shadow upon that plane lies at 45 degrees and is as long as the diagonal of the given line (Fig. 10 A and B). 19. In this figure the shadows of the extreme poinJ:s are found by twice laying off the distances of the points from the plane of projection, instead of by drawing the projection of the ray of shadow. The latter method is, however, generally preferable, as has been said, when both projections are given (Paragraph 8). j: 1-, ULX 3 3: X ^^^ r THE SHADOWS OF RIGHT LINES 29 20. Principal Normal Lines. — A right line which is parallel to two of the principal planes of projection and perpendicular to the third, casts shadows upon the first two which are equal and parallel to itself and parallel to the Ground Line, i. e. to the line of intersection of the two planes (Fig. 11 A). Its shadow upon the third plane, to which it is perpendicular, lies, as was shown in Fig. 11, at an angle of 45 degrees, and is equal in length to the diagonal of the given line (Fig. 11 B). The length and direction of these shadows being thus determined, their position is fixed by finding the shadow of a single point in each line,- presumably one of its terminal points, as was the case in Fig. 8. Such a line is parallel to one of the edges of a cube set parallel to the planes of projection. 21. The projection upon a principal plane of the shadow of a line perpendicular to that plane, is a 45° line, parallel to the projection of the Sun's rays, regardless of any unevenness in the surface upon which the shadow actually falls. The other projection of the shadow sfiows a true section of the surface. Thus, in Fig. 12, the auxiliary line 12 3 4 is- vertical, and the vertical projection of its shadow is a true section of the surface upon which the shadow falls. It is obviously parallel to the line at the comer, where the moldings miter, the lower section of the Line of S]iadow having the same slope as the beveled base. The horizontal projection of the shadow is a line at 45°, parallel with the projection of the Sun's rays. The reverse is true of the projection of the shadow of the horizontal line 5 6. 22. Principal Inclined Lines. — Inclined lines which are parallel to the diagonals of a cube whose edges are principal lines, or to the diagonals of its square faces, may be called Principal Inclined Lines. There are ten such, namely, four which are parallel to the four diagonals of the cube and six which are parallel to the diagonals of its square faces. /* DiD, DjD4 D,D, Fig, 13 A Fig. 13 B 23. The Diagonals, of a Cube.— II (Fig. 13 A) the four diagonals pf a cube are numbered D^, D^, D3, and D^, as in the figure, according to the corners of the near face from which they start, the shadow of D^ upon the vertical plane of projection will be a point; of D^, a horizontal line twice as long as the edge of the cube; of D^, a vertical line of the same length; and of D^ a line lying at 45 degrees, parallel to the projection of the ray of Ught and to the projection of the diagonal, and twice as long as this projection. The shadows of Dj, D3, and D^ 30 SHADES AND SHADOWS A 1 J5;r- — \ \ -^ V — :^^ extend from the points D/, D/ and D/, to the central point £>/, and an equal distance beyond it. Similar shadows are cast upon the other plane of projection (Fig. 13 B). Inclined lines parallel to the diagonals of such a cube have similar shadows. 24. The Diagonals of the Faces of a Cube. — Each of the six faces of such a cube is square and has two diagonals, one of which meets one end of D^, the first diagonal of the cube, the diagonal which is parallel to the direction of the light. The diagonals of the two faces that are parallel to the plane of projection are projected upon that plane in their true length and direction as lines lying at 45 degrees, and their shadows upon that plane are equal and parallel both to these projections and to the diagonals themselves. The four faces which are perpendicular to the plane of projection have eight diagonals, all of which are pro- jected upon it as right lines equal in length to one edge of the cube. The shadows upon that plane of the four diagonals which meet the first diagonal of the cube, are also as long as these projections and are at right angles to them, as in Fig. 14 A, where the diagonal ab throws its shadow upon the vertical plane at a'V at right angles to ab. The projection of this diagonal upon the horizontal plane is parallel to the projection of the Sun's rays. The shadows of the other four diagonals make with their projections an angle of 26 degrees and 34 minutes, the angle whose tangent is ^, and their length is equal to the square root of 5, the side of the square being taken as 1. This is shown in Fig. 14 B, where the diagonal cd throws its shadow upon the vertical plane in the line C d'. The projection of this diagonal upon the horizontal plane lies at right angles with the projection of the Sim's rays. Since the diagonals on opposite sides of the cube are similar, the eight diagonals of these four sides have only four different directions. The two pairs of diagonals which lie more nearly in the direction of the light naturally have shorter shadows than the two which lie across it. There are accordingly ten different diagonals in all, four of which are diagonals of the cube, and pass through its centre, and six are diagonals of its square faces. Inclined lines parallel to these diagonals have similar shadows. Other Inclined Lines.—All other inclined lines approximate more or less closely in direction to these ten diagonals, and their shadows more or less resemble these shadows in length and direction. Fig. 14 RECTILINEAR FIGURES 31 Chapter III THE SHADOWS OF RECTILINEAR FIGURES 25. The shadows of rectihnear figures upon a plane are bounded by the rectilinear shadows of the lines which enclose them, and can be determined by finding the shadows of the points where these lines meet (Fig. 15), 26. Rectangular Figures. — If a plane figure is bounded by principal lines, that is to say, by lines parallel to one plane of projection and either parallel or perpendicular to the others, it is, of course, a rect- angular parallelogram. Its shadow upon any principal plane can be determined directly from the projection upon that plane of the lines that bound it, if its dimensions and its distance from this plane are known. But its projection upon a second plane is often the best way of obtaining these data, as in the previous figures. 27. The Square. — The square (Fig. 16 A and B) is a figure of such frequent occurrence that it is worth while to make a particular study of its shadows in its three principal positions, to note the results, and to formulate them for future use. At A, the shadows are cast upon the vertical plane of projection; at B, upon the horizontal plane. 28. An inspection of these figures affords the following propositions: 1. If a square is parallel to a principal plane and its edges are per- pendicular to the two other planes ; its shadow on the first plane is, of course, a square, equal and similar to its projection on that plane, since all plane figures parallel to a plane cast upon it figures similar to themselves. (A 1.) 2: If the square is perpendicular to a principal plane so that its projection upon it is a right line, either parallel or perpendicular to the ground line, and equal in length to the side of the square; its shadow is a parallelogram, two of whose sides are equal and parallel to the given projection and whose other sides lie at 45 degrees, in the direction of the light, and are equal in length to the diagonal of the projection. The shadow is composed of two half squares, set back to back, and the area of the shadow is equal to the area of the square. (A 2 and 3, and B 2 and 3.) 29. Half Squares or Double Squares. — ^A rectangular halj square, which is the same as two small squares of half size, has for its shadow a similar small square, with half a square on either side, making four small triangular half -squares (Fig. 17). 30. Diagonals of Squares. — ^An inspection of Fig. 16 warrants the following propositions, which have been anticipated in paragraph 23 (Fig. 13): If a square is perpendicular to one principal plane and parallel to another, its projection on the first plane being equal to p^^ jg ■\ :Tr lLull I' li li| II-!- 1 1 ■ I i I 32 SHADES AND SHADOWS the side of the square; the shadow of the diagonal which hes in the direction of the light on the first plane, is at right angles with the projection of th'e square upon that plane, and is equal in length to the side of the square. The diagonal which lies across the light has a shadow equal in length to the square root of 5, the side of the square being 1 , and it makes with the projection an angle of 26 degrees and 34 minutes, the angle whose tangent is J. / L G L G / Fig. 17 / / / ^ €. L G y / y / y y y y y y ^' Fig. 18 Fig. 20 L G __. L ^ I I i \ j 31. Parallelograms. — Under similar conditions, the shadow of a parallelogram is (Pig. 18) a parallelogram two of whose sides are equal and parallel to the projection of the given parallelogram, and the other two lie at 45 degrees in the direction of light, and are equal in length to the diagonal of the other side of the parallelogram. (This is not the diagonal of the parallelogram itself.) _k 32. The Lozenge. — The rectangular lozenge, or diamond, is another figure with the shadow of which it is well to become familiar (Fig. 19). The student may formulate the result for himself. 33. Crosses.— '^h.Q diagonals of the square and of the loz- enge form two crosses. The diagonals of the square form an incHned cross, or X, like the cross of St. Andrew (Fig. 20 A, B, and C) ; those of the lozenge form a vertical cross like the St. George (Fig. 20 D, E, and F). The student may formulate these results for himself. Since the of a square is longer than the side, both the crosses and the shadows in A, B, and C are larger Fig. 19 cross of diagonal than in D, E, and F. RECTILINEAR FIGURES S3 •Va- 34. Octagons. — The Regular Octagon (Fig. 21) is formed from the square by cutting off the comers. On any side of the square, the distance from the comer of the square to the farther comer of the octagon is half the diagonal of the square. If the side of the square is taken as equal to 1, as in Fig. 21, this distance is equal to one-half the square root of 2, that is to say, to the square root of ^; since the square root of a reciprocal, the reciprocal of the square root, and the product of the reciprocal and the square root are equal ( Vf = — 1= = i V2). The remaining segment of the side of the square, which is the projection of the adjacent side of the octagon, is then equal to 1 — J V2, and the side of the octagon is equal to the diagonal of this line (1 — J a/2) V2, which is a/2 — 1; that is to say, |,. .^.^ ^^ +.]-^V2-l to the difference between the diagonal of the "" square and its side. (Half the side of the octa- gon is plainly equal to half the diagonal of the square minus half the side.) / "\ '>■*'//' \ / '■..^. 35. But it is generally more convenient to take half the side of the square, that is to say the radius of the inscribed circle, a,s a unit of measure, regarding the square as composed of four small squares, as in Fig. 22. In this case, the side of the square is 2 and all the linear dimensions come out twice as large as in Fig. 21, each side of the octagon being equal to "V ' "' y^*'. 2 ■V2 — 2, and its projection equal to 2 — a/2. Fig. 21 \-\. -2lg-Z.-- FiG. 22 •2.-VZ -Vl I ii. i, /' '' / .'' / / / / / / ^ ^ ■y / ^ > /' '-' / '-' // // Fig. 23 A Fig. 23 B 36 Fig. 23 A, B, and C shows the shadows cast by an octagon, in its three principal positions, upon a principal plane, 'in' the first position (A), the shadow is a regular octagon similar and equal to the one that casts it; in the other two, it is an irregular octagon, with right angles at the comers nearest and farthest from the light. The sides of the rectangle enclosing it measure 1 and a/2 if the diameter of the circle inscribed in the octagon is taken as the unit of measure, as in Fig. 21. (Fig. 23 B.) If the radius of the inscribed circle is taken as the unit of measure, as in Fig. 22, these dimensions become 2 and 2 a/2 They are shown in Fig. 23 C, which gives also the length of all the sides of the octagon of shadow. In either case, the circumscribing parallelogram is as wide as the side of the square from which the octagon is cut and as long as the diagonal of the square. 34 SHADES AND SHADOWS It appears then that of the eight sides of such a regular octagon the two which are parallel to the plane of projection have shadows which are equal and parallel to themselves and to their projections; while the two which are perpendicular to the plane of projection have shadows at 45°, as long as their diagonals. The two "corner" sides, which stand endwise to the light, have shadows shorter than themselves, being equal in length to their projections, and at right-angles to them (like the corresponding diagonals of a square) ; while the two which lie across the light have shadows whose slope is i (the angle being 24° 36') and which are longer than the sides themselves in the ratio of S/S to 1, and longer than the projections of the sides in the ratio of VS to V2. 37. Auxiliary Plane at ^5 Degrees. — If (Fig. 24) a horizontal square is set, as before, with its edges parallel and perpendicular to the vertical plane of projection, the projection upon this vertical plane of the shadow cast upon an auxiliary vertical plane set at 45 degrees, is a rectangular lozenge whose area is half the D V \ \ \ \ \ 1 \ \i \ ^^MdiMnyn 1^ h-->l \^Y / G [ .1 1. K r , ^^ x/ / • %/ 1 / /X 1 / / z / / Fig. 24 Fig. 25 area of the square; that is to say, the shadow is projected as a square standing on one corner, whose sides are to those of the original square in the ratio of J V2 to 1, and whose diagonal is equal to the sides of the original square, as in the figure. The true shape of the Shadow is shown by the dotted lines. 38. Shadows of Irregular Figures (Fig,. 25).— The shadows of plane figures bounded by curved or irregular lines are, in general, obtained by getting the shadows of a certain number of points on their perimeter and connecting them by curved or irregular lines analogous to those of the figure. If the figure is a rectilinear polygon, it suffices, as has been said, to obtain the shadows of the points at its corners and to connect them by straight lines, as in Fig. 15. RECTILINEAR SOLIDS OR POLYHEDRA 35 Chapter IV THE SHADOWS OF RECTILINEAR SOLIDS OR POLYHEDRA 39. Shade and Line of Shade. — The faces of a polyhedron are planes, and the edges by which they are separated are straight lines. The Line of Shade, separating the illuminated portion of the surface from the portion in shade, is made up of the successive edges which lie between the faces which are in light and those which are in shade (Fig. 26). 40. Shadow and Line of Shadow. — ^The shadow of a solid is the shadow of its illuminated surface. It is bounded by the shadow of its Line of Light and Shade. The shadow cast upon a plane surface by a polyhedron, or solid bounded by rectilinear plane figures, whethet regular or irregular, is a polygon which may be found by finding the shadow of those of its vertices, or solid angles, which occur in the Line of Light and Shade and connecting them by right lines as in Fig. 26. In this figure, the Line of Light and Shade is the broken line in which lie the points 1, ^, S, 4, 5, and 6. The shadows of these points determine the irregular hexagon of shadow. 41. Rectangular Solids. — If the faces of a soUd object are bounded by principal lines, parallel and perpendicular to the planes of pro- ♦ jection, the solid is a cube or a rectangular parallelo- piped, its faces are squares or rectangular parallel- ograms, and its Line of Light and Shade is com- posed of principal lines. If its position and dimen- sions are known, its shadow can easily be determined, on any principal plane, from its projection on that plane, without the aid of a second plane of projection. But its projection on a second plane is often the simplest way of furnishing these data. It is obvious that of the six faces of the cube, or parallelopiped, three are in shade, being turned away from the light, and three are in light, being turned toward it. The cube and parallelopiped are solids of such frequent occur- rence that it is worth while to make a special study of their shadows and to formulate propositions in regard to them for future use. 42. The Cube. — ^An inspection of Fig. 27 affords the following maxims, or rules for guidance in drawing: If the faces of a cube are parallel to the three planes of projection, two to each, the projection of the cube on either plane is a square equal and similar to one face of the cube, and its shadow on that plane is a symmetrical hexagon composed of two squares similar to the pro- jection of the cube, and two half squares; four of its edges are equal and parallel to the outlines of the projection, and the other two lie at 45 degrees, in the direction of the light, and are equal in length to the Fig. 26 / / / / Fio. 27 36 SHADES AND SHADOWS diagonal of the square. This hexagon is symmetrical about both axes, the longer one of which lies at 45 degrees, in the direction of the light, and is twice as long as the diagonal of the square, and the pther is half its length, being as long as the diagonal of the square ; this also lies at 45 degrees, being at right angles to the other axis. The shadow is twice as long as it is wide, and its area is half the surface of the cube. 43. The Diagonals of a Cube. — The four diagonals of a cube (Fig. 28) as has already been said in paragraph 22' (Fig. 12), are called D^, D^, D^, and D^, being numbered according to the front comers from which they start. £>] lies exactly in the direction of the light, and has for its shadow a point in the center of the hexagonal shadow of the cube. D^ lies across the light, extending from the front right-hand upper comer to the back left-hand lower comer. Its shadow is a horizontal line passing through the center of the hexagon and twice as long as the side of the cube. The shadow of the diagonal which starts at the front left-hand lower comer Dg is a similar line passing vertically through the center of the hexagon. The diagonal D^, whose projection lies in the direction of the light, has a shadow which lies in the same direction and also passes n .^ ^^:^ i-t / Fig. 28 Fig. 29 through the center of the hexagon, its length being twice the diagonal of the square, that is to say, twice its own projection. It is worth noting that if the shadow of the nearest face of a cube is cast upon a plane parallel to it in plan or elevation, as m the figure, the shadow of the diagonal which starts from the corner nearest the light will be a point, and the shadows of the other three will start from the shadows of the other three corners pass through the shadow of the first, and extend an equal distance beyond, toward the Sun. If the shadow 'of the farther face is taken, the shadows of the diagonals will extend away from the Suit, instead of toward it 44. The Vertical Cube.—li a cube stands on one corner, with one of its diagonals vertical as in Fig 29 Its projections present a somewhat unusual aspect. The horizontal projection is a regular hexagon the sides of which are equal to Vf , which is also the radius of the circumscribing circle, the edge of the cube bein? taken as 1. The radius of the inscribed circle is V| = i V2. One vertical projection is a parallelogram, whose sides are 1 and ^|2, and the diagonal is VS The other vertical projection is an irregular hexagon whose length is Vs and width \2. RECTILINEAR SOLIDS OR POLYHEDRA 37 \ Fig. 31 45. In this aspect, the cube may be regarded as formed by the - interpenetration of two triangular pyramids, the three faces of which are right triangles (Fig. 30). The shadows of the cube when in this position, are irregular hexagons, as in the figures. 46. The Parallelopiped. — The shadow of a right parallelopiped (Fig. 31), set parallel to the planes of projection, is an irregular hexagon, four of whose edges are equal and parallel to the four lines of the projection of the parallelopiped,, and the other two are at 45 degrees, and are equal in length to the diagonal of its third dimension. If either face is a square the area is equal to half the area of the parallelopiped . 47. The Octagonal Prism. — Triangular prisms so seldom occur in .architecture, that it is not worth while to give them special con- a_ sideration. Octagonal prisms are more common. The shadow of a regular octagonal prism, the elements and one side of which are parallel to the plane of projection on which the shadow falls (Fig. 32), is an irregular octagon right angled at the comers nearest and farthest from the light. The side parallel to the side of the prism is longer than the side of the prism by an amount equal to the projection of the oblique side of the octagon which forms the base of the prism. If the radius of the inscribed circle is taken as the unit of m;easure,l, this' dimension is equal to 2 — V2 (see para- graph '34, Fig. 21). The width of. the shadow is 2 a/2; that is to say, it is the diagonal of the width of the ./ Fio. 32 pnsm. 38 SHADES AND SHADOWS J ^ I \ / \ ' \ * 48. Polyhedra. — It is not always clear, upon mere inspection of the projections of a polyhedron which faces are in Ught and which in shade. But if, as in Fig. 33, and previously in Figs. J 1 and J 2, in the Introduction, the polyhedron is cut by a plane parallel to the light, it is easy to determine which of the sides of the polygon of intersection receive the Sun's rays, and therefore which of the faces of the polyhedron are in light, which in shade. For, if any line lying in a plane is in light, the whole plane is illuminated. In Fig. 33, a single secant plane suffices. It shows that only one of the faces shown in the vertical projection is in shade, the other four and the upper face being in light. It follows that the bottom face is in shade, and also that four of the faces on the other side of the polyhedron, opposite the four in light, are in shade. The ten lines composing the Line of Shade are now easily found, in both Tjlan and ele- vation, and the solid angles in which they meet. - The shadow of the dodecahedron is an irregular decagon, the angles of which are the shad- ows of these points. 49. Pyramids. — ^The Line of Light and Shade upon the surface of a pyramid is deter- mined by finding which of its faces is in light, which in shade. This may be done as Fig. 34 in the case of the dodeca- PiG. 33 IZ. JL A Fig. 35 ^V. Fig, 36 hedron by passing a plane through the pyramid perpendicular to its base and parallel to the ravs of U^ht Fig. 34 illustrates this m the case of a hexagonal pyramid. " 50. But in general, it is simpler to find the shadow of 'the vertex of the pyramid on the plane of its base and then to draw two lines of shadow from this point tangent to the perimeter of the base (Fig 35 A) The faces included between these two hnes will be in shade. If the shadow of the vertex falls within +L K=i" as at B, all the faces are in light; if it falls just upon the outline of the base as at S orZTl .r ' extended as in Fig^36 B, those faces of the pyramid will be planes of light" It A^f his apptX' all pyramids, of whatever number of sides and of whatever slope. • ^^ RECTILINEAR SOLIDS OR POLYHEDRA 39 \ 51. It follows that a right octagonal pyramid which has two sides of the base parallel to the planes of projection (Fig. 36) will have three faces in shade, if its slope is less than 45 degrees; and none, if it is less than the angle <^. If the slope is just 45 degrees, two of the faces will be planes of light and shade, and one will be in shade. Square and octagonal pyramids so set do not \ show any shaded face when seen in elevation. \ A 52. If an octagonal pyramid is inverted, that is to say, if the base is turned toward the light, as in Fig. 36, at D, E, and F, these phenomena are reversed; all the eight faces will ,be in shade if the slope is less than the angle <^; seven, if more than (ft and less than 45 degrees; five, if more than 45 degrees. If the slope is just 45 degrees, as in Fig. 36 B, two faces will be planes of light and shade. 53. Shadows Upon Rectilinear Solids. — ^The shadow of a principal line upon a rectilinear solid which is given by its projections (Fig. 37) , is easily fotuid by casting the shadow first on the plane of projection to which the line is normal, and then transferring to it the other plane. The projection of the shadow upon the first plane will be a 45-degree line, as in the elevation in the figure. In Fig. 38 A and B, one pyramid throws its shadow partly upon another. At A, the point X, which is a Point of Flight, is found by drawing a ray of shadow back from the point on the groimd where the shadows of the two pyramids meet. At B, where the two shadows cast upon the ground do not intersect, a line v^ V2 is drawn connecting the vertices of the pyramids and casting its shadow on the ground at s^, 5^. If this line is produced to s^ at the base of the right-hand pyramid, the point 53 will be the shadow of a point V3, upon v^ v^ produced. The line v^ V3 will then cast its shadow upon the face of the pyramid in the line v^ s^ and the shadow of v^ will fall at x. Fio. 37 Fig. 38 40 SHADES AND SHADOWS APPLICATIONS TO ARCHITECTURAL DETAILS CHAPTER I. POINTS There are of course no architectural details which consist only of points. CHAPTER, II. RIGHT LINES Example I A and B. A Post and Wall, with Cap and Base. A, Perspective; B, Plan and Elevation. This example illustrates the phenomena of discontinuous Lines with continuous Shadows, and of contmuous Lines with discontinuous Shadows. While the line of Cast Shadow V, 9,', 3\ 4', etc., is a continuous though broken line, the Imes which cast it, l,x; 2,8,4,6; y, 6, 7, 8; z, 9, 10, C\ are discontinuous, being cut up into four fragments. There are breaks between x and 2, 5 and y, and 8 and z. B ^/// y 4e^^/ PROJBcnOM OP THE ^ DrazCTION OPlJaHT Ex. I A Moreover, though the line AS is continuous, its shadow is discontinuous, being partly at A x, partly at 1^-3', and the continuous line y~B also casts a discontinuous shadow, partly from B' to 8', partly from B to z. The points x, y, and z, are Points of Flight, where the shadow jumps. 54. Here, as always, the edges which cast the shadows, and which are Lines of Light and Shade, lie between an illuminated surface which is turned toward the Sun, and a surface which is in shade and is turned away from the Sun. Moreover, the surfaces upon which shadows are cast are all turned toward the Sun. This illustrates the maxim that surfaces turned away from the Sun are always in Shade, and those turned toward the Sun, either in Light or in Shadow. This is seen in both figures. 55. It is to be noticed also in Example I B that the lines which are perpendicular to the Plane of Projection, like A, 2, 3 in the elevation, and 6, 7, 8, B, in the plan, cast shadows the projections of which, on those planes, are straight Hues, lying at an angle of 45 degrees, in spite of the irregularities of the surfaces on which they fall, as has already been seen in Fig. 14. Moreover, the shadow of the post falling upon the base of the wall gives in elevation at 6', 6', 7', and 8', a true section of the wall and base. Ex. II B 41 1-4 42 SHADES AND SHADOWS CHAPTER III. RECTILINEAR FIGURES Example II A. A Square Beacon, or Tower, for Electric Lighting. Plan and Elevation. — Shadows on the ground and on a wall. This tower is a skeleton of iron rods, carrying at the top eight electric lights. The shadows of the squares of which it is mainly composed are easily found, in plan and in elevation. Example II B. The Same; Perspective and Perspective Shadow. — The Perspective presents no difficulty. It is convenient to lay off the vertical dimensions upon the vertical line on the axis of the tower. One diagonal of each square is horizontal, the others are directed towards the center of the picture V^ as their common vanishing point. In the shadow upon the ground, the shadows of the right- and left-hand horizontal lines, R and L, are parallel to the lines themselves, and have the same vanishing points V^ and V^. The shadow of the vertical line at the axis of the tower, is horizontal, since the light falls parallel to the plane of the picture. The angle of the light is the angle ^, as usual. In the shadow upon the wall, the shadows of the left-hand lines L and of the vertical line Z, are parallel to the lines themselves, and have the same Vanishing Points, V^ and. V^. Since V is in the zenith, at an infinite distance, the shadow of the axis of the tower is vertical, parallel to the axis itself. The shadows of the right-hand lines R, which are in the elevation Normal Lines, easting their shadows in lines at 45 degrees, are directed toward the Vanishing Point V", their common vanishing point. This point is just over V^, at a distance equal to the diagonal of the line V^ V^. (See paragraph 37, Fig. Y Y, in the Appendix.) Example III. Five Square Lamp Posts; Plan and Elevation, With Shadows on Ground and Wall. — These lamp posts stand at a bend of the street, just so that the three numbered 1, 4, and 5 stand square, while 2 and 3, following the turn, are set comerwise. The shadows of 1, 2, and 5 fall on principal planes, while 3 and 4 throw their shadows upon an auxiliary 45° plane. The construction of these shadows follows the example set in Figs. 16, 19, and 24. I Wall p CHAPTER IV. RECTILINEAR SOLIDS Parallelopipeds . — Some architectural details, and even important features, are composed entirely of parallelopipeds. Such are fillets and dentils; the simpler kinds of entablatures, posts, pedestals and parapets; and square niches, steps, and seats. Example I, illustrating the Shadows of Straight lines, comes under this head. Dentils. — Dentils are commonly square in plan and a square and a half high, the three dimensions being in the ratio of 2, 2, and 3. They are generally placed half their width apart, so that their distance on centers, or on edges, is equal to their height, the dentil and interdentil, together, occupying a square in eleva- tion. But, in Pompeii, the dentils are generally narrower, as in Example VII. Example IV A. A Dentil; Plan and Two Elevations, With Cast Shadows. Example IV B. The Same; Perspective, Bird's-Eye View, and Toad's-Eye View. (See Appendix, Fig. Y Y). The vanishing points of horizontal lines, V and V^, are taken at a convenient distance apart, the center of the picture V, which is the vanishing point of normal lines, is half-way between them; the position of the spectator, or station point S, is in front of V^, in the air, at a distance equal to V V^ (or V^ V^). It is shown revolved into the plane of the picture^ at 5'. The vanishing point of the shadows of normal lines is at V", directly above V-, at a distance equal to the diagonal of the distance S V"; i. e., to the distance from S' to V-. The shadows of the other edges i side Eleytitlon Elevation Wall V///iy////My/M////y//////M MM'M'MM. Ex, Plan IV A THE SHADOWS OF RECTILINEAR FIGURES 43 Ex. Ill 44 SHADES AND SHADOWS of the dentil The vertical I Toads Eye View , being parallel to the edges themselves, have the same vanishing points, namely, V" and V^. lines, being parallel to the plane on which their shadows fall, have their shadows also vertical, and as both the lines and their shadows, are parallel to the picture, their perspectives are vertical. The Perspective Plan. — The vanishing points being — determined as before, the perspective plan, which is a square put into perspective, can be drawn anywhere, at any convenient scale (see Appendix, Fig. T T) . The point a being thus assumed, a r and a I are taken equal to the sides of the square, (or a r' and aV to its dia- gonal) . The same scale is used as in the orthographic plan, the sides of the square being laid off upon the ground line and then transferred to the perspective lines by means- of the Points of Distance £>■* and D'- (or the diagonal is laid off and transferred by means of the Vanishing Points F* and V'-'). Both methods are shown, as in Fig. VV, 3 and 4, in the Appendix. The length of the right-hand, side of the dentil, thus ascertained, determines the positioii of the wall, in plan. The Perspective.— The perspective can now be drawn above the perspective plan, and the bird's-eye view below it. The vertical dimension of the nearest comer is the same as in the orthographic pro- jection in Example IV A. The vertical and hori- zontal dimensions in the bird's-eye view and in the toad's-eye view are the same as those in the perspective. The Shadow. — In all three the rays of light are taken parallel to the plane of the picture, at the angle (^ (see Appendix, Fig. YY). shadow of the normal line 1-2, beginning at the point 1, lies in the direction of the vanishing point V" and terminates at the point 2 s, which is the shadow of the point S. A ray. of shadow drawn from the point 2 in the direction of the light, and parallel to the paper, intercepts the Hne of verti- cal shadow at this point. This ray makes the angle ir--T: I ■ i I .■|.;|.il' |;;,i;fi. !!!■:■. J; I : ' I . ■ ■ ■ -I I'll ' '^^ ... i^i ffl "■-. r. yi*-||TrT-| ■■ Y " f.:-i,-: 1!. "yi^M^^^Mii^jlM.^ I '' ' ■Lni'i li ■fi '.I ' i':i,lfi' /'i'l . . I 1I I ,' ' 'I i.i.i':i Ex. IX B 48 SHADES AND SHADOWS ^ iiiriiii|W Ex. X A Ex. X B y Diagonal Section M^1 — : / — r \/ ; \ 1 \ / Ex. XI ARCHITECTURAL APPLICATIONS 49 away from the light, but by the horizontal line on the side toward the light. The shadows of the cornice, and of the fillets, and Of , the abacus of the capital, are just what they would be if there were no moldings at all, and may be drawn as in Example V A. Example X B. The Same; Perspective Plan and Perspective With Shadows. — ^The farther comer of the capital of the post being itself in shadow, casts no shadow upon the wall. The beveled miter Hnes on the front comer, which slope downwards and backwards at the angle (/>, have their Vanishing Point at V* just below V^. Example XI. An Entablature; Plan and Elevation With Shadow. This is the same subject as in Example X, but it shows in plan, an internal angle, and three ex- ternal ones. Example XII A . A Doric Gutta , or Drop; Plan and Two Elevations. The Guttae, or Drops, which occur in Doric architecture, are usually either cylindrical or conical. But in the Roman Doric order, those which occur on the upper band of the architrave, beneath the triglyphs, are often rectangular in plan, being frusta of pyramids, or, rather, of wedges. They differ somewhat in size and shape in different examples, but never vary Ex. XII A Ex. XII B ^'"'wiliWf'ffil'IPffl i Horuen V" Ex. XIII A XIII B much from the type here shown, in which the depth of the base of the pyramid or wedge is three-fourths, and the height of the entire wedge is twice, the width of the base. The fmstum which constitutes the drop is half as high as the wedge from which it is cut, being as high as it is wide at the base. Example XII B. The Same; Perspective Plan and Perspective .—This presents no difficulty. The vanishing points of all the lines are easily ascertained, except for the inclined edges of the gutta, and for its shadow. These must be determined by finding the points at their extremities and connecting them. F^ is as far from V'^ as is F'^, and V" is directly over it, at the diagonal of this distance. (See Fig. YY in the Appendix.) 50 SHADES AND SHADOWS Example XIII A. Two GuttcB and Fillet; Plan and Two Elevations .—The point X is a Point of Flight, the shadow of the edge of the fillet passing from that point to the point a on the side of the second gutta, as is seen in the section. If, now, the two guttae were prolonged until their edges met, the point of meeting would obviously.be the initial point in the Line of Shadow cast by the first gutta upon^the side of the second. This point would fall at b, in the figure, and the line drawn from a, in the section, toward b completes the shadow. Example XIII B. The Same; Perspective Plan and Perspective. Example XIV A . A Roman Doric Architrave, with Tmnia, Regula, and Pyramidal Guttce and the Lower Part of the Frieze; Plan, Sections, and Elevation. — ^The height of the Roman Doric architrave is equal to half the width of the lower diameter of the Doric column, or J D. It is divided into two bands, or fascias, of which the lower one is one-third the height of the architrave, or J D. The upper band is twice as wide as the lower, or .J D., and has at 'its upper edge a square fillet, called the Taenia, which measures ■jVD. in height and projection. The frieze above it is ornamented by triglyphs, which are J D. wide, and are divided into three equal parts, called shanks. Each shank, or femur, is ^ D. in width, and has on each edge a beveled chamfer. This gives two channels on Frieze and Triglyph Frieze and Trig ly pin Ex. XIV A Ex. XIV B the face of the triglyph. These are cut in at an angle of 45 degrees, but the chamfers or half channels on the edges of the triglyph are shallower. The two channels and the three plain faces of the shanks between them, are all tV D. wide, the two chamfers making up the sixth twelfth. One side of each channel is a Surface of Light and Shade. The other side and both chamfers are in light Below the taenia, and directly beneatheach of the triglyphs which ornament the frieze is a little fillet * D in length, being just as long as the triglyphs are wide. This is called the Regula, or Listel. Under each regula are six gutts of the shape shown in Examples XII and XIII. They are ^V D. 6n centers, but they do not quite touch one another, measuring only ^ D. at the bottom (or A D.), so that the regula slightly overhangs them at the ends. The space between them is accordingly J^- D. The wedge from which they arc cut is twice as hieh as the base is wide, or i D., and the frustum is half the height of the wedge, as in the previous example ARCHITECTURAL APPLICATIONS 51 The regula is half as high as the guttae, and the two together are as high as the tasnia, or iV D. The three together occupy the upper half of the upper band and are just as wide as the lower band. The regula projects from the face of the upper band just \ D. The shadows of the taenia upon the regula and fascia, of the regula upon the guttae, and of the guttae upon the fascia, are found as in Example XIII A. JIliiiliElllllk '^=^"'°" Jliiliiilffl Ex. XV A Ex. XVI A Example XIV B. The Same; Per- spective Plan and Perspective. It is left for the student to cast the Shadows, , which are like those in Example XIII B. Example XV A. A Roman Doric Frieze, With Triglyphs and Upper Tcenia; Plan, Sections, and Elevation, With' Shadows. — The frieze, in the Roman Doric order is, in height, three quarters the width -of the lower diam- eter of the column, or |D, or i|D- Itis surmounted by abroad fillet, called the Upper Taenia, which is jig D, or -j^ D., in height, or one-twelfth the height of the frieze. The lower taenia meas- ures V2 D-. or A D- The channels and chamfers stop short of the top of the triglyph, terminating in faces sloping at 45' degrees. These faces, 'like one side of each channel, are accordingly Surfaces of Light and Shade, but the chamfers, as has been said, are both in light. „ , , i,t . t^ • 4.1 -u • a -n The plain face of the frieze, between the triglyphs, is called the Metope. It is exactly square, being | D.. wide The triglvphs are accordingly li diameters on centers, or on edges. They project Vt D. or A D. from the face of thfe metope, a dimension which is just the depth of the channels. Ex. XVI B 52 SHADES AND SHADOWS Example XV B. The Same; Perspective Plan and Perspective.— This is left for the strident to draw out The vanishing points of all the lines are easily determined, except those of the beveled line at the top of the chamfer. This must be obtained by points. Example XVI A. Steps and One Parapet; Plan, Section, and Elevation.— Th& shadow of the parapet is cast on the steps* and the shadow of the steps on the ground and on the wall. The Shadow of the Parapet on the Steps. — ^The Line of Shade of the parapet consists of the vertical line a h and the normal line b c. The shadow falls upon the tops of the steps which are numbered 1, 2, and 3, and upon the faces of those numbered 2, 3, and 4. The line a b, being in the same plane s the face of step No. ^, and the 1 ne 6 c in the same plane as the top f step No. 4> cast no shadows on 1 hem. The cast shadow is projected ] 1 discontinuous lines, in both plan nd elevation, as in Example I. The section shows that the shadow (f the point b falls on the upper surface of step No. 3, at the point b'. Since the projection of the shadow of a line perpendicular to a principal plane lies at 45 degrees, irrespective of the nature of the surface on which it falls, the shadows from a to b' in the plan and from c to b' in the ele- vation, are easily drawn. The shadow in one projection can then be com- pleted by means of that in the other. The lines ab' in the elevation and b'c in the plan give true sections of the steps. The section shows that the shadow of the two lower steps and part of the third falls upon the ground, the rest upon the wall. These shadows are easily cast, the lines parallel to the planes of projection having shadows equal and parallel to themselves, and those perpendicular to them having shadows equal in length to their diagonals, and lying at 45 degrees. Example XVI B. The perspective plan and perspective present no special difficulty. The shadows of lines that are parallel to the planes on which the shadows fall are parallel to the lines that cast them. The hori- zontal lines have the same vanishing Ex.'XVII B ARCHITECTURAL APPLICATIONS 63 -^\ Ex. XVIII A points, at V" and V^ or V^. The perspectives of vertical lines and of their shadows on vertical planes are vertical, and of their shadows on the ground plane horizontal. The shadows of lines normal to the vertical plane have their vanishing points at V^, which is directly over V^, as has been explained before, at a distance equal to the diagonal of the line V^ V^. The rays of light are, as before, parallel to the picture and make, with the ground line, the angle . Example XVII A . Steps With Two Parapets; Plan, Section, and Elevation. Example X VII B . Perspective Plan and Perspective. Example XVIII A. Steps and Ramps; Plan and Two Elevations; With Shadows. — The only lines which occur in this example which did not occur in Example XVII, are the inclined lines of the ramps and of their shadows. The, shadow of the right-hand ramp falls partly on the ground, partly on the wall, and is determined by the points at its ex- tremities; namely, 2' 3^ in the plan,_ and S' 4' in the elevation. The sec- tion shows how the point 3 may be determined. If the Line of Shade of the ramp Z-4 is extended at both ends until it pierces the plane of the ground at the point 7, and that of the wall at 8, these points will be their own shad- ows, in these planes, and the Line of Shadow can be drawn through 8 and 4', 7 and £', as in the figure. These lines will meet at the point 3', which is thus independently determined. The shadows of the left-hand ramp that fall upon the horizontal surfaces , of the steps are parallel to the shad- ows on the ground, and those which fall on the vertical faces of the steps are parallel to the shadow on the wall. Example XVIII B. The Same; Perspective Plan, Perspective, and Bird's-Eye View, With Shadows. — The shadows of the ramps on the ground and uppn the upper surfaces of the steps being all parallel to one another, have their vanishing point somewhere in the Horizon W^, since lines lying in or parallel to a plane have their vanishing points in the horizon of the plane (see Appendix, paragraph 19). This auxiHary Vanishing Point is easily found by producing the shadow XVIII B 54 SHADES AND SHADOWS Ex. XIX A Ex. XXI A Ex. XXI B Ex. XX A v" of the ramp on the ground 7 2' 3' until it cuts the Horizon; In the same way, the vanishing point of the shadows of the ramp on the vertical faces of the steps must lie in the vertical horizon H^^, at the point where it is cut by the shadow of the ramp on the wall S' 4" *, produced. Example XIX A. A Square Abacus on an Angular Post; Plan and Elevation, with Shadows. Example XIX B. Thp Same; Perspective Plan, Perspective, and Bird's- Eye View. Example XX A. A Square Abacus on an Octagonal Post; Plan and Elevation, with Shadows. Example XX B. The S<^i^^,' Perspective, Plan Perspective, and Bird's- Eye View, with Shadows. Example XXI A. An Octagonal Abacus on a Square Post; Plan and Elevation, with Shadows. Example XXI B. The Same; Perspective Plan, Perspective, and Bird's- Eye View, with Shadows. Exam-pie XXII A. An Octagonal Abacus on an Octagonal Post; Plan and Elevation, with Shadows. Example XXII B. The Same; Perspective Plan, Perspective, and Bird's-Eye View, with Shadows. In these four perspectives the left-hand Vanishing Point, V^, is not shown. It comes as far to the left of the Center V^ as V* comes to the right, and V" comes just above it, at the diagonal of that distance. Ex. XX B Ex. XXII A S Ex. XXII B ARCHITECTURAL APPLICATIONS 55 Ex. XXIIl A ^ > 1 v= i, I*- y1 Ex. XXIII B All these figures are set a good ways to the right, in order to show all their faces. They may be varied to advantage by setting them farther toward the left and setting the Vanishing Points V^ and Y^ farther apart. They will then show less distortion. Example XXIII A. A Stone Bench; Plan and Elevation, with Shadow on the Ground. Example XXIII B. The Same; Perspective Plan, Bird's-Eye View, With Shadows. — In the Perspective, but not in the Plan and Elevation or in the Bird's- eye view, the bench is set again.st a wall. Example XXIII C. The Same in a Different Aspect; Perspective Plan and Perspective. These examples may be varied by adding rect- angular arms at the ends of the seat. Ex. XXIII c 56 'SHADES AND SHADOWS e 6 5 1 7 1 3.4 Ex. XXIV A Ex. XXIV B Example XXIV A . A Bracket With an Inclined Face; Plan and Tivo Elevations, With Shadow on the Wall. — ^The plan is superfluous. The shadows of the points 2, 3, 4, 6, and 7 fall at 2\ S\ Jf, 6% and 7% as determined by means of the side elevation. 1, 5, and 8 are their own shadows, a; is a Point of Flight, and its shadow at xf determines the shadow of the inclined line. The other lines are either parallel to the wall and have shadows parallel to themselves and to their projection on the wall, or are normal to it and have shadows at 45 degrees. Example XXIV B . The Same; Perspective Plan, Perspective, and Bird's-Eye View, With Shadow on the Wall. This presents no new difficulty. The shadow of the inclined line is drawn from 7' to X'. A third point in this line of shadow may be found by extending the line x7 until it "^ pierces the wall, as in Fig. 7 C, Chapter II, and in Ex. XVIII. Example XX V A . The Same; Plan and Side Elevation,' with Shadow on the Wall. Example XXV B . The Same; Perspective Plan, and Perspec- tive, With Shadow. The Vanish- ing Points are omitted, but are easily found. V^ and V^ are as far from V^ as is S'. V" is above V^ at the distance V- D'-. Example XXVI A. A Roof and Chimney; Two Elevations. The shadow of the chimney on the roof is bounded by lines par- allel to the hip of the roof, and, like it, shows in elevation the true slope of the roof. (See Chapter II, paragraph 21.) ~ Ex. XXV A ARCHITECTURAL APPLICATIONS 57 Example XXVI B. The Same; Perspective Plan and Perspective. — In this drawing, the hip line of the roof is, as appears from the perspective plan, parallel to the plane of the picture. The shadow of the chimney and of all other vertical, lines is parallel to it. The perspectives of all these lines are, accordingly, parallel to the lines themselves, and are drawn at their true inclination. The horizontal lines on the top of the chimney which go back to the- left are parallel to the plane of the roof and their shadows in that plane are accordingly parallel to them, and have the same vanishing point V^. The direction of all the lines of the shadow are thus determined, except the shadows of the lines R, that is to say of the horizontal lines, at the top t f f 1 1 2 3 Ex. XXVII A Ex. XXVII B of the chimney, which go to the right. The shadows of these lines can be determined by obtaining the shadows of the points at their extremities, which are points in the vertical lines. But these shadows, namely, the shadows, upon the roof, of the right-hand horizontal lines R, are all parallel to one another. They must, then, have a common vanishing point, a,nd as they all lie in the plane of the roof, this auxiliary vanishing point VI must be on the horizon of that plane. (See Appendix, paragraph 21). This horizon is the line H^ drawn from V^ to V^, the vanishing point of the lines M, the steepest lines of the roof. One "such line is the Kne M, in which lies the intersection of the roof arid the chimney. Its vanishing point V" is in the vertical horizon H^ above V (see paragraph 35 of the Appendix, Fig. YY 3). 1—5 58 SHADES AND SHADOWS The iron rod, parallel to the right-hand edges of the chimney affords a better means of finding the Vanishing Point of these shadows than do the edges themselves, being longer. Its shadow begins where the rod touches the roof, and extends toward the point c where the shadow of the other end of the rod, on the- front of the chimney, would fall. This line, lying in the plane of the roof, finds its vanishing point in the line H'-"', the horizon of that plane, at V^. The right-hand edges of the chimney are parallel to the rod, and their shadows are parallel to its shadow, and are also directed toward the vanishing point V^, (See Appendix, paragraph 21). Example XXVII A. An Octagonal Bracket and Canopy; Plan and Two Elevations. Example XXVII B. The Same; Perspective, from above. This presents no new problems. The vanishing points of all the lines and of their shadows are known, since the shadows of the oblique lines of the octagons are either vertical, as at 3' /i', or have half the slope of the shadows of the normal lines, as at 1" 2\ In the latter case, F?, the Vanishing Point of the Line of Shadow, is half as far above the Horizon as is V", the ^ ^ D Ul , , lU D Front ^evation' Ex. XXVIII A Ex. XXIX. A Vanishing Point of the Shadows of Normal Lines. See Appendix, paragraph 38. Example XXVII C. It is left for the student to draw a second Perspective, showing the Bracket and Canopy as seen from below. Example XXVIII A. The Portal to a Tomb.— It is left to the student to cast the Shadows. Example XXVIII B. The Same; Perspective Plan and Perspective. —This is left for ftie student to draw. As the Plan is laid out in squares, the Perspective Plan is ea,sily constructed. Example XXIX A . A Square Tower and Octagonal St-eeple; Plan and Elevation.— Sh.a.dovfs cast but not rendered Example XXIX B. The Same; Perspective Plan and Perspective.— This is left for the student to draw Example XXX A. A Triangular Lamp Post With a Lantern in the Shape of a Vertical Cube, or Two Inter- secting Triangular Pyramids, and Hexagonal in Plan, like that shown in Fig. SO. Example XXX B. The Same in a Different Aspect, the Axis Being Revolved 60 or 180 Degrees .—This, Example is left for the student to work out. ' ■ ARCHITECTURAL APPLICATIONS 59 Example XXXI A. A Square Lamp Post With a Lantern in the Shape of Two Intersecting Square Pyramids, and Octagonal in Plan. Example XXXI B. The Same in a Different Aspect, the Axis Being Revolved 45 Degrees. — ^This Example is left for the student to work out. .Example XXXII. Pyramidal Voussoirs; Elevation and Two Plans. — It sometimes happens that the voussoirs of an arch are cut into the shape of trapezoidal pyra- mids, as here shown. The shadows in Example XXXII A and B, are cast as in Fig. 38 A and B. Example XXXII C requires a slightly different treatment, since the line s^ .s^, which is the shadow of the line Vj^ v., joining the vertices of the two pyramids does not inter- sect the base of the right-hand pyramid. In place of the line v^ v.^ is introduced, accordingly, the line w^ w^ parallel to v^ v^ and part way up the side of the pyramids. The point w^ throws its shadow at i,. Ex. XXXII A Ex. XXXII B \Wx/v\A4/ Ex. XXXII C Ex. XXX A Ex. XXXI A Ex. XXXII 1 on the line s p which is the shadow cast upon the wall by the edge of the pyramid v,p, and the line w,w^ throws its shadow on t, t, parallel to ., 5,. intersecting the base of the pyramid at ^3, which is the shadow of the point w upon the line w, w„ produced. The shadow of the hne w, w, upon the side of the pyramid falls . at it, and the shadow of w, at x. through which the required shadow can be drawn. 60 SHADES AND SHADOWS Ex. XXXV A Example XXXIIJ . A Gable in Plan and Elevation.— The simplest form of roof has two slopes -which descend from a horizontal ridge to horizontal eaves, and terminate in a gable, which generally projects beyond the wall at the end just as much as the eaves do at the sides. The underside of the eaves, or soffit, sometimes is horizontal, sometimes slopes down, and sometimes up, as in this figure. There are four angles between these soffits and the walls; first, a, the angle along the eaves; second, b, the angle of the rake, measured in a plane at right angles with the slope; third, c, the angle of the miter line, where the rake meets the eaves; and fourth, d, the angle at the ridge, where the two rakes meet. The slope at e is the same as at a ; those at b and d, steeper ; that at c, flatter. Example XXXIV A-T. Gables; Shades and Shadows. — ^The shape of the gable depends, of course, on the steepness of the roof, which is sometimes quite flat, reducing the gable to a horizontal line like the eaves, as in Example XXXIV D. The slope of the soffit upon the two rakes, as the sides of the gables are called, depends partly on the slope of the soffit along the eaves, and partly on the steepness of the roof. Under these varying conditions, either of the three soffits, that along the eaves, that of the ascending rake, and that of the descending rake, may be in light, or in shade, or in neither. It is convenient to call the eaves and the rake nearest the light, the first eaves, and the first rake, and the ones farthest from the light, the second. The first eaves and rake are ordinarily on the left, the second on the right. The Slope of the Soffit Greater Than 45 Degrees. — When the slope of the first soffit is greater than 45 degrees, the first eaves and both the rakes are in light, as at ^. Tlie Soffit at 4.5 Degrees. — ^When, as at B, C, and D, the slope of the eaves is 45 degrees, and the overhang of the gable is the same as that of the eaves, the miter lines at their inter- section have the direction of the first and second diagonals of a cube, D^ and £'2. But Di lies in the direction of the light, and any plane passing through it is a plane of light and shade. The- first eaves and the first rake are accordingly Planes of Light and Shade, whatever the slope of the roof, and the first rake casts no shadow either on the second rake, or on the wall. The second rake is in light, as at B and C. The Slope of the Soffit Less Than 45 Degrees. — When the slope of the first soffit is less than 45 degrees the first eaves as well as the second, and the first rake, are in shade, and the second rake is either in light, as at E, F, G, and H, or in shade, as in I, J, K, and L, or in neither, as in M, N, O, P, and D. This depends partly on the slope of the roof and partly on the slope of the soffit, whether it slopes up, as in E, I, and M, or is horizontal, as in F, J, and N, or slopes down, as in G and H, K and L, and O and P. The second rake is in neither light nor shade but is parallel to the rays of light, when, as at M, N, O, P, and D, the slope of the soffit and the slope of roof are such that a line lying in the raking soffit whose vertical' projection lies at 45 degrees has its horizontal projection also at 45 degrees. For such a line is parallel to the Ught, and so is the plane in which it lies. In that case, as appears from the plans, the point x, in the 45-degree line drawn through the comer, is as far from the comer of the building as is the point y. If the slope of the roof is given, this determines the slope of the soffit along the eaves, and vice versa. The steeper the soffit the flatter must be the gable, to produce this result, as in these figures. Ex. XXXV B ARCHITECTURAL APPLICATIONS 61 I/!--" I M /7 \ \/^irr^\\ 1 / i \ / :/ .'' \ \ y I I'-^ ^ Z- T i Ex. XXXIV 62 SHADES AND SHADOWS Ex. XXXV C Unequal Overhang. — In Figs. Q, R, S, and T, the slope of the roof is the same, and the eaves are horizontal. They either project more than the gables, or less, and the first rake is in light, as at Q, in shade, as at S and T, or in neither, as at R, accordingly. Pediments. — In classical architecture, the top of a wall is generally surmounted by a cornice, like the upper member of an entablature, consisting of three members, the most important of which is the Corona, which crowns the wall with a sort of shelf. This is generally supported by a series of moldings called the Bed Mold, among which often occur dentils or some kind of bracket, or both. Above the cornice is a gutter, or, as it is called, a Cymatium, which projects from it just as in the previous figures the eaves project from the wall. Its slope is about 45 degrees, more or less. The gable of classical buildings is called a Peditnent. The cymatiurn, corona, and bed-mold follow the slope of the roof, which is seldoni more than 30 degrees, and form a raking cornice, which projects in front as much as the cornice on the sides. But the corona and bed mold are also carried across, under the gable, forming what is called the Horizontal Cornice. This has no proper cymatium, though it generally carries the fillet or other small molding, which comes between the cymatium and the corona. Although the cymatium has always a curved cross-section, convex or concave, or more frequently, the curve of double curvature, called a cyma recta, it is convenient in studying . the shadows of pediments to replace these curves by right lines, that is to say, to replace the cylindrical forms of the cymatium by plane surfaces. Example XXXV, A, B, C, D, and E shows a number of similar pediments which differ in the slope of the cymatia. They exhibit the phenomena of the eaves and soffits discussed in the previous paragraphs. When, as at A, the cymatium slopes at 45 degrees, the first rake is a Surface of Light and Shade, the second is wholly in light, and the shadow of the cymatium on the vertical wall is horizontal, like that of the second diagonal of a cube, D^- When, as at B, the cymatium is -steeper than 45 degrees, both rakes are in light, and the shadow of the cymatium slopes. up. When, as at C, the slope is such that (as in Example XXXIV, M, N, O, and P) the second raking cymatium is a plane of Light and Shade, the sliadow of the cymatium coincides with that of the fillet above it and slopes down, being parallel to the pediment. When, as at D, the cymatium is steeper than at C, but still less than 45 degrees, its shadow still slopes downwards, but not so much. The second raking cymatium is in light and receives the shadow of the fillet that crowns the first raking cymatium, as in the figure. When, as at E, the cymatium slopes less than at C, the second raking cymatium is in shade, and the fillet above it throws upon the raking corona a shadow parallel to itself. A similar shadow of the same slope is also cast upon the wall, the second cymatium, which is not now a line of shade, casting no shadow. The Shades and Shadows of Curved Pediments are discussed in Chapter VIII. Ex. XXXV D Jll|!|llJi:2?"«:r,,. p iPiiiiiiiii 1 1 1'Micm ;, inlniijiiiiiii iL|| ■' i r ■ / /' 7 -7 / / \ / Ex. XXXV E APPENDIX PI an of Roof J .-\ \ V r^ A 1 1 / \.1 — r B n r J I G n ^ .. 1 \ / / 1 /Side E^.»vatit^n / / ! PLaK\ Fig. a a. Orthographic Projection V'-- Hofizott I I I I PeKspective Fig. B B. Perspective O <:^ Perspective Plan Fig. C C. Isometric Fig. DD 64 APPENDIX PROJECTIONS, PERSPECTIVE, AND TRIGONOMETRICAL TERMS PROJECTIONS Bvi^dings and details of buildings are drawn sometimes in Plan and Elevation, which show them as they would appear if seen at an infinite distance through a telescope of corresponding~power, and sometimes in Perspective, which shows them as they really look. Both are methods of Projection. Other methods of projection are sometimes employed, among which the most important is Isometric Projection. This shows all three dimensions of an object laid off at any convenient scale upon lines that diverge from each other at angles of 120 degrees. 1. Fig. A A shows a building and a couple of blocks of stone drawn in Plan and Elevation; Fig. B B shows them, and a third block, in Perspective; Fig. C C shows all these in Isometric Projection. In this last, the vertical and horizontal dimensions are laid off at the same scale, and the scale does not, as in Fig. B B, diminish as the lines recede. Fig. B B looks as the building itself might look, and this is the case also, in a less degree, in Fig. C C. But here, as in Fig. A A, the aspect presented is practically an impossible one. ORTHOGRAPHIC PROJECTION 2. The method of Plan and Elevations is called Orthographic Projection. (Fig. A A.) The Spectator is supposed to be so far away that the visual rays, or rays of light that pass from the object to the eye, are parallel, and are at right angles to the vertical and horizontal planes upon which the drawing is made, which are called Planes of Projection. The Plan is drawn on a Horizontal Plane of Projection. The Elevation is drawn on a Vertical Plane of Projection, and other vertical Planes of Projection are employed at right angles to the first and to the Horizontal Plane for Side Elevations or Sections, as in the figure. The line in which the Horizontal and Vertical Planes of Projection meet is called the Ground Line, and the Horizontal Plane is soihetimes called the Ground Plane. As may be seen in the figure, the Horizontal and Vertical Projections of a point always lie in a line at right angles to the ground line. 3. Principal Planes of Projection.^-These planes, at right angles to one another, one horizontal, one vertical, and a third (and sometimes a fourth), also vertical and normal to the other two, are called the Principal Planes of Projection. ion S\ii6 Elet/a1>ion Fig. M M 1 Fig. M M 2 \c../ Fig. N N 1 Fig. N N 2 meeting at its Vertex, as in Fig. M M 2. The points in which these elements are cut by the secant plane give the same ellipse as that found in the previous figure. The other Conic Sections, that is to say hyperbolas and parabolas, can be obtained by .similar processes. IS. Figs. N N 1 and 2 show a Sphere pierced by a Ver- tical Cylinder. The lines of intersection, which are not circles unless the axis of the cylinder passes through the center of the sphere, are obtained by similar processes. V. Given, two Planes by their traces, to find the projections of their line of intersection. (Fig. O 0.) ll. Let A and B be two planes given by their traces, A" and A'", B" and B"- The point C in the vertical plane where the vertical traces meet is one point in the required line of intersection ; its horizontal projection is at C". D, in the horizontal plane, is another point in the required line; its vertical projection is at Z?". The required line CD is projected on the vertical plane at C D" and on the horizontal plane at D C", since the projection of a line must pass through the projections of all the points in the line. 70 SHADES AND SHADOWS VI. Given, a Vertical Right Cylinder by its horizontal and vertical projections, to find the traces of a plane tangent to it along a given element. (Fig. P P). 15. The horizontal trace will be tangent to the horizontal projection of the cylinder, at the projection of the given element, and the vertical trace will be parallel to the vertical projection of the element and to the axis of the cylinder. VII. Given, a Vertical Right Cone by its vertical and horizontal projections, to find the traces of a plane tangent to the cone along a given element. (Fig. Q Q). 16. Prolong the given element in both directions until it pierces both planes of projection. The traces will pass through the points thus found. The horizontal trace will be perpendicular to the horizontal projection of the given element, and the vertical trace will meet the horizontal trace at the Ground Line. Fio. p p W kt \ \ \ \ \ \. \. \ \ \. \. ^ / ./ / / / / / / \v / / / Fig. Q Q PERSPECTIVE 71 PERSPECTIVE 17. Orthographic Projections, such as the Plans, Section, and Elevations drawn in Fig. A A, show such lines as are parallel to the plane of projection, in their true proportions and directions, and these are all drawn to the same scale, irrespective of any difference in their distances from the Spectator. The angles made by such lines are accordingly shown of their real dimensions and the surfaces enclosed by the lines are of their true shapes and relative sizes. Lines that are perpendicular to the plane of projection are indicated by points, and surfaces by lines. Lines and surfaces that are incHned to the plane of projection are foreshortened, being shown smaller than they are,, in the ratio of the cosine of the angle of inclination. All lines are drawn to the same scale in eveiy part, however, and parallel lines, whether parallel to the plane of projection or inclined to it, have their projections parallel to one another. This is also the case in Isometrical Projection as exemplified in Fig. C C, and in both cases the spectator is supposed to be at an infinite distance in front of the Vertical Plane of Projection. The aspect presented iSj accordingly, in both cases, an impossible one. IS. In Reality, however^ an object looks smaller at a distance than when near; parallel lines seem to converge toward an infinitely distant point (called their Vanishing Point), and parallel planes toward an infinitely distant line, which may be called their Horizon (just as horizontal planes seem to converge toward the real horizon). Angles also generally look more acute or more obtuse than they really are. There are as many Vanishing Points and Horizons as there are different systems of parallel Unes or planes. That element of a system of parallel lines which passes through the eye of the spectator is seen endwise, and appears as a point covering and coinciding with the Vanishing Point. In like manner the element of a system of parallel planes which passes through the eye of the spectator appears as a line covering and coinciding with the Horizon of the system. The plane upon which a perspective drawing is made is called the Plane of the Picture. It is situated not behind the object represented, as in Orthographic Projection, but between the object and the spectator, and to be parallel to the spectator's face, being vertical unless something is said to the contrary. It is supposed to be transparent, the lines of the drawing covering and coinciding with the lines of the object. The spectator's eye is supposed to occupy a definite position in front of the picture called the Station Point. The Point in the Plane of the Picture nearest the Station. Point is called the Center of the Picture, or, sometimes, the Point of Sight. The Station Point is just opposite the Center of the Picture, and on a level with it. 19. The element of a system of parallel lines that passes through the Station Point pierces the Plane of the Picture at a point which is the perspective of the Vanishing Point. This point is commonly called the Vanish- ing Point in the picture. So also, that element of a system of Parallel Planes which passes through the Station Point cuts the Plane of the Picture in a line which is the perspective of the Horizon of the system. This line is commonly called the Horizon in the picture. 20. Lines that are parallel to the Plane of the Picture obviously have their Vanishing Points at an infinite distance in that plane. Their perspectives are, accordingly, parallel to each other and to the hues themselves. They are parallel, also, to the perspective of the Horizons of any planes passed through them. All the lines of a system have the same Vanishing Point, and all the planes of a system have the same Horizon. Horizontal lines that make an angle with the Plane of the Picture are called Oblique Lines. Lines that ma"ke angles also with the Horizontal Plane are called Inclined Lines. 21 . Objects drawn in perspective on the Plane of the Picture are depicted not as they really are, but as they appear from the spectator's position. The more distant objects are drawn to a smaller scale than the nearer ones, and the parallel lines which seem to converge toward their Vanishing Point have perspectives which actually converge toward the perspective of the Vanishing Point. The Perspectives of any figures that are enclosed by lines parallel to the picture are similar to the figures themselves, and the angles that such lines make with each other are shown at their true value. Notation. — The Station Point is marked 5; ObHque lines going back to the right and left, R & L; Normal lines, going to the center, C; Vertical lines, going to the zenith, Z\ Inclined lines, M & M', N & N'; Vanishing Points, V; and Horizons, H. 72 SHADES AND SHADOWS 22. Oblique Perspective. — Figs. R R 1 and 2 show the Vanishing Points of the various lines and the Horizons of the planes that occur in the two buildings shown, and in their plans, which also are put into perspective. V^, V^, and V^, the Vanishing Points of the principal horizontal lines L, C, and R, are found by means of the orthographic plan, Fig. R R 1. The lines C make angles of 90 degrees, the lines R and L make 45 degrees, with the Plane of the Picture. Since C makes equal angles with L and with R, V^ comes half way between PLAN / i> "w" XStation Point Fig. RR 1 R/ y£ H"-" Fig. R R 2 PERSPECTIVE 73 V- and V*. The Plane of the Picture is perpendicular to the horizontal plane, which, as in Orthographic Projections, is called the Ground Plane, and intersects it in the line P P. Each building has one comer in this plane, and each plan touches this line, the right and left sides R and L making angles of 45 degrees with it. If the spectator at the Station Point S looks to the right and left in directions parallel with these lines, he sees the points V* and V^ , which are the Vanishing Points, in the Plane of the Picture, of the two systems of horizontal lines which are parallel to the sides of the buildings. V^ , the center of the picture, is in like manner the Vanishing Point of the horizontal lines C, which are normal to the picture. It is, as has been said, half way between V and V^. 23. Fig. R R 2 shows the Plane of the Picture, with these same Vanishing Points V, V^, and V^, and the perspectives of the two buildings drawn upon it. The Station Point S is now in the air in front of the picture. The real horizon if^*, in which all horizontal planes seem to meet, passes through the Vanishing Points y*, V^, and V^. The line G L is the Ground Line. This is the line in which the horizontal plane on which the buildings stand cuts the Plane of the Picture. Below it is a second Ground Line, G' U, in which the picture is cut by a similar horizontal plane, lower down, on which are drawn, in perspective, the plans of the two buildings shown in Fig. R R 1, as if at the bottom of a deep cellar. 24- If the spectator at S, in the air, opposite V^, raises or lowers his eyes so as to look in directions parallel to the gable lines of the two buildings, he will see the points V^, V^', V^, V^', just above and below F* and V^, and, since these gables have naturally been given the same slope, these Vanishing Points are at equal distances from the Horizon. The vertical line H"^, is the horizon of the plane of the right-hand side of the two buildings, and it passes through V^, V^, and V"', the vanishing points of the lines R, M, and M', parallel to that plane. In like manner the horizon if" is drawn through V^, V", and V^\ the vanishing points of the lines L, N, vi- VC ^ v" VM Vf.'- s Fig. S S 1 ^g^\ Fig. S S 2 and N' lying in the left-hand side. V^, at an infinite distance in the Plane of the Picture, in the Zenith, is the Vanishing Point of vertical lines, all of which, Uke the comers of the two houses, are parallel to that plane. 26. The inclined planes of the sloping roofs have their horizons at H^'^ and H^", passing through the Vanishing Points V^ and V, V^ and V. The valley lines in which the roofs intersect lie in these planes, and have their Vanishing Points at the intersection of these horizons. 45-Degree Perspective. — When objects in Oblique Perspective are thus set at angles of 45 degrees, they are said to be drawn in Oblique Perspective. This angle is the one most generally employed. It is used in all the perspective drawings in this book. The only reason for setting buildings at any other angle would be to make one of their sides more conspicuous than the other. But this is equally well secured by setting them over to the left, or to the right, as is done in Fig. R R 2. 26. Parallel, or 90-Degree, Perspective. — In Fig. S S, the buildings have their fronts and backs parallel to the Plane of the Picture, and their ends perpendicular to it, and are said to be drawn in Parallel, or 90-degree, Perspec- tive. The Vertical Lines, the Horizontal Lines of the front wall, and the Inclined Lines of the 'front gables, are parallel to the picture. Their Vanishing Points are accordingly at an infinite distance upon that plane, and the 74 SHADES AND SHADOWS perspectives of these lines are parallel to each other and to the lines themselves. The horizontal lines at the sides of the buildings and the ridges parallel to them are perpendicular to the Plane of the Picture, making with it an angle of 90 degrees, and are parallel to the line from the Station Point to the Center V^. V^ is accordingly their Vanishing Point. The Vanishing Points of inclined lines are found as in the previous figure. They are not lettered. 27. These figures illustrate the following Propositions: I. The Horizon of a Plane passes through the Vanishing Points of all the lines that lie in the plane, or are parallel to it. Conversely, II. All the lines that lie in a plane, or are parallel to it, have their Vanishing Points in its Horizon. Hence: III. The line at the intersection of two planes has its Vanishing Point at the intersection of their Horizons. IV. A Line that is parallel to the Plane of the Picture has its perspective not only parallel to the line itself, but to the Horizon of any plane that passes through the line. For, since it lies in this plane, it has its Vanishing Point in the Horizon of the Plane, and, since it is parallel to the picture, the Vanishing Point, in which it meets the Horizon, is at an infinite distance. But lines that meet at an infinite distance are parallel. Z8. The line in which a horizontal plane intersects the Plane of the Picture is called a Ground Line. When, as in Figs. R R and S S, there are several such v«- v'^ V ^ ~" *" planes, there are several Ground Lines. 29. Fig. TT shows such a plane, extending from the Ground Line to the Horizon, and several squares lying in it, some drawn in 45-degree Perspective and some in 90-degree Perspective. These figures illustrate the following Propositions: I. The Center V^, which is the Vanishing Point of lines perpendicular to the Plane of the Picture, lies half way between the point V*, which is the Vanishing Point of lines going back to the Right at an angle of 45 degrees, and the point V^, which is the Vanishing Point of the lines going back to the Left at 45 degrees. II. A horizontal square drawn in 90-degree Perspective has two sides horizontal, or parallel to the Horizon, and the other two directed toward- the Center V^. One diagonal is directed toward V^ and the other toward V^. III. A horizontal square drawn in 45-degree Perspective has two sides directed toward V, two toward V^, one diagonal toward V, and one horizontal, or parallel to the Horizon. This makes it easy to draw in perspective any plan that is composed of squares, as is illustrated in Figs. R R and S S. I 2 3 4 6 6 Jleififht of Gable Horizon ^^ ^^ Pig. TT Fig. UU 30. Scale.— Lines lying in the Plane of the Picture are, so to speak, their own perspectives, and any required dimensions can be laid off upon them by a convenient scale of equal parts. When, as in the Right-hand Plan at Fig. R R, the diagonal of a square, lying in the Ground Plane, comes upon the Ground Line, the scale used in other Hnes that are drawn in the Plane of the Picture may be applied to it. The size of the side of the square at that scale may be fouad, as in the figure, by constructing upon this diagonal, as a hypotenuse, a right-angled iSDSceles triangle. PERSPECTIVE 75 31. The Dwision of Parallel Lines. — Lines lying in a Ground Plane, which are parallel to the Plane of the Picture, have, as do all lines parallel to the picture, their perspectives parallel to them and proportional to themselves in every part. (Figs. U U i and 2.) Such a line can be divided proportionally by means of a scale of equal parts, as at 1, and dimensions taken upon the Ground Line, as at 2, can be transferred to it by Fig. VV means of lines drawn to any convenient point in the Horizon as a Vanishing Point — lines not shown in the figure. Such converging lines are the perspectives of parallel lines, and they cut off equal dimensions from the Ground Line and from the line parallel to it. The same treatment may be applied to any inclined or vertical lines parallel to the picture that touch a Grotmd Plane at one end, as at S and 4, by drawing a line in the Plane of the Picture parallel to them. The heights of the men at 6, and of the walls and roof of the building at 6, are determined in this way. 32. The Division of Oblique Lines. — The perspectives of horizontal lines, lying in the Ground Plane, which are inclined to the Plane of the Picture, as at Fig. VV ^, or perpendicular to it, as at VV .2 (being directed toward V^ as their Vanishing Point), are drawn at a constantly diminishing scale, so that a scale of equal parts cannot be applied to them. But such lines may be divided proportionally by laying off upon the Ground Line, or upon any other line in the Ground Plane parallel to the pictxire, the proportional parts reqxiired, at any convenient scale, drawing a third line as the base of a Scalene Triangle, finding the Van- ishing Point of this base upon the Horizon, as at v^ and v^, and drawing to this point the perspectives of lines parallel to this base. These will divide the given line in the required ratio, as in these figures. 33. Division of Lines by Scale. — Any required dimensions may be laid off upon a given line lying in the Ground Plane and perpendicular to the Plane of the Picture (being directed to V^ by taking the dimen- sions upon the Ground Line ' and then transferring them to the given line by drawing the perspectives of lines at 45 degrees, going to F* or F^ (Fig. V V ^). The two Hnes of 10 feet make a right angle. This was also illustrated in Fig. S S. 34. Any required dimensions may also be laid off upon a given line lying in the Ground Plane and at 45 degrees with the Plane of the Picture, being directed to V (or V'-), by taking upon the Ground Line the "diagonals" to these dimensions, and transferring them to the 45-degree line by lines directed to V- (or F^), Fig. V V 5. This is also illustrated in Fig. R R 5. 76 SHADES AND SHADOWS 35. A better way (Fig. W W 1) is to take upon the Horizon the points Z?* and D^ at distances from F* and V^, respectively, equal to the distance of V^ or V^ from the Station Point 5. Any dimensions taken upon the Ground Line may be transferred to the 46-degree lines directed to V^ (or V^) by drawing lines from the Ground Line, across the 45-degree line, to D^ (or D^). The points D"^ and D^ are called Points of Distance, or Measuring Points. This is also illustrated in Fig. V V 4 and in Fig. R R 2. The triangles formed are isosceles, their sides, as is shown in the Orthographic Plan, being parallel to the sides of the triangle S V^D^ip^ S F^Z>^), which are isosceles by construction. The two methods are shown together in Figs. W W 1 and 2, where they are seen to produce the same result, the real dimensions of R and L being laid off to n' and m' , and their diagonals to n and m. They make it easy to draw the Perspective Plan of a building, as it is generally well to do before drawing the Perspective itself, as it gives the position of all the vertical lines. v> P" '_^''\ \ J/V H X \ Fig. XXI H'RL- V- D" Fig. WW 1 D"- V '^> \ Fig. WW 2 Fig. XX 2 36. The Points of Distance may also, as in Fig. X X, be used to find the Vanishing Point of any given inclined line, such as M (or AO, lying in a vertical plane that makes an angle of 45 degrees with the Plane of the Picture, the given line making a known angle (a) with the Ground Plane. For the required Vanishing Point, V" (or Y") will be the vertex of a right-angled triangle lying in front of the picture, the base of which will be the horizontal line from the Station Point S to V* (or V-), the side will be the Vertical line from Y^ (or Y^) to the required Vanishing Point, and the angle at the base will be the known angle. If this triangle is constructed in the Plane of the Picture, as if revolved into it, the base will occupy the Horizon from Y^ (or Y"-) to D" (or D^)\ and if the known angle is then laid off at D^ (or Z)^), the required Vanishing Point Y" (or F^ will be found just above Y" (or F^). (Figs. X X 1 and 2.) In Fig. XXI, the given line M passes up and back to the right. Fig. X X 2 shows Fig. XXI perspective. m PERSPECTIVE 77 The distance V V" (or V^ V^ is proportional to the slope of the inclined line; that is to say, it is propor- tional to the tangent of the angle a. If the slope were twice, or half, as great, the point V would be twice, or half, as far from the Horizon. 37. If, as in Fig. Y Y 1, the angle a is 45 degrees, and the inclined line N goes up and back to the left, the Vanishing Point V^ will be as far above V^ as V^ is distant or from the Point of Distance D^, or from the Station Point S. When, therefore, as in Fig. Y Y 2, horizontal lines perpendicular to the Vertical Plane of Projection cast their shadows at 45 degrees, the Vanishing Point V", toward which the perspectives of these shadows is directed, is easily found, as appears in Fig. Y Y 3. The distance V^ V^ equals the distance V^ Z?^, and is the diagonal of the distance V^ V^ or of V^ S. V2 — 1 The angle V^ V* V^ is then the angle whose tangent equals -^ = ^V2=— =. which is the angle of ^ -V2 35° 15' 52", known in Shades and Shadows as the angle (P. S8. If the shadow cast upon the vertical plane makes with the Ground Line an angle of more or less than 45 degrees (that is to say, if the Slope, or Tangent of the angle is more or less than 1), then the distance V^ Y" is altered proportionally. (See paragraph 36.) If the slope is as 2 to 1, or as \ to 1, Y" is twice as far, or half as far, from Y"- as the distance V- D'-. (See Example XXVII B.) \ \ 45° V V<= ^ Vd"- / / s Fig. Y Y 1 Fig. Y Y 2 78 SHADES AND SHADOWS TRIQONOMETRICAL TERMS 39. The angle between two lines may be defined either by Degrees, Minutes, and Seconds, as measured upon the arc of a circle whose center is at the vertex of the angle, as in Fig. Z Z 1 ; or, by the ratios between the sides of a right-angled triangle formed by these lines and a line drawn perpendicular to one of them, as in Fig. Z Z 2. These sides are called the Altitude, Base, and Hypotenuse of the triangle, and are generally lettered a, b and h, as in the figure. The ratio a:b (=r), between the perpendicular line and the Fig. ZZ 1 Fig. ZZ 2 base, is called the slope of the inclined lines. The square of the hypptenuse is equal to the sum of the squares of the base and altitude: h' = a' + b', according to the Pytha- gorean Proposition. The angle between the base and the hypotenuse and opposite the side a is generally called alpha (a) and the one opposite the side b is called beta (^). These two angles are comple- ments, one of the other, their sum being 90°: a + /? = 90°; a = 90° - /9; j3 = 90 - a. 40. There are six ratios between these three lines ; namely, ^ . ti n ^^^ the reversed or reciprocal ratioSj -, -, -. h b b a a h To these the following names have been given: - = Sine of a (or Cosine of ^) ; = Tangent of a (or Cotangent of ^) ; b h b = Cosecant of a (or Secant of ^). = Cotangent of a (or Tangent of ^). Secant of a (or Cosecant of ^) ; - = Cosine of a (or Sine of ^) . The Cotangent, Cosine, and Cosecant of an angle are the Tangent, Sine, and Secant of its complement, thus: Cot a = Tan ^. Cos a = Sin ^. Cosec a = Sec ^. Cot y9 = Tan «. Cos ^ = Sin a. Cosec /9 = Sec a. The Sine and Cosecant of an angle, the Cosine and Secant, and the Tangent and Cotangent are reciprocals of one another. The Tangent of a is the slope of the inclined line. Since fractions with the same Denominator are to each other as their Numerators, we have also these six equations: Sin Cos Cos Sin = Tan; Cot; Tan a Si? = fe = ^^"•' Sec Tan Cosec; Cosec Cot Cot h b Sec. Cosec h h a Tan a These relations are exhibited in Fig. Z Z 3, in which the radius of the circle being taken as 1, the length of the lines gives the true value of these ratios. From this figure we obtain these three equations : Sin* + Cos" = 1; Sec' - Tan" = 1; Cosec" - Cot" = 1. 41- Since a and b are both smaller than h, the Sine and Cosine are always proper fractions, being less than 1; their reciprocals, the Secant and Cosecant, are improper fractions, being greater than 1; while Tangents and Cotangents may have any values from nothing to infinity. This appears in the figure. TRIGONOMETRICAL TERMS 79 4^. It follows that if, for any angle, any one of the six ratios is known, the other five can be obtained from it. For one of the five is the reciprocal of the given ratio, and the other four can be deduced at once from these two by means of the three equations just given. {If, for example, we have: then: fSec' = 1 + Tan' « = 1 + i = _ i- ICoseC a = 1 + Cot" a = 1 + 4 = 5; 1 Cos a = Sin a = Sec a 1 Cosec a 2__ V5' Tana Cot a = i- = 2. Sec a Cosec a Cos a = ivs: Sin a = iV5: 43. Reciprocals. — ^The Reciprocal of a quantity is the quotient of 1 divided by that quantity. The product of a quantity and its Reciprocal is 1. In dealing with these ratios, it is serviceable to remember the rule that the Square Root of a Reciprocal is equal to the Reciprocal of the Square Root, and to the Square Root multiplied by the Reciprocal, thus: /l 1 1 ,- „ , ^ 1 , ,7r rr I , r:r rr 1 \« 1 1 ,- = -= = -Vw. Hence we have: A/i=^=iV2; V^=4==i>/3; -^ = --= = \-iE, &tc. V2 V3 V5 44. Solution of Right Triangles. — If the angle at the base of a right-angle triangle is known, and the length of one side is also known, the length of the other two sides can be found by multiplying the given length by the ratio between the reqttired side and the given side. If the given side is: The Hypotenuse, this ratio is the Sine or Cosine of the given angle; The Base, this ratio is the Secant or Tangent of the given angle; The Altitude, this ratio is the Cosecant or Cotangent of the given angle. These relations are conveniently set forth in Fig. ZZ 4. Whichever side of the triangle is given, the adjacent side may be foimd by using the ratio written against the nearest end of the line required. Thus: Given b: Given a: Given h: h = b sec a h = a cosec a a = h sin a a = b tan a b = a cot a b = h cos a The two adjacent ratios are reciprocals. If the length of the given line is 1, the ratio indicated gives the required length, as in Fig. Z Z 3. Cosine Tangent - Projeotipn=h Cos.c Fig. ZZ 5 Cotangent Fig. Z Z 4 46. The length of the projection of a line upon a plane is found, as in Fig. Z Z 5, by multiplying the length of the line by the Cosine of the included angle, or angle of inclination, or by the Sine of its complement. 80 SHADES AND SHADOWS 46. The angle between two lines is most conveniently given in terms of the base and altitude -r, which are at right angles to one another, as in Fig. Z Z 2, and are easily measured; that is to say, the slope is most naturally stated in terms of the Tangent of the included angle. The simplest cases are shown in Fig. Z Z 6, in which are given both the relative dimensions of the altitude, base, and hypotenuse, and the size of the angles. It will be seen that except for angles of 30, 45, and 60 degrees, simple ratios between the sides do not correspond to simple degrees and minutes. 1 1 {;: 45° 45° Tan 45° = = 1. Cot 45° = 1. ., Sin45°=^=lV2 = 4 Cos45°=-^ = iV2 = 4 Tan 26° 30' = 1 2' Sin 26° 30' = :^ = ^^=4 Cos 26° 30' = -%. 2_ V5 a =30° c -<] ^= 60° .^ a P^-. Tan30°=i= = iV3 = 4 , Sin 30° = V3 1 2" fr Cos 30°=-^=iV3. a 35° 15' 52" d ■ xl /5 54° 44' 8" y < py^ ^^ Tan (p = Sin w = Cos FiG. ZZ 6 Cosec 45° = V2. Sec 45° = V2. Cot 26° 30' = 2. Cosec 26° 30' = V5. Sec 26° 30' = i* Cot 30° = a/3. Cosec 30° = 2. Sec 30° -fva. Cot (? = V2. Cosec = V3. Sec '4 47. The angle of 35° 15' 52" (35° 16', very nearly) is known in Shades and Shadows as the angle 0. It is the angle between the diagonal of a cube and either of its faces, and is the angle which the rays of light are generally assumed to make with either the vertical or the horizontal plane of projection. 48. Trigonometrical Tables.~The values of the Sines, Cosines, Tangents, and Cotangents of all the angles from to 90 degrees have been computed, and are published in Tabular Form. These are called the Natural Sines, Cosines, etc. For convenience in computation, tables of the Logarithms of these values are also published. These are called Logarithmic Sines, Cosines, etc. ' iM^jftte ' a i^a'i'gysjJ'