150 .,_ _^^ . Cornell University Library NC 750.L92 1921 Psrapeotive; an elementary text book, by B 3 1924 020 559 344 Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924020559344 PERSPECTIVE AN ELEMENTARY TEXT BOOK BY BEN J. LUBSCHEZ FELLOW OF THE AMERICAN INSTITUTE OF ARCHITECTS AUTHOR OF "over THE DRAWING BOARD, A draftsmen's hand book" THIRD EDITION, ENLARGED NEW YORK D. VAN NOSTRAND COMPANY 8 WARREN STREET 1921 B76 a.. Copyright, 1913, 1915, 1921 D. Van Nostrand Company THE DE VINNE PRESS PREFACE This book is intended principally for the strug- gling student who is endeavoring to better him- self by home study and who can get but little assistance, if any, either personal oi" from books, the latter usually being too difficult for him to read. It is intended to give him a beginning so that he may be able to solve the ordinary prob- lems of every-day practice, and, as stated in the introduction, to qualify him for the reading and study of the more profound books on the sub- ject. The general discussions may also be read with profit by laymen who wish merely a gen- eral knowledge of the science. The text is the result of many years' experience in making per- spective drawings in an architect's office and in jteaching the science to beginners. The manu- script was tried, with marked success, by sev- eral novices who had not before known any- thing about the subject, and who were enabled after a careful study of the book to lay out an ordinary perspective accurately and without help. If, in some measure, this book succeeds in do- ing this for others, if it lightens the task and straightens the road for but a few, the satisfac- PREFACE tion of the author will far exceed that of those helped. The student is assumed to have some know- ledge of plane geometry, but this, though de- sirable. Is not absolutely necessary. It is, of course, essential that he have some familiarity with architectural or mechanical drawing. B. J. L. Kansas City, Missouri, December tenth, nineteen twelve. Civ] PREFACE TO THIRD EDITION The encouraging success of the two previous editions of this little book has made it seem de- sirable to issue a third. The new material in the second edition was incorporated principally at the suggestion of the late Professor Frank Dempster Sherman of Columbia University, A short description of the use of circumscribed octagons as an aid in drawing perspective circles has been included in this edition, in view of a later suggestion by Pro- fessor Sherman for increasing the usefulness of the text. A chapter on the history of perspective draw- ing has also been added in this issue. The data for this chapter, short as it is, was gath- ered from scattered sources and after the examination of many old volumes. Usually this historical material would be placed first as an introduction, but in keeping with the original plan of the book, which was based on the natu- ral laboratory method of teaching, it has been placed last, on the assumption that the student would, after an acquaintance with the theory and principles of perspective, be much more in- terested in its history than before. The popularity of this little book has been the source of much gratification to the author, who hopes that its continued usefulness will more than justify this republication. B. J. L. New York City, May, nineteen twenty-one CONTENTS % PAOB I Introductory 1 Sc'ope and plan of the book. II Preliminary 3 Seeing in perspective — Picture plane — Defini- tion of perspective — Photograph and perspec- tive — ^Vanishing points — Horizon — Foreshort- ening — Size of image — ^Vanishing points of horizontal lines — Position of picture plane. III Making a Drawing in Perspective . . 10 Elevations and perspectives — The problem — Diagram of working points — Location of the point of station — Finding the vanishing points — Finding the measuring points — The per- spective plan — The perspective projection or picture — Redrawing without instructions. IV Second Problem in Perspective ... 23 To be solved in same general way as the first problem — Statement of problem and discus- sion of new and difficult points — The perspec- tive of points in front of the picture plane — Any problem in perspective may be solved if the perspective of any point in space can be found — Different methods of finding the per- spective of the same point. CONTENTS FACE V Vanishing Points and the Point of Station 30 Locating the point of station — Obliquity of view and foreshortening — Crowding of lines as they approach the horizon or vanishing points — Planes and their vanishing lines — Horizontal planes and the horizou — Oblique and inclined planes and their vanishing linesr— Vanishing lines and vanishing points — Van- ishing points of oblique lines — The vanishing point of any line and explanation of construc- tion for finding vanishing points — Two van- ishing points for every line — Conjugate and reciprocal vanishing points — Tri-conjugate vanishing points. VI Measuring Points and Scales ... 42 Scales — Front lines — Measuring points, what they are, how they are found and why — Spe- cial cases. vii Parallel or One-Point Perspective . 47 Conditions of — Vanishing points and measur- ing points — Definition and limits of — Special case of ordinary perspective with conjugate vanishing points — Problems in parallel per- spective — Use. A Table of Conjugate Vanishing Points WITH Their Corresponding Measur- ing Points for Different Angles of View facing page 53 viii Special Manipulations and Short-Cuts 53 Perspective centers — Plotting and craticula- tion — Circles and curves — Miter points and lines — When vanishing points are off the board — When measuring points are off the C viii 3 CONTENTS PAOK board — ^When the point of which we wish to find the perspective is off the board — Frac- tional measuring points — When the distance between vanishing points is limited — Discus- sion of so-called distortion. IX Variations 65 The method of direct projection — Compari- son of method of direct projection and method of perspective plan — ^Laying out per- spectives without recourse to vanishing points by direct projection and by ordinates and - parallel perspective — Other methods — Discus- sion. X Oblique and Inclined Lines and Planes 79 Definitions — Discussion — ^Vanishing points — Horizons and vanishing lines — Measuring points — Measuring lines — Problem — Explana- tion. XI The Perspective of Shadows (By Sunlight) 87 General conditions — Theory of shadows — Shadow of a point on any surface — Shadow of a straight line on a plane — Shadow of any line on any surface — Plotting shadows — Prob- lem — Light rays and light surfaces — ^Vanish- ing points of shadow lines by intersection of horizons or vanishing lines — Further solution of problem. XII The Perspective of Shadows (By Artificial Light) 95 Similarity to and differentiation from shadows by sunlight — Shadow lines drawn to point at center of source of light — Problem — Lines and their shadows on planes parallel to them — Further solution of problem. CONTENTS PAGI! xiii Who Discovered the Rules of Per- spective? 101 Early observation of perspective phenomena — Fundamental phenomenon of perspective — Vitruvius — Scenery for jEschylus by Agathar- cus— Albrecht Diirer — Leonardo da Vinci — Dr. Brook Taylor and the first complete treatise on perspective — Andrea Pozzo — Thomas Malton— Modern books. t^l INTRODUCTORY THERE are so many books on Perspective that an apology is necessary for inflicting another. The books, however, with which the writer is familiar, although excellent texts, usually do not begin at the very beginning, and the inexperienced student cannot read them un- derstandingly. Not so many years ago, the writer was a student-beginner himself, and he still remembers quite vividly the yearning for knowledge at that time. This yearning, in so far as concerned Perspective, was not for the knowledge of the science and theory of the sub- ject, but rather for the ability to make a per- spective drawing. So in this little book, after a preliminary talk and a few observations on Per- spective and its phenomena, we are going to make, you and I, a perspective drawing; we shall work together step by step and line by line. Then you will make another perspective draw- ing, using the same methods that we used to- gether, but without the explicit instructions. Having done this, we hope that your interest PERSPECTIVE will have been sufficiently aroused so that you will want to know why you did what you did, and then, again step by step and line by line, you will be told why. You will be told enough only, so that by deductive thinking you may be able to work out the ordinary problems encoun- tered, or, better yet, so that you may be able to read some of the admirable existing texts on the subject. It is not the intention to encroach on the field of such excellent books as Ware's "Modern Perspective" or Longfellow's "Applied Perspective." This, in brief, is the plan of this little book. It follows what may be called the "laboratory method" of modern teaching. Some of us have found that the way to learn, and the way we do learn, a new language is by speaking and read- ing it first, and then studying its grammar and rhetoric. So we shall do with Perspective. We shall learn first to make a perspective drawing — indeed, we shall make one or two — then we shall study its grammar and rhetoric. 1^1 II PRELIMINARY EVERYTHING we see, we see in perspective. Every image in the eye is in perspective, a perspective projection; every photograph is in perspective and a perspective projection. If the student should erect a glass screen between him- self and an object such as a building and run imaginary lines from his eye to various points on the object, say the corners, where these lines would pierce the glass screen would be the per- spective projections of the corresponding points on the object. If we now connect these projected points on the screen by lines, we shall have a picture or perspective projection of the object on the screen. We must always remember this : the perspective projection of an object is a pic- ture, an image of it on an assumed plane. It is with the construction of this image or projec- tion, without actually making the projections of the various defining points, that the science of Perspective has to deal. Let us get back for a moment to the glass screen. In the parlance of Perspective this screen is called the picture plane; the position of the observer's eye, the ZS2 PKTVRE PLA;1E BETWEEN OBJECT& T PJCTVEE PLANE BEHIND y A«AI.oa3linD CAMERA. 1:43 PRELIMINARY point of station. Although not necessarily so always, the picture plane is usually assumed to be vertical.' We are now ready for an explicit definition of a perspective (Figure 1). A perspective of an object is the projection of the defining lines and points (A, B, C . . .) of that object on a plane (PP) called the picture plane. This projection is formed by drawing lines from the defining points (A, B, C . . .) on the object to a point {$) called the point of sta- tion. The intersections (A', B', C . . .) of these lines {SA, SB, SC . . .) with the picture plane (PP) define the projection or so-caUed perspec- tive (A'.B'.C). A photograph is a true! perspective In Figure 1, if we continue the lines SA, SB, SC, etc., through the point of station, S, to a plane paral- lel to the picture plane, PP, as P'P', and inter- secting P'P" in A", B", C" . . ., the inverted im- age {A", B", C" . . .) thus formed is similar to the projection A', B', C . , t on PP, and is anal- ogous to the image formed on the ground glass or focussing screen of the camera. The ground glass of the camera corresponds to P'P', and the lens of the camera to S, the point of station. Having shown that a photograph is a true per- spective projection, we shall now study a photo- graph and note some of the peculiar phenomena of perspective (Figure 2). We see that vertical lines on the object or building remain vertical in the photograph. This we must remember. As long as the picture plane is vertical — and we PERSPECTIVE shall not consider cases where it is not — verti- cals in the object will remain vertical in the pic- ture. We see also that horizontal lines in tile object converge in the photograph. We should note that when these horizontal lines are paral- lel, such as those lying in the same or parallel sides of the building, they converge in the same point, as V and V. These points, V and V, are called the vanishing points for the lines which converge in then : Note that the line HH drawn through V and V is horizontal. This line is called the horizon of the picture. It is the neu- tral line of the perspective and is always on the level of the point of station or observer's eyefi All horizontal lines below the level of the hori-. zon incline upward toward it, and all horizontal lines above the level of the horizon incline downward toward it. The vanishing points C'V, or to locate the point B, the perspective of B', we draw a line from B' to M; where this line intersects C'Y' is B. Likewise, by drawing a line from A' to M' where it intersects C'Y we get A. The points M and M' are each really, as we shall prove later, the vanishing points of paral- lel lines which make equal angles with CB and CB' and CA and CA' of Figure 7. In other words, tlie triangle C'BB' of Figure 8 is the per- spective of the triangle CBB' of Figure 7, whose angles, B and B', are equal. The angles, B and B', of Figure 8 are therefore the perspectives of equal angles. Since this is true, the triangle C'BB', Figure 8, is the perspective of an isos- celes triangle, and therefore the lines C'B and PERSPECTIVE C'B' are the perspectives of equal lines. Like- wise, AC and A'C are the perspectives of equal lines. The proof of the construction for M and M' wiU be given later. We take actual scaled distances on P'P' and the line of measures because both these lines really lie in the picture plane, P'P' being the perspective of PP, Figure 7; and we must al- ways remember that a line which lies in the pic- ture plane does not change its magnitude in per- spective, because if it lies in the picture plane it is obviously its own projection in that plane. As we found B and B' in Figure 8, so can we find all the points defining horizontal dimen- sions, such as a, &, c . . . and p, q, r . . . We have but to locate a', b', c' . . . and p', q', r' . . . on P'P' at actual scaled distances either side of C, as per our plan, then draw lines to M and M' respectively and find their perspectives on C'B and C'A at the intersections a, 6, c . . . and p,q,T... From the points a, b, c . . . B and p, q, r . . . A we draw lines to the vanishing points V and V respectively. These lines and their in- tersections will give us the lines and points of the perspective plan, or the plan of Figure 6 in perspective. Before we leave the perspective plan, we must recall the following: a line drawn through, a point on P'P', a given distance from C, to M' will cut off the same distance from C in per- spective on C'A or its continuation; a line ni63 MAKING A DRAWING IN PERSPECTIVE drawn through a point on P'P', a given distance from C, to M will cut off the same distance from C in perspective on C'B or its continuation. To illustrate : In Figure 8, C'r ^perspective of C'r' C'c =perspective of C'c' C'a"=perspective of C'a' C'g'"=perspective of C'q' It is interesting to note that a" and q" lie in front of the picture plane. The finding of these points on the continuations of C'A and C'B then gives us a method of finding the perspectives of points when they lie in front of the picture plane. The perspective plan is now complete, and we must be sure that we understand how we drew every line before we proceed. Of the picture in Figure 8 we have already drawn the horizon, HH; we have located on HH the points V, V, M, M' and C; we have drawn through C the line of measures and have located on it the point G, 5 feet 6 inches below C and the horizon. The numbered points referred to below correspond to homologous points on the elevations (Fig- ure 6). Let us now project up from the perspective plan points a, g, B and A, the verticals 2-a", 3-3", 4-B" and 16-17-A". On the line of measures place point 1, eigh- teen inches above G. All points on the line of ni7a PERSPECTIVE measures to be laid off at the adopted scale, of course. From G and 1 draw lines to V. The inter- sections of these linfes with the perpendiculars locate the points 2, a", 3, g". 4, B". Locate on line of measures points 7 and 8, six and twelve inches respectively above G. From 7 and 8 draw lines to V, intersecting 3-flr" ^t 9 and 10. From 2 and 3 draw lines to V. Locate 5 over 5' of plan. From 5 draw line to V, intersecting 3-6 at 6. 6 should be over 6' of plan, a test for accuracy. Draw verticals 6-15, 13-14, and 11-12 over corresponding points of plan. From g", 10 and 9 draw lines to V, intersect- ing these verticals at 12, 11, 14, 13, 15. From 12, 11, 14, 13, 15 draw lines to V, which define the edges of steps. We have now com- pleted the steps and bulkheads. From 1 and 7 draw lines to V, intersecting the vertical 16-A" at 16 and 17. From 16 draw line to V, and from 4 draw line to V. Fill in curve of terrace, r"-'G, free-hand. This completes the steps and platform upon which the monument stands. On the line 1-2-3-4 locate 18' over 6 of plan, and draw vertical 18'-19'. On the line of measures locate 19" six inches above 1, and from 19" draw line to V, inter- secting 18'-19' at 19'. ni83 MAKING A DRAWING IN PERSPECTIVE On line 19"-19' locate 24' over c of plan, and draw vertical 24'-25'. On the line of measures place 25" twelve inches above 19", and through 25" draw line to V, intersecting 24'-25' at 25'. Locate 30" on line of measures six inches above 25", and through it draw hne to V. On this line place 30' above d of plan and draw vertical 30'-44'. Locate 44" on line of measures 9 feet 3 inches above 30", and draw line through it to V. On this Une, which intersects 30'-44' at 44', place 50' over e of plan and draw vertical 50'- 47'. On line of measures place 47" nine inches above 44" and draw line through it to V, inter- secting 50'-47' at 47'. Go back to 25' and through it draw a line to V. On this line place 33' above / of plan and draw vertical 33'-35'. On line of measures place 35" six feet above 19". Draw Une through 35" to V, intersecting 33'- 35' at 35'. Draw verticals 18-19 and 22-23 over corre- sponding points of plan. Draw lines from 18' and 19' to V, intereecting these verticals at 18, 19, 22, and 23. Draw vertical 20-21 over corresponding point of plan. From 18 and 19 draw lines to V, cutting 20-21 at 20 and 21. CIS] PERSPECTIVE From 21 draw line to V, and from 23 draw line to v. Draw verticals 24-25 and 28-29 over corre- sponding points of plan. Through 24' and 25' draw lines to V, cutting these verticals at 24, 25, 28 and 29. Through 24 and 25 draw lines to V. Draw vertical 26-27 over corresponding point of plan, cutting these lines in 26 and 27. Draw verticals 30-44, 31^5 and 32-46 over corresponding points of plan. Through 30' and 44' draw hues to V, cutting 30^44 and 3^-46 in 30, 44, 32 and 46. Through 30 and 44 draw lines to V, cutting 31-45 in 31 and 45. Through 47' draw line to V. Draw verticals down from this line at 47 and 49 over corresponding points of plan. From 47 draw line to V. Draw vertical down from this line at 48 over corresponding point of plan. Draw miter lines of bevel base, 30-25, 31-27 and 32-29. On line 25-27 locate 33 and 34, and on line 30-31 locate 38, over corresponding points of plan. Through 33, 34, and 38 draw verticals 33-35, 34-36, 38-37. From 35' draw line to V, cutting 33-35 and 38-37 at 35 and 37. From 35 draw line to V, cutting 34-36 at 36. Draw bevel line 33-38. 1:203 MAKING A DRAWING IN PERSPECTIVE Through 25" on line of measures draw line to V. On this Une place 25'" over w of plan and draw vertical 25'"-51'. Place 51 on line of measures, two and one- half feet above 19". Through 51 draw line to V, intersecting 25"'- 51' at 51'. On line 25-29 place 39 and 40, and on line 30- 32 place 52, over corresponding points of plan. At 39, 40 and 52 draw verticals. Through 51' draw line to V, cutting 39-41 and 52-43 at 41 and 43. Through 41 draw line to V, cutting 40-42 at 42. Draw lines from 43 to V and from 42 to V. Draw bevel line 39-52. This completes the perspective. This perspective illustrates in a striking way what was said before, that a perspective is the combination of contiguous perspective eleva- tions or their projections. This is readily seen in Figure 8. Of course, in practice it is not nec- essary to complete the perspective elevations; a few defining points only are necessary, and as you proceed and learn to solve perspective problems more readily, many short-cuts will occur to you. You must remember that the varied application of a few primary princi- ples is all that is necessary to solve most prob- lems in perspective. It is the variety and readiness of application rather than variety of principles that you must master. The sim- 1:213 PERSPECTIVE pie perspective you have just completed involves nearly all these necessary primary principles; therefore the foregoing must be thoroughly un- derstood in every step. Accuracy in drawing is also most essential. Usually it takes several pro- jections to find a point in perspective; a small error in the first place may be magnified into a large one by the time the final projection is made, and a very small discrepancy will often mar the appearance of a perspective picture. After the drawing like Figure 8 has been made, carefully following the instructions, it should be redrawn from memory, using Figiu-e 6, the plans and elevations only. This is very important. When this has been done readily and correctly you should be ready for the next problem, which is stated in the next chapter. 1:223 I IX, FIGVRi: 10. To face p. 3j — Luhschez *^ Perspective'^ ite/ IV SECOND PROBLEM IN PERSPECTIVE OUR second problem in perspective is much like the first, and except for a few points, may be drawn very easily by the student if he has mastered the other. The problem is to make a perspective drawing of the monument shown in Figure 6, but under different conditions than those of Figure 8. As stated below and shown in Figures 10 and lOA: The picture plane shall be drawn through the comer of the base-step of monument, and the point of station shall be thirty feet from this point; the axis shall make an angle of 115° with the long side of base-step; the point of station and the horizon to be twenty feet above the ground. Under these conditions will be pro- duced a so-called bird's-eye perspective. (See Figures 10 and lOA.) The making of this perspective involves no new principles in so far as the monument is con- cerned; this should be done in exactly the same way as formerly. The large platform and steps CSS] PERSPECTIVE upon which the monument stands will involve something new, however, for these are partly in front of the picture plane. The finding of the perspective of points lying in front of the pic- ture plane has been lightly touched upon in the finding of a" and q" in Figure 8. We must now take this matter up very carefully, for this prob- lem often occurs in perspective and is usually very puzzling to the beginner. We must first make a diagram, Figure lOA, similar to Figure 7, to find V, V, M, M', and C according to the new conditions. Having found these, we lay ofi" H~H for the picture and on it locate y, V, M, M' and C as before. We then draw through C, the line of measures, C-1&-G', and on it we place, according to hypothesis, G', twenty feet below C. We next locate the point 18, eighteen inches above C; this gives us the bottom point of base-step of monument. The perspective plan of the monument proper may be drawn by precisely the same methods as we used to draw the perspective plan in Fig- ure 8. After locating 18' conveniently on the line of measures and drawing through it the line P'P' and lines to V and V. 18'-X and 18'-Z, we can lay off the scaled distances on P'P', fore- shorten on 18'-X and 18'-Z to M and M', and draw lines through the newly found points to V and v. Our next concern is to find such points as S, C, b and B. We must remember two things : first, that we lay oif actual distances on P'P' and by foreshortening to M we get the 1:243 SECOND PROBLEM IN PERSPECTIVE perspectives of these distances on S-18'-X, by foreshortening to M' we get the perspectives of these distances on 6-18'-Z; second, that the measuring points M and M' may be used to find the perspective distances only on lines like S- 18'-X and 6-18'-Z, lines drawn to Y and V through 18', the projection or plan of C We can readily see by a glance at the perspective plan of Figure 10 that if we locate S and X, Z and h, and through them draw lines to V and Y', these lines will be the sides of the platform, and their intersections the corners. X and Z may be found in the usual way by locating X' and Z' on P'P', drawing lines through them to M and W, intersecting 18'-20'-X at X and 18'-22'-Z at Z. Point 5 lies in front of the picture plane to the left of 18'. We lay off S', five feet to the left of 18' on P'P'. Since S lies on S-18'-X, we draw a line through S' to M, intersecting S-18'-X at S. In the same way we find a", the perspective of a', three feet from 18'. We now find 6, which lies in front of the pic- ture plane and to the right of 18'. We lay off h', nine and one-half feet to the right of 18' on P'P': and since 6 lies on fc-18'-Z, we draw line through 6' to M', intersecting b-lS'-Z at b. Likewise we find p, q and r, the perspectives of p', q' and r', nine, eight and seven feet respec- tively from 18'. We place g' on P'P', two feet from X', and foreshorten to M on 18'-X to find g". We now draw lines through S, a", g" and CSS] PERSPECTIVE X to V, and through Z, r, q, p and b to V, to complete the perspective plan. On the perspective itself we already have 18 and G'. Through 18 and G' we draw lines to v. Draw the verticals I'-G" and 2'-fl"' over S and a" of plan and intersecting the lines through 18 and G in 1', 2', G" and a'". On the line through 18 to V locate 3' and 4' over g" and X of plan. Draw verticals through 3' and 4', locating 12' on line through G'. Six inches and twelve inches below 18 on line of measures, place 7' and 8', and through 7' and 8' draw lines to V, locating 9 and 10 on vertical 3'-12' and 7" on I'-G". Through 1' and 4' draw lines to V. Draw verticals 16-17, 1-G, 4-B' over A, C, B of plan, and through 16, 1 and G draw lines to v. Draw verticals 2-aa and 3-g'" over a and g of plan, and through 2 and 3 draw lines to V. Draw vertical 6-15 over 6' of plan, and through 6 draw line to V, intersecting 2-5 at 5; 5 should be over 5' of plan. Draw verticals 13-14 and 11-12 over 13' and 11' of plan. Through 9, 10 and 12' draw lines to V, locat- ing 13, 14, 11 and 12. The line through 12 should intersect S-g'" at g'". Through 15, 13, 14, 11 and 12 draw lines to V, the edges of the steps. Through 7" draw line to V, locating 17; draw SECOND PROBLEM IN PERSPECTIVE curve of terrace free-hand. This completes the platform and steps. The problem should be gone over thoroughly and drawn without the instructions as many times as necessary to fix thoroughly in the mind the method of finding the perspective of points when they lie in front of the picture plane. No other one thing in perspective seems to bother and puzzle the beginner as much as this. We locate a point in perspective by consider- ing the point as the intersection of two lines which are parallel to and at a known distance from two afready known lines. For example, C of the plan in Figure 10 is the intersection of C'A and C'B. parallel to 18'-Z and 18'-X and five feet and nine and one half feet from these lines respectively. 18'-Z and 18'-X are perpen- dicular to each other in reality, so S-18', five feet, is the perpendicular distance of S from 18'-Z. A line drawn through S to V is parallel to 18'-Z, which is also drawn to V. Likewise, &-18' is perpendicular to 18'-X and nine and one half feet from it. C'-b-B drawn to V is parallel to 18'-X. also drawn to V. The inter- section of these two lines gives us C. So we may find any point in perspective. The two lines 18'-Z and 18'-X may be called the funda- mental lines of the perspective. They are the starting lines, the lines upon which we build our perspective. To find any point in perspec- tive, let us look at our actual plan, and after determining the two fundamental lines, usually 1:273 PERSPECTIVE the lines of the two principal systems of lines used — that is, those going to the two principal vanishing points — ^passing through the point which is the plan of the line of measures, we draw lines through the point in question, paral- lel to the fundamental lines, and find the dis- tance of these lines from the fundamental lines; we can then, as we did a little while ago, draw the perspective of these lines, and their inter- section will be the perspective of the point de- sired. We can thus find the perspective of any point. By finding the perspective of any two points on it, we can find the perspective of any straight line. By finding the perspective of enough points on it to define it, the perspective of any line, whatever its curve, may be found. So, in fact, if we can find the perspective of any point, we can find the perspective of any line, and therefore any object, by the finding of its defining lines. When we determine a line by drawing through a known point on it to the van- ishing point, we determine the line by two points on it — one the point through which we draw, and the other the vanishing point. The whole science of perspective may be boiled down to the finding of the perspective of any point in space. The student should understand this very clearly, and by this time should be able to find the perspective of any assumed point under any conditions. The student should examine his drawings. Figures 8 and 10, very carefully and note the CSS] SECOND PROBLEM IN PERSPECTIVE different ways in which the same point may be found. Let us take the point 5, Figure 10. We found it by drawing from 6 to V and from 2' to V, intersecting at 5. We might have found it by going from r'" to V and from 2' to V, inter- secting at 5; or by drawing from 2 to V and plac- ing 5 on this line over 5' of plan, likewise by drawing from 6 to V and placing 5 over 5'. There are usually many ways of locating the same point, although all these ways amount to the same thing and give the same result. The student, with practice, will use the most con- venient way almost intuitively. It is excellent practice to analyze both Figures 8 and 10 to dis- cover the various ways in which the principal points may be projected and located. cas: VANISHING POINTS AND THE POINT OF STATION THE fundamental point of a perspective is obviously the point of station. This point, as its name implies, is the one where is the observ- er's eye. Upon its location depends, more or less, the position of every working point in the draw- ing. The I >int of station has, first, a plan posi- tion and, second, a position of elevation or height. Its plan position determines the distance apart of the vanishing points and measuring points. Its elevation determines the height of the horizon. As has been said before, the proper choice of the point of station determines the artistic value of the picture, and the ability to make this choice properly depends almost en- tirely on the judgment born of experience and observation of nature. As a hint for architec- tural perspective, it is usually best to avoid lo- cating the point of station so that the obliquity of the two sides of a building is the same; it is usually better to have one side — if there is any choice, the less important one — foreshortened VANISHING POINTS very perceptibly more than the other, giving pleasing contrast to the angles of the drawing. The height of the point of station, as has been said before, determines the height of the hori- zon. The horizon is always at the level of the eye of the observer. If he moves up, the horizon moves up with him; if he moves down, it moves down with him. This is obvious and a matter of daily experience to all of us. The point of station should not be taken too near the object. If it is, the vanishing points and measuring points are too close together and the whole per- spective looks forced and strained — we get the same artificial effect as we do in photographs taken with an extremely wide-angle lens. Nei- ther should the point of station be taken too far away, unless a telescopic or very distant effect is desired. Its distance should depend on the size of the object. A good way is to place the point of station at such distances as to make lines drawn from the extremities of the object to it about 50° or 60° apart. From the artistic standpoint, the point of station is not well located in Figure 10, It was placed as it was simply to bring out some new principles in per- spective, and to do so without sacrificing com- pactness of drawing. We must note that a line may decrease in per- spective from two causes: one, the angle it makes with the picture plane, and again on ac- count of its distance from the point of station. The greater the angle with the picture plane, the PERSPECTIVE shorter the line becomes in perspective; the greater its distance from the point of station, the less its perspective length. Both of these facts depend on the principle previously given, that the size of an image depends on the angle subtended at the eye (Figure 3). Both these facts are well illustrated in Figure 2. The prin- cipal windows in the photograpih of Figure 2 on both ends and front are the same width, yet the end ones are much narrower in the picture than the front ones because the end makes a much larger angle with the picture plane than the front. The windows on the front at the ex- treme right are narrower than those at the left because of their greater distance from the point of station. We should also notice an apparent crowding of lines as we approach the vanishing points or the horizon. This can be seen to some extent in Figure 2, but more easily in Figure 11, where FIGVILE 11. the first three squares on either side and the top occupy more space than the other five. There is an apparent crowding of lines on top towards VANISHING POINTS the horizon and at the sides towards the vanish- ing points. In Chapter II we explained that two parallel lines have a common vanishing point on ac- count of the apparent decrease of distance be- tween them, this distance becoming infinitesi- mal, or a point. Let us consider two parallel planes. Let us imagine in these parallel planes two parallel lines, one in each plane, having, of course, a common vanishing point. Let us now imagine another pair of lines oblique to the first but parallel to themselves; these will also have a common vanishing point. So we can imagine many pairs of lines and get a series of vanish- ing points. These vanishing points will define a line, and this line is the vanishing line of the two parallel planes. So each system of parallel planes has a common vanishing line, and all parallel lines lying in these planes have their vanishing points in the vanishing line of the planes. This is an important principle in per- spective. It is for this reason, as has already been explained in Chapter II, that all horizontal lines vanish in the horizon, the vanishing line of horizontal planes. A vertical plane, oblique to the picture plane, will have a vertical vanish- ing line. If in this vertical plane we draw a horizontal line, its vanishing point will be in the horizon. A vertical line drawn through this vanisidng point will be the vanishing line of the plane, because vertical planes must meet in a straight vertical line; the vanishing point of any ~ CSS] PERSPECTIVE line in these planes must be in the vanishing line of these planes; and since but one vertical line can be drawn through any given point, the vertical line drawn through the vanishing point of the horizontal line, and which point is in the vanishing line of the vertical plane, as above stated, must be the vanishing line of the vertical plane. By this principle we may find the van- ishing points of oblique lines lying in known vertical planes, such as gable lines, diagonals, inclined grade or sidewalk lines against build- ings, and so on. In Figure 12, 1-2-A-B-C-D is the perspective of a cube. A vertical line, V'd-V'-V'd', through V is the vanishing line of the plane ABCD. If we draw the diagonal BC in this plane, its vanishing point must lie in the vanishing line of its plane, or V'd-V'-V'd', so if we continue BC to its intersection, V'd, with this vanishing line, we get V'd, its vanishing point. By the use of this vanishing point we may draw the diagonal DE, which is parallel to BC, thus locating E; draw the vertical EF, then the diagonal FG, and so on. We draw any number of successive rectangles in perspective without the use of the perspective plan after we have drawn the first and its diagonal. Likewise, the point Vd may be found and the squares or rectangles on the other side drawn. It is also obvious how V'd' and Vd' may be found and used similarly. Figure 124 shows an applica- tion of this principle to architectural perspec- tive in drawing a triple-gabled house. We may ITS*: PERSPECTIVE find the distance AB by perspective plan in the ordinary way, draw the vertical BC, finding C from C, thus getting the height of the gable. We may now draw the gable line, AC. By draw- ing the vanishing line Vd-V'-V'd' and continu- ing AC, we find Y'd, the vanishing point of AC and its parallels. We may similarly find Vd', the vanishing point of CA' and its parallels. We may now draw EF and A'D, DG and so on. The same method is used in drawing such inclined planes as sidewalks, streets and roads when these are not horizontal (Figure 12B). One line can always be established, and its vanishing point, corresponding to Vd or Vd, found; the lines parallel to it may then be drawn at the proper distances. There are many other applications of this principle, and the occasions for its use are al- most innumerable. The vanishing point of any line may be found by looking along that line ; this is obvious, for if we look along a line its image is a point — ^its vanishing point. The line whose vanishing point we wish to find, however, may not pass through the point of station, so we cannot look along it. If, then, we look along a line passing through the point of station and parallel to the first line, we easily find the required vanishing point, for we find the vanishing point of its parallel, and parallel lines have common vanishing points. This is why, in Figures 7 and lOA, we draw lines through the point of station, parallel to the sides Z362 n373 PERSPECTIVE of the object, to get the vanishing points of lines in the sides of the object. This is further shown in Figure 13, where A-B-C-D-E-F-G is an ob- ject in space; hh, its actual horizon, and vv', the actual vanishing points of its two systems of lines; PPPP, the picture plane; and S, the point of station. The lines Sv and Sv' are drawn through S parallel to the systems of lines of the object, hence going to the vanishing points, v and v'. These lines intersect the picture plane (represented by the line PP' in Figure 7) in V and V, the vanishing points of the picture. These lines, obviously, lie in a horizontal plane through 5, which intersects the picture plane in the horizontal line HH, the horizon of the pic- ture. V and V, points in this horizontal plane and in the picture plane, must lie in the inter- section of the two planes, HH. This figure tells its own story quite plainly, and should be very carefully studied by the student; the more he studies it, the more procedures in his making of a perspective will be explained. When you looked along a line to see its van- ishing point, it perhaps did not occur to you that there was a vanishing point for that line behind you as well as in front. If you were looking north along a north and south line, you would see the vanishing point of the line in the north, but there would also be a corresponding vanish- ing point for the same line in the south, 180° away. This second vanishing point is almost never used, and is of practically no importance. cssn VANISHING POINTS In panoramic or curvilinear perspective — a sub- ject which we need not discuss here — an at- tempt has been made to utilize the two vanish- ing points of a line. Note, in a panoramic pic- ture when flattened out, the convergence or curvature of horizontal lines to points at each end. Special relations sometimes exist between vanishing points, and some of these we shall now discuss. When two points are the vanish- ing points of two systems of lines which are per- pendicular to each other — that is, when the di- rection of one system is perpendicular to the direction of the other — then the vanishing points are called conjugate vanishing points. Most cases of architectural perspective employ con- jugate vanishing points because the planes of '^buildings are usually perpendicular to each other, hence the line systems are perpendicular to each other. Of course this is not always the case, and there is really little use in differenti- ating conjugate vanishing points from others. There are also reciprocal vanishing points, which are the corners of an equilateral triangle of which the point of station is the apex; they are 60° apart and are sometimes used for the drawing of hexagons in perspective under spe- cial conditions. The reader at this stage of his study need not trouble himself about them. The most interesting of special cases is the one of tri-conjugate vanishing points. If we place a rectangular block on one of its corners. PERSPECTIVE so that none of its line systems is parallel to the picture plane, then there will be three systems of lines perpendicular to each other and haAdng three vanishing points and three horizons. The same will occur if the object or block is placed in the ordinary way and the picture plane is in- nCVRX 14. 1:403 VANISHING POINTS clined or out of plumb or vertical. The verti- cals on the object converge in the perspective; this happens in a distorted photograph when the camera has been tilted up or down and the ground glass has not been made vertical. We have all seen such photographs. The three van- ishing points in such cases are conjugate in pairs and the whole group is called tri-conju- gate. Tri-conjugate vanishing points are not of any great practical use, and need not be taken up by the student now. (See Figure 14.) 1:41: VI MEASURING POINTS AND SCALES As has already been shown, there are lines in . a perspective drawing which are their own perspectives and on which distances may be scaled. Such lines must lie in the picture plane and may be called the scales of the perspective. It may be mentioned here that lines which lie in the picture plane and are their own perspec- tives are called front lines, and all perspective scales are front lines. Such lines are the line of measures and P'P' in Figures 8 and 10. To throw scaled distances in perspective, we must draw to the vanishing points, or to certain points called meas^uring points (which are special van- ishing points), or to both. We stated in Chap- ter III that the measuring points were the van- ishing points of parallel lines which made equal a;ngles with a front line and another line passing through the center C, or, as shown in Figure 15, lines that make equal angles with CB' and CB or with CA' and CA. In Figure 15, which is a dia- gram similar to Figure 7, let us draw the paral- lels BB' and hh' so that the angles at B, B' and 1:423 MEASURING POINTS AND SCALES 6, b' are equal; this is easily done by making CB equal to CB' and Cb equal to Cb'. The triangles -.. FIGURE 15A BCB' and bCb' are then isosceles, and their base angles are equal. We now draw SM through S, the point of station, parallel to BB' and bb'; the point M will then be the vanishing point of BB'. bb' and their parallels. SV was drawn parallel to CB' and SM' is parallel to BB', therefore the two triangles CBB' and V'SM are similar geo- 1:433 PERSPECTIVE metrioally, their angles being homologously equal; that is, the angles MSV and SMV are equal to the angles BB'C and B'BC, but the an- gles BB'C and B'BC are equal to each other (drawn so), so the angles MSV and SMV must be equal to each other, and the triangle SVM is isosceles. The sides of this triangle, MV and SV, must therefore be equal, or M is the same distance from V as S is from V. Likewise, it may be shown that VM' is equal to VS, or that M' is the same distance from V as S is from V. Figure 15 is the plan or orthographic projection of the perspective plan. Figure 15 A, or rather the perspective plan is the perspective of Figure 15, Figure 15A, which is lettered similarly to Figure 15, illustrates this. In Figure 15A, AC'B is a front line, and the perspective plan of the picture plane; the line VCV is also in the pic- ture plane and is the horizon line of the per- spective. In the orthographic projection or plan of Figure 15, these two lines coincide. The tri- angle BC'B is the perspective of the isosceles triangle of Figure 15, and M is the vanishing point of BB' and its parallels, which cut off dis- tances perspectively equal on C'V and C'B. This has been explained before. The angles B, B', h and 6' are perspectively equal. The tri- angles formed by C'V and C'B and lines drawn to M (Figure lb A) are always perspectively isosceles, and so M is called a measuring point because a line drawn to it from a point on AC'B will cut off or measure the perspective of the MEASURING POINTS AND SCALES distance of the point from C on C'V. All this is similarly true of M'. It was proved in Figure 15 that MV equals SV and that M'V equals SV. This is always true, under all conditions; the measuring point for a line is the same distance from the vanishing point of that line as the van- ishing point is from the point of station. w. FIGVRZ: 16A. /,' FIGVKE 16 B -^ When the center, C, is midway between two conjugate vanishing points (Figure 16A), the measuring points are symmetrically placed; that is, MV equals M'V, and the distance of either measuring point from the corresponding vanishing point is equal to the diagonal of a square whose side equals one half the distance between vanishing points. When the center, C, is midway between two reciprocal vanishing points (Figure 1GB), then each vanishing point is the measuring point for the other vanishing point. This is obvious be- [453 PERSPECTIVE cause VSV is an equilateral triangle and VV equals V'5 or V5. These special cases of meas- uring points are of no great practical use, but are of interest as exemplifying some curious geometric relations in perspective. 1^6-} To face p. 4^^Lubschez "Perspective ' VII PARALLEL OR ONE-POINT PERSPECTIVE THERE are cases in perspective where the picture plane is assumed parallel to one side of the object. The vanishing point for lines in this system is at infinity; in other words, all the lines in the system retain their original di- rection, and horizontal lines remain horizontal, just as vertical lines remain vertical, aiid for the same reason. There is one finite vanishing point for the other system of lines. This van- ishing point is, of course, on the horizon oppo- site the point of station; it coincides with the center, C. This is evident from the diagram in Figure 17. The measuring point — ^we usually use but one — ^is located in the same way as pre- viously explained for two-point perspective; MC is equal to CS. The measuring point is al- ways as far away from the vanishing point as the vanishing point is from the point of station. The resultant perspective, made under the con- ditions just described, is called a parallel or one-point perspective. One-point perspective PERSPECTIVE is useful in many cases, but especially for the perspectives of interiors or street vistas. It must be remembered that it should not be used where the width of the picture is great; it then becomes forced or artificial. It should be confined to pic- tures where the angle of view, the angle between lines drawn from the extremities of the object to the point of station, is not over 50° or 60° — the less the better. Figures 18A and 18B show parallel perspectives. Figure ISA, a parallel perspective of a series of horizontal squares with their diagonals, very lucidly illustrates the principles of one-point perspective. It should be noticed in this drawing that the measuring PARALLEL OR ONE-POINT PERSPECTIVE points, M and M' (placed so thai MC or M'C equals CS), are the vanishing points of the di- agonals of the squares or lines at 45° with the picture ^lane. This will be readily seen if we look at one-point perspective simply as a special case of two-point perspective as shown in Fig- ure 19. FIGVItEia "^-"f^ In Figure 19 we draw the plan of the squares of Figure 184, and locate the point of station, S. PP', the plan or trace of the picture plane, coincides with the front of the first row of squares, one system of lines being by hypothesis parallel to the picture plane, and the other per- c*9n PERSPECTIVE pendicular. If we, then, proceed to find the van- ishing points, we draw through S a line parallel to the AA system of lines. We get no finite van- ishing points, for the line will intersect PV at infinity, being parallel to it. If, again, we draw a line through S parallel to the AB system, the line will be perpendicular to PP' and will inter- sect it at C, the center. The points M and M' being CS distant from C, the lines MS and MS' are at 45° with CS and VP'; they are parallel to the diagonals of the squares, which, of course, are at 45° with PV, therefore M and M' are the vanishing points of these diagonals. Now let us turn again to Figure 18A. Since M and M' are the vanishing points of the diagonals of the squares, we can easily see that if through any point, p, on PV, we draw a line to C, and to the right of p lay off a point, s, and if from s we draw a line to M, the distance s'p will be per- spectively equal to ps, for ss' is the diagonal of the square whose equal sides are s'p and ^s. Likewise, if we draw r to the left of p and draw a line to M', p'p will be perspectively equal to pr, for rp' is the diagonal of the square whose equal sides are j^t and p'-p. Also, if we draw lines to C through t and s and then draw hori- zontal lines through p' and s\ tt' will equal p'p and s"s will equal s'p perspectively, for We shall have merely completed the perspectives of the squares. This shows briefly the whole theory of one- point or parallel perspective, simply a special PARALLEL OR ONE-POINT PERSPECTIVE case of two-point perspective with conjugate vanishing points, but having one vanishing point at infinity on account of one system of lines being parallel to the picture plane and the other vanishing point coinciding with C on ac- count of its system of lines being perpendicular to the picture plane. Of course, lines whose vanishing points are at infinity are parallel. Figure ISB shows a one-point perspective of a beamed corridor, the elevations of which are shown in Figure 18C. This figure should be re- drawn by the student. In Figure 18B, the hori- zon, HH, is drawn and C and M located on it, then the rectangle ABDE is drawn to scale, a section of the corridor, and lines from A, B, D and E are drawn to C. Actual distances on the side wall are laid out on the ground line, AD, to scale and foreshortened on the perspective line through A by drawing lines to M. These distances may be transferred to the opposite side or any opposite vertical by horizontal lines. Vertical distances or heights are laid out on BA or ED and foreshortened to C. On the ground line, AD, we actually project a one-point per- spective plan, the floor plan, to locate all our verticals. This plan may be drawn at another level below or above our picture, just as we did in our other perspective problems, but in paral- lel perspectives of interiors it is usually more convenient to use the perspective of the floor it- self as the perspective plan. It must be understood that parallel or one- PERSPECTIVE point perspective is not confined to interiors. Any object may be drawn so; usually, however, interiors and street vistas, especially narrow ones, are the only pictures that appear at all well. Where the horizontal lines become inor- dinately long, the picture looks artificial. For interiors within reasonable limits, parallel per- spective is very convenient and considerably simpler than two-point. CSS 3 VIII SPECIAL MANIPULATIONS AND SHORT-CUTS THE perspective center of any rectangle may be found by drawing the diagonals of its perspective. This is quite obvious. Two straight lines can intersect in but one point; the diagonals of a rectangle intersect in its center, therefore the intersection of the perspectives of the diagonals is the perspective of the center. (See Figure 12.) FIGVRZ 20 Very often the perspectives of very irregular outlines, irregular curves, patterns of various 15*2 SPECIAL MANIPULATIONS sorts, landscape plans, and so on, have to be drawn. The simplest way to do this is to plot geometrically the orthographic plan or projec- tion of the outhne to be drawn, then throw the plotting diagram into perspective. We can then locate the intersections of the outline with the plotting reference lines and obtain enough points to determine the perspective. This pro- cess is called craticulation and is quite apparent from Figure 20. The drawing of circles and other curves in perspective is based on the process of craticu- lation explained above, but there are special cases where the defining points on a circle can be easily located. Usually we draw the circum- scribed square and its diagonals and f(nd the points of intersection of the diagonals and cir- cles; this will then give us the eight points on the circle — ^four tangent points and four inter- sections on the diagonals. A close inspection of Figure 21 will make this method of drawing circles and their arcs in perspective quite plain. It is very difficult for the beginner to draw the perspectives of circles neatly, especially in hori- zontal planes. The perspective of a circle is a Une of great beauty, being a true ellipse, but we must remember that the perspective of the di- ameter of the circle is not the axis of the ellipse. This is also shown in Figure 21. We should be very careful of the points of tangency or change of direction in our perspective circles. The ends of the ellipsci are usually a great deal sharper PERSPECTIVE than we are inclined to draw them. A close study of nature, painstaking analysis of correct photographs, and vigilant practice are neces- sary to acquire skill. CIKCVMJCJUBED OCTAGOH/ PLOTTING jariin po\nrj. 7n First drawing the circumscribed octagon often helps to determine the proper curve of the circle in perspective. Just as we found the eight points on the circle in Figure 21, so may we, by the same method, find the eight corners of the circumscribed octagon. The left side of Figure 21A shows this clearly. In addition to the cir- cumscribed square and its diagonals, we draw SPECIAL MANIPULATIONS the octagon inscribed in the square and circum- scribed about the circle. If we construct this in elevation on a line in or parallel to the picture plane as shown, the necessary points are easily found in perspective. The eight sides of the oc- tagon, all tangent to the circle, are a distinct aid in guiding the direction of curvature in the per- spective circle. By the same general method of using points and their ordinates in the picture plane or one parallel to it, sixteen or more points on the circle may be easily found, as shown at the right of Figure 214. This is useful when large circles are drawn in perspective, and may, of course, be used in combination with the circumscribed oc- tagon, if so desired. These methods can be much more plainly demonstrated graphically than described in words, and a study of Figure 21A should make everything quite plain. When the principal line systems of a per- spective are perpendicular to each other, as is usually the case, and the vanishing points are conjugate, it is very useful to locate the vanish- ing points of lines which bisect the right angles of the plan — lines which are at 45° with the principal systems. Lines drawn to these points are very useful in drawing the perspective plans of hip roofs, the projections of cornices, and for finding shadow projections. These special van- ishing points are called miter points, and lines drawn to them are called miter lines. When used in the casting of shades and shadows they TABLE (n C^/fJV6ATE VANl/HING ^OmV WITH iovajwmxm mea/viu/ig Yomv YOt dinillM hnGllS OY VIEW ALL DI/TA/tCEJ" AHE lAIDlCATED J/f 1/lCHE^ ANGLE Y-7 B D i«-A-*---B — J 2^ ^2 19 yg 5/I6 Z^8 3^ I5/16 29 I/k 1^ 4 ^16 4^? I 1/4 58 ^4 5^7 I 9/I6 48 _jfc J* !^Z lA 7 ^16 !»^.4 — B — t! 2^? 25^8 17 1^ 1 1/8 4 l/i 3^ 3>fe ZS ij^ I 1/I6 6 IJA ferf !*— B J-A-J 4^ 4^4 35 1/4 2 l/t 5^ 5iyi6 44 1/I6 2 1/I6 II 1/4 '*"A-* — »—•» zo 15 2 5/I6 3^ 7 1/2 22 1/2 7 \fi L'^ 4^ \o 50 A V8 1^ 5£? Mljl n iji 5 sfii i2iy& 2^ 1^ 1^ 4 1/8 4 ^/8 3^ 15 6 3/I6 6yi6 *-A- j~J 4^ 2^ 2(? 8 1/4 8 1/4 5 1 V'd', the vanishing points of these lines and r parallels, the gable lines. But the distance, ), from V, the vanishing point of the hori- 2 al line, to the point of station, S, is equal to V I, the distance from the vanishing point to the measuring point of the line. This, as is she vn in Chapter VI, is always true. If we re- vol e the triangles V'd,V',S and y'd',V',S on the axis, Y'd-V'-V'd', into the picture plane, the line Y'-S will coincide with the line V'-M, the point S will coincide with the point M and the angle at S with the angle at M, for they are each 30°. Therefore the triangles, V'd.V'M and V'd,V',S will coincide, as also will the triangles, V'd'.V'M and V'd'.V'.S. The line S-Y'd will coincide with M-Y'd and the line S-Y'd' with M-Y'd'. The construction for V'd or Y'd' by drawing the inclined lines from M is therefore exactly equivalent to actually drawing these in- clined lines from S. We will now consider the inclined roof plane 1, 2, 6, 7. The vanishing point of thejines 1-2 and 6-7 is Y'.d. The vanishing point of the lines 1-6 and 2-7 is V. A line, Hl-Hl, drawn through Y and Y'd is the vanishing line or the horizon line of the plane, 1, 2, 6, 7, and planes parallel to it. (See Chapter V.) Since the measuring point for distances on a line is always the same dis- tance from the vanishing point of the line as is the point of station, we may find Ml, the mea- suring point for 1-2, by laying off Ml-Y'd equal PERSPECTIVE to S-V'd or M-V'd, and we may find M'l, the measuring point for 1-6, by laying off M'l-V equal to S-V or M'-V. Since point 1 is in the picture plane, a line, Gl-Gl, drawn through it and parallel to Hl-Hl, the horizon line, which is also in the picture plane, will be in the picture plane (any two parallel lines lie in the same plane), and there- fore this line, Gl-Gl, is a front line and a mea- suring line. On this line we may lay off actual scaled distances and foreshorten them to 1-2 by the measuring point Ml. This will all look quite natural if we rotate the page until Hl-Hl becomes horizontal. The lines and points will then appear in their more familiar attitudes. The horizon line is the projection on the pic- ture plane of the actual horizon or vanishing line of a plane. As an infinite number of planes may pass through a single line, a line has an infinite number of horizons passing through its vanishing point. We usually use, for the sake of convenience, that horizon which passes through one or more other vanishing points of related lines in the same or parallel planes. In Figure 33, we use for 1-2 and 3-4, the horizon Hl-Hl, which is the vanishing line for the roof plane and its parallels and contains all vanish- ing points and measuring points for lines in this plane or its parallels. For measurements on 1-2, however, we might have drawn any horizon hne through V'd, its vanishing point, such as Hx-Hx, find Mx by making Mx-V'd equal to Vd-S or CSS] OBLIQUE LINES AND PLANES V'd-M, and lay ofiF scaled distances on a line Gx-Gx through 1 and parallel to Hx-Hx. All this is just as true for one point or parallel perspective, where one system of lines is paral- lel to the pictiu-e plane and has an infinite van- ishing point and the other system is oblique to the picture plane and has a finite, accessible vanishing point and measuring point. The pages just gone over must be thoroughly understood before we may go on to the fascinat- ing work in inclined lines and planes and the perspective of shadows. As was said in the be- ginning, a sure comprehension of Chapters V and VI is a necessary preliminary. Figure 34 shows a problem involving the per- spective drawing of inclined lines and planes. The drawing of the walls and the door-frame is done in the usual way. The unfamiliar points occur in the drawing of the transom, hinged at its top and swung out at an angle of 45° with the plane of the wall; and of the door, which is hung at the right and swung open at an angle of 45°. The diagram for the construction of the vari- ous points, Figure 344, is drawn similarly to the diagrams in Figures 7 and lOA. We find Vd, the vanishing point for the horizontal lines of the door, simply by drawing a line from S, the point of station, parallel to the door line and intersecting the line of the picture plane, P-P, at Vd. We find the measuring point for the door line, Md, by laying off Md-Vd equal to Vd-S, in the usual way; the measuring point Md being ess: PERSPECTIVE the same distance from Vd, the vanishing point; as Vd is from S. the point of station. The perspective plan. Figure 34B, is laid out similarly to others in previous problems except in so far as the door line is concerned. For this we first draw a line through a from Vd. We next locate 1' by drawing a line from Md through a. V is the actual point on the line of measures P-P, corresponding to the perspective, a. This is the reverse of the usual process when we take an actual point on the line of measures and find its perspective by means of the measur- ing point. Here we take the perspective of a point and by inverse projection from the mea- suring point, find its actual location on the line of measures. We now lay off the width of the door, 3% feet at the proper scale, on P-P from 1' to 2'. We draw a line from Md through 2' and find d, the perspective of 2'. The line a-d is the perspective plan of the door. In Figure 34, we draw lines from Vd through 1 and 1", and on these lines locate 2 and 2" over d of plan. 1, 2, 2", 1" is the perspective of the door. To draw the transom, we must first find the vanishing point of the line 3-4, which is inclined at 45° with the horizontal plane. We do this as we did in Figure 33. Since the line 3-4 is paral- lel to the plane c, c', k', k, its vanishing point must lie in the vertical line through V, this ver- tical line being the vanishing line of the plane c, c', k\ k and its parallels. We next draw a line ITS*: OBLIQUE LINES AND PLANES from M' at 45° (the inclination of 3-4), to H-H and intersecting V-Vt at Vt. the vanishing point of 3-4. We now draw lines of indeterminate length from Vt, through 3 and 3". Since one set of lines in the plane of the tran- som vanish in V, another set in Vt, a line drawn through Vt and V, Ht-Ht, is the horizon line for the transom plane. On this line, we lay off M't-Vt equal to M'-Vt, finding the m'easuring point, M't, for 3-4 and its parallels. We now continue 3-4, or one of its parallels, to its intersection with the picture plane. Through this intersection, we can then draw a front line and a line of measures. One way to do this is to continue 3"-3 to 3', its intersection with k-k'. Through 3', we now draw a line from Vt to X, its intersection with c-c\ Since c-c' is a front line lying in the picture plane (drawn so), X lies in the picture plane. We draw Gt-Gt through X and parallel to Ht-Ht. Gt-Gt is a front line and a line of measures for 3'-x or its paral- lel, 3-4. We now find 3a; on Gt-Gt by drawing a line from M't through 3. We lay off 3x-ix, two feet long at the given scale, and project 4 on 3-4 by drawing from M't through 4a;. 4-4", drawn to V, completes the transom. We might have just as easily projected 3" to Gt-Gt, scaled the distance for the width of transom and found 4". This problem and similar ones with varying angles should be practised until the principles and manipulations have become quite familiar. Mere studying of the text and illustrations is not PERSPECTIVE enough. The figures must be redrawn follow- ing the text and then without the assistance of' the text. Finally, variations of the problem should be drawn. 1862 XI THE PERSPECTIVE OF SHADOWS BY SUNLIGHT IN dravdng shadows cast by sunlight, the sun is assumed to be an infinite distance away, hence its rays of light are parallel. The shadow of a point on any surface is the intersection with the surface of the ray of light passing through the point. The shadow of a straight line on any surface is the intersection with the surface of the light plane generated by the light rays pass- ing through the points of the straight line. The shadow of any line on any surface is the inter- section with the surface, of the light surface generated by the light rays passing through the line. Although we call the lines of light which pass through a point, light rays, and the surfaces generated by these light rays passing through the points of a line, light planes and light sur- faces, we must remember that these lines and surfaces are light only from the source of light to the points or lines through which they pass. From these points or lines to the surface on Csrn PERSPECTIVE which the shadow is cast, they are lines and sur- faces of darkness. In the case of a straight line, if we know the direction of its shadow on a plane and can find the shadow of one of its points, we may draw the shadow through the one point known. In the case of the shadow of a limited straight line on a plane, which shadow is then a limited straight line, we may draw the complete shadow by connecting with a straight line the shadows of the points of extremity of the line. We can plot the shadow of any line on any surface by finding the shadows of points close together on the line. If then, we can cast the shadow of any point on any surface, we can cast the shadow of any line on any surface and then we can cast the shadow of any surface bound by lines or any solid bound by surfaces, on any surface. Figure 35A is a plan and elevations diagram of a barn and sign post of which Figure 35 is the perspective. To avoid the conventional angle of 45°, we assume that the sun's rays make an angle of 20° with the ground line of the nar- row side of the barn and an angle of 30° with the horizontal. The sign board makes an angle of 20° with the long side of the barn. The roof slope is 45°. In Figure 35i4., Vp, the vanishing point for the horizontal lines of the sign board, and YsG, the vanishing point for the ground lines or projec- tions on the ground plane, of the light rays, are found in the usual way by drawing lines from S, CSS] \«UfV* FIGVUL 35 To/mi f SS— /.•>/■!. if j: •Ttrt^.tivt" THE PERSPECTIVE OF SHADOWS the point of station, parallel to these lines on the plan. Mp, the measuring point for the horizon- tal lines of the sign board, is found in the usual way by laying off Mp-Vp equal to S-Vp. Ms is similariy found by making Ms-VsG equal to S-VsG. These points are then transferred to Figure 35. In Figure 35, V'd and V'd', the vanishing points for the roof slope lines, are found as ex- plained in Figure 33, by drawing from M, 45° lines to the vertical through V and intersecting the vertical, in V'd and V'd'. The perspective of the barn and sign post may now be drawn, when we are ready to cast the shadows. We shal| begin with the shadow of the build- ing on the ground at the right. First we must find VsV, the vanishing point of the sun's rays, lines which make an angle of 30° with the hori- zontal and whose horizontal projections make an angle of 20° with the ground line, 1-G. We have already found VsG, the vanishing point of these horizontal projections of the light rays. It is quite obvious that l-2s, the shadow on the ground of the vertical, 1-2, is the intersection with the ground plane of a plane generated by the light rays passing through the points of the vertical, 1-2. It is also quite obvious that l-2s makes the assumed angle of 20° with 1-G and vanishes in VsG. The vertical through VsG is the vanishing line of the vertical plane gener- ated by the light rays passing through 1-2 and in which plane lies 2-2«, a light ray making the 11893 PERSPECTIVE assumed angle of 30° with the horizontal. If then we draw from Ms, a line below the horizon and making an angle of 30° with it, it will inter- sect the vertical through YsG at VsV, the vanish- ing point of 2-2s and its parallels, the other light rays. This is simply a problem in inclined hues. If we draw the line 2-2s from 2 to VsV to its intersection with l-2s at 2s, 2s will be the shadow of 2, and l-2s the shadow on the ground of the vertical 1-2. The shadow of 2-3 on the ground, which is 2s-3s, is the intersection of a plane generated by the light rays passing through 2-3, the roof line. Since 2-3 lies in this light plane, its vanishing point, V'd, must lie in the vanishing line of the light plane. The light rays, 2-2s and 3-3s, lie in this same plane, hence their vanishing point, VsV, must lie in the vanishing line of the plane, and the line drawn through VsV and V'd, two points on the vanishing line, must be the vanish- ing line of the light plane. The line 2s-3s is a horizontal line (on the ground), hence its vanish- ing point must lie in the horizon line H-H. The line 2s-3s also lies in the light plane passing through 2-3 and whose vanishing line, V'd- VsV, we have just found. The vanishing point of 2s-3s must also lie, therefore, on V'd-VsV as well as on H-H, and must be the intersection of V'd-VsV and H-H or VsG2. This is one of the most interesting manipulations in perspective drawing and occurs repeatedly in the casting of THE PERSPECTIVE OF SHADOWS shadows. It should be thoroughly understood. Having found VsG2, we draw 2s-3s to it from 2s. A line drawn through 3 to VsV intersects 2s-3s at 3s, the shadow of 3. A line drawn from 3« to V gives the shadow of the ridge line and completes the shadow of the building at the right. The shadow of the ridge line is drawn to V because it is the intersection with the ground plane of the light plane generated by the light rays passing through the ridge line, which is parallel to the ground plane. The intersection or shadow is therefore parallel to the ridge line and drawn to the same vanishing point, V. Since the line 3-4 falls within the line 3-3s, the shadow of 3-4 falls within the shadow of the ridge line and does not show. By drawing from 5 to VsG, we get 5s and a vertical at 5s gives 5s' at the eaves. The point 5s (as also the points 6s and 7s) may be found more easily and accurately on the perspective plan, as shown. This determines the shadow of the chimney on the ground and on the side wall of the barn. We must next find the shadow of the line 5-13 on the roof, 5s'-13s. This line, 5s'-13s, is the intersection with the roof plane, of a light plane generated by the light rays passing through the points of 5-13, hence the vanishing point of this intersection, 5s'-13s, lies in the van- ishing line of the light plane, VsV-VsG con- tinued, and also in the vanishing line of the roof plane, Hd-Hd, aijd must be the intersection of 191-2 PERSPECTIVE these two lines at VsRl. We therefore draw 5«'- 13s to VsRl, and a line from 13 to VsV deter- mines 13s, the shadow of 13. The shadow of 13-14 on the roof is the inter- section with the roof of a light plane generated by the light rays passing through 13-14. The vanishing line for the light plane passing through 13-14, is the line passing through V, the vanishing point of 13-14, and VsV, the vanish- ing point of the light rays. This line continued intersects Hd-Hd, the vanishing line of the roof plane, in VsR2, the vanishing point of 13s-14s. A line from 14 to VsV determines 14s, the shadow of 14. A line drawn from 14s to V gives the line of the shadow of 14-15. The shadow of 14-15 on the roof plane, 14s-15s, is the intersection with the roof plane of a light plane passing through 14-15. Since 14-15 is parallel to the roof plane, the intersection or shadow, 14s-15s, is parallel to it and goes to V, the same vanishing point. A line from 15 to VsV gives 15s, the shadow of 15. A line from VsRl through 15s completes the shadow of the chimney on the roof. This shadow line is obviously parallel to 5s'-13s, being the shadow of a line parallel to the one of which 5s'-13s is the shadow and cast on the same plane. It therefore has the same vanish- ing point as 5s'-13s, VsRl. A line from 6 to VsG, intersecting the ground line of the chimney at 6s, where we draw a ver- 1:923 THE PERSPECTIVE OF SHADOWS tical on the face of the chimney, will give us the intersection with the ground and chimney of the light plane passing through 6-6'. A line from 6' to VsV, gives 6's, the shadow of 6' on the chim- ney. By connecting r and 6'«, we complete the shadow of the projecting wing on the ground and chimney. The line r-6'« may be drawn by finding its vanishing point as others were found — 'by intersecting vanishing lines — ^but this van- ishing point is inaccessible and in the present instance entirely unnecessary. A line from 7 to VsG gives 7s at the ground line of the extension or wing of the barn, 7-7s being the shadow of the sign post on the ground. A vertical at 7s to 7's, gives Is-Ts, the shadow of the sign post on the wall. By drawing from VsV, through 7's to 7' on the sign post, we find that 7's is the shadow of 7'. Finding 7' is un- necessary but interesting. A line from 7's to Vsi?l gives the shadow of the sign post on the roof, and a line to VsV from 8 gives 8s, the shadow of 8, the extremity of the sign post, on the roof. Lines from m and n to VsV give ms and ns, the shadows of m and n on the roof. The shadow of the line 9-10 on the roof, 9s-10s, is the intersection with the roof plane of a light plane generated by light rays passing through the line 9-10. The vanishing line of this light plane passes through Vp, the vanishing point of the line 9-10, one of its elements, and through VsV, the vanishing point of the light rays, also elements of the same plane. The vanishing PERSPECTIVE point of the line 95-lOs, which lies in the vanish- ing line of the light plane, VsV-Vp, and also lies in the vanishing line of the roof plane, Hd-Hd, must be the intersection of these two vanishing lines, or VpR. The line lls-12s, which is the shadow on the same roof plane, of the line 11- 12, a parallel to the line 9-10, is parallel to 9s-10s, the shadow of 9-10, and hence goes to the same vanishing point, VpR. We, therefore, draw lines through ms and ns to VpR and on these lines locate 9s, 10s, lis and 12s, the shad- ows of 9, 10, 11 and 12, by drawing from these points, 9, 10, 11 and 12, to VsV. We check this by finding that the lines 9s-lls and 10s-12s go to VsRl, the vanishing point for shadows, on the roof, of vertical lines. This completes this drawing, which involves every basic principle of the perspective of shadows. The student should redraw this several times, first by following carefully the instructions given and then with variations without the in- structions. toil FIGVR.E 36 To face p qs—Lubschez "Perspective' XII THE PERSPECTIVE OF SHADOWS BY ARTIFICIAL LIGHT THE general theory of shadows by artificial light is similar to that of shadows by sun- light. With shadows by artificial light, how- ever, the rays of light are not parallel but divergent, radiating from a point at the center of the source of light. As with sunlight, the shadow of a point on any surface is the intersection of the light ray passing through the point with the surface on which the shadow is cast. The shadow of a line on any surface is likewise the intersection of the surface generated by the light rays passing through the points of the line, with the surface on which the shadow is cast. Light rays, instead of being drawn to a van- ishing point, are drawn to the point at the center pf the source of light. In Figure 36, which is the perspective of an interior with table and lamp and of which Fig- ure S6A shows the plan and elevations, it is required to show all the shadows on the walls, ceihng and floor cast by the light at A. 195-2 PERSPECTIVE The drawing of the room and accessories should not present any difficulties. We may use the floor itself for the perspective plan as we did in the parallel or one point perspective in Figure 18B. To find the shadow of the table top on the floor, we must find where lines passing through the source of light. A, and the corners of the table top, 1, 2, 3, 4, pierce or intersect the floor at Is, 2s, 3s, 4s. The line A-h-c is drawn per- pendicular to the floor and c is the projection of A, the source of light, on the floor, while a is its projection on the ceiling. The point c is the per- spective plan, on the floor oi A. In fact c is drawn first and A is located at the proper height on a vertical at c. It is quite obvious that any plane passing through A-c will be perpendicu- lar to the floor. Since a plane may be drawn through any two intersecting lines, a plane, i4,c,ls, may be passed through A-c and A-l-ls. The plane A,c,\s must necessarily pass through c-ls, its intersection with the floor. Since V is the projection, on the floor, of 1, the line 1-1' is perpendicular to th« floor at 1' and lies in the plane ifl.c.ls, and 1' lies in the intersection of the plane A,c^s with the floor, the line c-ls. Therefore to find Is, we draw a Une, c-1' con- tinued, from c through 1' and another line, A-\ continued, the light ray from A through 1. The intersection of c-1' continued and A-\ contin- ued gives Is, the shadow of 1. Wemight proceed in thesame way we f oundls, 1:963 THE PERSPECTIVE OF SHADOWS to find2s, 3«and4s and connect these points for the shiadow of the table top, but this is unnecessary. The line ls-2s. the shadow of the line 1-2, is the intersection of the hght plane generated by the light rays passing through the Une 1-2, with the floor and is parallel to the line 1-2 since this line is parallel to the floor. We therefore sim- ply draw a line through Is to V, the vanishing point of 1-2 and its parallels. By drawing a line front A, the source of light, through 2, we find the intersection 2s, the shadow of 2. Similarly the line 2s-3s, the shadow of 2-3, is parallel to 2-3 and is drawn from 2s to V, the vanishing point of 2-3 and its parallels. The intersection, 3s, the shadow of 3, is located by drawing from A through 3. Just as we did before, we draw 3s-4s from 3s to V and draw Is^s from Is to V and thus complete the shadow of the table top. A similar process is employed to find the s]tadow of the lower edge of the lamp shade on the walls. Draw a perpendicular from A to the back wall at al. This is done by drawing from c to V, intersecting the ground line of the wall at cl, then drawing from A to V, intersecting this vertical at al. It is plain that a plane pass- ing through the lines A-al and al-cl will be perpendicular to the wall and to the lower edge of the lamp shade at its middle point, a2. The light ray A-a2 will lie in this plane and intersect al-cl at a2s, the shadow of a2. The shadow of the line 7-8 is the intersection with the wall of a light plane generated by the light rays passing 1:973 PERSPECTIVE through 7-8. This intersection is parallel to the line 7-8 since 7-8 is parallel to the wall, and may be drawn through a2s to V, the vanishing point for 7-8 and its parallels. By drawing lines from A through 7 and 8, the intersections on this line through a2s,7s and 8s, may be found; these are the shadows of 7 and 8 and the line 7s-a2s-Ss is the shadow of the line 7-8. The lines Ss-k and 7s-k', the shadows on the back wall of parts of the lines 8-5 and 7-6, are the intersections of the light planes generated by the light rays passing through 8-5 and 7-6. These light planes are parts of unlimited planes passing through A and through 8-5 and 7-6. These unlimited planes intersect the wall in al^s-k and al-7s-k\ The shadows 8s-k and 7s-k' may therefore be drawn by drawing through 8s and 7s, lines from al, the projection of A, the source of light, on the back wall. The point k in the corner, and therefore in both wall planes, is the shadow of a point on the line 8-5. The shadow of 8-5 on the side wall is parallel to 8-5 since 8-5 is parallel to the side wall, therefore a line drawn through k to V gives k-d'-a3s-m^n, the shadow of 8-5 on the side wall. The same construction that we used to find 7s-8s may be used to find k-d'-a3s-m-n and is also shown. In fact this construction is necessary to find d-dl, the deeper shadow on the door plane — d-dl is drawn through a3s' to v. By continuing d-d' on the door jamb to d", THE PERSPECTIVE OF SHADOWS the shadow lines on the door casing may be lo- cated and m' found. The point m is the shadow of m' and may be found on the wall shadow line by drawing through m' from A. A vertical through m, parallel to the door casing, gives the shadow of its edge. ,The shadow line on the ceiling of the upper edge of the shade may be found in the same way that the shadow of the table top on the floor was found. The point a is the projection on. the ceil- ing of the point A, the source of light. The point 5" is the projection on the ceiling of the point 5'. The line a-5" continued is the inter- section with the ceiling of a light plane perpen- dicular to the ceiling and passing through the comer 5', and contains the light ray, A-5'-5's. Point 5's is found by drawing from A through 5'. We now draw a line through 5's from V, locate 6's by drawing through 6' from A, draw through 6's to V, locate 7's by a line from A through 7' and draw a line from 7's to V. An- other line from 5's to V completes the outline. The shadows of the corner tubes are con- structed as shown at 6's — by drawing from a, through r and r', to rs and r's or from A through the lower end of the tube to rs and r's, and con- necting r with rs and r' with r's. In a similar way the other three shadows are drawn. This completes the problem. Like the others, this plate should be redrawn by carefully fol- lowing the instructions until the placing of every point and every line is thoroughly understood, 1991 . PERSPECTIVE then similar problems with variations of condi- tions should be solved. Like all other problems in perspective, we need to understand thor- oughly only a few basic principles and we shall have no trouble in building up the others. Cloon XIII WHO DISCOVERED THE RULES OF PERSPECTIVE? THE observation and study of the natural phenomena of perspective date back to an- cient times; the understanding of the graphic processes of perspective as known to-day dates back about two hundred years, although some of these processes were understood a century earlier. Much earlier than this, in the scratch- ings and drawings on bones of the Later Paleo- lithic period — 35,000 to 15,000 years ago, we find evidence of some cognizance of the simpler perspective phenomena. So the first observa- tion of perspective effect is lost in the dim ob- scurity of a very remote past. It is as old as attempts to draw from nature. The fundamental phenomenon of perspective is the formation of the image of an object on a real or imaginary transparent screen, the pic- ture plane, by points where it is pierced by lines from the eye to the various points on the object when it is looked at through the screen. In fact, early investigations of perspective, such as Dii- PERSPECTIVE rer's, were made by drawing on a sheet of glass or other transparent screen what could be seen through it. The word "perspective," meaning literally to see or look through, is derived from this phenomenon, which was recognized in the fifth century b.c. In the introduction to Book VII of Vitruvius we find that: Agatharcus, at the time when jEschylus taught at Athens the rules of tragic poetry, was the first who contrived scenery, upon which subject he left a treat- ise. This led Democritus and Anaxagorus, who wrote thereon, to explain how the points of sight and dis- tance ought to guide the lines, as in nature, to a centre; so that by means of pictorial deception, the real ap- pearances of buildings appear on the scene, which, painted on a flat vertical surface, seem, nevertheless, to advance and recede. Vitruvius is not clear as to whether the scenery itself or the treatise of Agatharcus in- spired the Greek philosopher, Anaxagorus, to write of perspective, but the latter is quoted as writing that, "... in drawing, the lines ought to be made to correspond, according to a natural proportion, to the figure which would be traced out on an imaginary intervening plane by a pen- cil of rays proceeding from the eye, as a fixed point of sight, to the several points of the object viewed." Thus it is seen that the basic phenom- enon of perspective was clearly understood. As far as we know, the treatise of Agatharcus referred to by Vitruvius was the first investiga- WHO DISCOVERED THE RULES? tion of the mathematics of perspective, and the scenes which Agatharcus painted for the trage- dies of iEschylus in the fifth century before Christ were the first perspective drawings, that is, the first drawings laid out according to rules of perspective rather than merely copied from nature. We are quite sure that these scenes were in parallel perspective and it is quite easy to see why even if anything about angular perspective were known, which is doubtful, it would not be wanted for these scenes. Being viewed from different parts of a theatre — ^from different points of station rather widely separated, paral- lel perspective offers less distortion, the planes parallel to the picture plane showing about as well from one station point as from another. Little about the theory of angular or two- point perspective was developed in the twenty- two centuries after Agatharcus (fifth century B.C. through the seventeenth century a.d.), although several elaborate treatises on parallel or one- point perspective appeared. Many paintings of the Renaissance, however, show a keen observa- tion of angular perspective in nature. A notable attempt to analyze the theory of angular per- spective was made by Albrecht Diirer, the fa- mous engraver, in 1525. His attempt, as well as others' of tiie period, was, as well illustrated in a book by a successor of Diirer's, Jan Vredeman de Vries, published in 1630, a scheme to lay out angular perspective on an elaborate network of 1:1033 PERSPECTIVE squares drawn in parallel perspective, an elabo- rate scheme of plotting or craticulation. (See pages 53-55.) Jan Vredeman de Vries, how- ever, employs an assumed horizon and gives the rule for finding vanishing points. Parallel or one-point perspective was well un- derstood by the Romans. The wall paintings of Pompeii (first century a.d.) were certainly not drawn directly from nature and were laid out in reasonably accurate one-point perspective. It is mystifying to think that the great painters and architects of the Renaissance in Italy did not evolve the rules of angular perspective; that the great mind of Leonardo da "Vinci, who under- stood the phenomena of perspective well and who also realized that perspective was a mathe- matical as well as a graphic process, did not dis- cover any rules for drawing in angular or two- point perspective. Leonardo had great regard for perspective. In the preamble to his dis- course on the subject, he says:^ "Perspective is the bridle and rudder of painting." Again later that: Perspective is a rational demonstration whereby ex- perience confirms how all things transmit their images to the eye by pyramidal lines. By pyramidal lines I mean those which start from the extremities of the surface of bodies and by gradually converging from a distance arrive at the same point; the said point being, as I shall show, in this particular case, located in the eye, which is the universal judge of all objects. 1 "Leonardo da Vinci's Note Books," McCurdy's translation, Clias. Scribner's Sons. WHO DISCOVERED THE RULES? Leonardo da Vinci's notes were made early in the sixteenth century. Other masters of the Renaissance wrote on and taught the subject, but all their theories were based on one-point perspective. In 1693, Andrea Pozzo wrote a profound treatise in Latin, elaborately and beautifully illustrated, but covering the theory of parallel perspective only. Pozzo's book was translated into English by John Stuart in 1807, and even this translation is a delightful book to peruse. Some of the illustrations, parallel per- spectives of the undersides of domes projected on horizontal picture planes, are really amazing. The Renaissance brought forth many treatises on perspective by Italian, French, Dutch, and ' other artists, but it remained for an English mathematician. Dr. Brook Taylor, a follower and ardent admirer of Sir Isaac Newton, to lay down concisely, in 1715, all the fundamental rules upon which the science of perspective is based. Dr. Taylor's treatise in its first edition contained only 42 pages, 12mo., and 18 small plates. It may be interesting, perhaps, to quote the heading and preface of Dr. Taylor's book: Linear Perspective or, a New Method of Represent- ing justly all manner of objects as they appear to the Eye in all Situations. In this Treatise I have endeavour'd to render the Art of Perspective more general, and more easy, than has yet been done. In order to do this, I find it necessary to lay aside the common Terms of Art, which have hitherto been used, such as Horizontal Line, Points of Distance Ec. and to use new ones of my own; such as PERSPECTIVE seem to be more significant of the Things they express, and more agreeable to the General Notion I have formed to my self of this Subject. Thus much I thought necessary to say by way of Preface; because it always needs an Apology to change Terms of Art, or any way to go out of the common Road. But I shall add no more, because the shortness of the Treatise it self makes it needless to trouble the Reader with a more particular Accoiint of it. Dr. Taylor evidently seemed to think that he wrote a treatise on art, but his book is a collec- tion of clean-cut mathematical theorems, brief and general, and not wasting itself over minor or easily deducible details. For this reason the book was criticized as too brief and too obscure, and it is easily conceived how dry and difficult it must have been to those who needed its les- sons most, — painters and draftsmen. Soon lengthy expositions, based on Taylor's funda- mentals, began to appear. Indirectly based on these fundamentals, they are still appearing, for every text-book on perspective published in nearly two centuries is a descendant of Taylor's "Essay on Linear Perspective." A most interest- ing elucidation of Taylor's book was published in England in 1774 by Thomas Malton. Instead of the original 42 12mo. pages, Malton's book contained 350 quarto pages and goes into mi- nute details. Interesting indeed are the engraved illustrations, many of which have flaps so pasted that they may be folded up to make actual mod- els of the planes involved. WHO DISCOVERED THE RULES? From the time of Taylor and Malton to fhe present day, there have appeared many score treatises on perspective, in all civilized coun- tries, in different languages, and of varying scope and value. Conspicuous among modern books are the French treatise of M. Joseph Ad- hemar, published in 1846, and the American book by Professor William R. Ware, first pub- lished in 1883. D07n INDEX A PAOE Accuracy, Necessity of 22 Adhemar, Joseph 107 Agatharcus 102-103 Anaxagorus 102 Angle governs size of image, Subtended. ... 7 Angle of view 30-31 "Applied Perspective," by William P. P. Longfellow 2 B Board limiting distance between vanishing points, Size of drawing- 62-63 Books on perspective. More elaborate 2 C Centers, Perspectiv.e 53 Circles in perspective 55-56 " " " Large 56a Circumscribed octagon as aid in drawing perspective circles 56-56a Conjugate vanishing points 39 Convergence of lines 6 Craticulation or plotting 53-54 Curves, Perspective of 53-55 Curves used when vanishing points are off the board 57 Curvilinear or Panoramic Perspective 39 1:1093 INDEX ■D PAQB Da Vinci, Leonardo 5, 104 Definition of perspective 5 " " picture plane 3 Diagonals, Measurement by 34 Diagram of working points 11-13 Diminution in perspective 7, 31-32 Direct projection method 65-69 Distortion in perspective 63 Drawing a perspective. Problem I 11-22 Problem II 24-29 Drawing a perspective in parallel or one- point perspective 51-52 Drawing a perspective with inclined lines and planes 80-86 Drawing a perspective with shadows by ar- tificial light 9&-100 Drawing a perspective with shadows by sun- light 89-94 Durer, Albrecht 101, 103 E Elevations and perspective 10-11 F Foreshortening 6 Fractional measuring points 61-62 Front lines 42 G Ground line 14 H Horizon as vanishing line of horizontal planes 13 Horizon line 6 Horizon lines of inclined and oblique planes 81-82 INDEX Ima^e in eye a perspective projection 3 Inclined and obligue lines 34-^6, 79-86 " Measuring lines for 81-«2 " Measuring points for 81-82 " Vanishing points of 81 planes, Horizons of. .81-82 " Vanishing lines of 81-82 line, Definition of 79 plane, " " 79 Large circles in perspective 56a Large perspective laid out within limits of picture 77-78 Leonardo da Vinci. (See Da Vinci, Leonardo) Light rays and light surfaces 87-88 Line, Perspective of any 28 Line of measures 14 Longfellow's "Applied Perspective" 2 M Malton, Thomas 106-107 Measures, Line of 14 Measuring point in parallel perspective .... 47 Measuring points 12-13, 42-46 " " as special vanishing points 15 Finding the 12-13 Fractional 61-€2 " " of inclined and oblique lines 81-82 off the board. Distant ... 61 " " Special cases of 45— If" INDEX PAGE Miter points and lines 56 "Modern Perspective," by William R. Ware 2, 65, 107 N Notation, Perspective 5-6 O Oblique and inclined lines 34-^6, 79-«6 '^ " " " Definition of . . 79 " " " Measuring lines for 82 " " " Measuring points of 81-82 " " " Vanishing points of 34,80 planes. Definition of. 79 Horizons of 81-82 Vanishing lines of . .81^2 Octagon as aid in drawing perspective circles. Circumscribed 56-56a One-point or parallel Perspective 49 as special case of two-point or conjugate perspec- tive 49 P Panoramic or curvilinear perspective 39 Parallel or one-point perspective 47-52 Parallel or one-point perspective, AU early perspectives in 103 Parallel or one-point perspective. Limita- tions of 51-52 Parallel or one-point perspective. Measur- ing point in 47 INDEX PAQX Perspective centers 53 II Definition of 5 First book containing all fun- damentals of 105 " drawings in 103 " investigations of 101-104 " laid out within limits of picture. Large 77-78 " notation 5 " of circles 55-56 " curves 52-53 " line, how found 28 " " object as image on assumed plane 3 " point, how found 27-28 " " shadows by artificial light 95-100 " shadows by sunlight 87-94 " Photograph as a 5 plan. The 14-16 " Seeing in 3 Perspective without use of vanishing points by direct projection .70-73 Perspective without use of vanishing points by ordinates and parallel perspective . . 73-74 Perspective without use of vanishing points by verticals in front plane 75-76 Photograph a perspective 5 Picture plane, Definition of 3 Position of 8 " " usually vertical 5 Plan of book 2 " The perspective 14-16 Planes and their vanishing lines 33 Plotting or craticulation 53-54 Point in perspective, how found 27-28 INDEX PAGB Point of station 5, 30-31 " " " Arbitrary location of 13 ' Height of 31 " " " Locating the 12 " " " Plan position of 30 " " " Significance of location of 12,30-31 Points in front of picture plane. Perspective of 24-25 Pozzo, Andrea 105 Proportional points 59 R Reciprocal vanishing points 39 Scales 42 Scenery for tragedies of iEschylus 102-103 Seeing in perspective 3 Shadow lines by artificial light 95 " " on ground, Vanishing points of 88-89 * * " of any line on any surface 87 " " " point on any surface 87 " " straight line on a plane 88 " " " " " any surface. 87 Shadows by artificial light, Perspective of 95-100 Shadows by sunlight. Perspective of 87-94 Shadows, Plotting of 88 Size of image governed by subtended angle 7 Station point. The 30-31 Stuart, John 105 Study of perspective phenomena in a photograph 5-7 Subtended angle governs size of image 7 nil*] PAGB INDEX T Table of conjugate vanishing points for dif- ferent angles of view, with their corre- sponding measuring points ....... .facing 57 Taylor, Brook 105 ICri-conjugate vanishing points 39-41 Two vanishing points for every line 38 V Vanishing line 8 " " of oblique and inclined planes 81-«2 Vanishing lines. Planes and their 33 Vanishing point of line, how found and why 36 " " shadow Une on ground. 88-89 " " sun's rays 89-90 " points. Conjugate 6-8, 39 Far off 57-60 Finding the 12 " " for different angles of view. Table of conju- gate facing 57 " " Limited distance be- tween 62-63 " " of oblique and inclined lines 33-38,80 off the board 57-60 Reciprocal 39 " " Special relations between 39 Tri-con jugate 39-41 " " Two for every line 38 Vitruvius 102 W Wilham R. Ware's "Modern Perspec- tive" 2, 65, 107 Working diagram at small scale 12 Working points. Diagram of 11-13 CHS]