P5RCP;jGTl¥£ t5>» fci rOR ART STUDENTS &* S» ^ .VGiiATTON Come IRewlOorftS fiaM.<=il 11 IDinive lariculturc Xibrari? OF THE tm doiiegc oti * " " I 3778 NC 750.H36""'""'"™'*>"-«'"1' S. "^ Cornell University WB 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/cu31924014453298 PERSPECTIVE FOR ART STUDENTS PE RSPECTIVE FOR ART STUDENTS BV RICHARD G. HATTON AUTHOR OP "elementary design," "figure drawing and COMrOSITION ■ With 208 Diagrams THIRD THOUSAND LONDON : CHAPMAN & HALL, LD. 1910 PREFACE "To pore over all these matters Paolo would remain alone, seeing scarcely any one, and remaining almost like a hermit for weeks and months in his house with- out suffering himself to be approached." So wrote Vasari of Paolo Uccello. To many since Uccello's time this " most elegant and agreeable art," as the author of the" Jesuit's Perspective " regarded it, has had fascinations, and much midnight oil has been burned by its votaries. The " Jesuit's Perspective," a seventeenth-century work, was reprinted for the seventh time late in the eighteenth century, and its errors and shortcomings seem to have stimulated the sounder mathematicians of the time to literary activity. These writers carried the art to its perfection, so far as theory was concerned. They found vanishing-points for lines in all possible positions, no matter how curiously inclined to the picture. To the author of the " Jesuit " this advanced part of the subject never once occurred ; he even marks as accidental, vanishing-points on the horizon other than the point of vi Preface sight. With only parallel perspective, it is small wonder that so tireless an author should seek to in- crease the usefulness of his science by facilitating, its manipulation, since he could not extend its bounds. Perhaps the most remarkable instance is that which he borrows from " the sieur Gr. D. L." His object is always to get the result quickly, without confusion, without long working lines, and with as few of them as possible. Hence, in this case, he avoids the use of a line elevation for his heights, and avoids also coming continually down to his ground-line for his dimensions He marks six feet on his ground line as a scale, and divides the first into inches. This scale he runs back to the point of sight. Provided thus with a regularly diminishing scale, if at any point on the ground he wishes to take or raise a measurement, he has his scale diminished to its due extent at the very point. He merely draws a level line at the point, and this, as it crosses the scale, indicates where he has to set his dividers. Then, to avoid having a distance-point off his paper, or at least a great way off, he measures his distances within by a scale only one-fourth of the scale for the widths and heights of which we have just been speaking. His distance-point is quite near. It is, in fact, a fractional distance-point measuring fourfold; and how accessible it is the student will very soon find. Furthermore, he uses the distance-point as it is used in Preface vii Fig. 52, page 76, which, in fact, is also to be placed to his credit. Nowadays, the architects are the only people who can make perspectiye drawings ; the art-student usually only draws impossible geometrical figures by a roundabout process. With the great change for the better in the ■ syllabus of the Examinations of the Board of Education there is a probability of more useful study being followed. The syllabus does not, however, touch the practical side, such as designers would employ, and the student would be advised not to omit that kind of study. The secret of success in such a subject as perspective lies in the student having the whole range planned out, so that he knows where his study actually comes to an end. Without such tabulation he must always have the feeling that with every new problem some unheard of difficulty will present itself. Under such a strain he never gets fairly face to face with what he has to do. This tabulation must be done by the student himself ; he must plan the whole ground out, taking, perhaps, the rules given on pages 267 to 270 as a rough guide. He must settle how many figures he regards as sufficient to cover all difficulties — a square, a triangle, a circle, a curve ; or, again, a cube, a pyramid, an object with legs. Then there are the difficulties of position — parallel, at an angle, near, remote, high up, low down. Then the difficulties of tilted position. viii Preface All these matters are capable of limitation, beyond whieb it is waste of time to travel. The following pages have been put together with a view to their being read, so that the student is advised to gain some general acquaintance with the subject before worhiTig out problems. Before he begins working, he should certainly sketch solutions, doing the work by free- hand instead of by rule and compass. CONTENTS PAGTB The New Syllabus of the Board of Education . . . xiii 1. What Perspective is 1 2. Cuboid Forms the most suitable for Perspective Problems . 2 3. Why things vanish 4 4. The Picture-plane .... ... 4 5. Objects protruding before the Picture-plane .... 8 6. Sketching and working Perspective Drawings . . 8 7. Sketching — ^the Three Lines 10 8. The Three Lines — their Directions 13 9. The Distance of the Object from the Spectator ; how Aspect is affected by it 14 10. The Distance of Objects on the Right or Left of the Spectator ; how Aspect is affected by it . . .17 11. Different Positions below or above the Eye; how they affect the Aspect 23 12. The Difference in Angular Position of the Object ; how it affects Aspect 24 13. Differences in the Inclination of the Plane on which the Object stands 25 14. Different InclinatioQS of Planes; Effect on the Aspect; Planes perpendicular to the Picture 29 15. Different Inclinations of Planes ; Effect on the Aspect ; Planes inclined to the Picture 30 16. Vanishing Points of Lines in the Various Planes ... 32 17. Guessing Measurements 48 18. Eadial Projection 50 19. Vanishing Lines and Picture-lines 53 20. The Horizon and the Ground-line 64 21. The Theory of the Horizon and of Vanishing Lines and Points generally .....'... 58 X Contents FAGB 22. Finding Vanishing Points on the Horizon .... 61 23. The Line of Heights 64- 24. Architects' Perspective 67 25. Measuring-points 70 26. Parallel Perspective 77 27. Objects in Angular Positions 83 28. Angular Vanishing Points ; Angular Perspective . ■ • 86 29. Examples of Angular Perspective 90 30. Inclined Perspective 94 31. To find a Vanishing Line 95 32. To find tiie V.L. of Horizontal Planes ^"i 38. To find the V.L. of Vertical Planes, perpendicular to the Picture 99 34. To find the V.L. of Inchned Planes, perpendicular to the Picture 99 35. Planes parallel to the Picture 100 36. To find the V.L. of Vertical Planes inclined to the Picture . 101 37. Finding the V.L. of directly Ascending or Descending Planes 104 38. Finding the V.L. of obliquely Ascending or Descending Planes 105 39. Laying down the Eye 108 40. The Scaffolding of a Problem 113 41. Finding Picture-lines 115 42. Picture-lines continued . 119 43. Vanishing Points and Lines vanishing to them . . . 122 44. Chess-board in Parallel Perspective shown in aU the Planes . 1 25 45. Perpendiculars to the Six Different Planes .... 129 46. A Few Hints to Draughtsmen 136 47. Projection of Shadows 142 48. Shadows cast by Artificial Light 144 49. Shadows cast upon a Vertical Plane by Artificial Light . 1 48 50. Shadows cast on an Oblique Plane by Artificial Light . .150 51. Shadows of and on Cylinders, etc., thrown by Artificial Light 152 52. Shadows cast by the Sun 154 53. To find the V.P. of the Sun's Bays 155 54. Shadows of Lines without V.P.'s when the Sun has no V.P 156 55. Shadows of Lines parallel to the Picture with the Sun behind the Picture 158 Contents xi PAGE 56. Shadows of Lines which vanish, and are cast by the Sun when its Bays vanish 161 67. Shadows of Lines vanishing, the Sun's Eays not vanishing 164 58. Shadows of Circular Objects 164 59. Reflections . . 173 60. Aerial Perspective 181 61. Solving Problems (Questions set at Examinations of the Board of Education) 182 A Set of Eules 267 Index 271 THE NEW SYLLABUS OF THE BOARD OF EDUCATION IS AS FOLLOWS : The Candidate will be expected to show — (a) Skill in using instruments and working out one or two problems accurately, and (b) Evidence of ability in the ready application of the rules of Perspective to the representation of objects views of buildings, landscapes, etc., by freehand sketches in pencil, ink, or sepia. There will be 1st and 2nd class passes. A 2nd class success will be accepted for the Elementary Drawing Certificate, and a 1st class for the Art Class Teacher Certificate. Candidates must qualify in (A) and (B). (A) Kepresenting in perspective from plan and elevation or from specification, simple solids or objects of plane or curved surfaces having one line or surface on, or parallel to, the ground-plane. Drawing and measuring lines inclined to the hori- zontal, and contained in vertical planes inclined to the picture-plane. Drawing figures or solids in perspective, some of whose leading construction lines are horizontal and the others contained in vertical planes at right angles to the horizontal lines, e.g. a cube with one xiv Syllabus edge horizontal, and one face making a given angle with the ground. Drawing solids having plane or curved surfaces in oblique positions, and all their constructive lines inclined to the ground, such representation being limited to problems which can be solved by the treatment of an oblique plane and perpendiculars thereto. Drawing reflections of solids in plane mirrors, hori- zontal or vertical. Drawing shadows of lines, surfaces or solids, recti- lineal or curved, upon any specified planes and ou surfaces of single curvature, by natural or artificial light. (B) Finding and describing from views given in per- spective the actual dimensions, positions, and other particulars respecting the objects represented under the conditions of any of the foregoing classes of subject (or in the case of shadows and reflections, ascertaining the position of the source of light, reflecting surface, etc.). Idicating how change of position, say of the spec- tator, of the object, or of the source of light, etc., aifects the representation of the given objects, etc. Indicating effects of distance on the appearance of objects, shadows, etc. Pointing out and correcting errors in perspective in respect of given subjects. ADDITIONAL NOTES TO THE EDITION OF 1910. I. Peaotical Application of Perspective. Erect a sheet of glass, say about 30 x 20 inches, vertically upon a table. Place on the table a strip of paper 1 foot wide and, say, 6 feet long, marked across into squares. Let the narrow end of the paper touch the glass ; 12 inches in front EYE' Fig. i. of the glass make a station point on the table, and erect a rod, say, 18 inches high, the summit of which is to be the eye. Placing your eye to the top of the rod, and looking at the paper, draw, in water-colour, upon the glass, the image of the squares on the paper. Fig. i. shows this arrangement. Fig. ii. shows the working upon the picture. What do we learn from these ? That our strip of six squares becomes reduced in vision to a form which i xvi Additional Notes has its sides converging to C.V. ; that the front edge of the 'first square A A' touches our picture, and is there actual 3 that the first square, AA'BB', is very distorted in the perspective view ; that if we place a Distance Point, D.P., as far along the horizon as the eye is from the C.V., we can measure the limits of the squares backwards. We find, in short, that actual and geometrical perspective correspond. The square CD is free from distortion, and we come to the conclusion that our picture need not begin lower than, Fig. ii. say, the line C. If we measure the line CC on our glass, we find it to be 4 inches ; the 12 inches of AA' have been reduced to 4 inches. If, therefore, we start our picture at the level of C, we have to adopt a scale of 4 inches to 1 foot. We find that this new picture line through C is 6 inches below the horizon on our glass, or 1 foot 6 inches according to the scale of 4 inches to 1 foot. Applying the same scale to the distance of the eye before the picture, we may now call it 3 feet, for 12 inches on the scale of 4 inches to 1 foot is 3 feet. This indicates a convention in ordinary geometrical perspective ; for the eye really remains only 12 inches before the picture. We see by our experiment how it comes about that the Additional Notes XVI 1 measurements upon the picture line are drawn to a scale, and are yet talked of as actual. Consider now Fig. iii. This is a landscape. The horizon is 6 inches above the lower edge of the picture. How far before us is this front edge, A ? To ascertain that we must decide how far the spectator is to stand before the picture to get properly into its perspective. The picture is 24 inches wide. Suppose we view it at twice that distance away = 4 feet. We must also decide whether the spectator is standing Fis. iii. on the level of the ground. Assume that he is, and that his eye is 5 feet above the ground. To find the distance that the middle of the lower edge of the picture, A, is from the spectator — ^how far A is along the road — we must continue the lines B and C till the space BC becomes 5 feet. We can calculate this readily. If 4 feet gives a drop of 6 inches from C.V. to A, it will take 40 feet to give a drop of 5 feet, or 36 feet to give a drop of 4 feet 6 inches, which is the space below the edge of the picture. Therefore, had we drawn the picture line 5 feet below C.V., we should have had to vanish a distance of 36 feet inwards to get to point A. We see, therefore, that the picture which we paint begins many feet away from us. But if we want the spectator to feel XVI 11 Additional Notes \ .,.<■?■■ ' in.'- -r' — o 1^ *-*^i A very much, in and among the scene we depict, we introduce objects or figures which are standing on nearer ground than that included in the picture (Fig. iv.). Many portraits, and all half or three-quarter figures are of this order. As a general rule, one should stand away from an object fully three times its height, or greatest dimension. From a picture we usually stand not nearer than twice its diagonal from it, perhaps not nearer than three times that measurement. Sometimes artists paint their pictures from points nearer to the picture than the spec- tators generally stand. "When this is the case there is some loss of reality on the spectator's part. If the picture is one of rigid perspective, as in archi- tectural subjects, rooms, streets, and the like, the outer parts of the picture look distorted, till the spectator gets into the correct position. For geome- trical perspective is a faultless science, and distorted pictures are only such so long as they are not viewed at the proper distance. It is consequently a great mistake to draw the perspective from a point impracticably near. Dlirer did so in his beautiful print St. Jerome in his Cell. The loss of reality is very great in that fine design. It is better to take a distance too remote than too near. Commonly it is said that comfortable vision is limited to mx^./^ HL Fig. iv. Additional Notes xix 30° around the central visual ray— 60° in all. I find 20° to 25° more nearly the correct range than 60°. II. Invekse Peespectite. Inverse perspective is very important to the artist. He generally sketches his design purely by freehand, leaving the laws of perspective alone. By so doing he can arrange his matter more artistically. But if he wants to test his work, he must do so inversely. Many of the inverse questions set at Examinations can be readily solved by persons who have a grasp of the subject such as is covered by this book, but some questions depend upon simple extensions of the subject, which, in former editions, were not dealt with. All one's troubles in perspective come from following stereotyped methods, instead of mastering principles. In the previous note it has been shown how the front edge of the picture can be taken back to any convenient position. In a word, we can have as many ground lines to the ground, or picture Hnes to other planes, as we see fit. Eefer to Fig. i. again. Upon the glass picture plane we can choose the line BB', or CO', or which- ^p cv/v\p ever we hke as our ground '^~~-^^ Vy^ line. In Fig. ii. we chose OC All that happens is that the scale alters. A simple question set in 1908 depended upon this : " Two lines of equal length are given, also Hor., C.Y., and G.L. find position of Eye." The eye wiU obviously be under O.V. The clue to its position is given by the slanting line B, whose V.P. can be found. Now one can only get to the eye by means of V.P.'s and XX Additional Notes M.P.'s. The ground line does not help us, because B is away from it, and what troubles us is the distance from B to the G-.L. We therefore assume a new or casual G.L. through the first point of B. What we want to get is the M.P. of B, for one V.P. alone will not enable us to get to the eye. We know B = A, so bring forward A to the new G.L. Take the size thus given, and set off against B, and N\P carry the measuring line through the end of B, and so find M.P. A similar question (April, 1906) is where a four-sided figure ABCD, • is given. We are told that AB = 12 feet, AD = 9 feet, and BC = 6 feet, and that the figure is at 60° to the left. We have to find horizon, C.V., and eye, and to get these we must find V.P. and M.P. of AB. Had the top line, CD, been parallel to AB, we could at once prolong AB and CD till they met. We have to find such a level line. BC is 6 feet, AD is 9 feet. Divided into six or nine parts, they both yield their scales. It will suffice, however, to divide AD into three equal parts. We thus find E 6 feet above A. EC is then parallel to AB, and we can find V.P. Now to find M.P. Draw a new ground line through A ; on it mark 12 feet, that is, double AE just found. Carry the line back through B, and find M.P. AD and BC are given vertical in order Fig. vi. Additional Notes XXI that we may know how to draw the horizon, for we have no other indication of the level. Other prohlems depend upon what we may call the casual measuring points. Lines are measured by obtaining other lines to cross their terminations. All this is dealt with on page 71. We find that one M.P. (really one pair, as shown in Fig. 50) wUl give us on the picture line the real size which we are getting on the line which is vanishing. This M.P. is the M.P. above all others. But any point will serve to repeat distances. Consider Fig. vii. There ABO are two equal cVa L a - H F ^\mp ^ E K ; — — h >SMi^ I Fig. vii. dimensions upon the same line. Their proper measuring lines wiU be AA', BB', OO. A'B' = AB, for AA'X is an isosceles triangle. But any parallel lines through A, B, and C will yield equal dimensions on PP. Thus we obtain c, h, and a, and though ai is less than AB, yet ab = Ic just as AB = BO. Problems to be worked in confined spaces are usually to be done by this casual M.P. Thus EFGH (Fig. vii.) is a rectangle : xxii Additional Notes find the horizon, no working lines to be beyond the limits given. If we could continue EF and GH till they met, we should at once get Y.P. and so the horizon through it. That not being allowed, we must find either another Y.P. or an M.P. We cannot find the true M.P. but we can find a casual one. By having EF and G-H equal, we are enabled to do this. We create two picture lines, one through E, the other through Gr. We mark my distance, EK, and make GrL equal. Draw KF and LH. These are over one another, and will meet at a casual M.P. on the horizon. Questions are set in which half of an object, a house, a desk, is given, and the other half has to be got. Here, again, the real M.P. not being obtainable or allowed, we use a casual M.P. An example of such a problem is also given in Fig. vi. If {he measuring point is wanted, some clue must be given. This generally is that a receding line is given as equal to one parallel to the picture. We had an example in Fig. v. But sometimes the line which gives the clue is vertical. A question (June, 1907) was similar to that of the rectangle EFGrH in Fig. vii. But it asked for the measuring point. The clue was given in there being a border to the rectangle. The border was equal all round, and so an actual measurement could be made. Among other means of working inverse questions, the use of the V.P. of diagonals must not be overlooked. A question of a table (June, 1907) depended upon the use of this V.P. Students who are afraid of the " other " planes — oblique, vertical ascending, etc. — lose valuable aids on occasion. Vertical planes should most certainly be thoroughly mastered. In all cases remember where the perpendkulcars to the planes go. Do they vanish ? and if so, where ? SET OF EXERCISES FORMING A COURSE OF i8 LESSONS. The references are to pages, or figures, in the book, where help may he gained in dealing with the problems. P. 66 means page 66, and f. 66 means fiigwre 66. Where a figure is referred to, it is intended that the whole comment upon that figure should be considered. Lesson 1. (i) Erect a sheet of glass in the manner described on page XV. above, and upon it draw a cube. The cube has 12 inch sides and stands on a horizontal plane. It is 2 feet from the glass, and one side of it is parallel to the glass. On the other side of the glass arrange the position of the eye, 18 inches above the horizontal plane on which the cube stands, and 12 inches before the glass, and opposite the middle of the cube. Place your eye at the position just found, and view the cube through the glass. Keeping the eye steady drawn on the glass the cube as seen. Mark on the glass the horizon and the C.V., noting that the receding lines of the cube are directed towards C.V., which is thus their V.P. , (ii) Make a geometrical drawing of the same subject, using the measurements full-size. [This drawing should give exactly the same result as the preceding experiment.] (References f. 35, f. 43, f. 53, f. 91, f. 97). (iii) Distance (of eye before P.P.) 3 feet. Height (of eye above horizontal plane) 18 inches. A cube 1 foot sides lies on H.P. immediately before the spectator with one side parallel to P.P. and touching it. Scale 4 inches = 1 foot. [This drawing should give exactly the same result as the preceding exercise. The two exercises differ only in the manner of the statement.] Lesson 2. (i) Using the glass as in Lesson 1 (i), draw the present book lying on a table before you. The book is 8 x 5 x 1 J inch thick. Place it 2 feet 6 inches from you. Place the glass P.P. 1 foot from you, and 1 foot 6 inches before the book. Let the book lie angularly, so that the nearest corner is immediately in front (that is neiflier to right nor left of the middle line) and the long and short edges are both at 45° to the glass P.P. Height of eye above the table, 14 inches. Add the curves and details by freehand. (ii) Scale | inch = 1 inch. Height 14 inches. Distance 32 inches. A book 8 X 5 X IJ inch thick lies on H.P., its nearest corner xxiv Set of Exercises touching .the P.P. immediately before the spectator. The edges of the book recede at 45° to right and left. Complete by adding curves, etc., by freehand. [This exercise will give the same result as the preceding experiment]. (References, f. 25, f. 35, f. 43, f. 45, f. 48, f. 50, f. 58, f. 107). Lesson 3. (i) Height, 5 feet. Distance, 12 feet. Scale, | inch = 1 foot. Pour strips of cloth lie on the ground. Bach is rectangular, 6x2 feet. They lie parallel to one another with spaces^ of 18 inches between them, but so placed that the second, third, and fourth successively have their nearer ends level with the middles of the first, second, and third. The long edges are to be perpendicular to the P.P., and that strip which is farthest on the left touches the P.P. with its nearest corner 4 feet on left of centre. (ii) The same as the preceding exercise, but having the long edges of the strips at 60° towards the left, and the piece farthest on the left touching the P.P. by one corner, 4 feet on the left. (References, f. 53, f. 54). LESsoiir 4. (i) Height, 5 feet. Distance, 12 feet. Find the locations of the following points. A is 5 feet on left, 2 feet within the picture. B is 2 feet left, 6 feet within. C is 3 feet right, 9 feet within. D is 6 feet right, 20 feet within. Upon each of these points raise perpendiculars 8 feet high, giving A', B', C and D'. Join AB, BC, CD and A'B', B'C, and CD'. Find the V.P.'s and M.P.'s of AB, BC, and CD, and measure them. Scale \ inch = 1 foot. (References, f. 49, f. 52). (ii) Distance, 12 feet. Scale, \ inch = 1 foot. Point E is 4 feet above the level of the eye ; it is 3 feet on right and 6 feet beyond the P.P. E is the upper end of a vertical line, EP 7 feet long. EF is axis of a pyramid whose base is a square 5 feet sides. One diagonal of the square base is parallel to P.P. (Referenoes, f. 79, f. 140). Lesson 5. (i) Subject given on p. 82. Height, 5 feet. Distance, 10 feet. Scale \ inch = 1 foot. The lowest step is 10 feet square. The subject being completed, consider the upper surface of the uppermost step to be the base of a pyramid 8 feet high. Draw the pyramid. This exercise is in setting back forms one beyond another. Many errors in the working of perspectives are due to coilfusion in this matter, which is really simple. (ii) Subject given on p. 87. Add the pyramid as in the preceding exercise. It would be a useful variation for the student to use a separate picture line for each step, and for the pyramid. Lesson 6. (i) Height, 5 feet. Distance, 12 feet. Scale, J inch = 1 foot. Point A on ground 6 feet on left, and in P.P. is beginning of a line at 35° with P.P. towards right. This line is near side of a road 20 feet wide. At a point 30 feet along the road from A, the road changes its direction, and then runs at 60° with P.P. towards Set of Exercises xxv right. _ After continuing another 30 feet, it again changes its direction and runs directly perpendicular to P.P. (Work the near side of the road first. The problem requires a, space extending from 11 inches on left of C.V. to 18 inches on right of _ it). Add cart ruts, etc. (Reference, f. 49). (ii) Add to the ahove a number of trees. Place anywhere 6 or 8 points. These are bases of trees all 30 feet high. (References, f. 44, f. 45). [Both these exercises deal with long measurements. The student will not attempt to use such a height as 30 feet, but will probably use 10 feet, and, having erected 10 feet at each tree, will merely triplicate the size obtained by using the dividers. In the case of the road, where the span from V.P. to eye is sometimes very great — beyond the span of the compasses — he will draw a line parallel to the horizon, say 2 inches above the eye, and wiU there strilie his arcs and get points like M.P.'s, carrying lines through these points, from the eye up to the horizon. This pro- cedure we follow also when a V.P. is inaccessible. We then know the direction of its vanishing 'pa/rallel, and so can get M.P. We, of course, merely use the geometrical method for dividing one line in the same proportion as another.] Lessoit 7. (i) Scale, full size. Distance, 18 inches. A large cotton- reel stands on one end, its centre 3 inches on left, and in the plane of the picture. The drum of the reel is 3 inches long, 1^ inch in diameter. The flange, at either end, may be repre- sented by a truncated cone, expanding from 1^ inch diameter to 2 inches diameter, the axis being half an inch. The reel has thus a total length of 4 inches. Draw the reel in perspective, standing on a plane 7 inches below the eye. (ii) Draw the same reel when its axis is parallel both to the horizontal plane and the P.P. and 3 inches below the level of the eye, the nearer end of the axis being 3 inches on right, the axis entirely in the plane of the picture. [Note. — This view wiU appear distorted except when viewed exactly from the correct position. The difBculty of viewing the picture from the right position leads draughtsmen to avoid such attitudes, in preference for angular positions which more readily look correct] (Refer- ences, f. 50, f. 162.) Lesson 8. (i) A millstone is 6 feet in diameter, 1 foot thick, and has a square hole in its centre, the diagonals of the square being 1 foot. Draw in perspective when the miUstone lies flat on the ground, the nearest point of its circumference to the P.P. being 4 feet left, 2 feet within. One diagonal of the square hole is parallel to the P.P. Scale ^ inch = 1 foot. Height 6 feet. Distance 12 feet, (ii) The same millstone stands on its edge on the groimd. Its nearer circular face is in a vertical plane receding towards the left xxvi Set of Exercises at 60° with P.P. The nearest point of the circmnference to the P.P. is 6 feet right, 2 feet within. One diagonal of the square hole recedes upwards at an angle of 40° to the Gr.P. (References, f. 60, f. 149.) Lesson 9. (i) Hitherto the exercises have been of subjects upon the ground-plane, or upon some other horizontal plane. Our working has been chiefly upon the horizontal planes, now and then, when getting a height we have used vertical planes, almost without knowing that we did so. But it is unwise for the student longer to neglect the study of the planes which are not horizontal. The use of V.P's " up in the air " and " down below the ground " is very serviceable, and saves a great deal of trouble. Moreover, the true foreshortening is so much more thoroughly exhibited. These V.P's are often called amdental V.P.'s (A.V.P.) — a term used in this book. They are really no more accidental than those on the horizon. In this lesson the student should learn that there are only seven possible positions of planes (p. 27), of which one, that which is parallel to P.P., has no V.P.'s, aad does not recede. There are thus six which have vanishing lines or " horizons," V.P.'s, M.P.'s, and so on. The lesson should be occupied in studying the six planes, or six positions of planes, using the sheet of glass, and noting the positions of the vanishing lines. Pages 29 to 48 deal with these matters. , Let the following be the Exercises. On six different papers prepare the following : Mark four points in any positions. Mark one C.V., and the others A, B, and C. Let the points not be over one another, nor directly right and left of one another. The positions need not be the same on all the problems. (i) Draw the V.L. of horizontal planes (the horizon). Joia AB and AO and BC. These are three Hues in a horizontal plane, and together form a triangle in the plane. Find the V.P.'s of AB. AC, and BC. \NoU — The points A, B, and C should fall by preference to one side of the line. If the V.L. pass between the points, the points will be on two planes, not one.] (ii) Draw the V.L. of vertical planes 'perpendicular to P.P. Form AB, AC, and BO as before. They are lines in a vertical plane. Find thek V.P's. (iii) Draw the V.L. of an inclined plane, at any angle to the horizontal, and do with the points as before. (iv) (a) Draw the V.L. of a directly ascending plane rising at any angle, and do the same with the points. (6) Draw the V.L. of directly descending plane, etc., as before. (v) Draw the V.L. of a vertical plane at any angle to P.P., etc., as before. (vi) (a) Draw the V.L. of a plane obliquely ascending towards right, et-c., as before. Set of Exercises xxvii (6) Draw the V.L. of a plane oUiqudy descending iowax&s right, etc., as before. In all these cases, if the V.L. is drawn between the points, the points cannot be on the same plane, but will be on two planes. Lesson 10. The position of the Eye in Belation to the Six Planes. — Make a paper or cardboard model of the relation of the eye to the P.P. and to the V.L.'s of all possible kinds of planes. (See pp. 62, 63, 108 to 112.) Make the following geometrical drawing : Mark C. V., draw horizon, place eye when 12 feet before P.P. Scale ^ inch to 1 foot. Draw V.L. of vertical planes, perpendicular to P.P., and find its eye. Draw V.L. of planes inclined at 30° to H.P., rising towards left, and find its eye. Draw V.L. of vertical planes making 40° with P.P. towards right, and find its eye. . Draw V.L. of planes ascending directly from spectator at angle of 50° with H.P., and find its eye. Draw V.L. of planes obhquely ascending towards right, having their intersections with H.P. at angle of 35° with P.P. towards left, and their angle of inclination with H.P. 40°, and find its eye. Find the O.V.L. in all cases. The C.V.L. is the end upon the P.P. of the C.V.E. (Central Visual Kay). (Reference, pp. 110 to 112.) Lesson 11. Height, 5 feet. Distance, 12 feet. Work 6 problems like those in Pigs. 85 to 90 taking the positions of the planes from the last lesson. Let the Picture-Lines in all cases be 5 feet from the V.L.'s. Get, in each case, a rectangle, as is done in Figs. 85 to 90, by using 3 feet and 6 feet to right and left of the middle of the P.L., and using 40° on right and 50° on left for the V.P.'s. Get the M.P.'s and measure the sides of the rectangle. [Page 125, line .3 — for square, read rectangle.'] Lesson 12. Height, 5 feet. Distance, 12 feet. Work 6 problems like those in Figs. 91 to 96, making each square 2 feet. Take 9 or 16 squares as one large square and within it draw a circle (Fig. 150). The positions of the planes to be the same as in Lesson 11. Lesson 13. The conditions being the same as in Lesson 11, find in each of the 6 planes (6 separate problems wUl be desirable) a square, 6 feet side, of which one side is paralled to the P.P. (as in the last lesson), and the nearest corner on a line which touches P.L. 4 feet left, and proceeds to C.V. or C.V.L., the nearest comer being 4 feet up that line. (In the case of the first problem, dealing with horizontal planes, the nearest comer will be 4 feet left and 4 feet within the picture, but in some of the eases the actual distance will not be 4 feet, owing to the plane being at an angle to the picture). Convert the squares into cubes, as is done in Figs. 97 to 102. This is a lesson on finding perpendiculars to the several planes — a most important matter. xxviii Set of Exercises Lesson 14. (i) Distance, 12 feet. Scale, \ inch = 1 foot. Find a point, A, in the P.P^ 5 feet left and 3 feet below level of eye. Join A to C.V. Find a point, B, 6 feet along the line. The line ACV, is the intersection of three planes — a horizontal, a vertical, and one inclined at 45° to the horizontal and descending from left to right. AU these planes are perpendicular to the P.P. At B draw 3 lines, one in each plane, and all parallel to the P.P., and each 2 feet long. Mark the extremities 0, D, and B. Join CA, DA, and E A, and find their V.P.'s in the V.L.'s of the three planes. Find their M.P.'s, and find their correct lengths. (References, f. 63, f. 64, p. 99, f. 65, f. 81, f. 88, f. 191 to 93). (ii) Height, 5 feet. Distance, 12 feet. Scale, \ inch = 1 foot. A point on G.P. is 6 feet left and 6 feet within. It is the nearest corner of a house of simplest form. The front of the house is 20 feet long and 9 feet high to the eaves. The front is in a vertical plane, making 40° with P.P. towards right. The side of the house is 12 feet wide and becomes the gable end of the house. The slopes of the gable are 45° with the horizontal. Add a door, windows, a chimney, and some railings parallel to the front of the house. Note the V.L.'s of all the planes of the house, and find all the P.L.'s. (References, f. 89, f. 94, f. 103, pp. 119 to 121). Lesson 15. Height, 5 feet. Distance, 12 feet. A small paved court is six-sided. Five of the sides are equal, 4 feet each, and form five sides of a regular octagon. The six comers of the court, reading from the left, are A, B, C, D, E, and P. The sixth side, AF, of the court touches the G.L., A being 6 feet on left. Each of the five sides AB, BC, CD, DE, and EF, is the base-line of a plane sloping down and touching the G.P. AB is the base-line of an inclined plane 30° with H.P. rising towards left. BC is base of an obliquely inclined plane rising towards left at the same angle to H.P. CD is base of a directly ascending plane also at 30° to H.P. DB is base of a vertical plane, and so is EF. Find the V.L. and P.L. for each plane. Mark the cenfre of each base line, that is, find points midway between A and B, B and C, etc., and mark them m. Run a line up each plane from m, perpendicular to the base-line. Upon these lines find points n, distant from m, in the case of the inclined planes, 2 feet, and in the case of the vertical planes 6 feet and 9 feet. At n in each case draw a line no, 9 feet long, perpendicular to the plane it arises from. The line no is the axis of an object consisting of a pyramid whose base is 2 feet square, and whose altitude is 4 feet, and whose axis is prolonged 5 feet to o by a rod, which at its uppermost foot carries a flag 1 foot square. Each of the five planes cames one of these objects, one corner of the square base being on nm, and the axis being in the same vertical plane as n and m. The flag is in all cases to be in a vertical plane (a little consideration will show that the flag is in the same Set of Exercises xxix vertical plane as n and m.) (References, p. 25, etc., p. 94, etc., f. 65, f. 68, f. 70, p. 108, etc., p. 129, etc.). Lesson 16. — The roof of a building is 78 feet long and 25 feet wide. Its pitch is 50°. The nearest comer, A, is 10 feet left, 30 feet within the picture, and 30 feet below the eye. One of the long sides passes away to the right at 50° to P.P. Along this side are six dormers, each 8 feet wide, rising 4 feet above the eaves, and surmounted by gable roofs of 50° pitch. The dormers are separated by 2-foot spaces, and the outermost are 10 feet from the ends. Draw in perspective, the eye being 12 feet from the picture. Scale, \ inch = 1 foot. [As 30 feet is a long distance for the eye above the H.P., take 10 feet first, and, having found the nearest corner, drop it down thrice the distance below the horizon. Then, through the point A thus found, draw a Picture Line, and use the scale which results from assuming the distance from A to horizon to be 30 feet. Use throughout V.P.'s in vertical and oblique planes as much as possible.] Lesson 17. — Height 5 feet. Distance 12 feet. Scale, \ inch = 1 foot. A cube stands in the position shown in Fig. 115. Its edges are 5 feet. Two sides are vertical, with one diagonal vertical. The nearest comer of the edge upon the ground is 4 feet right, 6 feet within. The edge vanishes to left at 40°. Find a point A, 4 feet left, 4 feet within. The exercise is upon dealing with different planes cutting through the cube, as if the cube were embedded in them. Take them in order, repeating the whole exercise if necessary. (i) A horizontal plane, 2 feet above the ground, cuts through the cube. Find the intersection. (ii) An inclined plane, whose intersection with G.P. includes point A, and whose inclination to G.P. is 30° towards right. (iii) An obliquely incKned plane, inclined at 30° to G.P. towards right, and whose intersection with G.P. is at 45° to left, and contains A. (iv) A directly ascending plane, at 30° to G.P., commencing at A. (v) A vertical plane, whose trace on G.P. includes A, and which passes through the centre of the cube. (In such problems, it is wisest to use oblique perspective as much as possible. Even to get the cube, a traer cube wiU be got by the oblique methods. The interpenetration of one plane with another can be found by the ordinary roundabout methods of finding the intersections of the plane with imaginary verticals from the comers of the cube. But the intersections wiU be best managed by oblique methods. Where the V.L.'s of two sets of planes cross is the V.P. of all intersections of those planes. Thus the V.L. of the inclined plane passes through the horizon at C. V. C.V. is therefore the V.P. of all intersections of all inclined planes at the same angle with all horizontal planes. XXX Set of Exercises As a rule, one commences with a vertical plane. Vertical planes are indeed valuable means of getting from one plane to another.) Lbssojs- 18. — The previous lesson provides us with subjects suflScient for the study of drawing shadows. We have the cube standing on its edge on the G.P., and we have the same cube partly embedded in other planes. Our only upright lines are the two vertical diagonals. These are not actually lines of the cube, but can stand as such and serve our purpose. Indeed, by casting the shadows of these verticals, we can prove whether our shadows of the slanting lines are true. (A) Shadows from arUJkial light— [Note— If AVP„ Pig. 115, were below a certain position, the ESi would be on the right of VPi, alove the horizon, but still on the line through seat and VPi. When Light and AVP are on a line parallel to the line through seat and VP, no ES is found, and the shadow is parallel to these lines. When Light and AVP coincide, the point is also BS.] Find a point 4 ft. left and 6 ft. within. At this point erect a line 8 ft. high. The summit is an artificial light. (i.) Cast shadows from the cube on the G.P. (Fig. 115). (ii.) Find the inclined plane as in Lesson 17, and cast the shadow of the cube upon it. (iii) Find the obliquely inclined plane as in Lesson 17, and cast the shadow of the cube upon it. (iv) Find the directly ascending plane, as in Lesson 17, and cast the shadow of the cube upon it. (v) A vertical plane, perpendicular to the picture, touches the comer of the cube which is farthest on the right. Find the shadow of the cube upon it, considering the ground-plane non- existent. (vi) A vertical plane parallel to the P.P. is behind the cube which touches it by its hindmost corner. Find the shadow of the cube upon the plane, the G.P. being considered non-existent. (B) Shadows cast hy the Sun — (vii) Bepeat all the problems, considering the sun to be before the spectator, its rays inclined at 40° to the ground and in planes 50°, with P.P. towards left. Problem (vi) is unsuitable for this lighting. [Note — Shadows on planes parallel to P.P. are parallel to lines joining Sun and V.P.] {viii) Repeat all the problems, considering the sun behind the spec- tator or left, its rays inclined at 30° to G.P., and in planes receding towards right at 40° to P.P. ^ix) Repeat all the problems, considering the sun in the plane ot the picture, its rays at 45° to the G.P. [In all these problems the main subject should be worked in strong line, and the shadows worked on tracing-paper.] PERSPECTIVE FOR ART STUDENTS I. What Perspective is. We all know that when we draw a view of an object we have to make its more distant parts smaller than its nearer. We all know that as things recede into the distance they become smaller in appearance. In short, Fig. 1. — An object dimiDishing as it recedes. we know that some such regular diminution of parts as is shown in Fig. 1 must take place. This regular diminution of receding parts is called the perspective of the objects ; and we draw in perspective when we pay due attention to this diminution. 2 Perspective The laws of perspective affect all objects, no matter how varied or curious their form, but the influence of these laws is most evident when the objects are of severe geometrical shape, such as railway-tracks, straight roads, buildings, aid the like. 2. Cuboid Forms the most suitable for Perspective Problems. It is no exaggeration to say that the parallelepiped (brick-form) and the cube are the most convenient solids for perspective representation. As a rule all forms are considered to be enclosed in one of these. Both the cube and the brick-form consist of lines in three directions only. These are lines to the right, to the left, and upwards. They are all perpendicular to one another. In Fig. 2 the influence of these cuboid forms is illustrated. Point A is the corner from which start the three lines — to right, to left, and upward. He who has mastered these three lines, especially if he can sketch them in by freehand, has practically mastered perspective. Of course the solids do not neces- sarily occur lying flat on a horizontal surface, such as the ground. He, too, who would master sketching in perspective, must certainly be able to draw from imagi- nation a cube in any position, but most assuredly when Cuboid forms most suitable 3 lying on the ground. He must be able to draw upon the sides of the cube stripes parallel with the edges. It is by this means that- all the form is obtained in Fig. 2. It is little use the student drawing cubes and such like from nature. One thing alone will help him, and that is the determination to make his cube, whether he draw from nature or no, look cubical, and practice till his Fig. 2. — Cuboid form prevailing. hand makes the necessary radiation of the lines auto- matically. The student will see how, as he progresses with the subject he gets into the habit of always finding, as practically the first thing to do in every case, how to get his lines to the right, his lines to the left, and his upright lines. Usually he finds he attacks his object best by getting its base on the plane the object stands on, and then raising perpendiculars to the plane for the upright edges. For this the different kinds of planes demand different treatment, as will appear in the proper place. Perspective Why Things vanish. We see by means of a lens collecting rays of light from the objects visible, and conducting them to a screen (retina) at the back of the eye. The lens and retina are the apparatus of vision. The image thus cast upon the retina is apprehended by the optic nerve. The lens and Fio. 3. — Objects of equal height and at different diatanoes from the lens giving images of different size on the retina. retina are in fixed positions, so that the nearest and the most distant objects must alike be imaged upon the same retina. A simple diagram, Fig. 3, serves to make clear that the rays of light from the more distant object, B, cannot expand to the extent reached by those from A. The Picture^plane. The image on the retina of the human eye is no more than half an inch across. A drawing of such dimensions The Picture-plane 5 would be quite useless if obtained. The idea of repro- ducing the actual image on the eye is therefore abandoned. Instead of the rays of light being regarded as piercing the lens and expanding upon the retina, they are held to be intercepted hefore they reach the lens, which is hence- forth called the eye, by a plane called the pictuke- PLANE. It will be seen by Fig. 4 how that this picture-plane may be placed at any distance from the eye with no other r: Fig, 4.— a, the object; B, the ball of the eye; L, the lens; C, a picture-plane between the object and the lens ; D, a plctnre-plane beyond the object. result than that of altering the size of the image on the plane. It will also be seen that the image is no longer upside-down as it was on the retina. All images on the picture-plane are produced by inter- sections. Thus in Fig. 5 rays of light are conyerging from the actual real points A to the eye. These four rays severally pierce the picture-plane, and produce there the points B. Each point signifies to the spectator (eye) a corresponding point A, for B is precisely in line with A. As far as the spectator is concerned, B and A are 6 Perspective both one ; in fact, the whole ray of light \% when viewed from the eye, merely a point. It is very important that Eye Fm. 5. — The image, B, produced on the picture-plane by rays inter- secting the plane. the student should understand this, for it is somewhat startling to have to accept a point on the picture-plane, Fig 6.— The artist touching a point some miles away. perhaps not above 12 feet from one, as actually a point miles in the distance. But of course, to the spectator, if a point on the picture-plane Nicies the point The Picture-plane 7 in the far distance, it is that point, for it takes its place in the view. The relation between picture-plane and spectator is therefore as follows. The picture-plane is some distance before the spectator, perhaps 12 feet, but the distance will vary according to the size required for the drawing, as has been sufSciently shown by Fig. 4. Innumerable Hon C V Fig. 7. — The eye and the picture-plane. rays of light will pierce the picture and converge to the spectator's eye. Of all these rays the one which occupies the shortest distance between the picture-plane and the eye is known as the Principal Visual Ray, P.V.R This shortest distance is, of course, the stated distance of the picture-plane from the eye, say 12 feet. The point where this P.V.R. pierces the picture-plane is the Centre of Vision, C.V. 8 Perspective 5- Objects protruding before the Picture'plane, The image remains true whether the object protrude before the picture-plane or not. For its image is obtained by radial projection from the eye, so that the rays as truly cast the images of objects before the picture upon it, as they bring up the images to it of objects behind. Precisely similar objects one before the other behind the picture are given in Pig. 55, p. 81. It is quite unusual for the object to project before the picture-plane in examination questions, but instances are given later on, among the hints for designers and architects, wherein it is an advantage to allow parts of the object to come through. In Question 5 (April Examination, 1901) solved on p. 205, the object protrudes before the picture-plane. 6. Sketching and Working Perspective Drawings. A capable draughtsman can make a satisfactory per- spective drawing merely by skilfully radiating the lines in a suitable manner. He must, however, know the main rules of perspective. He must know, that is — That parallel planes vanish in the same line ; and Perspective Drawings g That parallel lines vanish in the same point. These are the two great laws of perspective. He must know the following subsidiary fact : — Lines in a plane have their vanishing points on the vanishing line of the same plane. Just as the sketcher must guess where his yanishing points and vanishing lines are, so he must guess his measurements. A very great deal can be done in this way with care ; but everything depends upon skUl, and a good guess where a vanishing point should be will avail one little if the lines supposed to be converging thither go somewhere else instead. The difference between a sketched and a worked per- spective is one of accuracy, of accuracy due to the statement to the draughtsman of certain particulars which he can use mathematically to obtain his points and measurements with absolute precision. To the degree that these particulars are wanting the drawing must be sketched, and cannot be worked. What is the least information we can do with ? The least is a general instead of definite knowledge of the direction of the lines and planes. In making our sketch, we, however, convert this general or vague knowledge into precise and exact guesswork. If we know that the lines of a wall vanish somewhere on our right we do not vanish them "somewhere," but as definitely as we can in some direction which we guess to be about right. lO Perspective Sketching — The Three Lines. In paragraph 2 we hare seen that perspective deals most readily with objects of cuboid form, and that to deal with objects which are not cuboid it is advisable to imagine them enclosed in a rectangular framework. Fig. 2 illustrates this. He who sketches in perspective must master the three lines, the lines in three directions which bound the object. Suppose, therefore, we approach the representation of a cuboid object such as that in Fig. 8 — what have we to consider, what have we to do, to make our drawing perspectively correct? It is the object of this book to answer that and similar questions. Fig. 8. Let us exhaust first the knowledge we have which will affect this case. 1. We know our object has six sides, of which we are to see three. The uppermost surface, or top, is similar to the base, or side, on which the object stands. The base is represented in our Sketching — the Three Lines ii drawing at present only by the two edges nearest us, the two lowest lines of the drawing. 2. We know that it is the nature of a cuboid form to have the base and top precisely similar. We know that not only has the top four edges, of which those opposite one another are parallel, but we know that the base repeats these con- ditions. 3. The top and base are thus so similar, and so placed in relation to one another, that they must be always thought of together. Together they giye us eight lines (Fig. 9), of which four pass to ii, ■ T-i 3 J! Fw. 9. — The top and base of a the right, and four cuboid object. to the left. Of each of these two sets one will be invisible in the complete object. 4. We know already that the diminution of objects receding into the distance is regular and subject to law, and that we express this regularity and this law by a reference to parallel lines — we say that they vanish in the same point. All forms and lines, no matter how irregular, are subject to diminution according to distance, but the diminution is not expressible in words, except in such an instance as that of parallel lines or planes. 12 Perspective We know, then, that four of our eight lines in Fig. 9 will vanish to a point on the right, and that the other four will Tanish to another point on the left. This fact, which should be so self-eTident, is often overlooked by students, as all teachers of perspective will admit. Students remember to vanish the front lines properly, and then straightway vanish the back lines anywhere ! No perspective can be done without understanding, and the obvious fact here pointed out is the very first step. If it is properly taken, there is not much difficulty with the rest. 5. We know, also, that what is true of our top and base is true also of each of the two pairs of sides. The front and back correspond exactly. There are the eight lines as before. Four pass to the right, and four pass downward (Fig. 10). We know that "parallel lines vanish in the same point," but we must add to this that parallel lines which are also parallel to the spectator (the expression used in perspective is parallel to the picture) are not regarded as receding, and so do not vanish. Hence our four descending lines in Fig. 10, if supposed to be vertical in fact, are drawn vertically without any con- vergence. There is no need to separately illustrate and describe Fig. 10. — Two of the sides of a cuboid object. 6 The Three Lines — Their Directions 13 the remaining pair of sides, which correspond in every way, both of fact and treatment, to the front and back. The difficulties are thus centred in the three lines expressed in solid line in Fig. 11. We may take the central point at the base or at the top as we please, and \~~-f^'^^\ as is illustrated in the figure, L I 1 but each line must goTern ^**— -"^ ^. ^, ^. ,, " Fig. 11.— The three lines. three others. If, then, we know where each of these three lines goes, we know all* we need to know. 8. The Three Lines — Their Directions. The three lines indicate by their directions the posi- tion of the cuboid object. To the sketcher the position is what it appears to be, because the final judge in these matters is the eye. It is no use saying an object is lying flat on the ground if it looks tilted, nor saying its sides are equal if they do not look square. The worker of perspective, on the other hand, defends himself by showing that he has followed the proper rules, and if the result is wrong it is because perspective as a science is faulty. There is, no doubt, sometimes a 14 Perspective discrepancy between the result whicli satisfies a worker and that which satisfies a sketcher. The fact that per- spective, when worked by rules, does not permit any convergence of line parallel to the picture, is at the root of the discrepancies. We shall see later on where the two systems clash. The variations of the directions of the three lines are due to — 1. The distance of the object from the spectator. 2. The distance of it on his right or left. • 3. The distance of the object below or above the eye. 4. The angular position of the object. 5. The position or tilt of the plane on which the object stands. Each of these conditions will have to be treated separately. The Distance of the Object from the Spectator ; how Aspect is affected by it. At a later stage, when the working of perspective is explained, the student will find he has to recognize a front and a back edge to the ground. He will see that this front edge is very arbitrarily chosen. There is really no front edge to the ground in the sense that there is a back or distant edge. The back edge is the How Aspect is affected by Distance 15 horizon, with which all are conversant who hare seen the sea. If we like we can take the position of the spectator's feet as the beginning of the ground. But no one's picture ever conies so low down as to take in hisv own 'feet, so we may leave the matter in its natural vague state. What, however, is important to notice is that from the front edge of the ground, wherever that may be, to the back edge is an immense distance, expressed on a drawing by a very narrow ppace. The higher a point is, therefore, assuming it to be on the ground, up toward the horizon, the further away from the spectator it is. In Fig. 12 we have several squares lying on the ground-plane. It will be seen at once that the further away the square is, the less its lines are inclined. The Fig. 12.— Five squares of equal size placed sncoessively at greater distances from the spectator. square becomes in the drawing a very sharp dianrond, with its two front edges making so large an angle as to be very little removed from the straight. The two front edges of a square may thus be an indication of the distance the object is from the spectator. 1 6 Perspective Hence in Figs. 13 and 14 the difference of pitch of the two front lines, whether we take those made by the edge _^. ^iliiiWi Fio. 13. — A building at a considerable distande before the spectator. of the building on the ground or the edge of the building against the sky, indicates a difference of Fio. 14. — A building nearer the spectator than that shown in Fig. 13. distance, Fig. 13 representing a greater distance than Fig. 14. Another notable effect of the difference of distance is seen in the relative heights on the drawing of vertical lines of equal height actually. The difference is greater Distance Right or Left affects Aspect 17 when the distance is less. In Fig. 13 the columns on the building differ very little in height, in Fig. 14 considerably. Subjects arranged at a considerable distance have, therefore, an equality of size pervading the parts which gives to the work a monumental character, and fits it better for decorative purposes. I have already referred to this in "Figure-drawing and Composition." We are thus able to treat a considerable area of ground space without greatly changing the heights of our figures or buildings. On the other hand, a nearer view with its short range, and consequently boldly tilted lines, has a realistic vigour which is sometimes very attractive. 10. The Distance of Objects on the Right or Left of the Spectator; how Aspect is affected by it. Students of solid geometry (and none should attempt perspective who have not some knowledge of that science) know that the position of an object is described according to its relation to two planes, one horizontal, the other vertical, and together forming a corner such as is made by a half-open book. Where the two planes pass into or through one another is the intersection between them, and is commonly figured as "XT." If one attempts to describe the position occupied by a 1 8 Perspective thing, one has always to adopt some such bases of position as these planes and their intersection afford. If the student who knows nothing of solid geometry will take a circle of paper and a square of paper, and upon each place a mark, anywhere, and will then try to describe the position of the mark, he will find it impossible in the case of the circle, unless the mark occurs absolutely in the centre, and even then, if the mark be, say, a square, he will not be able to describe its angular position. One cannot hold a circle the right way up, or look at it from the "front." No such terms apply to it. But the square paper immediately provides bases from which measurements can be taken. If our mark is 2 inches from one side and 3 inches " from one of the sides adjoining the first side, there can be no doubt as to position of the point. In perspective (as an art or science) we describe positions in the same way. Thus in Fig. 15, AB and _ CD are two squares or , I cubes placed before the j ft spectator in the position . seen. The plan of the picture-plane is given I as a line, and it is this \ I line which is used as "^ jfKTKTOR the basis for the de- FiG. 15. — Plan of two objects, the picture- • ,• . t, -, m-r^ plane, and the spectator. SCriptlOn. AB and CD are both the same dis- tance beyond the line, an-d would be described as so far " beyond the picture-plane," or " within the picture." — \