CORNELL UNIVERSITY LIBRARY BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND GIVEN IN 1 89 1 BY HENRY WILLIAMS SAGE Cornell University Library NC 750.C67 1921 Perspective; the P^S ll ^SlS S 3 1924 020 581 041 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924020581041 _ 7? J^gsSi m^^^s^jmssci^^. . " ■£3B9EC»K s; " KSfe. SHBS Drawing by the Author A Lyoh Gate PERSPECTIVE THE PRACTICE & THEORY OF PERSPECTIVE AS APPLIED TO PICTURES, WITH A SECTION DEALING WITH ITS APPLICATION TO ARCHITECTURE BY REX VICAT COLE AUTHOR OF "THE ARTISTIC ANATOMY OF TKEE^ " BRITISH TREES," &°C. &*C. ILLUSTRATED BY 436 DRAWINGS & DIAGRAMS BY THE AUTHOR *> 36 PICTURES CHIEFLY BY OLD MASTERS PHILADELPHIA J. B. LIPPINCOTT COMPANY LONDON : SEELEY, SERVICE & CO., Ltd. I 9 2I The New Art Library Edited by M. H. Spielmann, f.s.a, cVP. G. 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By Albert Toft, a.r.c.a., m.s.b.s. With 119 Illustrations. 15s. nett. " Will be found an invaluable aid to the student. . . . Takes the student step by step through the various technical processes, the text being sup- plemented by over a hundred excellent illustra- tions."— Studio. Seeley, Service & Co., Ltd., 3$ Great Russell St. PREFACE IN our Art School days we looked upon Perspective with grave suspicion. We feared that cobwebbed in those entanglements of line there lurked our old enemies, Euclid and Geometry. My own distrust has never been wholly dispelled ; for which reason, out of sympathy for a new generation of art students, I have tried to set down the matter in plain words and to divest it of some problemat- ical exercises dear only to the mathematical mind. These, in truth, sometimes lead to a negative result — the " which is impossible " of Euclid — or they have but little bearing on our art. Dr. Johnson has said : " Long calculations or complex diagrams affright the timorous and inexperienced from a second view, but if we have skill sufficient to analyse them into simple principles, it will be discovered that our fear was groundless." A knowledge of Nature's perspective is essential to the artist. Her laws are not difficult to understand if they are taken one at a time, together with an explanation of the reasoning on which they are based. This is the method which I have followed in Part I. With the aid of some common sense on the part of the reader, it should be sufficient for all ordinary purposes. It is, however, necessary, in dealing with the drawing of archi- tectural details, to resort to some elaboration — elevations and ground plans must be used ; expedients for simplifying the work here come into play. In order to prevent these seeming to confuse the issue they have been kept together in Part III. I have known students to attend a course of well-delivered lectures on Perspective and yet say they did not understand a single word of what the lecturer was talking about. This confusion may arise from the fact that some knowledge of geometry has been taken for granted by the lecturer ; or because ground plans, station points, and a host of intricacies are commonly used as the starting-point for the building up of the object to be iv PREFACE drawn, instead of the object being first sketched and then put into correct perspective. The former method often leads to a most ungainly representation of cubes and circles that cannot but repel the student instead of interesting him from the very beginning. In most fine pictures which have stood the test of time, one sees a keen appreciation of the possibilities of perspective. As the struggles of the early masters in formulating the science are full of interest, I have tried in Part II to tell the tale of Perspective as we see it in the works of the great old painters. It is told of an early Italian painter that to his long-suffering wife's entreaty not to burn the midnight oil he simply murmured, " Oh, this perspective — this beautiful perspective." No doubt he had just discovered a new vanishing point. Although in our day few discoveries remain to be made in the science, several deductions might gain by revision ; and we can present the principles in a simple form. This being my aim I have cribbed without a blush from the teaching of my old friend, L. A. Pownall, and I hope I have remembered verbatim some of the expositions which he so pithily expressed. I have also referred to my well-thumbed copy of Cassagne's " Practical Perspective " and Wyllie's " Laws of Nature "—which affords delightful reading. More recently I have profited by the work of the late G. A. Storey, Professor of Perspective at the Royal Academy, and by Middleton's architectural essays on " The Principles of Architectural Perspective." CONTENTS PART I NATURE'S PERSPECTIVE AS SEEN AND USED DAILY BY PAINTERS CHAPTER PAGES I. The Principle of Perspective in Theory Visual rays — Tracing on glass — Perspective terms . 17-32 II. The Rules of Perspective and Their Application Receding lines ....... 33—43 III. Depths Division of lines — Inclined planes — Squares — Diagonals 44-53 IV. The Use op Plans in Sketching Foreshortened Surfaces Squares — Pavements — Rooms- — Concentric squares . 54-65 V. Inclined Planes Steps — Staircases ....... 66-82 VI. Inclined Planes — continued Roads — Doorways — Walls — Hayfields — Cornfields — Seashore 83-95 VII. The Circle From below — In a square — Parallel — Concentric . 96-102 VEIL The Circle — continued Wheels — Steps — Columns — Rooms — Niches — Towers 103-111 IX. Arches How to draw them — Bridges — Groined roofs . . 112-123 X. How to Draw Curves by Straight Lines Bridges — Flowers — Foliage — Water — Heads — Figures 124-133 XL Architecture op the Village Roofs — Gables — Windows — Chimneys — Church Towers — Battlements — Steeple — Composite pyramid 134-156 XII. Concerning Domes, Turrets and Steeples Octagonal steeple on square tower — Tower and steeple — Composite Domes ...... 157-164 XIII. Perspective of the Sktc and Sea Sky — Clouds — Smoke — Sea — -Distant objects . . 165-171 XIV. Perspective op Reflections Water — Inclined planes — Distant objects — A punt — Arches — Rippled water — In a mirror — Refraction . 172-184 vi • CONTENTS CHAPTER PAGES XV. Perspective of Shadows From the sun — Sun in front — Sun behind — On one side— Artificial light 185-199 XVI. Pavements Tiles : Square, Ornamental — Concentric squares — Octagonal — Hexagon — Lozenge .... 200-204 XVII. Perspective of Boats and Shipping Guiding points for curves — The sketch plan — Correct distances — Effect of distance .... 205-212 XVIII. Perspective from Unusual Points of Veew From a bridge— From an airship — Looking at the sky — Position of the painter — Mural painting — Objects near to, in confined spaces . . . 213-215 PART II PERSPECTIVE ASJPRACTISED BY OTHER NATIONS AND AT OTHER TIMES XIX. Perspective in "Greek and Roman and Other Paintings Greek— Roman — Egyptian — Early Italian — Italian and Dutch Schools 216-232 XX. Perspective in some French and English Paint- ings Engraving — Book illustrating — Painting'of ships . 233-244 XXI. Notes on the Perspective of the Japanese . 244-250 PART III MECHANICAL PERSPECTIVE XXII. Mechanical Perspective : Introduction Architecture- -Plan and elevation — Heights . . 251-258 XXIII. Mechanical Perspective — continued To find the'depth'of a house — Practical and mechanical perspective — Perspective of a ship] . . . 259-266 Appendix. Note I. — Enlarging a sketch. II. — To divide a line in given proportions. III. — To transfer the division of » line — To estimate the measure- ments of a canvas proportionately larger or smaller 267-269 Bibliography ... . ... 271—273 Index . 275-279 LIST OF ILLUSTRATIONS PAGE A Lych Gate. By the Author . . Frontispiece I. The Clubbed Foot. By Ribeea . . 23 II. COCHEM, ON THE MOSELLE. BY GeOEGE COLE 25 III. La Viebge au Donateue By Van Eyck . 27 IV. A Dbying Shed nsr a Bbickyabd. By the Authob ....... 29 ^iV. A Stile. By the Authob . . . .31 ' VI. A Dutch Inteeiob. By P. De Hooge . 35 VII. The Babn End. By the Authob . . 37 VIII. Room in the College or the Ducal Palace. By Guabdi 39 IX. Pencil Dbawing by E. W. Cooke, R.A. . 41 X. Sketch pbom Richmond Hill. By Vicat Cole, R.A 43 XI. Fbom an Engeaving by Domenico Peonti . 46 XII. Abcade at Bolton Abbey. By the Authob 49 XIII. Inteeiob of an Aet Galleby. Hans Joedaens 61 XIV. The Dead Waebioe By Velasquez . . 63 XV. A Lych Gate. By the Authoe ... 67 XVI. " This is the Heart the Queen Leant On " 69 XVII. Amy Robsabt. By Yeames .... 71 XVIII. The Malt-house Pump, Bubpham. By the Authob . .... 73 XIX. A Road. By the Authob .... 83 XX. A Curved Road bunning Uphill. By the Authob . . . . . .84 XXI. Poole, Doesetshibe. Afteb J. M. W. Tueneb 85 XXII: A COBNFIELD ON LEVEL GeOUND. By THE Authob ....... 91 XXIII. Sheaves. By the Authob ... . .92 XXIV. Sheaves. By the Authob .... 93 XXV. Columns in Bubpham Chuech. By the Authoe 97 XXVI. The Wickeb Cage. By the Authob . . 99 XXVII. The Wood-Waggon. By the Authob . . 104 XXVIII. The Fabm Caet. By the Authoe . .105 XXIX. The Sow. Fbom an Etching by W. H. Pyne 108 XXX. Some Types op Arches By the Authob . 113 XXXI. Doobway, Bubpham Chuech. By the Authoe 115 XXXII. Archway. Sketch by the Authob . . 118 XXXIII. Dbawing op a Beddge by the Authoe . .119 XXXIV. Gboined Roop in Bubpham Church. By the Authob ....... 122 XXXV. The Wooden Beidge. By the Authob . 125 vii viii LIST OF ILLUSTRATIONS XXXVI. Thirlmere Bridge, Cumberland. G. Cole XXXVII. Drawing by Rubens .... XXXVIII. Drawing by Credi .... XXXIX. Drawing by Maratti .... XL. Water-lilies. By Vioat Cole, R.A. . XLI. Groups of Roofs. By the Author XLII. Some Types of Hipped and Gabled Roofs By the Author XLIII. The Pump-house. By the Author XLP7. A Sussex Cottage. By the Author XLV. The Hipped Gable of a Cottage Roof. By the Author ..... XLVI. Farm Buildings. By the Author XL VII. A Sussex Brick-kiln. By the Author XLVIII. Types of Gables. By the Author XLIX. A Sussex Hovel. By the Author L. Some Types of Gables. By the Author LI. Dormer Windows. By the Author LII. Chimneys on Gabled Roofs. By the Author LIII. Burpham Church. By the Author LIV. Some Low-pitched Steeples. By the Author LV. Composite Octagon Spire. By the Author LVL The Bell Tower at Namur. By G. "Cole LVII. Study of a Sky. By the Author LVIII. Study of a Sky. By the Author LIX. Study of a Sky. By the Author LX. Beccles, 1918. By the Author . LXI. Drawing of a Punt. By the Author LXII. The Hill-top Pond. By the Author . LXIII. The Sunlit Sieve. By the Author . LXIV. The Harvester's Dinner. By the Author LXV. Lightship and Life-boat. E. Duncan LXVI. Swansea Pilot-boat. By E. Duncan . LXVII. Drawing by Louis Paul LXVIII. Drawing by Louis Paul LXIX. Painting on the Walls of Pompeii LXX. Rout of San Romano. By Uccello LXXI. A Drawing by Raphael LXXII. Interior of a Church. By H. Steenwyck LXXIII. St. Mark's. By Canaletto LXXIV. Interior of St. Pierre, Rome. By Panixi LXXV. Hogarth's Print for Perspective LXXVI. Ship of the Sixteenth Century . LXXVII. A Tiger. By Noami . LXXVIII. Crow and Heron. By Korpjsai LXXIX. The House of a Noble. By Yedsi LXXX. From a Coloured Indian Drawing LXXXI. Brick d'Anvers. Shipbuilder's Plan LXXXII. Brick d'Anvers. PAGE 126 127 129 129 131 134 136 138 139 141 143 145 146 147 148 149 151 153 155 161 163 166 167 168 173 177 179 187 193 207 207 210 211 219 221 225 227 229 231 235 243 247 248 249 250 264 265 PERSPECTIVE PART I NATURE'S PERSPECTIVE AS SEEN AND USED DAILY BY PAINTERS CHAPTER I THE PRINCIPLE OP PERSPECTIVE IN THEORY " If you do not rest on the good foundation of nature, you will labour with little honour and less profit." — Leonabdo da Vinci. LINEAR Perspective is a study that deals with the appearance of objects 1 as regards their size and the direction of their lines seen at varying distances and from any point of view. When practising it we are not concerned with their apparent changes of colour or tone, though those also help us to recognise the distance separating us, or that of one object from another. Visual rays. — -The Theory of Perspective is based on the fact that from every point of an object that we are looking at, a ray of light Fig. 1. is carried in a straight line to our eye. 2 By these innumerable rays we gain the impression of that object (Fig. 1). 1 " Objects " is a, mean word to use, because perspective laws also apply to the surface of the earth, the sea, and the sky, and all living things. It is used for convenience. 2 We see objects at a different angle according to whether we have both eyes open, the left shut, or the right shut. When drawing objects very close at hand look with one eye only. B 17 18 THE PRINCIPLE OF PERSPECTIVE IN THEORY Tracing on glass. — If we look at an object through, a sheet of glass we can trace on that glass the apparent height or width of that object (Fig. 2) ; in other words, we can mark off on the glass those Fig. 2. — Upright sheet of glass, object, and eye; showing the rays from the extremities of the object passing through the glass and marking its height on it. points where the rays from the extremities of the object on the way to our eye pass through it. Height oi objects at varying distances traced on glass. — If we now place two objects of similar height one behind the other (Fig. 3) our tracing of each discovers the one farthest off to appear on the glass shorter than the one close at hand. Fig. 3 makes it evident Fig. 3. — Side view (i.e. elevation) of posts, an upright glass, and painter's eye. f that this apparent difference in size is due to the fact that the converging rays from the further object have the longest distance to travel, and so are nearly together where they pass through the glass. On the other hand, the rays from the object close to the glass have only just started on their journey and so are still wide apart. Width of objects traced. — Let us repeat the experiment with two pencils of equal length lying on a flat surface, one behind the other. We shall be satisfied that their apparent length, as traced on the THE PRINCIPLE OF PERSPECTIVE IN THEORY 19 glass (Fig. 4), is also determined by these rays, and that the near one looks longer than the distant one. We have seen that the height and width of objects as they appear to us is determined by the converg- ing rays from their extremities to our eye ; that objects really equal in size appear shorter and narrower when further away. Depths of objects on a level surface traced.— It only remains to find out that the depths on a FlG . 4 ._Two pencils, glass, and receding surface are governed by eye, as seen from above (i.e. ground the same laws. plan). Fig. 5 represents three pinheads in a row, one behind the other, on the far side of the glass from the position of the eye. Notice, however, that the eye is above the pins (i.e. looking down on them), Fig. 5. — Side view (i.e. elevation) of the painter's eye, an upright glass, and a level board on which three pins are equally spaced. and so the points where their rays cut the glass are one above the other in regular order, the nearest pin (3) appearing the lowest down on the glass. Since the pins were placed at equal distances apart, their spacing, as shown on the glass, would also fix the depths of the ground surface between them^(Fig. 6). space n space m Fig. 6. — Same as Fig. 5, showing the spaces between the pins and as they appear on the glass. 20 THE PEINCIPLE OF PERSPECTIVE IN THEORY Theory o£ tracing applied to measurements on a canvas. — Up to now we have supposed ourselves tracing objects through a sheet of glass. In a perspective drawing our canvas is supposed to be a glass, and on it we trace only those objects that we can see through r Space! Fig. 7. — Tne glass (as in Fig. 5) with the pins traced on it, seen full-face. Fig. 8. — The glass (as in Fig. 6) showing the spaces traced on it. it without moving our heads. But painters should be practical ; so set up a canvas and prove to yourself that objects of equal size when far off appear narrower, shorter, and less deep than the near ii ] POST I ...1 POST H Fig. 9. — Posts the same height, Fig. 10. — Front view of one behind the other. canvas. Posts marked on edge and carried across to required position. ones, and that the spaces between them undergo a corresponding reduction. To do this, follow the instructions in Figs. 9 and 11. Fig. 11. — Four nails evenly spaced . Fig. 12. — Front view, show- ing nails marked on edge of canvas and carried across to required position. THE PRINCIPLE OF PEESPECTIVE IN THEORY 21 Fig. 9. Be yourself the " Painter." See that your canvas is vertical and so placed that the posts are just visible at one edge. With one eye shut and head still, mark off the heights of the posts where they seem to touch the canvas. Fig. 11. Behave as in the last exercise ; tick off the position of each nail on the edge of the canvas. These nails might represent the cracks between floor- boards, and our drawing shows that each board would appear one above the other (Fig. 13), necessarily narrower from No. 1 the nearest to 4 the farthest away. Theory of tracing to explain why parallel receding lines appear to meet. — Some people, when looking down a long straight length of railway track, have been curious to understand why the lines appear to get narrower and finally to meet in the far distance, though they know that the lines are actually parallel. It can be explained by tracing in this way — Fig. 14 shows that a piece of the railway line on the glass would appear as an upright line, the near end of the line being the lowest on the glass. The tracing on the glass would look like this (Fig. 15). Fio. 13. 1 Fig. 14. — Side view of painter, glass, and railway line (I-II) ■ Fig. 15. Fig. 16 shows how wide the near and far part of the track respec- tively would look on the glass, so that a front view of the glass (Fig. 17) would be like this. >e.. WIDTH BETWEEN RAILWAY ITNES U6.E g rt)I ■CANVAS -PAINTER o Fig. 16. — Bird's-eye view. The dots on the glassjshowing where the rays pass through it. Fig. 17. Fig. 19. Objects 22 THE PEINCIPLE OF PERSPECTIVE IN THEORY (In Fig. 14 we proved that the far end of the rail appeared higher up on the glass than the near end.) As we have secured the width of a piece of the track at its near and far end, we can join the ends on each side to make the rails. The tracing on the glass (Fig. 18) shows the rails getting closer and closer as they recede, and we see that they would, if longer, appear to meet. The spot on the glass where the receding lines appear to end or meet) is at the same level as your eye (Fig. 19). The trace of a level receding surface seen from below. — In former exercises we have had our eye higher than the objects we traced (so that we looked down on them) and the near objects became the lowest on the glass,' as in Figs. 4 and 5. The reverse happens when our eye is lower than the object (Fig. 20). Let us again draw our piece of railroad as in Fig. 18, it will serve as the floor of a room. Above it we can draw the ceiling if we still follow out Fig. 4, but remember to make the far end below the near end, as explained in Fig. 20. Our drawing (Fig. 21) shows that if we look down on level surfaces their receding lines must be drawn running up the canvas (from their near end to their far end). On the other hand, if we see their under side the lines must be made to run down the canvas as they recede. Fig. 20— Two ob- jects (I, II) above the painter's eye, with their position as seen through the glass. AN EXPLANATION OF TERMS USED IN PERSPECTIVE The Hobizok The visible horizon in Nature is that immense and distant circle that appears to be the extremity of our globe. Perched on a masthead we could, by moving round, examine the far distance where the sky and sea seem to meet, until piece by piece the whole circumference had come under our scrutiny (Fig. 22). On land we rarely get anything but r an interrupted view of a portion of the horizon. THE PRINCIPLE OF PERSPECTIVE IN THEORY 2a When standing or sitting on flat land the surface between us and the distant horizon is so foreshortened that mere hedges and bushes may hide miles of country and blot out the horizon. From a hill-top we look down upon the flat land, and the former Illus. I. Eibera. (Photo Manad.) The Clubbed Foot. Example of a low horizon. narrow strip between us and the horizon looks deeper ; also from this height we see more distant land. If the sea forms the boundary of our view the tips only of ship's masts may be seen beyond the apparent meeting of sky and sea. This explains that the curvature of the earth between us and the horizon is a real consideration for the landscape and sea painter, 24 THE PKINCIPLE OF PERSPECTIVE IN THEORY and it provides a reason for our seeing more distance from a height than from a slight rise. The horizon on our picture. — Though the horizon in Nature is a curved line, it is usual in pictures to represent it as perfectly level and straight ; the reason being that we can only see a small stretch of so large a circle at one time without moving our heads (Fig. 23). Fig. 24. Fig. 23. We draw this level line of the horizon straight across our picture. It is a matter of artistic taste and judgment whether we place it low, high, or in the centre of our canvas. It is essential, however, to understand that the position of the horizon line in relation to all the objects in our picture affords evidence of how high up we our- selves were when painting the scene. For instance, if we stand while we paint a whole length figure (on level ground) the horizon line would be cut by the head of our portrait (Fig. 24). But if we sit to paint him then the horizon line will pass through his waist (Fig. 25), and we should get a more fore- shortened view of the floor. It would be an absurd proceeding to begin a painting of an imaginary scene without first fixing the place where you suppose yourself to be painting it from. The relative position of the horizon line and the principal figures or other objects must be decided on at the onset ; after that the locality and size of all additional objects will be governed by the horizon. When you are painting an actual scene, you will draw the horizon line, whether it is visible or not in Nature, on your picture as soon as you have decided on the size of the principal objects. The horizon in Nature will be at the exact height of your eye, i.e. at that height where neither the upper nor lower surface of a level board (but only its near edge) would be visible. We may consider the horizon as a distant imaginary line parallel to the front of our face and stretching across the view at the actual 26 THE PEINCIPLE OF PEESPECTIVE IN THEORY height that our eye is from the ground 1 (Illus. II, III) ; so if stand- ing in a room our horizon would be roughly 5 ft. 6 in. from the flopr (Fig. 26), or if we are sitting 3 ft. 6 in. (Fig. 27). If our model is standing on a throne and Fig. 26. we g j^ ^ p a j n ^ on a i ow s tool, our horizon is ■ FlG - 27 ' about the level of his feet. Out of doors a slight rise on other- wise level ground would present a similar effect (see Illus. I). In a room we can find the horizon by measuring the height of our eye from the ground and chalk-marking that height on the wall facing us. Out of doors on hilly ground a stick with the height of our eye marked on it can be stuck in the ground just in front of us ; where that mark cuts the view will be the horizon. Note. — Michael Angelo de Carravaggio in his picture, " Christ carried to the tomb," used so low a horizon that no space of ground is seen between the standing figures, their feet being all on a hori- zontal line. Rubens painted his " Henry IV setting out for the war in Germany " with the spectator's eye level with the waist of the figures ; thus making the figures important though they occupy only half the height of the canvas, and leave room for the full height of the columns and archways overhead. Leonardo da Vinci con- ceived his picture, " The Virgin, Saint Anne, and the Infant Jesus," as though the spectator's eye was at the level of the seated figure ; but he chose to place the latter high up on the canvas. In this way the horizon, though a low one, is actually three-quarters of the way up the canvas. These examples should clear up any confusion in regard to the position of the horizon (representing the height of the spectator's eye) in the scene to be painted, and its position on the canvas. The Principal Vanishing Point (" P.V.P.") If you were to look down the barrel of a gun, holding it quite level (Fig. 28), the point you aimed at would in Perspective be called the Principal Vanishing Point (P.V.P.), and it would be on that imagined line that represents the horizon. It is the spot that a level line running directly away from us tends to. Fig. 28. 1 " From the ground," more correctly " from the sea-level" ; because if we went -up a mountain to see the view, the horizon would still be on a line with our eye, so it would be the height of our' eye plus the mountain, Illustration III makes this clear, Illus. III. Van Eych. La Viebge au Donateuk. (Photo Mans The horizon (as shown by the distant view) passes behind the heads of the figures. The painter must also have been sitting, because if he had been standing hi3 horizon would have been above their heads (see Chap. II). The lines of the pavements, capitals, and plinth of the columns all tend towards one spot on the horizon at its centre. For explanation of this, see Rules I, II, Chap. II. To obtain the depth fflt*. ION OK PICTURE The Theory or Perspective explained by a Diagram. Fig. 32. — A bird's-eye Tiew of the horizon, canvas, painter, and three vanishing points in Nature and on the canvas. It will be seen from this diagram that the distance along the horizon in Nature, from the P.V.P. to the V.P. for the diagonals of a square, is the same, as from the P.V.P. to " painter " ; consequently (as the diagram proves) the distance along the horizon on the picture, from P.V.P. to V.P. for diagonals, is the same as from P.V.P. to " painter." From this we learn that the V.P. for the diagonals of a square (provided that one side of the square is parallel to the horizon) is the same distance from the P.V.P. as the painter is from his canvas. (or set of parallel lines) has its own V.P. to which it runs. Besides these level lines there are others on inclined planes that run to vanishing points above or below the horizon, according to their position. Some of these level receding lines will be at such an angle that the points they run to will be further to the left or right than the width of ground we propose to paint, and consequently will be outside the extremities of our picture as V.P.+ in Fig. 32). As for some of the inclined planes, the lines of these will sometimes tend T5=="*'" m . Illus. V. A Stile. Lines receding to three separate V.P.'s on the horizon. Drawing by the Author. 32 THE PEINCIPLE OF PERSPECTIVE IN THEORY to points far above or below that portion of the ground or sky we wish to include in our picture. In practice we may draw the scene on a small scale in the middle of a large sheet of paper so as to have room for these outside V.P.'s on the margin. Then we can " square up " (see Note 1 in Appendix) the little picture and enlarge it on the selected canvas. Another way is to use the floor as an exten- sion of your canvas. CHAPTER II THE RULES OP PERSPECTIVE AND THEIR APPLICATION " Rules are to be considered as fences placed only where trespass is expected." — Sir Joshua Reynolds. RECEDING parallel lines. — We know that objects as they are .. seen closer at hand appear to be successively larger than others (of a similar size) that are further away. Test this state- ment by holding a flat ruler in front of you, with one end nearer to your face than the other, then the near end will seem to be wider than the far end, as Fig. 33a explains. If you join the near and far end by straight / lines, one on either side (Fig. 33b), you will com- / plete your representation of the outline of the <= ruler, and you will have satisfied yourself that Fig - 33a - Fig. 33b. the width of the ruler at any particular point is determined by the receding side lines. Again, if you continue those side lines until they meet, you will appreciate the truth of Rule I. Rule I. — All Receding Lines that are in Nature parallel TO ONE ANOTHER APPEAR (iF CONTINUED EAR ENOUGH) TO MEET AT ONE AND THE SAME POINT (Fig. 34). 7A« Sip* tmfj of £\.1£."> -"* Receding level lines.— Now hold SiStti'.s-r^ /~~\ the ruler with one end nearer to £., (.ook«.j-j ^ / i y ou ^ q U j te fe ve i ( ag t k e sur f ace fl L \ of water lies). Your drawing of it in this position will show that the point to which the sides tend is at the same height from the ground as your eye is ; in other words they would eventually meet on the horizon (Fig. 35). This makes Rule II easy to understand. j? la _ 35^ c 33 re 34 THE RULES OF PERSPECTIVE Receding lines that are parallel and level. Rule II. — All Receding Level Lines in Nature appear (if continued) to end on the Horizon. If they are also (in Nature) parallel to one another (Rule I) they appear to MEET AT THE SAME SPOT ON THE HORIZON. The gist of the matter is that because they (i.e. these particular receding lines) are in Nature parallel to one another (Rule I) they tend to the same spot, and if they happen to be level lines (Rule II) that spot must be on the horizon (Illus. IV, V, VI). Level planes.- — If you think of it, level planes must end at the horizon (not above or below it) ; because the horizon is at the height of your eye, and level surfaces are only visible so long as they are above or below your eye. At the exact height of your eye you could not see either their upper or under surface but only their edge (Fig. 36). Receding parallel lines inclined~upwards. — Now tip up the far end of the ruler, without turning it to one side or the other, and you notice that the receding Fig. 36. \i nes tend to a spot that is higher up than the former one, when the ruler was level (Fig. 37). horizon Line Fig. 37. Fig. 38. This merely means that if the ruler is inclined upwards you see it less foreshortened, but your horizon remains the same height. It can be remembered by Rule III (see Figs. 37, 38 ; Illus. VII). Rule III. — All Receding Lines that are in Nature parallel TO ONE ANOTHER IE INCLINED UPWARDS APPEAR EVENTUALLY TO MEET AT A SPOT THAT IS IMMEDIATELY ABOVE THAT SPOT WHERE THEY WOULD HAVE MET IF THEY HAD BEEN LEVEL LlNES. THE EULES OF PERSPECTIVE 35 Receding parallel lines inclined downwards. — If you tip the far end of the ruler downwards instead of up, just the reverse happens, and the spot to which the side lines tend will be below the horizon instead of above it ; write Rule IV thus : Illus. VI. Pieter de Hooge. A Dutch Interior. (Photo Mansd.) Lines of the sides of the room, and pavement meeting at the P.V.P. on the horizon. The horizon line cuts through the heads of the standing figures, showing that the painter was standing. The P.V.P. is just over the head of the woman showing her cards. So she was immediately opposite the painter, as we see by the lines of the pavement. Rule IV. — All Receding Lines that are in Nature parallel TO ONE ANOTHER IF INCLINED DOWNWARDS WOULD APPEAR (iF CONTINUED) TO MEET AT A SPOT IMMEDIATELY UNDER THAT SPOT WHERE THEY WOULD HAVE MET IF THEY HAD BEEN LEVEL LlNES. Of course, if you tip the far end of the ruler only a little down- ward you will still see its top side but very foreshortened (Fig. 40a) ; 36 THE RULES OP PERSPECTIVE if~you tip it still further you will see the under side, and the spot that the sides tend to will be still lower down. VP">.| WPHH-L, Fig. 39. for. Level. LttlE-S Fig. 40a. DOWNHILL i Fig. 40b. Level receding lines pointing to the right or left. — It is obvious that if you hold the ruler level but point it to the right — so that one corner of it is nearest to your face — the side lines of it will tend to a spot on the horizon on the right-hand side. You will, however, notice the two ends (because they recede) also tend to a spot on the left-hand side of the horizon (Fig. 41). HO ft i I ON HORIZON P-V-P Fig. 41. Note. — If the ruler points to the left side instead of the right, then exchange the words " left " and " right " in the above paragraph. Level lines receding to the principal vanishing point. — We have learnt that all lines that are in Nature level, and receding, appear to end somewhere on the horizon. If we stand on a line that is running directly away from us, such as the crack between two floor-boards, and look down its length we shall find it tends to the " principal vanishing point " on the horizon (Fig. 42). At this P.V.P. the floor-board and all other lines (in Nature) parallel to it — such as the sides of the room — meet (see Illus. VIII). If we were drawing such a room we should first mark on the far I P«.»T*H. Fig. 42. THE RULES OF PERSPECTIVE 37 wall itself a line showing the height of our eye (the horizon) as we stand or sit at our easel. Then we should sketch in the proportions of the room, including this horizon line. The next step would be to put a pin in our drawing where the IUus. VII. The Bars Exd. Drawing by the Author. The lines of the receding walls being parallel and level, meet at one spot on the horizon (P.V.P.). The sides of the road would also have met at the same spot if the road had also been level, but be- cause it runs uphill its V.P. is above the horizon and immediately over the P. V.P. P.V.P. would come on the horizon, and with one end of a ruler touching it, draw every line that in the room is parallel to the floor- board (Fig. 43). The end wall is directly facing us, and consequently not a receding 38 THE RULES OF PERSPECTIVE surface, so the lines, where the ceiling and floor touch it, are drawn parallel to the horizon line. PIT iff 11 n ■1 HORIZON Fro. 43. Fig. 44. Suppose we sketch the same room from a new position so that we look diagonally across the floor to the left-hand wall. The lines of the floor-boards will tend to a point on the right side of the horizon outside our picture. From this position the end wall becomes a receding surface (because one end is a bit nearer to us than the other), therefore the lines where the floor and ceiling meet it, will tend to a point far away on the left of the horizon. 1 We make our drawing by first sketching the proportions of the room, including the horizon line. That done the direction of one floor-board is copied, and its line continued till it meets the horizon line. To that point of junction (V.P.) draw all lines that are (in the room) parallel to the floor-board. Then copy one of the receding lines of the end wall and continue it till It meets the horizon. Make all lines that are parallel to it in the room meet at that point. Rectangular objects, such as boxes, tables, etc., would of course be drawn in the same way. Use of Receding Lines to Fokm a Scale Horizontal scale.- — We have learnt (Rule I) that receding lines, if they are parallel in Nature, appear to meet at a distant point. Therefore the converging lines we draw to represent them must also give us the apparent width between these lines at any point from the foreground to the far distance (Fig. 45). If the lines were on the side of a wall instead /~* : -l *•<- of lying on the floor, a corresponding height f iq . 45. 1 The perspective method for fixing the exact position of the two vanishing points for 0, rectangular object is detailed in Chap. XXII. When drawing from Nature we need no other aid than » hinged foot-rule. This is held at arm's length and the angle adjusted until one arm corresponds to the receding line we wish to draw, and the 3 other to a vertical or level line on the object. The angle gauged in this way is traced on our drawing. The foot-rule must not be held so that it is itself foreshortened. SAY s <* I««h«5 «*»»! ! IX « a O H 40 THE RULES OF PEESPECTIVE between them instead of • width would be represented as in Fig. 46. Two such receding scales in our picture — one vertical the other f*n*UeL Lines "* n^tuac- TK* SA-m.fr itt aua Say Sim inches a pa at- one j)RAwin&- P-ccrpmcr ABOVC Tki otkea. Ffto~\ u5 Fig. 46. horizontal — will enable us to decide how tall or wide objects should be just at the very spot vrhere we wish to introduce them. An easy way of remembering this vertical scale is. to think of it as a wall of a given height that stretches from us to the far distance. The horizontal scale might be a pavement or a railway line. The scale on level ground. — It does not matter how far to the right or left we place the V.P. for such a receding scale, we may use an existing V.P. or place a new one at a con- venient spot (Fig. 47). If the receding scale is to run on level ground, its V.P. of course must be somewhere on the horizon. The Height of Figures and the Receding Scale On inclined planes. — If the scale is required on a plane running uphill or downhill, then the V.P. will either be the one we used for Fig. 47. the lines on that plane, or a new one at the same level (Fig. 48) ; otherwise the scale would not be running on the ground. THE RULES OF PERSPECTIVE 41 In practice we draw a figure the height it should be in our picture, then we take receding lines from his head and feet to the V.P. we used for the ground he is standing on. When introducing additional figures we place their feet where they are to stand, then walk them across (keeping parallel to the horizon) to the scale to see how tall they should be (Pig. 49a). The height of figures with a low horizon would be found by the receding scale, and since the lower line of the scale and the horizon Illus. IX. Pencil Drawing. iJ I E. W. Cooke, R.A. Example of a low horizon with heads on horizon . For explanation of figures on sloping sand or wading, look up chapter on " Inclined Planes." line would run close together, some degree of precision in workman- ship becomes a necessity. When the feet of the figure are on the horizon the scale can be dispensed with, as the lower line would run on the horizon line (Pigs. 49b and 49c). If the head of the " painter " is the same height as the head of the figures he is painting, and they are on level ground, then the receding scale to find their height is unnecessary, since the top line of the scale would run along the horizon line (Pig. 49d). In such a case the height of figures is found by placing their feet where they are to stand and their heads on the horizon (Fig. 49e). 42 THE RULES OF PERSPECTIVE There is another way of finding the height of figures when looking down on them — thus, draw one figure the height it should be (com- pared with a doorway, for instance), then see what proportion of MO»llON „rfr** J* 1 SpOT- WNUtE ■ y" !}»« u vrr He Ml O Rtf w \\ [1 Fig. 49b. Fib. 49a. Fig. 49c. the space from his feet to the horizon is occupied "by the man. To introduce another figure place his feet on the correct spot, and then allot to the man the same proportion of the spaoe from his feet to the horizon as you gave to the first man (Figs. 49f and 49g). HORIZON \\\WT~%'"f- m, J Fig. 49d. Fig. 49e. A scale representing 1 ft. if drawn along the base of a picture and up one side will be found a convenience in architectural sub- jects. Usually a 6-ft scale (upright) will suffice. If we know how tall a 6-ft. man would look at a certain spot, we can calculate the height of any other object, such as a child, a horse and cart, or a cottage at that spot. In like manner we can calculate the width of a road, pavement, etc., by making our phantom man lie down at that spot (Fig. 50). :! Fig. 49f. horizon I- 1 Fig. 49g. Fig. BO. THE RULES OE PERSPECTIVE 43 . Yical Cole. R.A. Sketch fbom Richmond Hill. Example of a very high horizon. The " painter " was standing on ground about twelve feet above the terrace. We arrive at this conclusion by reasoning thus : If we stood on the terrace our head would be on the level of the horizon, and so would the head of the figure leaning against the balustrade. In this drawing the figure only occupies one-third of the space from his feet to the horizon, that is to say, another man could stand on this one's head, and yet another man on top of him; before the third man's head would be level with the horizon. If each man were six feet high the " painter's " head must have been eighteen feet above the terrace-level. The height of other figures on the terrace could be found by the receding scale as before. Another way would be to divide the space from the terrace (where the new man's feet are to be) to the horizon line, into three parts, and to give one part to the man (Kg. 49h). 1 1 r 1 Fig. 49h. CHAPTER III DEPTHS TWO very practical ways for finding depths on a receding surface are (A) by the use of diagonal lines with a receding scale ; (B) by a scale on the base-line or parallel to it. The former in particular is useful when sketching a foreshortened row of posts, trees, etc., for it marks the distance between each with sufficient accuracy, and all the lines can be drawn rapidly freehand. (A) Depths found by diagonals and a receding scale. — The construc- tion of a gate may include two uprights, some rails, and one or more cross-bars (Fig. 51). The centre is at the meeting of the cross-bars. Fig. 51. Fig. 52. Fig. 53. Therefore to find the centre of a gate or any other rectangular form we draw diagonal lines from corner to corner (Fig. 52). This applies equally well to any rectangular shape seen foreshortened (Fig. 53). If we add an upright where the diagonals meet (Fig. 54) we divide the form into two halves, of which the nearer appears to be the larger. Suppose you had a gate with one-cross bar (Fig. 55) and ' wished to add another half of a gate to its length, then (remembering RZ Fig. 54. Fig. 55. Fig. 56. that the diagonal line determined the centre) you could utilise half the existing gate for a new diagonal. This would start at the junction of the central upright and the top rail (Fig. 56) and would pass through the middle of the end upright, meeting the new bottom rail where the new upright is to come (Fig. 57). By the use of two diagonals you might also fix the meeting-place for the new upright with the top rail, but this is unnecessary. 44 DEPTHS 45 If you would like to add a whole gate to another, instead of half at a time (as we did), then continue the three bars and make a new diagonal (Fig. 58) from the top corner of one end-post and pass it through the centre of the other end-post. Where it touches the bottom rail make the new post. Our essay on the construction of a gate will enable us to draw a m ■-?^r.;" Fig. 57. Fig. 58. \ \ r— t r; Fig. 59. receding row of posts or trees. Proceed thus (Fig. 59) : Draw post 1. Fix height and centre of future posts by lines receding to V.P. from top, bottom, and middle of post 1. Draw post 2. From top of post 1 take a diagonal through centre of post 2 — where it touches receding line on ground plant post 3. Carry on by drawing another diagonal from the top of post 3 and each post as it is erected. Applied to a colonnade, etc. — When sketching an avenue of evenly-spaced trees or a colonnade make a receding scale of three lines, as in Fig. 59 — it would be most in- convenient to take in the tops of the trees (Fig. 60). 1 Our vertical scale will do equally well for level surfaces (Fig. 61) ; or we can make it on the ground (Fig. 62). If Fig. 59 is looked at from the side $ % $t>m')< Bps-, \j^ VP Fig. 60. it will lie flat and be similar to Fig. 62. VP Many examples of the x \ C -""'.'■" £■■"' i Ji Fig. 61. Fig. 62. 1 Turner delighted in the legiilarity of these receding spaces. His pictures show this not only in the architecture that he drew so beautifully, but also in his choice of subjects where trees line a roadway. He knew just where to break the monotony of too even a distribution without disturbing the feeling of sequence given by an avenue. One calls to mind his embankment on the Seine ("Rivers of France") and " Mortlake Terrace — Summer Evening." 46 DEPTHS use of diagonal lines in conjunction with the receding scale will crop up as we get on with our subject. These should be enough to explain the principle. But I wanted to make it clear that receding lines in perspective not only form a scale for measuring heights and widths, but with the addition of the diagonal line give the depth measurement. Illus. XI. From an Engraving hy Domenico Pronli. In this subject the width of the arches would be found by a receding scale on the pavement of the same width as one of the arches. The length of one arch and pier would then be drawn, and their comparative measurements obtained on the base-line, as in Fig. 65. (B) Depths found by a scale on the base-line. — If (Fig. 63) we saw a line (I— II) full-face, it being divided into equal proportions, and we carried receding lines from both ends (I-II) and from each divi- sion to a V.P. on the horizon 1 ; then a foreshortened line (I— III) would also be divided into similar proportions in perspective ; each division getting regularly shorter from the nearest to the fartlusst end of the line. The foreshortened line is more often the one we have to divide into certain proportions as in Fig. 64, line 1-2. To do so, take a line from the near end 1 to the horizon line, thus making a V.P. From Fig. 63. DEPTHS 47 the V.P. .take a line to the far end 2, continuing it sufficiently far to enable the line 1-3 to be drawn horizontally. Line 1-3 now repre- sents the foreshortened line seen full -face. Divide 1-3 into required proportions, and from each division take lines to the V.P. They will divide line 1-2 into like proportions. HOWiiON Fig. 64. The division of a receding line into equal or unequal spaces.- Unequal divisions are obtained in a similar way (as in Fig. 65). The divisions marked on the full-face line might represent the ground plan of the length of colonnade with the spaces between the columns ; then the same divi- sions would on the foreshortened line be the perspective ground plan of their width, columns would be raised (Fig. 66). On these the Another way of using Diagonal Lines to Find Depths (1) Of vertical spaces. — Fig. 67 represents a foreshortened rec- tangular form (say a window) in an upright position. If we draw a diagonal from the near bottom corner to the far top corner, then that diagonal is a receding line inclined upwards, and its vanishing point will be found by continuing its course until it meets a vertical line bhat starts from V.P. 1. (Rule III says that a line inclining upwards tends to a V.P. immediately above the point it would tend to if it were a level line.) It is evident (Fig. 68) that the diagonals of any number of shapes equally proportioned would in Nature be parallel -Fig. 68. Fig. 67. 21Z / 48 DEPTHS lines, and therefore when seen receding (Kule I) would meet at the same point, as in Figs. 69 and 70. Fig. 69. Pig. 70. / / / / ' 1 Fro. 71.— As in Kg. 68, but the dotted lines would have a V.P. as well as the other set of diagonal lines. In order to find the apparent depths of equally proportioned spaces on a receding row we must find the " uphill " V.P. for the first space, and from that V.P. draw per- spective diagonals to the far bottom corner of each space as it is made. The inter- section of the diagonals with the top re- ceding line will mark the depth for each succeeding space. The above Figs. 67-69 can be worked equally well with a " downhill " V.P. In some cases this may be more convenient. The diagonal of the first space must of course be taken from the top near corner to the " downhill " V.P. (immediately under the V.P. for the level lines). The depths of succeeding spaces will be fixed at those points where the diagonals cut the bottom receding line. If you find the V.P.'s for inclined lines come inconveniently high or low you can take an extra receding line to the level V.P., so as to cut off a portion of the height of the first space and others to come. Use that new line to run the diagonals to, instead of the top line (if an Fig. 72. " uphill " V.P. is being used) or bottom line (for a " downhill ' V.P.). Fig. 73. Ui!i±"l'lliS 49 In Fig. 74 we see that the diagonals taken to the bottom line and those taken to the new line both pass through the top receding line at the same points. Pan. ptAftOW'Lj' ■V* ..^' '< s s s s V S s Fig. 78. — Plan of any number of squares showing that the dia- gonal determines the length and width of each square. ^m$s\ Fig. 79.— The width of eight squares lyingjflat marked offjby receding lines and intersected by a diagonal. to pvp * ■$ f ^mi Fig. 81. 52 DEPTHS can find the depth of any number of rectangular spaces provided they are (in Nature) of equal dimensions (Figs. 78-81). Procedure. — Draw a line (1-5, 1 Fig. 80) parallel 'to horizon line. On it mark off the width of the divisions required. From each division (1, 2, 3, 4, 5) draw lines meeting at P.V.P. These fix the width of each space. Draw a diagonal. At the points (B, C, D, E, etc.) where the diagonal cuts the receding lines (1, 2, 3, 4, 5) draw lines parallel to line 1-5, and so obtain the depth of each row of spaces. The position of vanishing point for diagonals (D.V.P.) is deter- mined by the distance the painter is from his picture or from the object he paints. I have advised you to place a square lying level with one side facing you in the foreground of the scene you intend to paint, and to copy its depth in relation to some principal object ; this done, to draw its diagonal and continue it to the horizon to find the V.P. for diagonals D.V.P. Most painters prefer to fix this D.V.P. for diagonals first. In order to do so, you must decide at what distance you wish your finished picture to be seen. If you wish the owner to examine it at close quarters, you will suppose yourself to be just that short distance as he would be from »»? it, while you paint. If it is to be seen far away, you suppose yourself to be far off also when painting it. The actual distance from the D.V.P. (the vanishing point for diagonals) measured along the horizon line on your picture to the P.V.P., will be the same as the measured distance along the ground from " painter " (i.e. the place you suppose your- self to be while painting) to the P.V.P. on the picture (see Fig. 82). As a general rule, you consider that your picture should be viewed at a distance equal to twice its height or length — whichever is greatest — so that the whole picture may P I C T | r U R g t Fig. 82. — Bird's-eye view of the painter and his picture, with the dia- gonal V.P. on the horizon (continued beyond the length of his picture). See Appendix, Note 3, for methods of dividing a line. DEPTHS 53 be seen clearly and yet without moving the head — but we will talk of this later on. Fig. 83 represents the picture (3 ft. square) shown on the ground plan in Fig. 82. The V.P. for diagonals is 6 ft. from the P.V.P., just as the painter was 6 ft. from the picture. Horizon X*o/- CANSiAS HEPKMeNTlNG- Aj'of LANDSCAPE Fig. 83. If each foot length along the bottom edge (base line) of the picture represents 4 ft. of the nearest width of the scene painted, then the actual length of the nearest foreground represented is 12 ft. The oblong drawn on the picture was 4 ft. long and 8 ft. deep, and was 2 ft. to the right of the painter's ^line of sight. Similarly the same scale of 4 ft. of Nature 7to 1 ft. of canvas shows that • the painter's eye (the horizon) was 4 ft. above the ground level. Vanishing point for dia- gonals can be used above or below the horizon. — As all lines which are in Nature level and at an angle of 45° with the line of sight tend to the D.V.P. on the horizon, so also !-~r ■ VP !">■*■ Fio. 84. lines leaning upwards or downwards tend to a D.V.P. respectively above or below the horizon, provided they make an angle of 45° with level ground surface. The " uphill " or " downhill " D.V.P. will respectively be just as far above or below the horizon as the D.V.P. on it is from the P.V.P., as in Fig. 84. CHAPTER IV THE USE OF PLANS IN SKETCHING FOKESHORTENED SURFACES WHEN perspective is used in a purely mechanical way — that of constructing a pictorial view from the known dimensions of some object unseen — it is necessary to employ elevations and ground plans. On these the exact dimensions of each part are drawn to scale. This process is quite unsuitable to sketching, though we sometimes employ it in pictures, mainly when archi- tectural features are important. This scientific method is dealt with in Part II. As an aid to drawing, even though mathematical exactness is not aimed at, a ground plan is still useful ; partly because it fixes in one's mind the proportions of the object that is to be drawn foreshortened, partly because it stimulates a mental survey of the ground surface on which we propose to place forms. For example, a foreshortened square when all the sides are receding from us, is a troublesome thing to draw if we have to employ two vanishing points, because one of them, if not both, will probably be outside the dimension of our canvas. If, however, we conjure up the ground plan of a square seen at an angle, and place it in another square of which the front is parallel to the horizon, we can on the latter fix the position of the corners of the angled square. This done, we put the square with the parallel front into perspective, and with it the four corners of the angled square, and so arrive at the direction the receding sides take without any con- cern as to their vanishing points. How to draw a square when one corner is nearest you without using the vanishing points (1) when the corners on either side are on a line parallel to the horizon. Fig. 85 is the plan of two squares, one inside the ' other. The drawing is only given to show how to draw the small square B, by means of the square A. Draw square A, cross it with diagonals in order to find the Fig, 85. centre. Through the centre draw one line parallel to 54 SKETCHING FORESHORTENED SURFACES 55 the side and another parallel to the front. These will determine the centre-point on each of the four sides. Join these four points consecutively to make the square B. If we draw (Fig. 86) the foreshortened square B in the same way by means of the square A, whose receding sides and >""\' centre line will meet at the P.V.P., then the square B ■ : \ will be correctly foreshortened without the necessity of finding the V.P.'s for its sides ; but in Fig. 87 we have done so just to show we were right. F 10 - 86 - P-H P.A&o-.ltVP Fig. 89. Enclose it in Fig. 87. When we see a square lying flat and foreshortened in such a position that a line parallel to the horizon would cut both its left and right corners (Fig. 88), we know by the above figures that we need not find its V.P.'s in order to draw it correctly, and we get to work thus : Practice. — Sketch lightly (they will probably be wrong) the sides of the Fig. 88. square (Fig. 89), but take infinite pains to mark its comparative length and depth. a square as before and correct the sketch of its sides by repeating the working of Fig. 86. Application to a pavement. — Fig. 90 depicts a number of squares forming a pavement. If square I is first drawn, square II must be made of equal length on the line A-B. Their depth is obtained by continuing the back line of square 1 horizontally. A diagonal line continued from the front row will fix the depth of the back rows at those points where it cuts the lines that recede to the P.V.P., but this has already been ex- plained in Fig. 80. If more distant pavement has to be added one of the diagonal lines must be continued to fix again the depth of each row, as at C, D, E, F. Extra squares in the foreground could be added by lengthening /l\ 56 SKETCHING FORESHORTENED SURFACES towards us the lines that recede to the P.V.P., and also a diagonal to obtain the intersections marking their depth. If we know how many squares there are across the floor we can apply the easiei method of Fig. 80, by taking the whole floor-space first instead of one square. If the depth of the floor is greater than its width we can still apply the working of Fig. 80 by first making a square to represent the full width of the floor (of say twenty tiles), then we shall have twenty tiles also in depth, and we can add more, as we have seen in Fig. 90. 1 (2) How to draw a foreshortened square seen corner-ways, what- ever its position, without using vanishing points. It is sometimes a great convenience not to be obliged to use the vanishing points of a square or other rectangular form when one or both points are outside our picture, as in Fig. 91. Fig. 91. When a square is set at some other angle than that of Fig. 88 we can still, by varying our method, draw its sides that would meet at their respective V.P. without having to prolong them to it. Fig. 92 is the plan of a square enclosing another. Whatever angle the inner square may be set at, the following procedure can be used : — Practice. — From the corners 1 and 2 draw lines parallel to the side (A-B) of the enclosing square ; rule diagonal. From corners 3 and 4 draw horizontal lines (parallel to front of square B-C) until they meet the diagonal. To make a foreshortened view of this : — Draw the base of a fore- shortened square (Fig. 93) with the points B, 5, 2, C, ticked off upon it from the plan. From these carry lines to P. V.P. ; draw diagonal (to D.V.P. for diagonals), and where it cuts the side-line add back of square. Now place each corner of the inner square, by revers- ing the order in which you drew lines from them j^g. 93" 1 Pavements are worked out in detail in Chap. XVI. SKETCHING FORESHORTENED SURFACES 57 on the plan : i.e. to find corner 3, carry a horizontal line from the intersection of the line 5 with the diagonal. To find corner 4, take a horizontal from the meeting of the diagonal and the line 2. Place corner 1 at the far-end of line 5, and corner 2 at the near- end of line 2 ; join corners. To draw the angled square without using a plan. — Practice. — Draw the enclosing foreshortened square as before (Fig. 94) with its diagonal. Mark on it the position of one of the corners of the inner square (say 2). Make B-5 equal to 2-C, and carry on as before. The reason for this is that comer 1 is on the plan (see Fig. 92) as far from A as corner 2 is from C, therefore the line 5 is made the same dis- tance from B as 2 is from C. Also in the < * * * plan the corner 3 is as far removed from B as 4 is from D, and so horizontal lines from the intersection of the diagonal with the receding lines (5 and 2) will cut off a space 4-D equal to 3-B. Sometimes one has to sketch a square when there is not time to consider it as a perspective problem. It is well, then, to remember that its width must be longer than its depth ; that the angle formed by its near sides will be larger than a right angle ; and that this angle becomes flatter in proportion as the square is more distant. These rather obvious remarks are unnecessary if you will sup- „ pose yourself standing at T HORIZON VP r J O '•;■"-- ... ..._-- -..-•/ the corner of a square, '~**~~~"'\,.- ' / ("painter,"Fig.95). Since you would be right on top of the angle it would not be N e/*«GST ••' foreshortened, but would ■" *"- *iy y ^ look like it is — a right % lrrrt *i angle — -and its sides if t„«,t.on continued would decide the position of the two vanishing points V.P. But the next angle, if still seen from the same point, would be flatter (i.e. set at more than a right angle the one side with the other), because its sides also meet at V.P. The next angle 3 looks still more flat as it is more distant. In Fig. 96 1 is the top of a box, 2 is its side. The pattern on its side acts as a ground plan, helping us to draw the same pattern on the top. 58 SKETCHING FORESHORTENED SURFACES Practice. — First find out on what constructive lines the pattern (on the plan) was formed (they are shown in Fig. 96 by dotted lines). Draw those same lines of construction on the foreshortened to „ view. Remember that all receding parallel lines / * f \ \ on the plan must, when depicted on the fore- .f* ;' ^-^ i shortened top, be made to meet at the V.P. i Then shape the pattern in the foreshortened surface on these lines just as the full-face pattern was made. On the plan the constructive 1 upright lines A and B only formed the sides of the inner oblong, they had to be carried up to the edge that divides the plan from the fore- shortened top-surface before they could be carried across the latter. If we understand the constructive lines of the plan we need not always draw it under the foreshortened view. In Fig. 96 we can see that the corners of the diamond touch the centre of each line that forms respectively the top, bottom, and sides of the black oblong. Also that the black oblong is formed on the diagonal lines which cross the outer oblong. Therefore we can dispense with the plan and get the same result thus Practice. — Tick off on a piece of paper the length of the inner and outer oblong and its centre Transfer these to a line that will form the base of the foreshortened oblorig. From the outside dots draw lines receding to V.P. for the sides of the outer oblong. Decide on its depth to complete it. From the remaining dots (they show the length of the inner oblong and its centre) take lines also receding to V.P. Cross the foreshortened oblong with diagonals. Where they cut the lines that recede to V.P. will be the four corners and the centre of the inner oblong ; draw it. The centre of its near and far sides is formed by a horizontal line through the crossing point of the diagonals. Draw the diamond with the corners touching the centre of each side of the small oblong. It is just a matter of first drawing the constructive lines on the plan and then repeating them as they would look foreshortened. The plan need not be the same length as the surface we have to , draw foreshortened. If the base line of the latter is to be facing us the measurements ticked off the plan can be enlarged or reduced in the same proportion to make them the desired length. 1 If the 1 See Appendix. SKETCHING FORESHORTENED SURFACES 59 foreshortened surface is to be seen at an angle, the measurements can be transferred, as explained in Fig. 65. A roughly drawn plan, on which we draw a diagonal and a few upright lines, will often suffice as a guide for the spacing of objects on a receding surface. Take Fig. 97, A is the plan, B the foreshortened space. Where should the spot shown in A be placed in B ? Fig. 97. Fig. 98, Practice. — On plan draw diagonal (1). From spot draw the horizontal line (2) till it meets diagonal. Through that point and also through spot draw lines 3 and 4. Continue these lines down B (towards V.P. because they recede) ; and draw diagonal. Where it crosses line 3 draw horizontal line till it meets line 4 ; there place spot. All we have to remember is that the lines we draw on A we repeat on B, but in the reverse order ; the reason being that in A where the position of spot is fixed we draw lines from it, but in B we draw the lines first to find the position of s,pot. Our exercise with spot is elaborated in Fig. 98 by the addition of the little spots, but the family removal can be effected in the same way. If we look upon these spots as figures we regard them with greater interest. The advantage of finding their position in this way needs no advertisement. There used to be a good old-fashioned way 1 of finding standing- room for a crowd of figures. The floor-space was sectioned like a chess-board (Fig. 99) ; each square might represent 2 to 3 feet square, to allow for the shoulders. A diagonal line crossing the Since writing this I find that the custom was practised hy Raphael. 60 SKETCHING FORESHORTENED SURFACES receding lines determines the depth of each square, as we explained „. in Figs. 78 to 81. ,.---'.-' vp This method applies equally well for the placing of doorways, windows, or furniture, as we see in Figs. 100 and 101. Application to a room and the placing of figures. — Why not make a plan of a room, let us now say of the one Hans Jordaens painted (Illus. XIII), placing the chairs, pictures, tables, and figures, each at its angle, and in its proportion on the floor, and make it foreshortened by applying Figs. 97-101 1 You will not always be able to paint subjects just as you find them. Simple patterns can be sketched with sufficient accuracy free- hand by this means (see sketches of Chap. XVI). Fig. 100. □ Fig. 101. Fig. 102. Fig. 103. Applied to upright surfaces. — When drawing upright foreshortened surfaces we draw the plan on the near edge, just as if the pattern were on the front of a box, and we wished to draw it on one of the- sides (Fig. 102). The working is just the same, as will be seen by rotating page 58 until Fig. 96 is in an upright position. It would be hardly necessary to draw a plan of so simple an object as a diamond pane, yet one comes across inaccurate drawings of lead lights of diamond or lozenge pattern , SKETCHING FORESHORTENED SURFACES 61 The Jacobean panel sketched in Fig. 103 is much easier to draw if we use this method. It enables us with little effort to place the upper and lower mouldings in position and to give them their correct width ; and really this is not an easy thing to do without some guiding points. This might be put to a useful purpose in sketching the carved spaces on the front of a tomb, the panelling of a room, or the tracery of a cathedral window. Occasionally the lines of a plan interfere with those of the draw- Illus. XIII. Hans Jordaens. Photo Hansel. Interior of an Art Gallery. ing if placed side by side, so it is better to separate them, as Fig. 105 explains. Concentric squares. — Rectangular forms enclosed by other rec- tangular forms will all be of similar proportions so long as the same diagonal lines pass through their corners. How to draw the plan of concentric squares. Draw the outer square (Fig. 106). Cross it with diagonals. Mark off from both ends of the base the width that is to separate the inner from the outer square, such as I— II, III-IV. From these draw lines parallel to (IV-V) the sides of the square. Where they cut the diagonals will be the corners ; join them. If more squares are desired repeat the operation as from points VI and VII. Or you can simply start a square from one of the diagonal lines (say at A) and 62 SKETCHING FORESHORTENED SURFACES run it parallel to the outer square A to B, B to C, C to D, D to A, since the diagonal lines will determine its corners (see Appendix). pr ' " " T^ . M\\ / V"\ \ /---' \ \ -X TZX SI 3D Fig. 107. Fig. 105. The foreshortened view of concentric squares can be drawn without recourse to its plan. Practice. — (Fig. 107.) Draw the outer fore- shortened square and add the diagonals. On base tick off desired width between squares. From these carry lines to V.P. Where these lines cut the diagonals will be the corners of inner squares. Join them by horizontal lines. An Application of Fig. 107. Steps on four sides oi a hollow square. — -We sometimes have to draw a platform with a hollow centre ; in other words, steps which enclose four sides of a cavity, as in Fig. 108. Practice. — Draw the front of the platform, carry its side lines to the P.V.P. Judge its depth in relation to its height and make the .j a back line (1-2). Take a line from 2 till it meets the bottom receding line (at 3) to obtain its height at the back. Draw diagonals across top. The four corners of the hollow must Fig. 108. be on these diagonals, so fix the position of the near corner (4) and from this point carry a line to P.V.P. Where it cuts the 64 SKETCHING FORESHORTENED FIGURES Fig. 109. jE$ sr.Jt other diagonal (at 5) will be another corner. Carry horizontal lines from these corners (4 and 5) till they meet the diagonals (at 6 and 7). Join € i to 7. Fig. 109 explains how the same method . would apply if the base were seen at an angle instead of being parallel to the horizon, as in Fig. 108. The use of a plan in figure drawing. — I once asked my old friend, the late Byam Shaw, if he found any use for perspective when draw- <••>•<•-.-; ing foreshortened figures. His reply was that he never ; : drew one without thinking of it, and he sketched Figs. 110 and 111 in ex- planation. You will notice that the horizontal lines over the figure represent divisions across the back of him, so that when he ! is laid down these are on ■ the ground, while the up- • FlG - 111 - right lines from them give the height of the box he occupies. The foreshortening of the division is obtained in the ./J,X,.\ way explained in Chap. III. Fig. 110. Do not miss seeing that the last division on the chest overlaps and hides the neck. Shapes are often better explained by such contours than by the outline itself. Illus. XIV, of The Dead Warrior, of Velasquez, shows this admirably. (The study of Michelangelo's wonderful drawings will dispel any idea that drawing depends upon the representation of the outer rim alone.) '--'.' ---' l ~";r<.,_ ..., ^* _,**" ~~- ---__ "'-- -... ,,--' ^' — — - ■- .-;'■-..;. -- Fig. 118. s J =5 O h-1 s H 9 a W EH w a H 70 INCLINED PLANES ro r > - kemuwmg iikc ok base of ticture Fig. 119. proportionate measurement ; but you can draw it by per- spective rules if you wish to. Problem.— (Fig. 119.) To draw a step whose front is parallel to the horizon line 4 ft. long, 1 ft. in height, and 2 ft. in depth. Draw the front of the step, making its height one-fourth of its length. From the four corners take lines to the P.V.P. to form the sides. The length from to 2 represents one-half of the length, or two feet ; therefore from the point 2 take a line to the V.P. to which the diagonals of squares run (as explained in Chap. Ill, Fig. 77). The depth to A cut off by this line will be 2 ft., because it repre- sents the length to 2 seen fore- shortened. Eaise upright at A till it meets the line receding to the P.V.P. from the top corner of the step. Make the back of the step parallel to the front. "Uphill" V.P. found by hand- rail. — A staircase that is not built between walls will have its sides protected by a balustrade or hand-rail. If we copy the direction of the hand-rails and continue their lines (as they re- Key to Illustration XVII. uphill " V.P. at the point Fig. 120. cede) we shall find the " where they meet (Fig. 120). We then drop a vertical line from the " uphill " V.P. until it touches the horizon line and so find the " level " V.P. This is an easy way of getting round a difficulty, but it is as well also to use the receding scale as pre- viously explained, in order to save time Illus XVII. W. F. Yeames, R.A. (Tate, Gallery). (Photo Oassell & Go } Amy Eobsakt. 72 INCLINED PLANES and to ensure accuracy in obtaining the height of each step (see Illus. XVIII). (3) Steps seen at an angle. — Where steps are in such a position that you face one corner, the front and sides both recede from us, We must use two "level" V.P.'s when drawing the first step. The rest of the flight can be built by using the sloping scales and only one " level " V.P. (Illus. XXXII and Figs. 121 and 125). Practice. — (Fig. 121.) Draw the near end of step A, and continue the top and bottom lines till they meet on the horizon (V.P. "level"). Add front of step by measuring the angle (see note, p. 38 Chap. II), and continue one line to find V.P. 2. Find " uphill " V.P. by run- ning a diagonal (1-2) across the side of step, and continuing it till 'UPHILL V^ -^ .•tt? .■-■ :: :^ .-*'""-**' "\.-*V* .--•''.-'* '' s ,--»>.-• .-v*;' -i-'*-' --*.--' ~-T ,"' % v v ■■■■nmm ■ %. -^ i 7-— - _V- "-"-"-- '■'- "\ ~-~-~~ )IIIIHIIIININIImy|j[|H[|H|jj|l ' ---""""" J]HIIIIIIIIII!I~ ^^ __-*•'"" I Fig 12 L. it is immediately over V.P. "level." From "uphill" V.P. carry lines to the other three corners of the front of the step to form a scale — one at either end. Flush with the back of A step, raise the front of B step, its height at each end (2-3 and 5-4) will be found between the lines of the sloping scales. From front corners (4 or 3) of B step carry lines to " level " V.P. to form the tread. Where these lines cut the lower lines of the sloping scales raise uprights for C step. Join 7 to 6 to complete step B. Form other steps in similar fashion. (4) A flight of steps seen from the top. — Steps receding downhill can be drawn by the sloping scales whose use we are accustomed to, by drawing steps from below. The V.P. for the sloping scales will be at some point immediately under the " level " V.P. to which the side lines of the treads tend. Practice for Fig. 122. — Find horizon in Nature, 1 and copy its 1 As instructed in Chap. I. Illus. XVIII. Drawing by the Author. The Malt-House_JPumt, Bukpham. 74 INCLINED PLANES position on picture. Draw corner of top step and the next one (1 and 2), being careful to get their relative positions exactly, and to record how much the top step overlaps the one below it. Con- tinue line (1-3) of top step to horizon to find " level " V.P. Join ,-,7 I ^.^ //,f> I -VO- ' S / /'» I ' N ^^ "* ^ ^— - --■ ^0. __. \i^ ftXi&B of AMOTHBR S-rep Fig. 122. this with corner 2 and continue line till it is under corner 1. Line (2-4) so made will be the top of step 2 (as if you could see through the top step). Join 1 and 4 to find height of step. Draw a line touching corners (1 and 2) of both steps, continue it till it is under "level" V.P., in order to find "downhill" V.P. This line makes one of the sloping lines of the scale that the top corner of each step must touch. The other scale-line is made by joining the bottom corner (4) of the top step with the " downhill " V.P. Draw height of next step (line 2-5) between scale-lines. Join 5 to " level " V.P. to form top of third step. Draw 1-A the width of staircase. Join " level " V.P. with A and continue to B to complete top step* Join corner A with " downhill " V.P. Join 2 to C. Join " level " V.P. with C and continue to D. The rest of the flight can be quickly INCLINED PLANES 75 drawn in the same way with the scale on the left side to find the heights, and one scale-line on the right to find the corners. " Downhill " V.P. found by skirting-board. — If there is a skirting- board running down the stairway, its direction can be continued until it comes under the " level " V.P. to which the receding lines of the treads tend. In this way . V P . we can find the " downhill " V.P. A stairway that is not set be- tween walls will be provided with *.;, hand-rails or balustrades. It is a . *"'-V^ good plan to drawthese first to find \ "^V. tlle "downhill" V.P. (Fig. 123). uVmtlu ,VP , LEVEL "^"n HEIGHT Line IllUlllil | III III 1 inn : c mi|iiiiiiiii i liiiiii "VIS" Fio. 124. ,.;•' This will save time, but the operation described in Fig. 122 should be carried out. A tedious description will then be avoided in the ease of execution. (5) Steps on either side leading to a platform. — In important buildings the entrance is often flanked by nights of steps on either side of the terrace or portico that ornaments its front. In this case, if we were standing near one corner we should see the ascending flight ahead of us, and at the further end the corners only of the descending steps. We could apply the methods already explained (121-122) for drawing each flight. Fig. 124 speaks for itself, but note that the " downhill " V.P. for the far steps must be the same distance below the horizon as the " uphill " V.P. (for the near flight) is above it. VPS J>«>»«NKILL 76 INCLINED PLANES Height of steps found by measuring-staff. — Also notice that the height of each step on the far and near end is regulated by the upright on the comer of the nearest step. The upright is divided into spaces equal to the height of the nearest step ; from these divisions lines are carried to the " level " V.P. and determine the height of each course of steps. The upright line acts in the same capacity as a number of steps would, if placed one on top of another. (This method could have been applied to any of the previous examples.) This way of measuring the height of each step will be better understood by reference to Fig. 125, though in this case we are directly facing the flight of steps. Practice for Fig. 125. — Draw front of bottom step. Draw upright Fig. 125. and tick off divisions equal to height of bottom step. From these divisions draw lines to " level " V.P. (to form the top of each step). From bottom corner of step draw inclined line to the bottom corner of the next, taking care to make the top of the step the correct depth for its height. Continue the inclined line. It will be seen that the meeting point of the receding lines with the inclined line, determine not only the positions of the bottom corners, but also the depth of each step. This practice is only another application of the receding scale and inclined line which has been explained in Chap. II and used in Fig. 69. (6) Staircase with intervening landing leading to a gallery. — Begin by sketching its proportions ; then find the horizon, the " level " V.P., and the " uphill " V.P. (as in Fig. 130). Having found the height of the banister at the top of the first flight (by INCLINED PLANES 77 means of lines running to the " uphill " V.P.), you carry lines from the top and bottom of the newel-post and from the top of the hand- rail to the " level " V.P. in order to find the height of the banister along the landing, and the height of the next newel-post where the second staircase begins. Lines taken from the post (at the bottom of the second flight) to the '' uphill " V.P. determines the height of Fig. 127. the post at the top of the second flight. The height of the banisters and posts on the right-hand side is found by horizontal lines taken across at each junction of stairway with landing, from the left-hand banister and newel-posts. The stairs themselves are drawn as in previous examples. TO vp up«iU- ". Fig. 128. FigB. 126, 127, 128 explain three other ways of drawing the platform shown in Fig. 124. The height of figures on a flight of steps seen from below. — So long as the height of each step is visible and not partly hidden by the edge of a nearer one (as happens towards the top of the flight), we can estimate the height of a figure in relation to the height of 78 INCLINED PLANES the step lie is to stand on, a step ten times as tall, and so on. Failini Fig. 129. required height will be shown by- same height on the step above, and walk him along that step to the place where he is to stand (Fig. 132). Remember that the measure- ments as just described must be taken at the side of the flight, because the scale is directly under the side of the flight, though it does not always look so in the drawing. This would be a cumber- some bit of work when dealing with a simple flight ; it would be better to use a scale running up the edge of the steps to the " uphill " V.P., but a use will be found seven inches in height, a figure this we can use a receding scale thus, Fig. 131 : Along the ground on one side of the stairway carry a line to the "level" V.P. that you used for the sides of the steps ; determine the height of a figure at some near point (by comparison with the height of the step or the width of the stairs), stand him on the line and carry another from his head to the V.P. to com- plete the scale. On other steps where figures are to be placed, drop a vertical line down the side of the stairway until it touches the ground ; there the the scale ; make the figure the Fig. 130. INCLINED PLANES 79 for it where successive flights lead to intervening galleries. An alternative method, in the latter case, would be the one we used for finding the heights of the newel-posts in Pig. 130. " >The height of figures on a flight of steps seen from above. — The scale used in Fig. 122 for drawing the descending steps would serve equally well for fixing the height of a figure on any one of them. Fig. 131. Fig. 132. This would be done by marking the height of each step (as if those nearer to us were transparent), and drawing our figures their proper height in relation to the step they stand on. But it would save trouble in the end to make a scale representing the height of a man ; the bottom line of that scale would touch the end edge of each step (Fig. 134) at its centre (marked 1). Having marked the position of the feet of our figure on the step oa use the SCALE MADE BY STEPS Fig. 133. Fig. 134. where he is to be, we must move his feet along the step to the scale ; having there ascertained his height we are able to make the com- plete man walk back to where we indicated his feet. If the steps (Fig. 135) were so broad that we could not walk from one to another at a stride— let us call them platforms— then we should have to fix the height of a figure at the end edge of each 80 INCLINED PLANES platform (do this by a receding scale to the "downhill " V.P.). After that we must take a scale to the "level " V.P., in order to find the height of the figure on any particular platform on which we intro- duce him. Od two of the platforms I have lined up a squad of phantom-men, just to explain the idea more clearly. The operation is really the same as the preceding ones ; in those, however, we presumed that the man's feet would occupy the breadth of the step ; on these platforms we have added the scales to find the man's height as he walks from the front to the back of the platform. Fig. 135. (7) Steps on four sides of a square. — The base of a cross or sun-dial is often built on platforms set on larger ones, these forming stages of concentric steps. These can be drawn, a platform at a time, by the method described for concentric squares (Chap. IV, Figs. 108, 109). The height of each step (Fig. 136) would be found by a receding scale attached to the side of the lowest platform. The width of each step could be ticked off on the near edge of the lowest platform, as at A, D, E. The line from D (carried to the V.P.) would, at the points where it meets the- diagonal, determine the near and far corners of one side of the smaller platform. The line from A would also have to be carried to the V.P. Where it meets the base of the smaller (second) platform at B it would be carried up to the edge (to C). From C it acts for the third platform as the line at D acted for the second platform (the top surface of the second platform having been crossed with diagonals). Each succeeding platform would be raised in the same way as the previous one. The same steps (as Pig. 136) seen at an angle. — These present no other difficulties than those of obtaining the successive depths of the upper surface on each step and their height. Let us first draw the lowest platform with the base of the next one marked out on it. INCLINED PLANES 81 Practice. — Draw the lowest platform (Fig. 137) with diagonals crossing its top surface ; judge the distance between its near corner and that of the base of the second platform (1 to 2). From the near corner of the second platform (2) carry lines to V.P. 1 and Fig. 136. V.P. 2 ; where they cut the diagonals place two more corners ; from corner 3 take a line to V.P. 1 in order to find the fourth corner (5) ; join 5 to 4. To find the depths on the upper surface of each succeeding plat- form and their height. — Repeat the working of Fig. 137 to obtain Fig. 138. Continue the side line of the second platform (1 to 2) till it meets the edge of the lower platform (at 3). Continue it down the side (3 to 4). You can now use the side as a receding scale, marked off with divisions that appear proportionately smaller Pio. 137. (method explained in Chap. II, Fig. 59). Each division represents the depth between one platform and the next, but before we can use it we must find the height of the second platform ; to do so take the height of the platform below the point 3 and raise it above 3, so that it stands on the edge (line 5-3). From the top and bottom of that height take lines to V.P. 1 in order to find the height of the second platform at the corner 2. Draw the second platform like the first and draw diagonals. To find the near corner of the third platform carry a line from point 6 (towards V.P. 1) across the top of the first platform, up the side of the second platform and across 82 INCLINED PLANES its top (towards V.P. 1) till it meets the diagonals. The height of the third platform would be found in the same way by raising the height of the second platform at 7-8. The depth between the third and fourth platforms would be found by a line starting from point 9 and behaving as the line from 6 which we have just detailed. Any number of platforms could be raised likewise. —Jjf. Fig. 138. CHAPTER VI inclined planes — continued RULE II applied to the drawing o£ roads. — The sides and cart- ruts of a receding road on level ground tend to a V.P. on the horizon (call it " level " V.P.). If the road runs in the same direction, but uphill, its lines tend to a V.P. immediately above the " level " V.P. (call this point ?« m0T Illus. XIX. Sketch by the Author. A Road. " uphill " V.P.). The V.P. may be far above the horizon if the road is steep ; or only just over it if there is but a slight incline. The road, if it has the same direction but runs downhill, will have its V.P. immediately below the "level " V.P. The steepness of the road running up or down hill determines the height above or below the horizon for the V.P. to which its sides tend. 83 84 INCLINED PLANES If the road turns so that it takes a new direction, its V.P. will be more to the right or left accordingly, but it will not be higher up or lower down, unless the inclination of the road changes as well as its direction. So it comes about that a road, unless it is perfectly straight and lies in one plane, may have many vanishing points. Each section of it that takes a new direction or inclination has its own V.P. On hilly land a road may be seen in the foreground and again in , , - Illus. XX. Drawing by B. V. C. A Curved Road Running Uphill. the distance, but the intermediate stretch, where it runs downhill, may be hidden owing to the steepness of the ground. Suppose we have not Nature in front of us, we can still find the width of this distant road by drawing the foreground length and adding the connecting link just as if we could see through the hill down which it runs (Fig. 139). (1) Road running downhill. — Though a road runs downhill, we have to represent it running up our canvas. In order to make the illusion effective we must seize every feature that helps to give it a downward course. If its inclination is steep, the depth of the canvas occupied by its length will be slight. In this case we can -a a o c 86 INCLINED PLANES utilise the stones and unevenness of the contours. Those near at hand will hide parts of the roadway behind and will by their over- lapping suggest its steepness. Quite small banks at its sides may have their top lines running level or even downwards, and a piece of old fencing for the same reason may be valuable. Figures are even better for they are on the roadway itself — the head of a distant one level perhaps with the waist of another in the foreground explains that which the lines of the road with their upward direction might fail to do. Roadside trees from their greater height have more effect, the intermittent line of their tops running steeply down the canvas. If the road is precipitous, so that its sides if we could see them would be running down the canvas, our troubles end. Then the head and shoulders of a man or the shelvings of a cart and the horse's head, with all else cut off by the nearer part of the road, Fig. 139. Fig. 140. arrests our attention and cannot fail to explain the situation. We have but to introduce a figure towering over them on the top of the hill to make the deception perfect. With a road as precipitous as we are talking of it is advisable to insure correctness in the size of the figures by sketching in the sides of the road to the " downhill " V.P. Of course we cannot really see the road, so we have to think of it as if it were the under- surface of a plank reared up. This done, we guess or copy the height of a figure, either at the top or bottom of the hill, and then take lines from his head and feet to the V.P. to which the sides of the road tend. If a figure at the bottom of the hill is chosen to give the height for this receding scale, the lines must be continued up the road. Figures on roadways. — If many figures are to be introduced on a road that turns, or changes its inclination, a figure must be placed at each junction where the change occurs in order to carry on the INCLINED PLANES 87 3t»K scale for measuring their height in the next section. Each receding scale will in every instance tend to the same V.P. as serves for the piece of road it runs on. The scale 1 for the figures may be thought of as an imaginary railing, 6 ft. in height, bordering the road, conforming with it in each dip and rise, and using the same vanishing points that are necessary for the roadway. (2) A street on a hillside. View looking up. — It has been said that the thing to do in drawing is to make the lines run in the right direction. Follow this terse advice by remembering that level lines of doorsteps, lintels, flat roofs, window - frames, brickwork, or masonry, run towards a V.P. on the horizon ; but carry the lines of the pavement, roadway, and a scale for anything that is upon it to the " up- hill " V.P. above the horizon and in a direct line above the " level " V.P. If we look at a side-view of some houses of equal height (Fig. 141) we notice that each front forms as it were a step, and that lines drawn parallel to Fia. I* 1 - the road would touch each corner. We may not want to draw so formal a row for our view up the street, but lines to mark the corner of each house and meeting at the "uphill" V.P. would act as a receding scale to decide the height of distant houses and also the depth of their front walls that face the street, and these could be altered at pleasure. This is but an application of the sloping scale we used for drawing steps. Practice. — Sketch in roughly (Fig. 142) the height of the houses compared with the width of the road, so as to place the scene nicely on the canvas. Find horizon in Nature and draw it on the canvas as if it were an actual line in Nature. Copy direc- tion of top or some line on house that would in Nature be level (1-2), and continue it to the horizon to find the "level" V.P. Copy direction of the road B-A, and continue it till it comes immediately over " level " V.P. to find the " uphill " V.P. Copy accurately the depth of the same * The receding scale for figures, etc., is fully explained in Pigs. 46-49, Chap. II Fig. 142. 88 INCLINED PLANES TO UP t "UPHILL house B-C. You have now drawn a foreshortened square repre- senting the front of the house (1-2-C-B). From its near bottom and top corners (IB; 5 and 6) carry lines to the " uphill " V.P. to form scales ; they will give you the heights of each house and the depth of their frontage. Touching the far side of the near house raise the near side of the next house to the top line (1 to " uphill " V.P.) of the scale. From 5 run a line to "level" Y.P. to form top of the windows. Draw the rest of the. row in the same way as the first house. ^N^ IHSKII 1 ''!' ?1 I R IHS IB Carry lines (parallel to the horizon) ^ , JHnHIMIl, !: ! I B II i rL