§§ & § & : & & § § º º §§ §§ º § - gº ~ sº § & § - § § § sº º º gº Ş § - §º &N N § §§ § § § § § § sº N § .. & § Š § § § § w - §§§§ §§ § º º N YS § - § WN § SN - § 8. & §§ º sº º § º § § §§ § §§ - º§ § §§ wº § º § º §§ º § § § º & º º § & & SS § § -º-, §§ § § § & § § º §§ §§ § § § §§ N. N - § - § º d §§ º - §§ ºxº º - §§§ § § § º § § § § § º Ş § § § N N § º § § º º % Nº º §§ &N § $º Š § § § tºº * Ş § § § § §§ - § §§ § s ſº S. º § §§§ §§§ & º º º § § º § § º - § § § §§ § §§ - Ş § Š § § §§§ º . Sº § ſº Ş º Sº § N § Ş ºw $º Ş º §§ s §§ §§§ Ş ĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪĪİIĮĮ¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡¡ ∞∞∞∞∞∞Œ œŒ œŒ œſaeaeae，№ſ\\§©®ae --r-->… W.A z iz. 2^ Z - F- b 9 B OB9 ECT DRA WING. 29 formed by joining these points will be the plan of the second slab. At g and A erect perpendiculars, which, as the second slab stands a little back from the picture-plane, will be drawn rather shorter than a c. Draw a horizontal line for the edge, and complete the object as before. Set off, on the edge of this slab, the width which it pro- jects beyond the cube, which, as in the previous case, is the same as the height of the slab. Draw lines to the point of sight, cutting the diagonals, and this will give the plan of the cube and pyramid, which may now be com- pleted in the manner shown in the figure. The object in the distance (Fig. 2) is the one which has already been drawn in another position (Fig. 2 in the last plate), the front elevation being now parallel to the pic- ture. This view is so extremely simple that the student may fairly be expected to draw it without further instruc- tions. In the previous lessons the objects have been so placed, that their front and back surfaces have been parallel to the picture. Under such circumstances, the sides alluded to retain their original shape, however much they may be diminished in size by being moved into the distance. This has been exemplified in Plate III., in which the front of the cube in Fig. 4 is a square like that of Fig. 1, but reduced in size, in consequence of its being placed back in the picture; and similarly the side of the dis- tant upright in Fig. 3, Plate VI., is similar in shape to the side a b, but is diminished for the same reason. It now becomes necessary to consider the method of drawing objects when their sides are placed at different angles to the picture-plane. PLATE VIII. In order that the student may fully comprehend the exact difference between the positions of the objects now 3O OB% ECT DRA WING. to be considered, his attention is called to the two first figures in this plate. Fig. I is the plan of a cube, placed so that its front and back are parallel to the picture, which is supposed to stand on the line A. B. This position has already been explained in the figure on page 14; and several such subjects have been subsequently worked out. - Fig. 2 shows the plan of the same object when placed so that neither side is parallel to the picture-plane (A B), only the angle a is really in the foreground, the other sur- faces receding from it. In the present plan it will be seen that the object is placed at equal angles—that is, the side standing on a 5 recedes at the same angle as does the side standing on a d, and it will be seen from the plan that the side c d is parallel to a 3, and 6 c to a d. - We will now proceed (Fig. 3) to draw the cube so placed. Let a e be the vertical edge of the cube which is nearest to the spectator, and resting on the point a in the plan. Now we know that the edge a 'd in the plan and the corresponding edge of the top of the cube are hori- 2072.Éa/, but we have seen that horizontal lines when not parallel to the picture converge to a point in the dis- tance—that point being the point of sight when the lines in the object are at right angles to the picture; but the line a d and the corresponding edge of the upper surface, e d", are, as has been shown by the plan, not at right angles to the picture, and therefore they con- verge (not to the point of sight, but) to a point in the horizontal line called the Vanishing point. The lines a & and e & converge in a similar manner to a point on the left side. Care must be taken that these lines are not drawn up too obliquely, which makes the sketch appear as if the object were tilted up from the back. It must be under- stood that the vanishing points need not necessarily be on the paper; nor need the lines be drawn entirely to them. A little observation and practice will enable the OB% ECT DRA WING. 31 students to judge of the amount of inclination required, and to sketch the object with tolerable correctness. The rules for finding the exact position of the vanishing points, &c., do not fall within the province of this volume, but are fully treated of and practically worked out in another.” The student must use his judgment, too, in determining the positions of the perpendiculars à 5' and d d", bearing in mind that the width of the sides will vary according as the object is moved to the right or left, and that both sides will be the same when the object is placed immediately opposite the eye. The following principle will now be found useful to the student :— All lines which in the object are parallel to each other Vanish in the same point. Now it has already been shown in the plan, that 3 c is parallel to a d, and that c d is parallel to a 6. Therefore, according to the above principle, draw a line from e to the right-hand vanishing point V Pº, to which a d and b c have already been drawn, and from e draw a line to V P4, to which the lines a & and d c con- verge. The object being here drawn as if transparent, it will be seen that the same rule is carried out in relation to the distant lines of the base of the cube. Fig. 4 is a view of the same cube when placed above the level of the eye of the spectator, and the lines there- fore run down to the vanishing points on the horizontal line. Fig. 5 is an upright block which, being higher than the spectator, passes above the horizontal line ; and thus, although the lines from the nearest angle at the bottom are drawn upward, those from the top incline downwards. PLATE IX. Let us now apply the principles laid down in the last lesson. * “Practical Perspective”—Cassell’s Technical Series. 32 OB% ECT DRA WING. FIG. I.-a & is the edge of a cube which is nearest to the spectator, and this must therefore be drawn of its proper length. Now from a and & draw lines to the horizontal line, which, in this case, is above the object. The vanishing points would be outside of the paper. The cube is supposed to be placed at equal angles ; but, as the eye is a little towards the right side of the object, the line a d will be longer than a c and thus the right side of the model will be represented as it would appear— wider than the left. Having completed the base of the cube a c d e, and raised the perpendiculars c ſ and d g, from f and g draw lines to the opposite vanishing points, and their intersection will give the distant angle of the upper surface of the cube. In this quadrilateral, draw diagonals, and at their inter- section raise a perpendicular. On this, mark the height of the pyramid, which will be completed by drawing lines from the apex to the angles of the upper surface of the cube. FIG. 2.-This is the slab already drawn as the second step in Plate VII., and will require but little explanation. It stands on edge, its sides being parallel to those of the cube, Fig. I ; and therefore the lines which are horizontal in the model will converge to the vanishing points be- longing to Fig. I. It must, of course, be borne in mind that whatever may be the inclination of these surfaces to the picture-plane, so long as the edges are horizontal—that is, Žarallel to the ground—their vanishing points must be on the horizontal /ime. The student must understand that when the term “hori- zontal” is used in relation to the object, it is meant in a different sense from “horizontal" in the drawing. Thus, in Fig. 1, Plate III., the upper and lower edges of the front square of the cube are horizontal in the model, and are OB%ECT DRA WING. 33 rendered horizontal in the drawing because the surface is parallel to the picture-plane ; but the upper and lower edges of the right-hand side of the cube are horizontal also in the model, but as they are at right angles to the picture- plane they are drawn to the point of sight ; and again, the lines a' d', b c, č' e, and a' fin the model (Fig. 2) under consideration, are all in reality horizontal lines; but being inclined to the picture-plane, they converge to the vanish ing points as already shown. Fig. 3 is an equal-sided triangle or prism standing on its end, the one of its rectangular faces being parallel to the plane of the picture. Now it is evident, that if a sheet of glass were placed vertically so as to touch the perpendicular a 5 in Fig. 1, and a” b" in Fig. 3, the sides of the triangular prism would recede from it more suddenly than would those of the cube, because, as shown in Fig. 4, the angle between the face of the cube and the picture-plane would be 45°, whilst in the case of the triangular prism Fig. 5 it would be 60°. Therefore the vanishing points for the triangle will be nearer to the perpendicular than they would be if the object to be represented were a cube. The width of the sides must depend on the position of the spectator ; but however much the eye may be moved to the left or right, the points c d and e f must be on the same horizontal lines so long as the object is placed at equal angles. The student is again reminded that the present book is not by any means intended to Supersede, or to be a substitute for, the study of perspective proper. Its object is (1) to give general elementary notions of solid forms to beginners, and by showing them the absolute necessity for really scientific knowledge, lead them on gradually to the more severe studies of projection and perspective; and (2), further, it is hoped, that to those who have already acquired some knowledge of perspective as 34 OB%ZCT DRA WING. a science, the studies herein, and the models by which the book is accompanied, will afford opportunities for carrying out, by the hand and eye alone, the principles which they have previously worked out by rule and compass, and will Suggest to them the method of sketching the thousands of objects around them. Again and again it is earnestly impressed on all who would really derive from this book all the benefit intended, that they must not merely copy the plates, but that they should place the models, and draw directly from them, and further, when they have mastered the objects in the prescribed positions, they are advised to change their places and apply the principles which have been laid down. PLATE X. We now proceed to speak of the method of drawing circles and cylindrical bodies, and must at the onset re- mark that perspective does not deal with circles or other curves as such, but requires that they should be enclosed in rectangular forms ; these are then put into perspective, together with the points in them through which the curve passes. In the case of a circle, the nearest rectangular form which can enclose it is a square; and we will there- fore show the method of drawing a circle by this means. In Plate X., Fig. I is a circle which we require to draw when lying horizontally below the eye of the spectator. About the circle describe the square A B C D, and in it draw the diagonals A D, B C, and the diameters E F and G. H. Now proceed to the sketch (Fig. 2). From A and B' draw lines to the point of sight. Draw the line C D, representing the back of the square. Draw the diagonals Aſ D and B C, and the diameters E F G H. --- Having proceeded thus far, return to the original figure, and draw the lines e and f through the points where the circle passes through the diagonals—viz., g, h, ś, ź. Mark off on Fig. 2, from A and B, the distance A e or OB% ECT DRA WING. 35 y Bf-viz., A' e and B'ſ", and from these points draw lines to the point of sight. These lines passing through the diagonals, give the points g’, h", i, j. Eight points are thus obtained—viz., E', g’, G', 2', F, 7', H, and h'. Through these the curve which is the per- spective representation is to be drawn. PLATE XI. Now, as the circle just drawn is shown to be described in a plane square, it is clear that a cylinder would be con- tained in an oblong block, the ends of which are squares. Proceed therefore to sketch such a block (Fig. 1), and guided by the knowledge of the principles laid down in the previous plate, draw the elliptical figures representing the upper and lower ends. Great care must be taken in drawing the perpendiculars; they must join the curve in a smooth manner, so that no sharp point of junction is visible, and yet the object must not appear as if it were rounded off at the bottom, which gives a cylinder an unsafe or sack-like appearance. It is needless to say that a cylinder placed horizontally would be drawn in a similar method ; the position of the oblong being changed according to that of the cylinder. FIG. 2.-This is an octagonal prism. The end of the prism is parallel to the picture, and therefore retains its geometrical shape. Various methods for constructing polygons are given in the first volume of this series, and it is assumed that the student has already acquired this knowledge; if not, he is urgently advised to commence the study at once, as it is the basis of all other useful drawing. The figures in the present study are not, however, intended to be constructed geometrically, but the know- ledge of the principles will materially aid in the rapid and correct delineation. Having, then, drawn the octagonal end, draw lines from the angles to the point of sight; and it will be remembered C 2 36 OB%ECT DRA WING. (Plate VI., Fig. 1) that the distant end, since it is parallel to the near one retains its regular shape ; and thus no further instruction will be necessary to complete this object. Fig. 3 is a perspective view of a cone. Having drawn the figure containing the base, draw diameters, diagonals, &c., and raise a perpendicular from the intersection, Trace the curve in the quadrilateral, and draw the lines for the surface of the cone. It is needless to say that these straight lines, enclosing curved forms, are only to be used as guides in the early stages of study ; but a very little practice will soon enable the student to sketch the required form at once, using merely a horizontal for the diameter. OF SIEIADING.. IN order that the drawing of an object may resemble the original, it is necessary that, not only the shape but that the ever-varying effects caused by the rays of light falling upon it, should be imitated in our represen- tation. - It is not in this place intended to enter into the subject of the £7°ojection of shadows, which will be fully treated of in another volume of this series; but as a simple broad shade assists in “bringing out” a sketch of a solid form, a few hints are given to guide the student in shading from models. It cannot, however, be too strongly urged that no attempt at shading should be made until the outline has been examined in every way to test its correctness ; for it must be borne in mind that no amount of shading will make up for bad drawing, whereas a bold and clear out- line may, in most cases, be made independent of any shading at all. Only one light must be used when shading from a model, and this is to be placed in the manner which will P L AT E X. |' \, -- - -|-< FIG, §. >.< u } | |-/|\ <ſ##\ – !// | \\ Q- |//| | ·/!| \ ºſ | | | | } ! || |} ] | | { ~ ! | IN | ur) } | ] ! | |-F{G. 2, }ºo } 2?NQ F }| no - - - - -- * * OB% ECT DRA WING. 37 best bring out the form of the object, by throwing some portion of it into the shade. Now, if one of the rectangular solids be placed near the opposite edge of the table, the candle or lamp being situated on the side at which you are sitting, and on your left hand, then the light will be prevented falling on the fight side and back of the model, whilst the front and left side will be fully exposed to the rays: the back and right-hand side will then be in shade. But, in addition to the solidity of the object keeping the light from falling on the sides which are not opposite to it, it also hinders the light from falling on the table, which, near the back and right side of the model, will be darker still. This darker portion is called the shadow. The distinction, then, between these two terms is, that any part of an object which does not receive the rays of light is said to be shaded ; but when this object prevents the light falling on another surface, the part of that second object or surface which is thus obscured, is said to be in shadow. It may be taken as a general rule that, when the object and the surface on which it stands are of the same original colour, shadows are darker than shades. Rays of light falling upon any surface are reflected from it, according as the surface is more or less polished, and the reflection will be more or less intense as the re- flecting Surface is of a lighter or darker colour. Any surface, therefore, which is directed towards light, not only becomes itself illumined, but casts a certain amount of light on objects opposite to it. Thus, supposing a cube is placed so that the light may fall on one side, whilst the other is in shade ; if a sheet of drawing-paper is held up at a little distance from the shaded side, so that the light may strike directly on it, the rays will be reflected, and the shaded side will be visibly lighter than it was before. On this point, Mr. Butler Williams says: Although all surfaces that receive light do not reflect back an equal 38 OB% ECT DRAWING. quantity, yet all do so to some extent, and to a greater or less degree according as they are placed less or more obliquely with respect to the luminous body and to other surrounding objects. Were it not for reflected light, those objects or surfaces which are not directly illumined would be so totally immersed in shade as not to be seen, their exterior figure or outline only would be visible. If an object bounded by flat surfaces be relieved by a wall or other surface, and the light be supposed to proceed from the left, we may notice three prominent varieties of tint. The lightest will be on those surfaces most nearly opposed to, or facing the light; the second will be on the side of the object from which the direct rays of the light are interrupted by the substance of the object itself; the third will be the shadow cast by that object on a part of the surface facing the light, but of which part is deprived of the direct rays of light by the interposition of the object in relief. Now the shade on the side of the projecting object appears lighter than the shadow adjoining, because, from the adjacent surface of the wall, a certain portion of light is reflected ; and the shadow is the darkest because there is no surface near from which any strong light can be reflected to the place it covers. Shadows appear darker when cast on a surface in bright light, than when cast on a surface in a fainter light or in shade ; and the contras", in the first case, between the shade and the adjoining shadow is greater than in the latter case. Also, in the case of a shadow falling on a flat surface, that part of the shadow which is nearest to the object which causes it, is darker than the parts more distant ; the shadow becomes gradually less intense the further it recedes from the object whereby it is produced. As the application of the principles thus laid down will be shown in subsequent objects and groups. Some atten- tion will now be given to the manner in which shading is to be accomplished. OB% ECT DRA WING. 39 If the drawing be small, it will be sufficient to employ the pencil to shade it. For this purpose the B will be found the best for the general shading, HB for the lighter shades, and BB for the darkest shadows. The shading should not be done by rubbing the pencil up and down, but by clear lines, which must afterwards be softened by a pencil of a slightly lighter degree; but the lines should still remain visible, though not too dis- tinctly, and in order to obtain the crispness and brilliancy So necessary to the beauty of a drawing, the lines should, from time to time, be brought out, so that they may not be lost in the filling in. These lines are called “hatchings,” and though no absolutely universal rule can be given as to their direction, it will be found in most cases the best that the limes should follow the direction of the plane on which the shadow falls. Thus, let us suppose it were required to shade a cube suspended against the wall, on a higher level than our eye, which is situated on the right side, whilst the light comes from a point above and on the left side of the object. It is clear that, under these circumstances, we should see the right side and bottom of the cube, both of which would be shaded, and that there would be a “cast shadow ’’ on the wall. • The side of the cube should be shaded with vertical and the bottom with horizontal lines, whilst the shadow on the wall should again be done in vertical lines. Curved surfaces should be shaded by lines which par- take of the general curvature, and these may be crossed by others, but care must be taken that these hatchings do not cross each other at right angles, like the threads in a woven fabric; they should cross obliquely, so that the spaces between them may be lozenge or diamond shaped. None of the hatchings should be visible when the drawing is viewed from a short distance, but should form a uniform clear tint over the shaded surface. The largest and darkest shadow should be laid on first, 40 OB% ECT DRA WING. for if the opposite plan were adopted, the learner would have diſficulty in so graduating his tints that the darkest portions of the work should not become too dark and heavy. Still it is to be understood that the shadows and shades are not any of them at first to be made as dark as they are intended subsequently to be, for the constant retouching which the work necessarily receives would thus make the whole too dark, whereas by getting the shadows generally spread over the whole work, and then working on each in turn, the relation of one to the other is observed, and no one part looks faded whilst others may look fresh. It is, of course, necessary to avoid smearing the work, but this is easy with a certain amount of care. The outline, too, will require touching up as the draw- ing proceeds, but it must be particularly observed that in reality there is not a Zºne round objects, and that there- fore the boundary line which is necessary in a drawing must not be hard and darker than other parts, which causes the drawing to appear as if it were intended to re- present an object bound with a band of iron ; still the form must be clear and defined, and care must be taken that in the process of shading the outline may not become ragged. Drawings of objects when of a larger size should be exe- cuted in chalks. Those most generally used are “French conté crayons,” Nos. I, 2, and 3, and white chalk—also sold in sticks. The chalks are placed in chalk-holders, called portcrayons. In lieu of these, the crayon may be rolled spirally in a strip of drawing-paper, which has the great advantage of lightness, and, for economy’s sake, the small pieces of chalk may be fixed in a quill. To point these chalks a certain amount of practice is required. Having placed the larger end in the holder, scrape the other until it approaches a pointed form ; then turning it in the reverse way to that in which a pencil is cut, holding it between the thumb and middle finger of OB% ECT DRA WING. 4. I the left hand, and supporting the end on the forefinger, cut from the point towards the body, gradually turning the chalk round between the fingers ; by this means, with a sharp, broad knife, and a little care, a fine point may soon be obtained. For model drawing, very fine points are not generally required, therefore, when the point has once been made, it may be kept sharp enough for some time by a smail file or piece of Sand-paper, a strip of which may be glued on a piece of wood, like a small razor-strop. The three numbers on the chalks represent different degrees–No. I being the hardest. The outline is, in the first instance, to be drawn with sketching-charcoal. This should be properly pointed, and the sketch should be lightly made. Any parts which may be deemed incorrect can be dusted off with a cloth, clean handkerchief, or a piece of chamois-leather. This must not, however, be done too often, as the surface of the paper would thus become roughened, and the grain or “tooth * destroyed, which would seriously interfere with the manipulation of the chalk. Besides this, the habit of constantly rubbing out en- courages want of care in the sketching ; and therefore do not labour under the delusion that your first outline need necessarily be “only a rough sketch,” but aim at correctness from the beginning. When the outline in charcoal is satisfactory, it is to be dusted out, so as to leave merely a slight trace—just enough to guide the eye and hand. The lines are then to be repeated with crayon No. 1. As a rule, erasure of the chalk-lines ought not to be required, since all the corrections should have been made in the charcoal sketch. If, however, alteration be indispensable, the lines may be rubbed out with stale bread, either in its usual condition, or pinched between the fingers until it is kneaded into a paste. It must, however, be understood that the use of bread always more or less unfits the paper to receive the chalk, and thus 42 OB% ECT DRA WING. causes the shading to become spotty ; and, further, the frequent use of bread makes the paper become greasy. Vulcanised india-rubber will, in some degree, remove this, but again the surface of the paper will suffer, so that the old adage, “Prevention is better than cure,” must be borne in mind, especially so, since the cure is not an efficient one ; and thus again the absolute necessity for care is impressed on the student. The paper used when the model-drawing is to be exe- cuted in pencil, should be a good, firm, white cartridge, or any other which is not hot-pressed. If the paper be too smooth the pencil glides over it, and a level tint will not be obtained without much difficulty, whilst if it is too rough the drawing will be coarse and unsatisfactory. Tinted crayon paper is used for chalk-drawing; it may be had at various prices, but a cheap kind is now sold for use in drawing-classes and schools of art, which will be found quite good enough for general purposes. The paper generally used is of a pale grey or dull slate colour, or drab. The colour ought not to be too positive, but should serve as a middle tint between the white chalk and the palest tints in the shading. If the drawing be executed on white paper it will re- quire a background, and thus much time is absorbed, to a certain extent, unprofitably ; in fact, by far more work is required on white paper, for all the middle tints have to be worked in with the chalk, which is not the case with tinted paper, as already explained. The general ground for the shading may be laid on either with a piece of soft washleather, or a leather or paper “stump.” The “stump' is an implement made of chamois-leather or soft paper, closely rolled until it is about the thickness of your little finger ; it is then pointed at each end. The crayon to be used for stumping is No. 2, or, for very dark shadows, No. 3. It is to be very finely scraped, or it may be rubbed on Sand-paper, or filed, so as to OB% ECT DRA WING. 43 obtain an impalpable powder. A little of this powder is then placed on a piece of waste paper, and the point of the stump is dipped into and turned round in it, so that it may become charged with the chalk. Before, however, you touch the drawing with the stump, it must be rubbed on another piece of paper, so that the chalk may be evenly distributed over it, and that there may not be too much upon it. The quantity must, of course, depend on the depth of shade required. The stump is then to be passed lightly over the surface to be shaded. If the space be narrow, the point must be used ; if wide, the stump should be held almost horizon- tally, so that the side of the implement may be used, and for spreading the chalk over larger surfaces, the piece of washleather may be used for the same purpose as the Stump. The touch must be light and free, so as to spread an even tint over the paper. If you rub hard, the shade will become dark and streaky, and further, you will injure the surface of the paper. One end of the stump is to be kept free from chalk, to be used for smoothing and softening the work of the other. Should one part turn out spotty, the parts which are too light must be touched with the dark end of the stump, until a level appearance is obtained. The lights are produced by means of white chalk, rubbed on with the paper stump, but for the highest lights the white chalk is used directly to the drawing. You will find the white chalk by far more difficult to cut than the black, and the points break off very often. Both these difficulties may, however, be lessened by cutting the point flat like a chisel, and then drawing the lines with the Sharp edge. The shades and shadows being thus laid broadly in, the hatching is to be done according to the directions already given. 44 OB%ECT DRA WING. It is not deemed necessary here to give any further in- structions as to shading. The application of the prin- ciples will be further shown in the figures which are to follow. The student is urged to think for himself; to place two or three blocks of wood in various positions, and to move the light (or, if that be fixed, the models), so as to observe the varying effects of light and shade. The manipulative process is the result of practice, and this may be obtained by covering, at first, Small surfaces, and subsequently larger ones, with flat tints of various degrees of darkness, and hatching them in vertical and horizontal directions. The student must, however, bear in mind the axiom already laid down, that no amount of shading will remedy bad drawing, and that therefore by far greater importance must be attached to outlines than to shades. In all the lessons therefore the principles on which the outlines are based are fully given, so that this important point may not be lost sight of. PLATE XII. The group in this plate represents two cubes, the one of which is parallel to the picture, whilst the other is placed angularly, and surmounted by a pyramid com- posed of four equilateral triangles. Of course, the lower cube is to be sketched first, and in this view it has been so often drawn that it will not be necessary to give any instructions concerning it, and we will therefore proceed with our study of the upper objects. It will be clear that, when the sides of a cube are at equal angles to the plane of the picture, the one diagonal of the base will be parallel, and the other at right angles to that plane. This will be understood by Fig. 1, which is the plan of two cubes, placed in the manner described, A B C D being the plan of the lower, and E F G H the plan of the upper. From this it will be seen—(1) that the diagonal G H of the upper cube is yº, __º. OB%ECT DRA WING. S. ºš . . ... ºxi : : * -> *** * * << parallel to A D, the front edge of the lower one; (2) that the diagonal, E F, of the upper cube is parallel to A D and B C of the lower, and therefore at right angles to the plane of the picture ; and (3) that the intersection of the diagonals of the upper cube is in this position exactly on the intersection of the diagonals of the lower one, as if one axis penetrated the two. Therefore, let a b c d, Fig. 2, be the upper surface of the cube. Draw the diagonals a c and d b, and through their intersection draw g / parallel to a d, and e fin the direction of the point of sight, and projecting beyond a d and & c. From e draw lines to g and /, and produce them. From g and h draw lines converging towards e /, and e g. This will be accom- plished by making h f and g f rather shorter than e h and e g. This, then, will complete the plan of the upper cube, which is thus given in a separate figure in order to avoid confusion in the drawing. Having brought the sketch up to this stage, draw the front edge of the upper cube—which, it must be remarked, must be slightly longer than the edges of the lower cube, since it is rather nearer the eye, being, in fact, the most prominent line in the picture. From the upper extremity of this line draw the edges convergent with the lower ones, and in the same manner draw the back edges of the cube. Draw diagonals, and at their intersection raise a perpendicular, on which mark the apex of the pyramid. Join this point to the angles of the Cube, and thus com- plete the outline. Assuming that this has all been sketched in charcoal, and corrected with chalk (No. 1), the shading may now be proceeded with. The shadow cast by the projecting angle of the upper cube on the front of the lower one, may be rubbed in first ; then the shaded side of each of the cubes, and of the pyramid. In this the student will observe three different tints, the side of the upper cube not being as directly turned from the light as that of the lower (the 46 OB% ECT DRA WING. first being at 45° and the latter at 90°), will not be so entirely prevented receiving light, and will therefore be rather lighter in shade than the other, whilst the side of the pyramid will be still less shaded, owing to the slanting of its surface. The highest light of all will be at the front edge of the pyramid, gradually toning off into the whole surface. The next brightest light will be on the prominent edge of the cube, which similarly will gradually merge into the general tone of the whole side. The triangular shape of the cast shadow is caused by the prominent angle of the cube, and the variation of this with the slightest alteration in position is exceedingly interesting to observe. PLATE XIII. The group which forms the subject of the present lesson is composed of a cube, on which stands an oblong block covered by the Square pyramid. The following are the proportions of the objects—the cube, 6-inch side, the oblong block, 4-inch side and I2 inches high, and the pyramid, 4 inches Square at its base. These are the sizes of the models in the set on which the lessons in this manual are based, but of course any other proportions would do as well. It will thus be seen that when the oblong block stands on its end upon the cube, a margin is left, and the same width is also seen of the under surface of the base of the pyramid, which rests on, and overhangs, the oblong block. The front edge of the cube, being the most prominent line is, of course, to be drawn first, and the cube finished in the angular view as placed. Now it is evident, that since the sides of the two blocks are parallel, their diagonals will be coincident ; that is, the diagonals of the base of the oblong block will rest exactly on those of the surface of the cube, but they will not equal them in length (see Plate VII.). << 2. * * ------>;--><----- - * - {} N. . As 7°o º E Of Ais A É- Ääs | | | | | . yue =====E =- -et- rº-m- E- z- rº-E" *** ------ - i- ---------> -- ** * * * º : º “ º * , ºº º vº sº a - §§ º, ºy * T.." w; & º :E : §§ º ºr: tº: OB%ECT DR.4 WIAWG. 47 Having, then, completed the cube, draw diagonals in the upper surface, mark off on the diagonal which crosses from the most prominent angle, the apparent distance of the angle of the upright block, and from this point draw lines to the vanishing points of the edges of the cube, or at least convergent with them, so that, if produced, they would meet, for as the student advances, he is not expected, in hand-drawing, to really fix the vanishing points, and rule the lines ; a knowledge of the principles and observation of appearances will enable him to draw from objects with tolerable correctness; but, as said before, model-drawing is not intended to serve as a substitute for the study of perspective, but as an application by eye and hand of previously acquired rules, which have been accu- rately and carefully worked out : just as writing a letter or other composition is an application of the rules of grammar to which we have become so habituated that correctness comes almost by intuition. These lines, then, tending to the vanishing points, are to be drawn until they cut the diagonal which extends horizontally across the cube ; and from these points lines drawn in the opposite direction will meet on the first diagonal, and complete the base of the upright block. In the present position of the cube, the one diagonal is horizontal, whilst the other, being at right angles to the plane of the picture, is drawn to the point of sight. As already explained, this is because the object is placed at equal angles, but would not be the case if it were in the slightest degree rotated. Proceed now to draw the perpendicular edge of the oblong block which is nearest the eye, and having fixed its apparent height, draw lines to the vanishing points for the horizontal edges. In the present study the objects are placed so that the level of the eye is about the middle of the height of the group ; whilst, therefore, the lines of the cube and the base of the oblong block tend upward, those 48 OB% ECT DRA WING. of the top of the block and of the base of the pyramid incline downward. - Now, from the left and right angles of the base draw perpendiculars for the two other visible edges of the object, and from the fourth angle draw a perpendicular which will be terminated by lines drawn to the vanishing points from the extremities of the perpendiculars. The inner surface of the top of the object will thus be represented. Of course, this would not be visible unless the model were transparent ; but, as said before, it is best in the first sketch to assume this in order to account for lines which are not visible, and to find the places for others which depend on them. Through the near and distant angles of this inner surface draw a diagonal and produce it, remembering that, as the pyramid is exactly over the cube, the diagonals will be over each other, and therefore this one will con- verge to the point of sight—as does that of the cube—and the other diagonal will be horizontal, as already explained. On the first diagonal mark the most prominent angle of the base of the pyramid, which will, of course, be exactly over the front angle of the cube. From this point draw lines to the vanishing points, cutting the horizontal diagonal in points which will give the angles of the base, both of which again will be over the corresponding angles of the cube. From these points lines drawn to the opposite vanishing points will give the back lines of the base of the pyramid which are in the view only partially visible. The apex of the pyramid will, of course, be situated on a perpendicular raised on the intersection of the diagonals. It is advisable, in order to test the exactness of subjects such as this, to sketch the inner surface of the base and draw diagonals, then a line passing through the intersec- tions of each set of diagonals should be an absolutely vertical line—the axis, in fact, of the whole group. In shading this group, the light, as in the last case, has been placed on the left-hand side, and higher than the OB% ECT DRAWING, 49 pyramid; therefore it will be evident that the brightest light will be on the left side of the most prominent angle of the pyramid, and then on the left side of the prominent edge of the oblong block and of the cube. The right- hand side of the whole group will, of course, be shaded. The margin of the base of the pyramid, too, will be in shade, but its tone will be rather darker than that of the sides, whilst the shadow cast on the oblong block by the project- ing pyramid, and the cast shadows on the ground and on the upper surface of the cube, will be the darkest of all. It must be remembered that the most brilliant lights and deepest shadows are those nearest the eye, and that both of these diminish in intensity as their distance from us is increased. In shading, therefore, we must follow this natural effect. Thus, in the present group, the brightest light follows the most prominent angle. The surfaces. however, do not remain equally bright over their whole breadth, but the light gradually tones down as the surface recedes. Similarly, the shade on the right side is darkest where the surface is the most prominent, and the depth of colour is softened down as the distance increases. This is sometimes called arial perspective, or the per- spective appearance caused by the air, for the further an object or surface is removed from the eye, the greater will be the mass of atmosphere intervening, and thus a sort of medium is formed through which the object is seen more or less distinctly, according to the denseness of the air. Thus, in a clear day, distant objects appear much nearer than they do when the atmosphere is hazy ; and this accounts for the sharpness of outline and apparent flatness of objects seen in countries noted for the clearness of the air. - - Not only, however, is it found that the lights and shades diminish in intensity as they recede from the eye, but, as a necessary consequence, the contrast between surfaces becomes also less pronounced, and their outlines less dis- tinct, the more the distance is increased. Therefore out- |D 5o OB%5CT DRA it'ſ NC. lines themselves should vary in the thickness of the lines, and should become fainter and finer as they recede. The drawing on Plate XIV. represents an Octagonal prism. Before entering on this study it is necessary to preface it by an elementary geometrical construction, in order to show the reason for the disposition of the lines. Let A B C D E F G H * be an octagon of which a perspective view is required. C, —" D Now, as in the case - / of circles, it is necessary that polygons should be enclosed in rectangular H P T M C figures, and the one / which will best contain an octagon is the Square N formed by producing L A B ' four of its sides—viz., I, J, K, L. Draw the diagonals, J K and L J, also the lines A, F, B, E, G, D, H, C, cutting the diagonals in M, N, O, P. Put the square I J K L into perspective, and draw the diagonals, as shown in Fig. 2, Plate X. Mark off the spaces L A and I B. From A and B draw lines to the point of sight, cutting the diagonals in M, N, O, P. Through M, N, O, P draw horizontal lines which will cut the sides I J and K L in C, D, G, H. The points A, B, C, D, E, F, G, H will then be the angles of the octagon perspectively rendered as lying on the ground-plane with two of its sides parallel, and two at right angles to the picture, K F E PLATE XIV. Proceeding now to Plate XIV., it will be evident that, if a plane octagon is contained in a square, an octagonal A”sm will be contained in a solid oblong (called geome- oB3FCT DRAW/NG. 5 I trically Žaralle/offi/cd). In commencing, therefore, to draw the object, sketch this oblong, as shown by the Coºted lines which are retained to act as guides, but which may be rubbed out in the drawing when the re- quired figure has been correctly delineated. In the present study it will be evident that the eye of the spectator is on the right-hand side of the object, and on a higher level. The student is advised not to place his model exactly like that in the example, but to adopt the principles laid down instead of copying the drawing. Assuming however, that he has the exact model, his view will vary. according as his eye is above or below, on the left or right of the object ; and in determining, therefore, the position of the point of sight he will be guided by the instructions already given. Having, then, sketched the general form of the oblong block, rendering it as if transparent, the perspective representation of the octagon is to be drawn in the figure representing the square top of the block in the manner already shown on page 50. - From the angles of this figure perpendiculars are to be drawn which will cut the base in corresponding points, and these being joined, the view of the octagonal prism will be completed. - The shading of this object in its present position will be found extremely simple, the principles having already been explained in relation to the cubes on Plate XII. In accordance with these principles, since the light is supposed to fall on the front and left side of the prism, they will be fully illumined. The darkest shade will be on the right side, which, being a part of the square block in which the octagonal prism is contained, is at 90° to the picture-plane, whilst the side between this and the front being at only 45°, will be of a lighter tint. 52 OB9 ECT DRA WING. PLATE XV. The objects represented in this plate will possess some interest for carpenters and joiners, since they show the method of drawing pieces of wood which are to be “halved” together ; a process which has been described in “Building Construction,” page IOO, Fig. 106. Figs. I and 2 show the pieces separately. Out of the upper side of the one and the lower side of the other pieces are cut, the recess being in each case as wide as the wood to be sunk into it, and half as deep ; and thus, when the pieces are brought together, the thickness at A, fills up the depth of B, and the surface C becomes flush with D ; a square mortise is cut through both pieces, for a purpose to be subsequently explained. In drawing these pieces, no notice is, in the first instance, to be taken of the recesses, but the pieces of wood are to be drawn as if complete. The square end of the lower piece is to be drawn first, and from this the long edges converge to the point of sight, in the manner already explained. On one side of the square, mark E. the depth of the recess, and from E draw a line to the point C: Sight. Then, having marked F and G, draw horizontai lines across the upper surface, and from the extremity of each draw the vertical lines, as at B, as far as the line E. The rest will be easily understood from the illustration. In Fig. 2, which is parallel to the plane of the picture, the front is, of course, to be drawn first, of its true pro- portions, but slightly diminished, in consequence of being placed at a short distance back in the picture, so as to be immediately over the middle of Fig. I. The upper surface and end of this piece will be drawn as in previous cases; and the recess is next to be drawn, Care being taken that it corresponds in width with the line at F; the mortise will be easily drawn without further explanation. The object formed by the union of the pieces will be a cross, which will be the subject of a subsequent study. OB9 ECT DRA WING. 53 Figs. 3 and 4 are two pieces of wood of the same thick- ness as I and 2, and making together, including the length of the tenon, the same length as the other two pieces. The tenon on Fig. 3 is flat, that on Fig. 4 is Square, and has a space equal to one-third of its width cut away; the tenon on Fig. 2 is exactly equal in width to this space, into which it fits; and, thus, in making up the length, only one of the tenons is counted. When, therefore, the tenon a is inserted into the groove & the whole piece becomes the length of I and 2, and its purposes will be shown in a future lesson. . The method of drawing these two pieces is precisely the same as that already shown, and, therefore, no further ex- planation is deemed necessary. - Fig. 5, shows the cross lying horizontally, and serving as a stand for an upright, such as is shown at page 53 in “Building Construction.” Now it is clear that a cross having four equal arms, would be contained in a square slab, and this knowledge shows us the most simple method for drawing the object. Let a b c d be the front edge of such a slab, its length and thickness being those of the wood of which the cross is made. Complete the view of the block, rendering it as if trans- parent. Now, in the middle of the rectangle a b c d, draw the square if A / for the end of the one bar which is at right angles to the plane of the picture, and from the angles draw lines to point of sight, meeting the distant side of the slab in m, 7t, o, lines joining these points will complete this bar. - Draw the diagonal & e, which will cut the line 7 mt and i mt in Ž and q. - Through / and 7 draw horizontal lines, which meeting, a e in r s, and & fin t it, will give the upper surface of the second bar of the cross. From Ž, draw a perpendicular to meet the line / o, and this will give the point through which 54 OB9 ECT DA'A WING. the line v w is to be drawn ; the lines r v and f w will then complete the view of the cross. Now, at the points of junction of the two bars, raise perpendiculars, and finish the upright, as in Fig. 4. The shading of this object is exceedingly simple, and will be readily understood on reference to the illustra- tlun. PLATE XVI. The subject of the present lesson (Fig. 1) is the cross wherl placed vertically, its front elevation being at right angles to the plane of the picture. As already explained, this cross would be contained in a Square siab ; but it must be clearly understood that the principle of thus generalising the form would be applicable whatever might be the proportions of the arms. Having, then, sketched the slab, draw at the middle of the side which is parallel to the plane of the picture, the Square representing the end of the horizontal arm, and draw lines from the angles of this square to the point of sight ; the length and thickness of this end of the arm will be regulated by the distant side of the slab, which is, of course, parallel to the near side. Now draw a diagonal, a b, which, cutting the lines of the horizontal arm, will give the points c and d, through which the lines forming the face of the upright arm are to be drawn; the rest of the figure will be readily understood from the drawing. FIG. 2.-It having thus been shown that the cross which formed the subject of Fig. I is contained in a square slab, it will readily be seen that if the additional arms (Figs, 3 and 4, Plate XV.) are inserted, the cross will have six equal arms, and thus, instead of enclosing it in a square slab, a cube would be required for the purpose. We will proceed, therefore, to sketch a cube placed at an angle to the picture-plane—viz., a b c deſ. Between a and c, and between a and e, mark g and /, PLAT E XV. = «ff ſ-º ！！ ） ■ ■ ■ ■ ■ • q º ^ ！ 1ºs } \ ^ ^ «S; ·* \ •* , !ºs !\ ?\ £\● U |\ 0\●•^ < !}&�^ ） | | | |{^ < {«）, {}u� | ||！*• |\!∞ *^ ） &N |-||!�� !ıº, 1 | |||：*� 1 | |�� ●|||－Ď ۩ ✉ • • • • • • • • • • •’, |¡· , \\SQ | | |J• ,！ | ||●,， .| ! ! '�،}� ●）�● 1 | |：*!! ------~！$-_/„!°′ tae|� *} • ! !!,，！| • ſ|†∞U *aeQ ſºs•ę=========&• !£!� �§ș{） |●-->!∞ |● ae! |\ /！ !º.*●！ |* \ /r1 N|• 4| N/ • \f)|**_ſ=! S}afº._-=#Eſ|||||： |}：#`s）=----J E™AR!!!#--=============E=● ■=----=======S∞=== È\|||/ | _ =ã● ±*- - №=№， ſº -G ，\,, , , Qſ1 !«.^ º ） ;L ●*1 'l ,！ ●*• |:● ∞w ar J^ ^ſ',● �* • • •{ ） |－2°S）,! ）· , -: ,0 0 \ ， ！ſ sfº,Ģ· �•, . "● 0*！į•;}} 0�«»*1 |-£：！· ●•●！*S^~} §∞ ！/fi� ●، #！� ●u. º.+/}� �!/J\ * ,ſ.•T\ }•± *● • º,' ,0！ �：！{ �،J● * 1 *± w » ºSſae•Nº • • • • • • • • • •* ~~~~●ſ_ º, ， ,w º‘n! A\,\!± *！�¡ A`s±!! **！）● **{!| �*！!！ A\ ,|● ** ,1t. �• ,● *! ��| L *\!| �\ x,1 �ſº ºt|－ ！*1| �\ ^! �**G § �! 03 ！*， ●\!! \�§! SOEſae！ * ± *tº●*● *A.*） �！ * *\0*∞ *！t�∞ *�●*！ *�- *� *●| *1* �*|*！ **!�t *�*● *！t*● *!0�u *, !*L \ \º！ , !º.� `Nș ſe*& v \ \ ）|- vº• ? * • • • • • • • • • • • • • • •t| OB9 ECT DRA WING, 55 and i f, representing the perspective widths of the arms which touch the sides of the cube. From g h and 2 7 draw lines, which in the object would be parallel to the sides of the plan, and therefore in the drawing must tend to the same vanishing points—viz., gº, h /, 2 m, and 7 m. These lines give the perspective view of the plan of two slabs, intersecting each other at right angles; and it will be evident that in these two planes the arms of the cross will be contained. From these points erect perpendiculars, which, being joined on the upper surface of the cube, will complete the view of the two intersecting planes; the lozenge-shaped form caused by the crossing of the planes representing the Square upright, which is common to all the arms. It now remains to (as it were) hew the horizontal arms out of these two vertical slabs ; and it will be seen that these arms are in themselves portions of a third slab, inter- secting the other two horizontally; therefore, in the middle of the line a b, mark o, ø equal to the real thickness of the arms, and from o, ø draw lines to each of the vanishing points. These lines will cut the perpendicular g h in q, r, s, t, and 2 ſ, in u, v, w, v, these lozenges representing the ends of the arms on the sides of the cube nearest the spectator. From the points q, r, s, t and iſ, v, w, 4, therefore, lines are to be drawn to the vanishing points, and thus the object will be completed, care being taken that the distant ends of the arms are absolute continuations of those nearest the Spectator. PLATE XVII. The object (Fig. 1) in this plate contains the leading principles involved in drawing a table or stand for a machine, &c. It consists of four oblong blocks, the cross Section of which is a square, and of a square slab, the thickness of which is equal to that of the oblong blocks. 56 OB%ECT DRAWING. The oblong blocks are placed so as to form the corners of a square, therefore, in the first place, sketch the per- spective view of such a square, and draw diagonals. From a and & set off a c and 5 d, representing the width of the blocks which are to take the position of legs. Draw lines to the point of sight. These will cut the diagonals in four points, which will be the inner angles of the plans of the legs. The complete plan being thus prepared, it is to be left whilst the general form is proceeded with. Draw perpendiculars, a e and & y, equal to the length a 5, for, as in the models now before us, the length of the slab & fis fourteen and its thickness two inches; and as this slab rests on uprights twelve inches high, the front becomes a square, and the general form of the object is a cube. Now draw the line gh, representing the thickness of the slab, and from h draw a line to the point of sight. Perpendiculars raised from the points already marked in the plan will complete the general view of a square table. Fig. 2 shows how a stool may be similarly drawn. This object is placed so that its sides are at angles to the pic- ture, and will be easily understood from previous examples. The shading in these objects is obvious, and its general character having been glanced at in the drawing, should be studied from the actual group of models. PLATE XVIII. The subject of this lesson is an Octagonal prism lying on one of its long sides—its axis, and consequently the sides and ends, being at angles to the plane of the picture. As in the former case, the proper method is to imagine the prism to be enclosed in an oblong block having square ends. The figure on page 5o has shown the method of draw- ing a perspective view of an Octagon. It is not deemed necessary here to repeat that elementary figure, since, OB% ECT DRA WING. 57 although in the present study the octagonal end is vertical, and at an angle, and in the former view in Plate XIV. it is horizontal, the near and distant side being parallel with the picture-plane, the result is obtained in a similar manner, the lines, however, being drawn to the vanishing point instead of to the point of sight. Having, then, sketched the oblong block a b c d, efgh, and having marked upon a 5 the points if, draw lines from these to the vanishing points of the lines a d and & c. These lines will cut c d in Å and Z, then i ſ and A / will be two sides of the octagon. Now draw the diagonals a c and & d, cutting c & in 77, and 7t, and 7 / in o ż. Draw vertical lines through m o and n ſº, meeting 5 cin q r, and a d in s : ; then q r, s : will be two more sides of the octagon. Join q i, 7 AE, ! ?, and f s, by which means the view of the octagonal end will be completed. It is only now necessary to draw lines from each of the angles of this end to the vanishing point of 5 f and a e. These lines will cut the sides of the quadrilateral e fg h, and give the points for the distant end of the prism. This object is purposely rendered as if transparent, and without shading, so that the working may be clearly seen. It is needless to repeat, that the lines here shown would not all be necessary to the advanced student; but it is the know- ledge of the principles laid down, combined with practice, which gives facility in drawing either from the objects, from memory, or from imagination. FIG. 2.-This is a view of a Hexagonal prism, which is given to show the application of the same method to another polygon. In this case, however, the object is supposed to be hollow. Now if the side e of the hexagon were continued until it reached the horizontal line, the intersection would be the vanishing point for all lines parallel to e (the student will remember that all lines which in the object are parallel to each other vanish to the same point), therefore e and f 58 opyecT DRAWING. should converge to the vanishing point on the right side, whilst c and d should be drawn to the point on the left. Now it will be clear, that the lines forming the inner edge of the sides of this hollow prism would in the object be parallel to the outer lines, and that they would meet on the diagonals of the hexagon, as shown in the reduced figure in the margin, Fig. 3. Therefore, having drawn diagonals in the figure which is to represent the upper end of the prism, draw the line g parallel to a, and at Such a distance within it as may seem desirable. From the points where this line cuts the diagonals, draw a line to each vanishing point, meeting the diagonal, which is parallel to the plane of the picture. From these points lines are to be drawn to the opposite vanishing points meeting the next diagonals, then a line uniting these intersections will complete the figure, and it will be evident that this line will be parallel to a, g, and 6. PLATE XIX. The group shown in Fig. I consists of eight blocks of wood stacked at right angles to each other. It will be clear that the blocks thus placed, would form a group which could be contained in a cube ; and therefore, having determined the position of the nearest angle, a, proceed to draw lines to vanishing points, which will be fixed according to the inclination of the sides of the object in relation to the picture-plane. The points Å and c are next to be marked, and these two will be determined in the same manner. Now from 5 and c draw lines to the opposite vanishing points, and thus the ground-plan of the entire block will be perspectively rendered. From a mark a h, representing the apparent distance of the lowest block from the immediate foreground, and also a d for the same purpose on the other side of a. From 5 set off 5 g, and from c set off c Á, observing that these distances being, removed from the foreground, will be as ºr. - - - - - - - - * * l == * ... • f" f --...-- ** ... + ºr " y l y ---" • * * º -----, - - - - -- " A ... *** -- ** rº-------------g R w - * * * * * * * * * * * * º b i | r l f l t l f ! g l * ! º º t t I i ! 1 I I l r l I f l I n } l J b s * l t f ſ t ! ! || || t ! ! ~~~ t : ... • I t a ~~~ ; | L: : 1. *; . 2 * º f ... " g , iſ l * # * ! --- 11 iz: £- * i º l t t t t V. w v l A. i w i * * t - v I i \ | \ } ! \i ----------> * • * * << PLAT E XV I. --> -- * * -, * * ** * !---- * * * = --- : *-* * i - - * * • ... • * * - * - - - * * * - • * ,-, * -* • * \ - - - * • * * * * * ... -- ... * * • - ** *. ! ;* | f f f - *" wº - - .* * * y * * a * CZ f * * > .* 'J' * * t * I *~J l * ~ t * J l * t * ~ I I * | I * ~ - > - l - - - - - - - - - - -, s * * 2.2° 0. t * t e * t ~ * l ... " t ... " l ... " º * t l * PLAT E XV | 1. § - º - - - --- -E--- --- E-E- cº- -º: “ -º-º-º-º-º: :-r- E= | wº-ºº-º- *a-- OB% ECT DRAM//WG. 59 smaller than a d and a h, although representing the same Space. Again, from d set off d e, representing the thickness of the block, and also h i,j ż, and fg, the last two distances being diminished for the reason already explained. At this stage the student is referred to Fig. 2, which is a reduced copy of the ground-plan, and from this it will be seen that the ends of the blocks are portions of the planes which form the sides of the containing block; thus the lines d'é, fg, /, i, and 7 /ē are on the lines a 5 and a c. Proceed therefore to sketch the perspective view of this plan as already shown. . Draw the perpendicular a, and set off upon it the heights of the blocks—viz., to a/, lines drawn from these to the vanishing points will divide the whole containing block into four slabs lying horizontally. From h, ś, ź, and A draw perpendiculars, which, cutting the lines drawn to the vanishing points, will give the ends of the blocks, / i, / m, 7 k, n o, and all others immediately above them in the sanje plane. i From the angles of these ends draw lines to the vanish- ing points on the opposite side. The remaining lines will be readily seen from the drawing. º The shading is simple ; the light proceeds from a point on the left of, and higher than, the object, and thus the brightest light falls on the end of the block à h / m and those immediately above it, and on the left side of the blocks over the point d, the right side being, of course, in shadow. The cast shadows caused by the ends of the bars projecting beyond those immediately under them will best be studied from the objects, since they vary with every movement of the illuminating point, however slight that movement may be. \ It cannot, in fact, be too frequently impressed upon the student that the illustrations in this book are not by any means intended as drawing copies ; they are designed to Serve as guides—in placing the models, and in the method 6o OB% ECT DRA WING. of drawing. There is more to be learnt in one hour's Study from the merest blocks of wood than from the most careful work from copies, though it may extend over weeks. One of the most important features of object drawing is that a student who really wishes to work, need never be stopped by the want of subjects. Every block of wood or stone, however simple its form—and the more simple the better—will afford ample lessons in form, and in light and shade. As already mentioned a set of models has been specially designed to carry out the lessons given in this book, but in order to aid students who have not the opportunity of using these, a set of patterns are appended by which any one of common intelligence may be enabled to make a set of models of cardboard, which, although not very permanent, will still be found useful. In making these, however, the student will have to bring to bear a certain knowledge of practical geometry and projection. These subjects could not, without trenching upon the plan, be included in the present manual, and the student is therefore referred to the volumes in which they are specially treated of ; and this knowledge will not only be found useful in this particular study, but will be the foundation of all true notions of form. PLATE XX. FIG. I.-The group shown in Fig. I is made up of nine blocks similar to those used in the last plate. The end of the group being parallel to the plane of the picture, will be rendered geometrically—that is, of its correct form, since it is not altered by its position. Having, then, drawn the rectangle & Å i d, draw lines vertically and horizontally so as to divide it into the re- quired nine rectangles. Observe that, in the object under consideration, the whole rectangle will be in the propor- tion of 6 to 9; since in the model each block is 3 PLAT E XV || E J. C. 2 - F ( C | * === ~ ~=== = =， =：= *= *== *=== <== --→ → → → → → →= = => → → →= *= * PLAT E X | X. NS / \ \ \,}}> | （） * ſºT7AN«- ~）; `---##1º (}/\ . // } { ſº£/ / #W / / # №， （ / \/---- \!\, /\\ !\ ，7 {\ #wAA �~\\ ďw\!\ ،\\ ·\\\/# }* ：\,}{ \/ \ f \ V\/\/ \ /\,/ V\/\/ V// // \ / \ W f/ \ · · 7 ·A :A /\f\ f%,\ /WA /\�� {\：\\ \-\! #\/\\ ±\ ،*\\ f//*\ ae*\ /\\\\ 4�： www.\ }ſa- ----~---\\ f J،<% ·º/ \ ! |//~/ \ /}/\,/ }ſ.،S/\/ J！raeaeaeaeaeaeAff\ #;/\ •\ // Žeo • OB9 ECT DRA WING. 61 inches wide and 2 inches high, the whole rectangle is thus one third wider than its height. From b, c, and a draw lines to the point of sight, and complete the block by the distant vertical and horizontal lines. Then from the points where I and 4, 4 and 7, 7 and 8, 8 and 9 adjoin, draw lines to the point of sight, which will complete the view of the object. • FIG. 2.-This is an application of the foregoing figure. Thus it will be clear that, on removing the blocks 4, 7, and 8, the remainder, consisting of I, 2, 3, 5, 6, 9, will form three steps. Now (rom each angle draw a line to the point of sight, which will give the inner and outer angles of the steps— that is, the meeting of the risers and treads. The lines representing the distant edge of the steps, which are necessary to complete this portion of the sub- ject, require some care. It must be borne in mind that the risers of the steps are upright, and the treads horizontal, and that as their position is not altered by perspective, the ends of the steps being parallel to the picture-plane, the lines corresponding with these in the distant end must be vertical, and horizontal also. The special attention of the student is called to this point. We now proceed to employ the blocks 4, 7, and 8, which had been removed from the original block, by placing two (4 and 7) as the posts or jambs, and 8 as the lintel. The method of drawing this doorway has been given in Plate VI., &c., and need not therefore be repeated here. The whole of the front of the drawing, as well as the risers of the steps and the Soffit, are in shade, and the cast shadows fall on each step—on the jamb and on the ground. These shades and shadows should, as already stated, be effected by means of lines drawn in the direc- tion of the surface on which they fall. -- 62 OB% ECT DRA WING. PLATE XXI. In this illustration.we have a view of the same object when turned so that the steps are parallel to the plane of the picture. We will first consider the elementary form on which the present Subject is based. In Fig. I, Plate XX., a view of the compound block, made up of nine others, has been given. In the present case, however, the object is placed so that the ends are at right angles instead of being parallel to the plane of the picture. Draw, in the first place, the rectangle a â c d, which represents the front of the whole block, the length in the model being twice the height. Now from a, b, and c draw lines to the point of sight. Draw the back perpendicular, ef, and the distant horizon- tal, / g. This will complete the general view of the block. Now divide a 3 into three equal parts, by the points h, , and from these draw lines to the point of sight. Between a and e set off f and A (letters omitted for simplicity), thus dividing that line into three parts; but these are to decrease gradually as they recede from the foreground. In perspective, this—as will be remembered by those who have studied that subject—would be done by setting off f 4 on the picture-line, and drawing lines to the point of distance, cutting a e in the required points. In the present case the matter must be left to the judgment of such as have not studied the grammar, and it is hoped that the often-recurring question, “How am I to know the width P’ &c., to which in model drawing no true answer can be given, will lead all who wish to draw properly and Scientifically to take up the grammar of drawing on which all correct delineation must be based— object drawing being distinctly a freehand application of the scientific principles. - Having, then, marked the points, draw perpendiculars 2 º-r ſº- --------- º------- E---> -º- = --------- ==-----------sº a-º-º-º-º-º-º-º- ºr ~m--- ºu | | | | | ! - - - - - - - -- - -- |- -- * - * - --- --- - - * *R -- - sº &d* * -i* * --- - :* * * * * .•mº sº. gº * - * * *º sº a as sº * OB% ECT DRA WING. G 3 from them passing through the lines h and i drawn to the point of sight ; the quadrilateral a 5 e ſ will thus be divided into nine four-sided figures, which will re- spectively represent the blocks shown in the previous view, Plate XX. Horizontal lines drawn from h, &c., will complete the view of the block made up of nine oblong solids. Now it will be remembered that, from this composite block, three of the smaller solids are to be removed in order to transform the mere oblong into a flight of three Steps ; this having been done, the drawing can be pro- ceeded with. Draw the line h r, then the rectangle a h r d will be the front of the step in the immediate foreground. From 7, draw a line to the point of sight, and from / draw a line to meet this in s, thus completing the view of the first step. At 2 erect a perpendicular, l m, which will represent the height of the second step, / m. At s erect a perpen- dicular, and draw m f. From t draw a line to the point of sight, meeting a horizontal drawn from 7a in u. This will give the view of the second step. Draw we o and 24 v. Join o z by a horizontal line. Draw lines to the point of sight from o and v, and the whole block is then completed by the horizontal line already drawn from the distant angle. Now, on the line o v, erect perpendiculars for the front of the jambs of the doorway, and proceed to find their perspective height. It will be remembered that the blocks now used as the jambs, at first formed parts of the original block; their real height, then, is equal to the length a d. This height will, of course, be diminished by their distance from the foreground. To find the apparent height therefore, erect at c and 5 the perpendicular 5 ºr c w, and draw w ar. - Produce the perpendiculars beyond w and a to w and 64 OB% ECT DR4 H/YAG. +', equal to d r or a h. Draw the horizontal w x', then the rectangle, w w/ 4' ar, will be equal to a h r d, the front of one of the blocks which forms the lintel of the doorway resting on the jambs. Produce the perpendicular as v. Draw lines from w/ and w" to the point of sight. These cutting the perpendicular It w, will give the points y and 2. Draw a perpendicular at o, and draw lines to the point of sight from tº a, cutting the perpendicular in 2' and y. The horizontals, 2 2/ and y y', will then complete the rect- angle representing the front of the jamb. Now, from the bottom of the right-hand perpendicular of the jambs, draw lines to the point of sight ; and these cutting the back horizontal of the block will give the posi- tions for the distant perpendiculars forming the sides of the jambs, which are to be terminated by lines drawn from the tops of the left and right perpendiculars of the jambs. The perpendicular forming the distant edge of the block of steps will terminate these and give the end of the jamb, from the lower point of which a horizontal will complete the soffit. The shading, as in the last case, is extremely simple, the cast shadow on the ground following the form of the Steps. IPATTERNS FOR MAIKING INRAWING- IMIOIDIELS. IT has already been stated that a set of models has been specially designed to carry out the system laid down in this manual, which although of a size adapted for class teaching, are not too cumbrous for private study. With the view, however, of promoting that self-help which forms so great an element of success in life, the following set of patterns are introduced ; by these the principal of the models may be made in cardboard, and thus a useful set for home use will be acquired. The cardboard should be OB% ECT DRA WING, 6 5 sufficiently strong not to bulge out on the sides, but at the same time it must not be too thick, in which case it would not give sharp edges at the angles. Perfect skill in the construction of these models must depend on the student's knowledge of practical geometry and development (see vols. on “Linear Drawing” and “Pro- jection ”), but as it is desirable that each of the Technical Manuals should be as complete in itself as possible, the simplest methods of constructing the figures are shown ; as, however, the subjects named are of universal application, the student is strongly urged not to be satisfied with the small amount of instruction in those branches which the limits of this book permit, but to follow out the sys- tematic course laid down in the volumes referred to. The simplest model to make, and also the first required in object drawing, is the cube; it consists of six equal CR TD squares, and therefore the geometrical method of con-..." structing a square is here given. Let A B be the length of the side of the required square. R 66 OB% ECT DRA WING. At A draw the perpendicular A C. Make the perpendicular A C equal to A B. This may be done by placing the steel point of the compass in A, and drawing the quadrant B C with the length. A B. Now, from B and C, with the same length, draw arcs cut- ting each other in D. Draw a line from C to D, and another from B to D, which will complete the Square. The student is reminded that all the angles must be right angles, and that all the sides must be equal. If these two conditions are fulfilled, the diagonals A D and B C will be equal. PLATE XXII. Proceeding now to the construction of a cube, draw the square," a b c d (Fig. 1), and produce the sides indefi- nitely. From a set off a e and e g, equal to a 5, and from & set off & f and f h. T) Draw g h and ef. From c o set off c 2, and from d set off ::. 2 dj. Draw iſ. Now, on the lines crossing these, set off from a, a k and c /, and draw / /. On the other side, from 6, set off Ó m, and from a set off d n. Join d n, which will complete the six squares. Against the edges a #, AE /, / c, and the corresponding lines in the opposite side, and also against iſ, leave addi- .....tional strips for a purpose to be pointed out presently. :::... Now the figure must be cut out. This should be done ... with a sharp knife, so that the edges may not be ragged. … The knife should be guided by the edge (not the beviled ‘...? edge) of the rule. : • * * * * This should be done by the method shown in Fig. 2 in “Linear Draw- "***ing';” but in lieu of this may be done by placing the set-square against the T- square, as shown in the above drawing, the cross-head of the T-square being worked against the left side of the drawing-board. P – A T E X X 1. © OB% ECT DRAWING. 67 Cut entirely through the outer lines, but only penetrate the others to about half the depth of the cardboard. Now, inserting the point of the knife, peel off half the thickness of the strips left on the edges, the rest is to be used for the attachment of the sides. Turn the scored side of the figure downward, and turn up the sides ; it will then be seen that the square No. I will form the base, 2 3 4 5 the sides, and 6 the top of the cube, whilst the extra strips will be bent inward, and, having been deprived of half their original thickness, they will not cause the corners to appear clumsy. Fig. 2 is an object of the same character as the cube ——viz., an oblong block. To construct this, draw a line a 5 equal to the length of the intended object, and draw a c and & d at right angles to it, and of indefinite length. Now set off from a, a e, e /, / g, and g h, and from & set off & 2, t 7, 7 Å, and AE A, equal to the side of the square which is to form the end of the prism. Draw lines joining these points and the drawing will then present four equal rectangles. At each end of . one of these construct a square, observing to leave the strips on the edges, as shown in the drawing, and bear- ing in mind that the outer lines are to be cut entirely, and the others half through the cardboard. It will be readily understood that the block shown in Plate XXI., in which the end is an oblong and not s square, will be constructed in a precisely similar manner, PLATE XXIII. - To construct a triangular prism, the sides being equal. This object consists of two equilateral triangles which constitute the ends of the prism, and three rectangles, the width of which is equal to the side of the triangle. Let us, in the first place, show the method of construct- ing an equilateral triangle. A B is the given side. 68 OB% ECT DRA WING. From A with the length (or radius), A B describe arcs, cutting each other in C. Draw A C and c D, which will - complete the triangle. C It may be well in this place to remind the student that a triangle in which only two of the sides are equal is called an isosceles triangle, aS A. . When all three sides are A B of different lengths the figure is called a scalene triangle, as B. In a right-angled triangle (D) one of the angles, as c, is a right angle. A right-angled triangle may be either isoscles, as D, in which two of the sides are equal, or it may be scalene, as E, in which the sides are of different lengths. The longest side of a right-angled triangle, as F, is called the hypothenuse. Now to commence the prism. Draw the line a 5 equal to the required length, and at each extremity draw lines at right angles to it. - On these lines set off from a, a c, c d, and d e equal to the width of the sides, and from 5 set off the same lengths, & /, /g, and g h. Draw cy, d g, and ef, which will complete the sides. Now, on c d and f g construct equilateral triangles, leaving strips at the edges for attachment of the sides. Cut half through the inner lines and entirely through the outer ones, and, having turned the figure, the sides and ends are to be turned up and gummed or glued together. OB%ECT DRA WING. 69 Fig. 2 is the development of a square pyramid. Construct the square a b c d for the base. - From a and Ö, with the length of the slanting edge of the pyramid, describe arcs, cutting each in e. Draw e a and e à. Now, from e, with the radius” e a, draw an arc, and on it set off a ſ, a g. g. h. Join these points by straight lines; leave the necessary margins for attachment; cut the lines either half or entirely through as required, and complete the figure. PLATE XXIV. To construct a cylinder. With the required radius describe the circle which is to form one of the ends. (It will, of course, be under- stood that two such will be necessary.) Now it will be perfectly clear, even to the most casual observer, that when any cylindrical surface is unrolled it becomes a parallelogram ; for any sheet of paper, when rolled up, becomes a cylinder. The question, then, to be solved is, what size must the rectangle be, so that when rolled it may form a cylinder of the required diameter. To accomplish the result which may answer this question,-- Divide the circle which is to form one end of the cylinder into any number of equal parts ; and it must here be explained that the greater the number of these parts, the better; for it will be clear, that if these points were joined by straight lines, the straight lines would be slightly shorter than the curve ; and thus the greater the number of points, the less will that difference be. In the present example, which is intended only to show the method of construction, the circle is divided into twelve parts; but this would not in practice be found sufficient for a cylinder of any useful size, and the stu- * The radius is the length from the centre of a circle to the circum- ference; it is, in fact, the distance between the two points of the compass when describing a circle ; thus, if it were said, “With a radius of three inches,” it would mean open your compass to the distance of three inches. 7 o OB%ECT DRA WING. dent is therefore advised to divide into twice or thrice that number. - - Draw the horizontal line a, and on each side of any starting-point, as a, set off the divisions a, b, c, d, e, f, g, At g and g erect perpendiculars equal to the length of the cylinder. Join these by a horizontal line, and the rectangle thus formed will be the surface which, when rolled around the circular end, will give the required form. Of course, it must be remembered that margins must be left at the top and bottom of the rectangle by which the circular ends may be attached to it. One word as to the proper making of this object. When the two circles constituting the ends are made, and when the rectangle of its full size has been cut, this should be rolled up on a round stick or roller as closely as can be, like a roll of paper. This should be done again and again, so that the surface may become perfectly cylindrical, and may not, from the very strength of the cardboard, burst open, although they are glued together. Another good plan is (when the rectangle has been cut, and when the edges have been glued together, the edges having been previously shaved down with a sharp knife, and the margin having been split) to tie a thread around it during drying, in order to keep the cylindrical portion in shape, and to prevent its expanding. - This should have become firm before being affixed to the circular ends. This will then complete the object. Fig. 2 is the development of the cone shown in Plate XI. This should fit upon the cylinder, and thus the same circle which formed the ends of the cylinder, may be used as the base of the cone. Draw a line a dequal to the length of the Slanting side of the cone, and with this length as a radius, describe an arc. On this set off on each side of a the several divisions of the circle—viz., to 5 and c. Draw d ? and d c, which will complete the development of the cone. The necessary Imargin having been left, roll the shape and PLAT E XX | 1. (l, 6 3 l 2-—N € 1 1 f J 7%. 4 . g k 2 ºl C 2. * oč h 1 PLAT E - XXI I I. 1 &n - º p-r aſ as 2^ *s < * * \s 2 J. ºs. ( /TX/~/ \Z_\Z^^^^^^/~\4–A M- 62; N ſ: &l c!! Čſ & { • s: 2l * f 97 7NTZNL7NLZNLA-7NLZNTZNTZN /\ / 3 £ * P LAT E XXIV OB% ECT DRAWING. 7 I attach the edges & d and & 6, and the base having been prepared, the parts are to be fastened together, thus com- pleting the object. º PLATE XXV. Fig. I in this plate is the development of an octagonal prism, and we must therefore, in the first place, construct an octagon on a given line, A B. l– M C > Fl C A B E Produce A B on each side, Erect perpendiculars at A and B. From A and B, with radius A B, describe the quadrants C D and E F. - Bisect these quadrants (that is, divide each into two equal parts), viz., in G and H. Draw G and H, which will be two sides of the octagon. At H and G draw perpendiculars, G I and H K, equal to A B. Draw the horizontals G H and 1 K. Make the perpendiculars A and B equal to G H or I K —viz., A L and B M. Draw I L, L M, and M K, which will complete the Octagon. 72 OB%ECT DRA WING. ‘Now to make this prism, draw a 5 equal to the length of the prism, and draw a perpendicular at each end of it. From a set off Ö, c, d, e, ſ, g h, ś, equal to the side of the octagon, and draw lines across; then the rectangle a iſ & will be the development of the surface of the prism. At each end of one of the sides, as ef, construct an octagon, leave margins on the edges of the rectangle, which will complete the entire figure. When the whole has been cut out, the lines b, c, d, &c., are to be cut half through, so that the card may bend at the angles as required. Fig. 2 is the development of one of the square slabs, shown in Plate VII., and as this is to be constructed in a manner precisely similar to the cube, the proportions of the sides only being varied, no further explanation will be found necessary. The low pyramid shown in Plate VII., is also constructed like that in Plate XIII., the sides being formed of four equilateral triangles. PLATE XXVI. The object to be constructed is a cross, shown in Plate XXVI. The geometrical form of this will be easily un- derstood. Draw the lines for the horizontal bar, a, b, c, d of each side. Set off on these from a the width of the square end, e, a, f, c, of the whole cross, of the other Square end, g, h, , 7, and of the cross again. Draw the upright bar on each, and the Square ends at top and bottom of one of these, leaving the margins for attach- ment. The strip Fig. 2 will then be required to fill in the sides of the right angles, as g, Æ, l. The mode of finishing will now be obvious. The six-armed cross, shown in Plate XXVI., should be made of wood, and will be understood from the separate parts given in Plate XXV. THE END. P L A T E XXV. 2 ſ J A. 9. / f 63 1. N—- C % [. 3 —/ PLAT E XXVI. --" 2. l, 2—SL - --~~ *--- 6 # 9. Fº 62 Q. l J/ Å f_c NLZſ 1. —- CASSELL’S TECHNICAL MANUALS, CONSISTING OF A SERIES OF VOLUMES WHICH CONTAIN THE ESSENTIALS OF A COMPLETE TECHNICAL EDUCATION, COMPRISE— - Drawing for Carpenters and Joiners. With Ele- mentary Lessons in Freehand and Object Drawing. Containing 250 Illustrations and Drawing Copies. Cloth, 3s. 6d. Drawing for Bricklayers. 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