When the square is at an angle of 45° to the base line, then its sides are drawn respectively to the points of distance,DD, and one of its diagonals which is at right angles to the base is drawn to the point of sightS, and the otherab, is parallel to that base or ground line.
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Fig. 70.
To draw a pavement with its squares at this angle is but an amplification of the above figure. Mark off on base equal distances, 1, 2, 3, &c., representing the diagonals of required squares, and from each of these points draw lines to points of distanceDD´. These lines will intersect each other, and so form the squares of the pavement; to ensure correctness, lines should also be drawn from these points 1, 2, 3, to the point of sightS, and also horizontals parallel to the base, asab.
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Fig. 71.
Having drawn the square at an angle of 45°, as shown in the previous figure, we find the length of one of its sides,dh, by drawing a line,SK, throughh, one of its extremities, till it cuts the base line atK. Then, with the other extremitydfor centre anddKfor radius, describe a quarter of a circleKm; the chord thereofmKwill be the geometrical length ofdh. Atdraise verticaldCequal tomK, which gives us the height of the cube, then raise verticals ata,h, &c., their height being found by drawingCDandCD´to the two points of distance, and so completing the figure.
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Fig. 72.
The square at 45° will be found of great use in drawing pavements, roofs, ceilings, &c. In Figs. 73, 74 it is shown howhaving set out one square it can be divided into four or more equal squares, and any figure or tile drawn therein. Begin by making a geometrical or ground plan of the required design, as at Figs.73 and 74, where we have bricks placed at right angles to each other in rows, a common arrangement in brick floors, or tiles of an octagonal form as at Fig. 75.
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Fig. 73.
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Fig. 74.
The vanishing scale, which we shall find of infinite use in our perspective, is founded on the facts explained in Rule 10. We there find that all horizontals in the same plane, which are drawn to the same point on the horizon, are perspectively parallel to each other, so that if we measure a certain height or width on the picture plane, and then from each extremity draw lines to any convenient point on the horizon, then all the perpendiculars drawn between these lines will be perspectively equal, however much they may appear to vary in length.
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Fig. 76.
Let us suppose that in this figure (76)ABandA·B·each represent 5 feet. Then in the first case all the verticals, ase,f,g,h, drawn between AO and BO represent 5 feet, and in the second case all the horizontalse,f,g,h, drawn between A·O and B·O also represent 5 feet each. So that by the aid of this scale we can give the exact perspective height and width of any object in the picture, however far it may be from the base line, for of course we can increase or diminish our measurements atABandA·B·to whatever length we require.
As it may not be quite evident at first that the points O may be taken at random, the following figure will prove it.
FromAB(Fig. 77) drawAO,BO, thus forming the scale, raise verticalC. Now form a second scale fromABby drawingAO·BO·, and therein raise verticalDat an equal distance from the base. First, then, verticalCequalsAB, and secondly verticalDequalsAB, thereforeCequalsD, so that either of these scales will measure a given height at a given distance.
(See axioms of geometry.)
In this figure we have marked off on a level plain three or four pointsa,b,c,d, to indicate the places where we wish to stand our figures.ABrepresents their average height, so we have made our scaleAO, BO, accordingly. From each point marked we draw a line parallel to the base till it reaches the scale. From the point where it touches the lineAO, raise perpendicular asa, which gives the height required at that distance, and must be referred back to the figure itself.
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Fig. 78.
This is but a repetition of the previous figure, excepting that we have substituted these schoolgirls for the vertical lines. If we wish to make some taller than the others, and some shorter, we can easily do so, as must be evident (see Fig. 79).
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Fig. 79.Schoolgirls.
Note that in this first case the scale is below the horizon, so that we see over the heads of the figures, those nearest to us being the lowest down. That is to say, we are looking on this scene from a slightly raised platform.
To draw figures at different distances when their heads are above the horizon, or as they would appear to a person sitting on a low seat. The height of the heads varies according to the distance of the figures (Fig. 80).
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Fig. 80.Cavaliers.
How to draw figures when their heads are about the height of the horizon, or as they appear to a person standing on the same level or walking among them.
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Fig. 81.
In this case the heads or the eyes are on a level with the horizon, and we have little necessity for a scale at the side unless it is for the purpose of ascertaining or marking their distances from the base line, and their respective heights, which of course vary; so in all cases allowance must be made for some being taller and some shorter than the scale measurement.
In this example from De Hoogh the doorway to the left is higher up than the figure of the lady, and the effect seems to memore pleasing and natural for this kind of domestic subject. This delightful painter was not only a master of colour, of sunlight effect, and perfect composition, but also of perspective, and thoroughly understood the charm it gives to a picture, when cunningly introduced, for he makes the spectator feel that hecan walk along his passages and courtyards. Note that he frequently puts the point of sight quite at the side of his canvas, as at S, which gives almost the effect of angular perspective whilst it preserves the flatness and simplicity of parallel or horizontal perspective.
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Fig. 82.Courtyard by De Hoogh.
In an extended view or landscape seen from a height, we have to consider the perspective plane as in a great measure lying above it, reaching from the base of the picture to the horizon; but of course pierced here and there by trees, mountains, buildings, &c. As a rule in such cases, we copy our perspective from nature, and do not trouble ourselves much about mathematical rules. It is as well, however, to know them, so that we may feel sure we are right, as this gives certainty to our touch and enables us to work with freedom. Nor must we, when painting from nature, forget to take into account the effects of atmosphere and the various tones of the different planes of distance, for this makes much of the difference between a good picture and a bad one; being a more subtle quality, it requires a keener artistic sense to discover and depict it. (SeeFigs. 95and103.)
If the landscape painter wishes to test his knowledge of perspective, let him dissect and work out one of Turner's pictures, or better still, put his own sketch from nature to the same test.
In this figure the same principle is applied as in the previous one, but the chessmen being of different heights we have to arrange the scale accordingly. First ascertain the exact height of each piece, asQ, K, B, which represent the queen, king, bishop, &c. Refer these dimensions to the scale, as shown at QKB, which will give us the perspective measurement of each piece according to the square on which it is placed.
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Fig. 83.Chessboard and Men.
This is shown in the above drawing (Fig. 83) in the case of the white queen and the black queen, &c. The castle, the knight, and the pawn being about the same height are measured from the fourth line of the scale markedC.
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Fig. 84.
This is exemplified in the drawing of a fence (Fig. 84). Form scaleaS,bS, in accordance with the height of the fence or wall to be depicted. Letaorepresent the direction or angle at which it is placed, drawodto meet the scale atd, atdraise verticaldc, which gives the height of the fence atoo·. Draw linesbo·,eo,ao, &c., and it will be found that all these lines if produced will meet at the same point on the horizon. To divide the fence into spaces, divide base lineafas required and proceed as already shown.
It has already been shown that too near a point of distance is objectionable on account of the distortion and disproportion resulting from it. At the same time, the long distance-point must be some way out of the picture and therefore inconvenient. The object of the reduced distance is to bring that point within the picture.
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Fig. 85.
In Fig. 85 we have made the distance nearly twice the length of the base of the picture, and consequently a long way out of it. DrawSa,Sb, and fromadrawaDto point of distance, which cutsSbato, and determines the depth of the squareacob. Butwe can find that same point if we take half the base and draw a line from ½ base to ½ distance. But even this ½ distance-point does not come inside the picture, so we take a fourth of the base and a fourth of the distance and draw a line from ¼ base to ¼ distance. We shall find that it passes precisely through the same pointoas the other linesaD, &c. We are thus able to find the required pointowithout going outside the picture.
Of course we could in the same way take an 8th or even a 16th distance, but the great use of this reduced distance, in addition to the above, is that it enables us to measure any depth into the picture with the greatest ease.
It will be seen in the next figure that without having to extend the base, as is usually done, we can multiply that base to any amount by making use of these reduced distances on the horizontal line. This is quite a new method of proceeding, and it will be seen is mathematically correct.
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Fig. 86.
In Fig. 86 we have divided the base of the first square into four equal parts, which may represent so many feet, so thatA4 andBdbeing the retreating sides of the square each represents 4 feet. But we found point ¼Dby drawing 3D from ¼ base to ¼ distance, and by proceeding in the same way from each division,A, 1, 2, 3, we mark off onSBfour spaces each equal to 4 feet, in all 16 feet, so that by taking the whole base and the ¼ distance we find pointO, which is distant four times the length of the baseAB. We can multiply this distance to any amount by drawing other diagonals to 8th distance, &c. The same rule applies to this corridor (Fig. 87 and Fig. 88).
If we make our scale to vanish to the point of sight, as in Fig. 89, we can makeSB, the lower line thereof, a measuring line for distances. Let us first of all divide the baseABinto eight parts, each part representing 5 feet. From each division draw lines to 8th distance; by their intersections withSBwe obtainmeasurements of 40, 80, 120, 160, &c., feet. Now divide the side of the pictureBEin the same manner as the base, which gives us the height of 40 feet. From the sideBEdraw lines 5S, 15S, &c., to point of sight, and from each division on the base line also draw lines 5S, 10S, 15S, &c., to point of sight, and from each division onSB, such as 40, 80, &c., draw horizontals parallel to base. We thus obtain squares 40 feet wide, beginning at baseABand reaching as far as required. Note how the height of the flagstaff, which is 140 feet high and 280 feet distant, is obtained. So also any buildings or other objects can be measured, such as those shown on the left of the picture.
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Fig. 89.
A simple and very old method of drawing buildings, &c., and giving them their right width and height is by means of squares of a given size, drawn on the ground.
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Fig. 90.
In the above sketch (Fig. 90) the squares on the groundrepresent 3 feet each way, or one square yard. Taking this as our standard measure, we find the door on the left is 10 feet high, that the archway at the end is 21 feet high and 12 feet wide, and so on.
Fig. 91 is a sketch made at Sandwich, Kent, and shows a somewhat similar subject toFig. 84, but the irregularity and freedom of the perspective gives it a charm far beyond the rigid precision of the other, while it conforms to its main laws. This sketch, however, is the real artist's perspective, or what we might term natural perspective.
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Fig. 91.Natural Perspective.
In the drawing of Honfleur (Fig. 92) we divide the baseABasin the previous figure, but the spaces measure 5 feet instead of 3 feet: so that taking the 8th distance, the divisions on the vanishing lineBSmeasure 40 feet each, and at pointOwe have 400 feet of distance, but we require 800. So we again reduce the distance to a 16th. We thus multiply the base by 16. Now let us take a base of 50 feet atfand draw linefDto 16th distance; if we multiply 50 feet by 16 we obtain the 800 feet required.
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Fig. 92.Honfleur.
The height of the lighthouse is found by means of the vanishing scale, which is 15 feet below and 15 feet above the horizon, or 30 feet from the sea-level. AtLwe raise a verticalLM, which shows the position of the lighthouse. Then on that vertical measure the height required as shown in the figure.
Perspective of a lighthouse 135 feet high at 800 feet distance.
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Fig. 93.Key to Fig. 92, Honfleur.
The 800 feet could be obtained at once by drawing linefD, or 50 feet, to 16th distance. The other measurements obtained by 8th distance serve for nearer buildings.
The wonderful effect of distance in Turner's pictures is not to be achieved by mere measurement, and indeed can only be properly done by studying Nature and drawing her perspective as she presents it to us. At the same time it is useful to be able to test and to set out distances in arranging a composition. This latter, if neglected, often leads to great difficulties and sometimes to repainting.
To show the method of measuring very long distances we have to work with a very small scale to the foot, and in Fig. 94 I have divided the baseABinto eleven parts, each part representing 10 feet. First drawASandBSto point of sight.FromAdrawADto ¼ distance, and we obtain at 440 on lineBSfour times the length ofAB, or 110 feet × 4 = 440 feet. Again, taking the whole base and drawing a line from S to 8th distance we obtain eight times 110 feet or 880 feet. If now we use the 16th distance we get sixteen times 110 feet, or 1,760 feet, one-third of a mile; by repeating this process, but by using the base at 1,760, which is the same length in perspective asAB, we obtain 3,520 feet, and then again using the base at 3,520 and proceeding in the same way we obtain 5,280 feet, or one mile to the archway. The flags show their heights at their respective distances from the base. By the scale at the side of the picture,BO, we can measure any height above or any depth below the perspective plane.
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Fig. 94.larger view
Note.—This figure (here much reduced) should be drawn large by the student, so that the numbering, &c., may be made more distinct. Indeed, many of the other figures should be copied large, and worked out with care, as lessons in perspective.
An extended view is generally taken from an elevated position, so that the principal part of the landscape lies beneath the perspective plane, as already noted, and we shall presently treat of objects and figures on uneven ground. In the previous figure is shown how we can measure heights and depths to any extent. But when we turn to a drawing by Turner, such as the ‘View from Richmond Hill’, we feel that the only way to accomplish such perspective as this, is to go and draw it from nature, and even then to use our judgement, as he did, as to how much we may emphasize or even exaggerate certain features.
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Fig. 95.Turner's View from Richmond Hill.
Note in this view the foreground on which the principal figures stand is on a level with the perspective plane, while the river and surrounding park and woods are hundreds of feet below usand stretch away for miles into the distance. The contrasts obtained by this arrangement increase the illusion of space, and the figures in the foreground give as it were a standard of measurement, and by their contrast to the size of the trees show us how far away those trees are.
The three figures to the right markedf,g,b(Fig. 96) are on level ground, and we measure them by the vanishing scaleaS,bS. Those to the left, which are repetitions of them, are on an inclined plane, the vanishing point of which isS·; by the side of this plane we have placed another vanishing scalea·S·,b·S·, by which we measure the figures on that incline in the same way as on the level plane. It will be seen that if a horizontal line is drawn from the foot of one of these figures, sayG, to pointOon the edge of the incline, then dropped vertically too·, then again carried on too··where the other figuregis, we find it is the same height and also that the other vanishing scale is the same width at that distance, so that we can work from either one or the other. In the event of the rising ground being uneven we can make use of the scale on the level plane.
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Fig. 96.
LetPbe the given figure. Form scaleACS,Sbeing the point of sight andDthe distance. Draw horizontaldothroughP. FromAdraw diagonalADto distance point, cuttingdoino, throughodrawSBto base, and we now have a squareAdoBon the perspective plane; and as figurePis standing on the far side of that square it must be the distanceAB, which is one side of it, from the base line—or picture plane. For figures very far away it might be necessary to make use of half-distance.
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Fig. 97.
In previous problems we have drawn figures on level planes, which is easy enough. We have now to represent some above and some below the perspective plane.
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Fig. 98.
Form scalebS,cS; mark off distances 20 feet, 40 feet, &c. Suppose figureKto be 60 feet off. From point at his feet draw horizontal to meet verticalOn, which is 60 feet distant. At the pointmwhere this line meets the vertical, measure heightmnequal to width of scale at that distance, transfer this toK, and you have the required height of the figure in black.
For the figures under the cliff 20 feet below the perspective plane, form scaleFS,GS, making it the same width as the other, namely 5 feet, and proceed in the usual way to find the height of the figures on the sands, which are here supposed to be nearly on a level with the sea, of course making allowance for different heights and various other things.
Letabbe the height of a figure, say 6 feet. First form scaleaS,bS, the lower line of which,aS, is on a level with the base or on the perspective plane. The figure markedCis close to base, the group of three is farther off (24 feet), and 6 feet higher up, so we measure the height on the vanishing scale and also above it. The two girls carrying fish are still farther off, and about 12 feet below. To tell how far a figure is away, refer its measurements to the vanishing scale (seeFig. 96).
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Fig. 99.
In this case (Fig. 100) the same rule applies as in the previous problem, but as the road on the left is going down hill, the vanishing point of the inclined plane is below the horizon at pointS·;AS,BSis the vanishing scale on the level plane; andA·S·,B·S·, that on the incline.
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Fig. 100.
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Fig. 101.This is an outline of above figure to show the working more plainly.
Note the wall to the left markedWand the manner in which it appears to drop at certain intervals, its base corresponding with the inclined plane, but the upper lines of each division being made level are drawn to the point of sight, or to their vanishing point on the horizon; it is important to observe this, as it aids greatly in drawing a road going down hill.
In the centre of this picture (Fig. 102) we suppose the road to be descending till it reaches a tunnel which goes under a road or leads to a river (like one leading out of the Strand near Somerset House). It is drawn on the same principle as the foregoing figure. Of course to see the road the spectator must get pretty near to it, otherwise it will be out of sight. Also a level plane must be shown, as by its contrast to the other we perceive that the latter is going down hill.
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Fig. 102.
An extended view drawn from a height of about 30 feet from a road that descends about 45 feet.
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Fig. 103.Farningham.
In drawing a landscape such as Fig. 103 we have to bear in mind the height of the horizon, which being exactly opposite the eye, shows us at once which objects are below and which are above us, and to draw them accordingly, especially roofs, buildings, walls, hedges, &c.; also it is well to sketch in the different fields figures of men and cattle, as from the size of these we can judge of the rest.
LetKrepresent a frame placed vertically and at a given distance in front of us. If stood on the ground our foreground will touchthe base line of the picture, and we can fix up a standard of measurement both on the base and on the side as in this sketch, taking 6 feet as about the height of the figures.
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Fig. 104.Toledo.
If we are looking at a scene from a height, that is from a terrace, or a window, or a cliff, then the near foreground, unless it be the terrace, window-sill, &c., would not come into the picture, and we could not see the near figures atA, and the nearest to come into view would be those atB, so that a view from a window, &c., would be as it were without a foreground. Note that the figures atBwould be (according to this sketch) 30 feet from the picture plane and about 18 feet below the base line.
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Fig. 105.
Hitherto we have spoken only of parallel perspective, which is comparatively easy, and in our first figure we placed the cube with one of its sides either touching or parallel to the transparent plane. We now place it so that one angle only (ab), touches the picture.
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Fig. 106.
Its sides are no longer drawn to the point of sight as inFig. 7, nor its diagonal to the point of distance, but to some other points on the horizon, although the same rule holds good as regards their parallelism; as for instance, in the case ofbcandad, which, if produced, would meet atV, a point on the horizon called avanishing point. In this figure only one vanishing point is seen, which is to the right of the point of sightS, whilst the other is some distance to the left, and outside the picture. If the cube is correctly drawn, it will be found that the linesae,bg, &c., if produced, will meet on the horizon at this other vanishing point. This far-away vanishing point is one of the inconveniences of oblique or angular perspective, and therefore it will be a considerable gain to the draughtsman if we can dispense with it. This can be easily done, as in the above figure, and here our geometry will come to our assistance, as I shall show presently.
Let us place the given pointPon a geometrical plane, to show how far it is from the base line, and indeed in the exact position we wish it to be in the picture. The geometrical plane is supposed to face us, to hang down, as it were, from the base lineAB, like the side of a table, the top of which represents the perspective plane. It is to that perspective plane that we now have to transfer the pointP.
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Fig. 107.
FromPraise perpendicularPmtill it touches the base line atm. With centremand radiusmPdescribe arcPnso thatmnis now the same length asmP. As pointPis opposite pointm, somust it be in the perspective, therefore we draw a line at right angles to the base, that is to the point of sight, and somewhere on this line will be found the required pointP·. We now have to find how far frommmust that point be. It must be the length ofmn, which is the same asmP. We therefore fromndrawnDto the point of distance, which being at an angle of 45°, or half a right angle, makesmP·the perspective length ofmnby its intersection withmS, and thus gives us the pointP·, which is the perspective of the original point.
To do this we simply reverse the foregoing problem. Thus letPbe the given perspective point. From point of sightSdraw a line throughPtill it cutsABatm. From distanceDdraw another line throughPtill it cuts the base atn. Frommdrop perpendicular, and then with centremand radiusmndescribe arc, and where it cuts that perpendicular is the required pointP·. We often have to make use of this problem.