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Fig. 33.
This rule is useful in drawing steps, or roads going uphill and downhill.
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Fig. 34.
The farther a point is removed from the picture plane the nearer does its perspective appearance approach the horizontal line so long as it is viewed from the same position. On the contrary, if the spectator retreats from the picture planeK(which we suppose to be transparent), the point remaining at the same place, the perspective appearance of this point will approach the ground-line in proportion to the distance of the spectator.
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Fig. 35.
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Fig. 36.
The spectator at two different distances from the picture.
Therefore the position of a given point in perspective above the ground-line or below the horizon is in proportion to the distance of the spectator from the picture, or the picture from the point.
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Fig. 37.
Figures 38 and 39 are two views of the same gallery from different distances. In Fig. 38, where the distance is too short, there is a want of proportion between the near and far objects, which is corrected in Fig. 39 by taking a much longer distance.
Horizontals in the same plane which are drawn to the same point on the horizon are parallel to each other.
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Fig. 40.
This is a very important rule, for all our perspective drawing depends upon it. When we say that parallels are drawn to the same point on the horizon it does not imply that they meet at that point, which would be a contradiction; perspective parallels never reach that point, although they appear to do so. Fig. 40 will explain this.
SupposeSto be the spectator,ABa transparent vertical plane which represents the picture seen edgeways, andHSandDCtwo parallel lines, mark off spaces between these parallels equal toSC, the height of the eye of the spectator, and raise verticals 2, 3, 4, 5, &c., forming so many squares. Vertical line 2 viewed fromSwill appear onABbut half its length, vertical 3 will be only a third, vertical 4 a fourth, and so on, and if we multiplied these spacesad infinitumwe must keep on dividing the lineABby the same number. So if we supposeABto be a yard high and the distance from one vertical to another to be also a yard, then if one of these were a thousand yards away its representation atABwould be the thousandth part of a yard, or ten thousand yards away, its representation atABwould be the ten-thousandth part, and whatever the distance it must always be something; and thereforeHSandDC, however far they may be producedand however close they may appear to get, can never meet.
figure
Fig. 41.
Fig. 41 is a perspective view of the same figure—but more extended. It will be seen that a line drawn from the tenth uprightKtoScuts off a tenth ofAB. We look then upon these two linesSP, OP, as the sides of a long parallelogram of whichSKis the diagonal, ascefd, the figure on the ground, is also a parallelogram.
The student can obtain for himself a further illustration of this rule by placing a looking-glass on one of the walls of his studio and then sketching himself and his surroundings as seen therein.He will find that all the horizontals at right angles to the glass will converge to his own eye. This rule applies equally to lines which are at an angle to the picture plane as to those that are at right angles or perpendicular to it, as in Rule 7. It also applies to those on an inclined plane, as in Rule 8.
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Fig. 42.Sketch of artist in studio.
With the above rules and a clear notion of the definitions and conditions of perspective, we should be able to work out any proposition or any new figure that may present itself. At any rate, a thorough understanding of these few pages will make the labour now before us simple and easy. I hope, too, it may be found interesting. There is always a certain pleasure in deceiving and being deceived by the senses, and in optical and other illusions, such as making things appear far off that are quite near, in making a picture of an object on a flat surface to look as if it stood out and in relief by a kind of magic. But there is, I think, a still greater pleasure than this, namely, in invention and in overcoming difficulties—in finding out how to do things for ourselves by our reasoning faculties, in originating or being original, as it were. Let us now see how far we can go in this respect.
The rules here set down have been fully explained in the previous pages, and this table is simply for the student's ready reference.
All straight lines remain straight in their perspective appearance.
Vertical lines remain vertical in perspective.
Horizontals parallel to the base of the picture are also parallel to that base in the picture.
All lines situated in a plane that is parallel to the picture plane diminish in proportion as they become more distant, but do not undergo any perspective deformation. This is called the front view.
All horizontal lines which are at right angles to the picture plane are drawn to the point of sight.
All horizontals which are at 45° to the picture plane are drawn to the point of distance.
All horizontals forming any other angles but the above are drawn to some other points on the horizontal line.
Lines which incline upwards have their vanishing points above the horizon, and those which incline downwards, below it. In both cases they are on the vertical which passes through the vanishing point of their ground-plan or horizontal projections.
The farther a point is removed from the picture plane the nearer does it appear to approach the horizon, so long as it is viewed from the same position.
Horizontals in the same plane which are drawn to the same point on the horizon are perspectively parallel to each other.
In the foregoing book we have explained the theory or science of perspective; we now have to make use of our knowledge and to apply it to the drawing of figures and the various objects that we wish to depict.
The first of these will be a square with two of its sides parallel to the picture plane and the other two at right angles to it, and which we call
From a given point on the base line of the picture draw a line at right angles to that base. LetPbe the given point on the base lineAB, andSthe point of sight. We simply draw a line along the ground to the point of sightS, and this line will be at right angles to the base, as explained in Rule 5, and consequently angleAPSwill be equal to angleSPB, although it does not look so here. This is our first difficulty, but one that we shall soon get over.
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Fig. 43.
In like manner we can draw any number of lines at right angles to the base, or we may suppose the pointPto be placed at so many different positions, our only difficulty being to conceive these lines to be parallel to each other. See Rule 10.
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Fig. 44.
From a given point on the base line draw a line at 45°, or half a right angle, to that base. LetPbe the given point. Draw a line fromPto the point of distanceDand this linePDwill be at an angle of 45°, or at the same angle as the diagonal of a square. See definitions.
Draw a square in parallel perspective on a given length on the base line. Letabbe the given length. From its twoextremitiesaandbdrawaSandbSto the point of sightS. These two lines will be at right angles to the base (seeFig. 43). Fromadraw diagonalaDto point of distanceD; this line will be 45° to base. At pointc, where it cutsbS, drawdcparallel toabandabcdis the square required.
We have here proceeded in much the same way as in drawing a geometrical square (Fig. 47), by drawing two linesAEandBCat right angles to a given line,AB, and fromA, drawing the diagonalACat 45° till it cutsBCatC, and then throughCdrawingECparallel toAB. Let it be remarked that because the two perspective lines (Fig. 48)ASandBSare at right angles to the base, they must consequently be parallel to each other, and therefore are perspectively equidistant, so that all lines parallel toABand lying between them, such asad,cf, &c., must be equal.
figure
Fig. 48.
So likewise all diagonals drawn to the point of distance, whichare contained between these parallels, such asAd,af, &c., must be equal. For all straight lines which meet at any point on the horizon are perspectively parallel to each other, just as two geometrical parallels crossing two others at any angle, as at Fig. 49. Note also (Fig. 48) that all squares formed between the two vanishing linesAS,BS, and by the aid of these diagonals, are also equal, and further, that any number of squares such as are shown in this figure (Fig. 50), formed in the same way and having equal bases, are also equal; and the nine squares contained in the squareabcdbeing equal, they divide each side of the larger square into three equal parts.
From this we learn how we can measure any number of givenlengths, either equal or unequal, on a vanishing or retreating line which is at right angles to the base; and also how we can measure any width or number of widths on a line such asdc, that is, parallel to the base of the picture, however remote it may be from that base.
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Fig. 50.
As at first there may be a little difficulty in realizing the resemblance between geometrical and perspective figures, and also about certain expressions we make use of, such as horizontals, perpendiculars, parallels, &c., which look quite different in perspective, I will here make a note of them and also place side by side the two views of the same figures.
Of course when we speak ofPerpendicularswe do not mean verticals only, but straight lines at right angles to other lines in any position. Also in speaking oflinesa right orstraight lineis to be understood; or when we speak ofhorizontalswe mean all straight lines that are parallel to the perspective plane, such as those on Fig. 52, no matter what direction they take so long as they are level. They are not to be confused with the horizon or horizontal-line.
There are one or two other terms used in perspective which are not satisfactory because they are confusing, such as vanishing lines and vanishing points. The French term,fuyanteorlignes fuyantes, or going-away lines, is more expressive; andpoint de fuite, instead of vanishing point, is much better. I have occasionally called the former retreating lines, but the simple meaning is, lines that are not parallel to the picture plane; but a vanishing line implies a line that disappears, and a vanishing point impliesa point that gradually goes out of sight. Still, it is difficult to alter terms that custom has endorsed. All we can do is to use as few of them as possible.
Divide a vanishing line which is at right angles to the picture plane into any number of given measurements. LetSAbe the given line. FromAmeasure off on the base line the divisions required, say five of 1 foot each; from each division draw diagonals to point of distanceD, and where these intersect the lineACthe corresponding divisions will be found. Note that as linesABandACare two sides of the same square they are necessarily equal, and so also are the divisions onACequal to those onAB.
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Fig. 53.
The lineABbeing the base of the picture, it is at the same time a perspective line and a geometrical one, so that we can use it as a scale for measuring given lengths thereon, but should there not be enough room on it to measure the required number we draw a second line,DC, which we divide in the same proportion and proceed to dividecf. This geometrical figure gives, as it were, a bird's-eye view or ground-plan of the above.
Draw squares of given dimensions at given distances from the base line to the right or left of the vertical line, which passes through the point of sight.
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Fig. 55.
Letab(Fig. 55) represent the base line of the picture divided into a certain number of feet;HDthe horizon,VOthe vertical. It is required to draw a square 3 feet wide, 2 feet to the right of the vertical, and 1 foot from the base.
First measure fromV, 2 feet toe, which gives the distance from the vertical. Second, fromemeasure 3 feet tob, which gives the width of the square; fromeandbdraweS,bS, to point of sight. From eithereorbmeasure 1 foot to the left, toforf·. DrawfDto point of distance, which intersectseSatP, and gives the required distance from base. DrawPgandBparallel to the base, and we have the required square.
SquareAto the left of the vertical is 2½ feet wide, 1 foot from the vertical and 2 feet from the base, and is worked out in the same way.
Note.—It is necessary to know how to work to scale, especially in architectural drawing, where it is indispensable, but in workingout our propositions and figures it is not always desirable. A given length indicated by a line is generally sufficient for our requirements. To work out every problem to scale is not only tedious and mechanical, but wastes time, and also takes the mind of the student away from the reasoning out of the subject.
Divide a vanishing line into parts varying in length. LetBS·be the vanishing line: divide it into 4 long and 3 short spaces; then proceed as in the previous figure. If we draw horizontals through the points thus obtained and from these raise verticals, we form, as it were, the interior of a building in which we can place pillars and other objects.
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Fig. 56.
Or we can simply draw the plan of the pavement as in this figure.
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Fig. 57.
And then put it into perspective.
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Fig. 58.
On a given square raise a cube.
figure
Fig. 59.
ABCDis the given square; fromAandBraise verticalsAE,BF, equal toAB; joinEF. DrawES,FS, to point of sight; fromCandDraise verticalsCG,DH, till they meet vanishing linesES,FS, inGandH, and the cube is complete.
The transposed distance is a pointD·on the verticalVD·, at exactly the same distance from the point of sight as is the point of distance on the horizontal line.
It will be seen by examining this figure that the diagonals of the squares in a vertical position are drawn to this vertical distance-point, thus saving the necessity of taking the measurements first on the base line, as atCB, which in the case of distant objects, such as the farthest window, would be very inconvenient. Note that the windows atKare twice as high as they are wide.Of course these or any other objects could be made of any proportion.
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Fig. 60.
According to Rule 4, all lines situated in a plane parallel to the picture plane diminish in length as they become more distant, but remain in the same proportions each to each as the original lines; as squares or any other figures retain the same form. Take the two squaresABCD,abcd(Fig. 61), one inside the other; although moved back from squareEFGHthey retain the same form. Soin dealing with figures of different heights, such as statuary or ornament in a building, if actually equal in size, so must we represent them.
In this squareK, with the checker pattern, we should not think of making the top squares smaller than the bottom ones; so it is with figures.
This subject requires careful study, for, as pointed out in our opening chapter, there are certain conditions under which we have to modify and greatly alter this rule in large decorative work.
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Fig. 63.
In Fig. 63 the two statuesAandBare the same size. So if traced through a vertical sheet of glass,K, as atcandd, they would also be equal; but as the anglebat which the upper one is seen is smaller than anglea, at which the lower figure or statue is seen, it will appear smaller to the spectator (S) both in reality and in the picture.
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Fig. 64.
But if we wish them to appear the same size to the spectator who is viewing them from below, we must make the anglesaandb(Fig. 64), at which they are viewed, both equal. Then draw lines through equal arcs, as atcandd, till they cut the verticalNO(representing the side of the building where the figures are to be placed). We shall then obtain the exact size of the figure at that height, which will make it look the same size as the lower one,N. The same rule applies to the pictureK, when it is of large proportions. As an example in painting, take Michelangelo’s large altar-piece in the Sistine Chapel, ‘The Last Judgement’; here the figures forming the upper group, with our Lord in judgement surrounded by saints, are about four times the size, that is, about twice the height, of those at the lower part of the fresco. Thefigures on the ceiling of the same chapel are studied not only according to their height from the pavement, which is 60 ft., but to suit the arched form of it. For instance, the head of the figure of Jonah at the end over the altar is thrown back in the design, but owing to the curvature in the architecture is actually more forward than the feet. Then again, the prophets and sybils seated round the ceiling, which are perhaps the grandest figures in the whole range of art, would be 18 ft. high if they stood up; these, too, are not on a flat surface, so that it required great knowledge to give them their right effect.
Of course, much depends upon the distance we view these statues or paintings from. In interiors, such as churches, halls, galleries, &c., we can make a fair calculation, such as the length of the nave, if the picture is an altar-piece—or say, half the length; so also with statuary in niches, friezes, and other architectural ornaments. The nearer we are to them, and the more we have to look up, the larger will the upper figures have to be; but if these are on the outside of a building that can be looked at from a long distance, then it is better not to have too great a difference.
For the farther we recede the more equal are the angles at which we view the objects at their different stages, so that in each case we may have to deal with, we must consider the conditions attending it.
These remarks apply also to architecture in a great measure. Buildings that can only be seen from the street below, as pictures in a narrow gallery, require a different treatment from those out in the open, that are to be looked at from a distance. In the former case the same treatment as the Campanile at Florence is in some cases desirable, but all must depend upon the taste and judgement of the architect in such matters. All I venture to do here is to call attention to the subject, which seems as a rule to be ignored, or not to be considered of importance. Hence the many mistakes in our buildings, and the unsatisfactory and mean look of some of our public monuments.
In this double-page illustration of the wall of a picture-gallery, I have, as it were, hung the pictures in accordance with the style in which they are painted and the perspective adopted by their painters. It will be seen that those placed on the line level with the eye have their horizon lines fairly high up, and are not suited to be placed any higher. The Giorgione in the centre, the Monna Lisa to the right, and the Velasquez and Watteau to the left, are all pictures that fit that position; whereas the grander compositions above them are so designed, and are so large in conception, that we gain in looking up to them.
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Fig. 66.Larger View
Note how grandly the young prince on his pony, by Velasquez, tells out against the sky, with its low horizon and strong contrast of light and dark; nor does it lose a bit by being placed where it is, over the smaller pictures.
The Rembrandt, on the opposite side, with its burgomasters in black hats and coats and white collars, is evidently intended and painted for a raised position, and to be looked up to, which is evident from the perspective of the table. The grand Titian inthe centre, an altar-piece in one of the churches in Venice (here reversed), is also painted to suit its elevated position, with low horizon and figures telling boldly against the sky. Those placed low down are modern French pictures, with the horizon high up and almost above their frames, but placed on the ground they fit into the general harmony of the arrangement.
It seems to me it is well, both for those who paint and for those who hang pictures, that this subject should be taken into consideration. For it must be seen by this illustration that a bigger style is adopted by the artists who paint for high places in palaces or churches than by those who produce smaller easel-pictures intended to be seen close. Unfortunately, at our picture exhibitions, we see too often that nearly all the works, whether on large or small canvases, are painted for the line, and that those which happen to get high up look as if they were toppling over, because they have such a high horizontal line; and instead of the figures telling against the sky, as in this picture of the ‘Infant’ by Velasquez, the Reynolds, and the fat man treading on a flag, we have fields or sea or distant landscape almost to the top of the frame, and all, so methinks, because the perspective is not sufficiently considered.
Note.—Whilst on this subject, I may note that the painter in his large decorative work often had difficulties to contend with, which arose from the form of the building or the shape of the wall on which he had to place his frescoes. Painting on the ceiling was no easy task, and Michelangelo, in a humorous sonnet addressed to Giovanni da Pistoya, gives a burlesque portrait of himself while he was painting the Sistine Chapel:—
Now have I such a goitre ’neath my chin
That I am like to some Lombardic cat,
My beard is in the air, my head i’ my back,
My chest like any harpy’s, and my face
Patched like a carpet by my dripping brush.
Nor can I see, nor can I budge a step;
My skin though loose in front is tight behind,
And I am even as a Syrian bow.
Alas! methinks a bent tube shoots not well;
So give me now thine aid, my Giovanni.
At present that difficulty is got over by using large strong canvas, on which the picture can be painted in the studio and afterwards placed on the wall.
However, the other difficulty of form has to be got over also. A great portion of the ceiling of the Sistine Chapel, and notably the prophets and sibyls, are painted on a curved surface, in which case a similar method to that explained by Leonardo da Vinci has to be adopted.
In Chapter CCCI he shows us how to draw a figure twenty-four braccia high upon a wall twelve braccia high. (The braccia is 1 ft. 10⅞ in.). He first draws the figure upright, then from the various points draws lines to a pointFon the floor of the building, marking their intersections on the profile of the wall somewhat in the manner we have indicated, which serve as guides in making the outline to be traced.
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Fig. 68.Interior by de Hoogh.
To draw the interior of a cube we must suppose the side facing us to be removed or transparent. Indeed, in all our figures which represent solids we suppose that we can see through them,and in most cases we mark the hidden portions with dotted lines. So also with all those imaginary lines which conduct the eye to the various vanishing points, and which the old writers called ‘occult’.
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Fig. 69.
When the cube is placed below the horizon (as inFig. 59), we see the top of it; when on the horizon, as in the above (Fig. 69), if the side facing us is removed we see both top and bottom of it, or if a room, we see floor and ceiling, but otherwise we should see but one side (that facing us), or at most two sides. When the cube is above the horizon we see underneath it.
We shall find this simple cube of great use to us in architectural subjects, such as towers, houses, roofs, interiors of rooms, &c.
In this little picture by de Hoogh we have the application of the perspective of the cube and other foregoing problems.