The Theory and Practice of Perspective

by G. A. Storey

BOOK SECOND

[THE PRACTICE OF PERSPECTIVE]

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

[ IX]

The Square in Parallel Perspective

From a given point on the base line of the picture draw a line at right angles to that base. Let P be the given point on the base line AB, and S the point of sight. We simply draw a line along the ground to the point of sight S, and this line will be at right angles to the base, as explained in Rule 5, and consequently angle APS will be equal to angle SPB, although it does not look so here. This is our first difficulty, but one that we shall soon get over.

Fig. 43.

In like manner we can draw any number of lines at right angles to the base, or we may suppose the point P to be placed at so many different positions, our only difficulty being to conceive these lines to be parallel to each other. See Rule 10.

Fig. 44.

[ X]

The Diagonal

Fig. 45.

From a given point on the base line draw a line at 45°, or half a right angle, to that base. Let P be the given point. Draw a line from P to the point of distance D and this line PD will be at an angle of 45°, or at the same angle as the diagonal of a square. See definitions.

[ XI]

The Square

Draw a square in parallel perspective on a given length on the base line. Let ab be the given length. From its two

extremities a and b draw aS and bS to the point of sight S. These two lines will be at right angles to the base (see [Fig. 43]). From a draw diagonal aD to point of distance D; this line will be 45° to base. At point c, where it cuts bS, draw dc parallel to ab and abcd is the square required.

Fig. 46.	Fig. 47.

We have here proceeded in much the same way as in drawing a geometrical square (Fig. 47), by drawing two lines AE and BC at right angles to a given line, AB, and from A, drawing the diagonal AC at 45° till it cuts BC at C, and then through C drawing EC parallel to AB. Let it be remarked that because the two perspective lines (Fig. 48) AS and BS are at right angles to the base, they must consequently be parallel to each other, and therefore are perspectively equidistant, so that all lines parallel to AB and lying between them, such as ad, cf, &c., must be equal.

Fig. 48.

Fig. 49.

So likewise all diagonals drawn to the point of distance, which

are contained between these parallels, such as Ad, 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 lines AS, 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 square abcd being equal, they divide each side of the larger square into three equal parts.

From this we learn how we can measure any number of given

lengths, 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 as dc, that is, parallel to the base of the picture, however remote it may be from that base.

Fig. 50.

[ XII]

Geometrical and Perspective Figures Contrasted

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.

Fig. 51 A. The geometrical view.	Fig. 51 B. The perspective view.
Fig. 51 C. A geometrical square.	Fig. 51 D. A perspective square.
Fig. 51 E. Geometrical parallels.	Fig. 51 F. Perspective parallels.
Fig. 51 G. Geometrical perpendicular.	Fig. 51 H. Perspective perpendicular.
Fig. 51 I. Geometrical equal lines.	Fig. 51 J. Perspective equal lines.
Fig. 51 K. A geometrical circle.	Fig. 51 L. A perspective circle.

[ XIII]

Of Certain Terms made use of in Perspective

Fig. 52. Horizontals.

Of course when we speak of Perpendiculars we do not mean verticals only, but straight lines at right angles to other lines in any position. Also in speaking of lines a right or straight line is to be understood; or when we speak of horizontals we 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, fuyante or lignes fuyantes, or going-away lines, is more expressive; and point 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 implies

a 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.

[ XIV]

How to Measure Vanishing or Receding Lines

Divide a vanishing line which is at right angles to the picture plane into any number of given measurements. Let SA be the given line. From A measure off on the base line the divisions required, say five of 1 foot each; from each division draw diagonals to point of distance D, and where these intersect the line AC the corresponding divisions will be found. Note that as lines AB and AC are two sides of the same square they are necessarily equal, and so also are the divisions on AC equal to those on AB.

Fig. 53.

Fig. 54.

The line AB being 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 divide cf. This geometrical figure gives, as it were, a bird's-eye view or ground-plan of the above.

[ XV]

How to Place Squares in Given Positions

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.

Fig. 55.

Let ab (Fig. 55) represent the base line of the picture divided into a certain number of feet; HD the horizon, VO the 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 from V, 2 feet to e, which gives the distance from the vertical. Second, from e measure 3 feet to b, which gives the width of the square; from e and b draw eS, bS, to point of sight. From either e or b measure 1 foot to the left, to f or f·. Draw fD to point of distance, which intersects eS at P, and gives the required distance from base. Draw Pg and B parallel to the base, and we have the required square.

Square A to 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 working

out 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.

[ XVI]

How To Draw Pavements, &c.

Divide a vanishing line into parts varying in length. Let BS· 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.

Fig. 56.

Or we can simply draw the plan of the pavement as in this figure.

Fig. 57.

And then put it into perspective.

Fig. 58.

[ XVII]

Of Squares placed Vertically and at Different Heights, or the Cube in Parallel Perspective

On a given square raise a cube.

Fig. 59.

ABCD is the given square; from A and B raise verticals AE, BF, equal to AB; join EF. Draw ES, FS, to point of sight; from C and D raise verticals CG, DH, till they meet vanishing lines ES, FS, in G and H, and the cube is complete.

[ XVIII]

The Transposed Distance

The transposed distance is a point D· on the vertical VD·, 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 at CB, which in the case of distant objects, such as the farthest window, would be very inconvenient. Note that the windows at K are twice as high as they are wide.

Of course these or any other objects could be made of any proportion.

Fig. 60.

[ XIX]

The Front View of the Square and of the Proportions of Figures at Different Heights

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 squares ABCD, abcd (Fig. 61), one inside the other; although moved back from square EFGH they retain the same form. So

in dealing with figures of different heights, such as statuary or ornament in a building, if actually equal in size, so must we represent them.

Fig. 61.	Fig. 62.

In this square K, 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.

Fig. 63.

In Fig. 63 the two statues A and B are the same size. So if traced through a vertical sheet of glass, K, as at c and d, they would also be equal; but as the angle b at which the upper one is seen is smaller than angle a, at which the lower figure or statue is seen, it will appear smaller to the spectator (S) both in reality and in the picture.

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 angles a and b (Fig. 64), at which they are viewed, both equal. Then draw lines through equal arcs, as at c and d, till they cut the vertical NO (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 picture K, 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. The

figures 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.

Fig. 65.

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.

[ XX]

Of Pictures that are Painted according to the Position they are to Occupy

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.

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 in

the 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:—

“I’ho già fatto un gozzo in questo stento.”

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.

Fig. 67.

‘Draw upon part of wall MN half the figure you mean to represent, and the other half upon the cove above (MR).’ Leonardo da Vinci’s Treatise on Painting.

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 point F on 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.

[ XXI]

Interiors

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’.

Fig. 69.

When the cube is placed below the horizon (as in [Fig. 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.

[ XXII]

The Square at an Angle of 45°

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 sight S, and the other ab, is parallel to that base or ground line.

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 distance DD´. 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 sight S, and also horizontals parallel to the base, as ab.

Fig. 71.

[ XXIII]

The Cube at an Angle of 45°

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, through h, one of its extremities, till it cuts the base line at K. Then, with the other extremity d for centre and dK for radius, describe a quarter of a circle Km; the chord thereof mK will be the geometrical length of dh. At d raise vertical dC equal to mK, which gives us the height of the cube, then raise verticals at a, h, &c., their height being found by drawing CD and CD´ to the two points of distance, and so completing the figure.

Fig. 72.

[ XXIV]

Pavements Drawn by Means of Squares at 45°

Fig. 75.

The square at 45° will be found of great use in drawing pavements, roofs, ceilings, &c. In Figs. 73, 74 it is shown how

having 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.

Fig. 73.

Fig. 74.

[ XXV]

The Perspective Vanishing Scale

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.

Fig. 76.

Let us suppose that in this figure (76) AB and A·B· each represent 5 feet. Then in the first case all the verticals, as e, f, g, h, drawn between AO and BO represent 5 feet, and in the second case all the horizontals e, 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 at AB and A·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.

[ XXVI]

The Vanishing Scale can be Drawn to any Point on the Horizon

Fig. 77.

From AB (Fig. 77) draw AO, BO, thus forming the scale, raise vertical C. Now form a second scale from AB by drawing AO· BO·, and therein raise vertical D at an equal distance from the base. First, then, vertical C equals AB, and secondly vertical D equals AB, therefore C equals D, so that either of these scales will measure a given height at a given distance.

(See axioms of geometry.)

[ XXVII]

Application of Vanishing Scales to Drawing Figures

In this figure we have marked off on a level plain three or four points a, b, c, d, to indicate the places where we wish to stand our figures. AB represents their average height, so we have made our scale AO, 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 line AO, raise perpendicular as a, which gives the height required at that distance, and must be referred back to the figure itself.

Fig. 78.

[ XXVIII]

How to Determine the Heights of Figures on a Level Plane

First Case.

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).

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.

Second Case.

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).

Fig. 80. Cavaliers.

Third Case.

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.

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.

[ XXIX]

The Horizon above the Figures

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 me

more 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 he

can 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.

Fig. 82. Courtyard by De Hoogh.

[ XXX]

Landscape Perspective

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. (See [Figs. 95] and [103].)

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.

[ XXXI]

Figures of Different Heights

The Chessboard

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, as Q, 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.

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 marked C.

Fig. 84.

[ XXXII]

Application of the Vanishing Scale to Drawing Figures at an Angle when their Vanishing Points are Inaccessible or Outside the Picture

This is exemplified in the drawing of a fence (Fig. 84). Form scale aS, bS, in accordance with the height of the fence or wall to be depicted. Let ao represent the direction or angle at which it is placed, draw od to meet the scale at d, at d raise vertical dc, which gives the height of the fence at oo·. Draw lines bo·, 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 line af as required and proceed as already shown.

[ XXXIII]

The Reduced Distance. How to Proceed when the Point of Distance is Inaccessible

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.

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. Draw Sa, Sb, and from a draw aD to point of distance, which cuts Sb at o, and determines the depth of the square acob. But

we 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 point o as the other lines aD, &c. We are thus able to find the required point o without 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.

[ XXXIV]

How to Draw a Long Passage or Cloister by means of the Reduced Distance

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 that A4 and Bd being the retreating sides of the square each represents 4 feet. But we found point ¼ D by 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 on SB four spaces each equal to 4 feet, in all 16 feet, so that by taking the whole base and the ¼ distance we find point O, which is distant four times the length of the base AB. 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).

Fig. 87.	Fig. 88.

[ XXXV]

How to Form a Vanishing Scale that shall give the Height, Depth, and Distance of any Object in the Picture

If we make our scale to vanish to the point of sight, as in Fig. 89, we can make SB, the lower line thereof, a measuring line for distances. Let us first of all divide the base AB into eight parts, each part representing 5 feet. From each division draw lines to 8th distance; by their intersections with SB we obtain

measurements of 40, 80, 120, 160, &c., feet. Now divide the side of the picture BE in the same manner as the base, which gives us the height of 40 feet. From the side BE draw 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 on SB, such as 40, 80, &c., draw horizontals parallel to base. We thus obtain squares 40 feet wide, beginning at base AB and 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.

Fig. 89.

[ XXXVI]

Measuring Scale on Ground

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.

Fig. 90.

In the above sketch (Fig. 90) the squares on the ground

represent 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 to [Fig. 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.

Fig. 91. Natural Perspective.

[ XXXVII]

Application of the Reduced Distance and the Vanishing Scale to Drawing a Lighthouse, &c.

In the drawing of Honfleur (Fig. 92) we divide the base AB as

in the previous figure, but the spaces measure 5 feet instead of 3 feet: so that taking the 8th distance, the divisions on the vanishing line BS measure 40 feet each, and at point O we 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 at f and draw line fD to 16th distance; if we multiply 50 feet by 16 we obtain the 800 feet required.

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. At L we raise a vertical LM, 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.

Fig. 93. Key to Fig. 92, Honfleur.

The 800 feet could be obtained at once by drawing line fD, or 50 feet, to 16th distance. The other measurements obtained by 8th distance serve for nearer buildings.

[ XXXVIII]

How to Measure Long Distances such as a Mile or Upwards

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 base AB into eleven parts, each part representing 10 feet. First draw AS and BS to point of sight.

From A draw AD to ¼ distance, and we obtain at 440 on line BS four times the length of AB, 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 as AB, 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.

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.

[ XXXIX]

Further Illustration of Long Distances and Extended Views

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.

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 us

and 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.

[ XL]

How to Ascertain the Relative Heights of Figures on an Inclined Plane

The three figures to the right marked f, g, b (Fig. 96) are on level ground, and we measure them by the vanishing scale aS, bS. Those to the left, which are repetitions of them, are on an inclined plane, the vanishing point of which is S·; by the side of this plane we have placed another vanishing scale a·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, say G, to point O on the edge of the incline, then dropped vertically to o·, then again carried on to o·· where the other figure g is, 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.

Fig. 96.

[ XLI]

How to Find the Distance of a Given Figure or Point from the Base Line

Let P be the given figure. Form scale ACS, S being the point of sight and D the distance. Draw horizontal do through P. From A draw diagonal AD to distance point, cutting do in o, through o draw SB to base, and we now have a square AdoB on the perspective plane; and as figure P is standing on the far side of that square it must be the distance AB, 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.

Fig. 97.

[ XLII]

How to Measure the Height of Figures on Uneven Ground

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.

Fig. 98.

Form scale bS, cS; mark off distances 20 feet, 40 feet, &c. Suppose figure K to be 60 feet off. From point at his feet draw horizontal to meet vertical On, which is 60 feet distant. At the point m where this line meets the vertical, measure height mn equal to width of scale at that distance, transfer this to K, and you have the required height of the figure in black.

For the figures under the cliff 20 feet below the perspective plane, form scale FS, 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.

[ XLIII]

Further Illustration of the Size of Figures at Different Distances and on Uneven Ground

Let ab be the height of a figure, say 6 feet. First form scale aS, bS, the lower line of which, aS, is on a level with the base or on the perspective plane. The figure marked C is 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 (see [Fig. 96]).

Fig. 99.

[ XLIV]

Figures on a Descending Plane

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 point S·; AS, BS is the vanishing scale on the level plane; and A·S·, B·S·, that on the incline.

Fig. 100.

Fig. 101. This is an outline of above figure to show the working more plainly.

Note the wall to the left marked W and 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.

[ XLV]

Further Illustration of the Descending Plane

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.

Fig. 102.

[ XLVI]

Further Illustration of Uneven Ground

An extended view drawn from a height of about 30 feet from a road that descends about 45 feet.

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.

[ XLVII]

The Picture Standing on the Ground

Let K represent a frame placed vertically and at a given distance in front of us. If stood on the ground our foreground will touch

the 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.

Fig. 104. Toledo.

[ XLVIII]

The Picture on a Height

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 at A, and the nearest to come into view would be those at B, so that a view from a window, &c., would be as it were without a foreground. Note that the figures at B would be (according to this sketch) 30 feet from the picture plane and about 18 feet below the base line.

Fig. 105.