Chapter VI: Book IV (2)

Horizontals parallel to the base of the picture are also parallel to that base in the picture.

RULE 4

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.

RULE 5

All horizontal lines which are at right angles to the picture plane are drawn to the point of sight.

RULE 6

All horizontals which are at 45 deg to the picture plane are drawn to the point of distance.

RULE 7

All horizontals forming any other angles but the above are drawn to some other points on the horizontal line.

RULE 8

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.

RULE 9

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.

RULE 10

Horizontals in the same plane which are drawn to the same point on the horizon are perspectively parallel to each other.

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.

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.

X

THE DIAGONAL

From a given point on the base line draw a line at 45 deg, 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 deg, 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 deg to base. At point _c_, where it cuts _bS_, draw _dc_ parallel to _ab_ and _abcd_ is the square required.

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 deg 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.

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.

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.

XIII

OF CERTAIN TERMS MADE USE OF IN PERSPECTIVE

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_.

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.

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 2-1/2 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.

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

And then put it into perspective.

XVII

OF SQUARES PLACED VERTICALLY AND AT DIFFERENT HEIGHTS, OR THE CUBE IN PARALLEL PERSPECTIVE

On a given square raise a cube.

_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.

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.

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.

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.

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.

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.

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.

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 gia' 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.

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. 10-7/8 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.

'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_.]

XXI

INTERIORS

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'.

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 DEG.

When the square is at an angle of 45 deg 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.

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_.

XXIII

THE CUBE AT AN ANGLE OF 45 DEG.

Having drawn the square at an angle of 45 deg, 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.

XXIV

PAVEMENTS DRAWN BY MEANS OF SQUARES AT 45 DEG.

The square at 45 deg 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.

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.

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

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.

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).

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).

_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.

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.

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.

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_.

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.

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 1/2 base to 1/2 distance. But even this 1/2 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 1/4 base to 1/4 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

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 1/4 D by drawing 3D from 1/4 base to 1/4 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 1/4 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).

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.

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.

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.

XXXVII

APPLICATION OF THE REDUCED DISTANCE AND THE VANISHING SCALE TO DRAWING A LIGHTHOUSE, &C.

[Above illustration: Perspective of a lighthouse 135 feet high at 800 feet distance.]

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.

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.

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 1/4 distance, and we obtain at 440 on line _BS_ four times the length of _AB_, or 110 feet x 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.

_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.

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.

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.

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.

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).

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. 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.

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.

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.

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.

BOOK THIRD

XLIX

ANGULAR PERSPECTIVE

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.

Its sides are no longer drawn to the point of sight as in Fig. 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 of _bc_ and _ad_, which, if produced, would meet at _V_, a point on the horizon called a vanishing point. In this figure only one vanishing point is seen, which is to the right of the point of sight _S_, 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 lines _ae_, _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.

L

HOW TO PUT A GIVEN POINT INTO PERSPECTIVE

Let us place the given point _P_ on 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 line _AB_, 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 point _P_.

From _P_ raise perpendicular _Pm_ till it touches the base line at _m_. With centre _m_ and radius _mP_ describe arc _Pn_ so that _mn_ is now the same length as _mP_. As point _P_ is opposite point _m_, so must 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 point _P'_. We now have to find how far from _m_ must that point be. It must be the length of _mn_, which is the same as _mP_. We therefore from _n_ draw _nD_ to the point of distance, which being at an angle of 45 deg, or half a right angle, makes _mP_' the perspective length of _mn_ by its intersection with _mS_, and thus gives us the point _P'_, which is the perspective of the original point.

LI

A PERSPECTIVE POINT BEING GIVEN, FIND ITS POSITION ON THE GEOMETRICAL PLANE

To do this we simply reverse the foregoing problem. Thus let _P_ be the given perspective point. From point of sight _S_ draw a line through _P_ till it cuts _AB_ at _m_. From distance _D_ draw another line through _P_ till it cuts the base at _n_. From _m_ drop perpendicular, and then with centre _m_ and radius _mn_ describe arc, and where it cuts that perpendicular is the required point _P'_. We often have to make use of this problem.

LII

HOW TO PUT A GIVEN LINE INTO PERSPECTIVE

This is simply a question of putting two points into perspective, instead of one, or like doing the previous problem twice over, for the two points represent the two extremities of the line. Thus we have to find the perspective of _A_ and _B_, namely _a'b'_. Join those points, and we have the line required.

If one end touches the base, as at _A_ (Fig. 110), then we have but to find one point, namely _b_. We also find the perspective of the angle _mAB_, namely the shaded triangle mAb. Note also that the perspective triangle equals the geometrical triangle.

When the line required is parallel to the base line of the picture, then the perspective of it is also parallel to that base (see Rule 3).

LIII

TO FIND THE LENGTH OF A GIVEN PERSPECTIVE LINE

A perspective line _AB_ being given, find its actual length and the angle at which it is placed.

This is simply the reverse of the previous problem. Let _AB_ be the given line. From distance _D_ through _A_ draw _DC_, and from _S_, point of sight, through _A_ draw _SO_. Drop _OP_ at right angles to base, making it equal to _OC_. Join _PB_, and line _PB_ is the actual length of _AB_.

This problem is useful in finding the position of any given line or point on the perspective plane.

LIV

TO FIND THESE POINTS WHEN THE DISTANCE-POINT IS INACCESSIBLE

If the distance-point is a long way out of the picture, then the same result can be obtained by using the half distance and half base, as already shown.

From _a_, half of _mP_', draw quadrant _ab_, from _b_ (half base), draw line from _b_ to half Dist., which intersects _Sm_ at _P_, precisely the same point as would be obtained by using the whole distance.

LV

HOW TO PUT A GIVEN TRIANGLE OR OTHER RECTILINEAL FIGURE INTO PERSPECTIVE

Here we simply put three points into perspective to obtain the given triangle _A_, or five points to obtain the five-sided figure at _B_. So can we deal with any number of figures placed at any angle.

Both the above figures are placed in the same diagram, showing how any number can be drawn by means of the same point of sight and the same point of distance, which makes them belong to the same picture.