Note the way in which we find the measurements on base of pyramid and on line MN. By drawing AS and BS to point of sight we find Te, which measures 100 feet at a distance of 1,600 feet. We mark off seven of these lengths, and an additional 64 feet by the scale, and so obtain the required length. The position of the third corner of the base is found by dropping a perpendicular from K, till it meets the line eS.

Another thing to note is that the side of the pyramid that faces us, although an equilateral triangle, does not appear so, as its top angle is 382 feet farther off than its base owing to its leaning position.



CXXIII

THE PYRAMID IN ANGULAR PERSPECTIVE

In order to show the working of this proposition I have taken a much higher horizon, which immediately detracts from the impression of the bigness of the pyramid.



We proceed to make our ground-plan abcd high above the horizon instead of below it, drawing first the parallel square and then the oblique one. From all the principal points drop perpendiculars to the ground and thus find the points through which to draw the base of the pyramid. Find centres OO' and decide upon the height OP. Draw the sloping lines from P to the corners of the base, and the figure is complete.



CXXIV

TO DIVIDE THE SIDES OF THE PYRAMID HORIZONTALLY

Having raised the pyramid on a given oblique square, divide the vertical line OP into the required number of parts. From A through C draw AG to horizon, which gives us G, the vanishing point of all the diagonals of squares parallel to and at the same angle as ABCD. From G draw lines through the divisions 2, 3, &c., on OP cutting the lines PA and PC, thus dividing them into the required parts. Through the points thus found draw from V all those sides of the squares that have V for their vanishing point, as ab, cd, &c. Then join bd, ac, and the rest, and thus make the horizontal divisions required.





The same method will apply to drawing steps, square blocks, &c., as shown in Fig. 227, which is at the same angle as the above.



CXXV

OF ROOFS

The pyramidal roof (Fig. 228) is so simple that it explains itself. The chief thing to be noted is the way in which the diagonals are produced beyond the square of the walls, to give the width of the eaves, according to their position.



Another form of the pyramidal roof is here given (Fig. 229). First draw the cube edcba at the required height, and on the side facing us, adcb, draw triangle K, which represents the end of a gable roof. Then draw similar triangles on the other sides of the cube (see Fig. 159, LXXXIV). Join the opposite triangles at the apex, and thus form two gable roofs crossing each other at right angles. From o, centre of base of cube, raise vertical OP, and then from P draw sloping lines to each corner of base a, b, &c., and by means of central lines drawn from P to half base, find the points where the gable roofs intersect the central spire or pyramid. Any other proportions can be obtained by adding to or altering the cube.



To draw a sloping or hip-roof which falls back at each end we must first draw its base, CBDA (Fig. 230). Having found the centre O and central line SP, and how far the roof is to fall back at each end, namely the distance Pm, draw horizontal line RB through m. Then from B through O draw diagonal BA, and from A draw horizontal AD, which gives us point n. From these two points m and n raise perpendiculars the height required for the roof, and from these draw sloping lines to the corners of the base. Join ef, that is, draw the top line of the roof, which completes it. Fig. 231 shows a plan or bird's-eye view of the roof and the diagonal AB passing through centre O. But there are so many varieties of roofs they would take almost a book to themselves to illustrate them, especially the cottages and farm-buildings, barns, &c., besides churches, old mansions, and others. There is also such irregularity about some of them that perspective rules, beyond those few here given, are of very little use. So that the best thing for an artist to do is to sketch them from the real whenever he has an opportunity.







CXXVI

OF ARCHES, ARCADES, BRIDGES, &C.



For an arcade or cloister (Fig. 232) first set up the outer frame ABCD according to the proportions required. For round arches the height may be twice that of the base, varying to one and a half. In Gothic arches the height may be about three times the width, all of which proportions are chosen to suit the different purposes and effects required. Divide the base AB into the desired number of parts, 8, 10, 12, &c., each part representing 1 foot. (In this case the base is 10 feet and the horizon 5 feet.) Set out floor by means of 1/4 distance. Divide it into squares of 1 foot, so that there will be 8 feet between each column or pilaster, supposing we make them to stand on a square foot. Draw the first archway EKF facing us, and its inner semicircle gh, with also its thickness or depth of 1 foot. Draw the span of the archway EF, then central line PO to point of sight. Proceed to raise as many other arches as required at the given distances. The intersections of the central line with the chords mn, &c., will give the centres from which to describe the semicircles.



CXXVII

OUTLINE OF AN ARCADE WITH SEMICIRCULAR ARCHES

This is to show the method of drawing a long passage, corridor, or cloister with arches and columns at equal distances, and is worked in the same way as the previous figure, using 1/4 distance and 1/4 base. The floor consists of five squares; the semicircles of the arches are described from the numbered points on the central line OS, where it intersects the chords of the arches.





CXXVIII

SEMICIRCULAR ARCHES ON A RETREATING PLANE

First draw perspective square abcd. Let ae' be the height of the figure. Draw ae'f'b and proceed with the rest of the outline. To draw the arches begin with the one facing us, Eo'F enclosed in the quadrangle Ee'f'F. With centre O describe the semicircle and across it draw the diagonals e'F, Ef', and through nn, where these lines intersect the semicircle, draw horizontal KK and also KS to point of sight. It will be seen that the half-squares at the side are the same size in perspective as the one facing us, and we carry out in them much the same operation; that is, we draw the diagonals, find the point O, and the points nn, &c., through which to draw our arches. See perspective of the circle (Fig. 165).



If more points are required an additional diagonal from O to K may be used, as shown in the figure, which perhaps explains itself. The method is very old and very simple, and of course can be applied to any kind of arch, pointed or stunted, as in this drawing of a pointed arch (Fig. 235).





CXXIX

AN ARCADE IN ANGULAR PERSPECTIVE

First draw the perspective square ABCD at the angle required, by new method. Produce sides AD and BC to V. Draw diagonal BD and produce to point G, from whence we draw the other diagonals to cfh. Make spaces 1, 2, 3, &c., on base line equal to B 1 to obtain sides of squares. Raise vertical BM the height required. Produce DA to O on base line, and from O raise vertical OP equal to BM. This line enables us to dispense with the long vanishing point to the left; its working has been explained at Fig. 131. From P draw PRV to vanishing point V, which will intersect vertical AR at R. Join MR, and this line, if produced, would meet the horizon at the other vanishing point. In like manner make O2 equal to B2'. From 2 draw line to V, and at 2, its intersection with AR, draw line 2 2, which will also meet the horizon at the other vanishing point. By means of the quarter-circle A we can obtain the points through which to draw the semicircular arches in the same way as in the previous figure.





CXXX

A VAULTED CEILING

From the square ceiling ABCD we have, as it were, suspended two arches from the two diagonals DB, AC, which spring from the four corners of the square EFGH, just underneath it. The curves of these arches, which are not semicircular but elongated, are obtained by means of the vanishing scales mS, nS. Take any two convenient points P, R, on each side of the semicircle, and raise verticals Pm, Rn to AB, and on these verticals form the scales. Where mS and nS cut the diagonal AC drop perpendiculars to meet the lower line of the scale at points 1, 2. On the other side, using the other scales, we have dropped perpendiculars in the same way from the diagonal to 3, 4. These points, together with EOG, enable us to trace the curve E 1 2 O 3 4 G. We draw the arch under the other diagonal in precisely the same way.



The reason for thus proceeding is that the cross arches, although elongated, hang from their diagonals just as the semicircular arch EKF hangs from AB, and the lines mn, touching the circle at PR, are represented by 1, 2, hanging from the diagonal AC.



Figure 238, which is practically the same as the preceding only differently shaded, is drawn in the following manner. Draw arch EGF facing us, and proceed with the rest of the corridor, but first finding the flat ceiling above the square on the ground ABcd. Draw diagonals ac, bd, and the curves pending from them. But we no longer see the clear arch as in the other drawing, for the spaces between the curves are filled in and arched across.



CXXXI

A CLOISTER, FROM A PHOTOGRAPH

This drawing of a cloister from a photograph shows the correctness of our perspective, and the manner of applying it to practical work.





CXXXII

THE LOW OR ELLIPTICAL ARCH

Let AB be the span of the arch and Oh its height. From centre O, with OA, or half the span, for radius, describe outer semicircle. From same centre and oh for radius describe the inner semicircle. Divide outer circle into a convenient number of parts, 1, 2, 3, &c., to which draw radii from centre O. From each division drop perpendiculars. Where the radii intersect the inner circle, as at gkmo, draw horizontals op, mn, kj, &c., and through their intersections with the perpendiculars f, j, n, p, draw the curve of the flattened arch. Transfer this to the lower figure, and proceed to draw the tunnel. Note how the vanishing scale is formed on either side by horizontals ba, fe, &c., which enable us to make the distant arches similar to the near ones.







CXXXIII

OPENING OR ARCHED WINDOW IN A VAULT

First draw the vault AEB. To introduce the window K, the upper part of which follows the form of the vault, we first decide on its width, which is mn, and its height from floor Ba. On line Ba at the side of the arch form scales aa'S, bb'S, &c. Raise the semicircular arch K, shown by a dotted line. The scale at the side will give the lengths aa', bb', &c., from different parts of this dotted arch to corresponding points in the curved archway or window required.



Note that to obtain the width of the window K we have used the diagonals on the floor and width m n on base. This method of measurement is explained at Fig. 144, and is of ready application in a case of this kind.



CXXXIV

STAIRS, STEPS, &C.

Having decided upon the incline or angle, such as CBA, at which the steps are to be placed, and the height Bm of each step, draw mn to CB, which will give the width. Then measure along base AB this width equal to DB, which will give that for all the other steps. Obtain length BF of steps, and draw EF parallel to CB. These lines will aid in securing the exactness of the figure.







CXXXV

STEPS, FRONT VIEW

In this figure the height of each step is measured on the vertical line AB (this line is sometimes called the line of heights), and their depth is found by diagonals drawn to the point of distance D. The rest of the figure explains itself.





CXXXVI

SQUARE STEPS

Draw first step ABEF and its two diagonals. Raise vertical AH, and measure thereon the required height of each step, and thus form scale. Let the second step CD be less all round than the first by Ao or Bo. Draw oC till it cuts the diagonal, and proceed to draw the second step, guided by the diagonals and taking its height from the scale as shown. Draw the third step in the same way.





CXXXVII

TO DIVIDE AN INCLINED PLANE INTO EQUAL PARTS—SUCH AS A LADDER PLACED AGAINST A WALL



Divide the vertical EC into the required number of parts, and draw lines from point of sight S through these divisions 1, 2, 3, &c., cutting the line AC at 1, 2, 3, &c. Draw parallels to AB, such as mn, from AC to BD, which will represent the steps of the ladder.



CXXXVIII

STEPS AND THE INCLINED PLANE



In Fig. 248 we treat a flight of steps as if it were an inclined plane. Draw the first and second steps as in Fig. 245. Then through 1, 2, draw 1V, AV to V, the vanishing point on the vertical line SV. These two lines and the corresponding ones at BV will form a kind of vanishing scale, giving the height of each step as we ascend. It is especially useful when we pass the horizontal line and we no longer see the upper surface of the step, the scale on the right showing us how to proceed in that case.

In Fig. 249 we have an example of steps ascending and descending. First set out the ground-plan, and find its vanishing point S (point of sight). Through S draw vertical BA, and make SA equal to SB. Set out the first step CD. Draw EA, CA, DA, and GA, for the ascending guiding lines. Complete the steps facing us, at central line OO. Then draw guiding line FB for the descending steps (see Rule 8).





CXXXIX

STEPS IN ANGULAR PERSPECTIVE

First draw the base ABCD (Fig. 251) at the required angle by the new method (Fig. 250). Produce BC to the horizon, and thus find vanishing point V. At this point raise vertical VV'. Construct first step AB, refer its height at B to line of heights hI on left, and thus obtain height of step at A. Draw lines from A and F to V'. From n draw diagonal through O to G. Raise vertical at O to represent the height of the next step, its height being determined by the scale of heights at the side. From A and F draw lines to V', and also similar lines from B, which will serve as guiding lines to determine the height of the steps at either end as we raise them to the required number.







CXL

A STEP LADDER AT AN ANGLE



First draw the ground-plan G at the required angle, using vanishing and measuring points. Find the height hH, and width at top HH', and draw the sides HA and H'E. Note that AE is wider than HH', and also that the back legs are not at the same angle as the front ones, and that they overlap them. From E raise vertical EF, and divide into as many parts as you require rounds to the ladder. From these divisions draw lines 1 1, 2 2, &c., towards the other vanishing point (not in the picture), but having obtained their direction from the ground-plan in perspective at line Ee, you may set up a second vertical ef at any point on Ee and divide it into the same number of parts, which will be in proportion to those on EF, and you will obtain the same result by drawing lines from the divisions on EF to those on ef as in drawing them to the vanishing point.



CXLI

SQUARE STEPS PLACED OVER EACH OTHER



This figure shows the other method of drawing steps, which is simple enough if we have sufficient room for our vanishing points.

The manner of working it is shown at Fig. 124.



CXLII

STEPS AND A DOUBLE CROSS DRAWN BY MEANS OF DIAGONALS AND ONE VANISHING POINT

Although in this figure we have taken a longer distance-point than in the previous one, we are able to draw it all within the page.



Begin by setting out the square base at the angle required. Find point G by means of diagonals, and produce AB to V, &c. Mark height of step Ao, and proceed to draw the steps as already shown. Then by the diagonals and measurements on base draw the second step and the square inside it on which to stand the foot of the cross. To draw the cross, raise verticals from the four corners of its base, and a line K from its centre. Through any point on this central line, if we draw a diagonal from point G we cut the two opposite verticals of the shaft at mn (see Fig. 255), and by means of the vanishing point V we cut the other two verticals at the opposite corners and thus obtain the four points through which to draw the other sides of the square, which go to the distant or inaccessible vanishing point. It will be seen by carefully examining the figure that by this means we are enabled to draw the double cross standing on its steps.







CXLIII

A STAIRCASE LEADING TO A GALLERY

In this figure we have made use of the devices already set forth in the foregoing figures of steps, &c., such as the side scale on the left of the figure to ascertain the height of the steps, the double lines drawn to the high vanishing point of the inclined plane, and so on; but the principal use of this diagram is to show on the perspective plane, which as it were runs under the stairs, the trace or projection of the flights of steps, the landings and positions of other objects, which will be found very useful in placing figures in a composition of this kind. It will be seen that these underneath measurements, so to speak, are obtained by the half-distance.



CXLIV

WINDING STAIRS IN A SQUARE SHAFT

Draw square ABCD in parallel perspective. Divide each side into four, and raise verticals from each division. These verticals will mark the positions of the steps on each wall, four in number. From centre O raise vertical OP, around which the steps are to wind. Let AF be the height of each step. Form scale AB, which will give the height of each step according to its position. Thus at mn we find the height at the centre of the square, so if we transfer this measurement to the central line OP and repeat it upwards, say to fourteen, then we have the height of each step on the line where they all meet. Starting then with the first on the right, draw the rectangle gD1f, the height of AF, then draw to the central line go, f1, and 1 1, and thus complete the first step. On DE, measure heights equal to D 1. Draw 2 2 towards central line, and 2n towards point of sight till it meets the second vertical nK. Then draw n2 to centre, and so complete the second step. From 3 draw 3a to third vertical, from 4 to fourth, and so on, thus obtaining the height of each ascending step on the wall to the right, completing them in the same way as numbers 1 and 2, when we come to the sixth step, the other end of which is against the wall opposite to us. Steps 6, 7, 8, 9 are all on this wall, and are therefore equal in height all along, as they are equally distant. Step 10 is turned towards us, and abuts on the wall to our left; its measurement is taken on the scale AB just underneath it, and on the same line to which it is drawn. Step 11 is just over the centre of base mo, and is therefore parallel to it, and its height is mn. The widths of steps 12 and 13 seem gradually to increase as they come towards us, and as they rise above the horizon we begin to see underneath them. Steps 13, 14, 15, 16 are against the wall on this side of the picture, which we may suppose has been removed to show the working of the drawing, or they might be an open flight as we sometimes see in shops and galleries, although in that case they are generally enclosed in a cylindrical shaft.







CXLV

WINDING STAIRS IN A CYLINDRICAL SHAFT

First draw the circular base CD. Divide the circumference into equal parts, according to the number of steps in a complete round, say twelve. Form scale ASF and the larger scale ASB, on which is shown the perspective measurements of the steps according to their positions; raise verticals such as ef, Gh, &c. From divisions on circumference measure out the central line OP, as in the other figure, and find the heights of the steps 1, 2, 3, 4, &c., by the corresponding numbers in the large scale to the left; then proceed in much the same way as in the previous figure. Note the central column OP cuts off a small portion of the steps at that end.

In ordinary cases only a small portion of a winding staircase is actually seen, as in this sketch.





CXLVI

OF THE CYLINDRICAL PICTURE OR DIORAMA



Although illusion is by no means the highest form of art, there is no picture painted on a flat surface that gives such a wonderful appearance of truth as that painted on a cylindrical canvas, such as those panoramas of 'Paris during the Siege', exhibited some years ago; 'The Battle of Trafalgar', only lately shown at Earl's Court; and many others. In these pictures the spectator is in the centre of a cylinder, and although he turns round to look at the scene the point of sight is always in front of him, or nearly so. I believe on the canvas these points are from 12 to 16 feet apart.

The reason of this look of truth may be explained thus. If we place three globes of equal size in a straight line, and trace their apparent widths on to a straight transparent plane, those at the sides, as a and b, will appear much wider than the centre one at c. Whereas, if we trace them on a semicircular glass they will appear very nearly equal and, of the three, the central one c will be rather the largest, as may be seen by this figure.

We must remember that, in the first case, when we are looking at a globe or a circle, the visual rays form a cone, with a globe at its base. If these three cones are intersected by a straight glass GG, and looked at from point S, the intersection of C will be a circle, as the cone is cut straight across. The other two being intersected at an angle, will each be an ellipse. At the same time, if we look at them from the station point, with one eye only, then the three globes (or tracings of them) will appear equal and perfectly round.

Of course the cylindrical canvas is necessary for panoramas; but we have, as a rule, to paint our pictures and wall-decorations on flat surfaces, and therefore must adapt our work to these conditions.

In all cases the artist must exercise his own judgement both in the arrangement of his design and the execution of the work, for there is perspective even in the touch—a painting to be looked at from a distance requires a bold and broad handling; in small cabinet pictures that we live with in our own rooms we look for the exquisite workmanship of the best masters.



BOOK FOURTH

CXLVII

THE PERSPECTIVE OF CAST SHADOWS

There is a pretty story of two lovers which is sometimes told as the origin of art; at all events, I may tell it here as the origin of sciagraphy. A young shepherd was in love with the daughter of a potter, but it so happened that they had to part, and were passing their last evening together, when the girl, seeing the shadow of her lover's profile cast from a lamp on to some wet plaster or on the wall, took a metal point, perhaps some sort of iron needle, and traced the outline of the face she loved on to the plaster, following carefully the outline of the features, being naturally anxious to make it as like as possible. The old potter, the father of the girl, was so struck with it that he began to ornament his wares by similar devices, which gave them increased value by the novelty and beauty thus imparted to them.

Here then we have a very good illustration of our present subject and its three elements. First, the light shining on the wall; second, the wall or the plane of projection, or plane of shade; and third, the intervening object, which receives as much light on itself as it deprives the wall of. So that the dark portion thus caused on the plane of shade is the cast shadow of the intervening object.

We have to consider two sorts of shadows: those cast by a luminary a long way off, such as the sun; and those cast by artificial light, such as a lamp or candle, which is more or less close to the object. In the first case there is no perceptible divergence of rays, and the outlines of the sides of the shadows of regular objects, as cubes, posts, &c., will be parallel. In the second case, the rays diverge according to the nearness of the light, and consequently the lines of the shadows, instead of being parallel, are spread out.



CXLVIII

THE TWO KINDS OF SHADOWS

In Figs. 261 and 262 is seen the shadow cast by the sun by parallel rays.

Fig. 263 shows the shadows cast by a candle or lamp, where the rays diverge from the point of light to meet corresponding diverging lines which start from the foot of the luminary on the ground.





The simple principle of cast shadows is that the rays coming from the point of light or luminary pass over the top of the intervening object which casts the shadow on to the plane of shade to meet the horizontal trace of those rays on that plane, or the lines of light proceed from the point of light, and the lines of the shadow are drawn from the foot or trace of the point of light.





Fig. 264 shows this in profile. Here the sun is on the same plane as the picture, and the shadow is cast sideways.

Fig. 265 shows the same thing, but the sun being behind the object, casts its shadow forwards. Although the lines of light are parallel, they are subject to the laws of perspective, and are therefore drawn from their respective vanishing points.





CXLIX

SHADOWS CAST BY THE SUN

Owing to the great distance of the sun, we have to consider the rays of light proceeding from it as parallel, and therefore subject to the same laws as other parallel lines in perspective, as already noted. And for the same reason we have to place the foot of the luminary on the horizon. It is important to remember this, as these two things make the difference between shadows cast by the sun and those cast by artificial light.

The sun has three principal positions in relation to the picture. In the first case it is supposed to be in the same plane either to the right or to the left, and in that case the shadows will be parallel with the base of the picture. In the second position it is on the other side of it, or facing the spectator, when the shadows of objects will be thrown forwards or towards him. In the third, the sun is in front of the picture, and behind the spectator, so that the shadows are thrown in the opposite direction, or towards the horizon, the objects themselves being in full light.



CL

THE SUN IN THE SAME PLANE AS THE PICTURE

Besides being in the same plane, the sun in this figure is at an angle of 45 deg to the horizon, consequently the shadows will be the same length as the figures that cast them are high. Note that the shadow of step No. 1 is cast upon step No. 2, and that of No. 2 on No. 3, the top of each of these becoming a plane of shade.







When the shadow of an object such as A, Fig. 268, which would fall upon the plane, is interrupted by another object B, then the outline of the shadow is still drawn on the plane, but being interrupted by the surface B at C, the shadow runs up that plane till it meets the rays 1, 2, which define the shadow on plane B. This is an important point, but is quite explained by the figure.

Although we have said that the rays pass over the top of the object casting the shadow, in the case of an archway or similar figure they pass underneath it; but the same principle holds good, that is, we draw lines from the guiding points in the arch, 1, 2, 3, &c., at the same angle of 45 deg to meet the traces of those rays on the plane of shade, and so get the shadow of the archway, as here shown.





CLI

THE SUN BEHIND THE PICTURE

We have seen that when the sun's altitude is at an angle of 45 deg the shadows on the horizontal plane are the same length as the height of the objects that cast them. Here (Fig. 270), the sun still being at 45 deg altitude, although behind the picture, and consequently throwing the shadow of B forwards, that shadow must be the same length as the height of cube B, which will be seen is the case, for the shadow C is a square in perspective.



To find the angle of altitude and the angle of the sun to the picture, we must first find the distance of the spectator from the foot of the luminary.



From point of sight S (Fig. 270) drop perpendicular to T, the station-point. From T draw TF at 45 deg to meet horizon at F. With radius FT make FO equal to it. Then O is the position of the spectator. From F raise vertical FL, and from O draw a line at 45 deg to meet FL at L, which is the luminary at an altitude of 45 deg, and at an angle of 45 deg to the picture.

Fig. 272 is similar to the foregoing, only the angles of altitude and of the sun to the picture are altered.

Note.—The sun being at 50 deg to the picture instead of 45 deg, is nearer the point of sight; at 90 deg it would be exactly opposite the spectator, and so on. Again, the elevation being less (40 deg instead of 45 deg) the shadow is longer. Owing to the changed position of the sun two sides of the cube throw a shadow. Note also that the outlines of the shadow, 1 2, 2 3, are drawn to the same vanishing points as the cube itself.

It will not be necessary to mark the angles each time we make a drawing, as it must be seen we can place the luminary in any position that suits our convenience.





CLII

SUN BEHIND THE PICTURE, SHADOWS THROWN ON A WALL

As here we change the conditions we must also change our procedure. An upright wall now becomes the plane of shade, therefore as the principle of shadows must always remain the same we have to change the relative positions of the luminary and the foot thereof.

At S (point of sight) raise vertical SF', making it equal to fL. F' becomes the foot of the luminary, whilst the luminary itself still remains at L.



We have but to turn this page half round and look at it from the right, and we shall see that SF' becomes as it were the horizontal line. The luminary L is at the right side of point S instead of the left, and the foot thereof is, as before, the trace of the luminary, as it is just underneath it. We shall also see that by proceeding as in previous figures we obtain the same results on the wall as we did on the horizontal plane. Fig. B being on the horizontal plane is treated as already shown. The steps have their shadows partly on the wall and partly on the horizontal plane, so that the shadows on the wall are outlined from F' and those on the ground from f. Note shadow of roof A, and how the line drawn from F' through A is met by the line drawn from the luminary L, at the point P, and how the lower line of the shadow is directed to point of sight S.



Fig. 274 is a larger drawing of the steps, &c., in further illustration of the above.



CLIII

SUN BEHIND THE PICTURE THROWING SHADOW ON AN INCLINED PLANE



The vanishing point of the shadows on an inclined plane is on a vertical dropped from the luminary to a point (F) on a level with the vanishing point (P) of that inclined plane. Thus P is the vanishing point of the inclined plane K. Draw horizontal PF to meet fL (the line drawn from the luminary to the horizon). Then F will be the vanishing point of the shadows on the inclined plane. To find the shadow of M draw lines from F through the base eg to cd. From luminary L draw lines through ab, also to cd, where they will meet those drawn from F. Draw CD, which determines the length of the shadow egcd.



CLIV

THE SUN IN FRONT OF THE PICTURE



When the sun is in front of the picture we have exactly the opposite effect to that we have just been studying. The shadows, instead of coming towards us, are retreating from us, and the objects throwing them are in full light, consequently we have to reverse our treatment. Let us suppose the sun to be placed above the horizon at L', on the right of the picture and behind the spectator (Fig. 276). If we transport the length L'f' to the opposite side and draw the vertical downwards from the horizon, as at FL, we can then suppose point L to be exactly opposite the sun, and if we make that the vanishing point for the sun's rays we shall find that we obtain precisely the same result. As in Fig. 277, if we wish to find the length of C, which we may suppose to be the shadow of P, we can either draw a line from A through O to B, or from B through O to A, for the result is the same. And as we cannot make use of a point that is behind us and out of the picture, we have to resort to this very ingenious device.



In Fig. 276 we draw lines L1, L2, L3 from the luminary to the top of the object to meet those drawn from the foot F, namely F1, F2, F3, in the same way as in the figures we have already drawn.



Fig. 278 gives further illustration of this problem.



CLV

THE SHADOW OF AN INCLINED PLANE

The two portions of this inclined plane which cast the shadow are first the side fbd, and second the farther end abcd. The points we have to find are the shadows of a and b. From luminary L draw La, Lb, and from F, the foot, draw Fc, Fd. The intersection of these lines will be at a'b'. If we join fb' and db' we have the shadow of the side fbd, and if we join ca' and a'b' we have the shadow of abcd, which together form that of the figure.





CLVI

SHADOW ON A ROOF OR INCLINED PLANE

To draw the shadow of the figure M on the inclined plane K (or a chimney on a roof). First find the vanishing point P of the inclined plane and draw horizontal PF to meet vertical raised from L, the luminary. Then F will be the vanishing point of the shadow. From L draw L1, L2, L3 to top of figure M, and from the base of M draw 1F, 2F, 3F to F, the vanishing point of the shadow. The intersections of these lines at 1, 2, 3 on K will determine the length and form of the shadow.





CLVII

TO FIND THE SHADOW OF A PROJECTION OR BALCONY ON A WALL

To find the shadow of the object K on the wall W, drop verticals OO till they meet the base line B'B' of the wall. Then from the point of sight S draw lines through OO, also drop verticals Dd', Cc', to meet these lines in d'c'; draw c'F and d'F to foot of luminary. From the points xx where these lines cut the base B raise perpendiculars xa', xb'. From D, A, and B draw lines to the luminary L. These lines or rays intersecting the verticals raised from xx at a'b' will give the respective points of the shadow.



The shadow of the eave of a roof can be obtained in the same way. Take any point thereon, mark its trace on the ground, and then proceed as above.



CLVIII

SHADOW ON A RETREATING WALL, SUN IN FRONT

Let L be the luminary. Raise vertical LF. F will be the vanishing point of the shadows on the ground. Draw Lf' parallel to FS. Drop Sf' from point of sight; f' (so found) is the vanishing point of the shadows on the wall. For shadow of roof draw LE and f'B, giving us e, the shadow of E. Join Be, &c., and so draw shadow of eave of roof.



For shadow of K draw lines from luminary L to meet those from f' the foot, &c.

The shadow of D over the door is found in a similar way to that of the roof.



Figure 283 shows how the shadow of the old man in the preceding drawing is found.



CLIX

SHADOW OF AN ARCH, SUN IN FRONT

Having drawn the arch, divide it into a certain number of parts, say five. From these divisions drop perpendiculars to base line. From divisions on AB draw lines to F the foot, and from those on the semicircle draw lines to L the luminary. Their intersections will give the points through which to draw the shadow of the arch.





CLX

SHADOW IN A NICHE OR RECESS

In this figure a similar method to that just explained is adopted. Drop perpendiculars from the divisions of the arch 1 2 3 to the base. From the foot of each draw 1S, 2S, 3S to foot of luminary S, and from the top of each, A 1 2 3 B, draw lines to L as before. Where the former intersect the curve on the floor of the niche raise verticals to meet the latter at P 1 2 B, &c. These points will indicate about the position of the shadow; but the niche being semicircular and domed at the top the shadow gradually loses itself in a gradated and somewhat serpentine half-tone.





CLXI

SHADOW IN AN ARCHED DOORWAY



This is so similar to the last figure in many respects that I need not repeat a description of the manner in which it is done. And surely an artist after making a few sketches from the actual thing will hardly require all this machinery to draw a simple shadow.



CLXII

SHADOWS PRODUCED BY ARTIFICIAL LIGHT



Shadows thrown by artificial light, such as a candle or lamp, are found by drawing lines from the seat of the luminary through the feet of the objects to meet lines representing rays of light drawn from the luminary itself over the tops or the corners of the objects; very much as in the cases of sun-shadows, but with this difference, that whereas the foot of the luminary in this latter case is supposed to be on the horizon an infinite distance away, the foot in the case of a lamp or candle may be on the floor or on a table close to us. First draw the table and chair, &c. (Fig. 287), and let L be the luminary. For objects on the table such as K the foot will be at f on the table. For the shadows on the floor, of the chair and table itself, we must find the foot of the luminary on the floor. Draw So, find trace of the edge of the table, drop vertical oP, draw PS to point of sight, drop vertical from foot of candlestick to meet PS in F. Then F is the foot of the luminary on the floor. From this point draw lines through the feet or traces of objects such as the corners of the table, &c., to meet other lines drawn from the point of light, and so obtain the shadow.



CLXIII

SOME OBSERVATIONS ON REAL LIGHT AND SHADE

Although the figures we have been drawing show the principles on which sun-shadows are shaped, still there are so many more laws to be considered in the great art of light and shade that it is better to observe them in Nature herself or under the teaching of the real sun. In the study of a kitchen and scullery in an old house in Toledo (Fig. 288) we have an example of the many things to be considered besides the mere shapes of shadows of regular forms. It will be seen that the light is dispersed in all directions, and although there is a good deal of half-shade there are scarcely any cast shadows except on the floor; but the light on the white walls in the outside gallery is so reflected into the cast shadows that they are extremely faint. The luminosity of this part of the sketch is greatly enhanced by the contrast of the dark legs of the bench and the shadows in the roof. The warm glow of all this portion is contrasted by the grey door and its frame.



Note that the door itself is quite luminous, and lighted up by the reflection of the sun from the tiled floor, so that the bars in the upper part throw distinct shadows, besides the mystery of colour thus introduced. The little window to the left, though not admitting much direct sunlight, is evidence of the brilliant glare outside; for the reflected light is very conspicuous on the top and on the shutters on each side; indeed they cast distinct shadows up and down, while some clear daylight from the blue sky is reflected on the window-sill. As to the sink, the table, the wash-tubs, &c., although they seem in strong light and shade they really receive little or no direct light from a single point; but from the strong reflected light re-reflected into them from the wall of the doorway. There are many other things in such effects as this which the artist will observe, and which can only be studied from real light and shade. Such is the character of reflected light, varying according to the angle and intensity of the luminary and a hundred other things. When we come to study light in the open air we get into another region, and have to deal with it accordingly, and yet we shall find that our sciagraphy will be a help to us even in this bewilderment; for it will explain in a manner the innumerable shapes of sun-shadows that we observe out of doors among hills and dales, showing up their forms and structure; its play in the woods and gardens, and its value among buildings, showing all their juttings and abuttings, recesses, doorways, and all the other architectural details. Nor must we forget light's most glorious display of all on the sea and in the clouds and in the sunrises and the sunsets down to the still and lovely moonlight.

These sun-shadows are useful in showing us the principle of light and shade, and so also are the shadows cast by artificial light; but they are only the beginning of that beautiful study, that exquisite art of tone or chiaro-oscuro, which is infinite in its variety, is full of the deepest mystery, and is the true poetry of art. For this the student must go to Nature herself, must study her in all her moods from early dawn to sunset, in the twilight and when night sets in. No mathematical rules can help him, but only the thoughtful contemplation, the silent watching, and the mental notes that he can make and commit to memory, combining them with the sentiments to which they in turn give rise. The plein air, or broad daylight effects, are but one item of the great range of this ever-changing and deepening mystery—from the hard reality to the soft blending of evening when form almost disappears, even to the merging of the whole landscape, nay, the whole world, into a dream—which is felt rather than seen, but possesses a charm that almost defies the pencil of the painter, and can only be expressed by the deep and sweet notes of the poet and the musician. For love and reverence are necessary to appreciate and to present it.

There is also much to learn about artificial light. For here, again, the study is endless: from the glare of a hundred lights—electric and otherwise—to the single lamp or candle. Indeed a whole volume could be filled with illustrations of its effects. To those who aim at producing intense brilliancy, refusing to acknowledge any limitations to their capacity, a hundred or a thousand lights commend themselves; and even though wild splashes of paint may sometimes be the result, still the effort is praiseworthy. But those who prefer the mysterious lighting of a Rembrandt will find, if they sit contemplating in a room lit with one lamp only, that an endless depth of mystery surrounds them, full of dark recesses peopled by fancy and sweet thought, whilst the most beautiful gradations soften the forms without distorting them; and at the same time he can detect the laws of this science of light and shade a thousand times repeated and endless in its variety.

Note.—Fig. 288 must be looked upon as a rough sketch which only gives the general effect of the original drawing; to render all the delicate tints, tones and reflections described in the text would require a highly-finished reproduction in half-tone or in colour.

As many of the figures in this book had to be re-drawn, not a light task, I must here thank Miss Margaret L. Williams, one of our Academy students, for kindly coming to my assistance and volunteering her careful co-operation.



CLXIV

REFLECTION

[Transcriber's Note: In this chapter, [R] represents "R" printed upside-down.]

Reflections in still water can best be illustrated by placing some simple object, such as a cube, on a looking-glass laid horizontally on a table, or by studying plants, stones, banks, trees, &c., reflected in some quiet pond. It will then be seen that the reflection is the counterpart of the object reversed, and having the same vanishing points as the object itself.



Let us suppose R (Fig. 289) to be standing on the water or reflecting plane. To find its reflection make square [R] equal to the original square R. Complete the reversed cube by drawing its other sides, &c. It is evident that this lower cube is the reflection of the one above it, although it differs in one respect, for whereas in figure R the top of the cube is seen, in its reflection [R] it is hidden, &c. In figure A of a semicircular arch we see the underneath portion of the arch reflected in the water, but we do not see it in the actual object. However, these things are obvious. Note that the reflected line must be equal in length to the actual one, or the reflection of a square would not be a square, nor that of a semicircle a semicircle. The apparent lengthening of reflections in water is owing to the surface being broken by wavelets, which, leaping up near to us, catch some of the image of the tree, or whatever it is, that it is reflected.



In this view of an arch (Fig. 290) note that the reflection is obtained by dropping perpendiculars from certain points on the arch, 1, 0, 2, &c., to the surface of the reflecting plane, and then measuring the same lengths downwards to corresponding points, 1, 0, 2, &c., in the reflection.



CLXV

ANGLES OF REFLECTION

In Fig. 291 we take a side view of the reflected object in order to show that at whatever angle the visual ray strikes the reflecting surface it is reflected from it at the same angle.



We have seen that the reflected line must be equal to the original line, therefore mB must equal Ma. They are also at right angles to MN, the plane of reflection. We will now draw the visual ray passing from E, the eye, to B, which is the reflection of A; and just underneath it passes through MN at O, which is the point where the visual ray strikes the reflecting surface. Draw OA. This line represents the ray reflected from it. We have now two triangles, OAm and OmB, which are right-angled triangles and equal, therefore angle a equals angle b. But angle b equals angle c. Therefore angle EcM equals angle Aam, and the angle at which the ray strikes the reflecting plane is equal to the angle at which it is reflected from it.



CLXVI

REFLECTIONS OF OBJECTS AT DIFFERENT DISTANCES

In this sketch the four posts and other objects are represented standing on a plane level or almost level with the water, in order to show the working of our problem more clearly. It will be seen that the post A is on the brink of the reflecting plane, and therefore is entirely reflected; B and C being farther back are only partially seen, whereas the reflection of D is not seen at all. I have made all the posts the same height, but with regard to the houses, where the length of the vertical lines varies, we obtain their reflections by measuring from the points oo upwards and downwards as in the previous figure.



Of course these reflections vary according to the position they are viewed from; the lower we are down, the more do we see of the reflections of distant objects, and vice versa. When the figures are on a higher plane than the water, that is, above the plane of reflection, we have to find their perspective position, and drop a perpendicular AO (Fig. 293) till it comes in contact with the plane of reflection, which we suppose to run under the ground, then measure the same length downwards, as in this figure of a girl on the top of the steps. Point o marks the point of contact with the plane, and by measuring downwards to a' we get the length of her reflection, or as much as is seen of it. Note the reflection of the steps and the sloping bank, and the application of the inclined plane ascending and descending.





CLXVII

REFLECTION IN A LOOKING-GLASS

I had noticed that some of the figures in Titian's pictures were only half life-size, and yet they looked natural; and one day, thinking I would trace myself in an upright mirror, I stood at arm's length from it and with a brush and Chinese white, I made a rough outline of my face and figure, and when I measured it I found that my drawing was exactly half as long and half as wide as nature. I went closer to the glass, but the same outline fitted me. Then I retreated several paces, and still the same outline surrounded me. Although a little surprising at first, the reason is obvious. The image in the glass retreats or advances exactly in the same measure as the spectator.



Suppose him to represent one end of a parallelogram e's', and his image a'b' to represent the other. The mirror AB is a perpendicular half-way between them, the diagonal e'b' is the visual ray passing from the eye of the spectator to the foot of his image, and is the diagonal of a rectangle, therefore it cuts AB in the centre o, and AO represents a'b' to the spectator. This is an experiment that any one may try for himself. Perhaps the above fact may have something to do with the remarks I made about Titian at the beginning of this chapter.







CLXVIII

THE MIRROR AT AN ANGLE

If an object or line AB is inclined at an angle of 45 deg to the mirror RR, then the angle BAC will be a right angle, and this angle is exactly divided in two by the reflecting plane RR. And whatever the angle of the object or line makes with its reflection that angle will also be exactly divided.





Now suppose our mirror to be standing on a horizontal plane and on a pivot, so that it can be inclined either way. Whatever angle the mirror is to the plane the reflection of that plane in the mirror will be at the same angle on the other side of it, so that if the mirror OA (Fig. 298) is at 45 deg to the plane RR then the reflection of that plane in the mirror will be 45 deg on the other side of it, or at right angles, and the reflected plane will appear perpendicular, as shown in Fig. 299, where we have a front view of a mirror leaning forward at an angle of 45 deg and reflecting the square aob with a cube standing upon it, only in the reflection the cube appears to be projecting from an upright plane or wall.



If we increase the angle from 45 deg to 60 deg, then the reflection of the plane and cube will lean backwards as shown in Fig. 300. If we place it on a level with the original plane, the cube will be standing upright twice the distance away. If the mirror is still farther tilted till it makes an angle of 135 deg as at E (Fig. 298), or 45 deg on the other side of the vertical Oc, then the plane and cube would disappear, and objects exactly over that plane, such as the ceiling, would come into view.

In Fig. 300 the mirror is at 60 deg to the plane mn, and the plane itself at about 15 deg to the plane an (so that here we are using angular perspective, V being the accessible vanishing point). The reflection of the plane and cube is seen leaning back at an angle of 60 deg. Note the way the reflection of this cube is found by the dotted lines on the plane, on the surface of the mirror, and also on the reflection.





CLXIX

THE UPRIGHT MIRROR AT AN ANGLE OF 45 DEG. TO THE WALL

In Fig. 301 the mirror is vertical and at an angle of 45 deg to the wall opposite the spectator, so that it reflects a portion of that wall as though it were receding from us at right angles; and the wall with the pictures upon it, which appears to be facing us, in reality is on our left.



An endless number of complicated problems could be invented of the inclined mirror, but they would be mere puzzles calculated rather to deter the student than to instruct him. What we chiefly have to bear in mind is the simple principle of reflections. When a mirror is vertical and placed at the end or side of a room it reflects that room and gives the impression that we are in one double the size. If two mirrors are placed opposite to each other at each end of a room they reflect and reflect, so that we see an endless number of rooms.

Again, if we are sitting in a gallery of pictures with a hand mirror, we can so turn and twist that mirror about that we can bring any picture in front of us, whether it is behind us, at the side, or even on the ceiling. Indeed, when one goes to those old palaces and churches where pictures are painted on the ceiling, as in the Sistine Chapel or the Louvre, or the palaces at Venice, it is not a bad plan to take a hand mirror with us, so that we can see those elevated works of art in comfort.

There are also many uses for the mirror in the studio, well known to the artist. One is to look at one's own picture reversed, when faults become more evident; and another, when the model is required to be at a longer distance than the dimensions of the studio will admit, by drawing his reflection in the glass we double the distance he is from us.

The reason the mirror shows the fault of a work to which the eye has become accustomed is that it doubles it. Thus if a line that should be vertical is leaning to one side, in the mirror it will lean to the other; so that if it is out of the perpendicular to the left, its reflection will be out of the perpendicular to the right, making a double divergence from one to the other.



CLXX

MENTAL PERSPECTIVE

Before we part, I should like to say a word about mental perspective, for we must remember that some see farther than others, and some will endeavour to see even into the infinite. To see Nature in all her vastness and magnificence, the thought must supplement and must surpass the eye. It is this far-seeing that makes the great poet, the great philosopher, and the great artist. Let the student bear this in mind, for if he possesses this quality or even a share of it, it will give immortality to his work.

To explain in detail the full meaning of this suggestion is beyond the province of this book, but it may lead the student to think this question out for himself in his solitary and imaginative moments, and should, I think, give a charm and virtue to his work which he should endeavour to make of value, not only to his own time but to the generations that are to follow. Cultivate, therefore, this mental perspective, without forgetting the solid foundation of the science I have endeavoured to impart to you.



INDEX

[Transcriber's Note: Index citations in the original book referred to page numbers. References to chapters (Roman numerals) or figures (Arabic numerals) have been added in brackets where possible. Note that the last two entries for "Toledo" are figure numbers rather than pages; these have not been corrected.]

A Albert Durer, 2, 9. Angles of Reflection, 259 [CLXV]. Angular Perspective, 98 [XLIX] - 123 [LXXII], 133 [LXXX], 170. " " New Method, 133 [LXXX], 134 [LXXXI], 135 [LXXXII], 136 [LXXXIII]. Arches, Arcades, &c., 198 [CXXVI], 200 [CXXVII] - 208 [CXXIII]. Architect's Perspective, 170 [CVIII], 171 [197]. Art Schools Perspective, 112 [LXII] - 118 [LXVI], 217 [CXLI]. Atmosphere, 1, 74 [XXX].

B Balcony, Shadow of, 246 [CLVII]. Base or groundline, 89 [XLI].

C Campanile Florence, 5, 59. Cast Shadows, 229 [CXLVII] - 253 [CLXII]. Centre of Vision, 15 [II]. Chessboard, 74 [XXXI]. Chinese Art, 11. Circle, 145 [LXXXVIII], 151 [XCII] - 156 [XCVI], 159 [XCIX]. Columns, 157 [XCVII], 159 [XCIX], 161 [CI], 169 [CVI], 170 [CVII]. Conditions of Perspective, 24 [VII], 25. Cottage in Angular Perspective, 116 [LXV]. Cube, 53 [XVII], 65 [XXIII], 115 [LXIV], 119 [LXVIII]. Cylinder, 158 [XCVIII], 159 [CXIX]. Cylindrical picture, 227 [CXLVI].

D De Hoogh, 2, 62 [68], 73 [82]. Depths, How to measure by diagonals, 127 [LXXVI], 128 [LXXVII]. Descending plane, 92 [XLIV] - 95 [XLV]. Diagonals, 45, 124 [LXXIII], 125 [LXXIV], 126 [LXXV]. Disproportion, How to correct, 35, 118 [LXVII], 157 [XCVII]. Distance, 16 [III], 77 [XXXIII], 78 [XXXIV], 85 [XXXVII], 87 [XXXIX], 103 [LIV], 128 [LXXVII]. Distorted perspective, How to correct, 118 [LXVII]. Dome, 163 [CIII] - 167 [CV]. Double Cross, 218 [CXLII].

E Ellipse, 145 [LXXXIX], 146 [XC], 147 [168]. Elliptical Arch, 207 [CXXXII].

F Farningham, 95 [103]. figures on descending plane, 92 [XLIV], 93 [100], 94 [102], 95 [XLV]. " " an inclined plane, 88 [XL]. " " a level plane, 70 [79], 71 [XXVIII], 72 [81], 73 [82], 74 [XXX], 75 [XXXI]. " " uneven ground, 90 [XLII], 91 [XLIII].

G Geometrical and Perspective figures contrasted, 46 [XII] - 48. " plane, 99 [L]. Giovanni da Pistoya, Sonnet to, by Michelangelo, 60. Great Pyramid, 190 [CXXII].

H Hexagon, 177 [CXIV], 183 [CXVII], 185 [CXIX]. Hogarth, 9. Honfleur, 83 [92], 142 [163]. Horizon, 3, 4, 15 [II], 20, 59 [XX], 60 [66]. Horizontal line, 13 [I], 15 [II]. Horizontals, 30, 31, 36.

I Inaccessible vanishing points, 77 [XXXII], 78 [XXXIII], 136, 140 - 144. Inclined plane, 33, 118, 213 [CXXXVIII], 244 [XLV], 245 [XLVI]. Interiors, 62 [XXI], 117 [LXVI], 118 [LXVII], 128.

J Japanese Art, 11. Jesuit of Paris, Practice of Perspective by, 9.

K Kiosk, Application of Hexagon, 185 [XCIX]. Kirby, Joshua, Perspective made Easy (?), 9.

L Ladder, Step, 212 [CXXXVII], 216 [CXL]. Landscape Perspective, 74 [XXX]. Landseer, Sir Edwin, 1. Leonardo da Vinci, 1, 61. Light, Observations on, 253 [CLXIII]. Light-house, 84 [XXXVII]. Long distances, 85 [XXXVIII], 87 [XXXIX].

M Measure distances by square and diagonal, 89 [XLI], 128 [LXXVII], 129. " vanishing lines, How to, 49 [XIV], 50 [XV]. Measuring points, 106 [LVII], 113. " point O, 108, 109, 110 [LX]. Mental Perspective, 269 [CLXX]. Michelangelo, 5, 57, 58, 60.

N Natural Perspective, 12, 82 [91], 95 [103], 142 [163], 144 [164]. New Method of Angular Perspective, 133 [LXXX], 134 [LXXXI], 135 [LXXXII], 141 [LXXXVI], 215 [CXXXIX], 219. Niche, 164 [CIV], 165 [193], 250 [CLX].

O Oblique Square, 139 [LXXXV]. Octagon, 172 [CIX] - 175 [202]. O, measuring point, 110 [LX]. Optic Cone, 20 [IV].

P Parallels and Diagonals, 124 [LXXIII] - 128 [LXXVI]. Paul Potter, cattle, 19 [16]. Paul Veronese, 4. Pavements, 64 [XXII], 66 [XXIV], 176 [CXIII], 178 [CXV], 180 [209],181 [CXVI], 183 [CXVII]. Pedestal, 141 [LXXXVI], 161 [CI]. Pentagon, 186 [CXX], 187 [217], 188 [219]. Perspective, Angular, 98 [XLIX] - 123 [LXXII]. " Definitions, 13 [I] - 23 [VI]. " Necessity of, 1. " Parallel, 42 - 97 [XLVII]. " Rules and Conditions of, 24 [VII] - 41. " Scientific definition of, 22 [VI]. " Theory of, 13 - 24 [VI]. " What is it? 6 - 12. Pictures painted according to positions they are to occupy, 59 [XX]. Point of Distance, 16 [III] - 21 [IV]. " " Sight, 12, 15 [II]. Points in Space, 129 [LXXVIII], 137 [LXXXIII]. Portico, 111 [122]. Projection, 21 [V], 137. Pyramid, 189 [CXXI], 190 [224], 191 [CXXII], 193 [CXXIII] - 196 [CXXV].

R Raphael, 3. Reduced distance, 77 [XXXIII], 78 [XXXIV], 79 [XXXV], 84 [90]. Reflection, 257 [CLXIV] - 268 [CLXIX]. Rembrandt, 59 [XX], 256. Reynolds, Sir Joshua, 9, 60. Rubens, 4. Rules of Perspective, 24 - 41.

S Scale on each side of Picture, 141 [LXXXVII], 142 [163] - 144 [164]. " Vanishing, 69 [XXVI], 71 [XXVII], 81 [XXXVI], 84 [90]. Serlio, 5, 126 [LXXV]. Shadows cast by sun, 229 [CXLVII] - 252 [CLXI]. " " " artificial light, 252 [CLXII]. Sight, Point of, 12, 15 [II]. Sistine Chapel, 60. Solid figures, 135 [LXXXII] - 140 [LXXXV]. Square in Angular Perspective, 105 [LVI], 106 [LVII], 109 [120], 112 [LXII], 114 [LXIII], 121 [LXX], 122 [LXXI], 123 [LXXII], 133 [LXXX], 134 [LXXXI], 139 [LXXXV]. " and diagonals, 125 [LXXIV], 138 [LXXXIV], 139 [LXXXV], 141 [LXXXVI]. " of the hypotenuse (fig. 170), 149 [170]. " in Parallel Perspective, 42 [IX], 43 [X], 50 [XV], 53 [XVII], 54 [XIX]. " at 45 deg, 64 [XXII] - 66 [XXIV]. Staircase leading to a Gallery, 221 [CXLIII]. Stairs, Winding, 222 [CXLIV], 225 [CXLV]. Station Point, 13 [I]. Steps, 209 [CXXXIV] - 218 [CXLII].

T Taddeo Gaddi, 5. Terms made use of, 48 [XIII]. Tiles, 176 [CXIII], 178 [CXV], 181 [CXVI]. Tintoretto, 4. Titian, 59 [XX], 262 [CLXVII]. Toledo, 96 [104], 144 [164], 259 [259], 288 [288]. Trace and projection, 21 [V]. Transposed distance, 53 [XVIII]. Triangles, 104 [LV], 106 [LVII], 132 [148], 135 [151], 138 [158]. Turner, 2, 87 [95].

U Ubaldus, Guidus, 9.

V Vanishing lines, 49 [XIV]. " point, 119 [LXVIII]. " scale, 68 [XXV] - 72 [XXVIII], 74 [XXX], 77 [XXXII], 79 [XXXV], 84 [90]. Vaulted Ceiling, 203 [CXXX]. Velasquez, 59 [XX]. Vertical plane, 13 [I]. Visual rays, 20 [IV].

W Winding Stairs, 222 [CXLIV] - 225 [CXLV]. Water, Reflections in, 257 [CLXIV], 258 [CLXV], 260 [CLXVI], 261 [293].



* * * * *



Errors and Anomalies:

Missing punctuation in the Index has been silently supplied.

The name form "Albert Duerer" (for Albrecht) is used throughout. In all references to Kirby, Perspective made Easy (?), the question mark is in the original text.

Figure 66: Caption missing, but number is given in text ground plan of the required design, as at Figs. 73 and 74 text reads "Figs. 74 and 75" CV [Chapter head] "C" invisible

Index Durer, Albert umlaut missing Taddeo Gaddi text reads "Tadeo" Titian text reads Titien

THE END