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It is much easier to understand andremember a thing when a reason is given for it, than when we are merelyshown how to do it without being told why it is so done; for in thelatter case, instead of being assisted by reason, our real help in allstudy, we have to rely upon memory or our power of imitation, and to dosimply as we are told without thinking about it. The consequence is thatat the very first difficulty we are left to flounder about in the dark,or to remain inactive till the master comes to our assistance.

Now in this book it is proposed to enlist the reasoning faculty fromthe very first: to let one problem grow out of another and to bedependent on the foregoing, as in geometry, and so to explain each thingwe do that there shall be no doubt in the mind as to the correctness ofthe proceeding. The student will thus gain the power of finding out anynew problem for himself, and will therefore acquire a true knowledge ofperspective.

BOOK FIRST

THE NECESSITY OF THE STUDY OFPERSPECTIVE TO PAINTERS, SCULPTORS, AND ARCHITECTS

Leonardo da Vinci tells us in hiscelebratedTreatise on Painting that the young artist shouldfirst of all learn perspective, that is to say, he should first of alllearn that he has to depict on a flat surface objects which are inrelief or distant one from the other; for this is the simple art ofpainting. Objects appear smaller at a distance than near to us, so bydrawing them thus we give depth to our canvas. The outline of a ball isa mere flat circle, but with proper shading we make it appear round, andthis is the perspective of light and shade.

‘The next thing to be considered is the effect of theatmosphere and light. If two figures are in the same coloured dress, andare standing one behind the other, then they should be of slightlydifferent tone, so as to separate them. And in like manner, according tothe distance of the mountains in a landscape and the greater or lessdensity of the air, so do we depict space between them, not only makingthem smaller in outline, but less distinct.’1

Sir Edwin Landseer used to say that in looking at a figure in apicture he liked to feel that he could walk round it, and this exactlyexpresses the impression that the true art of painting should make uponthe spectator.

There is another observation of Leonardo’s that it is well Ishould here transcribe; he says: ‘Many are desirous of learning todraw, and are very fond of it, who are notwithstanding void of a properdisposition for it. This may be known by their want of perseverance;like boys who draw everything in a hurry, never finishing orshadowing.’ This shows they do not care for their work, and allinstruction is thrown away upon them. At the present time there is toomuch of this ‘everything in a hurry’,2and beginning in this way leads only to failure and disappointment.These observations apply equally to perspective as to drawing andpainting.

Unfortunately, this study is too often neglected by our painters,some of them even complacently confessing their ignorance of it; whilethe ordinary student either turns from it with distaste, or only enduresgoing through it with a view to passing an examination, little thinkingof what value it will be to him in working out his pictures. Whether themanner of teaching perspective is the cause of this dislike for it,I cannot say; but certainly most of our English books on thesubject are anything but attractive.

All the great masters of painting have also been masters ofperspective, for they knew that without it, it would be impossible tocarry out their grand compositions. In many cases they were eveninspired by it in choosing their subjects. When one looks at those sunnyinteriors, those corridors and courtyards by De Hooghe, with theirfigures far off and near, one feels that their charm consists greatly intheir perspective, as well as in their light and tone and colour. Or ifwe study those Venetian masterpieces by Paul Veronese, Titian,Tintoretto, and others, we become convinced that it was through theirknowledge of perspective that they gave such space and grandeur to theircanvases.

I need not name all the great artists who have shown their interestand delight in this study, both by writing about it and practising it,such as Albert Dürer and others, but I cannot leave out our own Turner,who was one of the greatest masters in this respect that ever lived;though in his case we can only judge of the results of his knowledge asshown in his pictures, for although he was Professor of Perspective atthe Royal Academy in 1807—over a hundred years ago—and tookgreat pains with the diagrams he prepared to illustrate his lectures,they seemed to the students to be full of confusion and obscurity; noram I aware that any record of them remains, although they must havecontained some valuable teaching, had their author possessed the art ofconveying it.

However, we are here chiefly concerned with the necessity of thisstudy, and of the necessity of starting our work with it.

3Before undertaking a large composition of figures, such as the‘Wedding-feast at Cana’, by Paul Veronese, or ‘TheSchool of Athens’, by Raphael, the artist should set out hisfloors, his walls, his colonnades, his balconies, his steps, &c., sothat he may know where to place his personages, and to measure theirdifferent sizes according to their distances; indeed, he must make hisstage and his scenery before he introduces his actors. He can thenproceed with his composition, arrange his groups and the accessorieswith ease, and above all with correctness. But I have noticed that someof our cleverest painters will arrange their figures to please the eye,and when fairly advanced with their work will call in an expert, to (asthey call it) put in their perspective for them, but as it does not formpart of their original composition, it involves all sorts ofdifficulties and vexatious alterings and rubbings out, and even then isnot always satisfactory. For the expert may not be an artist, nor insympathy with the picture, hence there will be a want of unity in it;whereas the whole thing, to be in harmony, should be the conception ofone mind, and the perspective as much a part of the composition as thefigures.

If a ceiling has to be painted with figures floating or flying in theair, or sitting high above us, then our perspective must take adifferent form, and the point of sight will be above our heads insteadof on the horizon; nor can these difficulties be overcome without anadequate knowledge of the science, which will enable us to work out forourselves any new problems of this kind that we may have to solve.

Then again, with a view to giving different effects or impressions inthis decorative work, we must know where to place the horizon and thepoints of sight, for several of the latter are sometimes required whendealing with large surfaces such as the painting of walls, or stagescenery, or panoramas depicted on a cylindrical canvas and viewed fromthe centre thereof, where a fresh point of sight is required at everytwelve or sixteen feet.

Without a true knowledge of perspective, none of these things can bedone. The artist should study them in the great compositions of themasters, by analysing their pictures and seeing4how and for what reasons they applied their knowledge. Rubens put lowhorizons to most of his large figure-subjects, as in ‘The Descentfrom the Cross’, which not only gave grandeur to his designs, but,seeing they were to be placed above the eye, gave a more naturalappearance to his figures. The Venetians often put the horizon almost ona level with the base of the picture or edge of the frame, and sometimeseven below it; as in ‘The Family of Darius at the Feet ofAlexander’, by Paul Veronese, and ‘The Origin of the“Via Lactea”’, by Tintoretto, both in our NationalGallery. But in order to do all these things, the artist in designinghis work must have the knowledge of perspective at his fingers' ends,and only the details, which are often tedious, should he leave to anassistant to work out for him.

We must remember that the line of the horizon should be as nearly aspossible on a level with the eye, as it is in nature; and yet one of thecommonest mistakes in our exhibitions is the bad placing of this line.We see dozens of examples of it, where in full-length portraits andother large pictures intended to be seen from below, the horizon isplaced high up in the canvas instead of low down; the consequence isthat compositions so treated not only lose in grandeur and truth, butappear to be toppling over, or give the impression of smallness ratherthan bigness. Indeed, they look like small pictures enlarged, which is avery different thing from a large design. So that, in order to see themproperly, we should mount a ladder to get upon a level with theirhorizon line (seeFig. 66, double-pageillustration).

We have here spoken in a general way of the importance of this studyto painters, but we shall see that it is of almost equal importance tothe sculptor and the architect.

A sculptor student at the Academy, who was making his drawings rathercarelessly, asked me of what use perspective was to a sculptor.‘In the first place,’ I said, ‘to reason outapparently difficult problems, and to find how easy they become, willimprove your mind; and in the second, if you have to do monumental work,it will teach you the exact size to make your figures according to theheight they are to be placed, and also the boldness with which theyshould be treated to give them their full effect.’5He at once acknowledged that I was right, proved himself an efficientpupil, and took much interest in his work.

I cannot help thinking that the reason our public monuments so oftenfail to impress us with any sense of grandeur is in a great measureowing to the neglect of the scientific study of perspective. As anillustration of what I mean, let the student look at a good engraving orphotograph of the Arch of Constantine at Rome, or the Tombs of theMedici, by Michelangelo, in the sacristy of San Lorenzo at Florence. Andthen, for an example of a mistake in the placing of a colossal figure,let him turn to the Tomb of Julius II in San Pietro in Vinculis, Rome,and he will see that the figure of Moses, so grand in itself, not onlyloses much of its dignity by being placed on the ground instead of inthe niche above it, but throws all the other figures out of proportionor harmony, and was quite contrary to Michelangelo’s intention.Indeed, this tomb, which was to have been the finest thing of its kindever done, was really the tragedy of the great sculptor’slife.

The same remarks apply in a great measure to the architect as to thesculptor. The old builders knew the value of a knowledge of perspective,and, as in the case of Serlio, Vignola, and others, prefaced theirtreatises on architecture with chapters on geometry and perspective. Forit showed them how to give proper proportions to their buildings and thedetails thereof; how to give height and importance both to the interiorand exterior; also to give the right sizes of windows, doorways,columns, vaults, and other parts, and the various heights they shouldmake their towers, walls, arches, roofs, and so forth. One of the mostbeautiful examples of the application of this knowledge to architectureis the Campanile of the Cathedral, at Florence, built by Giotto andTaddeo Gaddi, who were painters as well as architects. Here it will beseen that the height of the windows is increased as they are placedhigher up in the building, and the top windows or openings into thebelfry are about six times the size of those in the lower story.

WHAT IS PERSPECTIVE?

Perspective is a subtle form ofgeometry; it represents figures and objects not as they are but as wesee them in space, whereas geometry represents figures not as we seethem but as they are. When we have a front view of a figure such as asquare, its perspective and geometrical appearance is the same, and wesee it as it really is, that is, with all its sides equal and all itsangles right angles, the perspective only varying in size according tothe distance we are from it; but if we place that square flat on thetable and look at it sideways or at an angle, then we become consciousof certain changes in its form—the side farthest from us appearsshorter than that near to us,7and all the angles are different. ThusA (Fig. 2) is a geometrical square andB is the same square seen inperspective.

The science of perspective gives the dimensions of objects seen inspace as they appear to the eye of the spectator, just as a perfecttracing of those objects on a sheet of glass placed vertically betweenhim and them would do; indeed its very name is derived fromperspicere, to see through. But as no tracing done by hand couldpossibly be mathematically correct, the mathematician teaches us how bycertain points and measurements we may yet give a perfect image of them.These images are called projections, but the artist calls them pictures.In this sketchK is the verticaltransparent plane or picture,O is acube placed on one side of it. The young student is the spectator on theother side of it, the dotted lines drawn from the corners of the cube tothe eye of the spectator are the visual rays, and the points on thetransparent picture plane where these visual rays pass through itindicate the perspective position8of those points on the picture. To find these points is the main objector duty of linear perspective.

Perspective up to a certain point is a pure science, not dependingupon the accidents of vision, but upon the exact laws of reasoning. Noris it to be considered as only pertaining to the craft of the painterand draughtsman. It has an intimate connexion with our mentalperceptions and with the ideas that are impressed upon the brain by theappearance of all that surrounds us. If we saw everything as depicted byplane geometry, that is, as a map, we should have no difference of view,no variety of ideas, and we should live in a world of unbearablemonotony; but as we see everything in perspective, which is infinite inits variety of aspect, our minds are subjected to countless phases ofthought, making the world around us constantly interesting, so it isdevised that we shall see the infinite wherever we turn, and marvel atit, and delight in it, although perhaps in many cases unconsciously.

In perspective, as in geometry, we deal with parallels, squares,triangles, cubes, circles, &c.; but in perspective the same figuretakes an endless variety of forms, whereas in geometry it has but one.Here are three equal geometrical squares: they are all alike. Here arethree equal perspective squares, but all varied9in form; and the same figure changes in aspect as often as we view itfrom a different position. A walk round the dining-room table willexemplify this.

It is in proving that, notwithstanding this difference of appearance,the figures do represent the same form, that much of our work consists;and for those who care to exercise their reasoning powers it becomes notonly a sure means of knowledge, but a study of the greatestinterest.

Perspective is said to have been formed into a science about thefifteenth century. Among the names mentioned by the unknown but pleasantauthor ofThe Practice of Perspective, written by a Jesuit ofParis in the eighteenth century, we find Albert Dürer, who has left ussome rules and principles in the fourth book of hisGeometry;Jean Cousin, who has an express treatise on the art wherein are manyvaluable things; also Vignola, who altered the plans of St.Peter’s left by Michelangelo; Serlio, whose treatise is one of thebest I have seen of these early writers; Du Cerceau, Serigati, Solomonde Cause, Marolois, Vredemont; Guidus Ubaldus, who first introducedforeshortening; the Sieur de Vaulizard, the Sieur Dufarges, JoshuaKirby, for whoseMethod of Perspective made Easy(?) Hogarth drewthe well-known frontispiece; and lastly, the above-namedPractice ofPerspective by a Jesuit of Paris, which is very clear and excellentas far as it goes, and was the book used by Sir Joshua Reynolds.2 But nearly allthese authors treat chiefly of parallel perspective, which they do withclearness and simplicity, and also mathematically, as shown in the shorttreatise in Latin by Christian Wolff, but they scarcely touch upon themore difficult problems of angular and oblique perspective. Of modernbooks, those to which I am most indebted are theTraité Pratique dePerspective of M. A. Cassagne (Paris, 1873), which isthoroughly artistic, and full of pictorial examples admirably done; andto M. Henriet’sCours Rational de Dessin. There aremany other foreign books of excellence, notably M. Thibault'sPerspective, and some German and Swiss books, and yet,notwithstanding this imposing array of authors, I venture to saythat many new features and original10problems are presented in this book, whilst the old ones are notneglected. As, for instance, How to draw figures at an angle withoutvanishing points (see p. 141,Fig. 162,&c.), a new method of angular perspective which dispenses withthe cumbersome setting out usually adopted, and enables us to drawfigures at any angle without vanishing lines, &c., and is almost, ifnot quite, as simple as parallel perspective (see p. 133,Fig. 150, &c.). How to measure distances by the squareand diagonal, and to draw interiors thereby (p. 128,Fig. 144). How to explain the theory of perspective byocular demonstration, using a vertical sheet of glass with strings,placed on a drawing-board, which I have found of the greatest use (seep. 29,Fig. 29). Then again, I show howall our perspective can be done inside the picture; that we can measureany distance into the picture from a foot to a mile or twenty miles (seep. 86,Fig. 94); how we can draw the GreatPyramid, which stands on thirteen acres of ground, by putting it 1,600feet off (Fig. 224), &c., &c. And whilepreserving the mathematical science, so that all our operations can beproved to be correct, my chief aim has been to make it easy ofapplication to our work and consequently useful to the artist.

The Egyptians do not appear to have made any use of linearperspective. Perhaps it was considered out of character with theirparticular kind of decoration, which is to be looked upon as picturewriting rather than pictorial art; a table, for instance, would berepresented like a ground-plan and the objects upon it in elevation orstanding up. A row of chariots with their horses and drivers sideby side were placed one over the other, and although the Egyptians hadno doubt a reason for this kind of representation, for they were grandartists, it seems to us very primitive; and indeed quite young beginnerswho have never drawn from real objects have a tendency to do very muchthe same thing as this ancient people did, or even to emulate themathematician and represent things not as they appear but as they are,and will make the top of a table an almost upright square and theobjects upon it as if they would fall off.

No doubt the Greeks had correct notions of perspective, for thepaintings on vases, and at Pompeii and Herculaneum, which were either byGreek artists or copied from Greek pictures,11show some knowledge, though not complete knowledge, of this science.Indeed, it is difficult to conceive of any great artist making hisperspective very wrong, for if he can draw the human figure as theGreeks did, surely he can draw an angle.

The Japanese, who are great observers of nature, seem to have got attheir perspective by copying what they saw, and, although they are notquite correct in a few things, they convey the idea of distance and maketheir horizontal planes look level, which are two important things inperspective. Some of their landscapes are beautiful; their trees,flowers, and foliage exquisitely drawn and arranged with the greatesttaste; whilst there is a character and go about their figures and birds,&c., that can hardly be surpassed. All their pictures are lively andintelligent and appear to be executed with ease, which shows theirauthors to be complete masters of their craft.

The same may be said of the Chinese, although their perspective ismore decorative than true, and whilst their taste is exquisite theirwhole art is much more conventional and traditional, and does not remindus of nature like that of the Japanese.

We may see defects in the perspective of the ancients, in themediaeval painters, in the Japanese and Chinese, but are we always rightourselves? Even in celebrated pictures by old and modern masters thereare occasionally errors that might easily have been avoided, if a readymeans of settling the difficulty were at hand. We should endeavour thento make this study as simple, as easy, and as complete as possible, toshow clear evidence of its correctness (according to its conditions),and at the same time to serve as a guide on any and all occasions thatwe may require it.

To illustrate what is perspective, and as an experiment that any onecan make, whether artist or not, let us stand at a window that looks outon to a courtyard or a street or a garden, &c., and trace with apaint-brush charged with Indian ink or water-colour the outline ofwhatever view there happens to be outside, being careful to keep the eyealways in the same place by means of a rest; when this is dry, place apiece of drawing-paper over it and trace through with a pencil. Now wewill rub out the tracing on the glass, which is sure to be ratherclumsy, and, fixing12our paper down on a board, proceed to draw the scene before us, usingthe main lines of our tracing as our guiding lines.

If we take pains over our work, we shall find that, without troublingourselves much about rules, we have produced a perfect perspective ofperhaps a very difficult subject. After practising for some little timein this way we shall get accustomed to what are called perspectivedeformations, and soon be able to dispense with the glass and thetracing altogether and to sketch straight from nature, taking littlenote of perspective beyond fixing the point of sight and thehorizontal-line; in fact, doing what every artist does when he goes outsketching.

Fig. 6. This is a much reducedreproduction of a drawing made on my studio window in this way sometwenty years ago, when the builder started covering the fields at theback with rows and rows of houses.

THE THEORY OF PERSPECTIVE

Definitions

Fig. 7. In this figure,AKBrepresents the picture or transparent vertical plane through which theobjects to be represented can be seen, or on which they can be traced,such as the cubeC.

The lineHD is theHorizontal-line orHorizon, the chief line in perspective,as upon it are placed the principal points to which our perspectivelines are drawn. First, thePoint of Sight and nextD, thePoint of Distance. The chief vanishingpoints and measuring points are also placed on this line.

Another important line isAB, theBase orGround line, as it is on this that we measure thewidth of any object to be represented, such asef, the base ofthe squareefgh, on which the cubeC is raised.E isthe position of the eye of the spectator, being drawn in perspective,and is called theStation-point.

Note that the perspective of the board, and the lineSE, is not14the same as that of the cube in the pictureAKB, and also that so much of the board which isbehind the picture plane partially represents thePerspective-plane, supposed to be perfectly level and to extendfrom the base line to the horizon. Of this we shall speak further on. Innature it is not really level, but partakes in extended views of therotundity of the earth, though in small areas such as ponds theroundness is infinitesimal.

Fig. 8. This is a side view of the previous figure, the picture planeK being represented edgeways, and thelineSE its full length. It also showsthe position of the eye in front of the point of sightS. The horizontal-lineHD and the base or ground-lineAB are represented as receding from us, and in thatcase are called vanishing lines, a not quite satisfactory term.

It is to be noted that the cubeCis placed close to the transparent picture plane, indeed touches it, andthat the squarefj faces the spectatorE, and although here drawn in perspective it appearsto him as in the other figure. Also, it is at the same time aperspective and a geometrical figure, and can therefore be measured withthe compasses. Or in other words, we can15touch the squarefj, because it is on the surface of the picture,but we cannot touch the squareghmb at the other end of the cubeand can only measure it by the rules of perspective.

The Point of Sight, the Horizon, and the Pointof Distance

There are three things to be considered and understood before we canbegin a perspective drawing. First, the position of the eye in front ofthe picture, which is called theStation-point, and of course isnot in the picture itself, but its position is indicated by a point onthe picture which is exactly opposite the eye of the spectator, and iscalled thePoint of Sight, orPrincipal Point, orCentre of Vision, but we will keep to the first of these.

If our picture plane is a sheet of glass, and is so placed that wecan see the landscape behind it or a sea-view, we shall find that thedistant line of the horizon passes through that point of sight, and wetherefore draw a line on our picture which exactly corresponds with it,and which we call theHorizontal-line orHorizon.3 The height ofthe horizon then depends entirely upon the position of the eye of thespectator: if he rises, so does the horizon; if he stoops or descends tolower ground, so does the horizon follow his movements. You may sit in aboat on a calm sea, and the horizon will be as low down as you are, oryou may go to the top of a high cliff, and still the horizon will be onthe same level as your eye.

16This is an important line for the draughtsman to consider, for theeffect of his picture greatly depends upon the position of the horizon.If you wish to give height and dignity to a mountain or a building, thehorizon should be low down, so that these things may appear to towerabove you. If you wish to show a wide expanse of landscape, then youmust survey it from a height. In a composition of figures, you selectyour horizon according to the subject, and with a view to help thegrouping. Again, in portraits and decorative work to be placed high up,a low horizon is desirable, but I have already spoken of thissubject in the chapter on the necessity of the study of perspective.

III

Point of Distance

Fig. 11. The distance of the spectator from the picture is of greatimportance; as the distortions and disproportions arising from too neara view are to be avoided, the object of drawing being to make thingslook natural; thus, the floor should look level, and not as if it wererunning up hill—the top of a table flat, and not on a slant, as ifcups and what not, placed upon it, would fall off.

In this figure we have a geometrical or ground plan of two squares atdifferent distances from the picture, which is represented by the lineKK. The spectator is first atA, the corner of the near squareAcd. If fromA we draw a diagonal of that square and produce itto the lineKK (which may representthe horizontal-line in the picture), where it intersects that line atA· marks the distance that thespectator is from the point of sightS. For it will be seen that lineSA equals lineSA·. In like manner, if the spectator is atB, his distance from the pointS is also found on the horizon by means of thediagonalBB´, so that all lines ordiagonals at 45° are drawn to the point of distance (seeRule 6).

Figs. 12 and 13. In these two figures the difference is shown betweenthe effect of the short-distance pointA· and the long-distance pointB·; the first,Acd, does not appear to lie so flat on theground as the second square,Bef.

From this it will be seen how important it is to choose the17right point of distance: if we take it too near the point of sight, asin Fig. 12, the square looks unnatural and distorted. This, I maynote, is a common fault with photographs taken with a wide-angle lens,which throws everything out of proportion, and will make the east end ofa church or a cathedral appear higher than the steeple or tower; but assoon as we make our18line of distance sufficiently long, as at Fig. 13, objects take theirright proportions and no distortion is noticeable.

In some books on perspective we are told to make the angle of vision60°, so that the distanceSD (Fig. 14)is to be rather less than the length or height of the picture, as atA. The French recommend an angle of28°, and to make the distance about double the length of the picture, asatB (Fig. 15), which is far moreagreeable. For we must remember that the distance-point is not only thepoint from which we are supposed to make our tracing on the verticaltransparent plane, or a point transferred to the horizon to make ourmeasurements by, but it is also the point in front of the canvas that weview the picture from, called the station-point. It is ridiculous, then,to have it so close that we must almost touch the canvas with our nosesbefore we can see its perspective properly.

Now a picture should look right from whatever distance we19view it, even across the room or gallery, and of course in decorativework and in scene-painting a long distance is necessary.

We need not, however, tie ourselves down to any hard and fast rule,but should choose our distance according to the impression of space wewish to convey: if we have to represent a domestic scene in a smallroom, as in many Dutch pictures, we must not make our distance-point toofar off, as it would exaggerate the size of the room.

20The height of the horizon is also an important consideration in thecomposition of a picture, and so also is the position of the point ofsight, as we shall see farther on.

In landscape and cattle pictures a low horizon often gives space andair, as in this sketch from a picture by Paul Potter—where thehorizontal-line is placed at one quarter the height of the canvas.Indeed, a judicious use of the laws of perspective is a great aidto composition, and no picture ever looks right unless these laws areattended to. At the present time too little attention is paid to them;the consequence is that much of the art of the day reflects in a greatmeasure the monotony of the snap-shot camera, with its everyday andwearisome commonplace.

Perspective of a Point, Visual Rays,&c.

We perceive objects by means of the visual rays, which are imaginarystraight lines drawn from the eye to the various points of the thing weare looking at. As those rays proceed from the pupil of the eye, whichis a circular opening, they form themselves into a cone called theOptic Cone, the base of which increases in proportion to itsdistance from the eye, so that the larger the view which we wish to takein, the farther must we be removed from it. The diameter of the base ofthis cone, with the visual rays drawn from each of its extremities tothe eye, form the angle of vision, which is wider or narrower accordingto the distance of this diameter.

Now let us suppose a visual rayEAto be directed to some small object on the floor, say the head of anail,A (Fig. 17). If we interposebetween this nail and our eye a sheet of glass,K, placed vertically on the floor, we continue tosee the nail through the glass, and it is easily understood that itsperspective appearance thereon is the pointa, where the visualray passes through it. If now we trace on the floor a lineAB from the nail to the spotB, just under the eye, and from the pointo,where this line passes through or under the glass, we raise aperpendicularoS, thatperpendicular passes through the precise point that the visual ray21passes through. The lineAB traced onthe floor is the horizontal trace of the visual ray, and it will be seenthat the pointa is situated on the vertical raised from thishorizontal trace.

Trace and Projection

If from any lineA orB orC (Fig. 18),&c., we drop perpendiculars from different points of those lines onto a horizontal plane, the intersections of those verticals with theplane will be on a line called the horizontal trace or projection of theoriginal line. We may liken these projections to sun-shadows when thesun is in the meridian, for it will be remarked that the trace does notrepresent the length of the original line, but only so much of it aswould be embraced by the verticals dropped from each end of it, andalthough lineA is the same length aslineB its horizontal22trace is longer than that of the other; that the projection of a curve(C) in this upright position is astraight line, that of a horizontal line (D) is equal to it, and the projection of aperpendicular or vertical (E) is apoint only. The projections of lines or points can likewise be shown ona vertical plane, but in that case we draw lines parallel to thehorizontal plane, and by this means we can get the position of a pointin space; and by the assistance of perspective, as will be shown fartheron, we can carry out the most difficult propositions of descriptivegeometry and of the geometry of planes and solids.

The position of a point in space is given by its projection on avertical and a horizontal plane—

Thuse· is the projection ofE on the vertical planeK, ande·· is the projection ofE on the horizontal plane;fe·· is thehorizontal trace of the planefE, ande·f is the trace of the same plane onthe vertical planeK.

Scientific Definition of Perspective

The projections of the extremities of a right line which passesthrough a vertical plane being given, one on either side of it, to findthe intersection of that line with the vertical plane.AE (Fig. 20) is the right line. The projection ofits extremityA on the vertical planeisa·, the projection ofE, theother extremity, ise·.AS isthe horizontal trace ofAE, anda·e· is its trace23on the vertical plane. At pointf, where the horizontal traceintersects the baseBc of thevertical plane, raise perpendicularfP till it cutsa·e· at pointP, which is the point required. For it is at thesame time on the given lineAE and thevertical planeK.

This figure is similar to the previous one, except that the extremityA of the given line is raised from theground, but the same demonstration applies to it.

And now let us suppose the vertical planeK to be a sheet of glass, and the given lineAE to be the visual ray passing from24the eye to the objectA on the otherside of the glass. Then ifE is theeye of the spectator, its projection on the picture isS, the point of sight.

If I draw a dotted line fromE tolittlea, this represents another visual ray, ando, thepoint where it passes through the picture, is the perspective of littlea. I now draw another line fromg toS, and thus form the shaded figurega·Po, which is the perspective ofaAa·g.

Let it be remarked that in the shaded perspective figure the linesa·P andgo are bothdrawn towardsS, the point of sight,and that they represent parallel linesAa· andag, which are at right anglesto the picture plane. This is the most important fact in perspective,and will be more fully explained farther on, when we speak of retreatingor so-called vanishing lines.

RULES

VII

The Rules and Conditions of Perspective

The conditions of linear perspective are somewhat rigid. In the firstplace, we are supposed to look at objects with one eye only; that is,the visual rays are drawn from a single point, and not from two. Of thiswe shall speak later on. Then again, the eye must be placed in a certainposition, as atE (Fig. 22), at agiven height from the ground,S·E, andat a given distance from the picture, asSE. In the next place, the picture or picture planeitself must be vertical and perpendicular to the ground or horizontalplane, which plane is supposed to be as level as a billiard-table, andto extend from the base line,ef, of the picture to the horizon,that is, to infinity, for it does not partake of the rotundity of theearth.

We can only work out our propositions and figures in space withmathematical precision by adopting such conditions as the above. Butafterwards the artist or draughtsman may modify and suit them to a moreelastic view of things; that is, he can make his figures separate fromone another, instead of their outlines coming close together as they dowhen we look at them25with only one eye. Also he will allow for the unevenness of the groundand the roundness of our globe; he may even move his head and his eyes,and use both of them, and in fact make himself quite at his ease when heis out sketching, for Nature does all his perspective for him. At thesame time, a knowledge of this rigid perspective is the sure andunerring basis of his freehand drawing.

Rule 1

Rule 2

Vertical lines remain vertical in perspective, and are divided in thesame proportion asAB (Fig. 24), theoriginal line, anda·b·, the perspective line, and if the one isdivided atO the other is divided ato· in the same way.

It is not an uncommon error to suppose that the vertical lines of ahigh building should converge towards the top; so they would if we stoodat the foot of that building and looked up, for then we should alter theconditions of our perspective, and our point of sight, instead of beingon the horizon, would be up in the sky. But if we stood sufficiently faraway, so as to bring the whole of the building within our angle ofvision, and the point of sight down to the horizon, then these samelines would appear perfectly parallel, and the different stories intheir true proportion.

Rule 3

Horizontals parallel to the base of the picture are also parallel tothat base in the picture. Thusa·b· (Fig. 25) is parallel toAB,27and toGL, the base of the picture.Indeed, the same argument may be used with regard to horizontal lines aswith verticals. If we look at a straight wall in front of us, its topand its rows of bricks, &c., are parallel and horizontal; but if welook along it sideways, then we alter the conditions, and the parallellines converge to whichever point we direct the eye.

This rule is important, as we shall see when we come to theconsideration of the perspective vanishing scale. Its use may beillustrated by this sketch, where the houses, walls, &c., areparallel to the base of the picture. When that is the case, then objects28exactly facing us, such as windows, doors, rows of boards, or of bricksor palings, &c., are drawn with their horizontal lines parallel tothe base; hence it is called parallel perspective.

Rule 4

All lines situated in a plane that is parallel to the picture planediminish in proportion as they become more distant, but do not undergoany perspective deformation; and remain in the same relation andproportion each to each as the original lines. This is called the frontview.

Rule 5

All horizontals which are at right angles to the picture plane aredrawn to the point of sight.

Thus the linesAB andCD (Fig. 28) are horizontal or parallel to theground plane, and are also at right angles to the picture planeK. It will be seen that the perspectivelinesBa·,Dc·, must, according to the laws ofprojection, be drawn to the point of sight.

This is the most important rule in perspective (seeFig. 7 at beginning of Definitions).

An arrangement such as there indicated is the best means ofillustrating this rule. But instead of tracing the outline of the squareor cube on the glass, as there shown, I have a hole drilled throughat the pointS (Fig. 29), which Iselect for the point of sight, and through which I pass two loosestringsA andB, fixing their ends atS.

AsSD represents the distance thespectator is from the glass or picture, I make stringSA equal in length toSD. Now if the pupil takes this string in one handand holds it at right angles to the glass, that is, exactly in front ofS, and then places one eye at the endA (of course with the stringextended), he will be at the proper distance from the picture. Let himthen take the other string,SB, in theother hand, and apply it to pointb´ where the square touches theglass, and he will find that it exactly tallies with the sideb´f30of the squarea·b´fe. If he applies the same string toa·,the other corner of the square, his string will exactly tally or coverthe sidea·e, and he will thus have ocular demonstration of thisimportant rule.

In this little picture (Fig. 30) in parallel perspective it will beseen that the lines which retreat from us at right angles to the pictureplane are directed to the point of sightS.

Rule 6

All horizontals which are at 45°, or half a right angle to thepicture plane, are drawn to the point of distance.

We have already seen that the diagonal of the perspective square, ifproduced to meet the horizon on the picture, will mark on that horizonthe distance that the spectator is from the point of sight (seedefinition, p. 16). This point of distance becomesthen the measuring point for all horizontals at right angles to thepicture plane.

31Thus in Fig. 31 linesAS andBS are drawn to the point of sightS, and are therefore at right angles to thebaseAB.AD being drawn toD (the distance-point), is at an angle of 45° to thebaseAB, andAC is therefore the diagonal of a square. The line1C is made parallel toAB, consequentlyA1CB is a squarein perspective. The lineBC,therefore, being one side of that square, is equal toAB, another side of it. So that to measure a lengthon a line drawn to the point of sight, such asBS, we set out the length required, sayBA, on the base-line, then fromA draw a line to the point of distance, and where itcutsBS atC is the length required. This can be repeated anynumber of times, say five, so that in this figureBE is five times the length ofAB.

Rule 7

All horizontals forming any other angles but the above are drawn tosome other points on the horizontal line. If the angle is greater thanhalf a right angle (Fig. 32), asEBG,the point is within the point of distance, as atV´. If it is less, asABV´´, then32it is beyond the point of distance, and consequently farther from thepoint of sight.

In Fig. 32, the dotted lineBD,drawn to the point of distanceD, isat an angle of 45° to the baseAG. Itwill be seen that the lineBV´ is at agreater angle to the base thanBD; itis therefore drawn to a pointV´,within the point of distance and nearer to the point of sightS. On the other hand, the lineBV´´ is at a more acute angle, and is thereforedrawn to a point some way beyond the other distance point.

Note.—When this vanishing point is a long way outsidethe picture, the architects make use of a centrolinead, and the paintersfix a long string at the required point, and get their perspective linesby that means, which is very inconvenient. But I will show you later onhow you can dispense with this trouble by a very simple means, withequally correct results.

Rule 8

Lines which incline upwards have their vanishing points above thehorizontal line, and those which incline downwards, below it. In bothcases they are on the vertical which passes through the vanishing point(S) of their horizontalprojections.

Rule 9

The farther a point is removed from the picture plane the nearer doesits perspective appearance approach the horizontal line so long as it isviewed from the same position. On the contrary, if the spectatorretreats from the picture planeK(which we suppose to be transparent), the point remaining at the sameplace, the perspective appearance of this point will approach theground-line in proportion to the distance of the spectator.

35Therefore the position of a given point in perspective above theground-line or below the horizon is in proportion to the distance of thespectator from the picture, or the picture from the point.

Figures 38 and 39 are two views of the same gallery from differentdistances. In Fig. 38, where the distance is too short, there is a wantof proportion between the near and far objects, which is corrected inFig. 39 by taking a much longer distance.

Rule 10

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

This is a very important rule, for all our perspective drawingdepends upon it. When we say that parallels are drawn to the same pointon the horizon it does not imply that they meet at that point, whichwould be a contradiction; perspective parallels never reach that point,although they appear to do so. Fig. 40 will explain this.

SupposeS to be the spectator,AB a transparent vertical plane whichrepresents the picture seen edgeways, andHS andDC twoparallel lines, mark off spaces between these parallels equal toSC, the height of the eye of the spectator,and raise verticals 2, 3, 4, 5, &c., forming so many squares.Vertical line 2 viewed fromSwill appear onAB but half its length,vertical 3 will be only a third, vertical 4 a fourth, and soon, and if we multiplied these spacesad infinitum we must keepon dividing the lineAB by the samenumber. So if we supposeAB to be ayard high and the distance from one vertical to another to be also ayard, then if one of these were a thousand yards away its representationatAB would be the thousandth part ofa yard, or ten thousand yards away, its representation atAB would be the ten-thousandth part, and whateverthe distance it must always be something; and thereforeHS andDC, howeverfar they may be produced37and however close they may appear to get, can never meet.

Fig. 41 is a perspective view of the same figure—but moreextended. It will be seen that a line drawn from the tenth uprightK toScuts off a tenth ofAB. We look thenupon these two linesSP, OP, as thesides of a long parallelogram of whichSK is the diagonal, ascefd, the figure onthe ground, is also a parallelogram.

The student can obtain for himself a further illustration of thisrule by placing a looking-glass on one of the walls of his studio andthen sketching himself and his surroundings as seen therein.38He will find that all the horizontals at right angles to the glass willconverge to his own eye. This rule applies equally to lines which are atan angle to the picture plane as to those that are at right angles orperpendicular to it, as in Rule 7. It also applies to those on aninclined plane, as in Rule 8.

39With the above rules and a clear notion of the definitions andconditions of perspective, we should be able to work out any propositionor any new figure that may present itself. At any rate, a thoroughunderstanding of these few pages will make the labour now before ussimple and easy. I hope, too, it may be found interesting. There isalways a certain pleasure in deceiving and being deceived by the senses,and in optical and other illusions, such as making things appear far offthat are quite near, in making a picture of an object on a flat surfaceto look as if it stood out and in relief by a kind of magic. But thereis, I think, a still greater pleasure than this, namely, ininvention and in overcoming difficulties—in finding out how to dothings for ourselves by our reasoning faculties, in originating or beingoriginal, as it were. Let us now see how far we can go in thisrespect.

VIII

A Table or Index of the Rules ofPerspective

The rules here set down have been fully explained in the previouspages, and this table is simply for the student's ready reference.

Rule 1

Rule 2

Rule 3

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

Rule 4

All lines situated in a plane that is parallel to the picture planediminish in proportion as they become more distant, but do not undergoany perspective deformation. This is called the front view.

Rule 5

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

Rule 6

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

Rule 7

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

Rule 8

Lines which incline upwards have their vanishing points above thehorizon, and those which incline downwards, below it. In both cases theyare on the vertical which passes through the vanishing point of theirground-plan or horizontal projections.

Rule 9

The farther a point is removed from the picture plane the nearer doesit appear to approach the horizon, so long as it is viewed from the sameposition.

Rule 10

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

BOOK SECOND

THE PRACTICE OFPERSPECTIVE

In the foregoing book we have explained the theory or science ofperspective; we now have to make use of our knowledge and to apply it tothe drawing of figures and the various objects that we wish todepict.

The first of these will be a square with two of its sides parallel tothe picture plane and the other two at right angles to it, and which wecall

The Square in Parallel Perspective

From a given point on the base line of the picture draw a line atright angles to that base. LetP bethe given point on the base lineAB,andS the point of sight. We simplydraw a line along the ground to the point of sightS, and this line will be at right angles to thebase, as explained in Rule 5, and consequently angleAPS will be equal to angleSPB, although it does not look so here. This is ourfirst difficulty, but one that we shall soon get over.

43In like manner we can draw any number of lines at right angles to thebase, or we may suppose the pointP tobe placed at so many different positions, our only difficulty being toconceive these lines to be parallel to each other. See Rule 10.

The Diagonal

From a given point on the base line draw a line at 45°, or half aright angle, to that base. LetP bethe given point. Draw a line fromP tothe point of distanceD and this linePD will be at an angle of 45°, or atthe same angle as the diagonal of a square. See definitions.

The Square

Draw a square in parallel perspective on a given length on the baseline. Letab be the given length. From its two44extremitiesa andb drawaS andbS tothe point of sightS. These two lineswill be at right angles to the base (seeFig.43). Froma draw diagonalaD to point of distanceD; this line will be 45° to base. At pointc,where it cutsbS, drawdc parallel toab andabcd is the squarerequired.

We have here proceeded in much the same way as in drawing ageometrical square (Fig. 47), by drawing two linesAE andBC at rightangles to a given line,AB, and fromA, drawing the diagonalAC at 45° till it cutsBC atC, and thenthroughC drawingEC parallel toAB.Let it be remarked that because the two perspective lines (Fig. 48)AS andBS are at right angles to the base, they mustconsequently be parallel to each other, and therefore are perspectivelyequidistant, so that all lines parallel toAB and lying between them, such asad,cf, &c., must be equal.

So likewise all diagonals drawn to the point of distance, which45are contained between these parallels, such asAd,af, &c., must be equal. Forall straight lines which meet at any point on the horizon areperspectively parallel to each other, just as two geometrical parallelscrossing two others at any angle, as at Fig. 49. Note also (Fig. 48)that all squares formed between the two vanishing linesAS,BS, and by theaid of these diagonals, are also equal, and further, that any number ofsquares such as are shown in this figure (Fig. 50), formed in the sameway and having equal bases, are also equal; and the nine squarescontained in the squareabcd being equal, they divide each sideof the larger square into three equal parts.

From this we learn how we can measure any number of given46lengths, either equal or unequal, on a vanishing or retreating linewhich is at right angles to the base; and also how we can measure anywidth or number of widths on a line such asdc, that is, parallelto the base of the picture, however remote it may be from that base.

XII

Geometrical and Perspective FiguresContrasted

As at first there may be a little difficulty in realizing theresemblance between geometrical and perspective figures, and also aboutcertain 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 twoviews of the same figures.

XIII

Of Certain Terms made use of in Perspective

Of course when we speak ofPerpendiculars we do not meanverticals only, but straight lines at right angles to other lines in anyposition. Also in speaking oflines a right orstraightline is to be understood; or when we speak ofhorizontals wemean all straight lines that are parallel to the perspective plane, suchas those on Fig. 52, no matter what direction they take so long as theyare level. They are not to be confused with the horizon orhorizontal-line.

There are one or two other terms used in perspective which are notsatisfactory because they are confusing, such as vanishing lines andvanishing points. The French term,fuyante orlignesfuyantes, or going-away lines, is more expressive; andpoint defuite, instead of vanishing point, is much better. I haveoccasionally called the former retreating lines, but the simple meaningis, lines that are not parallel to the picture plane; but a vanishingline implies a line that disappears, and a vanishing point implies49a point that gradually goes out of sight. Still, it is difficult toalter terms that custom has endorsed. All we can do is to use as few ofthem as possible.

XIV

How to Measure Vanishing or Receding Lines

Divide a vanishing line which is at right angles to the picture planeinto any number of given measurements. LetSA be the given line. FromA measure off on the base line the divisionsrequired, say five of 1 foot each; from each division drawdiagonals to point of distanceD, andwhere these intersect the lineAC thecorresponding divisions will be found. Note that as linesAB andAC are twosides of the same square they are necessarily equal, and so also are thedivisions onAC equal to those onAB.

The lineAB being the base of thepicture, it is at the same time a perspective line and a geometricalone, so that we can use it as a scale for measuring given lengthsthereon, but should there not be enough room on it to measure therequired number we draw a second line,DC, which we divide in the same proportion andproceed to dividecf. This geometrical figure gives, as it were,a bird's-eye view or ground-plan of the above.

How to Place Squares in Given Positions

Draw squares of given dimensions at given distances from the baseline to the right or left of the vertical line, which passes through thepoint of sight.

Letab (Fig. 55) represent the base line of the picturedivided into a certain number of feet;HD the horizon,VOthe vertical. It is required to draw a square 3 feet wide,2 feet to the right of the vertical, and 1 foot from thebase.

First measure fromV, 2 feettoe, which gives the distance from the vertical. Second, frome measure 3 feet tob, which gives the width of thesquare; frome andb draweS,bS, topoint of sight. From eithere orb measure 1 foot tothe left, tof orf·. DrawfD to point of distance, which intersectseS atP, and gives the required distance from base. DrawPg andB parallel to the base, and we have the requiredsquare.

SquareA to the left of thevertical is 2½ feet wide, 1 foot from the vertical and 2 feetfrom 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 inworking51out our propositions and figures it is not always desirable.A given length indicated by a line is generally sufficient for ourrequirements. To work out every problem to scale is not only tedious andmechanical, but wastes time, and also takes the mind of the student awayfrom the reasoning out of the subject.

XVI

How To Draw Pavements, &c.

Divide a vanishing line into parts varying in length. LetBS· be the vanishing line: divide it into4 long and 3 short spaces; then proceed as in the previousfigure. If we draw horizontals through the points thus obtained and fromthese raise verticals, we form, as it were, the interior of a buildingin which we can place pillars and other objects.

XVII

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

ABCD is the given square; fromA andB raise verticalsAE,BF, equal toAB; joinEF. DrawES,FS, to point of sight; fromC andD raiseverticalsCG,DH, till they meet vanishing linesES,FS, inG andH,and the cube is complete.

XVIII

The Transposed Distance

The transposed distance is a pointD· on the verticalVD·, at exactly the same distance from the point ofsight as is the point of distance on the horizontal line.

It will be seen by examining this figure that the diagonals of thesquares in a vertical position are drawn to this verticaldistance-point, thus saving the necessity of taking the measurementsfirst on the base line, as atCB,which in the case of distant objects, such as the farthest window, wouldbe very inconvenient. Note that the windows atK are twice as high as they are wide.54Of course these or any other objects could be made of anyproportion.

XIX

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

According to Rule 4, all lines situated in a plane parallel tothe picture plane diminish in length as they become more distant, butremain in the same proportions each to each as the original lines; assquares or any other figures retain the same form. Take the two squaresABCD,abcd (Fig. 61), oneinside the other; although moved back from squareEFGH they retain the same form. So55in dealing with figures of different heights, such as statuary orornament in a building, if actually equal in size, so must we representthem.

In this squareK, with the checkerpattern, we should not think of making the top squares smaller than thebottom ones; so it is with figures.

56This subject requires careful study, for, as pointed out in our openingchapter, there are certain conditions under which we have to modify andgreatly alter this rule in large decorative work.

In Fig. 63 the two statuesA andB are the same size. So if tracedthrough a vertical sheet of glass,K,as atc andd, they would also be equal; but as the angleb at which the upper one is seen is smaller than anglea,at which the lower figure or statue is seen, it will appear smaller tothe spectator (S) both in reality andin the picture.

But if we wish them to appear the same size to the spectator who isviewing them from below, we must make the anglesa andb(Fig. 64), at which they are viewed, both equal. Then draw lines throughequal arcs, as atc andd, till they cut the verticalNO (representing the side of thebuilding where the figures are to be placed). We shall then obtain theexact size of the figure at that height, which will make it look thesame size as the lower one,N. Thesame rule applies to the pictureK,when it is of large proportions. As an example in painting, takeMichelangelo’s large altar-piece in the Sistine Chapel, ‘TheLast Judgement’; here the figures forming the upper group, withour Lord in judgement surrounded by saints, are about four times thesize, that is, about twice the height, of those at the lower part of thefresco. The58figures on the ceiling of the same chapel are studied not only accordingto their height from the pavement, which is 60 ft., but to suit thearched form of it. For instance, the head of the figure of Jonah at theend over the altar is thrown back in the design, but owing to thecurvature in the architecture is actually more forward than the feet.Then again, the prophets and sybils seated round the ceiling, which areperhaps 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 itrequired great knowledge to give them their right effect.

Of course, much depends upon the distance we view these statues orpaintings 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 alsowith statuary in niches, friezes, and other architectural ornaments. Thenearer we are to them, and the more we have to look up, the larger willthe upper figures have to be; but if these are on the outside of abuilding that can be looked at from a long distance, then it is betternot to have too great a difference.

59For the farther we recede the more equal are the angles at which we viewthe objects at their different stages, so that in each case we may haveto deal with, we must consider the conditions attending it.

These remarks apply also to architecture in a great measure.Buildings that can only be seen from the street below, as pictures in anarrow gallery, require a different treatment from those out in theopen, that are to be looked at from a distance. In the former case thesame treatment as the Campanile at Florence is in some cases desirable,but all must depend upon the taste and judgement of the architect insuch matters. All I venture to do here is to call attention to thesubject, which seems as a rule to be ignored, or not to be considered ofimportance. Hence the many mistakes in our buildings, and theunsatisfactory and mean look of some of our public monuments.

Of Pictures that are Painted according to thePosition 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 stylein 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 havetheir horizon lines fairly high up, and are not suited to be placed anyhigher. The Giorgione in the centre, the Monna Lisa to the right, andthe Velasquez and Watteau to the left, are all pictures that fit thatposition; 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, tellsout against the sky, with its low horizon and strong contrast of lightand dark; nor does it lose a bit by being placed where it is, over thesmaller pictures.

The Rembrandt, on the opposite side, with its burgomasters in blackhats and coats and white collars, is evidently intended and painted fora raised position, and to be looked up to, which is evident from theperspective of the table. The grand Titian in60the centre, an altar-piece in one of the churches in Venice (herereversed), is also painted to suit its elevated position, with lowhorizon and figures telling boldly against the sky. Those placed lowdown are modern French pictures, with the horizon high up and almostabove their frames, but placed on the ground they fit into the generalharmony of the arrangement.

It seems to me it is well, both for those who paint and for those whohang pictures, that this subject should be taken into consideration. Forit must be seen by this illustration that a bigger style is adopted bythe artists who paint for high places in palaces or churches than bythose who produce smaller easel-pictures intended to be seen close.Unfortunately, at our picture exhibitions, we see too often that nearlyall the works, whether on large or small canvases, are painted for theline, and that those which happen to get high up look as if they weretoppling over, because they have such a high horizontal line; andinstead of the figures telling against the sky, as in this picture ofthe ‘Infant’ by Velasquez, the Reynolds, and the fat mantreading on a flag, we have fields or sea or distant landscape almost tothe top of the frame, and all, so methinks, because the perspective isnot sufficiently considered.

Note.—Whilst on this subject, I may note that thepainter in his large decorative work often had difficulties to contendwith, which arose from the form of the building or the shape of the wallon which he had to place his frescoes. Painting on the ceiling was noeasy task, and Michelangelo, in a humorous sonnet addressed to Giovannida Pistoya, gives a burlesque portrait of himself while he was paintingthe Sistine Chapel:—

“I’ho già fatto un gozzo in questostento.”

61At present that difficulty is got over by using large strong canvas, onwhich the picture can be painted in the studio and afterwards placed onthe wall.

However, the other difficulty of form has to be got over also.A great portion of the ceiling of the Sistine Chapel, and notablythe prophets and sibyls, are painted on a curved surface, in which casea similar method to that explained by Leonardo da Vinci has to beadopted.

In Chapter CCCI he shows us how to draw a figure twenty-four bracciahigh upon a wall twelve braccia high. (The braccia is 1 ft.10⅞ in.). He first draws the figure upright, then from thevarious points draws lines to a pointF on the floor of the building, marking theirintersections on the profile of the wall somewhat in the manner we haveindicated, which serve as guides in making the outline to be traced.

XXI

Interiors

To draw the interior of a cube we must suppose the side facing us tobe removed or transparent. Indeed, in all our figures which representsolids we suppose that we can see through them,63and in most cases we mark the hidden portions with dotted lines. So alsowith all those imaginary lines which conduct the eye to the variousvanishing points, and which the old writers called‘occult’.

When the cube is placed below the horizon (as inFig. 59), we see the top of it; when on the horizon, as inthe above (Fig. 69), if the side facing us is removed we see both topand bottom of it, or if a room, we see floor and ceiling, but otherwisewe should see but one side (that facing us), or at most two sides. Whenthe cube is above the horizon we see underneath it.

We shall find this simple cube of great use to us in architecturalsubjects, such as towers, houses, roofs, interiors of rooms, &c.

In this little picture by de Hoogh we have the application of theperspective of the cube and other foregoing problems.

XXII

The Square at an Angle of 45°

When the square is at an angle of 45° to the base line, then itssides are drawn respectively to the points of distance,DD, and one of its diagonals which is at rightangles to the base is drawn to the point of sightS, and the otherab, is parallel to that baseor ground line.

To draw a pavement with its squares at this angle is but anamplification of the above figure. Mark off on base equal distances, 1,2, 3, &c., representing the diagonals of required squares, and fromeach of these points draw lines to points of distanceDD´. These lines will intersect each other, and soform the squares of the pavement; to ensure correctness, lines shouldalso be drawn from these points 1, 2, 3, to the point of sightS, and also horizontals parallel to thebase, asab.

XXIII

The Cube at an Angle of 45°

Having drawn the square at an angle of 45°, as shown in the previousfigure, we find the length of one of its sides,dh, by drawing aline,SK, throughh, one of itsextremities, till it cuts the base line atK. Then, with the other extremityd forcentre anddK for radius,describe a quarter of a circleKm; the chord thereofmK will be the geometrical length ofdh. Atd raise verticaldCequal tomK, which gives us theheight of the cube, then raise verticals ata,h, &c.,their height being found by drawingCDandCD´ to the two points of distance,and so completing the figure.

XXIV

Pavements Drawn by Means of Squares at 45°

The square at 45° will be found of great use in drawing pavements,roofs, ceilings, &c. In Figs. 73, 74 it is shown how67having set out one square it can be divided into four or more equalsquares, and any figure or tile drawn therein. Begin by making ageometrical 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 formas at Fig. 75.

XXV

The Perspective Vanishing Scale

The vanishing scale, which we shall find of infinite use in ourperspective, is founded on the facts explained in Rule 10. We there findthat all horizontals in the same plane, which are drawn to the samepoint on the horizon, are perspectively parallel to each other, so thatif we measure a certain height or width on the picture plane, and thenfrom each extremity draw lines to any convenient point on the horizon,then all the perpendiculars drawn between these lines will beperspectively equal, however much they may appear to vary in length.

Let us suppose that in this figure (76)AB andA·B· eachrepresent 5 feet. Then in the first case all the verticals, ase,f,g,h, drawn between AO and BOrepresent 5 feet, and in the second case all the horizontalse,f,g,h, drawn between A·O and B·O alsorepresent 5 feet each. So that by the aid of this scale we can givethe 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 increaseor diminish our measurements atAB andA·B· to whatever length werequire.

As it may not be quite evident at first that the points O may betaken at random, the following figure will prove it.

XXVI

The Vanishing Scale can be Drawn to any Point onthe Horizon

FromAB (Fig. 77) drawAO,BO, thusforming the scale, raise verticalC.Now form a second scale fromAB bydrawingAO·BO·, and therein raise verticalD at an equal distance from the base. First, then,verticalC equalsAB, and secondly verticalD equalsAB,thereforeC equalsD, so that either of these scales will measure agiven height at a given distance.

XXVII

Application of Vanishing Scales to DrawingFigures

In this figure we have marked off on a level plain three or fourpointsa,b,c,d, to indicate the placeswhere we wish to stand our figures.ABrepresents their average height, so we have made our scaleAO, BO, accordingly. From each point marked we drawa line parallel to the base till it reaches the scale. From the pointwhere it touches the lineAO, raiseperpendicular asa, which gives the height required at thatdistance, and must be referred back to the figure itself.

XXVIII

How to Determine the Heights of Figures on aLevel Plane

First Case.

This is but a repetition of the previous figure, excepting that wehave substituted these schoolgirls for the vertical lines. If we wish tomake 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 thatwe see over the heads of the figures, those nearest to us being thelowest down. That is to say, we are looking on this scene from aslightly raised platform.

Second Case.

To draw figures at different distances when their heads are above thehorizon, or as they would appear to a person sitting on a low seat. Theheight 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 thehorizon, or as they appear to a person standing on the same level orwalking 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 forthe purpose of ascertaining or marking their distances from the baseline, and their respective heights, which of course vary; so in allcases allowance must be made for some being taller and some shorter thanthe scale measurement.

XXIX

The Horizon above the Figures

In this example from De Hoogh the doorway to the left is higher upthan the figure of the lady, and the effect seems to me73more pleasing and natural for this kind of domestic subject. Thisdelightful painter was not only a master of colour, of sunlight effect,and perfect composition, but also of perspective, and thoroughlyunderstood the charm it gives to a picture, when cunningly introduced,for he makes the spectator feel that he74can walk along his passages and courtyards. Note that he frequently putsthe point of sight quite at the side of his canvas, as at S, which givesalmost the effect of angular perspective whilst it preserves theflatness and simplicity of parallel or horizontal perspective.

XXX

Landscape Perspective

In an extended view or landscape seen from a height, we have toconsider the perspective plane as in a great measure lying above it,reaching from the base of the picture to the horizon; but of coursepierced here and there by trees, mountains, buildings, &c. As a rulein such cases, we copy our perspective from nature, and do not troubleourselves much about mathematical rules. It is as well, however, to knowthem, so that we may feel sure we are right, as this gives certainty toour touch and enables us to work with freedom. Nor must we, whenpainting from nature, forget to take into account the effects ofatmosphere and the various tones of the different planes of distance,for this makes much of the difference between a good picture and a badone; being a more subtle quality, it requires a keener artistic sense todiscover and depict it. (SeeFigs. 95 and103.)

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 scaleaccordingly. First ascertain the exact height of each piece, asQ, K, B, which represent the queen, king,bishop, &c. Refer these dimensions to the scale, as shown at QKB,which will give us the perspective measurement of each piece accordingto the square on which it is placed.

76This is shown in the above drawing (Fig. 83) in the case of the whitequeen and the black queen, &c. The castle, the knight, and the pawnbeing about the same height are measured from the fourth line of thescale markedC.

XXXII

Application of the Vanishing Scale to DrawingFigures at an Angle when their Vanishing Points are Inaccessible orOutside the Picture

This is exemplified in the drawing of a fence (Fig. 84). Form scaleaS,bS, in accordance with the height of the fence orwall to be depicted. Letao represent the direction or angle atwhich it is placed, drawod to meet the scale atd, atd raise verticaldc, which gives the height of the fenceatoo·. Draw linesbo·,eo,ao, &c., andit will be found that all these lines if produced will meet at the samepoint on the horizon. To divide the fence into spaces, divide base lineaf as required and proceed as already shown.

XXXIII

The Reduced Distance. How to Proceed when thePoint of Distance is Inaccessible

It has already been shown that too near a point of distance isobjectionable on account of the distortion and disproportion resultingfrom it. At the same time, the long distance-point must be some way outof the picture and therefore inconvenient. The object of the reduceddistance is to bring that point within the picture.

In Fig. 85 we have made the distance nearly twice the length of thebase of the picture, and consequently a long way out of it. DrawSa,Sb, and froma drawaD to point of distance, which cutsSb ato, and determines thedepth of the squareacob. But78we can find that same point if we take half the base and draw a linefrom ½ base to ½ distance. But even this ½ distance-point does not comeinside the picture, so we take a fourth of the base and a fourth of thedistance and draw a line from ¼ base to ¼ distance. We shall find thatit passes precisely through the same pointo as the other linesaD, &c. We are thus able tofind the required pointo without going outside the picture.

Of course we could in the same way take an 8th or even a 16thdistance, but the great use of this reduced distance, in addition to theabove, is that it enables us to measure any depth into the picture withthe greatest ease.

It will be seen in the next figure that without having to extend thebase, as is usually done, we can multiply that base to any amount bymaking use of these reduced distances on the horizontal line. This isquite a new method of proceeding, and it will be seen is mathematicallycorrect.

XXXIV

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

In Fig. 86 we have divided the base of the first square into fourequal parts, which may represent so many feet, so thatA4 andBdbeing the retreating sides of the square each represents 4 feet.But we found point ¼D by drawing 3Dfrom ¼ base to ¼ distance, and by proceeding in the same way from eachdivision,79A, 1, 2, 3, we mark off onSB four spaces each equal to4 feet, in all 16 feet, so that by taking the whole base and the ¼distance we find pointO, which isdistant four times the length of the baseAB. We can multiply this distance to any amount bydrawing other diagonals to 8th distance, &c. The same rule appliesto this corridor (Fig. 87 and Fig. 88).

XXXV

How to Form a Vanishing Scale that shall givethe 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 makeSB, the lower linethereof, a measuring line for distances. Let us first of all dividethe baseAB into eight parts, eachpart representing 5 feet. From each division draw lines to 8thdistance; by their intersections withSB we obtain80measurements of 40, 80, 120, 160, &c., feet. Now divide the side ofthe pictureBE in the same manner asthe base, which gives us the height of 40 feet. From the sideBE draw lines 5S, 15S, &c.,to point of sight, and from each division on the base line also drawlines 5S, 10S, 15S, &c.,to point of sight, and from each division onSB, such as 40, 80, &c., draw horizontalsparallel to base. We thus obtain squares 40 feet wide, beginning at baseAB and reaching as far as required.Note how the height of the flagstaff, which is 140 feet high and 280feet distant, is obtained. So also any buildings or other objects can bemeasured, 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., andgiving them their right width and height is by means of squares of agiven size, drawn on the ground.

In the above sketch (Fig. 90) the squares on the ground84represent 3 feet each way, or one square yard. Taking this as ourstandard measure, we find the door on the left is 10 feet high, that thearchway 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 somewhatsimilar subject toFig. 84, but theirregularity and freedom of the perspective gives it a charm far beyondthe rigid precision of the other, while it conforms to its main laws.This sketch, however, is the real artist's perspective, or what we mightterm natural perspective.

XXXVII

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

In the drawing of Honfleur (Fig. 92) we divide the baseAB as85in the previous figure, but the spaces measure 5 feet instead of3 feet: so that taking the 8th distance, the divisions on thevanishing lineBS measure 40 feeteach, and at pointO we have 400 feetof distance, but we require 800. So we again reduce the distance to a16th. We thus multiply the base by 16. Now let us take a base of 50 feetatf and draw linefD to16th distance; if we multiply 50 feet by 16 we obtain the 800 feetrequired.

The height of the lighthouse is found by means of the vanishingscale, which is 15 feet below and 15 feet above the horizon, or 30 feetfrom the sea-level. AtL we raise averticalLM, which shows the positionof the lighthouse. Then on that vertical measure the height required asshown in the figure.

The 800 feet could be obtained at once by drawing linefD, or 50 feet, to 16th distance. The othermeasurements obtained by 8th distance serve for nearer buildings.

XXXVIII

How to Measure Long Distances such as a Mile orUpwards

The wonderful effect of distance in Turner's pictures is not to beachieved by mere measurement, and indeed can only be properly done bystudying Nature and drawing her perspective as she presents it to us. Atthe same time it is useful to be able to test and to set out distancesin arranging a composition. This latter, if neglected, often leads togreat difficulties and sometimes to repainting.

To show the method of measuring very long distances we have to workwith a very small scale to the foot, and in Fig. 94 I have divided thebaseAB into eleven parts, each partrepresenting 10 feet. First drawASandBS to point of sight.86FromA drawAD to ¼ distance, and we obtain at 440 on lineBS four times the length ofAB, or 110 feet × 4 = 440 feet. Again, takingthe whole base and drawing a line from S to 8th distance we obtain eighttimes 110 feet or 880 feet. If now we use the 16th distance we getsixteen times 110 feet, or 1,760 feet, one-third of a mile; by repeatingthis process, but by using the base at 1,760, which is the same lengthin perspective asAB, we obtain 3,520feet, and then again using the base at 3,520 and proceeding in the sameway we obtain 5,280 feet, or one mile to the archway. The flags showtheir heights at their respective distances from the base. By the scaleat the side of the picture,BO, we canmeasure any height above or any depth below the perspective plane.

Note.—This figure (here much reduced) should be drawnlarge by the student, so that the numbering, &c., may be made moredistinct. Indeed, many of the other figures should be copied large, andworked out with care, as lessons in perspective.

XXXIX

Further Illustration of Long Distances andExtended Views

An extended view is generally taken from an elevated position, sothat the principal part of the landscape lies beneath the perspectiveplane, as already noted, and we shall presently treat of objects andfigures on uneven ground. In the previous figure is shown how we canmeasure heights and depths to any extent. But when we turn to a drawingby Turner, such as the ‘View from Richmond Hill’, we feelthat the only way to accomplish such perspective as this, is to go anddraw it from nature, and even then to use our judgement, as he did, asto how much we may emphasize or even exaggerate certain features.

Note in this view the foreground on which the principal figures standis on a level with the perspective plane, while the river andsurrounding park and woods are hundreds of feet below us88and stretch away for miles into the distance. The contrasts obtained bythis arrangement increase the illusion of space, and the figures in theforeground give as it were a standard of measurement, and by theircontrast to the size of the trees show us how far away those treesare.

How to Ascertain the Relative Heights of Figureson an Inclined Plane

The three figures to the right markedf,g,b(Fig. 96) are on level ground, and we measure them by the vanishingscaleaS,bS. Those to the left, which are repetitions of them,are on an inclined plane, the vanishing point of which isS·; by the side of this plane we have placed anothervanishing scalea·S·,b·S·, by which we measure thefigures on that incline in the same way as on the level plane. It willbe seen that if a horizontal line is drawn from the foot of one of thesefigures, sayG, to pointO on the edge of the incline, then droppedvertically too·, then again carried on too·· where theother figureg is, we find it is the same height and also thatthe other vanishing scale is the same width at that distance, so that wecan work from either one or the other. In the event of the rising groundbeing uneven we can make use of the scale on the level plane.

XLI

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

LetP be the given figure. FormscaleACS,S being the point of sight andD the distance. Draw horizontaldo throughP. FromA draw diagonalADto distance point, cuttingdo ino, througho drawSB to base, and we now have a squareAdoB on the perspective plane; and as figureP is standing on the far side of that squareit must be the distanceAB, which isone side of it, from the base line—or picture plane. For figuresvery far away it might be necessary to make use of half-distance.

XLII

How to Measure the Height of Figures on UnevenGround

In previous problems we have drawn figures on level planes, which iseasy enough. We have now to represent some above and some below theperspective plane.

91Form scalebS,cS; mark off distances 20 feet, 40 feet,&c. Suppose figureK to be 60 feetoff. From point at his feet draw horizontal to meet verticalOn, which is 60 feet distant. At the pointm where this line meets the vertical, measure heightmnequal to width of scale at that distance, transfer this toK, and you have the required height of the figure inblack.

For the figures under the cliff 20 feet below the perspective plane,form scaleFS,GS, making it the same width as the other, namely5 feet, and proceed in the usual way to find the height of thefigures on the sands, which are here supposed to be nearly on a levelwith the sea, of course making allowance for different heights andvarious other things.

XLIII

Further Illustration of the Size of Figures atDifferent Distances and on Uneven Ground

92Letab be the height of a figure, say 6 feet. First formscaleaS,bS, the lower line of which,aS, is on a level with the base or on the perspectiveplane. The figure markedC is close tobase, the group of three is farther off (24 feet), and 6 feethigher up, so we measure the height on the vanishing scale and alsoabove it. The two girls carrying fish are still farther off, and about12 feet below. To tell how far a figure is away, refer its measurementsto the vanishing scale (seeFig. 96).

XLIV

Figures on a Descending Plane

In this case (Fig. 100) the same rule applies as in the previousproblem, but as the road on the left is going down hill, the vanishingpoint of the inclined plane is below the horizon at pointS·;AS,BS is the vanishing scale on the levelplane; andA·S·,B·S·, that on the incline.

Note the wall to the left markedWand the manner in which it appears to drop at certain intervals, itsbase corresponding with the inclined plane, but the upper lines of eachdivision being made level are drawn to the point of sight, or to theirvanishing point on the horizon; it is important to observe this, as itaids greatly in drawing a road going down hill.

XLV

Further Illustration of the DescendingPlane

In the centre of this picture (Fig. 102) we suppose the road to bedescending till it reaches a tunnel which goes under a road or leads toa river (like one leading out of the Strand near Somerset House). It isdrawn on the same principle as the foregoing figure. Of course to seethe road the spectator must get pretty near to it, otherwise it will beout of sight. Also a level plane must be shown, as by its contrast tothe 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 roadthat descends about 45 feet.

96In drawing a landscape such as Fig. 103 we have to bear in mind theheight of the horizon, which being exactly opposite the eye, shows us atonce which objects are below and which are above us, and to draw themaccordingly, especially roofs, buildings, walls, hedges, &c.; alsoit 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

LetK represent a frame placedvertically and at a given distance in front of us. If stood on theground our foreground will touch97the base line of the picture, and we can fix up a standard ofmeasurement both on the base and on the side as in this sketch, taking6 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 theterrace, window-sill, &c., would not come into the picture, and wecould not see the near figures atA,and the nearest to come into view would be those atB, so that a view from a window, &c., would beas it were without a foreground. Note that the figures atB would be (according to this sketch) 30 feet fromthe picture plane and about 18 feet below the base line.

BOOK THIRD

XLIX

Angular Perspective

Hitherto we have spoken only of parallel perspective, which iscomparatively easy, and in our first figure we placed the cube with oneof its sides either touching or parallel to the transparent plane. Wenow place it so that one angle only (ab), touches thepicture.

Its sides are no longer drawn to the point of sight as inFig. 7, nor its diagonal to the point of distance, butto some other points on the horizon, although the same rule holds goodas regards their parallelism; as for instance, in the case ofbcandad, which, if produced, would meet atV, a point on the horizon called a99vanishing point. In this figure only one vanishing point is seen, whichis to the right of the point of sightS, whilst the other is some distance to the left,and outside the picture. If the cube is correctly drawn, it will befound that the linesae,bg, &c., if produced, willmeet on the horizon at this other vanishing point. This far-awayvanishing point is one of the inconveniences of oblique or angularperspective, and therefore it will be a considerable gain to thedraughtsman if we can dispense with it. This can be easily done, as inthe above figure, and here our geometry will come to our assistance, asI shall show presently.

How to put a Given Point into Perspective

Let us place the given pointP on ageometrical plane, to show how far it is from the base line, and indeedin the exact position we wish it to be in the picture. The geometricalplane is supposed to face us, to hang down, as it were, from the baselineAB, like the side of a table, thetop of which represents the perspective plane. It is to that perspectiveplane that we now have to transfer the pointP.

FromP raise perpendicularPm till it touches the base line atm. With centrem and radiusmP describe arcPn so thatmn is now the same lengthasmP. As pointP is opposite pointm, so100must it be in the perspective, therefore we draw a line at right anglesto the base, that is to the point of sight, and somewhere on this linewill be found the required pointP·.We now have to find how far fromm must that point be. It must bethe length ofmn, which is the same asmP. We therefore fromn drawnD to the point of distance, which being atan angle of 45°, or half a right angle, makesmP· the perspective length ofmn by itsintersection withmS, and thusgives us the pointP·, which is theperspective of the original point.

A Perspective Point being given, Find itsPosition on the Geometrical Plane

To do this we simply reverse the foregoing problem. Thus letP be the given perspective point. From pointof sightS draw a line throughP till it cutsAB atm. From distanceD draw another line throughP till it cuts the base atn. Frommdrop perpendicular, and then with centrem and radiusmndescribe arc, and where it cuts that perpendicular is the required pointP·. We often have to make use of thisproblem.

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 thetwo points represent the two extremities of the line. Thus we have tofind the perspective ofA andB, namelya·b·. Join those points,and we have the line required.

If one end touches the base, as atA (Fig. 110), then we have102but to find one point, namelyb. We also find the perspective ofthe anglemAB, namely theshaded trianglemAb.Note also that the perspective triangle equals the geometricaltriangle.

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

LIII

To Find the Length of a Given PerspectiveLine

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

This is simply the reverse of the previous problem. LetAB be the given line. From distanceD throughA drawDC, and fromS, point of sight, throughA drawSO. DropOP at right angles to base, making itequal toOC. JoinPB, and linePB isthe actual length ofAB.

103This problem is useful in finding the position of any given line orpoint on the perspective plane.

LIV

To Find these Points when the Distance-Point isInaccessible

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

104Froma, half ofmP·,draw quadrantab, fromb (half base), draw line fromb to half Dist., which intersectsSm atP,precisely the same point as would be obtained by using the wholedistance.

How to put a Given Triangle or other RectilinealFigure into Perspective

Here we simply put three points into perspective to obtain the giventriangleA, or five points to obtainthe five-sided figure atB. So can wedeal with any number of figures placed at any angle.

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

It is to be noted that the figures appear reversed in theperspective. That is, in the geometrical triangle the base atabis uppermost, whereas in the perspectiveab is lowermost, yetboth are nearest to the ground line.

LVI

How to put a Given Square into AngularPerspective

LetABCD (Fig. 115) be the givensquare on the geometrical plane, where we can place it as near or as farfrom the base and at any angle that we wish. We then proceed to find itsperspective on the picture by finding the perspective of the four pointsABCD as already shown. Note that thetwo sides of the perspective squaredc andab beingproduced, meet at pointV on thehorizon, which is their vanishing point, but to find the point on thehorizon where sidesbc andad meet, we should have to go along way to the left of the figure, which by this method is notnecessary.

LVII

Of Measuring Points

We now have to find certain points by which to measure thosevanishing or retreating lines which are no longer at right angles to thepicture plane, as in parallel perspective, and have to be measured in adifferent way, and here geometry comes to our assistance.

Note that the perspective squarePequals the geometrical squareK, sothat sideAB of the one equals sideab of the other. With centreAand radiusAB describe arcBm· till it cuts the base line atm·. NowAB =Am·, and if we joinbm· then triangleBAm· is an isosceles triangle.So likewise if we joinm·b in the perspective figure willm·Ab be the sameisosceles triangle in perspective. Continue linem·b till it cutsthe horizon inm, which point will be the measuring point for thevanishing lineAbV. For if in an isosceles triangle we draw linesacross it, parallel to its base from one side to the other, we divideboth sides in exactly the same quantities and proportions, so that if wemeasure on the base line of the picture the spaces we require, such as1, 2, 3, on the lengthAm·, and then from these divisions draw linesto107the measuring point, these lines will intersect the vanishing lineAbVin the lengths and proportions required. To find a measuring point forthe lines that go to the other vanishing point, we proceed in the sameway. Of course great accuracy is necessary.

Note that the dotted lines 1,1, 2,2, &c., are parallel in theperspective, as in the geometrical figure. In the former the lines aredrawn to the same pointm on the horizon.

LVIII

How to Divide any Given Straight Line into Equalor Proportionate Parts

LetAB (Fig. 117) be the givenstraight line that we wish to divide into five equal parts. DrawAC at any convenient angle, and measure offfive equal parts with the compasses thereon, as 1, 2, 3, 4, 5. From5C draw line to 5B. Now from each division onAC draw lines 4, 4, 3, 3, &c.,parallel to 5,5. ThenAB will bedivided into the required number of equal parts.

LIX

How to Divide a Diagonal Vanishing Line into anyNumber of Equal or Proportional Parts

In a previous figure (Fig. 116) we have shownhow to find a measuring point when the exact measure of a vanishing lineis required, but if it suffices merely to divide a line into a givennumber of equal parts, then the following simple method can beadopted.

108We wish to divideab into five equal parts. Froma,measure off on the ground line the five equal spaces required.From 5, the point to which these measures extend (as they are takenat random), draw a line throughb till it cuts the horizon atO. Then proceed to draw lines fromeach division on the base to pointO,and they will intersect and divideab into the required number ofequal parts.

The same method applies to a given line to be divided into variousproportions, as shown in this lower figure.

Further Use of the Measuring Point O

One square in oblique or angular perspective being given, draw anynumber of other squares equal to it by means of this pointO and the diagonals.

LetABCD (Fig. 120) be the givensquare; produce its sidesAB,DC till they meet at pointV. FromD measureoff on base any number of equal spaces of any convenient length, as 1,2, 3, &c.; from 1, through corner of squareC, draw a line to meet the horizon atO, and fromO drawlines to the several divisions on base line. These lines will divide thevanishing lineDV into the requirednumber of parts equal toDC, the sideof the square. Produce the diagonal of the squareDB till it cuts the horizon atG. From the divisions on lineDV draw diagonals to pointG: their intersections with the other vanishing lineAV will determine the direction of thecross-lines which form the bases of other squares without the necessityof drawing them to the other vanishing point, which in this case is somedistance to the left of the picture. If we produce these cross-lines tothe horizon we shall find that they all meet at the other vanishingpoint, to which of course it is easy to draw them when that point isaccessible, as in Fig. 121; but if it is too far out of the picture,then this method enables us to do without it.

Figure 121 corroborates the above by showing the two vanishing pointsand additional squares. Note the working of the diagonals drawn to pointG , inboth figures.

LXI

Further Use of the Measuring Point O

Suppose we wish to divide the side of a building, as in Fig. 123, orto draw a balcony, a series of windows, or columns, or what not,or, in other words, any line above the horizon, asAB. Then fromA wedrawAC parallel to the horizon, andmark thereon111the required divisions 5, 10, 15, &c.: in this case twenty-five(Fig. 122). FromC draw a line throughB till it cuts the horizon atO. Then proceed to draw the other lines fromeach division toO, and thus dividethe vanishing lineAB as required.

112In this portico there are thirteen triglyphs with twelve spaces betweenthem, making twenty-five divisions. The required number of parts to drawthe columns can be obtained in the same way.

LXII

Another Method of Angular Perspective, beingthat Adopted in our Art Schools

In the previous method we have drawn our squares by means of ageometrical plan, putting each point into perspective as required, andthen by means of the perspective drawing thus obtained, finding ourvanishing and measuring points. In this method we proceed in exactly theopposite way, setting out our points first, and drawing the square (orother figure) afterwards.

Having drawn the horizontal and base lines, and fixed upon theposition of the point of sight, we next mark the position of thespectator by dropping a perpendicular,S ST, from that point of sight, making it thesame length as the distance we suppose the spectator to be from thepicture, and thus we makeST thestation-point.

113To understand this figure we must first look upon it as a ground-plan orbird’s-eye view, the lineV²V¹ or horizon line representing thepicture seen edgeways, because of course the station-point cannot be inthe picture itself, but a certain distance in front of it. The angle atST, that is the angle which decidesthe positions of the two vanishing pointsV¹,V², is always a right angle, and the tworemaining angles on that side of the line, called the directing line,are together equal to a right angle or 90°. So that in fixing upon theangle at which the square or other figure is to be placed, we say‘let it be 60° and 30°, or 70° and 20°’, &c. Havingdecided upon the station-point and the angle at which the square is tobe placed, drawTV¹ andTV², till they cut thehorizon atV¹ andV². These are the two vanishingpoints to which the sides of the figure are respectively drawn. But westill want the measuring points for these two vanishing lines. Wetherefore take first,V¹ ascentre andV¹T as radius, and describe arc of circle till it cutsthe horizon inM¹, which isthe measuring point for all lines drawn toV¹. Then with radiusV²Tdescribe arc from centreV²till it cuts the horizon inM², which is the measuring point for allvanishing lines drawn toV². We have now set out our points. Letus proceed to draw the squareAbcd. FromA, the nearest angle (in this instance touching thebase line), measure on each side of it the equal lengthsAB andAE, whichrepresent the width or side of the square. DrawEM² andBM¹ from the two measuring points, whichgive us, by their intersections with the vanishing linesAV¹ andAV², the perspective lengths of the sidesof the squareAbcd. Joinb andV¹ anddV², which intersecteach other atC, thenAdcb is the square required.

This method, which is easy when you know it, has certain drawbacks,the chief one being that if we require a long-distance point, and asmall angle, such as 10° on one side, and 80° on the other, then thesize of the diagram becomes so large that it has to be carried out onthe floor of the studio with long strings, &c., which is a veryclumsy and unscientific way of setting to work. The architects in suchcases make use of the centrolinead, a clever mechanical contrivancefor getting over the difficulty of the far-off vanishing point, but bythe method I have shown you, and shall further illustrate, you will findthat you can dispense with114all this trouble, and do all your perspective either inside the pictureor on a very small margin outside it.

Perhaps another drawback to this method is that it is notself-evident, as in the former one, and being rather difficult toexplain, the student is apt to take it on trust, and not to troubleabout the reasons for its construction: but to show that it is equallycorrect, I will draw the two methods in one figure.

LXIII

Two Methods of Angular Perspective in oneFigure

115It matters little whether the station-point is placed above or below thehorizon, as the result is the same. In Fig. 125 it is placed above, asthe lower part of the figure is occupied with the geometrical plan ofthe other method.

In each case we make the squareKthe same size and at the same angle, its near corner being atA. It must be seen that by whichever methodwe work out this perspective, the result is the same, so that both arecorrect: the great advantage of the first or geometrical system being,that we can place the square at any angle, as it is drawn withoutreference to vanishing points.

LXIV

To Draw a Cube, the Points being Given

As in a previous figure (124) we found thevarious working points of angular perspective, we need now merelytransfer them to the horizontal line in this figure, as in this casethey will answer our purpose perfectly well.

LetA be the nearest angle touchingthe base. DrawAV¹,AV². FromA, raise verticalAe, the height of the cube. FromedraweV¹,eV², from the otherangles raise verticalsbf,dh,cg, to meeteV¹,eV²,fV², &c., and the cube iscomplete.

LXV

Amplification of the Cube Applied to Drawing aCottage

Note that we have started this figure with the cubeAdhefb. We have taken three timesAB, its width, for the front of our house, andtwiceAB for the side, and have madeit two cubes high, not counting the roof. Note also the use of themeasuring-points in connexion with the measurements on the base line,and the upper measuring lineTPK.

LXVI

How to Draw an Interior at an Angle

Here we make use of the same points as in a previous figure, with theaddition of the pointG, which is thevanishing point of the diagonals of the squares on the floor.

FromA draw squareAbcd, and produce its sides in alldirections; again fromA, through theopposite angle of the squareC, draw adiagonal till it cuts the horizon atG. FromG drawdiagonals throughb andd, cutting the base ato,o, make spaceso,o, equal toAo all along the base, and from them drawdiagonals toG; through the pointswhere these diagonals intersect the vanishing lines drawn in thedirection ofAb,dc andAd,bc, draw lines tothe other vanishing pointV¹, thus completing the squares, and socover the floor with them; they will then serve to measure width ofdoor, windows, &c. Of course horizontal lines on wall 1 aredrawn toV¹, and those onwall 2 toV².

In order to see this drawing properly, the eye should be placed about3 inches from it, and opposite the point of sight; it will thenstand out like a stereoscopic picture, and appear as actual space, butotherwise the perspective seems deformed, and the118angles exaggerated. To make this drawing look right from a reasonabledistance, the point of distance should be at least twice as far off asit is here, and this would mean altering all the other points andsending them a long way out of the picture; this is why artists usethose long strings referred to above. I would however, advise themto make their perspective drawing on a small scale, and then square itup to the size of the canvas.

LXVII

How to Correct Distorted Perspective by Doublingthe Line of Distance

Here we have the same interior as the foregoing, but drawn withdouble the distance, so that the perspective is not so violent and theobjects are truer in proportion to each other.

To redraw the whole figure double the size, including thestation-point, would require a very large diagram, that we could not getinto this book without a folding plate, but it comes to the same thingif we double the distances between the various119points. Thus, if fromS toG in the small diagram is 1 inch, in thelarger one make it 2 inches. If fromS toM²is 2 inches, in the larger make it 4, and so on.

Or this form may be used: makeABtwice the length ofAC (Fig. 130), orin any other proportion required. OnAC mark the points as in the drawing you wish toenlarge. MakeAB the length that youwish to enlarge to, drawCB, and thenfrom each division onAC draw linesparallel toCB, andAB will be divided in the same proportions, as Ihave already shown (Fig. 117).

There is no doubt that it is easier to work direct from the vanishingpoints themselves, especially in complicated architectural work, but atthe same time I will now show you how we can dispense with, at allevents, one of them, and that the farthest away.

LXVIII

How to Draw a Cube on a Given Square, using onlyOne Vanishing Point

ABCD is the given square (Fig.131). AtA raise verticalAa equal to side of squareAB·, froma drawab to the vanishingpoint. RaiseBb. ProduceVD toE totouch the base line. FromE raiseverticalEF, making it equal toAa. FromF drawFV. RaiseDd andCc, their heights being determined by thelineFV. Joinda and the cubeis complete. It will be seen that the verticals raised at each corner ofthe square are equal perspectively, as they are drawn between parallelswhich start from equal heights, namely, fromEF andAato the same pointV, the vanishingpoint. Any other120line, such asOO·, can be directed tothe inaccessible vanishing point in the same way asad,&c.

Note. This is only one of many original figures and problemsin this book which have been called up by the wish to facilitate thework of the artist, and as it were by necessity.

LXIX

A Courtyard or Cloister Drawn with one VanishingPoint

In this figure I have first drawn the pavement by means of thediagonalsGA,Go,Go, &c., and the vanishing pointV, the square atA being given. FromA draw diagonal through opposite corner till it cutsthe horizon atG. From this same pointG draw121lines through the other corners of the square till they cut the groundline ato,o. Take this measurementAo and mark it along the base right and leftofA, and the lines drawn from thesepointso to pointG will givethe diagonals of all the squares on the pavement. Produce sides ofsquareA, and where these lines areintersected by the diagonalsGodraw lines from the vanishing pointVto base. These will give us the outlines of the squares lying betweenthem and also guiding points that will enable us to draw as many more aswe please. These again will give us our measurements for the widths ofthe arches, &c., or between the columns. Having fixed the height ofwall or dado, we make use ofV pointto draw the sides of the building, and by means of proportionatemeasurement complete the rest, as inFig.128.

LXX

How to Draw Lines which shall Meet at a DistantPoint, by Means of Diagonals

This is in a great measure a repetition of the foregoing figure, andtherefore needs no further explanation.

I must, however, point out the importance of the pointG. In angular perspective it in a measure takes theplace of the point of distance in parallel perspective, since it is thevanishing point of diagonals at 45° drawn between parallels such asAV,DV, drawn to a vanishing pointV. The method of dividing lineAV into a number of parts equal toAB, the side of the square, is also shown in aprevious figure (Fig. 120).

LXXI

How to Divide a Square Placed at an Angle into aGiven Number of Small Squares

ABCD is the given square, and onlyone vanishing point is accessible. Let us divide it into sixteen smallsquares. Produce sideCD to base atE. DivideEA into four equal parts. From each division drawlines to vanishing pointV. DrawdiagonalsBD andAC, and produce the latter till it cuts the horizoninG. Draw the three cross-linesthrough the intersections made by the diagonals and the lines drawn toV, and thus divide the square intosixteen.

This is to some extent the reverse of the previous problem. It alsoshows how the long vanishing point can be dispensed with, and theperspective drawing brought within the picture.

LXXII

Further Example of how to Divide a Given ObliqueSquare into a Given Number of Equal Squares, say Twenty-five

Having drawn the squareABCD, whichis enclosed, as will be seen, in a dotted square in parallelperspective, I divide the line123EA into five equal parts instead offour (Fig. 135), and have made use of the device for that purpose bymeasuring off the required number on lineEF, &c. Fig. 136 is introduced here simply toshow that the square can be divided into any number of smaller squares.Nor need the figure be necessarily a square; it is just as easy to makeit an oblong, asABEF (Fig. 136); foralthough we begin with a square we can extend it in any direction weplease, as here shown.

LXXIII

Of Parallels and Diagonals

To find the centre of a square or other rectangular figure we havebut to draw its two diagonals, and their intersection will give us thecentre of the figure (see 137A). Wedo the same with perspective figures, as atB. In Fig.C isshown how a diagonal, drawn from one angle of a squareB through the centreO of the opposite side of the square, will enable usto find a second square lying between the same parallels, then a third,a fourth, and so on. At figureKlying on the ground, I have divided the farther side of the squaremn into ¼, ⅓, ½. If I draw125a diagonal fromG (at the base)through the half of this line I cut off onFS the lengths or sides of two squares; if throughthe quarter I cut off the length of four squares on the vanishing lineFS, and so on. In Fig. 137D is shown how easily any number of objects at anyequal distances apart, such as posts, trees, columns, &c., can bedrawn by means of diagonals between parallels, guided by a central lineGS.

LXXIV

The Square, the Oblong, and their Diagonals

Having found the centre of a square or oblong, such as Figs. 138 and139, if we draw a third line through that centre at a given angle andthen at each of its extremities draw perpendicularsAB,DC, we dividethat square or oblong into three parts, the two outer portions beingequal to each other, and the centre one either126larger or smaller as desired; as, for instance, in the triumphal arch wemake the centre portion larger than the two outer sides. When certainarchitectural details and spaces are to be put into perspective,a scale such as that in Fig. 123 will be found of greatconvenience; but if only a ready division of the principal proportionsis required, then these diagonals will be found of the greatest use.

LXXV

Showing the Use of the Square and Diagonals inDrawing Doorways, Windows, and other Architectural Features

This example is from Serlio'sArchitecture (1663), showingwhat excellent proportion can be obtained by the square and diagonals.The width of the door is one-third of the base of square, the heighttwo-thirds. As a further illustration we have drawn the same figure inperspective.

LXXVI

How to Measure Depths by Diagonals

If we take any length on the base of a square, say fromA tog, and fromg raise aperpendicular till it cuts the diagonalAB inO, then fromO draw horizontalOg·, we form a squareAgOg·, and thus measure on one side of thesquare the distance or depthAg·. So can we measure any other length, suchasfg, in like manner.

To do this in perspective we pursue precisely the same method, asshown in this figure (143).

128To measure a lengthAg on theside of squareAC, we draw a line fromg to the point of sightS, andwhere it crosses diagonalAB atO we draw horizontalOg, and thus find the required depthAg in the picture.

LXXVII

How to Measure Distances by the Square andDiagonal

It may sometimes be convenient to have a ready method by which tomeasure the width and length of objects standing against the wall of agallery, without referring to distance-points, &c.

129In Fig. 144 the floor is divided into two large squares with theirdiagonals. Suppose we wish to draw a fireplace or a piece of furnitureK, we measure its baseef onAB, as far fromB as we wish it to be in the picture; draweoandfo to point of sight, and proceed as in the previous figureby drawing parallels fromOo,&c.

Let it be observed that the great advantage of this method is, thatwe can use it to measure such distant objects asXY just as easily as those near to us.

There is, however, a still further advantage arising from it,and that is that it introduces us to a new and simpler method ofperspective, to which I have already referred, and it will, I hope,be found of infinite use to the artist.

Note.—As we have founded many of these figures on agiven square in angular perspective, it is as well to have a ready andcertain means of drawing that square without the elaborate setting outof a geometrical plan, as in the first method, or the more cumbersomeand extended system of the second method. I shall therefore showyou another method equally correct, but much simpler than either, whichI have invented for our use, and which indeed forms one of the chieffeatures of this book.

LXXVIII

How by Means of the Square and Diagonal we canDetermine the Position of Points in Space

Apart from the aid that perspective affords the draughtsman, there isa further value in it, in that it teaches us almost a new science, whichwe might call the mystery of aspect, and how it is that the objectsaround us take so many different forms, or rather appearances, althoughthey themselves remain the same. And also that it enables us, with,I think, great pleasure to ourselves, to fathom space, to work outdifficult problems by simple reasoning, and to exercise those inventiveand critical faculties which give strength and enjoyment to mentallife.

130And now, after this brief excursion into philosophy, let us come down tothe simple question of the perspective of a point.

Here, for instance, are two aspects of the same thing: thegeometrical squareA, which is facingus, and the perspective squareB,which we suppose to lie flat on the table, or rather on the perspectiveplane. LineA·C· is the perspective oflineAC. On the geometrical square wecan make what measurements we please with the compasses, but on theperspective squareB· the only line wecan actually measure is the base line. In both figures this base line isthe same length. Suppose we want to find the131perspective of pointP (Fig. 146), wemake use of the diagonalCA. FromP in the geometrical square drawPO to meet the diagonal inO; throughO drawperpendicularfe; transfer lengthfB, so found, to the base of the perspective square;fromf drawfS to pointof sight; where it cuts the diagonal inO, draw horizontalOP·, which gives us the point required. In the sameway we can find the perspective of any number of points on any side ofthe square.

LXXIX

Perspective of a Point Placed in any Positionwithin the Square

Let the pointP be the one we wishto put into perspective. We have but to repeat the process of theprevious problem, making use of our measurements on the base, thediagonals, &c.

Indeed these figures are so plain and evident that furtherdescription of them is hardly necessary, so I will here give twodrawings of triangles which explain themselves. To put a triangle intoperspective we have but to find three points, such asfEP, Fig. 148A, and then transfer these points to the perspectivesquare 148B, as there shown, and formthe perspective triangle; but these figures explain themselves. Anyother triangle or rectilineal132figure can be worked out in the same way, which is not only the simplestmethod, but it carries its mathematical proof with it.

LXXX

Perspective of a Square Placed at an Angle NewMethod

As we have drawn a triangle in a square so can we draw an obliquesquare in a parallel square. In Figure 150A we have drawn the oblique squareGEPn. We find the points on the baseAm, as in the previous figures, whichenable us to construct the oblique perspective squaren·G·E·P· in the parallel perspective squareFig. 150B. But it is not necessary toconstruct the geometrical figure, as I will show presently. It is hereintroduced to explain the method.

Fig. 150B. To test the accuracy ofthe above, produce sidesG·E· andn·P· of perspective square tillthey touch the horizon, where they will meet atV, their vanishing point, and again produce theother sidesn·G· andP·E· till they meet on the horizon at theother vanishing point, which they must do if the figure is correctlydrawn.

In any parallel square construct an oblique square from134a given point—given the parallel square at Fig. 150B, and given pointn· on base. MakeA·f· equal ton·m·, drawf·S andn·S to point of sight. Where these lines cut thediagonalAC draw horizontals toP· andG·,and so find the four pointsG·E·P·n· through which to draw thesquare.

LXXXI

On a Given Line Placed at an Angle to the BaseDraw a Square in Angular Perspective, the Point of Sight, and Distance,being given.

LetAB be the given line,S the point of sight, andD the distance (Fig. 151, 1). ThroughA drawSCfrom point of sight to base (Fig. 151, 2 and 3). FromC drawCDto point of distance. DrawAoparallel to base till it cutsCD ato, througho drawSP, fromB markoffBE equal toCP. FromE drawES intersectingCD atK, fromK drawKM, thus completing the outer parallel square.ThroughF, wherePS intersectsMK,drawAV till it cuts the horizon inV, its vanishing point. FromV drawVBcutting sideKE of outer square inG, and we have the four points135AFGB, which are the four angles of thesquare required. JoinFG, and thefigure is complete.

Any other side of the square might be given, such asAF. First throughA andF drawSC,SP,then drawAo, then througho drawCD. FromC draw base of parallel squareCE, and atMthroughF drawMK cutting diagonal atK, which gives top of square. Now throughK drawSE,givingKE the remaining side thereof,produceAF toV, fromV drawVB. JoinFG,GB, andBA, and the square required is complete.

The student can try the remaining two sides, and he will find theywork out in a similar way.

LXXXII

How to Draw Solid Figures at any Angle by theNew Method

As we can draw planes by this method so can we draw solids, as shownin these figures. The heights of the corners of the triangles areobtained by means of the vanishing scalesAS,OS, which havealready been explained.

In the same manner we can draw a cubic figure (Fig. 154)—a box,for instance—at any required angle. In this case, besides thescaleAS,OS, we have made use of the vanishing linesDV,BV,136to corroborate the scale, but they can be dispensed with in these simpleobjects, or we can use a scale on each side of the figure asa·o·S, should both vanishingpoints be inaccessible. Let it be noted that in the scaleAOS,AO is madeequal toBC, the height of thebox.

By a similar process we draw these two figures, one on the square,the other on the circle.

LXXXIII

Points in Space

The chief use of these figures is to show how by means of diagonals,horizontals, and perpendiculars almost any figure in space can be setdown. Lines at any slope and at any angle can be drawn by thisdescriptive geometry.

The student can examine these figures for himself, and willunderstand their working from what has gone before. Here (Fig. 157) inthe geometrical square we have a vertical planeAabBstanding on its baseAB. We wish toplace a projection of this figure at a certain distance and at a givenangle in space. First of all we transfer it to the side of the cube,where it is seen in perspective, whilst at its side is anotherperspective square lying flat, on which we have to stand our figure. Bymeans of the diagonal of this flat square, horizontals from figure onside of cube, and lines drawn from point of sight (as alreadyexplained), we obtain the direction of base lineAB, and also by means of linesaa· andbb· we obtain the two points in spacea·b·. JoinAa·,a·b· andBb·, and we have the projection required, andwhich may be said to possess the third dimension.

In this other case (Fig. 158) we have a wedge-shaped figure standingon a triangle placed on the ground, as in the previous figure, its threecorners being the same height. In the vertical geometrical square wehave a ground-plan of the figure, from which we draw lines to diagonaland to base, and notify by numerals 1, 3,1382, 1, 3; these we transfer to base of the horizontal perspectivesquare, and then construct shaded triangle 1, 2, 3, and raiseto the height required as shown at 1·, 2·, 3·. Although we maynot want to make use of these special figures, they show us how we couldwork out almost any form or object suspended in space.

LXXXIV

The Square and Diagonal Applied to Cubes AndSolids Drawn Therein

As we have made use of the square and diagonal to draw figures atvarious angles so can we make use of cubes either in parallel or angularperspective to draw other solid figures within139them, as shown in these drawings, for this is simply an amplification ofthat method. Indeed we might invent many more such things. But subjectsfor perspective treatment will constantly present themselves to theartist or draughtsman in the course of his experience, and while Iendeavour to show him how to grapple with any new difficulty or subjectthat may arise, it is impossible to set down all of them in thisbook.

LXXXV

To Draw an Oblique Square in Another ObliqueSquare without Using Vanishing Points

It is not often that both vanishing points are inaccessible, still itis well to know how to proceed when this is the case. We first draw thesquareABCD inside the parallelsquare, as in previous figures. To draw the smaller squareK we simply draw a smaller parallel squareh h hh, and within that, guided by the intersections of the diagonalstherewith, we obtain the four points through which to draw squareK. To raise a solid figure on these squareswe can make use of the vanishing scales as140shown on each side of the figure, thus obtaining the upper square1 2 3 4, then by means of the diagonal 1 3 and 2 4and verticals raised from each corner of squareK to meet them we obtain the smaller upper squarecorresponding toK.

It might be said that all this can be done by using the two vanishingpoints in the usual way. In the first place, if they were as far off asrequired for this figure we could not get them into a page unless itwere three or four times the width of this one, and to use shorterdistances results in distortion, so that the real use of this system isthat we can make our figures look quite natural and with much lesstrouble than by the other method.

LXXXVI

Showing How a Pedestal can be Drawn by the NewMethod

This is a repetition of the previous problem, or rather theapplication of it to architecture, although when there are many detailsit may be more convenient to use vanishing points or thecentrolinead.

LXXXVII

Scale on Each Side of the Picture

As one of my objects in writing this book is to facilitate theworking of our perspective, partly for the comfort of the artist, andpartly that he may have no excuse for neglecting it, I will hereshow you how you may, by a very simple means, secure the generalcorrectness of your perspective when sketching or painting out ofdoors.

Let us take this example from a sketch made at Honfleur (Fig. 163),and in which my eye was my only guide, but it stands the test of therule. First of all note that lineHH,drawn from one side of the picture to the other, is the horizontal line;below that is a wall and a pavement markedaV, also going from one side of the picture to theother, and being lower down ata than atV it runs up as it were to meet the horizon at somedistant point. In order to form our scale I take first the length ofHa, and measure it above andbelow the horizon, along the side to our left as many times as required,in this case four or five. I now take the lengthHV on the right side of the picture and measure itabove and below the horizon, as in the other case; and then from thesedivisions obtain dotted lines crossing the picture from one side to theother which must all meet at some distant point on the horizon. Theseact as guiding lines, and are sufficient to give us the direction of anyvanishing lines going to the same point. For those that go in theopposite direction we proceed in the same way, as fromb on theright toV· on the left. They are hereput in faintly, so as not to interfere with the drawing. In the sketchof Toledo (Fig. 164) the same thing is shown by double lines on eachside to separate the two sets of lines, and to make the principle moreevident.

LXXXVIII

The Circle

If we inscribe a circle in a square we find that it touches thatsquare at four points which are in the middle of each side, as ata bc d. It will also intersect the two diagonals at the four pointso (Fig. 165). If, then, we put this square and its diagonals,&c., into perspective we shall have eight guiding points throughwhich to trace the required circle, as shown in Fig. 166, which has thesame base as Fig. 165.

LXXXIX

The Circle in Perspective a True Ellipse

Although the circle drawn through certain points must be a freehanddrawing, which requires a little practice to make it true, it issufficient for ordinary purposes and on a small scale, but to bemathematically true it must be an ellipse. We will first draw an ellipse(Fig. 167). Letee be its long, or transverse, diameter, anddb its short or conjugate diameter. Now take half of the longdiametereE, and from pointd withcE for radiusmark onee the two pointsff, which are the foci of theellipse. At each focus fix a pin, then make a loop of fine string thatdoes not stretch and of such a length that when drawn out the double146thread will reach fromf toe. Now place this doublethread round the two pins at the fociff· and distend it with thepencil point until it forms trianglefdf·, then push the pencilalong and right round the two foci, which being guided by the threadwill draw the curve, which is a true ellipse, and will pass through theeight points indicated in our first figure. This will be a sufficientproof that the circle in perspective and the ellipse are identicalcurves. We must also remember that the ellipse is an oblique projectionof a circle, or an oblique section of a cone. The difference between thetwo figures consists in their centres not being in the same place, thatof the perspective circle being atc, higher up thane thecentre of the ellipse. The latter being a geometrical figure, its longdiameter is exactly in the centre of the figure, whereas the centrec and the diameter of the perspective are at the intersection ofthe diagonals of the perspective square in which it is inscribed.

Further Illustration of the Ellipse

In order to show that the ellipse drawn by a loop as in the previousfigure is also a circle in perspective we must reconstruct around it thesquare and its eight points by means of which it was drawn in the firstinstance. We start with nothing but147the ellipse itself. We have to find the points of sight and distance,the base, &c. Let us start with baseAB, a horizontal tangent to the curve extendingbeyond it on either side. FromA andB draw two other tangents so that theyshall touch the curve at points such asTT· a little above the transverse diameter and on alevel with each other. Produce these tangents till they meet at pointS, which will be the point of sight.Through this point draw horizontal lineH. Now draw tangentCD parallel toAB.Draw diagonalAD till it cuts thehorizon at the point of distance, this will cut through diameter ofcircle at its centre, and so proceed to find the eight points throughwhich the perspective circle passes, when it will be found that they alllie on the ellipse we have drawn with the loop, showing that the twocurves are identical although their centres are distinct.

XCI

How To Draw a Circle in Perspective Without aGeometrical Plan

Divide baseAB into four equalparts. AtB drop perpendicularBn, makingBn equal toBm, or one-fourth of base. Joinmn andtransfer this measurement to each side ofd on base line; thatis, makedf anddf· equal tomn. DrawfS andf·S, and the intersections of these lines with thediagonals of square will give us the four pointso oo o.

The reason of this is thatff· is the measurement on the baseAB of another squareo o o owhich is exactly half of the outer square. For if we inscribe a circlein a square and then inscribe a second square in that circle, thissecond square will be exactly half the area of the larger one; for itsside will be equal to half the diagonal of the larger square, as can beseen by studying149the following figures. In Fig. 170, for instance, the side of smallsquareK is half the diagonal of largesquareo.

In Fig. 171,CB represents half ofdiagonalEB of the outer square inwhich the circle is inscribed. By taking a fourth150of the basemB and drawingperpendicularmh we cutCB ath in two equal parts,Ch,hB. It will be seen thathB is equal tomn, one-quarter of thediagonal, so if we measuremn on each side ofD we getff· equal toCB, or half the diagonal. By drawingff,f·f passing through the diagonals we get the four pointso o oo through which to draw the smaller square. Without referring togeometry we can see at a glance by Fig. 172, where we have simply turnedthe squareo o o o on its centre so that its angles touch thesides of the outer square, that it is exactly half of squareABEF, since each quarter of it, such asEoCo, is bisected by its diagonaloo.

XCII

How to Draw a Circle in Angular Perspective

TakemD, the fourth of thebase. Findmn as in Fig. 171, measure it on each side ofE, and so obtainEf andEf·, and proceed to drawfV,EV,f·V and the diagonals, whoseintersections with these lines will give us the eight points throughwhich to draw the circle. In fact the process is the same as in parallelperspective, only instead of making our divisions on the actual baseAD of the square, we make them onGD, the base line.

To obtain the central linehh passing throughO, we can make use of diagonals of the half squares;that is, if the other vanishing point is inaccessible, as in thiscase.

XCIII

How to Draw a Circle in Perspective moreCorrectly, by Using Sixteen Guiding Points

First draw squareABCD. FromO, the middle of the base, draw semicircleAKB, and divide it into eight equalparts. From each division raise perpendiculars to the base, such as2 O, 3 O, 5 O,&c., and from divisionsO,O,O drawlines to point of sight, and where these lines cut the diagonalsAC,DB,draw horizontals parallel to baseAB.Then through the points thus obtained draw the circle as shown in thisfigure, which also shows us how the circumference of a circle inperspective may be divided into any number of equal parts.

XCIV

How to Divide a Perspective Circle into anyNumber of Equal Parts

This is simply a repetition of the previous figure as far as itsconstruction is concerned, only in this case we have divided thesemicircle into twelve parts and the perspective into twenty-four.

154We have raised perpendiculars from the divisions on the semicircle, andproceeded as before to draw lines to the point of sight, and have thusby their intersections with the circumference already drawn inperspective divided it into the required number of equal parts, to whichfrom the centre we have drawn the radii. This will show us how to drawtraceries in Gothic windows, columns in a circle, cart-wheels,&c.

The geometrical figure (177) will explain the construction of theperspective one by showing how the divisions are obtained on the lineAB, which represents base of square,from the divisions on the semicircleAKB.

XCV

How to Draw Concentric Circles

First draw a square with its diagonals (Fig. 178), and from itscentreO inscribe a circle; in thiscircle inscribe a square, and in this again inscribe a second circle,and so on. Through their intersections with the diagonals draw lines tobase, and155number them 1, 2, 3, 4, &c.; transfer these measurementsto the base of the perspective square (Fig. 179), and proceed toconstruct the circles as before, drawing lines from each point on thebase to the point of sight, and drawing the curves through theinter-sections of these lines with the diagonals.

Should it be required to make the circles at equal distances, as forsteps for instance, then the geometrical plan should be madeaccordingly.

Or we may adopt the method shown at Fig. 180, by taking quarter baseof both outer and inner square, and finding the measurementmn oneach side ofC, &c.

XCVI

The Angle of the Diameter of the Circle inAngular and Parallel Perspective

The circle, whether in angular or parallel perspective, is always anellipse. In angular perspective the angle of the circle's diametervaries in accordance with the angle of the square in which it is placed,as in Fig. 181,cc is the diameter of the circle andeethe diameter of the ellipse. In parallel perspective the diameter of thecircle always remains horizontal, although the long diameter of theellipse varies in inclination according to the distance it is from thepoint of sight, as shown in Fig. 182, in which the third circle is muchelongated and distorted, owing to its being outside the angle ofvision.

XCVII

How to Correct Disproportion in the Width ofColumns

The disproportion in the width of columns in Fig. 183 arises from thepoint of distance being too near the point of sight, or, in other words,taking too wide an angle of vision. It will be seen that column 3is much wider than column 1.

158In our second figure (184) is shown how this defect is remedied, bydoubling the distance, or by counting the same distance as half, whichis easily effected by drawing the diagonal fromO to ½-D, insteadof fromA, as in the other figure,O being at half base. Here the squareslie much more level, and the columns are nearly the same width, showingthe advantage of a long distance.

XCVIII

How to Draw a Circle over a Circle or aCylinder

First construct square and circleABE, then draw squareCDF with its diagonals. Then find the various pointsO, and from these raise perpendicularsto meet the diagonals of the upper square at pointsP, which, with the other points will be sufficientguides to draw the circle required. This can be applied to towers,columns, &c. The size of the circles can be varied so that the upperportion of a cylinder or column shall be smaller than the lower.

XCIX

To Draw a Circle Below a Given Circle

Construct the upper square and circle as before, then by means of thevanishing scalePOV, which should bemade the depth required, drop perpendiculars from the various pointsmarkedO, obtained by the diagonals,making them the right depth by referring them to the vanishing scale, asshown in this figure. This can be used for drawing garden fountains,basins, and various architectural objects.

Application of Previous Problem

That is, to draw a circle above a circle. In Fig. 187 can be seen howby means of the vanishing scale at the side we obtain the height of theverticals 1, 2, 3, 4, &c., which determine the directionof the upper circle; and in this second figure, how we resort to thesame means to draw circular steps.

Doric Columns

It is as well for the art student to study the different orders ofarchitecture, whether architect or not, as he frequently has tointroduce them into his pictures, and at least must know theirproportions, and how columns diminish from base to capital, as shown inthis illustration.

CII

To Draw Semicircles Standing upon a Circle atany Angle

Given the circleACBH, on diagonalAB draw semicircleAKB, and on the same lineAB draw rectangleAEFB, its height being determined by radiusOK of semicircle. From centreO drawOF tocorner of rectangle. Throughf·, where that line intersects thesemicircle, drawmn parallel toAB. This will give intersectionO· on the verticalOK, through which all such horizontals asm·n·, level withmn, must pass. Now take any otherdiameter, such asGH, and thereonraise rectangleGghH, the same height as the other. The manner ofdoing this is to produce diameterGHto the horizon till it finds its vanishing point atV. FromV through163K drawhg, and throughO· drawn·m·. FromO draw the two diagonalsog andoh,intersectingm·n· atO,O, and thus we have the five pointsGOKOH through which to draw the requiredsemicircle.

CIII

A Dome Standing on a Cylinder

This figure is a combination of the two preceding it. A cylinderis first raised on the circle, and on the top of that we drawsemicircles from the different divisions on the circumference of the164upper circle. This, however, only represents a small half-globularobject. To draw the dome of a cathedral, or other building high aboveus, is another matter. From outside, where we can get to a distance, itis not difficult, but from within it will tax all our knowledge ofperspective to give it effect.

We shall go more into this subject when we come to archways andvaulted roofs, &c.

CIV

Section of a Dome or Niche

First draw outline of the nicheGFDBA (Fig. 193), then at its base draw square andcircleGOA,S being the point of sight, and divide thecircumference of the circle into the required number of parts. Then drawsemicircleFOB, and over that anothersemicircleEOC. The manner of drawingthem is shown in Fig. 192. From the divisions on the circleGOA raise verticals to semicircleFOB, which will divide it in the same way. Dividethe smaller semicircleEOC into thesame number of parts as the others,165which divisions will serve as guiding points in drawing the curves ofthe dome that are drawn towardsD, butthe shading must assist greatly in giving the effect of the recess.

166In Fig. 192 will be seen how to draw semicircles in perspective. Wefirst draw the half squares by drawing from centresO of their diameters diagonals to distance-point, asOD, which cuts the vanishing lineBS atm, and gives us the depthof the square, and in this we draw the semicircle in the usual way.

A Dome

First draw a section of the domeACEDB (Fig. 194) the shape required. DrawAB at its base andCD at some distance above it. Keeping these ascentral lines, form squares thereon by drawingSA,SB,SC,SD,&c., from point of sight, and determining their lengths by diagonalsfh,f·h· from point of distance, passing throughO. Having formed the two squares, drawperspective circles in each, and divide their circumferences into twelveor whatever number of parts are needed. To complete the figure draw fromeach division in the lower circle curves passing through thecorresponding divisions in the upper one, to the apex. But as these arefreehand lines, it requires some taste and knowledge to draw themproperly, and of course in a large drawing several more squares andcircles might be added to aid the draughtsman. The interior of the domecan be drawn in the same way.

CVI

How to Draw Columns Standing in a Circle

In Fig. 195 are sixteen cylinders or columns standing in a circle.First draw the circle on the ground, then divide it into sixteen equalparts, and let each division be the centre of the circle on which toraise the column. The question is how to make each one the right widthin accordance with its position, for it is evident that a near columnmust appear wider than the opposite one. On the right of the figure isthe vertical scaleA, which gives theheights of the columns, and at its foot is a horizontal scale, or ascale of widthsB. Now, according tothe line on which the column stands, we find its apparent width markedon the scale. Thus take the small square and circle at 15, without itscolumn, or the broken column at 16; and note that on each side of itscentreO I have measuredoa,ob, equal to spaces marked 3 on the same horizontal in the scaleB. Through these pointsa andb I have drawn lines towards point of sightS. Through their intersections with diagonale, which is directed to point of distance, draw the farther andnearer sides of the square in which to describe the circle and thecylinder or column thereon. I have made all the squares thusobtained in parallel perspective, but they do not represent the bases ofcolumns arranged in circles, which should converge towards the centre,and I believe in some cases are modified in form to suit thatdesign.

CVII

Columns and Capitals

This figure shows the application of the square and diagonal indrawing and placing columns in angular perspective.

CVIII

Method of Perspective Employed byArchitects

The architects first draw a plan and elevation of the building to beput into perspective. Having placed the plan at the required angle tothe picture plane, they fix upon the point of sight, and the distancefrom which the drawing is to be viewed. They then draw a lineSP at right angles to the picture planeVV·, which represents that distance sothatP is the station-point. The eyeis generally considered to be the station-point, but when lines aredrawn to that point from the ground-plan, the station-point171is placed on the ground, and is in fact the trace or projection exactlyunder the point at which the eye is placed. From this station-pointP, draw linesPV andPV·parallel to the two sides of the planba andad (whichwill be at right angles to each other), and produce them to the horizon,which they will touch at pointsV andV·. These points thus obtained will bethe two vanishing points.

The next operation is to draw lines from the principal points of theplan to the station-pointP, such asbP,cP,dP,&c., and where these lines intersect the picture plane (VV· here represents it as well as the horizon),drop perpendicularsb·B,aA,d·D, &c., to meet the vanishing linesAV,AV·, whichwill determine the pointsA,B,C,D, 1, 2, 3, &c., and also theperspective lengths of the sides of the figureAB,AD, and thedivisionsB, 1, 2, &c. Taking theheight of the figureAE from theelevation, we measure it onAa;as in this instanceA touches theground line, it may be used as a line of heights.

Fig. 197. A method of angularPerspective employed by architects.
[To face p. 171]

I have here placed the perspective drawing under the ground plan toshow the relation between the two, and how the perspective is workedout, but the general practice is to find the required measurements ashere shown, to mark them on a straight edge of card or paper, andtransfer them to the paper on which the drawing is to be made.

This of course is the simplest form of a plan and elevation. It iseasy to see, however, that we could set out an elaborate building in thesame way as this figure, but in that case we should not place thedrawing underneath the ground-plan, but transfer the measurements toanother sheet of paper as mentioned above.

CIX

The Octagon

To draw the geometrical figure of an octagon contained in a square,take half of the diagonal of that square as radius, and from each cornerdescribe a quarter circle. At the eight points where they touch thesides of the square, draw the eight sides of the octagon.

To put this into perspective take the base of the squareAB and thereon form the perspective squareABCD. From either extremity of that base(sayB) drop perpendicularBF, draw diagonalAF, and then fromB with radiusBO,half that diagonal, describe arcEOE.This will give us the measurementAE.MakeGB equal toAE. Then draw lines fromG andE towardsS, and by means of the diagonals findthe transverse linesKK,hh,which will give us the eight points through which to draw theoctagon.

How to Draw the Octagon in AngularPerspective

Form squareABCD (new method),produce sidesBC andAD to the horizon atV, and produceVAtoa· on base. Drop perpendicular fromB toF the samelength asa·B, and proceed asin the previous figure to find the eight points on the oblique squarethrough which to draw the octagon.

It will be seen that this operation is very much the same as inparallel perspective, only we make our measurements on the base linea·B as we cannot measure thevanishing lineBA otherwise.

CXI

How to Draw an Octagonal Figure in AngularPerspective

In this figure in angular perspective we do precisely the same thingas in the previous problem, taking our measurements on the base lineEB instead of on the vanishing lineBA. If we wish to raise a figure onthis octagon the height ofEG we formthe vanishing scaleEGO, and from theeight points on the ground draw horizontals toEO and thus find all the points that give us theperspective height of each angle of the octagonal figure.

CXII

How to Draw Concentric Octagons, withIllustration of a Well

The geometrical figure 202A showshow by means of diagonalsAC andBD and the radii 1 2 3, &c.,we can obtain smaller octagons inside the larger ones. Note how theseare carried out in the second figure (202 B), and their application to this drawing of anoctagonal well on an octagonal base.

CXIII

A Pavement Composed of Octagons and SmallSquares

To draw a pavement with octagonal tiles we will begin with an octagoncontained in a squareabcd. Produce diagonalac toV. This will be the vanishing point for thesides of the small squares directed towards it. The other sides aredirected to an inaccessible point out of the picture, but theirdirections are determined by the lines drawn from divisions on base toV² (see back,Fig. 133).

I have drawn the lower figure to show how the squares which containthe octagons are obtained by means of the diagonals,177BD,AC, and the central lineOV². Given the squareABCD. FromD drawdiagonal toG, then fromC through centreo drawCE, and so on all the way up the floor untilsufficient are obtained. It is easy to see how other squares on eachside of these can be produced.

CXIV

The Hexagon

The hexagon is a six-sided figure which, if inscribed in a circle,will have each of its sides equal to the radius of that circle (Fig.206). If inscribed in a rectangleABCD, that rectangle will be equal in length to twosides of the hexagon or two radii of the circle, asEF, and its width will be twice the height of anequilateral trianglemon.

To put the hexagon into perspective, draw base of quadrilateralAD, divide it into four equal parts, andfrom each division draw lines to point of sight. Fromh dropperpendicularho, and form equilateral trianglemno. Takethe heightho and measure it twice along the base fromA to 2. From 2 draw line178to point of distance, or from 1 to ½ distance, and so find lengthof sideAB equal toA2. DrawBC, andEF through centre o·, andthus we have the six points through which to draw the hexagon.

CXV

A Pavement Composed of Hexagonal Tiles

In drawing pavements, except in the cases of square tiles, it isnecessary to make a plan of the required design, as in this figurecomposed of hexagons. First set out the hexagon as atA, then draw parallels 1 1, 2 2, &c.,to mark the horizontal ends of the tiles and the intermediate linesoo. Divide the base into the required number of parts, each equalto one side of the hexagon, as 1, 2, 3, 4, &c.; from thesedraw perpendiculars as shown in the figure, and also the diagonalspassing through their intersections. Then mark with a strong line theoutlines of the hexagonals, shading some of them; but the figureexplains itself.

It is easy to put all these parallels, perpendiculars, and diagonalsinto perspective, and then to draw the hexagons.

First draw the hexagon onAD as inthe previous figure, dividing179AD into four, &c., set off rightand left spaces equal to these fourths, and from each division drawlines to point of sight. Produce sidesme,nf till theytouch the horizon in pointsV,V·; these will be the two vanishing pointsfor all the sides of the tiles that are receding from us. From eachdivision on base draw lines to each of these vanishing points, then drawparallels through their intersections as shown on the figure. Having allthese guiding lines it will not be difficult to draw as many hexagons asyou please.

Note that the vanishing points should be at equal distances fromS, also that the parallelogram inwhich each tile is contained is oblong, and not square, as alreadypointed out.

We have also made use of the triangleomn to ascertain thelength and width of that oblong. Another thing to note is that we havemade use of the half distance, which enables us to make our pavementlook flat without spreading our lines outside the picture.

CXVI

A Pavement of Hexagonal Tiles in AngularPerspective

This is more difficult than the previous figure, as we only make useof one vanishing point; but it shows how much can be done by diagonals,as nearly all this pavement is drawn by their aid. First make ageometrical planA at the anglerequired. Then draw its perspectiveK.Divide line 4b into four equal parts, and continue thesemeasurements all along the base: from each division draw lines toV, and draw the hexagonK. Having this one to start with we produce itssides right and left, but first to the left to find pointG, the vanishing point of the182diagonals. Those to the right, if produced far enough, would meet at adistant vanishing point not in the picture. But the student should studythis figure for himself, and refer back toFigs.204 and205.

CXVII

Further Illustration of the Hexagon

To draw the hexagon in perspective we must first find the rectanglein which it is inscribed, according to the view we take of it. That atA we have already drawn. We will nowwork out that atB. Divide the baseAD into four equal parts and transferthose measurements to the perspective figureC, as atAD,measuring other equal spaces along the base. To find the depthAn of the rectangle, makeDK equal to base of square. DrawKO to distance-point, cuttingDO atO, and thusfind lineLO. Draw diagonalDn, and through its intersectionswith the183lines 1, 2, 3, 4 draw lines parallel to the base, and we shallthus have the framework, as it were, by which to draw the pavement.

CXVIII

Another View of the Hexagon in AngularPerspective

Given the rectangleABCD in angularperspective, produce sideDA toE on base line. DivideEB into four equal parts, and from each divisiondraw lines to vanishing point, then by means of diagonals, &c., drawthe hexagon.

184In Fig. 214 we have first drawn a geometrical plan,G, for the sake of clearness, but the one aboveshows that this is not necessary.

To raise the hexagonal figureK wehave made use of the vanishing scaleOand the vanishing pointV. Anothermethod could be used by drawing two hexagons one over the other at therequired height.

CXIX

Application of the Hexagon to Drawing aKiosk

This figure is built up from the hexagon standing on a rectangularbase, from which we have raised verticals, &c. Note how the juttingportions of the roof are drawn fromo·. But the figure explainsitself, so there is no necessity to repeat descriptions already given inthe foregoing problems.

CXX

The Pentagon

The pentagon is a figure with five equal sides, and if inscribed in acircle will touch its circumference at five equidistant points. With anyconvenient radius describe circle. From half this radius, marked 1,draw a line to apex, marked 2. Again, with 1 as centre and 1 2as radius, describe arc 2 3. Now with 2 as centre and 2 3 asradius describe arc 3 4, which will cut the circumference atpoint 4. Then line 2 4 will be one of the sides of thepentagon, which we can measure round the circle and so produce therequired figure.

To put this pentagon into parallel perspective inscribe the circle inwhich it is drawn in a square, and from its five angles4, 2, 4, &c., drop perpendiculars to base and number themas in the figure. Then draw the perspective square (Fig. 217) andtransfer these measurements to its base. From these draw lines to pointof sight, then by their aid and the two diagonals proceed to constructthe pentagon in the same way that we did the triangles and otherfigures. Should it be required to place this187pentagon in the opposite position, then we can transfer our measurementsto the far side of the square, as in Fig. 218.

188Or if we wish to put it into angular perspective we adopt the samemethod as with the hexagon, as shown at Fig. 219.

Another way of drawing a pentagon (Fig. 220) is to draw an isoscelestriangle with an angle of 36° at its apex, and from centre of each sideof the triangle draw perpendiculars to meet ato, which will bethe centre of the circle in which it is inscribed. From this centre andwith radiusOA describe circleA 3 2, &c. Take base oftriangle 1 2, measure it round the circle, and so find the fivepoints through which to draw the pentagon. The angles at 1 2 willeach be 72°, double that atA, whichis 36°.

CXXI

The Pyramid

Nothing can be more simple than to put a pyramid into perspective.Given the base (abc), raise from its centre a perpendicular(OP) of the required height, then drawlines from the corners of that base to a pointP on the vertical line, and the thing is done. Thesepyramids can be used in drawing roofs, steeples, &c. The cone isdrawn in the same way, so also is any other figure, whether octagonal,hexangular, triangular, &c.

CXXII

The Great Pyramid

This enormous structure stands on a square base of over thirteenacres, each side of which measures, or did measure, 764 feet. Itsoriginal height was 480 feet, each side being an equilateral triangle.Let us see how we can draw this gigantic mass on our little sheet ofpaper.

In the first place, to take it all in at one view we must put it veryfar back, and in the second the horizon must be so low down that wecannot draw the square base of thirteen acres on the perspective plane,that is on the ground, so we must draw it in the air, and also to a verysmall scale.

Divide the baseAB into ten equalparts, and suppose each of these parts to measure 10 feet,S, the point of sight, is placed on the left of thepicture near the side, in order that we may get a long line of distance,S ½D; but even this line is only half the distance werequire. Let us therefore take the 16th distance, as shown in ourprevious illustration of the lighthouse (Fig. 92), which enables us tomeasure sixteen times the length of baseAB, or 1,600 feet. The baseef of the pyramidis 1,600 feet from the base line of the picture, and is, according toour 10-foot scale, 764 feet long.

The next thing to consider is the height of the pyramid. We make ascale to the right of the picture measuring 50 feet fromB to 50 at point whereBP intersects base of pyramid, raise perpendicularCG and thereon measure 480 feet. As wecannot obtain a palpable square on the ground, let us draw one 480 feetabove the ground. Frome andf raise verticalseM andfN, making them equal to perpendicularG, and draw lineMN, which will be the same length as base, or 764feet. On this line form squareMNKparallel to the perspective plane, find its centreO· by means of diagonals, andO· will be the central height of the pyramid andexactly over the centre of the base. From this pointO· draw sloping linesO·f,O·e,O·y,&c., and the figure is complete.

192Note the way in which we find the measurements on base of pyramid and onlineMN. By drawingAS andBS to pointof sight we findTe, whichmeasures 100 feet at a distance of 1,600 feet. We mark off seven ofthese lengths, and an additional 64 feet by the scale, and so obtain therequired length. The position of the third corner of the base is foundby dropping a perpendicular fromK,till it meets the lineeS.

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 angleis 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 muchhigher horizon, which immediately detracts from the impression of thebigness of the pyramid.

We proceed to make our ground-planabcd high above the horizoninstead of below it, drawing first the parallel square and then theoblique one. From all the principal points drop perpendiculars to theground and thus find the points through which to draw the base of thepyramid. Find centresOO· and decideupon the heightOP. Draw the slopinglines fromP to the corners of thebase, and the figure is complete.

CXXIV

To Divide the Sides of the PyramidHorizontally

Having raised the pyramid on a given oblique square, divide thevertical lineOP into the requirednumber of parts. From194A throughC drawAG tohorizon, which gives usG, thevanishing point of all the diagonals of squares parallel to and at thesame angle asABCD. FromG draw lines through the divisions 2, 3,&c., onOP cutting the linesPA andPC,thus dividing them into the required parts. Through the points thusfound draw fromV all those sides ofthe squares that haveV for theirvanishing point, asab,cd, &c. Then joinbd,ac, and the rest, and thus make the horizontal divisionsrequired.

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 areproduced 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). Firstdraw the cubeedcba at the required height, and on the sidefacing us,adcb, draw triangleK, which represents the end of a gable roof. Thendraw similar triangles on the other sides of the cube (seeFig. 159, LXXXIV). Join the opposite triangles196at the apex, and thus form two gable roofs crossing each other at rightangles. Fromo, centre of base of cube, raise verticalOP, and then fromP draw sloping lines to each corner of basea,b, &c., and by means of central lines drawn fromP to half base, find the points wherethe gable roofs intersect the central spire or pyramid. Any otherproportions can be obtained by adding to or altering the cube.

To draw a sloping or hip-roof which falls back at each end we mustfirst draw its base,CBDA (Fig. 230).Having found the centreO and centrallineSP, and how far the roof is tofall back at each end, namely the distancePm, draw horizontal lineRB throughm. Then fromB throughO drawdiagonalBA, and from197A draw horizontalAD, which gives us pointn. From these twopointsm andn raise perpendiculars the height requiredfor the roof, and from these draw sloping lines to the corners of thebase. Joinef, that is, draw the top line of the roof, whichcompletes it. Fig. 231 shows a plan or bird's-eye view of the roof andthe diagonalAB passing through centreO. But there are so many varieties ofroofs they would take almost a book to themselves to illustrate them,especially the cottages and farm-buildings, barns, &c., besideschurches, old mansions, and others. There is also such irregularityabout 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 tosketch them from the real whenever he has an opportunity.

CXXVI

Of Arches, Arcades, Bridges, &c.

199For an arcade or cloister (Fig. 232) first set up the outer frameABCD according to the proportions required.For round arches the height may be twice that of the base, varying toone and a half. In Gothic arches the height may be about three times thewidth, all of which proportions are chosen to suit the differentpurposes and effects required. Divide the baseAB into the desired number of parts,8, 10, 12, &c., each part representing 1 foot. (Inthis case the base is 10 feet and the horizon 5 feet.) Set outfloor by means of ¼ distance. Divide it into squares of1 foot, so that there will be 8 feet between each column orpilaster, supposing we make them to stand on a square foot. Draw thefirst archwayEKF facing us, and itsinner semicirclegh, with also its thickness or depth of1 foot. Draw the span of the archwayEF, then central linePO to point of sight. Proceed to raise as many otherarches as required at the given distances. The intersections of thecentral line with the chordsmn, &c., will give the centresfrom which to describe the semicircles.

CXXVII

Outline of an Arcade with SemicircularArches

This is to show the method of drawing a long passage, corridor, orcloister with arches and columns at equal distances, and is worked inthe same way as the previous figure, using ¼ distance and ¼ base. Thefloor consists of five squares; the semicircles of the arches aredescribed from the numbered points on the central lineOS, where it intersects the chords of thearches.

CXXVIII

Semicircular Arches on a Retreating Plane

First draw perspective squareabcd. Letae· be theheight of the figure. Drawae·f·b and proceed with the rest ofthe outline. To draw the arches begin with the one facing us,Eo·F enclosed in the quadrangleEe·f·F.With centreO describe the semicircleand across it draw the diagonalse·F,Ef·, andthroughnn, where these lines intersect the semicircle, drawhorizontalKK and alsoKS to point of sight. It will be seen that thehalf-squares at the side are the same size in perspective as the onefacing us, and we carry out in them much the same operation; that is, wedraw the diagonals, find the pointO,and the pointsnn, &c., through which to draw our arches. Seeperspective of the circle (Fig. 165).

If more points are required an additional diagonal fromO to202K 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, asin this drawing of a pointed arch (Fig. 235).

CXXIX

An Arcade in Angular Perspective

First draw the perspective squareABCD at the angle required, by new method. ProducesidesAD andBC toV. Drawdiagonal BD and produce to pointG,from whence we draw the other diagonals tocfh. Make spaces1, 2, 3, &c., on base line equal toB 1 to obtain sides of squares. Raise verticalBM the height required. ProduceDA toO onbase line, and fromO raise verticalOP equal toBM. This line enables usto dispense with the long vanishing point to the left; its working hasbeen explained at Fig. 131. FromPdrawPRV to vanishing pointV, which will intersect verticalAR atR. JoinMR, and this line, if produced, wouldmeet the horizon at the other vanishing point.203In like manner makeO2 equal toB2·. From 2 draw line toV, and at 2, its intersection withAR, draw line 2 2, which will also meet thehorizon at the other vanishing point. By means of the quarter-circleA we can obtain the points throughwhich to draw the semicircular arches in the same way as in the previousfigure.

CXXX

A Vaulted Ceiling

From the square ceilingABCD wehave, as it were, suspended two arches from the two diagonalsDB,AC,which spring from the four corners of the squareEFGH, just underneath it. The curves of thesearches, which are not semicircular but elongated, are obtained by meansof the vanishing scalesmS,nS. Take any two convenientpointsP,R, on each side of the semicircle, and204raise verticalsPm,Rn toAB,and on these verticals form the scales. WheremS andnScut the diagonalAC dropperpendiculars to meet the lower line of the scale at points 1, 2.On the other side, using the other scales, we have droppedperpendiculars in the same way from the diagonal to 3, 4. Thesepoints, together205withEOG, enable us to trace the curveE 1 2 O 3 4 G. We draw the arch under the other diagonal inprecisely the same way.

The reason for thus proceeding is that the cross arches, althoughelongated, hang from their diagonals just as the semicircular archEKF hangs fromAB, and the linesmn, touching the circle atPR, are represented by 1, 2,hanging from the diagonalAC.

206Figure 238, which is practically the same as the preceding onlydifferently shaded, is drawn in the following manner. Draw archEGF facing us, and proceed with the rest ofthe corridor, but first finding the flat ceiling above the square on thegroundABcd. Draw diagonalsac,bd, and the curves pending from them. But we no longersee the clear arch as in the other drawing, for the spaces between thecurves are filled in and arched across.

CXXXI

A Cloister, from a Photograph

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

CXXXII

The Low or Elliptical Arch

LetAB be the span of the arch andOh its height. From centreO, withOA, or half the span, for radius, describe outersemicircle. From same centre andoh for radius describe the innersemicircle. Divide outer circle into a convenient number of parts,1, 2, 3, &c., to which draw radii from centreO. From each division drop perpendiculars. Wherethe radii intersect the inner circle, as atgkmo, drawhorizontalsop,mn,kj, &c., and208through their intersections with the perpendicularsf,j,n,p, draw the curve of the flattened arch. Transfer thisto the lower figure, and proceed to draw the tunnel. Note how thevanishing scale is formed on either side by horizontalsba,fe, &c., which enable us to make the distant arches similarto the near ones.

CXXXIII

Opening or Arched Window in a Vault

First draw the vaultAEB. Tointroduce the windowK, the upper partof which follows the form of the vault, we first decide on its width,which ismn, and its height from floorBa. On lineBa at the side of the arch form scalesaa·S,bb·S, &c. Raise the semicircular archK, shown by a dotted line. The scale at the sidewill give the lengthsaa·,bb·, &c., from differentparts of this dotted arch to corresponding points in the curved archwayor window required.

Note that to obtain the width of the windowK we have used209the diagonals on the floor and widthm n on base. This method ofmeasurement is explained at Fig. 144, and is of ready application in acase of this kind.

CXXXIV

Stairs, Steps, &c.

Having decided upon the incline or angle, such asCBA, at which the steps are to be placed, and theheightBm of each step, drawmn toCB, which will give thewidth. Then measure along baseAB thiswidth equal toDB, which will givethat for all the other steps. Obtain lengthBF of steps, and drawEF parallel toCB.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 verticallineAB (this line is sometimes calledthe line of heights), and their depth is found by diagonals drawn to thepoint of distanceD. The rest of thefigure explains itself.

CXXXVI

Square Steps

Draw first stepABEF and its twodiagonals. Raise verticalAH, andmeasure thereon the required height of each step, and thus form scale.Let the second stepCD be less allround than the first byAo orBo. DrawoC till it cuts the diagonal, and proceed to draw thesecond step, guided by the diagonals and taking its height from thescale as shown. Draw the third step in the same way.

CXXXVII

To Divide an Inclined Plane into EqualParts—such as a Ladder Placed against a Wall

Divide the verticalEC into therequired number of parts, and draw lines from point of sightS through these divisions 1, 2, 3,&c., cutting the lineAC at1, 2, 3, &c. Draw parallels toAB, such asmn, fromAC toBD, whichwill 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 inclinedplane. Draw the first and second steps as in Fig. 245. Then through1, 2, draw 1V,AV toV, thevanishing point on the vertical lineSV. These two lines and the corresponding ones atBV will form a kind of vanishingscale, giving the height of each step as we ascend. It is especiallyuseful when we pass the horizontal line and we no longer see the uppersurface of the step, the scale on the right showing us how to proceed inthat case.

214In Fig. 249 we have an example of steps ascending and descending. Firstset out the ground-plan, and find its vanishing pointS (point of sight). ThroughS draw verticalBA, and makeSAequal toSB. Set out the first stepCD. DrawEA,CA,DA, andGA, for the ascending guiding lines. Complete thesteps facing us, at central lineOO.Then draw guiding lineFB for thedescending steps (see Rule 8).

CXXXIX

Steps in Angular Perspective

First draw the baseABCD (Fig. 251)at the required angle by the new method (Fig. 250). ProduceBC to the horizon, and thus find vanishing pointV. At this point raise verticalVV·. Construct215first stepAB, refer its height atB to line of heightshI on left, and thus obtain height of step atA. Draw lines fromA andF toV·. Fromn draw diagonal throughO toG. Raise vertical atO to represent the height of the next step, itsheight being determined by the scale of heights at the side. FromA andFdraw lines toV·, and also similarlines fromB, which will serve asguiding lines to determine the height of the steps at either end as weraise them to the required number.

CXL

A Step Ladder at an Angle

First draw the ground-planG at therequired angle, using vanishing and measuring points. Find the heighthH, and width at topHH·, and draw the sidesHA andH·E. NotethatAE is wider thanHH·, and also that the back legs are not at the sameangle as the front ones, and that they overlap them. FromE raise verticalEF, and divide into as many parts as you requirerounds to the ladder. From these divisions draw lines 1 1,2 2, &c., towards the other vanishing point (not in thepicture), but217having obtained their direction from the ground-plan in perspective atlineEe, you may set up asecond verticalef at any point onEe and divide it into the same number ofparts, which will be in proportion to those onEF, and you will obtain the same result by drawinglines from the divisions onEF tothose onef 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 simpleenough if we have sufficient room for our vanishing points.

CXLII

Steps and a Double Cross Drawn by Means ofDiagonals and one Vanishing Point

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

Begin by setting out the square base at the angle required. FindpointG by means of diagonals, andproduceAB toV, &c. Mark height of stepAo, and proceed to draw the steps as alreadyshown. Then by the diagonals and measurements on base draw the secondstep 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 lineK from its centre. Throughany219point on this central line, if we draw a diagonal from pointG we cut the two opposite verticals of the shaftatmn (see Fig. 255), and by means of the vanishing pointV we cut the other two verticals at theopposite corners and thus obtain the four points through which to drawthe other sides of the square, which go to the distant or inaccessiblevanishing point. It will be seen by carefully examining the figure thatby this means we are enabled to draw the double cross standing on itssteps.

CXLIII

A Staircase Leading to a Gallery

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

CXLIV

Winding Stairs in a Square Shaft

Draw squareABCD in parallelperspective. Divide each side into four, and raise verticals from eachdivision. These verticals will mark the positions of the steps on eachwall, four in number. From centreOraise verticalOP, around which thesteps are to wind. LetAF be theheight of each step. Form scaleAB,which will give the height of each step according to its position. Thusatmn we find the height at the centre of the square, so if wetransfer this measurement to the central lineOP and repeat it upwards, say to fourteen, then wehave the height of each step on the line where they all meet. Startingthen with the first on the right, draw the rectanglegD1f, the height ofAF, then draw to the central linego,f1, and 1 1, and thus complete the first step. OnDE, measure heights equal toD 1. Draw 2 2 towards central line, and2n towards point of sight till it meets the second verticalnK. Then drawn2 tocentre, and so complete the second step. From 3 draw 3a to thirdvertical, from 4 to fourth, and so on, thus obtaining the height of eachascending step on the wall to the right, completing them in the same wayas numbers 1 and 2, when we come to the sixth step, the other endof which is against the wall opposite to us. Steps 6, 7, 8, 9are all on this wall, and are therefore equal in height all along, asthey are equally distant. Step 10 is turned towards us, and abuts on thewall to our left; its measurement is taken on the scaleAB just underneath it, and on the same line to whichit is drawn. Step 11 is just over the centre of basemo, and istherefore parallel to it, and its height ismn. The widths ofsteps 12 and 13 seem gradually to increase as they come towards us, andas they rise above the horizon we begin to see underneath them. Steps13, 14, 15, 16 are against the wall on this side of the picture, whichwe may suppose has been removed to show the working of the drawing, orthey might be an open flight as we sometimes see in shops and galleries,although in that case they are generally enclosed in a cylindricalshaft.

CXLV

Winding Stairs in a Cylindrical Shaft

First draw the circular baseCD.Divide the circumference into equal parts, according to the number ofsteps in a complete round, say twelve. Form scaleASF and the larger scaleASB, on which is shown the perspective measurementsof the steps according to their positions; raise verticals such asef,Gh, &c. Fromdivisions on circumference measure out the central lineOP, as in the other figure, and find the heights ofthe steps 1, 2, 3, 4, &c., by the corresponding numbers inthe large scale to the left; then proceed in much the same way as in theprevious figure. Note the central columnOP cuts off a small portion of the steps at thatend.

226In ordinary cases only a small portion of a winding staircase isactually 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 nopicture painted on a flat surface that gives such a wonderful appearanceof truth as that painted on a cylindrical canvas, such as thosepanoramas of ‘Paris during the Siege’, exhibited some yearsago; ‘The Battle of Trafalgar’, only lately shown at Earl'sCourt; and many others. In these pictures the spectator is in the centreof a cylinder, and although he turns round to look at the scene thepoint of sight is always in front of him, or nearly so. I believeon the canvas these points are from 12 to 16 feet apart.

228The reason of this look of truth may be explained thus. If we placethree globes of equal size in a straight line, and trace their apparentwidths on to a straight transparent plane, those at the sides, asa andb, will appear much wider than the centre one atc. Whereas, if we trace them on a semicircular glass they willappear very nearly equal and, of the three, the central onecwill be rather the largest, as may be seen by this figure.

We must remember that, in the first case, when we are looking at aglobe or a circle, the visual rays form a cone, with a globe at itsbase. If these three cones are intersected by a straight glassGG, and looked at from pointS, the intersection ofC will be a circle, as the cone is cut straightacross. The other two being intersected at an angle, will each be anellipse. 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) willappear equal and perfectly round.

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

In all cases the artist must exercise his own judgement both in thearrangement of his design and the execution of the work, for there isperspective even in the touch—a painting to be looked at from adistance requires a bold and broad handling; in small cabinet picturesthat we live with in our own rooms we look for the exquisite workmanshipof the best masters.

BOOK FOURTH

CXLVII

The Perspective of Cast Shadows

There is a pretty story of two lovers which is sometimes told as theorigin of art; at all events, I may tell it here as the origin ofsciagraphy. A young shepherd was in love with the daughter of apotter, but it so happened that they had to part, and were passing theirlast evening together, when the girl, seeing the shadow of her lover'sprofile cast from a lamp on to some wet plaster or on the wall, took ametal point, perhaps some sort of iron needle, and traced the outline ofthe face she loved on to the plaster, following carefully the outline ofthe 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 hebegan to ornament his wares by similar devices, which gave themincreased value by the novelty and beauty thus imparted to them.

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

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

CXLVIII

The Two Kinds of Shadows

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

The simple principle of cast shadows is that the rays coming from thepoint of light or luminary pass over the top of the intervening objectwhich casts the shadow on to the plane of shade to meet the horizontaltrace of those rays on that plane, or the231lines of light proceed from the point of light, and the lines of theshadow 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 asthe picture, and the shadow is cast sideways.

Fig. 265 shows the same thing, but the sun being behind the232object, casts its shadow forwards. Although the lines of light areparallel, they are subject to the laws of perspective, and are thereforedrawn from their respective vanishing points.

CXLIX

Shadows Cast by the Sun

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

The sun has three principal positions in relation to the picture. Inthe first case it is supposed to be in the same plane either to theright or to the left, and in that case the shadows will be233parallel with the base of the picture. In the second position it is onthe other side of it, or facing the spectator, when the shadows ofobjects will be thrown forwards or towards him. In the third, the sun isin front of the picture, and behind the spectator, so that the shadowsare thrown in the opposite direction, or towards the horizon, theobjects themselves being in full light.

The Sun in the Same Plane as the Picture

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

When the shadow of an object such asA, Fig. 268, which would fall upon the plane, isinterrupted by another objectB, thenthe234outline of the shadow is still drawn on the plane, but being interruptedby the surfaceB atC, the shadow runs up that plane till it meets therays 1, 2, which define the shadow on planeB. This is an important point, but is quiteexplained by the figure.

Although we have said that the rays pass over the top of the objectcasting the shadow, in the case of an archway or similar figure theypass underneath it; but the same principle holds good, that is, we drawlines from the guiding points in the arch, 1, 2, 3, &c.,at the same angle of 45° to meet the traces of those rays on the planeof 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° theshadows on the horizontal plane are the same length as the height of theobjects that cast them. Here (Fig. 270), the sun still being at 45°altitude, although behind the picture, and consequently throwing theshadow ofB forwards, that shadow mustbe the same length as the height of cubeB, which will be seen is the case, for the shadowC is a square in perspective.

236To 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 theluminary.

From point of sightS (Fig. 270)drop perpendicular toT, thestation-point. FromT drawTF at 45° to meet horizon atF. With radiusFTmakeFO equal to it. ThenO is the position of the spectator. FromF raise verticalFL, and fromOdraw a line at 45° to meetFL atL, which is the luminary at an altitude of45°, and at an angle of 45° to the picture.

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

Note.—The sun being at 50° to the picture instead of45°, is nearer the point of sight; at 90° it would be exactly oppositethe spectator, and so on. Again, the elevation being less (40° insteadof 45°) the shadow is longer. Owing to the changed position of the suntwo sides of the cube throw a shadow. Note also that the outlines of theshadow, 1 2, 2 3, are drawn to the same vanishing points asthe cube itself.

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

CLII

Sun Behind the Picture, Shadows Thrown on aWall

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

AtS (point of sight) raiseverticalSF·, making it equal tofL.F· becomes the foot of the luminary, whilst theluminary itself still remains atL.

We have but to turn this page half round and look at it from theright, and we shall see thatSF·becomes as it were the horizontal line. The luminaryL is at the right side of pointS instead of the left, and the foot thereof is, asbefore, the trace of the luminary, as it is just underneath it. We shallalso see that by239proceeding as in previous figures we obtain the same results on the wallas we did on the horizontal plane. Fig.B being on the horizontal plane is treated asalready shown. The steps have their shadows partly on the wall andpartly on the horizontal plane, so that the shadows on the wall areoutlined fromF· and those on theground fromf. Note shadow of roofA, and how the line drawn fromF· throughA ismet by the line drawn from the luminaryL, at the pointP,and how the lower line of the shadow is directed to point of sightS.

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

CLIII

Sun Behind the Picture Throwing Shadow on anInclined Plane

The vanishing point of the shadows on an inclined plane is on avertical dropped from the luminary to a point (F) on a level with the vanishing point (P) of that inclined plane. ThusP is the vanishing point of the inclined planeK. Draw horizontalPF to meetfL (the line drawn from the luminary to the horizon).ThenF will be the vanishing point ofthe shadows on the inclined plane. To find the shadow ofM draw lines fromF through the241baseeg tocd. From luminaryL draw lines throughab, also tocd,where they will meet those drawn fromF. DrawCD, whichdetermines the length of the shadowegcd.

CLIV

The Sun in Front of the Picture

When the sun is in front of the picture we have exactly the oppositeeffect to that we have just been studying. The shadows, instead ofcoming towards us, are retreating from us, and the objects throwing themare in full light, consequently we have to reverse our treatment. Let ussuppose the sun to be placed242above the horizon atL·, on the rightof the picture and behind the spectator (Fig. 276). If we transport thelengthL·f· to the oppositeside and draw the vertical downwards from the horizon, as atFL, we can then suppose pointL to be exactly opposite the sun, and if we makethat the vanishing point for the sun's rays we shall find that we obtainprecisely the same result. As in Fig. 277, if we wish to find the lengthofC, which we may suppose to be theshadow ofP, we can either draw a linefromA throughO toB, or fromB throughO toA, for theresult is the same. And as we cannot make use of a point that is behindus and out of the picture, we have to resort to this very ingeniousdevice.

In Fig. 276 we draw linesL1,L2,L3from the luminary to the top of the object to meet those drawn from thefootF, namelyF1,F2,F3, in the same way as in the figures wehave already drawn.

CLV

The Shadow of an Inclined Plane

The two portions of this inclined plane which cast the shadow arefirst the sidefbd, and second the farther endabcd. Thepoints we have to find are the shadows ofa andb. FromluminaryL drawLa,Lb, and fromF, the foot, drawFc,Fd. The intersection of these lines will beata·b·. If we joinfb· anddb· we have the shadowof the sidefbd, and if we joinca· anda·b· wehave the shadow ofabcd, which together form that of thefigure.

CLVI

Shadow on a Roof or Inclined Plane

To draw the shadow of the figureMon the inclined planeK (or a chimneyon a roof). First find the vanishing pointP of the inclined plane and draw horizontalPF to meet vertical raised fromL, the luminary. ThenF will be the vanishing point of the shadow. FromL drawL1,L2,L3 to top of figureM, and from the base ofM draw 1F, 2F, 3F toF, the vanishing point of the shadow.The intersections of these lines at 1, 2, 3 onK will determine the length and form of theshadow.

CLVII

To Find the Shadow of a Projection or Balcony ona Wall

To find the shadow of the objectKon the wallW, drop verticalsOO till they meet the base lineB·B· of the wall. Then from the point of sightS draw lines throughOO, also drop verticalsDd·,Cc·, to meet these lines ind·c·; drawc·F andd·F to foot of luminary. From the pointsxxwhere these lines cut the baseB raiseperpendicularsxa·,xb·. FromD,A, andB draw lines to the luminaryL. These lines or rays intersecting the verticalsraised fromxx ata·b· will give the respective points ofthe 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 proceedas above.

CLVIII

Shadow on a Retreating Wall, Sun in Front

LetL be the luminary. RaiseverticalLF.F will be the vanishing point of the shadows on theground. DrawLf· parallel toFS. DropSf· from point of sight;f· (so found)is the vanishing point of the shadows on the wall. For shadow of roofdrawLE andf·B, giving use, the shadow ofE. JoinBe,&c., and so draw shadow of eave of roof.

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

CLIX

Shadow of an Arch, Sun in Front

Having drawn the arch, divide it into a certain number of parts, sayfive. From these divisions drop perpendiculars to base line. Fromdivisions onAB draw lines toF the foot, and from those on the semicircledraw lines toL the luminary. Theirintersections will give the points through which to draw the shadow ofthe 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 tothe base. From the foot of each draw 1S, 2S, 3S to foot of luminaryS, and from the top of each,A 1 2 3 B, draw lines toLas before. Where the former intersect the curve on the floor of theniche raise verticals to meet the latter atP 1 2 B, &c. These points will indicate about theposition of the shadow; but the niche being semicircular and domed atthe top the shadow gradually loses itself in a gradated and somewhatserpentine half-tone.

CLXI

Shadow in an Arched Doorway

252This is so similar to the last figure in many respects that I need notrepeat a description of the manner in which it is done. And surely anartist after making a few sketches from the actual thing will hardlyrequire 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, arefound by drawing lines from the seat of the luminary through the feet ofthe objects to meet lines representing rays of light drawn from theluminary itself over the tops or the corners of the objects; very muchas in the cases of sun-shadows, but with253this difference, that whereas the foot of the luminary in this lattercase is supposed to be on the horizon an infinite distance away, thefoot in the case of a lamp or candle may be on the floor or on a tableclose to us. First draw the table and chair, &c. (Fig. 287), and letL be the luminary. For objects on thetable such asK the foot will be atf on the table. For the shadows on the floor, of the chair andtable itself, we must find the foot of the luminary on the floor. DrawSo, find trace of the edge ofthe table, drop verticaloP,drawPS to point of sight, dropvertical from foot of candlestick to meetPS inF. ThenF is the foot of the luminary on thefloor. From this point draw lines through the feet or traces of objectssuch as the corners of the table, &c., to meet other lines drawnfrom 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 onwhich sun-shadows are shaped, still there are so many more laws to beconsidered in the great art of light and shade that it is better toobserve them in Nature herself or under the teaching of the real sun. Inthe 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 mereshapes of shadows of regular forms. It will be seen that the light isdispersed in all directions, and although there is a good deal ofhalf-shade there are scarcely any cast shadows except on the floor; butthe light on the white walls in the outside gallery is so reflected intothe cast shadows that they are extremely faint. The luminosity of thispart of the sketch is greatly enhanced by the contrast of the dark legsof the bench and the shadows in the roof. The warm glow of all thisportion is contrasted by the grey door and its frame.

Note that the door itself is quite luminous, and lighted up by thereflection of the sun from the tiled floor, so that the bars in theupper part throw distinct shadows, besides the mystery of colour thusintroduced. The little window to the left, though not admitting muchdirect sunlight, is evidence of the brilliant glare outside; for thereflected light is very conspicuous on the255top and on the shutters on each side; indeed they cast distinct shadowsup and down, while some clear daylight from the blue sky is reflected onthe window-sill. As to the sink, the table, the wash-tubs, &c.,although they seem in strong light and shade they really receive littleor no direct light from a single point; but from the strong reflectedlight re-reflected into them from the wall of the doorway. There aremany 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 thecharacter of reflected light, varying according to the angle andintensity of the luminary and a hundred other things. When we come tostudy light in the open air we get into another region, and have to dealwith it accordingly, and yet we shall find that our sciagraphy will be ahelp to us even in this bewilderment; for it will explain in a mannerthe innumerable shapes of sun-shadows that we observe out of doors amonghills and dales, showing up their forms and structure; its play in thewoods and gardens, and its value among buildings, showing all theirjuttings and abuttings, recesses, doorways, and all the otherarchitectural details. Nor must we forget light's most glorious displayof all on the sea and in the clouds and in the sunrises and the sunsetsdown to the still and lovely moonlight.

These sun-shadows are useful in showing us the principle of light andshade, and so also are the shadows cast by artificial light; but theyare only the beginning of that beautiful study, that exquisite art oftone orchiaro-oscuro, which is infinite in its variety, is fullof the deepest mystery, and is the true poetry of art. For this thestudent must go to Nature herself, must study her in all her moods fromearly dawn to sunset, in the twilight and when night sets in. Nomathematical rules can help him, but only the thoughtful contemplation,the silent watching, and the mental notes that he can make and commit tomemory, combining them with the sentiments to which they in turn giverise. Theplein air, or broad daylight effects, are but one itemof the great range of this ever-changing and deepeningmystery—from the hard reality to the soft blending of evening whenform almost disappears, even to the merging of the whole landscape, nay,the whole world, into a dream—which is felt256rather than seen, but possesses a charm that almost defies the pencil ofthe painter, and can only be expressed by the deep and sweet notes ofthe poet and the musician. For love and reverence are necessary toappreciate 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—electricand otherwise—to the single lamp or candle. Indeed a whole volumecould be filled with illustrations of its effects. To those who aim atproducing intense brilliancy, refusing to acknowledge any limitations totheir 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 mysteriouslighting of a Rembrandt will find, if they sit contemplating in a roomlit with one lamp only, that an endless depth of mystery surrounds them,full of dark recesses peopled by fancy and sweet thought, whilst themost beautiful gradations soften the forms without distorting them; andat the same time he can detect the laws of this science of light andshade a thousand times repeated and endless in its variety.

Note.—Fig. 288 must be looked upon as a rough sketchwhich only gives the general effect of the original drawing; to renderall the delicate tints, tones and reflections described in the textwould require a highly-finished reproduction in half-tone or incolour.

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

CLXIV

Reflection

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

Let us supposeR (Fig. 289) to bestanding on the water or reflecting plane. To find its reflection makesquare[R] equal to the original squareR. Complete the reversed cube by drawing its othersides, &c. It is evident that this lower cube is the reflection ofthe one above it, although it differs in one respect, for whereas infigureR the top of the cube is seen,in its reflection [R] it is hidden,&c. In figureA of a semicirculararch we see the258underneath portion of the arch reflected in the water, but we do not seeit in the actual object. However, these things are obvious. Note thatthe reflected line must be equal in length to the actual one, or thereflection of a square would not be a square, nor that of a semicircle asemicircle. The apparent lengthening of reflections in water is owing tothe 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 isreflected.

In this view of an arch (Fig. 290) note that the reflection isobtained by dropping perpendiculars from certain points on the arch,1, 0, 2, &c., to the surface of the reflecting plane, andthen 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 toshow that at whatever angle the visual ray strikes the reflectingsurface it is reflected from it at the same angle.

We have seen that the reflected line must be equal to the originalline, thereforemB must equalMa. They are also at rightangles toMN, the plane of reflection.We will now draw the visual ray passing fromE, the eye, toB,which is the reflection ofA; and justunderneath it passes throughMN atO, which is the point where the visualray strikes the reflecting surface. DrawOA. This line represents the ray reflected from it.We have now two triangles,OAmandOmB, which are right-angled triangles and equal,therefore anglea equals angleb. But anglebequals anglec. Therefore angleEcM equalsangleAam, and the angle atwhich the ray strikes the reflecting plane is equal to the angle atwhich it is reflected from it.

CLXVI

Reflections of Objects at DifferentDistances

In this sketch the four posts and other objects are representedstanding on a plane level or almost level with the water, in order toshow the working of our problem more clearly. It will be seen that thepostA is on the brink of thereflecting plane, and therefore is entirely reflected;B andC beingfarther back are only partially seen, whereas the reflection ofD is not seen at all. I have made allthe posts the same height, but with regard to the houses, where thelength of the vertical lines varies, we obtain their reflections bymeasuring from the pointsoo upwards and downwards as in theprevious figure.

Of course these reflections vary according to the position they areviewed from; the lower we are down, the more do we see of thereflections of distant objects, and vice versa. When the figures are ona higher plane than the water, that is, above the plane of reflection,we have to find their perspective position,261and drop a perpendicularAO (Fig. 293)till it comes in contact with the plane of reflection, which we supposeto run under the ground, then measure the same length downwards, as inthis figure of a girl on the top of the steps. Pointo marks thepoint of contact with the plane, and by measuring downwards toa·we get the length of her reflection, or as much as is seen of it. Notethe reflection of the steps and the sloping bank, and the application ofthe inclined plane ascending and descending.

CLXVII

Reflection in a Looking-glass

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

Suppose him to represent one end of a parallelograme·s·, andhis imagea·b· to represent the other. The mirrorAB is a perpendicular half-way between them, thediagonale·b· is the visual ray263passing from the eye of the spectator to the foot of his image, and isthe diagonal of a rectangle, therefore it cutsAB in the centreo, andAO representsa·b· to the spectator. This isan experiment that any one may try for himself. Perhaps the above factmay have something to do with the remarks I made about Titian at thebeginning of this chapter.

CLXVIII

The Mirror at an Angle

If an object or lineAB is inclinedat an angle of 45° to the mirrorRR,then the angleBAC will be a rightangle, and this angle is exactly divided in two by the reflecting planeRR. And whatever the angle of theobject or line makes with its reflection that angle will also be exactlydivided.

Now suppose our mirror to be standing on a horizontal plane and on apivot, so that it can be inclined either way. Whatever angle the mirroris to the plane the reflection of that plane in the mirror will be atthe same angle on the other side of it, so that if the mirrorOA (Fig. 298) is at 45° to the planeRR then the265reflection of that plane in the mirror will be 45° on the other side ofit, or at right angles, and the reflected plane will appearperpendicular, as shown in Fig. 299, where we have a front view of amirror leaning forward at an angle of 45° and reflecting the squareaob with a cube standing upon it, only in the reflection the cubeappears to be projecting from an upright plane or wall.

If we increase the angle from 45° to 60°, then the reflection of theplane and cube will lean backwards as shown in Fig. 300. If we place iton a level with the original plane, the cube will be standing uprighttwice the distance away. If the mirror is still farther tilted till itmakes an angle of 135° as atE (Fig.298), or 45° on the other side of the verticalOc, then the plane and cube would disappear,and objects exactly over that plane, such as the ceiling, would comeinto view.

In Fig. 300 the mirror is at 60° to the planemn, and theplane itself at about 15° to the planean (so that here we areusing angular perspective,V being theaccessible vanishing point). The reflection of the plane and cube isseen leaning back at an266angle of 60°. Note the way the reflection of this cube is found by thedotted lines on the plane, on the surface of the mirror, and also on thereflection.

CLXIX

The Upright Mirror at an Angle of 45° to theWall

In Fig. 301 the mirror is vertical and at an angle of 45° to the wallopposite the spectator, so that it reflects a portion of that wall asthough it were receding from us at right angles; and the wall with thepictures upon it, which appears to be facing us, in reality is on ourleft.

268An endless number of complicated problems could be invented of theinclined mirror, but they would be mere puzzles calculated rather todeter the student than to instruct him. What we chiefly have to bear inmind is the simple principle of reflections. When a mirror is verticaland placed at the end or side of a room it reflects that room and givesthe impression that we are in one double the size. If two mirrors areplaced opposite to each other at each end of a room they reflect andreflect, 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 picturein front of us, whether it is behind us, at the side, or even on theceiling. Indeed, when one goes to those old palaces and churches wherepictures are painted on the ceiling, as in the Sistine Chapel or theLouvre, or the palaces at Venice, it is not a bad plan to take a handmirror with us, so that we can see those elevated works of art incomfort.

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

The reason the mirror shows the fault of a work to which the eye hasbecome accustomed is that it doubles it. Thus if a line that should bevertical is leaning to one side, in the mirror it will lean to theother; so that if it is out of the perpendicular to the left, itsreflection will be out of the perpendicular to the right, making adouble divergence from one to the other.

CLXX

Mental Perspective

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

To explain in detail the full meaning of this suggestion is beyondthe province of this book, but it may lead the student to think thisquestion out for himself in his solitary and imaginative moments, andshould, I think, give a charm and virtue to his work which heshould endeavour to make of value, not only to his own time but to thegenerations that are to follow. Cultivate, therefore, this mentalperspective, without forgetting the solid foundation of the science Ihave endeavoured to impart to you.

Footnotes

2.There is another book calledThe Jesuit's Perspective which Ihave not yet seen, but which I hear is a fine work.

3.In a sea-view, owing to the rotundity of the earth, the real horizontalline is slightly below the sea line, which is noted inChapter I.

4.Some will tell us that Nature abhors a straight line, that all longstraight lines in space appear curved, &c., owing to certain opticalconditions; but this is not apparent in short straight lines, so if ourdrawing is small it would be wrong to curve them; if it is large, like ascene or diorama, the same optical condition which applies to the linein space would also apply to the line in the picture.

INDEX

Index citations in the original book referred to page numbers. Wherepossible, links will lead directly to a chapter header or illustration.Note that the last two entries for Toledo are figure numbers rather thanpages; these have not been corrected.

AlbertDürer,2,9.

Angles of Reflection,259.

Angular Perspective,98-123,133,170.

Ang"larPersp"ctive, New Method,133,134,135,136.

Arches, Arcades, &c.,198,200-208.

Architect's Perspective,170,171.

Art Schools Perspective,112-118,217.

Atmosphere,1,74.

Balcony, Shadow of,246.

Base or groundline,89.

Campanile Florence,5,59.

Cast Shadows,229-253.

Centre of Vision,15.

Chessboard,74.

Chinese Art,11.

Circle,145,151-156,159.

Columns,157,159,161,169,170.

Conditions of Perspective,24,25.

Cottage in Angular Perspective,116.

Cube,53,65,115,119.

Cylinder,158,159.

Cylindrical picture,227.

De Hoogh,2,62,73.

Depths, How to measure by diagonals,127,128.

Descending plane,92-95.

Diagonals,45,124,125,126.

Disproportion, How to correct,35,118,157.

Distance,16,77,78,85,87,103,128.

Distorted perspective, How to correct,118.

Dome,163-167.

Double Cross,218.

Ellipse,145,146,147.

Elliptical Arch,207.

Farningham,95.

Figures on descending plane,92,93,94,95.

Fig"res"n an inclined plane,88.

Fig"res"n a level plane,70,71,72,73,74,75.

Fig"res"n uneven ground,90,91.

Geometrical and Perspective figures contrasted,46-48.

Geom"trical plane,99.

Giovanni da Pistoya, Sonnet to, by Michelangelo,60.

Great Pyramid,190.

Hexagon,177,183,185.

Hogarth,9.

Honfleur,83,142.

Horizon,3,4,15,20,59,60.

Horizontal line,13,15.

Horizontals,30,31,36.

271

Inaccessible vanishing points,77,78,136,140-144.

Inclined plane,33,118,213,244,245.

Interiors,62,117,118,128.

Japanese Art,11.

Jesuit of Paris, Practice of Perspective by,9.

Kiosk, Application of Hexagon,185.

Kirby, Joshua, Perspective made Easy (?),9.

Ladder, Step,212,216.

Landscape Perspective,74.

Landseer, Sir Edwin,1.

Leonardo da Vinci,1,61.

Light, Observations on,253.

Light-house,84.

Long distances,85,87.

Measure distances by square and diagonal,89,128,129.

Mea"ure vanishing lines, How to,49,50.

Measuring points,106,113.

Meas"ring point O,108,109,110.

Mental Perspective,269.

Michelangelo,5,57,58,60.

Natural Perspective,12,82,95,142,144.

New Method of Angular Perspective,133,134,135,141,215,219.

Niche,164,165,250.

Oblique Square,139.

Octagon,172-175.

O, measuring point,110.

Optic Cone,20.

Parallels and Diagonals,124-128.

Paul Potter, cattle,19.

Paul Veronese,4.

Pavements,64,66,176,178,180,181,183.

Pedestal,141,161.

Pentagon,186,187,188.

Perspective, Angular,98-123.

Persp"ctive, Definitions,13-23.

Persp"ctive, Necessity of,1.

Persp"ctive, Parallel,42-97.

Persp"ctive,Rules and Conditions of,24-41.

Persp"ctive,Scientific definition of,22.

Persp"ctive, Theory of,13-24.

Persp"ctive, What is it?6-12.

Pictures painted according to positions they are to occupy,59.

Point of Distance,16-21.

Po"nt"f Sight,12,15.

Points in Space,129,137.

Portico,111.

Projection,21,137.

Pyramid,189,190,191,193-196.

Raphael,3.

Reduced distance,77,78,79,84.

Reflection,257-268.

Rembrandt,59,256.

Reynolds, Sir Joshua,9,60.

Rubens,4.

Rules of Perspective,24-41.

272

Scale on each side of Picture,141,142-144.

Sc"le Vanishing,69,71,81,84.

Serlio,5,126.

Shadows cast by sun,229-252.

Sha"owsca"st"y artificial light,252.

Sight, Point of,12,15.

Sistine Chapel,60.

Solid figures,135-140.

Square in Angular Perspective,105,106,109,112,114,121,122,123,133,134,139.

Sq"are and diagonals,125,138,139,141.

Sq"are of the hypotenuse (fig. 170),149.

Sq"are in Parallel Perspective,42,43,50,53,54.

Sq"are at 45°,64-66.

Staircase leading to a Gallery,221.

Stairs, Winding,222,225.

Station Point,13.

Steps,209-218.

Taddeo Gaddi,5.

Terms made use of,48.

Tiles,176,178,181.

Tintoretto,4.

Titian,59,262.

Toledo,96,144,259,288.

Trace and projection,21.

Transposed distance,53.

Triangles,104,106,132,135,138.

Turner,2,87.

Ubaldus, Guidus,9.

Vanishing lines,49.

Vani"hing point,119.

Vani"hingscale,68-72,74,77,79,84.

Vaulted Ceiling,203.

Velasquez,59.

Vertical plane,13.

Visual rays,20.

Winding Stairs,222-225.

Water, Reflections in,257,258,260,261.

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BOOK I
		page
The Necessity of the Study of Perspective To Painters, Sculptors, andArchitects		1
What Is Perspective?		6
The Theory of Perspective:
I.	Definitions	13
II.	The Point of Sight, the Horizon, and the Point of Distance.	15
III.	Point of Distance	16
IV.	Perspective of a Point, Visual Rays, &c.	20
V.	Trace and Projection	21
VI.	Scientific Definition of Perspective	22
Rules:
VII.	The Rules and Conditions of Perspective	24
VIII.	A Table or Index of the Rules of Perspective	40
BOOK II
The Practice of Perspective:
IX.	The Square in Parallel Perspective	42
X.	The Diagonal	43
XI.	The Square	43
XII.	Geometrical and Perspective Figures Contrasted	46
XIII.	Of Certain Terms made use of in Perspective	48
XIV.	How to Measure Vanishing or Receding Lines	49
XV.	How to Place Squares in Given Positions	50
XVI.	How to Draw Pavements, &c.	51
XVII.	Of Squares placed Vertically and at Different Heights, or the Cube inParallel Perspective	53
XVIII.	The Transposed Distance	53
XIX.	The Front View of the Square and of the Proportions of Figures atDifferent Heights	54
XX.	Of Pictures that are Painted according to the Position they are toOccupy	59
XXI.	Interiors	62
XXII.	The Square at an Angle of 45°	64
XXIII.	The Cube at an Angle of 45°	65
XXIV.	Pavements Drawn by Means of Squares at 45°	66
XXV.	The Perspective Vanishing Scale	68
viii XXVI.	The Vanishing Scale can be Drawn to any Point on the Horizon	69
XXVII.	Application of Vanishing Scales to Drawing Figures	71
XXVIII.	How to Determine the Heights of Figures on a Level Plane	71
XXIX.	The Horizon above the Figures	72
XXX.	Landscape Perspective	74
XXXI.	Figures of Different Heights. The Chessboard	74
XXXII.	Application of the Vanishing Scale to Drawing Figures at an Angle whentheir Vanishing Points are Inaccessible or Outside the Picture	77
XXXIII.	The Reduced Distance. How to Proceed when the Point of Distance isInaccessible	77
XXXIV.	How to Draw a Long Passage or Cloister by Means of the ReducedDistance	78
XXXV.	How to Form a Vanishing Scale that shall give the Height, Depth, andDistance of any Object in the Picture	79
XXXVI.	Measuring Scale on Ground	81
XXXVII.	Application of the Reduced Distance and the Vanishing Scale to Drawing aLighthouse, &c.	84
XXXVIII.	How to Measure Long Distances such as a Mile or Upwards	85
XXXIX.	Further Illustration of Long Distances and Extended Views.	87
XL.	How to Ascertain the Relative Heights of Figures on an InclinedPlane	88
XLI.	How to Find the Distance of a Given Figure or Point from the BaseLine	89
XLII.	How to Measure the Height of Figures on Uneven Ground	90
XLIII.	Further Illustration of the Size of Figures at Different Distances andon Uneven Ground	91
XLIV.	Figures on a Descending Plane	92
XLV.	Further Illustration of the Descending Plane	95
XLVI.	Further Illustration of Uneven Ground	95
XLVII.	The Picture Standing on the Ground	96
XLVIII.	The Picture on a Height	97
BOOK III
XLIX.	Angular Perspective	98
L.	How to put a Given Point into Perspective	99
LI.	A Perspective Point being given, Find its Position on the GeometricalPlane	100
ix LII.	How to put a Given Line into Perspective	101
LIII.	To Find the Length of a Given Perspective Line	102
LIV.	To Find these Points when the Distance-Point is Inaccessible	103
LV.	How to put a Given Triangle or other Rectilineal Figure intoPerspective	104
LVI.	How to put a Given Square into Angular Perspective	105
LVII.	Of Measuring Points	106
LVIII.	How to Divide any Given Straight Line into Equal or ProportionateParts	107
LIX.	How to Divide a Diagonal Vanishing Line into any Number of Equal orProportional Parts	107
LX.	Further Use of the Measuring Point O	110
LXI.	Further Use of the Measuring Point O	110
LXII.	Another Method of Angular Perspective, being that Adopted in our ArtSchools	112
LXIII.	Two Methods of Angular Perspective in one Figure	115
LXIV.	To Draw a Cube, the Points being Given	115
LXV.	Amplification of the Cube Applied to Drawing a Cottage	116
LXVI.	How to Draw an Interior at an Angle	117
LXVII.	How to Correct Distorted Perspective by Doubling the Line ofDistance	118
LXVIII.	How to Draw a Cube on a Given Square, using only One VanishingPoint	119
LXIX.	A Courtyard or Cloister Drawn with One Vanishing Point	120
LXX.	How to Draw Lines which shall Meet at a Distant Point, by Means ofDiagonals	121
LXXI.	How to Divide a Square Placed at an Angle into a Given Number of SmallSquares	122
LXXII.	Further Example of how to Divide a Given Oblique Square into a GivenNumber of Equal Squares, say Twenty-five	122
LXXIII.	Of Parallels and Diagonals	124
LXXIV.	The Square, the Oblong, and their Diagonals	125
LXXV.	Showing the Use of the Square and Diagonals in Drawing Doorways,Windows, and other Architectural Features	126
LXXVI.	How to Measure Depths by Diagonals	127
LXXVII.	How to Measure Distances by the Square and Diagonal	128
LXXVIII.	How by Means of the Square and Diagonal we can Determine the Position ofPoints in Space	129
x LXXIX.	Perspective of a Point Placed in any Position within the Square	131
LXXX.	Perspective of a Square Placed at an Angle. New Method	133
LXXXI.	On a Given Line Placed at an Angle to the Base Draw a Square in AngularPerspective, the Point of Sight, and Distance, being given	134
LXXXII.	How to Draw Solid Figures at any Angle by the New Method	135
LXXXIII.	Points in Space	137
LXXXIV.	The Square and Diagonal Applied to Cubes and Solids DrawnTherein	138
LXXXV.	To Draw an Oblique Square in Another Oblique Square without UsingVanishing-points	139
LXXXVI.	Showing how a Pedestal can be Drawn by the New Method	141
LXXXVII.	Scale on Each Side of the Picture	143
LXXXVIII.	The Circle	145
LXXXIX.	The Circle in Perspective a True Ellipse	145
XC.	Further Illustration of the Ellipse	146
XCI.	How to Draw a Circle in Perspective Without a Geometrical Plan	148
XCII.	How to Draw a Circle in Angular Perspective	151
XCIII.	How to Draw a Circle in Perspective more Correctly, by Using SixteenGuiding Points	152
XCIV.	How to Divide a Perspective Circle into any Number of EqualParts	153
XCV.	How to Draw Concentric Circles	154
XCVI.	The Angle of the Diameter of the Circle in Angular and ParallelPerspective	156
XCVII.	How to Correct Disproportion in the Width of Columns	157
XCVIII.	How to Draw a Circle over a Circle or a Cylinder	158
XCIX.	To Draw a Circle Below a Given Circle	159
C.	Application of Previous Problem	160
CI.	Doric Columns	161
CII.	To Draw Semicircles Standing upon a Circle at any Angle	162
CIII.	A Dome Standing on a Cylinder	163
CIV.	Section of a Dome or Niche	164
CV.	A Dome	167
CVI.	How to Draw Columns Standing in a Circle	169
CVII.	Columns and Capitals	170
CVIII.	Method of Perspective Employed by Architects	170
xi CIX.	The Octagon	172
CX.	How to Draw the Octagon in Angular Perspective	173
CXI.	How to Draw an Octagonal Figure in Angular Perspective	174
CXII.	How to Draw Concentric Octagons, with Illustration of a Well	174
CXIII.	A Pavement Composed of Octagons and Small Squares	176
CXIV.	The Hexagon	177
CXV.	A Pavement Composed of Hexagonal Tiles	178
CXVI.	A Pavement of Hexagonal Tiles in Angular Perspective	181
CXVII.	Further Illustration of the Hexagon	182
CXVIII.	Another View of the Hexagon in Angular Perspective	183
CXIX.	Application of the Hexagon to Drawing a Kiosk	185
CXX.	The Pentagon	186
CXXI.	The Pyramid	189
CXXII.	The Great Pyramid	191
CXXIII.	The Pyramid in Angular Perspective	193
CXXIV.	To Divide the Sides of the Pyramid Horizontally	193
CXXV.	Of Roofs	195
CXXVI.	Of Arches, Arcades, Bridges, &c.	198
CXXVII.	Outline of an Arcade with Semicircular Arches	200
CXXVIII.	Semicircular Arches on a Retreating Plane	201
CXXIX.	An Arcade in Angular Perspective	202
CXXX.	A Vaulted Ceiling	203
CXXXI.	A Cloister, from a Photograph	206
CXXXII.	The Low or Elliptical Arch	207
CXXXIII.	Opening or Arched Window in a Vault	208
CXXXIV.	Stairs, Steps, &c.	209
CXXXV.	Steps, Front View	210
CXXXVI.	Square Steps	211
CXXXVII.	To Divide an Inclined Plane into Equal Parts—such as a LadderPlaced against a Wall	212
CXXXVIII.	Steps and the Inclined Plane	213
CXXXIX.	Steps in Angular Perspective	214
CXL.	A Step Ladder at an Angle	216
CXLI.	Square Steps Placed over each other	217
CXLII.	Steps and a Double Cross Drawn by Means of Diagonals and one VanishingPoint	218
CXLIII.	A Staircase Leading to a Gallery	221
CXLIV.	Winding Stairs in a Square Shaft	222
CXLV.	Winding Stairs in a Cylindrical Shaft	225
CXLVI.	Of the Cylindrical Picture or Diorama	227
xii	BOOK IV
CXLVII.	The Perspective of Cast Shadows	229
CXLVIII.	The Two Kinds of Shadows	230
CXLIX.	Shadows Cast by the Sun	232
CL.	The Sun in the Same Plane as the Picture	233
CLI.	The Sun Behind the Picture	234
CLII.	Sun Behind the Picture, Shadows Thrown on a Wall	238
CLIII.	Sun Behind the Picture Throwing Shadow on an Inclined Plane	240
CLIV.	The Sun in Front of the Picture	241
CLV.	The Shadow of an Inclined Plane	244
CLVI.	Shadow on a Roof or Inclined Plane	245
CLVII.	To Find the Shadow of a Projection or Balcony on a Wall	246
CLVIII.	Shadow on a Retreating Wall, Sun in Front	247
CLIX.	Shadow of an Arch, Sun in Front	249
CLX.	Shadow in a Niche or Recess	250
CLXI.	Shadow in an Arched Doorway	251
CLXII.	Shadows Produced by Artificial Light	252
CLXIII.	Some Observations on Real Light and Shade	253
CLXIV.	Reflection	257
CLXV.	Angles of Reflection	259
CLXVI.	Reflections of Objects at Different Distances	260
CLXVII.	Reflection in a Looking-glass	262
CLXVIII.	The Mirror at an Angle	264
CLXIX.	The Upright Mirror at an Angle of 45° to the Wall	266
CLXX.	Mental Perspective	269
	Index	270


Fig. 51 A. The geometrical view.	Fig. 51 B. The perspective view.
47
Fig. 51 C. A geometrical square.	Fig. 51 D. A perspective square.

Fig. 51 E. Geometrical parallels.	Fig. 51 F. Perspective parallels.

Fig. 51 G. Geometricalperpendicular.	Fig. 51 H. Perspectiveperpendicular.

Fig. 51 I. Geometrical equallines.	Fig. 51 J. Perspective equallines.
48
Fig. 51 K. A geometrical circle.	Fig. 51 L. A perspective circle.

The picture at two different distances from the point.

Fig. 38.	Fig. 39.


Fig. 14.	Fig. 15.


Fig. 46.	Fig. 47.


Fig. 61.	Fig. 62.


Fig. 87.	Fig. 88.

Movatterモバイル変換

The Project Gutenberg eBook ofThe Theory and Practice of Perspective

HENRY FROWDE, M.A.

PUBLISHER TO THE UNIVERSITY OF OXFORDLONDON, EDINBURGH, NEW YORKTORONTO AND MELBOURNE

THE

THEORY AND PRACTICEOF PERSPECTIVE

BY

G. A. STOREY, A.R.A.

TEACHER OF PERSPECTIVE AT THE ROYAL ACADEMY

OXFORDAT THE CLARENDON PRESS

1910

OXFORD

PRINTED AT THE CLARENDON PRESSBY HORACE HART, M.A.PRINTER TO THE UNIVERSITY

DEDICATED

TO

SIR EDWARD J. POYNTER

BARONET

PRESIDENT OF THE ROYAL ACADEMY

IN TOKEN OF FRIENDSHIP

AND REGARD

PREFACE

CONTENTS

BOOK FIRST

Definitions

The Point of Sight, the Horizon, and the Pointof Distance

Point of Distance

Perspective of a Point, Visual Rays,&c.

Trace and Projection

Scientific Definition of Perspective

The Rules and Conditions of Perspective

A Table or Index of the Rules ofPerspective

Rule 1

Rule 2

Rule 3

Rule 4

Rule 5

Rule 6

Rule 7

Rule 8

Rule 9

Rule 10

BOOK SECOND

The Square in Parallel Perspective

The Diagonal

The Square

Geometrical and Perspective FiguresContrasted

Of Certain Terms made use of in Perspective

How to Measure Vanishing or Receding Lines

How to Place Squares in Given Positions

How To Draw Pavements, &c.

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

The Transposed Distance

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

Of Pictures that are Painted according to thePosition they are to Occupy

“I’ho già fatto un gozzo in questostento.”

Interiors

The Square at an Angle of 45°

The Cube at an Angle of 45°

Pavements Drawn by Means of Squares at 45°

The Perspective Vanishing Scale

The Vanishing Scale can be Drawn to any Point onthe Horizon

Application of Vanishing Scales to DrawingFigures

How to Determine the Heights of Figures on aLevel Plane

First Case.

Second Case.

Third Case.

The Horizon above the Figures

Landscape Perspective

Figures of Different Heights

The Chessboard

Application of the Vanishing Scale to DrawingFigures at an Angle when their Vanishing Points are Inaccessible orOutside the Picture

The Reduced Distance. How to Proceed when thePoint of Distance is Inaccessible

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

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

Measuring Scale on Ground

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

How to Measure Long Distances such as a Mile orUpwards

Further Illustration of Long Distances andExtended Views

How to Ascertain the Relative Heights of Figureson an Inclined Plane

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

PUBLISHER TO THE UNIVERSITY OF OXFORD
LONDON, EDINBURGH, NEW YORK
TORONTO AND MELBOURNE

THEORY AND PRACTICE
OF PERSPECTIVE

OXFORD
AT THE CLARENDON PRESS

PRINTED AT THE CLARENDON PRESS
BY HORACE HART, M.A.
PRINTER TO THE UNIVERSITY