Part 4

The rectilinear limitation of the sides, top, and base of the picture is of course quite arbitrary, as the space of a landscape would be which was seen through a window; less or more being seen at the spectator’s pleasure, as he retires or advances.

The distance of the station-point is not so arbitrary. In ordinary cases it should not be less than the intended greatest dimension (height or breadth) of the picture. In most works by the great masters it is more; they not only calculate on their pictures being seen at considerable distances, but they like breadth of mass in buildings, and dislike the sharp angles which always result from station-points at short distances.[31]

Whenever perspective, done by true rule, looks wrong, it is always because the station-point is too near. Determine, in the outset, at what distance the spectator is likely to examine the work, and never use a station-point within a less distance.

There is yet another and a very important reason, not only for care in placing the station-point, but for that accurate calculation of distance and observance of measurement which have been insisted on throughout this work. All drawings of objects on a reduced scale are, if rightly executed, drawings of the appearance of the object at the distance which in true perspective reduces it to that scale. They are not _small_ drawings of the object seen near, but drawings the _real size_ of the object seen far off. Thus if you draw a mountain in a landscape, three inches high, you do not reduce all the features of the near mountain so as to come into three inches of paper. You could not do that. All that you can do is to give the appearance of the mountain, when it is so far off that three inches of paper would really hide it from you. It is precisely the same in drawing any other object. A face can no more be reduced in scale than a mountain can. It is infinitely delicate already; it can only be quite rightly rendered on its own scale, or at least on the slightly diminished scale which would be fixed by placing the plate of glass, supposed to represent the field of the picture, close to the figures. Correggio and Raphael were both fond of this slightly subdued magnitude of figure. Colossal painting, in which Correggio excelled all others, is usually the enlargement of a small picture (as a colossal sculpture is of a small statue), in order to permit the subject of it to be discerned at a distance. The treatment of colossal (as distinguished from ordinary) paintings will depend therefore, in general, on the principles of optics more than on those of perspective, though, occasionally, portions may be represented as if they were the projection of near objects on a plane behind them. In all points the subject is one of great difficulty and subtlety; and its examination does not fall within the compass of this essay.

Lastly, it will follow from these considerations, and the conclusion is one of great practical importance, that, though pictures may be enlarged, they cannot be reduced, in copying them. All attempts to engrave pictures completely on a reduced scale are, for this reason, nugatory. The best that can be done is to give the aspect of the picture at the distance which reduces it in perspective to the size required; or, in other words, to make a drawing of the distant effect of the picture. Good painting, like nature’s own work, is infinite, and unreduceable.

I wish this book had less tendency towards the infinite and unreduceable. It has so far exceeded the limits I hoped to give it, that I doubt not the reader will pardon an abruptness of conclusion, and be thankful, as I am myself, to get to an end on any terms.

[30] As in algebraic science, much depends, in complicated perspective, on the student’s ready invention of expedients, and on his quick sight of the shortest way in which the solution may be accomplished, when there are several ways.

[31] The greatest masters are also fond of parallel perspective, that is to say, of having one side of their buildings fronting them full, and therefore parallel to the picture plane, while the other side vanishes to the sight-point. This is almost always done in figure backgrounds, securing simple and balanced lines.

APPENDIX.

I.

PRACTICE AND OBSERVATIONS.

II.

DEMONSTRATIONS.

I.

PRACTICE AND OBSERVATIONS ON THE PRECEDING PROBLEMS.

PROBLEM I.

An example will be necessary to make this problem clear to the general student.

The nearest corner of a piece of pattern on the carpet is 4½ feet beneath the eye, 2 feet to our right and 3½ feet in direct distance from us. We intend to make a drawing of the pattern which shall be seen properly when held 1½ foot from the eye. It is required to fix the position of the corner of the piece of pattern.

Let _AB_, Fig. 51., be our sheet of paper, some 3 feet wide. Make _ST_ equal to 1½ foot. Draw the line of sight through _S_. Produce _TS_, and make _DS_ equal to 2 feet, therefore _TD_ equal to 3½ feet. Draw _DC_, equal to 2 feet; _CP_, equal to 4 feet. Join _TC_ (cutting the sight-line in _Q_) and _TP_.

Let fall the vertical _QP′_, then _P′_ is the point required.

If the lines, as in the figure, fall outside of your sheet of paper, in order to draw them, it is necessary to attach other sheets of paper to its edges. This is inconvenient, but must be done at first that you may see your way clearly; and sometimes afterwards, though there are expedients for doing without such extension in fast sketching.

It is evident, however, that no extension of surface could be of any use to us, if the distance _TD_, instead of being 3½ feet, were 100 feet, or a mile, as it might easily be in a landscape.

It is necessary, therefore, to obtain some other means of construction; to do which we must examine the principle of the problem.

In the analysis of Fig. 2., in the introductory remarks, I used the word “height” only of the tower, _QP_, because it was only to its vertical height that the law deduced from the figure could be applied. For suppose it had been a pyramid, as _OQP_, Fig. 52., then the image of its side, _QP_, being, like every other magnitude, limited on the glass _AB_ by the lines coming from its extremities, would appear only of the length _Q′S_; and it is not true that _Q′S_ is to _QP_ as _TS_ is to _TP_. But if we let fall a vertical _QD_ from _Q_, so as to get the vertical height of the pyramid, then it is true that _Q′S_ is to _QD_ as _TS_ is to _TD_.

Supposing this figure represented, not a pyramid, but a triangle on the ground, and that _QD_ and _QP_ are horizontal lines, expressing lateral distance from the line _TD_, still the rule would be false for _QP_ and true for _QD_. And, similarly, it is true for all lines which are parallel, like _QD_, to the plane of the picture _AB_, and false for all lines which are inclined to it at an angle.

Hence generally. Let _PQ_ (Fig. 2. in Introduction, p. 6) be any magnitude _parallel to the plane of the picture_; and _P′Q′_ its image on the picture.

Then always the formula is true which you learned in the Introduction: _P′Q′_ is to _PQ_ as _ST_ is to _DT_.

Now the magnitude _P_ dash _Q_ dash in this formula I call the “SIGHT-MAGNITUDE” of the line _PQ_. The student must fix this term, and the meaning of it, well in his mind. The “sight-magnitude” of a line is the magnitude which bears to the real line the same proportion that the distance of the picture bears to the distance of the object. Thus, if a tower be a hundred feet high, and a hundred yards off; and the picture, or piece of glass, is one yard from the spectator, between him and the tower; the distance of picture being then to distance of tower as 1 to 100, the sight-magnitude of the tower’s height will be as 1 to 100; that is to say, one foot. If the tower is two hundred yards distant, the sight-magnitude of its height will be half a foot, and so on.

But farther. It is constantly necessary, in perspective operations, to measure the other dimensions of objects by the sight-magnitude of their vertical lines. Thus, if the tower, which is a hundred feet high, is square, and twenty-five feet broad on each side; if the sight-magnitude of the height is one foot, the measurement of the side, reduced to the same scale, will be the hundredth part of twenty-five feet, or three inches: and, accordingly, I use in this treatise the term “sight-magnitude” indiscriminately for all lines reduced in the same proportion as the vertical lines of the object. If I tell you to find the “sight-magnitude” of any line, I mean, always, find the magnitude which bears to that line the proportion of _ST_ to _DT_; or, in simpler terms, reduce the line to the scale which you have fixed by the first determination of the length _ST_.

Therefore, you must learn to draw quickly to scale before you do anything else; for all the measurements of your object must be reduced to the scale fixed by _ST_ before you can use them in your diagram. If the object is fifty feet from you, and your paper one foot, all the lines of the object must be reduced to a scale of one fiftieth before you can use them; if the object is two thousand feet from you, and your paper one foot, all your lines must be reduced to the scale of one two-thousandth before you can use them, and so on. Only in ultimate practice, the reduction never need be tiresome, for, in the case of large distances, accuracy is never required. If a building is three or four miles distant, a hairbreadth of accidental variation in a touch makes a difference of ten or twenty feet in height or breadth, if estimated by accurate perspective law. Hence it is never attempted to apply measurements with precision at such distances. Measurements are only required within distances of, at the most, two or three hundred feet. Thus it may be necessary to represent a cathedral nave precisely as seen from a spot seventy feet in front of a given pillar; but we shall hardly be required to draw a cathedral three miles distant precisely as seen from seventy feet in advance of a given milestone. Of course, if such a thing be required, it can be done; only the reductions are somewhat long and complicated: in ordinary cases it is easy to assume the distance _ST_ so as to get at the reduced dimensions in a moment. Thus, let the pillar of the nave, in the case supposed, be 42 feet high, and we are required to stand 70 feet from it: assume _ST_ to be equal to 5 feet. Then, as 5 is to 70 so will the sight-magnitude required be to 42; that is to say, the sight-magnitude of the pillar’s height will be 3 feet. If we make _ST_ equal to 2½ feet, the pillar’s height will be 1½ foot, and so on.

And for fine divisions into irregular parts which cannot be measured, the ninth and tenth problems of the sixth book of Euclid will serve you: the following construction is, however, I think, more practically convenient:—

The line _AB_ (Fig. 53.) is divided by given points, _a_, _b_, _c_, into a given number of irregularly unequal parts; it is required to divide any other line, _CD_, into an equal number of parts, bearing to each other the same proportions as the parts of _AB_, and arranged in the same order.

Draw the two lines parallel to each other, as in the figure.

Join _AC_ and _BD_, and produce the lines _AC_, _BD_, till they meet in _P_.

Join _aP_, _bP_, _cP_, cutting _cD_ in _f_, _g_, _h_.

Then the line _CD_ is divided as required, in _f_, _g_, _h_.

In the figure the lines _AB_ and _CD_ are accidentally perpendicular to _AP_. There is no need for their being so.

Now, to return to our first problem.

The construction given in the figure is only the quickest mathematical way of obtaining, on the picture, the sight-magnitudes of _DC_ and _PC_, which are both magnitudes parallel with the picture plane. But if these magnitudes are too great to be thus put on the paper, you have only to obtain the reduction by scale. Thus, if _TS_ be one foot, _TD_ eighty feet, _DC_ forty feet, and _CP_ ninety feet, the distance _QS_ must be made equal to one eightieth of _DC_, or half a foot; and the distance _QP′_, one eightieth of _CP_, or one eightieth of ninety feet; that is to say, nine eighths of a foot, or thirteen and a half inches. The lines _CT_ and _PT_ are thus _practically_ useless, it being only necessary to measure _QS_ and _QP_, on your paper, of the due sight-magnitudes. But the mathematical construction, given in Problem I., is the basis of all succeeding problems, and, if it is once thoroughly understood and practiced (it can only be thoroughly understood by practice), all the other problems will follow easily.

Lastly. Observe that any perspective operation whatever may be performed with reduced dimensions of every line employed, so as to bring it conveniently within the limits of your paper. When the required figure is thus constructed on a small scale, you have only to enlarge it accurately in the same proportion in which you reduced the lines of construction, and you will have the figure constructed in perspective on the scale required for use.

PROBLEM IX.

The drawing of most buildings occurring in ordinary practice will resolve itself into applications of this problem. In general, any house, or block of houses, presents itself under the main conditions assumed here in Fig. 54. There will be an angle or corner somewhere near the spectator, as _AB_; and the level of the eye will usually be above the base of the building, of which, therefore, the horizontal upper lines will slope down to the vanishing-points, and the base lines rise to them. The following practical directions will, however, meet nearly all cases:—

Let _AB_, Fig. 54., be any important vertical line in the block of buildings; if it is the side of a street, you may fix upon such a line at the division between two houses. If its real height, distance, etc., are given, you will proceed with the accurate construction of the problem; but usually you will neither know, nor care, exactly how high the building is, or how far off. In such case draw the line _AB_, as nearly as you can guess, about the part of the picture it ought to occupy, and on such a scale as you choose. Divide it into any convenient number of equal parts, according to the height you presume it to be. If you suppose it to be twenty feet high, you may divide it into twenty parts, and let each part stand for a foot; if thirty feet high, you may divide it into ten parts, and let each part stand for three feet; if seventy feet high, into fourteen parts, and let each part stand for five feet; and so on, avoiding thus very minute divisions till you come to details. Then observe how high your eye reaches upon this vertical line; suppose, for instance, that it is thirty feet high and divided into ten parts, and you are standing so as to raise your head to about six feet above its base, then the sight-line may be drawn, as in the figure, through the second division from the ground. If you are standing above the house, draw the sight-line above _B_; if below the house, below _A_; at such height or depth as you suppose may be accurate (a yard or two more or less matters little at ordinary distances, while at great distances perspective rules become nearly useless, the eye serving you better than the necessarily imperfect calculation). Then fix your sight-point and station-point, the latter with proper reference to the scale of the line _AB_. As you cannot, in all probability, ascertain the exact direction of the line _AV_ or _BV_, draw the slope _BV_ as it appears to you, cutting the sight-line in _V_. Thus having fixed one vanishing-point, the other, and the dividing-points, must be accurately found by rule; for, as before stated, whether your entire group of points (vanishing and dividing) falls a little more or less to the right or left of _S_ does not signify, but the relation of the points to each other _does_ signify. Then draw the measuring-line _BG_, either through _A_ or _B_, choosing always the steeper slope of the two; divide the measuring-line into parts of the same length as those used on _AB_, and let them stand for the same magnitudes. Thus, suppose there are two rows of windows in the house front, each window six feet high by three wide, and separated by intervals of three feet, both between window and window and between tier and tier; each of the divisions here standing for three feet, the lines drawn from _BG_ to the dividing-point _D_ fix the lateral dimensions, and the divisions on _AB_ the vertical ones. For other magnitudes it would be necessary to subdivide the parts on the measuring-line, or on _AB_, as required. The lines which regulate the inner sides or returns of the windows (_a_, _b_, _c_, etc.) of course are drawn to the vanishing-point of _BF_ (the other side of the house), if _FBV_ represents a right angle; if not, their own vanishing-point must be found separately for these returns. But see Practice on Problem XI.

Interior angles, such as _EBC_, Fig. 55. (suppose the corner of a room), are to be treated in the same way, each side of the room having its measurements separately carried to it from the measuring-line. It may sometimes happen in such cases that we have to carry the measurement _up_ from the corner _B_, and that the sight-magnitudes are given us from the length of the line _AB_. For instance, suppose the room is eighteen feet high, and therefore _AB_ is eighteen feet; and we have to lay off lengths of six feet on the top of the room wall, _BC_. Find _D_, the dividing-point of _BC_. Draw a measuring-line, _BF_, from _B_; and another, _gC_, anywhere above. On _BF_ lay off _BG_ equal to one third of _AB_, or six feet; and draw from _D_, through _G_ and _B_, the lines _Gg_, _Bb_, to the upper measuring-line. Then _gb_ is six feet on that measuring-line. Make _bc_, _ch_, etc., equal to _bg_; and draw _ce_, _hf_, etc., to _D_, cutting _BC_ in _e_ and _f_, which mark the required lengths of six feet each at the top of the wall.

PROBLEM X.

This is one of the most important foundational problems in perspective, and it is necessary that the student should entirely familiarize himself with its conditions.

In order to do so, he must first observe these general relations of magnitude in any pyramid on a square base.

Let _AGH′_, Fig. 56., be any pyramid on a square base.

The best terms in which its magnitude can be given, are the length of one side of its base, _AH_, and its vertical altitude (_CD_ in Fig. 25.); for, knowing these, we know all the other magnitudes. But these are not the terms in which its size will be usually ascertainable. Generally, we shall have given us, and be able to ascertain by measurement, one side of its base _AH_, and either _AG_ the length of one of the lines of its angles, or _BG_ (or _B′G_) the length of a line drawn from its vertex, _G_, to the middle of the side of its base. In measuring a real pyramid, _AG_ will usually be the line most easily found; but in many architectural problems _BG_ is given, or is most easily ascertainable.

Observe therefore this general construction.

Let _ABDE_, Fig. 57., be the square base of any pyramid.

Draw its diagonals, _AE_, _BD_, cutting each other in its center, _C_.

Bisect any side, _AB_, in _F_.

From _F_ erect vertical _FG_.

Produce _FB_ to _H_, and make _FH_ equal to _AC_.

Now if the vertical altitude of the pyramid (_CD_ in Fig. 25.) be given, make _FG_ equal to this vertical altitude.

Join _GB_ and _GH_.

Then _GB_ and _GH_ are the true magnitudes of _GB_ and _GH_ in Fig. 56.

If _GB_ is given, and not the vertical altitude, with center _B_, and distance _GB_, describe circle cutting _FG_ in _G_, and _FG_ is the vertical altitude.

If _GH_ is given, describe the circle from _H_, with distance _GH_, and it will similarly cut _FG_ in _G_.

It is especially necessary for the student to examine this construction thoroughly, because in many complicated forms of ornaments, capitals of columns, etc., the lines _BG_ and _GH_ become the limits or bases of curves, which are elongated on the longer (or angle) profile _GH_, and shortened on the shorter (or lateral) profile _BG_. We will take a simple instance, but must previously note another construction.

It is often necessary, when pyramids are the roots of some ornamental form, to divide them horizontally at a given vertical height. The shortest way of doing so is in general the following.

Let _AEC_, Fig. 58., be any pyramid on a square base _ABC_, and _ADC_ the square pillar used in its construction.

Then by construction (Problem X.) _BD_ and _AF_ are both of the vertical height of the pyramid.

Of the diagonals, _FE_, _DE_, choose the shortest (in this case _DE_), and produce it to cut the sight-line in _V_.

Therefore _V_ is the vanishing-point of _DE_.

Divide _DB_, as may be required, into the sight-magnitudes of the given vertical heights at which the pyramid is to be divided.

From the points of division, 1, 2, 3, etc., draw to the vanishing-point _V_. The lines so drawn cut the angle line of the pyramid, _BE_, at the required elevations. Thus, in the figure, it is required to draw a horizontal black band on the pyramid at three fifths of its height, and in breadth one twentieth of its height. The line _BD_ is divided into five parts, of which three are counted from _B_ upwards. Then the line drawn to _V_ marks the base of the black band. Then one fourth of one of the five parts is measured, which similarly gives the breadth of the band. The terminal lines of the band are then drawn on the sides of the pyramid parallel to _AB_ (or to its vanishing-point if it has one), and to the vanishing-point of _BC_.

If it happens that the vanishing-points of the diagonals are awkwardly placed for use, bisect the nearest base line of the pyramid in _B_, as in Fig. 59.

Erect the vertical _DB_ and join _GB_ and _DG_ (_G_ being the apex of pyramid).

Find the vanishing-point of _DG_, and use _DB_ for division, carrying the measurements to the line _GB_.

In Fig. 59., if we join _AD_ and _DC_, _ADC_ is the vertical profile of the whole pyramid, and _BDC_ of the half pyramid, corresponding to _FGB_ in Fig. 57.

We may now proceed to an architectural example.

Let _AH_, Fig. 60., be the vertical profile of the capital of a pillar, _AB_ the semi-diameter of its head or abacus, and _FD_ the semi-diameter of its shaft.

Let the shaft be circular, and the abacus square, down to the level _E_.

Join _BD_, _EF_, and produce them to meet in _G_.

Therefore _ECG_ is the semi-profile of a reversed pyramid containing the capital.

Construct this pyramid, with the square of the abacus, in the required perspective, as in Fig. 61.; making _AE_ equal to _AE_ in Fig. 60., and _AK_, the side of the square, equal to twice _AB_ in Fig. 60. Make _EG_ equal to _CG_, and _ED_ equal to _CD_. Draw _DF_ to the vanishing-point of the diagonal _DV_ (the figure is too small to include this vanishing-point), and _F_ is the level of the point _F_ in Fig. 60., on the side of the pyramid.

Draw _Fm_, _Fn_, to the vanishing-points of _AH_ and _AK_. Then _Fn_ and _Fm_ are horizontal lines across the pyramid at the level _F_, forming at that level two sides of a square.